Benthic Bulldozers and Pumps: Laboratory and Modelling Studies of Bioturbation and Bioirrigation

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1 Benthic Bulldozers and Pumps: Laboratory and Modelling Studies of Bioturbation and Bioirrigation Nicola Jane Grigg February 23 A thesis submitted for the degree of Doctor of Philosophy of The Australian National University.

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3 Declaration The work in this thesis is my own, unless stated otherwise. My research was conducted in the Pye Laboratory, CSIRO Land and Water. None of the material has been submitted towards a degree in any other institution. Nicola Grigg i

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5 Acknowledgments She tells herself that thought takes time and there s nothing you can do about it. Or what in general they stupidly call creation. That doesn t much resemble streams. Ponds, rather. You flounder in them. It goes nowhere, it s not happy, not communicative. Postmodern Fables, Jean-François Lyotard. Unhappy floundering is a common experience among PhD students, and I ve been no exception. I wish I could say that my slow thinking was focussed on my PhD research; unfortunately I was embroiled in the usual issues that trouble us in our mid-twenties how to live, and why. Growing environmental problems at local to global scales concern me greatly, as do the glaring differences in quality of life around the world. I feel useless in the face of such enormous issues. It was against this rather gloomy backdrop that all the following people provided invaluable support during my research. Ian Webster and Phillip Ford are wonderfully generous people, in spite of the considerable pressures on them. Phillip and Ian have been the most patient and helpful of supervisors, and I am grateful for all the time and energy they have spent guiding me through my work. I particularly appreciated their sharp scientific insights and calm confidence in my ability. Ian White at ANU was always enthusiastic about my work and I welcomed his encouragement and guidance, as well as his assistance with the administrative hoops that needed to be jumped along the way. The workshop staff (Peter Morphett and Craig Webber) and Pye Lab technicians (Garry Miller, Dale Hughes, Mark Kitchen and Chris Drury) were a pleasure to work with, and I was always impressed at their ability to transform my hazy descriptions into useful experimental equipment or advice. I m grateful to the IT staff for keeping my computer functional, and for making it so easy for me to work on my thesis from home, yet still remain in contact with the lab. My office is situated in the Pye Lab, which is home to a wonderful collection of people, many of whom have helped me with my work and my ever-changing world view. Countless other people aided me in surprising ways throughout my project. As just one example, Mike Grace sat and discussed PhD blues with me around a group campfire at an ASL conference in the early stages of my PhD. My friends, mostly met through UWA, ANU, CSIRO and birthing classes, have all been invaluable, and I m grateful for the fun, encouragement and advice they ve provided so consistently throughout my PhD. My daughter, Zoe, was born 3 ½ years into my project and she s been a wonderful addition to my life. Pregnancy and looking after a young baby has reinforced in my mind the value of good sleep, food and exercise. Without Zoe I would not have had the determination to give these simple needs the attention they deserve. Thanks to Zoe, I also discovered the joys of working part-time. Warmest thanks to my parents. They were generous with their love and support, and provided financial help once my scholarship had ended and savings had been diminished. Brett, my husband, somehow managed to set up and run our vegie gardens, work part-time, look after Zoe part-time and supply an endless stream of lovingly prepared meals throughout my PhD. Thanks B :-) I d like to thank my funding sources. My living allowance was funded by an Australian Postgraduate Award. Supervision, technical support, computers, lab and office space were all provided by CSIRO Land and Water, Canberra. The Centre for Resource and Environmental Studies at ANU and the Water Research Foundation provided supervision and funding for field and laboratory expenses. ii

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7 Abstract Aquatic sediments are the recipients of a continual rain of organic debris from the water column. The decomposition reactions within the sediment and the rates of material exchange between the sediment and water column are critically moderated by the transport processes within the sediment. The sediment and solute movement induced by burrowing animals bioturbation and bioirrigation far exceed abiotic transport processes such as sedimentation burial and molecular diffusion. Thalassinidean shrimp are particularly abundant burrowing animals. Living in high density populations along coastlines around the world, these shrimp build complex burrow networks which they actively maintain and irrigate. I used a laser scanner to map thalassinidean shrimp (Trypaea australiensis) mound formation. These experiments measured rapid two-way exchange between the sediment and depth. Subduction from the sediment surface proved to be just as important as sediment expulsion from depth, yet this is not detected by conventional direct entrapment techniques. The experiments demonstrated that a daily sampling frequency was needed to capture the extent of the two-way exchange. I derived a one-dimensional non-local model accounting for the excavation, infill and collapse (EIC) of burrows. Maximum likelihood analyses were used to test the model against 21 Pb and 228 Th profiles taken from sediment cores in Port Phillip Bay, Melbourne. The maximum likelihood approach proved to be a useful technique for quantifying parameter confidence bounds and allowing formal comparison with a comparable biodiffusion model. The EIC model generally outperformed the biodiffusion model, and in all cases best EIC model parameter estimates required some level of burrow infill with surface material. The EIC model was expanded to two and three dimensions, which allowed the representation of lateral heterogeneity resulting from the excavation, infill and collapse of burrow structures. A synthetic dataset generated by the two-dimensional model was used to demonstrate the effects of heterogeneity and core sampling on the mixing information that can be extracted from one-dimensional sediment core data. Burrow irrigation brings oxygenated water into burrow depths, and can affect the nitrogen cycle by increasing the rates of coupled nitrification and denitrification reactions. I modelled the nitrogen chemistry in the annulus of sediment surrounding an irrigated burrow using a radially-symmetrical diffusion model. The model was applied to three published case studies involving thalassinidean shrimp experiments and to field data from Port Phillip Bay. The results highlighted divergences between current theoretical understanding and laboratory and field measurements. The model further demonstrated potential limitations of measurements of burrow characteristics and animal behaviour in narrow laboratory tanks. Activities of burrowing animals had been hypothesised to contribute to high denitrification rates within Port Phillip Bay. Modelling work in this thesis suggests that the model burrow density required to explain these high denitrification rates is not consistent with the sampled density of thalassinidean shrimp in the Bay, although dense burrows of other animals are likely to be important. Limitations of one-dimensional representations of nitrogen diagenesis were explored via comparisons between one-dimensional models and the full cylinder model. iii

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9 Table of Contents 1 INTRODUCTION PORT PHILLIP BAY ENVIRONMENTAL STUDY ANIMAL-SEDIMENT INTERACTIONS The Thalassinidea Thalassinidean bioturbation Modelling bioturbation Thalassinidean bioirrigation Bioirrigation modelling AVERAGING IN HETEROGENEOUS ENVIRONMENTS Altered reaction zonation Diffusion in one, two and three dimensions Uncertainty due to the random nature of sediments Summary THESIS STRUCTURE AND AIMS LASER SCANNER MEASUREMENTS OF TRYPAEA AUSTRALIENSIS MOUND TOPOGRAPHY INTRODUCTION EXPERIMENT DESCRIPTION DATA ANALYSIS Calibration Map construction Individual mound analysis RESULTS AND DISCUSSION General Observations Volume estimates CONCLUSIONS ONE-DIMENSIONAL SEDIMENT MODELLING: EXCAVATION, INFILL AND COLLAPSE INTRODUCTION MODEL DERIVATION PHYSICAL INTERPRETATION AND TYPICAL BEHAVIOUR Effect of parameter values in the EIC model Comparison between the EIC and diffusion models CASE STUDY: PORT PHILLIP BAY 21 PB AND 228 TH PROFILES Problem description and parameter estimation method Results of Maximum Likelihood analysis Model Assumptions CONCLUSIONS HIGHER-DIMENSIONAL EIC SEDIMENT MODEL INTRODUCTION TWO-DIMENSIONAL MODEL EQUATIONS SAMPLE MODEL OUTPUT AND COMPARISONS WITH 1-D MODEL PORT PHILLIP BAY PROFILES THREE-DIMENSIONAL MODEL OTHER CONSIDERATIONS Nonlinear reaction terms More realistic burrow maps More realistic descriptions of burrow collapse Scale issues CONCLUSIONS MODELLING NITROGEN DYNAMICS AROUND A BURROW INTRODUCTION BACKGROUND MODEL DESCRIPTION...19 iv

10 5.4 GENERAL MODEL BEHAVIOUR THALASSINIDEAN SHRIMP CASE STUDIES Case 1: modelling Callianassa truncata field data (Ziebis et al. (1996)) Case 2: modelling Callianassa subterranea laboratory data (Forster and Graf (1995)) Case 3: modelling Callianassa japonica and Upogebia major laboratory measurements (Koike and Mukai (1983)) Summary of thalassinidean shrimp cases PORT PHILLIP BAY One-dimensional radial model species cylinder model Summary CONCLUSIONS CONCLUSIONS AND FUTURE WORK LASER SCANNER EXPERIMENTS D EIC MODEL HIGHER DIMENSIONAL EIC MODEL BURROW IRRIGATION MODELLING FUTURE WORK APPENDIX: A NOTE ABOUT NON-LOCAL IRRIGATION MODELS BACKGROUND TO WHAT EXTENT IS EQUATION (7.6) VALID? VALIDITY OF USING A CONSTANT VALUE FOR α CAN A NON-LOCAL MODEL DESCRIBE A MULTIPLE-SPECIES SYSTEM WELL? SUMMARY REFERENCES v

11 1 Introduction Coastal ecosystems have had huge changes forced on them by human development. In the 2 years since European settlement in Australia, land use practices (particularly European agricultural methods) have dramatically altered the hydrology and nutrient budgets of the landscape, so changing the nature of the run-off into estuarine and coastal systems. Sewage from population centres, pollutants from industrial zones, the formation of harbours and marinas, aquaculture, professional and recreational fishing, tourist developments and introduced species have also had their impacts. These impacts are well recognised, and have prompted many studies into how our aquatic ecosystems function and respond to these forcing factors. The motivation for my work stemmed from one such study, the Port Phillip Bay Environmental Study (Harris et al. (1996)), which found that the nitrogen budget of the Bay was critically moderated by sediment processes. Given the low flushing rates with the ocean, high denitrification rates in the Bay sediments were found to provide the main nitrogen sink in the system, thus preventing eutrophication. An active benthic population exists in Port Phillip Bay, and one hypothesis advanced was that these burrowing animals may be significant in creating the right environment for the observed high denitrification rates. From this starting information, I conducted a study that aimed to address quite general questions about the interactions between burrowing animals and aquatic sediments. My emphasis has been on quantitative analyses of these processes, and on applying appropriate modelling approaches. This introductory chapter provides background on the main themes, aims and structure of the thesis. The emphasis here is on creating a context in which to consider the subsequent chapters. 1.1 Port Phillip Bay Environmental Study The Port Phillip Bay Environmental Study (PPBES) was a large study of Port Phillip Bay (PPB) in Victoria, Australia (Figure 1.1). The study began in 1992 and finished in Its aim was to address questions of management and sustainable use of the Bay. Many of the study results were published in a special issue of the journal Marine and Freshwater Research (Volume 5, Issue 6). The following description is based on descriptions in the final report (Harris et al. (1996))

12 Figure 1.1 Map of Port Phillip Bay (taken from Harris et al. (1996)) The city of Melbourne (population of over 3 million) sprawls over the Yarra River and spreads along the northern shores of PPB. The Bay itself is relatively shallow (the deepest part is 24m deep, and half the volume is in water under 8m deep), and its flushing time with Bass Strait is approximately one year. The emphasis of the study was to quantify the important processes occurring within the Bay what are the energy and material flows through the system? A substantial effort was invested in identifying all nutrient flows into the system, understanding the mixing processes and flushing with Bass Strait, and unravelling the nutrient transformations occurring within the Bay. As the system is nitrogen limited, attention was devoted to the nitrogen dynamics of the Bay. The flushing rate with Bass Strait proved to be relatively unimportant, and the fate of incoming nitrogen is determined by processes occurring within the Bay. Ammonia and nitrate entering the Bay are taken up rapidly by phytoplankton, and the dead organic matter that falls to the sediment floor is remineralised and released back - 2 -

13 into the water column, where it is again available to life in the water column. This cycling between the water column and sediment may be repeated many times very rapidly. Despite the poor flushing with Bass Strait, nitrogen does not accumulate in the system, and PPB remains oligotrophic to mesotrophic. The PPBES found that bacterial denitrification reactions occurring within the Bay sediments are responsible for large losses of nitrogen from the system, so that 7-9% of the nitrogen that enters the system eventually escapes to the atmosphere as nitrogen gas (N 2). These conclusions were inferred both from attempts to close the Bay nitrogen budget and by direct measurement of nitrogen gas fluxes from the sediment during benthic chamber studies. The PPBES suggested that a very active benthic population of burrowing animals could be contributing to the high denitrification rates. Burrows introduce oxygen to sediment depths, and can be considered to extend the area of the sediment-water interface. By increasing the area of oxic-anoxic interface within sediments, a large burrow surface area provides the perfect environment for coupled nitrification and denitrification: ammonium is oxidised to nitrate in the oxygenated sediment immediately surrounding the burrow, and then the nitrate is reduced to nitrogen gas in the surrounding anoxic sediment. The hypothesis of the importance of burrowing organisms in PPBES was an underlying motivation behind my research. The study provided much of the data I used to test my models. Chapters 3 and 4 use sediment core data collected during the PPBES to test how well non-local bioturbation model parameters could be estimated from radionuclide data. A section of Chapter 5 is devoted to modelling the effects of burrow irrigation on denitrification rates in Port Phillip Bay. 1.2 Animal-sediment interactions Sediment water interactions critically determine water quality, as indicated by the PPBES findings on the importance of sediment denitrifcation in preventing nitrogen accumulation in the poorly-flushed bay. The ability to manage and restore estuarine systems can be improved with a better understanding of the processes involved and of the fate of nutrients and pollutants. Sediment biogeochemists wish to unravel sediment transport processes to better understand and even predict the nutrient and pollutant pathways through aquatic ecosystems

14 Burrowing animals are responsible for a range of alterations to their sediment surroundings. They move sediment, pump water into their burrows, eat, excrete and secrete organic material, and interact with each other and resident microbial populations (Aller (1982)). In this work, only the physical effects of sediment movement and burrow flushing bioturbation and bioirrigation and their impact on sediment diagenesis processes are considered. A comprehensive overview of animalsediment relations can be found in McCall and Tevesz (1982), and overviews of sediment diagenesis modelling include Berner (198), Boudreau (1997) and Boudreau (2). A useful metaphor is to consider the sediments as a chemical reactor receiving fuel in the form of organic matter from the water column. Transport processes within the sediment govern the dispersal of organic matter and the removal of decomposition products to the overlying water column. In the absence of animals dwelling in the sediments, solid material can be transported only by burial due to sedimentation from the water column. Solute species can be transported by advection due to compaction, hydrological flows and bubble movement, however molecular diffusion is usually the dominate transport process. Bioturbation (animal-induced sediment movement) mixes surface material to large depths in short time scales that cannot be achieved by sedimentation burial alone, it brings material from depth to the sediment surface and it moves material across reaction zones within the sediment. Often viewed simply as a mixing or smoothing function, bioturbation is an important mechanism for introducing heterogeneity to sediments. Bioirrigation (solute transport by animals) can introduce reactive solutes from the overlying water column to depths that might not be reached if molecular diffusion were the only transfer process because the solutes would be consumed by reactions while being transported by slower diffusive processes. By actively pumping water in and out of the sediments, animals introduce oxidants and flush organic material and decomposition products out of the sediments and into the overlying water. Again, bioirrigation patterns are spatially and temporally heterogeneous and so they create patchy and dynamic reaction zones. The interactions between bioturbation and bioirrigation are important. Bioturbation can introduce pockets of labile organic matter to depth at the same time as bioirrigation - 4 -

15 introduces oxidants which will react rapidly with it. This speeds up the decomposition reactions considerably (D'Andrea et al. (22)). In this thesis I ve measured and modelled aspects of bioturbation and bioirrigation. Much of the work in this thesis concentrates on, but is not necessarily limited to, the guinea pig of bioturbation and bioirrigation studies, thalassinidean shrimp (Figure 1.2) The Thalassinidea Thalassinidean shrimp (Crustacea: Decapoda) exist in marine and brackish water bodies all around the world (excluding Arctic and Antarctic regions). Approximately 516 species have been described so far. 95% of these occur in water shallower than 2 m, and they burrow in sediments ranging from coarse coral rubble to sand and mud (Dworschak (2)). Along the eastern Australian coastline, they are referred to as marine yabbies and are considered excellent bait (Hailstone (1962)). Other common names include bait shrimp, ghost shrimp and ghost nippers. The most common species is Trypaea australiensis, the subject of my experiments. The animals are routinely extracted from sediment with yabby pumps for bait purposes, and Hailstone (1962) stated that this method can extract over 2 shrimp an hour in a densely populated bed. They further suggested that very dense populations can have in the order of 1 burrow openings/m 2 (where each animal typically maintains at least two burrow openings at the sediment surface). More recently, McPhee and Skilleter (22) found that the density of burrow openings is a poor indicator of population density. In their study, located near North Stradbroke Island (Queensland, Australia), burrow densities never exceeded 5 burrow openings/m 2 and the population density rarely exceeded 2 individuals/m 2. Little is known about the impact of extracting large numbers of these animals out of the sediment for bait, although Wynberg and Branch (1997) conducted experiments with a similar species of burrow shrimp which suggested that just the trampling of shrimp beds associated with this practice has the potential to cause lasting damage. Similar results have been found in an (as yet unpublished) study of Trypaea australiensis last year (Bird, personal communication). Dittmann (1996) showed Trypaea australiensis has promotive interactions with the surrounding infaunal community; in particular both meiofaunal and macrofaunal densities were reduced in areas where Trypaea australiensis was excluded in experimental plots

16 It is believed that each animal builds a burrow as a juvenile (after a planktonic larval stage), and remains in this burrow for the rest of its life, extending the burrow as it grows. Its lifespan is unknown, but generally assumed to be approximately two years. All feeding and reproduction is conducted within the burrow, and adults are very rarely observed on the sediment surface. Trypaea australiensis, like most thalassinidean shrimp, is a deposit feeder (collects food from the burrow wall), however Stapleton et al. (22) demonstrated considerable plasticity in Trypaea australiensis trophic modes, suggesting that it engages in suspension feeding as well. Grain size selectivity in feeding and sediment transport is well-documented among many thalassinidean shrimp species (eg. Stapleton et al. (22), Ziebis et al. (1996) and Stamhuis et al. (1998)). Thalassinidean shrimp beat three pleopods (paddle-like appendages on the underside of the abdomen) in a metachronous fashion to generate continuous ventilation currents through their burrows. More vigorous beating of the pleopods is used to flush sediment out of the burrow to the sediment surface. Figure 1.2 Trypaea australiensis, a common thalassinidean shrimp along the Eastern Australian coastline (approximately 5 cm in length) Thalassinidean bioturbation When gazing out over a tidal flat inhabited by thalassinidean shrimp, it is immediately obvious that these animals move sediment about. In a highly dynamic environment, burrow openings are kept clear and the mounds retain their characteristic conical shape (Figure 1.3). It s common to find dark grey mounds scattered among the population, clearly indicating that anoxic sediment material from depth is routinely transported to the sediment surface. These mounds retain their grey colour for a matter of hours only, becoming rapidly oxidised to a beach-sand cream colour

17 Figure 1.3 Trypaea australiensis mounds on an exposed tidal flat (Corunna Lake, NSW) (as a guide to scale, burrow holes are roughly.5 to 1 cm in diameter). Resin casts of their burrows reveal the extent of their subterranean activities. For the particular species in my work (Trypaea australiensis), it is common to find burrows extending deeper than one metre below the sediment surface (Figure 1.4). The question arises why do they build such deep burrows? Common answers include: quest for food, safety from predators and preventing desiccation during low tide (for species inhabiting intertidal regions). The world-wide mean mixed depth of sediments is 98 mm, and Boudreau (1998) demonstrated that this depth is consistent with burrowing activities being controlled by food availability. The fact that these particular animals burrow down to ten times that depth suggests that in this particular case the other, protective, purposes are important factors determining burrow depth. Figure 1.4 Resin casts of Trypaea australiensis burrows. I made these casts in Lake Nagundga, NSW, but was unable to unearth the full depth of the burrows. (The long side of the scale object located between the two casts is 1 cm. The plastic cups were used to pour the resin into the burrow. Each cup was positioned over a burrow opening.) Chapter 2 describes laboratory experiments I conducted involving Trypaea australiensis. A laser scanner was used to created detailed maps of mound evolution over a period of - 7 -

18 several months. Even in the quiescent laboratory environment, the shrimp shifted substantial quantities of sediment rapidly between the surface and sediment depths. Sediment mixing is not limited to thalassinidean shrimp. Extensive literature exists describing the sediment shifting behaviour for a wide range of deposit-feeding benthic invertebrates (eg. reviews by Cammen (198) and Thayer (1983)) Modelling bioturbation Given the importance of organic matter in fuelling sediment decomposition reactions, any sediment diagenesis model needs underlying solid transport assumptions to simulate the introduction and dispersal of organic matter (and other particle-bound species) in the sediment. The nature of the transport processes has a profound effect on the decomposition pathways in the sediment. For example, from the above description of thalassinidean shrimp burrows, we know that sediment is rapidly transported from depth to the surface. This process transfers an anoxic parcel of sediment to an oxic environment, so changes the diagenesis reactions occurring within the sediment parcel. 21Pb is a naturally-occurring radioactive tracer commonly used to infer sediment mixing processes, however 21Pb profiles are notoriously difficult to un-mix. Boudreau (1986b) and Boudreau and Imboden (1987) demonstrated that very different mixing mechanisms can produce similarly-shaped tracer profiles. A common approach is to fit a diffusion equation to the profile, producing a convenient biodiffusion coefficient, D b. While this approach may yield a convincing fit to the 21 Pb data, it completely fails to capture the rapid upward (non-local) transport of anoxic sediment observed in thalassinidian (and other animals ) burrowing activities. These problems have been recognised for decades, and in a comprehensive trilogy of papers, Bernard Boudreau detailed the limited conditions under which a diffusion analogy is appropriate. He formulated a general non-local mixing model that is well able to capture a whole variety of mixing behaviour and derived a range specific mixing models from his more general equation (Boudreau (1986a), Boudreau (1986b) and Boudreau and Imboden (1987)). The parameters in these non-local models are measurable, at least in theory, making them far more satisfactory than the contrived diffusion coefficient. While such non-local models are used, by far the most common approach is still to apply a diffusion model. A difficulty is that while non-local model parameters are based in reality (unlike the biodiffusion coefficient), measuring them is difficult, so - 8 -

19 parameter estimation remains a data-fitting exercise. Under these circumstances there is a strong argument that where data are scarce the most honest approach is to choose the model with the fewest parameters that provides the best fit to the data. There is little point in increasing the complexity of a model if it does not provide an improved explanation of the observed data (eg. Soetaert et al. (1996), Meile et al. (21) and Hilborn and Mangel (1997)). As a consequence, the advection-diffusion equation, with very few parameters and a long pedigree of application to these problems, remains the most commonly adopted modelling approach, even though its flaws are well known. Hilborn and Mangel (1997) specifically stated that an obviously-wrong process model is preferable to a more correct model containing more parameters, if the fit to the data is just as good. I would argue that the situation is not so clear-cut, and this approach could be just as dangerous as applying over-parameterised models. In diagenesis modelling, the transport processes interact with the chemistry in surprising ways, particularly if material is moved rapidly across reaction zones. If there is uncertainty in the transport processes, simply adopting the model that creates the best fit to inert tracer data is surely inadequate, and a safer approach is to test a range of transport assumptions. This was the approach recommended by Boudreau and Imboden (1987) and followed by Middelburg et al. (1996), who applied a diffusion equation for the bulk of their modelling (due to lack of data on non-local processes), but assessed the effect of including non-local transport in their sensitivity analysis. An alternative approach to such continuum models is to collect individual-scale mixing data (eg. ingestion and egestion rates) from laboratory studies and use a transition matrix approach to modelling bioturbation. A transition matrix defines the probability that sediment at one location will be shifted to another location during one time step. Thus the model is founded on measured quantities and accounts for stochasticity and a range of mixing behaviour (both local and nonlocal). Foster (1985) developed an early example of such a model, and this model was in turn modified and extended by Trauth (1998). The work by David Shull (Shull (21), Shull and Yasuda (21)) is the most recently published and comprehensive example of transition matrix modelling. There is a formal equivalence between the discrete and continuum models; Shull (21) demonstrated how the advection term in the transition matrix can be derived from Boudreau's (1997) equation for conservation of solids. Both non-local model and transition matrix approaches have the ability to represent very complicated - 9 -

20 or very simple transport assumptions (where the level of complexity is usually governed by the level of knowledge about the mixing behaviour). All these considerations motivated the modelling work in Chapter 3. Having observed and quantified rapid exchanges between the surface and depth in laboratory laser scanner experiments (Chapter 2), I used Boudreau's (1986b) general equation to derive a one-dimensional non-local model that would capture these exchanges yet remain as easy to use as the diffusion equation. While bioturbation models are typically restricted to one dimension, the drawbacks of this approach are well known to the modelling community (see further discussion in Section 1.3), and research efforts are focussing more and more on the need for two or three-dimensional modelling. Transition matrix approaches have been mainly used to describe one-dimensional mixing processes, however they could easily be applied to more dimensions. Similarly, I demonstrate in Chapter 4 the ease with which a nonlocal model derived from Boudreau s general equation can be extended to more dimensions. François et al. (1997) developed a model that relies on mixing descriptions defined at the organism and the community level. These authors defined four different functional groups (biodiffusor, downward-conveyor, upward-conveyor and regenerator) and their model was designed with the intent of exploring interactions between co-existing functional groups. Choi et al. (22) and Boudreau et al. (21) pointed out that all the above approaches are limited by the fact that the large-scale transport processes are all predetermined by the user. By contrast, these authors have developed a Lattice Automaton Bioturbation Simulator (LABS) that allows individual agents within the sediment to be programmed to follow a set of simple rules that might be expected to guide a burrowing organism. The resulting large-scale burrow structures are not determined by the user, but emerge from the cumulative impact of these simple behavioural rules. Their model has the further advantage that transformations that occur within the animal s gut are easily incorporated into this modelling approach. This last approach is highly appealing and no doubt will become more common, particularly as such computationally-intensive methods become more accessible. In the short-term, however, coupling a LABS type model to a full diagenesis model is unlikely to become routine. In the sediment modelling component of my project (Chapters 3 and 4), I ve demonstrated a less complicated, but quite flexible, non-local - 1 -

21 transport description that can be applied in one, two or three dimensions. It remains amenable to one-dimensional tracer core fitting approaches (appropriate for situations where tracer cores are the only data with which to characterise transport descriptions), but can accommodate more complicated mixing prescriptions based on field or laboratory measurements of animals activities or burrow structures if these data are available. Higher dimensional versions of the model can be compared with laterallyaveraged one-dimensional equivalents, so allowing the modeller to determine the circumstances under which the extra dimension(s) are necessary Thalassinidean bioirrigation Thalassinidean shrimp and other animals actively pump water into their burrows to meet their requirements for oxygen in an anoxic environment. The impacts on the surrounding sediment are plainly visible if one digs into the sediment and intersects an irrigated burrow; a thin annulus of light-coloured sediment separates the burrow lumen from the surrounding dark, anoxic sediment. Figure 1.5 is a photograph of a Trypaea australiensis burrow in a thin laboratory tank. The burrow extends to a depth of 5cm, and light-coloured sediment surrounds the burrow to this depth, clearly indicating the effects of the individual s pumping behaviour on the surrounding sediment chemistry. Figure 1.5 A Trypaea australiensis burrow formed in a thin (5cm) laboratory aquarium. The sediment depth is 5 cm

22 Burrow irrigation can be thought of as an effective increase in the area of the sedimentwater interface. A logical conclusion from this observation would be that burrows increase fluxes between the sediment and overlying water. Further, the increase factor would simply be the burrow surface area per unit sediment surface area, perhaps decreased by a factor accounting for partial burrow flushing. Unfortunately, this proves not to be the case, and the influence of burrow irrigation is more subtle when diagenesis reactions are involved. The magnitude and even direction of the changes in flux can be counter-intuitive. A conservative assertion is that burrow irrigation alters fluxes between the sediment and overlying water (Aller (1982)), and bioirrigation models or measurements are necessary to make any more specific conclusions. Furukawa et al. (21) stressed the importance of the burrow walls as an extension of the water-sediment interface, and discussed how its chemical mass transfer properties are likely to differ from the water-sediment interface. In particular, oscillating irrigation conditions, the addition of biochemical compounds excreted by animals and different microbial dynamics are all likely to contribute to these differences Bioirrigation modelling The most convincing bioirrigation modelling has been based on burrow microenvironment models devised by Robert Aller in the late 197s. Aller (198) envisaged the sediment as being a densely packed array of irrigated vertical cylinders. Assuming no lateral exchange between these cylinders, the porewater concentrations in the annulus of sediment surrounding the irrigated core can be modelled using a cylindrical-polar diffusion model. The boundary conditions in the model are simply that the burrow wall is held at a constant concentration (usually the concentration in the overlying water) and that a no-flux boundary condition exists on the outer radius of the sediment annulus. With basic assumptions about species kinetics, analytical solutions to this model are possible. The cylinder model represents a powerful technique for modelling two-dimensional pore-water distributions in irrigated sediments. Cylinder model results are highly dependent on the nature of the reaction kinetics. Aller (198) modelled ammonium as an example of a zeroth order reaction. In this case, radially-averaged concentrations are substantially altered by irrigated burrows, and are most sensitive to inter-burrow spacing. Total fluxes across the sediment water interface are not affected by the presence of irrigated burrows. Silica was modelled as an example of first order kinetics. In contrast to ammonium, radially averaged profiles

23 showed little change from profiles created with a diffusion-only model. Silica fluxes were greatly affected by the presence of irrigated burrows. Thus the model has been very useful in highlighting the importance of the interplay between reaction kinetics and geometry. Numerical solutions to the cylinder model are required for coupled multiple-species, non-linear reactions and oscillating boundary conditions. These further steps have been developed in Marinelli and Boudreau (1996), Boudreau and Marinelli (1994) and Boudreau and Marinelli (1993), where the impacts of oscillating irrigation were considered for single species and for multiple-species coupled reactions. Again, using ammonium and silica as examples, Boudreau and Marinelli (1994) demonstrated that the response to oscillating burrow irrigation depends on the nature of the reaction kinetics. They demonstrated that the time-averaged fluxes for both species are very similar to fluxes calculated for continuous irrigation (although the flux variations within an irrigation cycle are substantial). Radially-averaged ammonium profiles are very sensitive to oscillating irrigation, whereas radially-average silica profiles are not. Added complications were analysed by Marinelli and Boudreau (1996). They coupled the oscillating irrigation cylinder model to a diagenetic model of oxygen, sulphide and ph. Their modelling and experimental work suggested that oscillating irrigation behaviour significantly affects diagenesis processes. Furukawa et al. (2) used the cylinder model to model oxygen fluxes and concentrations within irrigated sediment, demonstrating how deep oxygen penetration in sediments may be significant yet not apparent in measured oxygen profiles from the sediment surface. Furukawa et al. (21) coupled a sediment diagenesis model to a more sophisticated microenvironment model (based on the cylinder model, but allowing for burrow tilt angles and higher burrow density nearer the sediment surface) and found it was a good tool for aiding the interpretation of their data from bioirrigated sediments in laboratory mesocosms. Boudreau (1984) demonstrated that a one-dimensional non-local model can be derived by radially averaging Aller s cylinder model. This one-dimensional formulation is now routinely used in diagensis modelling (eg. Boudreau (1996), Wang and Van Cappellen (1996) and Middelburg et al. (1996)). The equivalence between the one-dimensional and cylinder models is not exact, however. Aller (21) pointed out that a non-local model is only an adequate replacement for a full cylinder model under quite specific circumstances, not usually met by diagenesis models. This issue is explored in the

24 Appendix, where I demonstrate that a non-local model would be inappropriate to tackle the specific modelling questions addressed in Chapter 5. These cautions about the one-dimensional, non-local irrigation model should not detract from its importance. This model has significant advantages over the full cylinder model in that multiple runs required for robust statistical analysis and parameter estimation are possible. This level of rigour would be prohibitively timeconsuming with a full cylinder model. For example, Meile et al. (21) used an inverse modelling approach to infer non-local irrigation coefficient profiles from measured solute concentration and reaction rate profiles. The inverse modelling approach required many thousands of model runs (a simplex optimisation fitting routine embedded in hundreds of Monte Carlo simulations). Their approach ensured that the resulting irrigation coefficient profiles were no more complicated than could be justified by the data. These kinds of analyses are important, particularly when a better understanding of model certainty and confidence are sought, and yet these approaches remain beyond reach for two-dimensional modelling. Koretsky et al. (22) used detailed three-dimensional burrow structure information to calculate irrigation coefficients. This too yielded confidence bounds on irrigation coefficient estimates, which could then be propagated through diagenesis models and multiple diageneis model runs. Multi-dimensional irrigation modelling is needed for many reasons, yet the statistical rigour that can be achieved by multiple model runs is not presently feasible in other than one-dimensional models. This conundrum ensures that innovative use of onedimensional descriptions of irrigation behaviour will continue to be needed, although this should not prevent further exploration with two and three-dimensional models. I experienced this dilemma directly in Chapter 5, where I used burrow irrigation models to investigate nitrogen dynamics surrounding an irrigated burrow. Using a onedimensional radial model I was able to conduct many model runs and thoroughly explore the solution space. Using the slower cylinder model I was only able to investigate a limited number of scenarios. 1.3 Averaging in heterogeneous environments In earlier decades, limited computer power required continuum assumptions and much spatial averaging to make modelling exercises amenable to analytical techniques

25 or simple numerical techniques. Increased computational power is now allowing closer investigation of process interactions in heterogeneous environments. In other fields, agent-based models are becoming a powerful tool in this regard, and it is apparent from these models that the emergent outcomes from small-scale interactions can differ substantially from equivalent calculations based on aggregates or averages (eg. Bonabeau (22)). The Lattice Automaton Bioturbation Simulator (LABS) model (Choi et al. (22), Boudreau et al. (21)) is the first attempt to introduce agent-based techniques to sediment modelling. Individual-scale processes, such as chemical transformations within animals gut, are possible in this model, and animals activities are controlled by local-scale rules. Regarding the issue of overparameterisation, Boudreau et al. (21) stressed that their automaton rules were constructed from individual-scale data. There was no attempt to manipulate parameters to find a good fit with data, yet their simulated 21 Pb distributions produced similar D b values to those generated from field data. The LABS model should allow further investigation of issues of averaging and aggregation, particularly when non-linear coupled diagenesis reactions are incorporated in its structure. There are strong indications that incorporating more dimensions, and individual scale mixing processes and reactions will have an impact on the large scale emergent behaviour in diagenesis models. There are at least three important considerations: 1. Heterogeneity creates altered reaction zonation and allows incompatible reactions to occur in close proximity; 2. Diffusive transport gradients around small point sources and sinks take very different shapes in one, two and three dimensions; 3. Heterogeneity, randomness and stochasticity in sediments lead to greater uncertainty when attempting to use sediment data to infer underlying transport processes. These topics are considered in turn in the following sections Altered reaction zonation As a simple illustrative example, consider heterogeneous model sediment, comprising mostly oxic sediment interspersed with small anoxic patches. If the model assumes that denitrification occurs in anoxic conditions and is suppressed by oxic conditions, the

26 heterogeneous model will predict some denitrification will occur within the sediment. If, however, only the spatial average of the sediment is considered, the sediment is oxic on average and so no denitrification will be predicted to occur. In heterogeneous sediments, reactions which are not supposed to co-occur in an area can do so quite readily due to fine-scale spatial chemical zonation. Brandes and Devol (1995) modelled nitrate and oxygen profiles measured in Puget Sound sediments, Washington. The measured nitrate and oxygen profile shapes and sediment penetration distances were similar, and could only be explained by a one-dimensional model which allowed denitrification to occur at all oxygen levels. The authors found that a two-dimensional discrete diagenesis model in which the bulk of the diagenesis occurs in highly reactive microsite was able to match the field profiles, while remaining consistent with oxic inhibition of denitrification. Sakita and Kusuda (2) developed a model incorporating particulate organic matter (POM) microsites and used it to model nitrate and oxygen profiles. They found that the shape of the profiles was sensitive to the size and distribution of the POM microsites, and that an equivalent conventional onedimensional diffusion model would underestimate the anoxic and anaerobic reaction rates in the sediment, due to the absence of microsites. More complex reaction geometry produces a greater surface area between reaction zones. Again, using denitrification as an example, a complicated pattern of interspersed oxic and anoxic patches will have a greater area of oxic-anoxic interface than a simple plane separating two blocks of different sediment. The oxic-anoxic boundary is ideal for coupled nitrification/denitrification reactions, so increasing this surface area can substantially increase the denitrification rates in the sediment. A related issue is that of temporal heterogeneity. Aller (1994) concluded that oscillating redox conditions may be among the most important factors affecting organic matter remineralisation in bioturbated sediments: Geometrically and temporally complex redox mosaics are the rule. Experimental evidence and theoretical considerations indicate that even brief, periodic re-exposure to O 2 results in more complete and sometimes rapid decomposition than is possible under constant conditions or unidirectional redox change Diffusion in one, two and three dimensions Harper et al. (1999) investigated some of the ramifications of modelling in only one dimension. Their work was motivated by experimental observations that trace metal

27 profiles are spiky on a small (1µm) scale. Modelling such small-scale concentration peaks in one dimension requires artificially high rates of supply or removal to sustain the steep concentration gradients. They demonstrated this point by considering diffusion from a point source in one, two and three dimensions. One-dimensional diffusion from a source (with fixed concentration boundary conditions) creates a linearly varying concentration with distance from the source. By contrast, the concentration change is proportional to 1/x in two dimensions and to ln(x) in three dimensions. One result is that sharper concentration gradients exist around small sources diffusing into large volumes in 2-D or 3-D models than in 1-D models. In other words, 2-D and 3-D sources more effectively transport material away by diffusion. Consequently, if a 1-D model is used to infer the magnitude of sources and sinks from a profile exhibiting small-scale heterogeneity, then calculated source strengths will be underestimated, and the sink strengths will be overestimated. Harper et al. (1999) pointed out that the effect is most severe when dealing with smallscale heterogeneity in the measured data. Reasons for a spiky measured profile include: high spatial resolution measurements; considerable separation between microniche sources (ie. non-overlapping sources); slow homogenising rate (due to local-scale bioturbation, for example); and fast sink time scales. The authors concluded that the application of 1-D reaction transport models to 3-D systems will result in biased estimates of process rates Uncertainty due to the random nature of sediments Recognising that sediment transport is influenced by randomness, Nicolis (1995) modelled tracer transport by including stochastic forcing in the equation of motion for the tracer. A one-dimensional model was used to demonstrate that the inclusion of stochastic effects has a marked impact on the distribution of tracer arrival times at a particular depth and tracer depths at a particular time. A two-dimensional model incorporating a random walk into the transport equation demonstrated the formation of heterogeneity, due to the appearance of a fractal diffusion front which is complex in shape and broadens with time. These findings point to another reason to include higher dimensions and heterogeneity into sediment models. The inherently stochastic nature of sediments and their transport processes means that the usual simplistic assumptions linking a space axis to a time axis are questionable. If some of this stochasticity were to be included in

28 sediment models, we would be better able to ascertain how much information can be reasonably extracted from tracer cores Summary These considerations of heterogeneity suggest that the incorporation of local-scale interactions for both solid and porewater species is worthwhile. This is particularly true when threshold concentrations are important (i.e. nothing significant happens above or below a critical concentration) and where there are local hotspots inhabited by particular combinations of chemical species and microbial populations. It could be useful also for determining when traditional tracing and dating methods are able to extract meaningful information in environments heavily influenced by random processes. This theme of averaging heterogeneity appears in most chapters of this thesis. The laser scanner experiments in Chapter 2 demonstrate a highly heterogeneous sediment transport pattern. Sediment transport occurs at a range of time-scales, and while some sediment particles oscillate rapidly between surface and depth, other particles away from burrow shafts are subject to much slower burial. Measuring or reporting only areal and temporal averages of sediment turnover excludes much sediment mixing information. Chapter 4 expands the non-local bioturbation model used in Chapter 3 to two and three dimensions, and demonstrates the ease with which this model could be used to introduce heterogeneity to model sediments (while retaining all the advantages of a continuum modelling approach). The burrow irrigation modelling in Chapter 5 draws on Aller s cylinder model approach and specifically accounts for spatial heterogeneity. Some of the limitations to using a radially-averaged version of the cylinder model (a one-dimensional non-local model) are discussed in the Appendix. 1.4 Thesis structure and aims The preceding themes of animal-sediment interactions and heterogeneity permeate all chapters. The broadest aim of my work was to gain a better understanding of the interactions between animals and sediments, with a particular emphasis on burrowing animals impact on sediment diagenesis. This aim necessitated investigations into both bioturbation and bioirrigation. Chapter 2 reports on laser scanner experiments which measured high-resolution maps of thalassinidean shrimp mound topography. The aim of the experiments was to

29 quantify the expulsion and subduction of sediment by these burrowing animals, as apparent in mound changes. The use of a laser scanner is a new approach to this problem, and the results were able to demonstrate potential errors associated with standard sediment turnover measurement techniques. In Chapter 3 the experimental observations from Chapter 2 formed the underlying basis for the construction of a non-local bioturbation model derived from Boudreau's (1986b) general model. The model has analytical solutions, which make it amenable to statistical techniques requiring thousands of model solutions. A maximum likelihood approach was used to assess how well field measurements of tracer cores can parameterise such a model. In Chapter 4 the non-local model is expanded to two and three dimensions to demonstrate the ease with which the model could be used to explore issues of heterogeneity in sediment models (and laterally-averaged two-dimensional results can be compared directly with results from the model s equivalent 1D formulation). The emphasis shifts to bioirrigation in Chapter 5. I modelled the impacts of burrow irrigation on sediment nitrogen dynamics using a cylinder model approach. Port Phillip Bay data were used as a specific case study, as were three other published studies on thalassinidean shrimp burrow chemistry. Relative costs and benefits of one and two-dimensional descriptions were made clear through this work. A discussion about non-local irrigation models is included as an Appendix. Here the circumstances under which a laterally-averaged cylinder model can be considered equivalent to the full cylinder model are explored. An over-arching aim throughout this work was to assess the performance of bioturbation and bioirrigation models, and in particular their ability to explain experimental and field measurements

30 2 Laser scanner measurements of Trypaea australiensis mound topography 2.1 Introduction Decomposition reactions in the sediment are fuelled by the supply of organic matter. An active population of burrowing animals will ensure that organic matter is not simply a thin surface layer, accumulated through sedimentation from the water column. Rather, material may be subducted to considerable depths, or buried rapidly due to the expulsion of sediment from depth. The nature of thalassinidean shrimp burrowing points to both possibilities: sediment expulsion rates from mounds are considerable, so creating high burial rates at the surface; and their deep, permanent burrow structures trap organic matter drawn in by bioirrigation currents or mound collapse. An understanding of the animal mixing processes and rates can aid the estimation of organic matter distribution in the sediment. Typically, measurements of animal mixing are described in terms of a sediment turnover rate, for which there are a number of estimation methods. Rowden and Jones (1993) identified the three most common methods for measuring sediment turnover in thalassinidean shrimp populations, and discussed the relative advantages and disadvantages of each: Direct entrapment. A sediment trap positioned over a burrow opening collects ejected material. Rates are expressed as mass (wet or dry) or volume per mound, individual or area per unit time. This is the most commonly used method. Levelling. An area of mounds is levelled and left for a known period of time. Any new material deposited on the surface is collected or the mound geometry is measured. Rates are expressed as mass or volume per unit area per unit time. Tracer particles. Labelled sediment (eg. fluorescent dye or paint) is placed at known depth(s) in the sediment. Cores are taken after a known time and the depth distribution of labelled particles measured. Rates are expressed as a deposition depth per unit time. Rowden and Jones (1993) concluded that these methods are not directly comparable, and can yield vastly different sediment turnover estimates for the same species. Their main recommendations were to use the direct entrapment method, quote rates in terms of dry mass (rather than wet mass or volume), and to use the characteristics of the

31 measurement to constrain the space and time units (eg. if experiments were conducted over one week, do not extrapolate to yearly estimates). Problems exist with all three methods. Most importantly, they are all destructive methods in some way. The direct entrapment method removes ejected material and the levelling method has an unknown effect on sediment ejection rates. The tracer particle method needs a sectioned core to obtain just one estimate of turnover rates, so turnover rates as a function of time in one place are not possible and this method is most appropriate over long time scales. A significant advantage of the tracer particle method is that it can measure downward transport. The present study made estimates of Trypaea australiensis (Figure 1.2) sediment turnover rates in the laboratory by using a laser scanner. The laser scanner data allowed the creation of high resolution maps of the sediment surface, and the changes in time could be used to estimate rates of mound volume change. This technique has the advantage of being non-destructive, and it is able to detect both upward and downward transport at the sediment surface. 2.2 Experiment description A laser scanner was constructed in the Pye Laboratory, CSIRO in the early 199s (J.H. Taylor, personal communication 1997). Its design was based on the description in Huang et al. (1988), with the main difference being the use of a two-dimensional traversing frame rather than a one-dimensional frame. It had been used to measure high-resolution elevation maps of soil surfaces in erosion studies (unpublished), and had been dismantled. I re-assembled the equipment for these experiments. The equipment consisted of a low power helium-neon laser, a 35 mm format single lens reflex (SLR) camera with a 5 mm lens, a 256 element photodiode array, a highprecision two-dimensional traversing frame and associated electronics and software. The laser was mounted on the traversing frame and directed vertically downwards onto the sediment, creating a light point on the surface. The camera was mounted at a small angle to the laser and the image of the laser point was focussed onto the photodiode array, which was fitted to the back of the camera (Figure 2.1)

32 photodiode array Laser 35mm SLR camera Figure 2.1 Positions of laser and camera relative to the sediment surface. Three possible sediment heights are illustrated to demonstrate the change in image location on the photodiode array. Diagram is not to scale. The position of the image on the diode array changes if the sediment surface height changes, and so can be used to determine sediment surface elevation. The shape of the image on the diode array could be observed using a cathode ray oscilloscope (CRO). The image was always a bell curve spread across several diodes, and so a method was needed to specify the location of the image centre. Following Huang et al. (1988), a threshold intensity was used to find the extent of the image on the array, and the centre of the image was assumed to lie midway between the first and second threshold crossing. The image position is referred to as the scan count. In the original article by Huang et al. (1988), analytical equations were given to determine elevation from the scan count, assuming that the absolute positions of the camera and laser are known. In practice, however, better accuracy is obtained by calibrating image positions against a known sediment surface. The calibration approach is better because the geometry of the camera and laser arrangement cannot be determined to the accuracy required. In these experiments, the camera is in air while the sediment is under water, so there is the further complication of refraction at the water surface. The calibration technique avoids these difficulties. The high precision traversing frame was driven by stepper motors, controlled by a Pascal program (authored by Dale Hughes). I modified this program to read in the scan count while traversing

33 Four aquaria, each 37 mm 57 mm, were positioned beneath the traversing frame and filled with sediment sourced from Corunna Lake, a coastal lake near Narooma, NSW, in south-eastern Australia. This lake has a narrow opening to the ocean, and at the time of sampling its tidal sand flats were inhabited by dense populations of thalassinidean shrimp (mainly Trypaea australiensis). The sediment and the animals used in the experiments were taken from these tidal flats. Artificial seawater was made up according to standard seawater composition, and a combined pump and filtration system was used to maintain the water quality. The sediment depth was between 25 and 27 cm and the water depth was between 7 and 9 cm in each aquarium. An air conditioner maintained the room at approximately 2 C. A calibration ramp was mounted on the inside of one of the walls of each aquarium. The ramp was simply a solid length of plastic, 15 mm in length and 2 mm wide, positioned at an angle to the sediment surface; the uppermost point of each ramp extended out of the water, and the lowermost point extended below the sediment surface level. The ramp elevation was measured relative to the top of each aquarium. Calibration involved traversing the camera and laser over the length of the calibration ramp, so that the image position on the diode array was known over the height range of interest. The scanner, aquaria and calibration ramps are illustrated in Figure 2.2. Figure 2.2 Photograph of the laser scanner experimental setup

34 Three of the aquaria contained three adult Trypaea australiensis individuals each, while the fourth aquarium was left without animals. Each populated aquarium contained one male and two females (sizes given in Table 2.1). The animals were constrained to build their burrows in the centre of each aquarium (a 3 cm by 4 cm area) by the insertion of solid aluminium fences. The fences were needed so that the animals would not burrow near the calibration ramps. Further, measurements could not be made at the edges of the aquaria (aquarium walls blocked the camera s line of sight), so it was more convenient to ensure that the animals were burrowing well away from the walls. The diet of these animals is unknown, but zooplankton was added to the water regularly (usually fortnightly) to ensure fresh organic matter would make its way into the animals burrows. Table 2.1 Description of individuals released into each tank. Diameter was measured at the second joint of the abdomen. Length was measured from the eyes to the posterior end of the abdomen. Sex Diameter (mm) Length (mm) Wet Mass (g) Aquarium 1 Male Female Female Aquarium 2 Male Female Female Aquarium 3 Male Female Female Each aquarium scan comprised 161 lines, 42 mm long and separated by 2 mm. The spatial resolution was 1 mm along a scan line. The detection resolution was actually much finer than 1 mm. The scanner on average detected 18 points over each millimetre section (standard deviation of < 4 points), but only the average scan count over each 1 mm section and the number of points forming this average were saved. One scan produced two files, each containing elements. Scanning time for each aquarium was 1.5 hours, so at least 6 hours were needed to scan all four aquaria. Before each scan, the camera focus was altered to get the sharpest signal on the CRO. Calibration was required prior to each scan and a post-scan calibration was conducted so that an estimate of drift over the experiment could be obtained. Scans were usually separated by approximately a fortnight, but there were periods of more frequent scanning on Aquarium 1. The sediment surface was cleared of all mounds on Day 111 so that recovery from surface disturbance could be observed

35 Analysis was divided into two time periods. Time Period 1 is Day 1 to Day 11, and Time period 2 is Day 111 to Day Data Analysis All data analysis was conducted in Matlab. Analysis comprised calibration, map construction and individual mound analysis, as described in the following three sections Calibration Calibration ramp elevation was measured at known y-values by traversing the laser to a y-location, and then measuring the vertical distance from the top of the aquarium to the laser point on the ramp. Five or six points were measured on each ramp, and linear relationships were found to describe ramp elevation (z) as a function of y (R 2 >.9995). The locations of the calibration ramps were measured either two or three times during the first two weeks of experiments, and again at the end of all experiments just under a year later. Measured values within the first two weeks differed by at most.5 mm, and after a year the ramp locations had changed by at most 1.5 mm, and were more typically under.5 mm from original measurements. At the beginning and end of each aquarium scan, the calibration ramps were scanned five times, producing scan count values for known y locations. (In four cases, one of the ten ramp scans produced a spurious scan count. These four values were replaced with an average of their two neighbours. In one case, a bump was clear on all 1 scans of a calibration ramp, and so seven points were removed from all 1 scans to remove the bump.) The values across the ten scans were averaged, the elevations at these locations were calculated from the linear relationship described above, and a third order polynomial fit was found relating elevation (z) to scan count (c). There were 53 scans of Aquarium 1 and 33 of the other three aquaria, so 152 calibration polynomials were produced using this technique. Residuals were calculated by taking the absolute value of the difference between ramp elevations and elevations calculated from the third order polynomial. The mean residual was.18 mm (n = 14559, standard deviation =.17 mm), and the maximum residual was 1.9 mm Map construction Raw scan count files were converted to elevation using the calibration polynomials. Spurious points (elevations outside the calibration range) were identified and replaced

36 with interpolated functions (Matlab s interp2 function, using the bicubic interpolation method). The worst scan (requiring many such replacements with interpolated values) is shown in Figure 2.3. In this case, specks of dirt were floating on the water surface, and so interfered with the scan (the laser point would either intersect the particles, or it would be refracted when passing the meniscus surrounding the floating particles). The surface water was checked and cleared of debris in all subsequent scans, so this problem was an isolated event. Figure 2.3 Left hand image: surface reconstruction from raw elevation data (red patches mark out-of-range elevations). Right hand image: filtered elevation data (outof-range values replaced with interpolated values). This particular example was the worst case. Most out of range values were due to particles floating on the water surface, and this problem was eradicated in subsequent scans. The resulting elevation files are readily viewed as three-dimensional maps and animation sequences. Surface maps for each aquarium are shown in Figure 2.4 to Figure 2.7. The figures have been generated in Matlab using the surf function. The shadows in the images (due to a simulated light source) are to assist three-dimensional mound visualisation. Elevation is quoted relative to the aquarium top in all cases

37 Figure 2.4 Aquarium 1 elevation maps. Some maps excluded for ease of comparison with other aquaria, and are shown in Figure

38 Figure 2.5 Aquarium 2 elevation maps

39 Figure 2.6 Aquarium 3 elevation maps - 3 -

40 Figure 2.7 Aquarium 4 elevation maps

41 2.3.3 Individual mound analysis Although three individuals were added to each aquarium, it was uncertain how many animals were alive throughout the experiment. In the field, too, mounds can be identified easily but the relationship between the number of mounds and the number of individuals is unknown (McPhee and Skilleter (22)). For these reasons, volumes and rates of change of volumes were calculated for each mound without attempting to assign them to individual animals. Mound regions were identified by tagging locations that experienced elevation changes of >15 mm over the measurement period of interest. Matlab was used to generate lines showing the 15 mm contour (Figure 2.8), and then contour lines were edited manually to close polygons and to split some regions into separate mounds, where possible (Figure 2.9). All mound regions were defined to lie within the fenced area. Volume changes were calculated within the mound regions using trapezoid numerical integration. Some mounds spilled material over the fence, and so volume estimates for these mounds are an underestimate. Figure 2.8 The left-hand image shows the maximum change in elevation in Aquarium 1, over Time Period 1 (Day 1 to Day 11). The right-hand image shows the same data, with the 15 mm contour line in red. All data values below 15 mm are coloured dark blue ( mm) to accentuate mound regions

42 Figure 2.9 The left hand side shows the raw mound regions calculated from Figure 2.8 (Aquarium 1, Time Period 1), and the right hand figure shows the final mound regions after manual editing. Polygons have been closed (eg. lower edge has been added to Mound 1), and mound regions have been split (eg. compare Mound 2 on LHS and Mounds 2 and 4 on RHS). Small regions not obviously associated with a mound have been removed (eg. Mound 26 on LHS). Once mound regions had been identified for each time period, the locations of burrow openings were found for each scan. A Matlab script was used to loop through images of each mound at each scan, and the mouse was used to identify burrow openings for each scan image. All mounds and their openings for each time period are shown in Figure 2.1. The mound area and average (mode) number of openings for all mounds are given in Table

43 Time period Aquarium 1 5A 3A 6A 2A 4A 1A Aquarium 2 2A 1A 4A 5A 3A Aquarium 3 6A 2A 5A 7A 4A 1A 3A Time period Aquarium 1 5B 3B 7B 2B 4B 1B Aquarium 2 2B 1B 6B 3B Aquarium 3 6B 2B 5B 4B 1B 3B Figure 2.1 Mound regions for each aquarium and time period. Dark grey circles mark all hole locations that existed during the lifetime of each mound. (Time period 1: day 1 to day 11; Time period 2: day 111 to day 359.) Axis dimensions are mm. Table 2.2 The area and the average (mode) number of openings for each mound. Letters A and B denote Time Period 1 and 2 respectively. Mound region Aquarium 1 Aquarium 2 Aquarium 3 Average number of openings (mode) Area (cm 2 ) Average number of openings (mode) Area (cm 2 ) Average number of openings (mode) Area (cm 2 ) 1A A A A A A A B B B B B B B

44 2.4 Results and discussion General Observations The sediment was not sieved prior to filling the aquaria, so other small life-forms were present in the sediment. The sediment was sorted by hand to remove any obvious macrofauna (mainly molluscs), but small worms and meiofauna remained. Their presence explains the changes in sediment surface observed in Aquarium 4, and away from mound regions in the other aquaria. The burrow openings to the shrimp burrows are unmistakable, and are easy to distinguish from the smaller surface changes caused by other organisms. Burrow openings, once formed, tended to remain in place in undisturbed conditions. Clearing the surface on Day 111 prompted more dramatic changes, but the general position of the burrow openings remained largely unchanged. Individual mounds were observed in Aquaria 1 and 3, however Aquarium 2 was quite different. Many burrow openings were concentrated into a relatively small area, creating a giant mound comprising many holes (Figure 2.11). Further, the holes in this mound were less permanent. Figure 2.11 Example of a giant mound comprising many burrow openings. (Aquarium 2, Day 62, Mound 1A. Axis units are mm.) Burrow openings were observed to open and close regularly, and considerable downward transport of sediment surface into burrows was apparent. Some mounds were observed to collapse and change into funnels over time (eg. Figure 2.12). Some openings never formed a mound, and so were sites of downward transport only

45 Where a mound and a funnel were close to one another, mound material was observed to move down into the neighbouring funnel (eg. Figure 2.13). Figure 2.12 Example of a mound changing into a funnel. (Mound 4A in Aquarium 3). Figure 2.13 Example of mound material falling into neighbouring funnel. (Aquarium 1, Mounds 2B & 4B, Time Period 2). The two left hand figures show the surface maps, and the two right hand figures show the difference in elevation between two maps. White regions are regions of negligible elevation change, green regions show a rise in surface elevation and red regions show a fall in surface elevation. It is clear from these observations that the burrow openings are sites of relatively rapid two-way exchange of solid material between the surface and depth. In the following

46 section, the scan data is used to calculate rates of transport associated with these surface changes Volume estimates The volume change from scan to scan was calculated for each mound region. The rate of volume change could then be expressed as a volume per mound per unit time. Elevation rises at some coordinates and falls at locations within any mound region, so volume changes have been grouped into three categories: 1. volume expulsion (calculated from points experiencing an elevation increase between adjacent scans); 2. volume subduction (calculated from points experiencing an elevation decrease between adjacent scans); and 3. net elevation change (volume expulsion + volume subduction). Figure 2.14 shows the cumulative volume change for each mound region. The mean rates of change of volume (cm 3 /day/mound region) are given in Table 2.3. These rates were calculated by dividing the final cumulative volume changes for each mound (expulsion, subduction and net) by the number of days, and taking the mean of all mounds. Table 2.3 Mean rates of volume expulsion, volume subduction and net volume change per mound for each aquarium and each time period (cm 3 /day/mound region). Time Period 1 Time Period 2 Aquarium 1 Aquarium 2 Aquarium 3 Aquarium 1 Aquarium 2 Aquarium 3 Mean rate of volume expulsion (std dev) cm 3 /day/mound region Mean rate of volume subduction (std dev) cm 3 /day/mound region Mean rate of volume change (std dev) cm 3 /day/mound region 1.38 (1.11) -.6 (.4).78 (.76) 2.98 (5.56) -.58 (.73) 2.41 (4.86) 1.27 (.86) -.56 (.38).71 (.7).39 (.29) -.36 (.25).3 (.14).83 (1.39) -.8 (1.12).3 (1.25).51 (.36) -.39 (.23).11 (.45)

47 cumulative volume change (cm 3 ) Aquarium 1, Time period Day number Aquarium 2, Time period 1 cumulative volume change (cm 3 ) Aquarium 1, Time period Day number Aquarium 2, Time period 2 cumulative volume change (cm 3 ) Day number Aquarium 3, Time period 1 cumulative volume change (cm 3 ) Day number Aquarium 3, Time period 2 cumulative volume change (cm 3 ) Day number cumulative volume change (cm 3 ) Day number Figure 2.14 Cumulative volume graphs for each mound region within each aquarium and time period (Table 2.2). Each line within a colour group is a single mound region. Green lines show the cumulative volume expulsion, red lines show the cumulative volume subduction and the blue line is the cumulative net volume change (the sum of expulsion and subduction). The marker shape indicates mound region (Mound 1: circle; Mound 2: square; Mound 3: plus; Mound 4: diamond; Mound 5: star; Mound 6: triangle (down); Mound 7: triangle (up)). This straightforward computation neglects many other considerations. Firstly, the giant mound in Aquarium 2 (Mound 1) was an agglomeration of many mounds, so the rate of volume change for this mound far exceeded the rate for other mounds. Defining a robust set of criteria for splitting the region into individual mounds is not possible,

48 and so the best that can be done is to estimate the number of mounds in the giant by counting the average (mode) number of openings that exist and normalising the mound volume rate estimate by the number of openings. Another problem is that it is clear that there was a start-up effect at the beginning of the experiment. The rate of sediment ejection in the first weeks is far greater than in subsequent weeks. The start-up effect is presumably due to the initial construction of a new burrow. Following this initial burrow construction, volume changes are most likely due to routine burrow maintenance, feeding and burrow extension. Forster and Graf (1995) reported that Callianassa subterranea specimens were observed to extend their burrows continuously for periods of over a year. Stamhuis et al. (1996) measured behaviour and time allocation of Callianassa subterranea in laboratory aquaria and found that 27% of animals time was spent burrowing (defined as any interaction with the sediment and transport activities stirring, lifting, carrying, dropping,tamping, bulldozering, pumping and would include feeding). Stamhuis et al. (1997) measured the total length of Callianassa subterranea tunnel systems over time, and found that initial burrowing velocity (increase in total burrow length per hour) was much higher for the first approximately four days of new burrow construction. Figure 2.15 shows the same data split into three time periods (Time period 1 split into a start-up period (three weeks) and a subsequent lower growth period), and the data has been normalised by the average (mode) number of openings in each mound at each time step. The data from Day 359 have been excluded; 77 days elapsed between the second last and last scan (Day 282 and Day 359), whereas the usual interval between scans had been no longer than three weeks. These data were used to calculate mean rates of volume change per burrow opening for three time periods (Start-up period, remainder of Time Period 1 and Time Period 2). The results are shown in Table

49 cumulative volume change (cm 3 /opening) cumulative volume change (cm 3 /opening) cumulative volume change (cm 3 /opening) Aquarium 1, Time period Day number Aquarium 2, Time period Day number Aquarium 3, Time period Day number volume change (cm 3 /opening) volume change (cm 3 /opening) volume change (cm 3 /opening) Aquarium 1, Time period Day number Aquarium 2, Time period Day number Aquarium 3, Time period Day number Figure 2.15 Same data as Figure 2.14, normalised by the number of holes in each mound region. Time period 1 has been split into a start-up period (three weeks) and a subsequent period. The colour code and markers are the same as Figure

50 Table 2.4 Mean rates of volume expulsion, volume subduction and net volume change per opening for each aquarium and each time period (cm 3 /day/burrow opening). Mean rate of volume expulsion (std dev) cm 3 /day/opening Mean rate of volume subduction (std dev) cm 3 /day/ opening Mean rate of volume change (std dev) cm 3 /day/ opening Aquarium ( 4.5) -.45 (.41) 4.51 (4.29) Time period 1a Aquarium (3.36) -.29 (.21) 2.12 (3.24) (start-up) Aquarium (2.15) -.34 (.21) 3.18 (2.1) Aquarium 1.63 (.58) -.65 (.46) -.2 (.5) Time period 1b Aquarium 2.79 (.91) -.31 (.17).47 (.89) Aquarium 3.9 (1.13) -.64 (.47).26 (1.15) Aquarium 1.52 (.42) -.48 (.29).4 (.26) Time period 2 Aquarium 2.36 (.38) -.83 (1.54) -.47 (1.47) (excluding Day 359) Aquarium 3.68 (.52) -.53 (.34).14 (.68) Strong sampling frequency dependence is apparent in the data. Aquarium 1 was scanned more frequently than the other aquaria between Day 184 and 219 (Time Period 2), and the cumulative volume change graph is markedly steeper during this time period (see Figure 2.15, Aquarium 1, Time Period 2). I calculated the mean rate of volume change per unit area between scans in all mound regions, and plotted them against the time interval between scans (Figure 2.16). Rates of volume change increase with decreasing time interval

51 mean rate of volume change per unit area between scans (cm/day) time interval between scans (days) Figure 2.16 Mean rate of volume change per unit area between scans. Circles are sediment accumulation data, squares are sediment subduction data. Blue and red markers are data from mound regions in Aquaria 1 to 3, black markers are data from the unpopulated Aquarium 4. Error bars indicate ± 1 standard deviation. This pattern is consistent with an aliasing effect. For example, if the surface elevation h were to rise and fall in a sinusoidal pattern according to the following equation: h = A sin 2π T t (2.1) where t is time (days), A is the amplitude (cm) and T is the period of oscillation (days), the rate of volume expulsion per unit area would be 2A/T cm/day, the rate of volume subduction per unit area would be 2A/T cm/day, and the net rate would be cm/day (Figure 2.17). surface elevation (cm) A expulsion subduction expulsion A T/4 T/2 3T/4 T time (days) Figure 2.17 Surface elevation rising and falling according to equation (2.1). Sediment is being expelled during the first and last quarter, and subducted during the middle two quarters, as indicated by shading

52 If this surface were to be measured infrequently, the calculated volume rates would be a strong function of sampling interval. For example, Figure 2.18 shows the calculated volume expulsion and subduction rates per unit area for different sampling intervals (surface elevation is given by equation (2.1), with A = 1 mm and T = 2 days). The behaviour is the same as that observed in Figure rate of volume change per unit area (cm/day) expulsion subduction average sampling interval (days) Figure 2.18 The rate of volume expulsion and subduction calculated from sampling an oscillating surface randomly in time over 1 days at a range of sampling frequencies. The surface elevation is given by equation (2.1), with A = 1 mm and T = 2 days. The impact of sampling interval on rates of volume change is substantial. Figure 2.19 shows the same scan data sampled at different frequencies (Aquarium 1, Days 184 to 219). During this period of time, Aquarium 1 was sampled more frequently than the other aquaria (typically daily, with some exceptions). The other aquaria were only sampled on Days 184, 199, 25 and 219. The rates of volume change from these two scenarios are given in Table 2.5. The calculated rates are 3 or 4 times higher than those calculated from the same data sub-sampled on days 184, 199, 25 and 219. (The rates of net volume change remain the same as this is determined only by the first and last scan of the sampling period.)

53 cumulative volume change (cm 3 ) Day number cumulative volume change (cm 3 ) Day number Figure 2.19 Cumulative volume changes for Aquarium 1, Days 184 to 219, sampled at different frequencies. Left hand graph: all data; Right hand graph: the same data sampled less frequently (at the same frequency as the other aquaria). The colour code and markers are the same as Figure Table 2.5 Influence of sampling frequency on calculated mean rates of volume change. Mean rate of volume expulsion (std dev) cm 3 /day/opening Mean rate of volume subduction (std dev) cm 3 /day/ opening Mean rate of volume change (std dev) cm 3 /day/ opening Frequent sampling 1.12 (.84) (.9) -.14 (.42) Infrequent sampling.28 (.25) -.42 (.37) -.14 (.42) Figure 2.2 shows the maps constructed from the frequent scans of Aquarium 1. It is clear from these maps that the mound holes open and close on a daily basis, creating an oscillating change in elevation near the opening of each mound. The results in Table 2.5 demonstrate that if this oscillation isn t sampled adequately, much of the upward and downward transport is missed and calculated rates of volume expulsion and subduction are underestimates

54 Figure 2.2 Maps of Aquarium 1 sediment surface during period of frequent scanning. Matters are further complicated because it is likely that in the vicinity of the mound openings, the same sediment is getting shifted up and down. Some particles are oscillating rapidly between the surface and depth, while others nearby remain unmoved over the same period of time. It s clear that there is a distribution of sediment turnover rates in the sediment, and for some applications valuable information is lost if the sediment turnover is described only as an average quantity. The nature of these experiments does not allow a good estimate of the dry mass of sediment expelled per annum, simply because volumes of sediment were measured rather than mass. As a rough guide, however, if we assume a porosity of.4 and a sand density of 2.65 g/cm 3, expulsion rates range from 16 to 64 g/opening/year and subduction rates range from 17 to 74 g/opening/year. The population density of Trypaea australiensis varies enormously. Assuming 2 burrow openings/m 2 and the

55 lower expulsion rate, sediment expulsion rates could be as low as 3kg/m 2 /year. If there are 2 burrow openings/m 2 and the higher expulsion rate, sediment expulsion rates could be as high as 127 kg/m 2 /year. Subduction rates would cover a similar range. There are good reasons for believing that the volume rates calculated from this experiment are far lower than experienced in the field. Comparisons between field and laboratory measurements for other species of thalassinidean shrimp have revealed large differences. The following examples all used direct entrapment measurement techniques. Bird (1997) estimated sediment turnover rates for Biffarius arenosus in the field and found ejection rates of 4.26 g dry mass sediment/individual/day. Her earlier measurements conducted in the laboratory (using the same species) had produced a sediment turnover estimate of.79g dry mass/individual/day (Bird et al. (1997)). Laboratory-derived estimates for Callianassa subterranea are 15 kg/m 2 /year (Stamhuis et al. (1997)) and 11 kg/m 2 /year (Rowden et al. (1998)), yet the same methods used in the field for Callianassa filholi yielded a sediment turnover rate of 96 kg/m 2 /year (Berkenbusch and Rowden (1999)), even though the density of Callianassa filholi individuals was 2.5 times lower than the density of Callianassa subterranea and the animals are deposit feeders of a similar size. Berkenbusch and Rowden (1999) suggested that the large difference between the two measurements reflects the fact that the Callianassa subterranea measurements were made in constant laboratory conditions, while the Callianassa filholi measurements were made in situ and accounted for a number of physical factors (including temperature variation, animal size and burrow position relative to the shore). Other burrowing animals in the field would encounter and damage thalassinidean burrows, so creating more burrow maintenance work for the inhabitants. Other contributing factors could be the presence of juveniles and postlarval shrimp in field studies (they were absent from the laboratory studies), and the fact that the Callianassa filholi measurements were made on an intertidal sandflat (the authors believe there may be a significant difference in sediment turnover rates between intertidal and subtidal populations). Berkenbusch and Rowden (1999) noted a strong seasonal variation in Callianassa filholi sediment turnover rates. By maintaining the laboratory at approximately 2 degrees, I removed a seasonal influence from my experiments. There has been speculation that thalassinidean sediment turnover rates are influenced by sediment nutrient content, however measurements reported by Stamhuis et al. (1997) were unable to confirm a significant effect

56 2.5 Conclusions The techniques employed in these experiments are more accurate than direct entrapment or mound volume estimates based on cone volume formulae. A further advantage is that the scanner detected downward as well as upward transport. Unfortunately, the animals in these experiments were restricted to laboratory aquaria which were shallower than their native sediment, and there were none of the disturbances typically experienced in the field (other burrowing macroinvertebrates, waves, tides, storms and fishing folk armed with yabby pumps). For these reasons it would be desirable to undertake a similar experiment in the field. Nevertheless, the experiments yielded useful insights into the creatures burrowing habits, and highlighted many issues regarding turnover measurement techniques. The experiments have demonstrated substantial downward transport of sediment, particularly once a burrow has been established. Burrow openings function as chimneys of rapid exchange between the sediment surface and depth. This mechanism is important as it would facilitate rapid transport of fresh organic matter to considerable depths in the sediment, so affecting diagenesis pathways at depth. The oscillating transport between surface and depth has not been detected by any of the conventional methods for estimating sediment transport. Further, a high sampling rate needs to be employed to detect the oscillating transport; failure to do so produces significant underestimates in the calculated expulsion and subduction rates. These experiments have done little to constrain Trypaea australiensis sediment turnover estimates. Rather, they have highlighted the potential for enormous variation in estimates. Similar difficulties have been encountered in other studies. For example, Stamhuis et al. (1997) recalculated sediment turnover rates for Callianassa subterranea by employing additional knowledge about temperature and population density effects, and nearly trebled an earlier estimate using the same data. Work by Berkenbusch and Rowden (1999) and Rowden et al. (1998) have highlighted the sensitivity of turnover estimates to body size, temperature and overlying tidal behaviour. The most striking effect highlighted in my experiments has been the effect of downward transport and sampling frequency. I have found very little discussion on these points in the literature. We surmise from these experiments that some sediment particles are oscillating rapidly between surface and depth, while other particles nearby are buried far more slowly, to return to the surface only after a much longer time scale. Problems in assessing the distribution of sediment turnover times and depths within the sediment

57 will not be resolved using any present techniques. There may be some scope for nondestructive tracer particle experiments (allowing continuous measurements rather than taking destructive cores). In particular, experiments with fluorescent particles are possible. For example, particles could be buried at depth and their arrival at the surface recorded by multiple photographs under ultra-violet radiation. Alternatively, such particles could be distributed as a layer on the surface and their subduction monitored by photographs and subsequent coring at experiment completion. Another possibility is the use of tungsten carbide labels, which can be detected under x-ray radiation

58 3 One-dimensional sediment modelling: excavation, infill and collapse 3.1 Introduction Diffusion models are commonly used to model the transport of solid species in sediment, despite the fact that the drawbacks of this approach are well known. Over 15 years ago, Boudreau (1986a) showed that the diffusion analogy is appropriate under only very specific circumstances: the mixing needs to be rapid relative to the tracer decay rate, and the transport distances need to be small relative to the size of the mixing layer and the scale of the profile. Boudreau (1986b) developed a general model framework that allows the construction of any number of more physically realistic non-local models. Non-local models include variations on the conveyor belt model theme (eg. Robbins (1986), Boudreau (1986b)), and less commonly regenerator or burrow-and-fill models (eg. Gardner et al. (1987), Sharmer et al. (1987)). Transition matrix approaches (discussed in Section 1.2.3) are another form of discrete non-local model, and have been used in several studies (eg. Foster (1985), Trauth (1998), Shull (21) and Shull and Yasuda (21)). All these models have the advantage that they are firmly linked to measurable physical processes (eg. worm feeding and excretion rates and depths), as opposed to a biodiffusion coefficient. My laboratory measurements of thalassinidean shrimp sediment transport (Chapter 2) demonstrated rapid excavation of sediment from depth, and considerable downward transport of surface material back into the burrow. A diffusion model would be quite inappropriate for modelling these sediment transport processes, and in this chapter I propose a non-local model which is a hybrid between a conveyor belt and burrow-andfill model (Boudreau (1997)). I refer to it as the EIC model (Excavate, Infill and Collapse). One view that I ve encountered during my work is that a more complicated description of bioturbation is unnecessary since the diffusion model is not over-parameterised and it usually gives a good fit to the data. A more realistic transport process description is desirable if the aim is to overlay sediment diagenesis reactions onto the transport model. In practice, getting the data for a more physically realistic transport process is rarely possible. For example, Boudreau (1986a) concluded For isotopes outside the

59 window of sensitive parameter values, the data may never be of sufficient accuracy to warrant the use of realistic but more complex mixing functions. In this chapter I demonstrate that the EIC model can be applied with the same ease as a diffusion model and can produce very similarly shaped profiles. Its value may be in testing diagenesis model results. The same diagenesis model could be executed with two very different underlying transport assumptions to see whether results differ significantly. Further significant advantages of the EIC model are that the parameters can be linked to measurable quantities, and that it can be readily expanded into two or three dimensions (discussed in Chapter 4). 3.2 Model derivation Boudreau (1986b) described a general one dimensional model that allows descriptions of bioturbation at a range of scales. Diffusion, conveyor belt, burrow-and-fill (or regenerator) models and many other bioturbation models can be derived from the general equation (Boudreau (1986b), Boudreau and Imboden (1987)). The tracer conservation equation allows for transport between different locations within the sediment column, and for exchange across the sediment water interface: φ sz, t p z, tl2 = φ sz', t K z', z; t p z', tdz' t + K z, t f t φ φ I p L sz p z t1 sz, t p z, tkez, t,, K z, z' ; t dz' φ s z, t w z, t p z, t z +R z, t, p z, t transport to z from elsewhere in the sediment column; injection of material at z from the overlying water column; removal from z to elsewhere in the sediment column; removal from z to the overlying water column; transport by advection production or decay (3.1) where φ s is the solid fraction (φ s = 1 φ, where φ is porosity), p(z,t) is the property of interest (eg. tracer concentration), t is time, K(z,z';t) is the fraction of material removed from depth z per unit depth and per unit time and transported to depth z', K I(z,t) is the fraction of the external flux injected per unit depth in the sediment, f p(t) is the external flux of the property of interest (units of property mass per unit area and time), K E(z,t) is the fraction of material removed from depth z and transported to the water column per unit time, R is the rate of production or decay due to chemical reaction or radioactivity, L 2 is the deepest position of material delivery and L 1 is the deepest point of material removal

60 For burrowing animals such as thalassinidean shrimp, I have assumed that the dominant transport processes are: burrow excavation (material transported from depth to the surface), burrow infill (surface material dropping down a burrow shaft and filling the burrow) and burrow collapse (material directly above a burrow tunnel shifting downwards to close the tunnel). The EIC model is a one-dimensional description of these processes derived from equation (3.1). Only four terms in equation (3.1) have been used: removal from depth to the overlying water column, transport to depth from elsewhere in the sediment column, transport by advection, and the reaction term. The model is split into two layers so that a uniformly mixed surface layer of thickness z r overlies a region containing burrows extending to depth L. It is assumed that the surface layer is mixed rapidly due to waves, and surface dwelling animals. The lower region loses material to the surface (burrow excavation) and receives well-mixed material back from the surface layer (burrow infill) and from the sediment directly above the burrow structures (burrow collapse). In other words, burrowed material is transported to the surface and the resulting burrow void is refilled with either surface sediment, sediment collapsed from above or a mixture of the two. There is no diffusive mixing. Diagrams representing the model assumptions are given in Figure 3.1. Removal from the lower layer is described by a removal function K E(z) (the fraction of sediment volume removed at depth z per unit time, units of time -1 ). The excavated material returns to the sediment column via the sediment surface, and is characterised by a surface flux boundary condition in the model. The burrows immediately receive uniformly mixed surface material. In Boudreau s burrow and fill model (Boudreau and Imboden (1987)), the burrows are filled completely through infill. In the EIC model, the extent to which the burrow is filled is set by a parameter, f whose values lie between and 1. When f = 1, the burrow is completely filled with surface material and there is no burrow collapse. When f =, no surface material fills the burrow and the burrow cavity closes due to collapse. Intermediate values represent a combination of infill and burrow collapse

61 L (a) (b) (c) L (d) (e) (f) Figure 3.1 An idealised representation of infill-only ((a), (b) and (c)) and collapse-only ((d), (e) and (f)) burrowing scenarios, with zero sedimentation velocity. The uniformlymixed surface layer is represented by a diagonal-striped pattern. In both cases, a burrow is excavated from a horizontal tunnel at depth (red rectangle of sediment in (a) and (d)). The excavated material is removed from the sediment column to the overlying water leaving a void at depth (black rectangle in (b) and (e)). The excavated material is deposited on the surface and uniformly mixed through the surface layer (change represented by shift from black to red diagonal stripes, indicating that excavated material has been included in the surface layer). The void is filled either with surface sediment only (infill, (c)), sediment directly above the void only (collapse, (f)) or a mixture of the two (not illustrated). In this chapter the EIC model is onedimensional and assumes a uniform burrowing rate with depth. This diagram represents non-uniform burrowing in two dimensions for ease of conceptualisation only. The four terms from equation (3.1) used in the derivation of the EIC model are as follows. L φ s z ', t K z ', z ; t p z ', t 2 dz ' represents transport from elsewhere in the sediment column. This term captures the transport from the surface layer (z* < z r*) to the burrowing zone (z r* < z* < 1). φ s z, t p z, tk E z, t is removal of material from location z to the overlying water column. In other words, I assume that the material excavated in the burrowing process

62 is pumped to the water column and its return to the sediment surface is represented by a flux boundary condition (described later). φ s z, t w z, t p z, t represents transport by advection. z is the reaction/production term. R z, t, p z, t Putting these terms together yields: φ sz, t p z, t L2 = φ sz', t K z', z; t p z', td z' φ sz, t p z, tkez, t t φ sz, t w z, t p z, t + Rz, t, pz, t z (3.2) is the fraction of material removed from depth z' per unit depth and K z', z; t p z', t per unit time and transported to z. In the EIC model K is zero if z' > z r (no material is moved from the burrowing zone to elsewhere in the sediment column, other than by advection, as all excavated material is expelled to the water column). Material is taken from the surface mixed layer (z' < z r) and distributed over a range of z, by a function defined here as K z', z; t = a s z if z' < z s f r if z' > z r where a s is a constant and s f(z) L is a distribution function defined so that sf z d z = 1. (3.3) Now substituting in the definition for K and setting φ σ (z) = φ s1 and p(z) = p 1 when z < z r (both constants because the surface layer has uniform properties) the first term of equation (3.2) becomes: L z r φ s z', t K z', z; t p z', t d z' = φ s z', t assf z p z', t d z' (3.4) = φ s1p1aszrsf z So we now have: φ sz, t p z, t t = φ s1p1 aszrsf z φ s z, t p z, t KE z, t (3.5) φ s z, t w z, t p z, t + R z, t, p z, t z

63 The next assumption is that the volume returned to the burrow per unit time cannot exceed the volume excavated from the burrow per unit time. The excavated volume per unit time is: L KE d z = Kav L (3.6) so K av (dimension time -1 ) is defined to be the depth-averaged excavation rate over the sediment column. The fraction of this excavated volume that is returned as burrow infill is denoted f, so the infill volume per unit time is fk avl. Using equation (3.5) and substituting the sediment density ρ s for p, the mass of solid L infill is φ s1 z r a sρ s s fd z z = φ s1ρ s a s z r and the bulk density of the solid infill is φ sρ s, so the volume of solid infill is z ra s. Thus we infer that z ra s = fk avl and so φ sz, t p z, t t = φ 1p1 fk Ls z φ z, t p z, t K z, t (3.7) s av f s E φ s z, t w z, t p z, t + R z, t, p z, t z The lower layer equations for total solids and tracer concentration are then: φ s t = wφ s sf z fkavlφ s1 KE z φ s z (3.8) φ sc = wφ sc sf z fkavlφ s1c1 KE z φ sc λφ sc t z (3.9) where λ is the decay constant for the tracer, C is the concentration of the tracer (mass per unit volume of solids), C 1 is the tracer concentration in the surface layer and φ s1 is the solid fraction in the surface layer. The terms on the right hand side of equation (3.9) represent infill, excavation, advection and decay (from left to right respectively)

64 The surface layer equations for total solids and for tracer concentration are: dφ s dt 1 wφ s wrφ s1 φ s1fkavl sf z d z zr = z wφ s wrφ s1 φ s1fkavl = z r r L (3.1) dφ s C dt 1 1 wφ sc wrφ s1c1 φ s1c1fkavl sf z dz zr = λφ s C z wφ sc wrφ s1c1 φ s1c1fkavl = λφ s C z r r L (3.11) where φ s and C are the solid fraction and tracer concentration of the material arriving at the sediment water interface, w r is the velocity at the base of the surface mixed layer. w is the velocity at the sediment water interface, and is comprised of the sedimentation velocity (w E) and the excavated material being returned to the sediment via the sediment surface. The flux boundary conditions at the sediment surface are: L s = se E + s Ed z φ w φ w φ K z z r (3.12) L s = se in + s E d z φ w C φ F φ K z C z r (3.13) where φ se is the solid fraction of sedimenting material and φ sef in is the concentration flux of material arriving at the sediment water interface from the water column. The terms on the right hand side of each equation represent the fluxes to the surface due to sedimentation and excavation respectively. Note that φ SEF in could be represented as φ SEw EC E (where C E is the concentration of incoming material), however I wanted to be able to model finite concentration fluxes with zero volume flux. Boudreau (1986a) non-dimensionalised equations before further analysis. Nondimensionalisation simplifies the analysis somewhat and allows the identification of important parameter combinations. Characteristic length, concentration flux and time scales have been chosen from L, F in and K av respectively. Note that Boudreau (1986a) used L/w as a characteristic time scale, which presents difficulties if dealing with a system with zero sedimentation velocity. A more meaningful measure in cases where sedimentation velocity is negligible is to normalise quantities against a measure of the

65 excavation rate, K av. Boudreau used the tracer concentration of the sedimenting material to normalise tracer concentration. Here I have normalised by the incoming concentration flux to allow for the possibility that there may be a negligible volume flux (w E = ) but significant concentration flux to the surface. The non-dimensionalised variables are defined as: z CLKav w λ z* = ; C* = ; t* = Kavt; w* = ; sf * = sf L; λ* = ; K L F KavL Kav in The lower layer equations become: E K * = K E av φ s t * = w * φ s φ s1 sf * f KE * φ s z * (3.14) φ sc * = φ φ w * φ sc * s1sf * fc1 * KE * sc * λ * φ sc * t * z * (3.15) The surface layer equations are: φ s w * φ s wr * φ s fφ s = t * z * r (3.16) φ s1c1 * w * φ sc * wr * φ s1c1 * fφ s1c1 * = λ * φ s1c1 * t * z * r (3.17) The surface boundary conditions are: φ w * φ w * φ K * d z * = + s se E s z * φ w * C * φ φ K * C * d z * s se s z * 1 = + r 1 r E E (3.18) (3.19) It should be noted that there is scope for temporal variation in many of the functions and parameters. For example, the proportion of infill and collapse, f could be timevarying, as could the excavation and infill functions (K E and s f). These possibilities have not been explored in this work. After making some simplifying assumptions, I derived steady state analytical solutions to these equations. The assumptions are by no means justifiable on biophysical

66 grounds, and are made to allow the convenience of an analytical solution (Mulsow et al. (1998) referred to this as an ostrich-like strategy, and I agree). It is assumed that porosity is constant, that K E is a constant over all depths, and that surface material is distributed back into the burrow uniformly, so s f * = K * = E, z* z * 1, 1 z * r r z* > z * r (3.2) Constant porosity and steady state assumptions reduce the lower equations to w * f 1 = z * 1 * z r (3.21) w * C * = z * fc1 * C * λ * C * (3.22) 1 z * r and the surface layer equations become w * f w r * = (3.23) w * C * fc1 * wr * C1 * λ * C * = z * r 1 (3.24) and the surface boundary conditions are w * w E * 1 = + (3.25) w 1 1 * C * = 1 + C *d z * 1 z * z r * r (3.26) Analytical solutions to these equations are: wr * = w * f (3.27) w * 1 zr * + zr * f + f 1 z * w* = 1 z * r (3.28)

67 fc1 * C* = f + λ * 1 z * r λ * 1 zr * C * + f + λ * 1 z * 1 r w * 1 zr * + zr * f + f 1 z * w * f1 zr * r f + λ* 1 z * 1 f (3.29) 1 f λ * 1 zr * A3 C1* = 1 + A A A A A 1+ λ * 1 z * where r w * 1 A λ* 1 zr * (3.3) 1 f 1 A = w * + λ * z * 1 r 2 λ r A = f + * 1 z * A = w * f 3 These solutions are valid for f < 1. f = represents the collapse-only case (which is equivalent to an upward conveyor belt model with uniform removal with depth), and f = 1 represents the infill only case (which is equivalent to Boudreau s burrow-and-fill model). The analytical solution for f = 1 needs to be derived separately. For f = 1 the lower layer equations are: w * = z * (3.31) w * C * C1 * C * = λ * C * z * 1 z * r (3.32) and the surface layer equations are: w * 1 w r * = (3.33) w * C * C1 * wr * C1 * λ * C * = z * r 1 (3.34)

68 If the sedimentation velocity is zero, the analytical solution is simply two layers of uniform concentration. The lower layer tracer concentration is C1 * C* = z r * λ * (3.35) and the upper layer tracer concentration is C * = 1 r r 1+ 1 zr * λ * 1+ λ * z * 1+ 1 z * λ * 1 1 (3.36) For a non-zero sedimentation velocity, the analytical solution is: C * z * zr * 1+ 1 zr * λ * (3.37) 1 C* = 1 λ * 1 zr * exp z * λ * w * 1 z * C * = 1 w r 1 * + λ * zr * 1 + λ * 1 z * λ * 1 zr * we * z * λ * r r exp r z * λ * w * E E 1 r 1 (3.38) I constructed a numerical version of the model (equations (3.14) to (3.19)) using the method of lines in Matlab. The advection terms were discretised using an upwind differencing scheme, and Matlab s stiff ordinary differential equation integrator, ode15s was used to integrate the resulting ordinary differential equation. There was excellent agreement between numerical solutions and the analytical solutions (visually indistinguishable). The numerical model is used in Section to model more complicated porosity and excavation assumptions. 3.3 Physical interpretation and typical behaviour Effect of parameter values in the EIC model The key parameters in the EIC model are z r*, f, λ* and w E*. The non-dimensional thickness of the surface uniformly mixed layer is given by z r*. f represents the relative magnitude of infill compared with excavation. λ* represents the relative magnitude of decay compared with excavation rate. The normalised sedimentation rate w E*

69 represents the relative magnitude of the sedimentation rate compared with excavation. The relative influence of each of these processes is illustrated in the following examples. The effects of varying f can be seen in Figure 3.2 for w E* =.1, λ* = 1 and z r* =.1. The burrowing zone is located in the region.1 < z* < 1. Beneath the burrowing zone tracers are subject to decay and burial only. For f = (collapse only), the tracer profile decreases linearly from the surface mixed layer. For f = 1 (infill only), the tracer profile decays rapidly to a relatively high constant value which persists to the base of the burrowing zone. Surface material is being directly injected uniformly with depth to the base of the burrowing layer in this case, so it is not surprising that relatively high concentrations are seen at depth. The f =.5 profile (equal amounts of collapse and infill) generally lies between the two extremes of the f = and f = 1 case. Below the base of the burrowing zone at z* =1 the tracer is subject only to natural sedimentation velocity and decay. λ* = 1 z r *=.1 w E * =.1 Normalised depth, z*.5 1 f= f=.5 f= Normalised concentration, C* Figure 3.2 Comparison between EIC model solutions with different f values. Increasing w E* has the effect of reducing the differences between the three models (Figure 3.3). This would suggest that in areas where the natural sedimentation rate is high, quantifying the relative amounts of burrow collapse and infill would appear to be unimportant

70 Normalised depth, z* λ* = 1 z r *=.1 w E * = f= 1.8 f=.5 f= Normalised concentration, C* Figure 3.3 EIC model solutions with the same parameters as Figure 3.2, except w E = 1. Decreasing w E* to zero maximises the differences between the three models (Figure 3.4). Note that the f = 1 profile is simply a three-layer profile. There is a surface peak overlying a uniformly mixed sediment column to the base of the burrowing zone, and no tracer is transported below the base of the burrowing zone. Clearly, if sediment turnover due to bioturbation is so high that natural sedimentation rates are negligible, the specific mechanisms for exchange play a more important role in determining the shape of tracer profiles

71 .2 Normalised depth, z* λ* = 1 z r *=.1 w E * = 1.2 f= f=.5 f= Normalised concentration, C* Figure 3.4 Model solutions with the same parameters as Figure 3.2, except w E = Comparison between the EIC and diffusion models For the purposes of comparison, the diffusion model can be formulated in a similar framework to the EIC model by assuming that there is a uniformly mixed surface layer of thickness z r overlying a burrowing zone (with biodiffusion coefficient D b) extending to depth L. The sedimentation rate is w E and the material flux at the surface is F in. Again, constant porosity is assumed. Non-dimensionalising in a similar manner to before (although D b is used instead of K av) z CDb Dbt wl λ z* = ;C* = ;t* = L LF L 2 ;w* = ;λ* = L in D b 2 D b The governing equation (non-dimensionalised, steady state) for the burrowing zone is 2 d C * * d C w * E λ * C* = d z * d z * 2 (3.39)

72 For the inventory, I = Cd z to be at steady state, the incoming flux balances the loss due to decay so forming one condition that must be met when solving equation (3.39). d I = Fin I = dt λ λ Cd z = F in λ * C *d z* = 1 (3.4) Another condition at steady state is that the diffusive flux out of the uniformly mixed surface layer needs to balance remaining material fluxes in this layer (surface flux entering the layer, decay and advection out of the layer). The condition is expressed in the following equation * * * * d C * 1 λ * C1zr wec1 + = (3.41) d z * * z r Other conditions are that concentrations need to be continuous at boundaries between layers (ie. at z* = z r * and z* = 1). The analytical solution equations describing the diffusion model profiles are then C* = 1 2 E * * * 1 r 2 r r Aexp β z + Bexp β z z* z Aexp β z * + Bexp β z * z < z* 1 λ * Aexp β 1 + Bexp β 2 exp w z * 1 z * > 1 * * r (3.42) where 1 A = γ 2 B; B = γ * * * * 1 1 r E 2 2 r E γ = β λ * z w ; γ = β λ * z w 1 α = z we exp β 1zr + + β 1 β 1 β 1 λ * exp * * r γ 1 + α 1λ * γ α γ α λ * * * 2 * E E E w + w + 4λ * w β 1 = ; β 2 = 2 * 1 * 1 we α 2 = zr exp β 2zr + + β 2 β 2 β 2 λ * exp * * * 2 w E + 4λ *

73 For the purposes of comparison, the following examples have been converted back to dimensional variables. Selected fixed values of L, z r, F in, λ, Κ av, f and w E were used and then values of D b were found that produce the best diffusion model match to the EICgenerated profile (while retaining the same L, z r, F in and λ values), and vice versa. Figure 3.5 illustrates an example that is the equivalent to the scenario in Figure 3.2 (λ* = 1, w E* =.1, z r* =.1, f =.5 in the EIC model). Values of L = 7 cm, F in = 1 and λ =.31 y -1 were chosen (so K av =.31 y -1 ). The value of D b that provides the closest match to the EIC model (least squares difference between the two profiles) is D b = 4. cm 2 y λ =.31y 1 L = 7 cm z r = 7 cm F in = 1 4 depth (cm) EIC model K av =.31y 1, f =.5, w E =.22cmy 1 diffusion model Db = 39.99cm 2 y 1, w E =.22cmy Concentration Figure 3.5 Comparison between EIC and diffusion model profiles. The EIC model parameters are the same as those in Figure 3.2 (f =.5 case), and diffusion model parameters have been found to produce the least squares difference between the two models. The two profiles in Figure 3.5 are indistinguishable, which agrees with Boudreau s comment that similar profiles can be generated from very different processes. For other parameter values, however, the match is not so close. In Figure 3.6(a) the value of λ* for the EIC model is increased to 5 (so the corresponding value of K av is.62) and the value of. In this example the diffusion model provides a relatively poor match; in particular the non-zero values at depth resulting from surface material infill cannot be

74 simulated using a diffusion approach. In this case I forced the diffusion model to use the correct sedimentation velocity, w E =.43 cm/yr. If w E is allowed to vary, the match is no better (Figure 3.6(b)). 1 λ =.31y 1 L = 7 cm z r = 7 cm F in = 1 1 λ =.31y 1 L = 7 cm z r = 7 cm F in = depth (cm) depth (cm) EIC model K av =.62y 1, f =.5, w E =.43cmy 1 diffusion model Db = 4.72cm 2 y 1, w E =.43cmy Concentration (a) 6 7 EIC model K av =.62y 1, f =.5, w E =.43cmy 1 diffusion model Db = 5.7cm 2 y 1, w E =.16cmy Concentration Figure 3.6 Two diffusion model fits to EIC model profiles: (a) the diffusion model has been forced to use the same sedimentation velocity as the EIC model; (b) all diffusion model parameters are allowed to vary to fit the EIC model profile. (b) The differences are further accentuated if f is closer to 1 in the EIC model, as illustrated in Figure 3.7. The diffusion model is unable to produce a profile that is generated by such a high level of infill from the surface

75 1 λ =.31y 1 L = 7 cm z r = 7 cm F in = 1 2 depth (cm) EIC model K av =.62y 1, f =.8, w E =.43cmy 1 diffusion model Db = 7.45cm 2 y 1, w E =.23cmy Concentration Figure 3.7 Comparison between EIC and diffusion model profiles. EIC model parameters are the same as in Figure 3.6, with the exception of f which has been increased to.8. The comparison of the diffusion and EIC models demonstrates that the EIC model produces a more diverse range of profile shapes. While some of its profiles are very similar to diffusion profiles, it is capable of generating a range of profiles that cannot be mimicked by a diffusion model (Figure 3.7). The EIC model presented here has one more parameter than a similar formulation of the diffusion model, so it is not surprising that it is able to produce a greater variety of profile shapes. It is important to determine whether the extra model complexity (the inclusion of another parameter) provides enough of an improvement for it to be considered a better model. The next section uses a maximum likelihood approach to applying and testing the EIC model, including making the comparison with the diffusion model. 3.4 Case study: Port Phillip Bay 21 Pb and 228 Th profiles Problem description and parameter estimation method Hancock and Hunter (1999) used excess 21Pb and 228Th profiles to quantify sedimentation and bioturbation rates in Port Phillip Bay (PPB). 21 Pb has a half-life of 22.3 years. It is generated in the sediments (the supported 21 Pb) and also arrives at

76 the sediment surface (the excess 21 Pb). The excess 21 Pb is fallout from the atmosphere, and it is scavenged by particles in the water column which then deposit on the sediment surface. The depositional flux of 21 Pb is usually assumed to be constant over time. 228Th has a half-life of 1.9 years. It is generated in the water column, where is it scavenged by suspended sediment and deposited on the sediment surface. It is generated in the sediment also. Again the excess 228 Th is used, and it is assumed to be a constant depositional flux. All 21 Pb and 228 Th data referred to in this chapter are the excess concentrations. Hancock and Hunter (1999) employed an advection diffusion equation with two unknown parameters, the sedimentation rate and the biodiffusion coefficient. As there were profiles from two tracers of different half lives, a least squares fit to the profile pairs could be used to determine the two parameters. The cores were taken from three sites within Port Phillip Bay, and the core names were SC1, FS1, SC3, LC3 and SC2 (short core 1, frozen slab core 1, short core 3, long core 3 and short core 2). The numbers refer to the site location (sites 1 and 3 were located in the centre of the Bay, and site 2 was in the far north of the Bay). In this work modelling has been restricted to the cores from sites 1 and 3 because the porosity variation with depth at site 2 was much higher than in the other sites (.34 to.77 at site 2, compared with.72 to.87 at sites 1 and 3). The analytical solution to the EIC model requires constant porosity, an assumption that is violated in all cores, but most severely in SC2 and so this core is excluded from the analysis. Mulsow et al. (1998) pointed out that it s not the variation in porosity that matters, so much as the variation in solid fraction. Although there is a large difference between the ratios of highest/lowest porosity (1.2 for sites 1 and 3 compared to 2.3 for SC2), the ratios of the highest/lowest solid fractions are closer (2.2 for sites 1 and 3 compared to 2.9 for SC2), so it may have been valid to include SC2 in my analysis. Hancock and Hunter (1999) determined the slope of the profiles plotted in log-linear space. They observed three distinct layers of different slope in the profiles: a fairly uniform surface layer (Layer 1, ~2 cm thick) overlying two layers of differing slope (Layer 2 between 2 cm and 2 cm and Layer 3 between 21 cm and 45 cm). The slopes were shallowest in Layer 3, and the biodiffusion coefficient calculated for this lowest layer exceeded the biodiffusion coefficient determined for Layer 2 by a factor of between 2.5 and 1. The authors commented that this result goes against expectations; biodiffusion coefficients are expected to decrease with depth, reflecting the reduced

77 density of animals. Their explanation for their seemingly anomalous result was that burrowing shrimps and worms inhabit PPB sediments to depths exceeding 4 cm. These animals activities could allow rapid transport of surface material to depth (either through inverse conveyor feeding or infilling of vacant tunnels). This observation suggests that the EIC model may be more appropriate for modelling the PPB sediment profiles. There were a number of assumptions embedded in the analysis by Hancock and Hunter (1999). They split the profiles into three layers for analysis, choosing the section boundaries by eye. This effectively fixed the parameters L and z r. They made no assumptions about F in as they only fitted the slopes of the curve in log-space. However, to match the core profiles (and not just slopes) their D b and w E values would need to be combined with estimates of F in. So while they only estimated two parameters (D b and w E), their results fixed particular values of L, z r and F in. Thus there were effectively five parameters in their system. The EIC model requires one more parameter, replacing D b with f and K av. A further comment about estimating F in values: where a sediment core is sufficiently long, F in can be estimated well from the inventory and λ for each tracer. Only one core (LC3) was deep enough to do this, leaving much greater uncertainty in F in for the other cores. Rather than analysing the profiles by layer, as Hancock and Hunter (1999) did, I used the above formulations of the diffusion and EIC models to model the whole profile. Parameters were fitted using a maximum likelihood approach (Hilborn and Mangel (1997)) and 9% confidence intervals were found for all parameters. Following Hilborn and Mangel (1997), the likelihood of the data given the hypothesis is expressed as {data hypothesis} and the negative log-likelihood is L{data hypothesis} = -log( {data hypothesis}). The likelihood of the data given the hypothesis is assumed to be proportional to the probability of the data given the hypothesis (and is further assumed to be identical,

78 without loss of generality). This form of analysis requires the hypotheses under test to be stochastic models, however the EIC and diffusion models have been presented as deterministic models only. I assumed that the set of observed tracer concentrations, C obs depend on the deterministic values, C det (the values predicted by the EIC or diffusion model) and the measurement uncertainty U: C obs = C det + U (3.43) It follows that the probability distribution for the difference between the observed and deterministic values (C obs - C det) is the same as the probability distribution for U. The tracer counts are Poisson distributed, and the standard deviation for each data point (σ i) is known. As counts are high, the Poisson distribution can be well approximated by a Normal distribution. Consequently, in the likelihood estimation procedure I assumed that U is Normally distributed. The negative log-likelihood for the i th data point is then the log of the Normal distribution function: 1 obs, i det, i L i = ln σ i + ln 2π σ i C C 2 (3.44) and the negative log-likelihood for a tracer profile containing n points is: n L = ln 1 + ln Cobs, i C, σ i 2π σ i= 1 i det i 2 (3.45) The set of parameters that minimises equation (3.45) are those that provide the closest fit between the model and the observed data. (Note that if σ i were the same for all i, this approach would yield the same set of parameters as a least squares fit). The advantage of the negative log-likelihood approach is that the values of L can be used to construct likelihood profiles (defined below) and prescribed confidence intervals on the parameter values. The MLE (maximum likelihood estimate) can be used to compare different models with differing numbers of parameters, so allowing a comparison of the EIC model with the diffusion model. The EIC model has one more parameter than the diffusion model, so we need to ensure that any improved fit is not simply due to the addition of one more parameter

79 Model parameters for each core were estimated using Matlab s fminsearch function, using all the observed 21 Pb and 228 Th data for that core. This is an unconstrained nonlinear optimisation function which uses the simplex search method, and it was applied to find the parameters that minimise the value of L. An unconstrained optimisation function is not ideal, as in this system there are constraints on all parameters (eg. all parameters have to be nonzero). To include these constraints in the optimisation routine, transformed parameters were used (for example, squaring parameters to ensure positive values, or mapping to a sin function to ensure they remained within a desired range). I limited L to a range of 4 8 cm, w E to a range of.5 cm/yr and allowed z r to be no greater than 3% of the profile depth. The 21 Pb and 228Th profiles were fitted together. Another common problem with optimisation routines is their tendency to dwell in local minima as global minima are notoriously difficult to locate, and results are likely to be sensitive to the initial parameter guesses supplied to the optimisation routine. This problem was addressed by conducting hundreds of optimisations (>5) starting from randomly chosen starting points. Once model parameters had been estimated for a core, likelihood profiles were constructed for each parameter, which is a form of sensitivity analysis. One parameter was fixed at a particular value, and the remaining parameters were estimated using the optimisation routine. The fixed parameter was then changed to another value and the process repeated. Where the resulting plot of L against the fixed parameter range is a steeply-sided U-shape, the parameter is well constrained; a flat curve indicates that the observed data are insufficient to estimate the parameter well. Quantified confidence bounds can be derived from the likelihood profiles using the likelihood ratio test. If allowed to vary, and 1 is the maximum likelihood value when all parameters are is the maximum likelihood value when one value has been fixed and the others allowed to vary, χ 2 = -2ln( / 1) has an approximate Chi-Square distribution with one degree of freedom. When we convert to negative log-likelihoods, χ 2 = 2(L L 1). So, for example, at the 9% confidence boundaries L = L Results of Maximum Likelihood analysis Example likelihood profiles are presented in Figure 3.8. Solid lines are the likelihood profiles and dashed lines indicate the negative log-likelihood value corresponding to - 7 -

80 9% confidence for each model (1.35 above the minimum negative log-likelihood value). The 9% confidence bounds are the intersection points between the likelihood profile and the dashed line associated with that profile. An example of a wellconstrained parameter is the 21 Pb flux value estimated for SC1 by the diffusion model (9% confidence bounds of 7.4 and 8.1 mbq/cm 2 /yr). At the other extreme, the maximum burrow depth, L, is poorly constrained by the data, and values ranging from 42.5 cm to 8 cm are all equally likely for the EIC model. Another important observation is that the two models can make quite different predictions about a parameter. For example, for LC3 the diffusion model estimates a lower limit of.32 cm/yr for the sedimentation velocity (with.46 cm/yr as the most likely estimate), whereas the EIC model for the same data estimates a range of.8 cm/yr to.32 cm/yr (with.18 cm/yr as the most likely estimate). The confidence bounds for all parameters and all cores are given in Table 3.1 to Table 3.4. The 9% confidence ranges were much broader than I had expected, and in many cases the data were unable to infer very much about the parameters values at all. Hancock and Hunter (1999) calculated a diffusion coefficient of 5. ±.1 cm 2 /yr and a sedimentation velocity of.15 cm/yr for the 2-2 cm region of sediment. Extending the diffusion analysis to the full profile and conducting a more rigorous uncertainty analysis produced a far greater range of diffusion coefficients and sedimentation velocities. Viable D b estimates range from cm 2 /yr, and sedimentation velocities range from.5 cm/yr. Sedimentation velocity had to be manually constrained to.5 cm/yr; in the absence of this constraint, even higher sedimentation velocities would fall within the 9% confidence range (regardless of model choice). This imposed constraint can be justified on the grounds that the depositional flux in the bay was calculated to be.42 cm/yr, which would be an over-estimate as it includes re-suspended sediment (Hancock and Hunter (1999)). The core data were therefore inadequate to estimate the sedimentation velocity in most cases. The results underline the need to capture the full profile in a sediment core. LC3 was the only core deep enough to reasonably constrain estimates of L and 21 Pb F in in the EIC model. The incoming flux estimates can only be estimated well when the full tracer inventory is captured by the core. LC3 was also different in that the estimates for z r were much higher than in the other cores (up to 2 cm, compared with up to 5.9 cm for other cores). These values are arguably too high, as it is physically unrealistic to have

81 uniform, instant mixing to depths of 2 cm. I re-ran the parameter estimation scripts with a constraint that z r < 1 cm in LC3, however the negative log-likelihood values were far higher, and visual inspection of the core plainly shows uniform mixing to this depth. The original report providing the 21 Pb and 228 Th data (Hancock et al. (1997)) stated that LC3 was a longer and narrower core than the others and that the top 1-2cm may have been artificially mixed during core collection and freezing. Given this knowledge, good information about the mixing behaviour in LC3 cannot be inferred; however it remains a valuable core as it is the only one that captures the full 21 Pb inventory

82 L L (cm) L Pb φse F (mbq/cm 2 /yr) in L 55 L Th φse F (mbq/cm 2 /yr) in f L L w (cm/yr) E zr (cm) L K av (yr 1 ) L D b (cm 2 /yr) Figure 3.8 Likelihood profiles for EIC and diffusion model fits to SC1 data. Solid lines are the likelihood profiles and dashed lines indicate the negative log-likelihood value corresponding to 9% confidence for each model (1.35 above the minimum negative log-likelihood value). The red plots are results from the EIC model and the black plots are results from the diffusion model

83 Table 3.1 Parameter estimates and 9% confidence intervals for EIC and diffusion model fits to SC1 core data. EIC model Diffusion model 9% lower 9% upper 9% lower 9% upper Parameter Best fit Best fit bound bound bound bound L (cm) Pb φ SEF in (mbq/cm 2 /yr) Th φ SEF in (mbq/cm 2 /yr) f w E (cm/yr) z r (cm) K av (yr -1 ) D b (cm 2 /yr) Table 3.2 Parameter estimates and 9% confidence intervals for EIC and diffusion model fits to FS1 core data. EIC model Diffusion model 9% lower 9% upper 9% lower 9% upper Parameter Best fit Best fit bound bound bound bound L (cm) Pb φ SEF in (mbq/cm 2 /yr) Th φ SEF in (mbq/cm 2 /yr) f w E (cm/yr) z r (cm) K av (yr -1 ) D b (cm 2 /yr) Table 3.3 Parameter estimates and 9% confidence intervals for EIC and diffusion model fits to SC3 core data. EIC model Diffusion model 9% lower 9% upper 9% lower 9% upper Parameter Best fit Best fit bound bound bound bound L (cm) Pb φ SEF in (mbq/cm 2 /yr) Th φ SEF in (mbq/cm 2 /yr) f w E (cm/yr) z r (cm) K av (yr -1 ) D b (cm 2 /yr)

84 Table 3.4 Parameter estimates and 9% confidence intervals for EIC and diffusion model fits to LC3 data. EIC model Diffusion model 9% lower 9% upper 9% lower 9% upper Parameter Best fit Best fit bound bound bound bound L (cm) Pb φ SEF in (mbq/cm 2 /yr) Th φ SEF in (mbq/cm 2 /yr) f w E (cm/yr) z r (cm) K av (yr -1 ) D b (cm 2 /yr)

85 SC1 FS1 5 5 z (cm) z (cm) tracer concentration (mbq/g dry weight) tracer concentration (mbq/g dry weight) SC3 LC3 5 5 z (cm) z (cm) tracer concentration (mbq/g dry weight) tracer concentration (mbq/g dry weight) Figure 3.9 EIC model fits to core data. Red markers are 21 Pb data, blue markers are 228Th data. Black lines are the best model fits to the data. The lighter red and blue lines demonstrate a range of possible fits constructed from parameters values lying at the extremes of the 9% confidence bounds

86 SC1 FS1 5 5 z (cm) z (cm) tracer concentration (mbq/g dry weight) tracer concentration (mbq/g dry weight) SC3 LC3 5 5 z (cm) z (cm) tracer concentration (mbq/g dry weight) tracer concentration (mbq/g dry weight) Figure 3.1 Diffusion model fits to core data. Markers and line colours are the same as in Figure 3.9. The EIC model fits have been plotted against the data in Figure 3.9 and the diffusion model fits in Figure 3.1. Included in these plots are model fits that lie at the extremes of the 9% confidence parameter range. The aim of this figure is to demonstrate visually that the wide ranges calculated for parameters really do produce viable fits to the data (a visual check on the more formal likelihood calculations). Figure 3.9 demonstrates the drawbacks of short cores quite clearly. There is a large range of possible EIC model concentrations at depths greater than the deepest measured point for all cores other than LC3. Another point to note is the impact of the range of

87 sedimentation velocity values on profile shape. For example, in the EIC model fits for SC3 we can see some profiles exhibit a sharp concentration drop to zero at the base of the burrowing zone (indicating w E is close to or at zero) while other profiles are characterised by a more gradual decrease to zero below the burrowing zone (indicating a much higher w E). For all cores the EIC model fits produced a higher maximum likelihood value, so by this measure, provided a better fit to the data. Whether the improvement is significant or not can be judged using the Akaike Information Criterion (AIC), A i, for comparing non-nested models. This criterion adds a penalty for each additional parameter. The AIC for model M i with p i parameters is: (3.46) A = 2L Y M * + 2p i i i where Y is the data and L(Y M i *) is the minimum value of the negative log-likelihood evaluated at the best choice of parameters for model M i (Hilborn and Mangel (1997), with printing error corrected). The best model is the one with the lowest AIC. The AIC values for each core and model are listed in Table 3.5, and demonstrate that in each case the EIC model s improvement in fit is significant for all cores except FS1, where effectively neither model is better than the other. Table 3.5 AIC values calculated for each core and model using equation (3.46). Core EIC Model AIC Diffusion model AIC SC FS SC LC An advantage of the EIC model is that even this simplest version relies on parameters that have a physical meaning, and are measurable. The rate of volume expelled to the surface per unit area, Q ex is simply Q ex = K L (3.47) av

88 The estimates for Q ex for each core are.36,.82,.78 and.46 cm 3 /cm 2 /year (SC1, FS1, SC3 and LC3 respectively). The laser scanner experiments in Chapter 2 suggested excavation rates of the order of 1 cm 3 /burrow opening/day. At this rate, densities of 1-24 openings/m 2 would yield the Q ex values inferred from the cores. For the purposes of comparison, actual thalassinidean shrimp densities in PPB ranged from 1-21 individuals/m 2 in 1995 (Bird et al. (1997)). Assuming two openings per individual, this corresponds to 2 42 openings/m 2. A direct comparison is not strictly valid, as the measurements in Chapter 2 were of a considerably larger species, Trypaea australiensis. Trypaea australiensis exists within Port Phillip Bay, but is far less common than smaller thalassinidean shrimp species. Nevertheless, the comparison serves to demonstrate that the proposed mixing mechanism and rates are consistent with measured excavation rates from a related species. The laboratory measurements suggested that burrow infill rates were comparable to sediment expulsions rates, which implies that f ~ 1. While f = 1 was the best estimate for SC1, SC3 and LC3 and in all cases, f=1 lay within the 9% confidence bounds, f was generally poorly constrained in the maximum likelihood analysis. Even so, the lower bound was never greater than.27, indicating that some infill was always needed to provide the best fit Model Assumptions In this section I address the simplifying assumptions made to allow an analytical solution to the EIC model. The assumptions were: uniform constant excavation and infill, steady state conditions and constant porosity. These assumptions greatly reduced the number of parameters which could be varied in the model; there was no option to change depth dependence, non-steady state solutions and different porosity assumptions. The data are too sparse even to constrain the remaining parameters very well, so there is certainly insufficient data to resolve even further detailed inclusions in the model structure. However, there is still merit in experimenting with different model structures to provide an idea of the latitude in these sediment transport descriptions. The numerical version of the EIC model requires none of the above assumptions and so allows the investigation of alternative model assumptions. We firstly consider the steady state assumption. European settlement dramatically altered the land cover in the catchments. Run-off from cleared catchments can be substantially higher than vegetated catchments, and carry more sediment. The

89 assumption of a constant burial velocity with time ignores the possibility of recent spikes of sediment arriving in the Bay following large-scale land clearing. Given that the bulk of the catchment changes happened over a century ago, any 21 Pb from that period would have long since decayed, and the existing 21 Pb reflects the sediment mixing processes of only the last few decades. Consequently, I ve not experimented with such scenarios with the numerical version of the EIC model. Further steady state assumptions include constant values of f, K E and s f over time. These parameters are all likely to be seasonally influenced, but affected also by relatively sudden changes in surrounding conditions. Temporal variations in weather, tidal velocities and disturbances by other animals are likely to create temporally varying excavation and infill characteristics. Long time-scale variations, such as seasonally-imposed changes, are likely to have the greatest impact on model results and could be included in the model, however I did not conduct any model runs to explore these issues. The constant porosity assumption is clearly violated in all of the cores. When porosity variation is to be included in a diffusion model, a decision has to be made about the nature of the mixing: is it intraphase or interphase mixing (eg. Boudreau (1997))? The EIC model requires no such distinction to be made, however it does of course need equivalent assumptions defining how porosity is altered by the excavation, infill and collapse process. The simplest approach that retains depth-varying porosity information is to assume the porosity profile is invariant with time, so the left hand side of equation (3.8) is set to zero and can be used to solve for w as a function of depth. The best parameter estimates from the analytical solution were applied to the numerical version of the EIC model, together with this additional porosity assumption. The results are plotted with the core data and compared with the best analytical fit in Figure While the numerical model profiles fit the data approximately, the negative log-likelihood values calculated from these profiles are substantially higher than the values for the analytical version. Differences from minimum negative loglikelihood values calculated previously were 5, 4, 33 and 7 for SC1, FS1, SC3 and LC3 respectively (note that a difference of 1.35 was used to calculate 9% confidence boundaries)

90 SC1 SC1 FS1 FS1 2 2 z (cm) z (cm) φ s Concentration (mbq/g dry) φ s Concentration (mbq/g dry) SC3 SC3 LC3 LC3 z (cm) 5 5 z (cm) φ s Concentration (mbq/g dry) φ s Concentration (mbq/g dry) Figure 3.11 Comparison between analytical EIC model best fits (which assume constant porosity), and the numerical model (which assumes a time-invariant depth-varying porosity profile) results if the same parameters are applied. The porosity profiles for each core are shown, along with the quadratic fit used to represent porosity variation in the numerical model. The blue lines are the analytical model best fits and the red lines are the numerical model profiles. The difference in negative log-likelihood values between the two profiles are 5, 4, 33 and 7 for SC1, FS1, SC3 and LC3 respectively. Uniform burrowing and infill with depth (constant K E and s f) is another large assumption underlying this work. Clearly this assumption is physically implausible. It would be relatively easy to use more detailed field data to supply more realistic burrow volumes as a function of depth. To demonstrate the potential impact of such a change, the best parameter estimates from the analytical model were substituted into a numerical version of the EIC model with a linearly decreasing excavation and infill rate (decreasing to zero at the base of the burrowing zone, z = L). The results are plotted in Figure Again, the visual comparisons of the data and model results appear reasonable, but the negative log-likelihood values calculated for each profile differ substantially from those for the analytical version (differences of 17, 19, 25 and 13 for cores SC1, FS1, SC3 and LC3 respectively)

91 SC1 FS1 SC3 LC Concentration Concentration Concentration Concentration (mbq/g dry) (mbq/g dry) (mbq/g dry) (mbq/g dry) 5 5 Figure 3.12 Comparison between analytical model best fits (which assume constant porosity and uniform excavation and infill rates with depth), and the numerical model (which assumes constant porosity and linearly decreasing excavation and infill rates with depth) results if the same parameters are applied. The blue lines are the analytical model best fits and the red lines are the numerical model profiles. The difference in negative log-likelihood values between the two profiles are 17, 19, 25 and 13 for SC1, FS1, SC3 and LC3 respectively. Depth axis units are cm. Parameter confidence bounds were not calculated for either of the above numerical models. The calculations would be considerably more time consuming than for the analytical solution, as tens of thousands of model runs are required and numerical model solutions are substantially slower to calculate than analytical model solutions. If such calculations were to be conducted, new sets of best parameter estimates and confidence bounds would be found. The various sets of best parameters generated from different model assumptions (analytical model; numerical depth-varying porosity model; and numerical depth-varying excavation and infill model) would differ and the model producing the lowest overall negative log-likelihood value would win a competition between these models (as each model contains the same number of parameters). Whether the confidence intervals should be calculated from within the winning model only, or calculated from negative log-likelihood values across all models, is a more difficult question. The conservative approach would be to choose the latter option. It is unclear how such an analysis would alter the existing calculated confidence bounds, although given the large differences in negative log-likelihood values for the model profiles presented in Figure 3.11 and Figure 3.12, the potential increase in parameter bounds is enormous. Certainly the best fits to the two numerical models will produce different parameter sets to the set calculated from the analytical model

92 A final comment needs to be made about the representation of uncertainty in the stochastic model. In equation (3.43) the uncertainty term, U, was taken to be the measurement uncertainty based on Poisson distributed counting data. This is not the only source of uncertainty in the model, as there is uncertainty about the mixing process itself. Hence the confidence bounds on error estimates have been calculated from the idealised assumption of measurement uncertainty only, and would be broader if measures of process uncertainty were to be included. 3.5 Conclusions The EIC model developed here has been shown to be a good alternative to the diffusion model as it can produce similarly-shaped profiles yet its parameters (burrow excavation and infill rates) have more physical meaning than a biodiffusion coefficient. The EIC model is particularly suitable for modelling sediment with large persistent burrows, such as those created by thalassinidean shrimp. The model contains one more parameter than a diffusion equation (D b is replaced by K av and f), but a maximum likelihood parameter estimation approach allows a formal comparison between the two models using the Akaike Information Criterion (AIC). Such an approach can determine if a better fit by the EIC model is due to the extra parameter or to the model itself. The EIC and diffusion models were applied to 21 Pb and 228 Th cores from Port Phillip Bay, where thalassinidean shrimp are known to be important bioturbators. This established that the EIC model gave a better fit to the data for all but one core (where the models performed equally well). The diffusion modelling approach is particularly suspect for PPB cores, as much of the data suggests a substantially higher D b at depth than at shallower locations. While the EIC model was able to give a better fit to the data, little information on the mixing parameter ranges could be gleaned from the model results. The parameter estimates were poorly constrained, as demonstrated by a sensitivity analysis. In particular, burial velocities, burrow excavation rates and burrow infill were all subject to large 9% confidence intervals. Further, cores from the same site yielded very different parameter estimates. It was clear from all cores, however, that some degree of burrow infill was required to produce the best EIC model fits (f >.27 in all cases). This is strong evidence that nonlocal downward transport from the sediment surface is an important mechanism. For the bulk of my PhD research I was unaware of the work being carried out by David

93 Shull (Shull (21), Shull and Yasuda (21)), and I read his publications only towards the end of my work. It s interesting to see that we drew similar conclusions about the importance of downwards non-local transport and the need to incorporate it into models. Where I chose a continuum model approach, he developed a discrete transition matrix model. Hancock and Hunter (1999) suggested that burial velocities and biodiffusion coefficients could be well resolved from pairs of radionuclide profiles with differing decay constants. My analysis using the diffusion models and a maximum likelihood approach to estimate parameters and confidence intervals found that estimates for burial velocity and biodiffusion coefficients were relatively poorly constrained by the data. Nevertheless, the excavation rates inferred from each core were consistent with measured excavation rates reported in Chapter 1. The analytical solution of the EIC model relies on a number of assumptions. Assumptions about constant porosity and uniform burrowing are unrealistic. The computationally expensive requirements of the maximum likelihood approach favoured analytical solutions to the EIC and diffusion models. The maximum likelihood approach was not repeated for the slower numerical solution. However, best parameter estimates from the constant-porosity analytical versions of the model were applied to variable-porosity and linearly decreasing burrowing numerical versions to assess whether these considerations could be important. While visually the comparisons to data appeared reasonable, the difference was enough to suggest that, were a rigorous statistical approach applied, the confidence intervals for the variableporosity version could be even larger than those estimated using the analytical model. A possible approach to decide whether to investigate these assumptions further is to use the analytical model to estimate parameters and quantify confidence bounds. If the confidence bounds prove to be quite tight, it would be worth extending the analysis and going through a more computationally intense process of including different porosity assumptions and excavation and infill functions. In the present case, such extra effort is not warranted as the parameters are relatively poorly constrained by the data. The EIC model suffers the same problems as any one-dimensional description of a three-dimensional process. In particular, heterogeneity introduced by burrowing is

94 obliterated by the lateral averaging. In the next chapter, the EIC model is extended to two and three dimensions to investigate the impacts of dimensionality

95 4 Higher-dimensional EIC sediment model 4.1 Introduction The one-dimensional EIC model described in Chapter 3 can be expanded to two or three dimensions. The model still assumes that there are three forms of sediment transport - excavation, infill and collapse but the excavation and infill functions (K E and s f) are two or three-dimensional burrow maps. The main advantage of expanding the EIC model to more dimensions is that it can provide a mechanism for introducing heterogeneity to model sediments. High resolution measurements have demonstrated the heterogeneous nature of sediments (eg. Davison et al. (1997) and Shuttleworth et al. (1999)). Some of the issues surrounding heterogeneous sediment distributions were outlined in Section 1.3. Heterogeneous sediments behave differently to laterally homogenised sediments for a range of reasons. The existence of microniches can allow incompatible reactions to cooccur in close proximity (eg. anoxic microniches can allow denitrification to occur in a patch of sediment that, on average, is oxic). Further, Harper et al. (1999) demonstrated that modelling diffusion reactions in one dimension does not accurately capture the three-dimensional diffusion gradients surrounding small sources and sinks. Bioturbation clearly plays an important role in generating heterogeneity in sediments, but one-dimensional descriptions of sediment mixing typically represent bioturbation as a smoothing or an averaging process. Representing bioturbation as a diffusion process in any number of dimensions certainly limits the role of bioturbation to that of a smoothing function. Non-local models and the LABS model (Choi et al. (22)) have the capacity to generate heterogeneity (see Section 1.2.3). The aim of 2-D and 3-D diagenesis modelling is not to produce results that recreate a specific heterogeneous concentration distribution. Rather, there are circumstances under which such fine-scale model resolution is necessary to model the emergent large-scale properties well. While the need for a higher-dimensional representation is well-recognised, means for introducing heterogeneity into sediments have not been explored in detail. In conversations with Mike Harper (1999) and Bill Davison (1998), they both suggested that their work demonstrating the importance of 3-D modelling has not been widely adopted for (at least) three reasons: computational expense, lack of 3-D data and a lack of a mechanism for introducing heterogeneity into a sediment model. Harper et al. (1999) imposed a heterogeneous snapshot for their modelling

96 work, without devising a transport model to introduce the heterogeneity. This chapter aims to address this last point, and to demonstrate such a mechanism that retains all the advantages of traditional transport equations. In particular, material arrives at the sediment surface according to a prescribed flux, and its subsequent distribution through the sediment obeys mass conservation equations. Before higher-dimensional diagenesis modelling becomes the norm (as is surely inevitable), the model proposed in this chapter may provide a first simple method for describing two and three dimensional bioturbation transport processes. Alternatives such as the LABS model (Choi et al. (22)) will be superior in the long run, however the 2-D and 3-D EIC model requires few parameters and has the advantage of being firmly linked to the one-dimensional EIC model for which there are analytical solutions. Consequently, there is the possibility of deriving model parameters from one-dimensional tracer cores and then applying the mixing parameters to the higherdimensional model. These characteristics may be an advantage when applied to higher-dimensional diagenesis modelling. 4.2 Two-dimensional model equations The equations are substantially the same as those in the previous chapter, but are repeated here for completeness. The excavation function, K E is now a two-dimensional function of horizontal distance x and vertical distance z. The horizontal extent of the model is defined by x max. Infill material is returned to the burrows by the distribution function s f, which is now a two-dimensional function and has units of length -2. It is defined so that xmax L sfx, zd zdx = 1 zr (4.1) where x max is the maximum x value. The spatially-averaged excavation function, K av is defined as: K av 1 = Lx xmax L max K x, z d zdx E (4.2) So that the volume rate of sediment excavation in burrow formation is K avlx max. Again, the constant parameter f represents the fraction of this volume that is returned to the

97 burrow as infill from the surface. The lower layer equations for total solids and for tracer concentration are: φ x, z s t = w x, z φ s x, z sf x, z fkavlx φ s x, z KE x, z φ s x, z (4.3) max 1 z φ s x, z C x, z = sf x, z fkavlx φ s C KE x, z φ s x, z C x, z (4.4) max 1 1 t wx, zφ sx, z C x, z λφ sx, z C x, z z where λ is the decay constant for the tracer, C is the concentration of the tracer (mass per unit volume of solids), C 1 is the tracer concentration in the surface layer and φ s1 is the solid fraction in the surface layer. The terms on the right hand side of (3.9) represent infill, excavation, advection and decay. The surface layer equations for total solids and for tracer concentration are: dφ s dt 1 dφ s C dt wφ sxmax wrφ s1xmax φ s1fkavlxmax sf x, z dzdx zr = x z wφ s wrφ s1 φ s1fkavl = z 1 1 r max r xmax L (4.5) wφ scxmax wrφ s1c1xmax φ s1c1 fkavlxmax sf x, z d zdx zr = x z λφ C s1 1 max wφ sc wrφ s1c1 φ s1c1fkavl = λφ s C z r r 1 1 xmax L (4.6) where φ s and C are the solid fraction and tracer concentration of the material arriving at the sediment water interface from sedimentation and w r is the velocity at the base of the surface mixed layer. The flux boundary conditions are: = + 1 xmax L φ sw φ sewe φ ske x, z d zd x x zr max = + E 1 xmax L φ swc φ sefin φ s x, z K x, z C x, z d zdx x zr max (4.7) (4.8)

98 where w E is the sedimentation velocity, φ SE is the solid fraction of sedimenting material and φ SEF in is the concentration flux of material arriving at the sediment water interface from the water column. The terms on the right hand side of each equation represent the fluxes to the sediment surface due to sedimentation and excavation respectively. Periodic boundary conditions are applied at x = and x = x max. Non-dimensional variables are defined as in the Chapter 3, with the addition of x*: z x CLKav w λ z* = ; x* = ; C* = ; t* = Kavt; w* = ;sf * = sf Lxmax ; λ* = ; K L x F KavL Kav max in E K * = K E av And the resulting non-dimensional equations for the lower layer are: φ s t * = w * φ s sf * fφ s1 KE * φ s z * (4.9) φ sc * = φ φ w * φ sc * sf * f s1c1 * KE * sc * λ * φ sc * t * z * (4.1) and for the upper layer: dφ s1 w * φ s wr * φ s1 φ s1f (4.11) = d t * z * r d φ s1c1 * w * φ sc * wr * φ s1c1 * φ s1c1 * f = d t * z * r (4.12) d φ s1c1 * w * φ sc * wr * φ s1c1 * φ s1c1 * f = d t * z * r λφ C s1 1 The flux boundary conditions at the sediment surface are: + = φ w * φ w * φ K * d z * d x * s se E s zr * 1 + = φ w * C * φ φ K * C * d z *d x * s se s zr * E E (4.13) (4.14) A simple way to implement the model is to prescribe a diagonal burrow extending to the base of the burrowing zone and across the horizontal extent of the model (Figure - 9 -

99 4.1). Laterally averaged, this represents uniform excavation over the extent of the sediment column, for which we have 1-D analytical steady state solutions. K E surface mixed layer z* burrow zone burial only, no burrowing x* Figure 4.1 Diagram of a simple diagonal burrow map. The burrowing zone is contained within the two horizontal dotted lines (located at z = z r* and z = 1). Laterally averaged, this burrow function represents uniform burrow excavation with depth in the burrowing zone (an assumption used with the one-dimensional EIC model in the previous chapter). The 2-D model equations were solved numerically using Matlab. Assuming constant porosity allowed the use of equations (4.9), (4.11) and (4.13) to calculate the velocity, which was then applied to the remaining equations describing the tracer movement. A full differencing method, employing a forward-time and upwind differencing scheme (with the time-step chosen to ensure stability) was used. The burrow surface location and burrow direction was randomly changed at each time step, but the burrow angle was fixed. 4.3 Sample model output and comparisons with 1-D model The two dimensional distributions for three different f values are shown in Figure 4.2 (other parameters were λ*=1, z r* =.1, w E* = ). The model was run to reach a laterallyaveraged steady state (specifically, t*=2). When f =, no surface material falls into the burrow cavity, and transport is solely due to burial and collapse. As a consequence, the sediment maintains a fairly laterally homogeneous tracer distribution. As soon as some surface infill is produced, lateral heterogeneity becomes more important. At the

100 extreme case, where burrows are totally filled by surface material, we see that pockets of high tracer concentration can be found at depth. Figure 4.2 Example 2-D EIC model output for three different f values. Other parameters were set at λ*=1, z r* =.1 and w E* =. Burrows were relocating at each time step, t*=.5. The laterally-averaged concentration distributions are shown in Figure 4.3, along with the one-dimensional analytical solutions derived in the previous chapter. These graphs demonstrate that for diagonal, continually relocating burrows (of the kind illustrated in Figure 4.1), the one and two-dimensional models produce the same laterally-averaged profiles (as expected). It should be noted that if the burrows are set to relocate less frequently, the profiles can be considerably different to the one-dimensional solutions. The difference is because the same material is oscillating between surface and depth for an extended period of time, while surrounding material remains unmoved. Profiles demonstrating this effect are included in Figure 4.3. The one-dimensional EIC model cannot provide a good fit to profiles generated in this way. The time between burrow relocation was set to t* =.1 (in Figure 4.2 the burrow relocated each time step, t* =.5). The 2-D distributions from these model runs are shown in Figure 4.4. Increasing burrow permanence has most effect where burrow infill is significant (f > ), and the consequences are higher surface-layer values, lower average concentrations at depth and greater lateral variation in concentrations at depth (particularly as f approaches 1). Given that rapid two-way exchange was found to occur in semi-permanent burrow

101 structures in Chapter 2, the effect of burrow permanence has to be considered when applying the 2-D EIC model. f= f=.5 f= z* C* 2 4 C* 5 1 C* Figure 4.3 Comparison between 1-D EIC model solutions and the laterally-averaged 2- D solutions for three different f values. There are four profiles in each diagram. The blue, green and red lines are basically indistinguishable, and represent the 1-D analytical, 1-D numerical and laterally averaged 2-D solutions respectively. The aqua line laterally averages 2-D model runs, however in these cases the burrows have been semi-permanent, relocating with a period of t* =.1. Figure 4.4 Sample 2-D EIC model output demonstrating the potential effects of burrow permanence. The parameters are the same as those used in Figure 4.2. The only difference is that burrows were set to relocate less frequently, with a time interval of t*=

102 It is instructive to observe the changes made to a buried layer of inert tracer under the different mixing regimes. Such concentration distributions are shown in Figure 4.5, and were generated using the same mixing parameters as above. Note that this inert tracer scenario requires F in =, which prevents the calculation of C* (as it would require division by zero). This problem is avoided simply by using C instead of C* in the equations and omitting the first term on the right hand side of equation (4.14). Where there is no burrow infill, the buried material is brought to the surface, mixed and then migrates downwards due to collapse advection. The tracer layer also migrates downwards due to this collapse advection. None of the tracer can make its way below this migrating tracer layer. At the other extreme, where there is no burrow collapse and only burrow infill, the buried material is brought to the surface, mixed and returned to all depths in the burrowing zone. There is no advection (as w E* = and there is no collapse advection) and so the original layer of tracer remains unmoved and still contains pockets of high-concentration material. The f =.5 case represents a mixture of these two extremes. These examples serve to demonstrate the kind of mixing information that could be ascertained through an in situ tracer experiment

103 Figure 4.5 The fate of a buried layer of inert tracer after 2-D EIC mixing (constantly moving diagonal burrows) for a duration of t*=1. The initial tracer concentration within the buried layer was set at 1, its location is represented by a white bar in the top left-hand plot. The model parameters were z r* =.1, w E* = and three different f values. 4.4 Port Phillip Bay profiles Using the best parameter estimates for each PPB sediment core found in the previous chapter, two-dimensional concentration distributions were generated using the 2-D EIC model. A random core of the resulting distribution was then taken (by taking an average over a 3cm lateral distance) and coarser 21 Pb and 228 Th profiles were generated by slicing the core at prescribed depths and taking an average of each slice (producing one data point for each z-location in the corresponding field cores). The maximum likelihood estimation (MLE) techniques used in Chapter 3 were used to estimate the EIC parameters and their corresponding confidence bounds (assuming the

104 σ value attached to each data point is simply the mean of all the σ values from the field core). The original EIC parameters used to generate the data should be recovered by this process and the purpose of this exercise was to test whether this would be the case. The MLE analysis was repeated so that there were six scenarios in total. Variations were: to sample the core to a greater depth (to ensure the full tracer profile was captured only LC3 satisfied this criterion in the original core datasets); reduce the σ values (equivalent to reducing the measurement error of the concentration data); and apply the diffusion model instead of the EIC model. The scenario numbers and descriptions are given in Table 4.1. Table 4.1 Scenario numbers and descriptions Scenario Model Core length σ 1 EIC Field core length Mean field core σ 2 EIC Deep core Mean field core σ 3 EIC Field core length Half the mean field core σ 4 EIC Deep core Half the mean field core σ 5 Diffusion Field core length Mean field core σ 6 Diffusion Deep core Mean field core σ The 21 Pb and 228 Th distributions generated for SC1 are shown in Figure 4.6. Spikes at depth were recorded in some of the 228 Th cores from PPB and this figure demonstrates how burrow infill could produce such spikes. z (cm) 21 Pb concentration (mbq/g dry mass) x (cm) 228 Th concentration (mbq/g dry mass) x (cm) Figure Pb ex and 228 Th distributions generated using the 2-D EIC model. The parameters used were those estimated in the previous chapter for SC1 using the 1-D EIC model. Units x and z axes are cm

105 The likelihood profiles for all scenarios for SC1 are shown in Figure 4.7. Either lengthening the core or reducing the measurement uncertainty tightened the confidence bounds for L and F in in particular. The likelihood profiles for SC1 are for the most part quite similar to those generated for the actual field data (Figure 3.8). The results for the remaining cores are given in Table 4.2 to Table 4.5. This exercise proved to be surprisingly useful. For all cores, the parameter 9% confidence bounds estimated from the synthetic core data are very similar to the 9% bounds estimated from the actual field data. This demonstrates that even when the EIC mechanism is used to generate the data, we do not get any tighter confidence bounds than those calculated in Chapter 3 for this level of sampling and measurement uncertainty. The exception to this finding is for deep core or reduced σ scenarios, where confidence bounds were considerably tighter for L and F in. The results also showed that in some cases a tight incorrect parameter range will be estimated. For example, the Scenario 3 fit for L in SC1 estimated a range of cm, where the actually L value used to generate the data was 74.5 cm. In general, however, the actual values used to create the data are recovered and their associated confidence bounds match those found in Chapter

106 4 L L (cm) L Pb φse F (mbq/cm 2 /yr) in L L Th φse F (mbq/cm 2 /yr) in f 3 3 L 2 L w (cm/yr) E zr (cm) 3 3 L 2 L K av (yr 1 ) D b (cm 2 yr 1 ) Figure 4.7 Likelihood profiles for SC1 for the scenarios given in Table 4.1. Legend (scenario numbers in brackets): red solid line (1 ), red dotted line (2), blue solid line (3), blue dotted line (4), green solid line (5), green dotted line (6)

107 Table 4.2 Parameter estimates and 9% confidence intervals for EIC and diffusion model fits to core SC1 (9% confidence values in parentheses). Parameter Actual value Scenario 1 fit Scenario 2 fit Scenario 3 fit Scenario 4 fit Scenario 5 fit Scenario 6 fit used L (cm) (4, 8) 74.2 (62.3, 8) 4 (4, ) 74.2 (68.3, 8) 4 (4, 8) 57.4 (56., 61.1) 21 Pb φ SEF in 8.5 (7.9, 11.4) 1.4 (9.5, 11.3) 8.52 (8.2, 8.98) 1.35 (9.87, 1.87) 8.31 (7.8, 8.9) 8.68 (8.4, 8.9) 1.4 (mbq/cm 2 /yr) 228 Th φ SEF in 11.7 (9.7, 13.9) 12.5 (1.2, 14.7) 11.7 (1.6, 12.8) (11.3, 13.6) 13.3 ( 1.1, 16.) 12.4 (1.1, 14.2) 12.5 (mbq/cm 2 /yr) f (.3, 1).99 (.38, 1).59 (.36, 1) 1. (.52, 1) - - w E (cm/yr) (.97,.47).41 (,.47).28 (.16,.44).41 (.14,.44).32 (.11,.5).46 (.45,.47) z r (cm) (2.8, 4.2) 3.31 (2.78, 4.13) 3.18 (2.99, 3.53) 3.31 (2.96, 3.66) 2.39 (2,3.4) 2.23 (2, 2.15) K av (yr -1 ) (.3,.16) (.32,.13).37 (.3, - - (.3,.12).17) D b (cm 2 /yr) (.18, 5.8) 1.19 ( 1.16, 1.26) Table 4.3 Parameter estimates and 9% confidence intervals for EIC and diffusion model fits to core FS1 (9% confidence values in parentheses). Parameter Actual value Scenario 1 fit Scenario 2 fit Scenario 3 fit Scenario 4 fit Scenario 5 fit Scenario 6 fit used L (cm) (4, 8) 4 (4, 66.8) 73.9 (4, 8) 4 (4, 48.5) 78.5 (4, 8) 4 (4, 8) 21 Pb φ SEF in (7.7,13.6) 8.39 (7.7, 9.3) 11.5 (8.1, 13.5) 8.4 (8., 8.8) 8.97 (8.19,9.97) 8.9 (8.2, 9.6) 8.3 (mbq/cm 2 /yr) 228 Th φ SEF in 24.1 (7.7,28.7) (18.7, 27.6) 24.7 (2.71, 27.79) (2.93, 25.38) 21.5 (17.15, 26.) 21.4 (17.2, 25.7) 22.3 (mbq/cm 2 /yr) f (, 1).4 (.23,.55).77 (.36, 1).4 (.31,.48) - - w E (cm/yr)..35 (,.5).3 (,.37).35 (,.5).3 (,.17).5 (.3,.5).5 (,.5) z r (cm) ( 3.5,5.4) 4.4 (3.4, 5.1) 4.51 (4.5, 4.89) 4.42 (3.99, 4.75) 3.9 (2.15,5.) 3.98 (2., 5.) K av (yr -1 ) (.72,.74 (.2,.24).21 (.15,.24) - - (.1,.25).25) D b (cm 2 /yr) (.26, 4.98).67 (.26,9.57) Table 4.4 Parameter estimates and 9% confidence intervals for EIC and diffusion model fits to core SC3 (9% confidence values in parentheses).

108 Parameter Actual value Scenario 1 fit Scenario 2 fit Scenario 3 fit Scenario 4 fit Scenario 5 fit Scenario 6 fit used L (cm) (4, 8) (58.1, 74.9) 66.8 (42.4, 8) 62.9 (6.6, 67.7) 43.4 (4, 45.9) 53.5 (5.5, 65.5) 21 Pb φ SEF in 11.8 ( 8.5, 15.2) 12.1 (11.3, 12.9) 11.8 (8.3, 14.6) 12.1 (11.7, 12.5) 9.3 ( 8.7, 9.9) 1.5 (9.9,11.2) 12.1 (mbq/cm 2 /yr) 228 Th φ SEF in 16.5 (12.5, 2.3) 16.1 (14.15, 18.1) 16.5 (13.4, 19.) 16.1 (15.1, 17.1) 14.6 (13.2,16.4) 16.1 (14.2,18.11) 15.3 (mbq/cm 2 /yr) f (.44, 1).9 (.63, 1).73 (.5, 1).9 (.79, 1) - - w E (cm/yr) (,.5).39 (.21,.5).19 (,.5).39 (.29,.49).5 (.35,.5).5 (.44,.5) z r (cm) (2.3, 3.2) 2.75 (2.38, 3.4) 2.74 (2.51, 2.96) 2.75 (2.59, 2.9) 2 (2,2.3) 2 (2,2.24) K av (yr -1 ) (.1,.3).14 (.1,.18).17 (.1,.27).14 (.11,.16) - - D b (cm 2 /yr) (2.9,7.3) 8.75 (6.39,12.6) Table 4.5 Parameter estimates and 9% confidence intervals for EIC and diffusion model fits to core LC3 (9% confidence values in parentheses). Parameter Actual Scenario 1 fit Scenario 3 fit Scenario 5 fit value used L (cm) (43.5, 56.9) 51.6 (48.4, 56.5) 46.5 (4, 72.8) 21 Pb φ SEF in 12.2 (11.4, 12.9) 12.2 (11.8, 12.6) 11.8 (11.1,12.7) 11.9 (mbq/cm 2 /yr) 228 Th φ SEF in 25.7 (14., 35.5) 25.7 (19.8, 31.6) 23.6 (13., 34.4) 25.7 (mbq/cm 2 /yr) f (.5, 1).99 (.63, 1) - w E (cm/yr) (,.39).22 (,.29).34 (,.5) z r (cm) (15.9, 2) 19.1 (17.9, 19.7) 14.9 (13.,17.2) K av (yr -1 ) (.58,.18).91 (.75,.16) - D b (cm 2 /yr) (1.64,13.61)

109 The non-dimensional time step in the model, t*, ranged from.3 to.5 and burrows were set to re-locate at each time step. The K av values used to generate the 2-D data were.48,.2,.13 and.87. Assuming burrows remain in place for two years, the non-dimensional burrow re-locating time (2*K av) would range from.96 to.4. Figure 4.4 demonstrates that a burrow relocation time of.1 makes a considerable change to laterally averaged profiles. The construction of a synthetic 2-D data set including this modification was not tested, but is another permutation that could be explored. 4.5 Three-dimensional model The model is just as easily converted to a three-dimensional version, although with the obvious cost of increased computation time. Again, a simple assumption was made about the 3-D burrow map it is a constantly relocating diagonal burrow that excavates uniformly to the base of the burrowing zone (so its lateral average is the same as the uniform-burrowing example used in the 1-D modelling in the previous chapter). Examples of 3-D burrows are presented in Figure z y x.5 1 Figure 4.8 Positions of five randomly located 3-D diagonal burrow maps (each burrow is represented by a different colour and marker)

110 2-D slices of the 3-D distributions generated for three values of f are presented in Figure 4.9. The three values of f were f =, f =.5 and f = 1, and other parameters were set to λ*=1, z r* =.1 and w E* =. A coarser grid than the 2-D model was used (a nondimensional grid size of.2 instead of.1), while still ensuring that the laterally averaged results matched the analytical solutions. The shift to three dimensions could be important if such a solid concentration distribution is interacting with diffusing porewaters (eg. adsorption-desorption reactions). Because the 3-D distribution is much spikier than the 2-D distribution (compare the 3-D f = 1 case in Figure 4.9 with its equivalent 2-D case in Figure 4.2) the issues that Harper et al. (1999) raised will be important (see discussion in Section 1.3.2). C*, f = C*, f =.5 C*, f = z* x* x* x* C* Figure D slices from 3-D EIC model output. Parameter values were λ*=1, z r* =.1 and w E* =. The model was run until t*=5, and the laterally averaged results match the 1-D analytical solutions. 4.6 Other considerations Nonlinear reaction terms Considering only linear, first order kinetics ensured that the laterally averaged 2-D and 3-D model results match the 1-D model results. The real interest in applying higherdimensional models lies in the fact that non-linear interactions between species in a heterogeneous environment lead to rather different outcomes than the same species reacting in a homogenised environment

111 Consider two substances, A and B with coupled reaction rates: dc dt dc dt A B = kc C + transport terms A B = kc C + transport terms A B (4.15) where C A and C B are the concentration of A and B respectively and k is a rate constant. In Figure 4.1 a 1 1 table shows 2-D concentration distributions of A and B which are spatially correlated, along with the 1-D laterally-averaged concentration profiles. 2-D distribution a, b a, b a, b a, b a, b a, b a, b a, b a, b a, b 1-D distribution (lateral mean of the 2- D distribution) a b, 1 1 a b, 1 1 a b, 1 1 a b, 1 1 a b, 1 1 a b, 1 1 a b, 1 1 a b, 1 1 a b, 1 1 a b, 1 1 Figure 4.1 Left hand side: 2-D concentration distributions for two spatially-correlated species A and B. Right-hand side: 1-D profiles of A and B generated by laterally averaging the 2-D distributions. The total reaction rate per unit surface area in each system is simply R 1 = A A z A B kc C d zd A (where A is lateral area of sediment, and z is the depth coordinate; units of R are mass/area/time). In the one-dimensional system ab abk z ab R = 1 k z =, but in the two-dimensional system R = 1 k z = abk z. In other words, the calculated total reaction rate is ten times higher in the twodimensional system. This example demonstrates how the total reaction rate calculated from a higher-dimensional model will not necessarily be the same as that generated

112 from a one-dimensional model, even if the reactions, rate constants and laterallyaveraged concentrations of A and B are the same in both. Sediment diagenesis models explore the fate of labile and refractory organic matter, accounting for the fact that the rate of organic matter mineralisation is directly related to the nature and amount of available oxidants. Neither the organic matter distribution nor the distribution of oxidants is uniform, and in the case of burrows, there may be pockets of high organic matter and high oxidant concentrations where diagenesis occurs at a greatly enhanced rate relative to the surrounding sediments. Kristensen (2) presented evidence which suggests that aged organic matter degrades up to 1 times faster under aerobic conditions than anaerobic conditions. The question of organic matter decay in heterogeneous sediments is of great interest to the sediment modelling community, and could be explored further using the EIC model as the underlying transport process in a full sediment diagenesis model More realistic burrow maps I have only considered the simplest of burrow maps: a diagonal burrow that, laterally averaged, is equivalent to uniform excavation and infill with depth. Given the many years of resin casting of thalassinidean shrimp burrows, it would be possible to create more realistic maps based on actual measurements. Risk (1978) demonstrated a simple burrow simulation model which automatically generated three dimensional burrow structures. The model was derived to generate burrow structures for Axius serratus, a burrowing shrimp for which the authors had little information other than polyester casts. Their model proved to be very useful as it captured the key burrow features while requiring few parameters. Such an approach could be easily applied to generate burrow maps for the EIC model. An even better approach would be to follow the method used by Koretsky et al. (22). These authors identified a set of end member shapes commonly observed in burrow networks (eg. vertical, angled, u-shaped and y-shaped tunnels). Koretsky et al. (22) used these shapes, combined with a set of parameters and probability distributions (defining total burrow density, occurrence rates of each burrow shape and the radius and lengths of burrow segments for each shape) to automatically generate stochastic burrow networks. Their purpose was to calculate bioirrigation coefficients, but their automatic burrow network simulation model would also be ideal for defining burrow

113 maps for the EIC model. It should be noted, however, that the representation of vertical tubes using the existing model would be problematic unless f = More realistic descriptions of burrow collapse The representation of burrow collapse is crude in the EIC model description. Only material directly above the burrow cavity moves downwards when a burrow collapses, while other material remains stationary. In reality, when a burrow collapses, other material from the sides will slump in to fill the cavity. Clearly the adoption of more realistic collapse dynamics would be desirable, however it is questionable whether the extra investment in model development would be warranted at this stage. There are other simplifying assumptions underlying the model (particularly the porosity assumptions and issues of grain size selection) that would warrant further attention before turning to better descriptions of collapse Scale issues The 2-D model was used to generate synthetic cores that were over 3 cm long. Modelled burrow thicknesses were simply the grid size (a non-dimensional grid size of.1, equating to dimensional grid sizes of approximately 4 to 6 mm depending on L). These scales are appropriate for thalassinidean shrimp burrows. The model could be equally well applied to other scales. Brandes and Devol (1995), Harper et al. (1999) and Sakita and Kusuda (2) modelled the effects of microsites that were 5 µm to 5 µm in size, within model sediments no more than 3 mm deep. In each of these cases the microsites were assigned a random spatial distribution by the modellers. The 2-D EIC model could be used to perform this function, so providing an explicit transport description for the creation of microsites in these models. 4.7 Conclusions It has been demonstrated that the one-dimensional EIC model developed in Chapter 3 can be expanded to two or three dimensions, so providing a simple non-local transport model that can be employed in a higher-dimensional sediment diagenesis model. The model can be constructed so that the laterally-averaged concentration distributions are identical to those produced by the 1-D analytical solution derived in the previous chapter, so providing a useful link to 1-D analyses. A synthetic dataset was generated using the 2-D model. A core was taken from this data and maximum likelihood estimation techniques were employed (using the

114 analytical solution to the 1-D model) to try and recover the mixing parameters which generated the data in the first place. The best estimates generally matched the parameters used to generate the data, but the confidence intervals were comparable to those generated by the 1-D parameter estimation method (using actual core data). The most effective ways of reducing confidence intervals were not surprising: ensure that core depths are sufficient to capture the entire concentration distribution and endeavour to reduce measurement uncertainty as much as possible. Nevertheless, the heterogeneity introduced by the 2-D process creates sufficient concentration variation at depth to make unravelling the mixing parameters from core data problematic. It is possible that the EIC model could be used to infer sediment transport mixing parameters from tracer cores using the 1-D EIC model. These parameters could then be applied to a diagenesis model which employs a 2-D EIC model as its solid transport process. The 1-D parameter estimation process allows the rigorous analysis of parameter confidence bounds, and so the parameter extremes could be applied to the 2-D model allowing the sensitivity to the underlying transport processes to be assessed. The comparison between 1-D and 2-D diagenesis models would also be a useful exercise, as it would help improve our understanding of the conditions under which 1-D modelling is inadequate

115 5 Modelling nitrogen dynamics around a burrow 5.1 Introduction Burrow irrigation by thalassinidean shrimp serves several functions, including sediment and waste expulsion and, in some cases feeding (eg. Stapleton et al. (22)). A key role is to draw oxygenated water into the burrow system. While many of these animals can tolerate low oxygen levels, and even prolonged periods of anoxia (Paterson and Thorne (1995)), they do need to respire aerobically within their burrows. The introduction of oxygen into sediment depths profoundly influences the surrounding sediment chemistry, with the effects plainly visible in many cases (eg. Figure 1.5). The modelling work in this chapter aimed at investigating the effect of burrow irrigation on nitrogen dynamics, and to make some more specific predictions about the impact of thalassinidean shrimp behaviour. A further aim was to model the sediments of Port Phillip Bay to investigate the role that irrigated burrows could play in observed denitrification rates. Section 5.3 describes the model development, Section 5.4 outlines some general model behaviour, Section 5.5 investigates three thalassinidean shrimp case studies, and Section 5.6 describes the Port Phillip Bay modelling. 5.2 Background Burrows serve to increase the area of the oxic-anoxic interface, thus altering the redox zonation geometry (Figure 5.1) and potentially increasing the number of sites suitable for coupled nitrification-denitrification reactions. Figure 5.1 Diagram from Aller (1982) demonstrating the potential impact of burrow irrigation on redox zonation in sediments

116 Aller (198) used a cylindrical burrow model to illustrate the likely effect of irrigated burrows on distributions of porewater species. The cylinder model takes the following form (assuming zero advection velocity): 2 C = D C Ds + t z r r r C s r + R C, z, r 2 (5.1) where z is vertical distance from the sediment surface, r is the horizontal distance from the burrow centre, C is the concentration, D s is the molecular diffusion coefficient corrected for tortuosity and R is a reaction term. Typical boundary conditions are C = C z = C = C r = r 1 C r r r = = 2 C B z L z = = (5.2) where r 1 is the radius of the burrow, r 2 is the outer radius of the sediment annulus surrounding the burrow and L is the depth of the burrow. B is a prescribed flux. Aller (1988) used a modified version of this approach to model the nitrogen dynamics surrounding an irrigated burrow. The main modification was to represent two zones in the sediment: an oxic annulus of sediment surrounding the burrow (Zone 1), surrounded by an anoxic annulus of sediment (Zone 2). Analytical solutions were investigated for two species: NH + 4 and NO 3. The NH + 4 case assumed that NH + 4 was produced only in Zone 2 according to zeroth order kinetics (and no nitrification in the oxic zone). The NO 3 case assumed zeroth order nitrification in Zone 1 and first order denitrification in Zone 2, as well as zeroth order nitrification exponentially decreasing with depth, to avoid having to represent two layers in the vertical direction as well as radial direction. Some of the main findings from the work are briefly mentioned here. Firstly, irrigated burrows profoundly influenced the distribution of species with zeroth order production terms, such as NH 4 +. Burrow separation distance was a particularly important control on NH 4 + concentration in the model. Unless nitrification regions directly overlapped, denitrification was increased when burrows were present. The ratio of denitrification to nitrification was also altered by the presence of burrows, usually increasing

117 The approach used in this chapter was influenced by Aller s work. I coupled a simple nitrogen dynamics model (based on Blackburn and Blackburn (1993)) to a radiallysymmetrical burrow irrigation model. Advantages of this approach include the fact that the oxic and anoxic zones are no longer user-prescribed but determined by the model itself (oxygen is one of the species modelled), and nitrification and denitrification are represented by more realistic reaction terms. The aim of this work was to apply a burrow irrigation model to some specific case studies. The main findings from Aller s founding work remain unchanged, and in general his analytical solutions are the most elegant approach for investigating fundamental issues of interactions between model geometry and reaction kinetics. 5.3 Model description The aim of this work was to develop irrigation models that allow for time-varying, incomplete burrow flushing and coupled chemical reactions. Most of the cases considered in this chapter are modelled using a one-dimensional radial model. The model simulates a one-dimensional transect radiating out from the burrow centre, assuming radial symmetry. The simplification is justified because the burrow lengths of interest are long and along-burrow variation is expected to be minor relative to variations in the radial direction. This implicitly assumes that the sediment surface is sufficiently far away so that it does not influence the reactions. This assumption is investigated in Section Modelling more complicated irrigation behaviour presents many challenges, particularly as measurements of burrow chemistry and irrigation behaviour are scarce. The temptation is to include a number of processes that common sense suggest must be occurring within the burrow, however in the absence of constraining data these inclusions merely add convenient parameters which can be tweaked endlessly to achieve novel results. Every attempt has been made to minimise this possibility. Where possible, parameters are constrained by published measurements and are not allowed to deviate from these. The remaining unconstrained parameters are discussed in each case. The model was constructed in Matlab, using the method of lines. The one-dimensional diffusion equation in cylindrical polar coordinates was represented by a central difference finite difference approximation in space, hence reducing the system of coupled partial differential equations to a set of coupled ordinary differential equations

118 (ODEs). Matlab s stiff ODE solver, ode15s, was used to solve the system of equations. The solver s performance was improved by specifying a sparse analytical Jacobian matrix for the system of equations, and modifying ode15s to perform a column permutation prior to any LU decompositions (matrix decomposition to a product of upper and lower triangular matrices). The Jacobian matrix of a system of ordinary differential equations, d y = f y is defined as: dt J = f1 f1 y1 y2 f2 f2 y1 y2 where y = [y 1, y 2, y 3, ] and f(y) = [f 1(y), f 2(y), f 3(y), ]. A dissolved species at grid location i is modelled by the following equation: Ci t = D 2 C i D Ci + r r r + R (5.3) 2 i and is represented by the following finite difference approximation in the model: dc dt D C 2 = C + C 2 r D C + r C 2 r i i+ 1 i i 1 i+ 1 i 1 i + R( C ) i (5.4) where C i is the porewater concentration at grid location i, r is the grid step, D is the rate of diffusion and R is the reaction term. This formulation assumes constant porosity in the sediment. The model allows for a finer grid to be prescribed on both sides of the burrow wall, so allowing greater spatial resolution at this interface. At the interface between two grid sizes, second order finite difference formulas for an uneven grid were used (Boudreau (1997) p. 326): C r ri ri + 1 = r r + r i i i+ 1 C i 1 ri + 1 ri ri + Ci + r r r r + r i i+ 1 i+ 1 i i+ 1 C i+ 1 (5.5) and

119 2 C = C + 2 i 1 Ci r r r + r r r r r + r r i i i+ 1 i i i+ 1 i+ 1 i i+ 1 C i+ 1 (5.6) where r = r r i+ 1 i+ 1 i. Seven chemical species, oxygen (O 2), nitrate (NO 3 ), sulfate (SO 2-4 ), dissolved organic carbon (DOC), ammonium (NH + 4 ), nitrogen gas (N 2) and sulfide (HS ) were modelled using the kinetics based on those described in Blackburn and Blackburn (1993). The model assumes that reactions only occur within the sediment, so R = for all species in the burrow water. Twelve species were represented in their model: carbon dioxide (CO 2), dissolved organic carbon (DOC), dissolved organic nitrogen (DON), HS, NH + 4, exchangeable ammonium (NH 4_ex), NO 3, N 2, particulate organic carbon (POC), particulate organic nitrogen (PON), O 2 and SO 2-4. The reasons for omitting five of the species are varied. The distributions of nitrogen species take the same shape as the distributions of carbon species, varying only by a known factor (the carbon to nitrogen ratio), so PON and DON can be omitted. POC was omitted and its function was replaced with a production rate of DOC (P DOC) which specifies the rate and distribution of DOC production. CO 2 was omitted, as its exclusion does not affect the distributions of other species. A model version was constructed which included exchangeable ammonium, however there was negligible difference in results when it was excluded from the model. The original authors of the model have also routinely omitted NH 4_ex from their model formulation (eg. Blackburn and Henriksen (1983), Blackburn et al. (1994)). The equations for the remaining species reaction rates are: + O 2 3 2_ stim 6 2_ stim 4 7 2_ stim R = k O DOC 2k O NH 2k O HS + NO 4 3_ stim 2_ inhib 6 2_ stim 4 R - = k NO O DOC + k O NH 3 R 2- = k SO O DOC + k O HS SO 5 4_ stim 2_ inhib 7 2_ stim 4 R = P k O DOC 2k SO O DOC DOC DOC 3 2_ stim 5 4_ stim 2_ inhib 1. 25k NO O DOC 4 3_ stim 2_ inhib RN =.5 k NO stimo inhib DOC 2 4 3_ 2_ R DOC + = k O k NO 4 λ NH 3 2_ stim 4 3_ stim 2_ inhib CN O DOC + 2k SO O k O λ DOC CN + 5 4_ stim 2_ inhib 6 2_ stim NH4 CN R = k SO O DOC k O HS - HS 5 4_ stim 2_ inhib 7 2_ stim λ (5.7)

120 where square brackets represent porewater concentration. λ CN is the carbon to nitrogen ratio (assumed to follow Redfield C:N stoichiometry of 16:16). The rate constants are the same as those specified by Blackburn and Blackburn (1993): k 3 = 3 d -1 for DOC oxidation by O 2; k 4 = 3 d -1 for DOC oxidation by NO 3 ; k 5 = 5 d -1 for DOC oxidation by SO 2-4 ; k 6 = 3 d -1 for NH + 4 oxidation by O 2 (nitrification); k 7 = 2 d -1 for HS oxidation by O 2. The _stim and _inhib functions are: C O _ stim = 2_ inhib C 1 = C crit crit O2 1 O2 < C C crit C < C otherwise crit otherwise (5.8) where C crit is 3 nmol/cm 3. Molecular diffusion coefficients were corrected for temperature and salinity, where known. In the absence of this information, the diffusion coefficients given in Blackburn and Blackburn (1993) were used. These molecular diffusion coefficients were further corrected for tortuosity within the sediment. The choice of diffusion coefficient in the burrow itself varied. In some cases simply the molecular diffusion coefficients in water were used, but in other cases it made more sense to assume a higher rate of mixing within the burrow (due to animal movement). In these cases diffusion rates were set to be a factor of 1 higher than molecular diffusion rates. These details were found to make little difference to model results. A no-flux boundary condition applies at the far boundary (r = r 2), which is equivalent to assuming that there is no flux between burrows: C = (5.9) r r r2 = Radial symmetry allows us to assume that concentration distributions have a local maximum or minimum at the burrow centre, so C = (5.1) r r =

121 Three forms of irrigation behaviour were modelled: 1) Constant, fully flushed burrow (burrow water concentrations identical to surface water concentrations). In this case, the burrow region is not included in the model (as it is simply a uniform concentration) and a constantconcentration boundary condition applies at r = r 1. 2) Oscillating irrigation conditions. Irrigation behaviour is specified using a period (T i) and an irrigation fraction (p i). The irrigation fraction p i is the fraction of time that is spent flushing the burrow. For example, a burrow irrigated for 15 minutes every hour has T i = 1/24 days and p i =.25. During periods of burrow irrigation, the burrow water concentrations were held constant. The concentrations were calculated by assuming that the burrow water is mixed with an injection of surface water. For example, if C q is the burrow concentration at the end of a quiescent period, the concentration assumed during the subsequent irrigation period is C = fc + ( 1 f ) C (5.11) o q where C o is the concentration in the overlying water and f is a mixing efficiency; it is the fraction of burrow volume that is replaced with overlying water during a flushing event. 3) Constant, partially flushed burrow. The equation within the burrow zone includes a sink term: Ci t = D 2 C i D Ci + + F C C 2 l i r ri r (5.12) where F l is a flushing time constant and C is the overlying water concentration. The sink term represents the injection of overlying water and the expulsion of burrow water at a rate specified by F l. The residence time of water in the burrow is 1/F l. By specifying irrigation in this way, we avoid the complication of oscillating conditions, yet still capture the fact that burrow concentrations differ from overlying water conditions

122 Fluxes across the burrow surface, J bur, can be calculated from the model, using the following equation: J r D C bur = π φ s r 2 1 r= r1 (5.13) Note that the above expression produces a rate per unit burrow length (eg. nmol/cm/day). It is easily converted to a flux per unit of planar sediment surface (referred to as J r) by multiplying the burrow flux by burrow length (L) and number of burrows per unit area (N): Jr = LNJbur (5.14) When the system is at steady state, J bur can be calculated from the following integration: r2 Jbur = 2πφ Rr d r r1 (5.15) Equation (5.15) was used in preference to equation (5.13) as it is more accurate to numerically integrate across a large number of cells than to numerically differentiate using relatively few cells at the burrow wall. Key model outputs of interest are the rates of nitrification and denitrification. Again, due to the model geometry, these are rates per unit burrow length (nmol/cm/day). The carbon mineralisation rate is: r2 RC = 2πφ PDOC r d r (5.16) r1 The nitrification rate is r2 RNit = 2πφ k6o2_ stim NH4 r1 r dr (5.17) and the denitrification rate is R Dnit = 2πr φ D 1 s 2N r 2 r= r1 (5.18)

123 or if the system is at steady state it is preferable to calculate the denitrifcation rate using the integral form, r2 RDnit = 2 2RN r r 2 r1 π φ d. (5.19) It is often convenient to express nitrification and denitrification rates as a percentage efficiency, which is the proportion of nitrogen mineralisation which is occurring through the nitrification and denitrification pathways. If overlying water concentrations of NH + 4 and NO 3 are zero (ie mineralisation of sediment organic matter is the only source of nitrogen), the nitrification efficiency is: E Nit CN R = λ R C Nit (5.2) and the denitrification efficiency is: E Dnit CN R = λ R C Dnit (5.21) 5.4 General model behaviour This section outlines some general model behaviour with reference to key parameters, P DOC, r 1, r 2 and F l. The model outputs of interest are the concentrations of O 2, NO 3 and NH + 4 and the rates of nitrification and denitrification. Typical profiles for each of these are presented in Figure 5.2. In this example, the burrow has a radius of 3mm and is fully flushed with overlying water. Oxygen is totally consumed within a 7 mm of the burrow wall, and this defines the zone of nitrification (which requires oxygen and NH 4 + ). NH 4 + is zero at the burrow wall, as it is fully flushed with overlying water (which is set to have an NH 4 + concentration of zero). The steady state distribution of NH 4 + represents a balance between production from organic matter decay, reaction with oxygen and diffusion across the burrow wall. NO 3 produced within the nitrification zone diffuses out in two directions, where it is either lost from the system through diffusion across the burrow wall or by reaction with organic matter in anoxic zones (denitrification). The zone of denitrification exists just beyond the zone of nitrification, and its size depends on the rate of supply of NO 3 and the volume of anoxic sediment. There is some overlap between the nitrification and denitrification zones, so that denitrification occurs in the presence of very low oxygen concentrations;

124 real sediments are heterogeneous, and there will be anaerobic microsites existing in low oxygen areas. O 2 (µmol/l) 2 1 NO 3 NH 4 + (µmol/l) (µmol/l) nitrification rate (nmol/cm 3 /day) denitrification rate (nmol/cm 3 /day) r r r r r r (cm) Figure 5.2 Typical profiles generated from the one-dimensional radial burrow model. The 3 mm radius burrow is fully flushed with overlying water ([O 2] = 23 µmol/l, [NO 3 ]=[NH 4 + ]=). P DOC was set at 5 µmol/l/day. The effect of varying P DOC for this model configuration is illustrated in Figure 5.3 and Figure 5.4. At very low values of P DOC the oxygen penetrates to r 2, and so there are no anoxic regions in the sediment; no denitrification is possible under these circumstances. Nitrification rates are also low, as the rate of supply of NH + 4 is low (the rate of supply of NH 4 + is directly proportional to P DOC). Nitrification efficiencies are 1% in these circumstances; no nitrogen leaves the system as DON or NH + 4 because oxygen is readily available to react with both species. As P DOC increases, an anoxic zone forms and denitrification becomes possible. For high oxygen penetration distances, the total denitrification rate is limited by the volume of anoxic sediment, and for low oxygen penetration distances the denitrification rate is limited by the amount of NO 3 diffusing from the nitrification zone. The denitrification efficiencies peak at low P DOC values, where nitrification rates are high but there is a small outer

125 anoxic zone where denitrification can occur. As P DOC increases beyond this value, the nitrification and denitrification efficiencies decrease (Figure 5.4). This is because the rate of nitrification is overwhelmed by the rate of production of NH + 4, and significant quantities of NH + 4 leave the system through diffusion across the burrow wall. The distance between burrows (2r 2) determines the amount of NH + 4 that can build up in the sediment. This in turn affects the distributions of other species, and in particular + it shifts the location of the oxic/anoxic boundary (Figure 5.5). Higher NH 4 concentrations fuel higher nitrification (and so denitrification) rates, however the efficiency of nitrification and denitrification decreases with increasing r 2 (Figure 5.6). O 2 (µmol/l) NO 3 (µmol/l) P DOC (µmol/l/day) P DOC (µmol/l/day) N 2 (µmol/l) NH 4 + (µmol/l) P DOC (µmol/l/day) P DOC (µmol/l/day) P DOC (µmol/l/day) nitrification (nmol/cm 3 /day) r (cm) 4 2 P DOC (µmol/l/day) denitrification (nmol/cm 3 /day) r (cm) Figure 5.3 The effect of changing P DOC on concentration and rate distributions

126 1 nitrification denitrification efficiency (%) P DOC (µmol/l/day) Figure 5.4 Changes in nitrification and denitrification efficiencies with changing P DOC (all other parameters the same as example given in Figure 5.2). O 2 (µmol/l) NO 3 (µmol/l) r 2 (cm) r 2 (cm) N 2 (µmol/l) NH 4 + (µmol/l) r 2 (cm) r 2 (cm) r 2 (cm) nitrification (nmol/cm 3 /day) r (cm) denitrification (nmol/cm 3 /day) r 2 (cm) r (cm) 2 1 Figure 5.5 Effect of r 2 values on concentration and rate distributions

127 1 nitrification denitrification efficiency (%) r (cm) 2 Figure 5.6 Changes in nitrification and denitrification efficiencies with changing r 2 (all other parameters the same as example given in Figure 5.2). O 2 (µmol/l) NO 3 (µmol/l) r 1 (cm).3 1 r 1 (cm) N 2 (µmol/l) NH 4 + (µmol/l) r 1 (cm) r 1 (cm) r 1 (cm) nitrification (nmol/cm 3 /day) r (cm) r 1 (cm) denitrification (nmol/cm 3 /day) r (cm) Figure 5.7 Effect of r 1 values on concentration and rate distributions for a fully flushed burrow

128 The effect of changing the burrow radius, r 1 is not so pronounced for a fully-flushed burrow (Figure 5.7). Oxygen penetration increases with increasing r 1, creating a wider nitrification zone, and so slightly increasing nitrification (and denitrification) efficiencies (Figure 5.8). The effect is more pronounced for a partially flushed burrow. Under these circumstances, NH 4 + and NO 3 remain trapped in the burrow after diffusing across the burrow wall. The amount trapped in the burrow strongly depends on the burrow volume (and so r 1), and this in turn affects the nitrification and denitrification efficiencies. For the same range of r 1 values, the partially flushed burrow produces a greater range of nitrification and denitrification efficiencies (although they are all lower than the fully flushed efficiencies) nitrification (fully flushed) denitrification (fully flushed) nitrification (F l =24day 1 ) denitrification (F l =24day 1 ) efficiency (%) r (cm) 1 Figure 5.8 Changes in nitrification and denitrification efficiencies with changing r 1 for two burrow flushing rates (all other parameters the same as example given in Figure 5.2). The effects of flushing rate are shown in Figure 5.9 and Figure 5.1. Nitrification efficiencies decrease with decreasing flushing rate as oxygen supply to the sediment is lowered. Denitrification efficiencies follow the same trend, as denitrification rates are limited by nitrification rates. A general summary of the model response to changing parameters is given in Table

129 O 2 (µmol/l) fully flushed F l =48 day 1 F l =24 day 1 F l =24/5 day 1 F l =1 day 1 NO 3 (µmol/l) fully flushed F l =48 day 1 F l =24 day 1 F l =24/5 day 1 F l =1 day 1 r r (cm) r r (cm) nitrification (nmol/cm 3 /day) fully flushed F l =48 day 1 F l =24 day 1 F l =24/5 day 1 F l =1 day 1 r r (cm) denitrification (nmol/cm 3 /day) fully flushed F l =48 day 1 F l =24 day 1 F l =24/5 day 1 F l =1 day 1 r r (cm) Figure 5.9 Effect of flushing rates on concentration and rate distributions. 1 nitrification denitrification efficiency (%) fully flushed F l (day 1 ) Figure 5.1 Changes in nitrification and denitrification efficiencies with changing flushing rates (all other parameters the same as example given in Figure

130 Table 5.1 Directions of model response to parameter changes Maximum NH 4 + concentration O 2 penetration distance Nitrification efficiency Increasing P DOC Increases Decreases Decreases Denitrification efficiency Generally decreases Increasing r 2 Increases Decreases Decreases Decreases Increasing r 1 Decreases Increases Increases Increases Increasing F l Decreases Increases Increases Increases In summary, nitrification efficiencies are high when the nitrification rate is high enough to keep up with NH 4 + production. When the NH 4 + production rate overwhelms the nitrification rate, ammonium builds up in the sediment surrounding the burrow, creating a steep gradient to the burrow wall and a high ammonium diffusive flux across the burrow wall. NH 4 + production is controlled by P DOC, and the distance r 2 is an important control on the amount of NH 4 + that can accumulate in the system. Nitrification efficiency in turn is the main control on denitrification efficiency. Where nitrification efficiency is high, denitrification efficiencies will be high also, except for the situation when the nitrification zone approaches r 2, so there is little or no anoxic zone in which denitrification can occur. 5.5 Thalassinidean shrimp case studies The aim of this section is to model some specific case studies drawn from the literature. There have been some excellent studies of thalassinidean shrimp burrow structure and burrow chemistry, and these provide an opportunity to test the modelling approach and compare theoretical understanding with actual measurements Case 1: modelling Callianassa truncata field data (Ziebis et al. (1996)) The measurements in Callianassa truncata burrows by Ziebis et al. (1996) are the only in situ measurements of thalassinidean shrimp burrow chemistry. A Callianassa truncata population was allowed to colonise the sediment surrounding a buried observatory in the Bay of Campese, Italy. My aim was to use their data to constrain model parameters in order to investigate the likely effect on nitrogen dynamics at depth. The population was estimated at 12 burrow structures/m 2. Burrows consist of tubes (4-5 mm diameter) and chambers (14 mm diameter), and typically extend to depths of 5cm beneath the sediment surface. The authors estimated that the area of the sediment-water interface was increased by roughly 4%

131 The sediment was sandy (porosity of.4) and poor in organic matter. The authors found that at a depth of 48 cm, burrow-water oxygen concentrations oscillated between 3% and 12% of air saturation. The period of oscillation was approximately one hour. The oxygen record at a depth of 26 cm showed no obvious periodicity, and oxygen levels were generally maintained between 1% and 4% of air saturation. Irrespective of sediment depth, oxygen penetration depths in the tube walls and spherical burrow chambers were approximately 3 mm and 7 mm respectively. Oxygen profiles measured in the laboratory suggested that oxygen penetration was 4mm at the sediment surface (away from burrow mounds). NH + 4 levels were low in the burrow (2-14 µmol L -1 ) and NH + 4 was also depleted (<15 µmol L -1 ) in the surrounding sediment to a radius of up to 3-4 cm. Depth profiles of NH + 4 levels were made in inhabited and uninhabited sediment. The uninhabited areas of sediment were established in mesh cages attached to the side of the observatory, and maximum values of approximately 1µmol/L were measured at depth (~5cm). Measurements were made two months after the observatory was deployed. Given that the sediment would have been mixed to remove animals and install the observatory, it seems reasonable to assume a uniform distribution of organic matter in the sediment column. I used a vertical (1-D planar) version of the nitrogen model to find a value of P DOC that would produce NH + 4 concentrations of ~1µmol/L after two months (P DOC = 15 µmol/l/d). The surface oxygen penetration predicted by this scenario is large (~ 3cm), and far exceeds the 4mm penetration depth suggested by the authors. It should be noted that oxygen penetration at the surface was not measured in situ, and the penetration depth was inferred from measurements in a laboratory flume. Further, the NH + 4 concentrations are zero to a depth of 1cm in the field measurements (Figure 5.11), which suggests even deeper oxygen penetration than that predicted by the model. In short, there is a contradiction between the laboratory measurements of oxygen (which suggest a 4 mm oxygen penetration depth) and the field measurements of NH 4 + (which imply a 1 cm oxygen penetration depth) that cannot be explained by my model (which predicts a 3 cm oxygen penetration depth). A possible explanation for these differences is high levels of pore-water advection in the upper section of the field sediment

132 Figure 5.11 Redox (solid line) and NH 4 + profiles (filled circles) in sediment inhabited and uninhabited (dotted line and open circles) sediment, taken from Ziebis et al. (1996). Having found a P DOC value that produced the required NH + 4 levels in the absence of burrows, I had planned to substitute this value directly into the oscillating irrigation (radial) model, using burrow radii inferred from the resin casts and population density. To fit the oscillating model to the data, irrigation parameters would need to be found that produce an oxygen penetration depth of 3mm at burrow oxygen levels of 17% and 38% of air saturation. The oscillating model is very time-consuming to run, and it is best used to infer the burrow concentrations using known irrigation parameters. In this case we do not know the irrigation behaviour, but we do know the burrow concentrations for two important species: O 2 and NH + 4. A further complication is that the burrow oxygen records (particular at 26cm depth) show dramatic changes in oxygen levels over a very short space of time (eg a drop from 43% to 17% in the space of minutes, Figure 5.12). Figure 5.12 Continuous oxygen measurements within a burrow at 26 cm and 48 cm depth, taken from Ziebis et al. (1996)

133 This kind of behaviour cannot be reproduced in the model; the sediment oxygen demand would need to be extremely high to produce such rapid drops in oxygen, yet this would then be incompatible with the measured oxygen penetration depths. The most likely cause of the rapid changes in burrow oxygen levels is that water of varying oxygen concentrations was advected along the burrow lumen. For these reasons constant, partially flushed conditions were assumed within the burrow. As the flushing rate increases, the burrow oxygen level increases and the burrow NH 4 + level decreases. By running the model for a range of flushing rates, I was able to select rates that provided a good match to measured burrow oxygen levels (eg. Figure 5.13). The burrow geometry settings were r 1 =.2cm and r 2 = 5cm (derived from the population density of 12 individuals/m 2, 12πr 2 2 = 1 4 cm 2 ) burrow O 2 concentration (µm) residence time (hr) Figure 5.13 Modelled burrow oxygen concentrations as a function of burrow flushing rates (represented here as burrow residence time). Asterisks mark the measured oxygen concentrations in Callianassa truncata burrows at depths of 2cm and 4cm below the sediment surface. Using the above parameters, the predicted oxygen penetration distances from the burrow wall, 7 mm and 1 mm, were greater than the measured values, both 3 mm, and the modelled NH + 4 levels 3-4 cm from the burrow, >5 µmol/l, were much higher than the measured values at that distance, < 15 µmol/l. These results present a seemingly irresolvable situation. Either increasing the P DOC value, or decreasing the burrow flushing rate, would decrease the oxygen penetration distance, however both these alterations to the model would increase the porewater NH + 4 concentrations. The modelled NH + 4 concentrations are already much higher than the measured values, and so a further increase would only worsen the match between modelled and measured data

134 Further runs were conducted experimenting with the radial distribution of P DOC. It is widely believed that burrow walls have a higher mineralisation rate than the surrounding sediment. In earlier chapters we saw that burrows can function to subduct surface material to considerable depths where it can be incorporated into the burrow wall. Mucus linings needed to maintain burrow structural integrity can also be a source of organic matter. Kristensen (2) suggested a higher mineralisation rate at the burrow wall could be due to old, partly degraded organic matter being exposed to oxygen (estimated to increase the mineralisation rate by a factor of ten). Other suggestions include microbial stimulation by grazing and oscillating redox conditions (Aller (1994)). While the precise mechanisms remain unclear, there appears to be wide consensus that enhanced mineralisation is likely to occur at the burrow wall. To test the effect of this in model I altered the P DOC term to be a step function, with a higher value within a band of sediment immediately surrounding the burrow (the region r 1 r < r w ): P DOC = P = f P r r < r c P2 rw r r2 w (5.22) where f c is simply a prescribed factor. To allow direct comparison with the uniformly distributed P DOC case, P DOC = P c (uniformly distributed between r 1 and r 2), P 1 and P 2 were set so that the total mineralisation rate (obtained by integrated between r 1 and r 2) was the same in each case. This requirement sets P 2 to be: P 2 = Pc r 2 2 r f r r + r r 2 2 c w 1 2 w (5.23) Model runs were repeated with altered P DOC distributions as specified in equations (5.22) and (5.23). Values of f c = 1 and f c = 1 were tested. The region of enhanced mineralisation was set to be a 3mm band of sediment immediately surrounding the burrow. Comparisons for all three model runs are presented in Figure 5.14 and Figure

135 oxygen concentration (µm) f c = 1 f c = 1 f c = 1 measurements oxygen concentration (µm) f c = 1 f c = 1 f c = 1 measurements distance from burrow centre (cm) distance from burrow centre (cm) Figure 5.14 Modelled and measured oxygen profiles radiating from burrow tubes. Measurements taken at 2cm depth (left) and 4cm depth (right) (Ziebis et al. (1996)). 6 7 ammonium concentration (µm) f c = 1 f c = 1 f c = 1 ammonium concentration (µm) f c = 1 f c = 1 f c = distance from burrow centre (cm) distance from burrow centre (cm) Figure 5.15 Modelled NH 4 + profiles radiating from burrow tubes. The two scenarios represent the 2cm depth (left) and 4cm depth (right) simulations. Three differences to the measurements are obvious. Firstly, in all model runs the oxygen profiles were markedly different in shape to those measured by Ziebis et al. (1996). The modelled oxygen profile shapes were typical of exponential decline with distance, but measured tube profiles were roughly linear. Secondly, the measurements suggest that oxygen penetration distance is roughly 3 mm, regardless of the burrow oxygen concentration, yet the model always predicts deeper oxygen penetration + associated with higher burrow oxygen levels. Thirdly, in all model runs, the NH 4 concentrations away from the burrow far exceed the measured values, although the modelled burrow NH + 4 concentrations fall within the range of measured field values

136 The model results only come close to matching the oxygen penetration distance and far-field NH 4 + concentrations when the rate of DOC production in the burrow wall exceeds the rate in the surrounding sediment by a factor of 1. Clearly there is no one elegant model configuration which explains all the data. My + inability to match both NH 4 data and O 2 data simultaneously highlights the fact that the model is a serious simplification of reality, and does not capture the full complexity of the field processes. Likely explanations include: 1. advection: the sediment is highly permeable, so diffusion and reaction may not be the only processes influencing concentration distributions. Complicated advection patterns may dramatically alter porewater distributions (Huettel and Webster (21)). In the lower sections of the burrow (>1 cm from the sediment surface) the burrow is basically a single shaft, and not a tube connecting two surface openings. Conservation of volume may require that the fluid the shrimp displaces as it wanders along such a shaft is forced through the permeable wall into the surrounding sediment, hence the mere act of roaming its burrow may induce a transitory pumping effect in the near vicinity of the burrow. 2. sediment sorting: Ziebis et al. (1996) described preferential ejection of fine material and incorporation of larger grains into chamber walls, so it is likely that sediment packing is substantially altered at the burrow wall. Changes to sediment packing alter the tortuosity, and so the porewater diffusion coefficient of dissolved species. 3. time-varying system: the history of the concentration distributions and transport processes is unknown, yet could significantly impact on the measured distributions. The field system was almost certainly not at steady state. I ran the model for 2 months, starting from an assumption of uniformly mixed sediment, and this was insufficient to reach steady state conditions. The authors took measurements 2 months after observatory installation, and the extent of mixing due to installation is unknown. 4. reaction kinetics: the model assumes first order reaction kinetics for oxygen consumption when oxygen levels are below 3 µmol/l and zeroth order kinetics for oxygen concentrations greater than this. The profile shape emerging from the combination of species reaction and diffusive transport vary with the type of

137 reaction kinetics. The exponential decay shape is typical of first order reactions. If the value of C crit is altered in equation (5.8) different shaped profiles will result. Later sections of this chapter investigate the impact of burrow irrigation on nitrification and denitrification rates, as it has been widely suggested that irrigated burrows can increase both these rates. For this reason I calculated the nitrification and denitrification rates associated with the above model runs, and present them in Table 5.2 for completeness. Table 5.2 Denitrification efficiencies and denitrification:nitrification ratios for C. truncata model runs. Scenario Denit efficiency (at 2 months) Denit/Nit (at 2 months) Denit efficiency (Steady state) Denit/nit (Steady state) 2 cm depth, f c = cm depth, f c = cm depth, f c = cm depth, f c = cm depth, f c = cm depth, f c = Case 2: modelling Callianassa subterranea laboratory data (Forster and Graf (1995)) Forster and Graf (1995) measured irrigation currents and oxygen levels in C. subterranea burrows. The specimens were taken from silty fine sand in the North Sea and housed in flat aquaria (5cm x 4cm x 5cm) where burrow development, oxygen levels and irrigation velocities could be measured. When compared to the C. truncata study, a very different picture emerges. Only the upper sections of C. subterranea burrows were studied (oxygen measurements were made no deeper than 17 cm below the sediment surface), whereas Ziebis et al. (1996) report only oxygen measurements made at depths greater than 2 cm from the surface. At these depths C. truncata burrows contained significant oxygen concentrations; by contrast, for most of the time there was no detectable oxygen in C. subterranea burrows 17 cm from the surface. Oxygen penetration distances into the surrounding sediment were typically 3 and 7 mm for C. trunctata, yet Forster and Graf (1995) reported that oxygen penetration was less than.25 mm into the surrounding sediment at 17mm from the surface, and around 1mm at 16mm from the surface

138 Forster and Graf (1995) described two forms of burrow irrigation: regular and irregular. Regular irrigation was characterised by quiescent periods lasting 4 minutes interspersed with brief irrigation periods lasting 15 seconds. Irregular irrigation occurred when the animal was observed rushing rapidly around the upper sections of the burrow. Irregular irrigation events lasted less than a minute, but the flow velocity could be substantially higher than the regular irrigation flow velocity (up to 2 cm/s for irregular irrigation,.3 cm/s for regular irrigation). Oxygen measurements at 17cm below the sediment surface suggest that oxygen is only present in this section of the burrow during periods of irregular irrigation. The study described burrow geometry, irrigation behaviour and oxygen measurements, and so there ought to be sufficient information to parameterise the model. In particular, the measurements taken in the upper section of the burrow seem conducive to modelling as there are simultaneous irrigation and oxygen data. Away from any burrows, the oxygen penetration distance was 4 mm from the sediment surface. A vertical version of the nitrogen model was used to determine the value of P DOC that produces this penetration distance, and this was P DOC = 4 µmol/l/d. This same value was then substituted into a radial version of the model, with r 1 set at.3 cm and r 2 = 2.5cm. The animals were burrowing in aquaria that were 5 cm thick, which places a constraint on the size of r 2. The value of r 1 simply followed from the authors measurements of burrow diameters. The irrigation parameters were set up so that the irrigation frequency was 4 minutes, the irrigation duration was 15 seconds, again the measured values. Forster and Graf (1995) measured oxygen levels 16 mm from the sediment surface and 1mm from the burrow wall within the sediment surface. During irrigation periods they measured oxygen levels of 8% saturation, which then dropped at a rate of 25.2 µmol/l/min (quoted as.42 nmol O 2 /cm 3 /s in the paper). Matching model results to measurements again proved to be difficult. Firstly, the authors reported an instantaneous large rise in oxygen levels (% to 8% saturation) 1mm from the burrow wall within the sediment (Figure 5.16)

139 Figure 5.16 Changes in oxygen concentration (measured by an oxygen sensor position 1mm from burrow wall, within the sediment) registered with simultaneous measurements of flow above the inhalant funnel opening, taken from Forster and Graf (1995). Diffusion through the sediment is a slow process, even across a distance of only 1mm. To illustrate the point, we can make the generous assumption that oxygen is diffusing through water (no sediment) and it is not being consumed by any reactions. Initial conditions are zero oxygen at all locations and for t > the concentration at x = is C o. Further assumptions include a no flux boundary at a distance L and cartesian, rather than radial, coordinates have been used for simplicity. The governing equations are: 2 C = D C 2 t x C x, = C, t = C C = x L, t (5.24) and the solution is (Boudreau (1997)): 2C 2 C = C + sin β nx exp Dβ nt Lβ β = n n= 1 n 1 2 π L n (5.25) Assuming L = 1 cm (an arbitrary choice, set to be sufficiently far away from the oxygen penetration distance) and D = 1.8 cm 2 /d the concentration will reach C =.8C at x =.1 cm when t ~ 6 minutes. In the scenario described by Forster and Graf (1995) the time required would be even longer as the oxygen is diffusing through sediment and it

140 is being consumed by reactions in the sediment. Without knowing further details about the experimental setup, it could be possible that the sensor placement induced some additional transport, so burrow water was advected to the sensor location during burrow irrigation periods. If this is so, the oxygen time series is maybe more representative of oxygen levels within the burrow, at the burrow wall. Unfortunately, this too seems unlikely. The rate of oxygen depletion once irrigation has stopped is remarkably high. Unlike in the C. truncata study, where we could speculate that alongburrow advection currents could rapidly transport patches of oxygen depleted water past the oxygen sensor, thus explaining the large, rapid changes in oxygen levels, Forster and Graf (1995) reported simultaneous velocity measurements which confirm that the burrow water was quiescent. If oxygen levels are at 8% saturation at the burrow wall, the model can only explain such a rapid drop to % saturation if mineralisation rates in the surrounding sediment are extremely high. This possibility is excluded by the fact that the modelled oxygen distribution would penetrate a much smaller distance than the measured 1 mm penetration distance in the burrow wall and ~4mm at the sediment surface. The vertical model suggested that the P DOC needs to be around 4 µmol/l/d to produce a 4mm oxygen penetration at the surface. As an aside, I did find a model configuration that produced a reasonable match to the time-series given by Forster and Graf (1995) (Figure 5.17). It was a spherical model with small values of r 1 (.5mm) and r 2 (4mm), and all other parameters were kept unchanged (P DOC = 4 µmol/l/d; 15s irrigation period in each 4 minute irrigation cycle). This model was constructed to test the following scenario: during burrow irrigation, burrow water is advected to the sensor tip embedded in the sediment so triggering an instantaneous rise in the oxygen time series; once irrigation stops the tip is surrounded by oxygen consuming sediment and the oxygen at the sensor tip is rapidly depleted (Figure 5.18). The spherical model result in Figure 5.17 lends credibility to this possibility

141 18 16 spherical model Data from Forster & Graf (1995) O 2 concentration (µm) time (minutes) Figure 5.17 Oxygen data from Figure 5.16 (assuming the O 2 concentration in the overlying water is 215 µmol/l) compared with spherical model result (see text for details.) Oxygen probe sediment burrow Burrow water is advected into this space during burrow irrigation so the sensor measures a rapid increase in oxygen. Oxygen is consumed rapidly in quiescent times due to the surrounding volume of sediment. Figure 5.18 Possible explanation for high rate of oxygen increase and decline measured by Forster and Graf (1995) 1mm from burrow wall

142 A conflict remains: assuming P DOC = 4 µmol/l/d, there should still be some oxygen present 1mm from the burrow wall, yet the measurements suggest it is anoxic for the majority of the time. This characteristic of the data is curious, and conflicts with standard expectations. For example, Fenchel (1996) measured oxygen penetration depths at the sediment surface and in the sediment surrounding worm burrows. He found that the measurements matched well with a theoretically derived relationship between the sediment and burrow oxygen penetration depths: 2 L = r + r + 2 r L (5.26) b s where L b and L s are the oxygen penetration distances at the burrow wall and sediment surface respectively. Using this relationship and assuming r 1 = 3 mm and L s = 4 mm, equation (5.26) predicts a burrow oxygen penetration distance of 2.7 mm (compared to <1mm measured). The derivation of (5.26) did not include confounding influences such as differing diffusive boundary layer thicknesses between burrow and the surface, and it assumed the same oxygen concentration in the burrow and at the surface. Yet Fenchel (1996) found that measurements from worm burrows matched the theoretical estimates very well. Another possible explanation for the unexpectedly low burrow oxygen penetration is that the burrow wall is lined with material with a dramatically different mineralisation rate to the surface sediment (and consumes all the oxygen within 1mm from the burrow). I used the model to investigate both a homogeneous and heterogeneous P DOC distribution: Case 1: Used a uniform value of P DOC = 4 µmol/l/d (obtained from matching oxygen penetration distances at the surface). Case 2: Assumed there is a 2 mm band surrounding the burrow with a higher P DOC value (details in Table 5.3). The oxygen time series data and corresponding flow meter measurements provide good information about the irrigation behaviour, so supplying well-constrained irrigation parameters that can be used in both cases. The results for the two cases are listed in Table

143 Table 5.3 Model results simulating two laboratory scenarios and a field scenario. Case 1 Case 2 Field case P DOC 1 r1 r < r1 + 2mm (µmol/l/d) r + 2mm r r Oxygen penetration depth (mm) Oxygen flux per unit burrow length (µmol O 2 cm -1 day -1 ) Nitrification rate (per unit burrow length) (µmol N cm -1 day -1 ) Denitrification rate (per unit burrow length) (µmol N cm -1 day -1 ) Denitrification efficiency Denitrification : nitrification Burrow measurements made by Forster and Graf (1995) 17cm from the sediment surface recorded zero oxygen levels during regular irrigation events. Regular irrigation pumped overlying water into the burrow at a rate of 6 ml h -1. The oxygen concentration of the overlying water in the laboratory was 215 µmol L -1. For the oxygen to be zero in the burrow at a distance of 17cm from the burrow opening, all the oxygen must have been consumed by the walls. These measurements suggest a per unit burrow length oxygen consumption rate of at least 1.8 µmol O 2 cm -1 day -1 in the upper 17 mm of the burrow. Case 1 predicts this oxygen flux well (1.7 µmol O 2 cm -1 day -1 ). Case 2 overestimates the flux (2.8 µmol O 2 cm -1 day -1 ), however this result is not necessarily inconsistent with the inferred flux of 1.8 µmol O 2 cm -1 day -1, as this is minimum flux that is necessary to consume all the oxygen in the upper 17cm of the burrow. Nitrification and denitrification rates are substantially higher for Case 1, demonstrating the significant influence mineralisation rates (and distribution) can have on the nitrogen dynamics. These model runs simulated the laboratory conditions. Forster and Graf (1995) also measured oxygen profiles in situ. The oxygen gradient at the surface was much steeper, suggesting a higher sediment mineralisation rate in the field. In addition, the oxygen concentration in the overlying water was markedly different (presumably due to different temperatures). The field oxygen concentration was 292 µmol/l, whereas the oxygen concentration in the laboratory water was around 215 µmol/l

144 Assuming that the irrigation behaviour remains the same as in the laboratory, the impact of this behaviour on the field sediments can be assessed using the model. As before, the P DOC value was found by matching the oxygen penetration distance measured at the sediment surface (2.5 mm), yielding a value of P DOC ~ 25 µmol/l/d. The results are listed in Table 5.3. The per unit burrow length oxygen consumption rate was 3.1 µmol O 2 cm -1 day -1, which is substantially higher than the above estimate of 1.8 µmol O 2 cm -1 day -1. If, as the authors suggested, oxygen is advected into the burrow at a rate of 1.75 µmol O 2 cm -1 day -1, then at this rate of burrow wall oxygen consumption the oxygen would be completely consumed within 13.5cm of the burrow opening (compared to 17 cm in the laboratory). It is conceivable that the shrimp matches its behaviour to the surrounding sediment conditions. Given the higher sediment oxygen consumption rates in the field sediments, it would not be surprising to find that the shrimp increases its burrow irrigation rate accordingly Case 3: modelling Callianassa japonica and Upogebia major laboratory measurements (Koike and Mukai (1983)) Koike and Mukai (1983) measured oxygen, nitrate plus nitrite and NH 4 + concentrations in burrows in laboratory aquaria occupied by Callianassa japonica and Upogebia major. These measurements were made with constant overlying water conditions (water temperature 2.5 C), and further oxygen measurements were taken during periods of simulated low tide, where the overlying water level was lowered to become level with the sediment surface. The authors inferred burrow irrigation rates using an oxygen mass balance. The authors also measured individual shrimp respiration rates. The radial nitrogen model developed here was used to simulate the two scenarios (high tide and low tide) described in the experiments. No measurements of burrow radius were provided by Koike and Mukai (1983), although typical callianassid burrow radii range from.2.5 cm. The authors stated that the natural population density is ~ 2 individuals m -2, which corresponds to r 2 = 12.6 cm. However, the laboratory aquaria were only 3 cm wide, so limiting the size of r 2 to 1.5 cm High tide During high tide conditions the shrimp can maintain a flow through their burrow, frequently flushing the burrow with overlying water. Koike and Mukai (1983) did not

145 investigate the irrigation behaviour, however they did infer irrigation rates from their measurements. Their oxygen measurements were only taken every 3 minutes, too low a sampling frequency to infer the irrigation patterns. The oxygen measurements, burrow volumes and irrigation rates were used to infer the irrigation parameters required in the model. The value of f in equation (5.11) was estimated by assuming that the maximum and minimum oxygen concentrations represent the burrow concentrations during irrigation and at the end of quiescent periods, respectively. The calculation for C. japonica is: C = fc + ( 1 f ) C o. 45C = fc +. 12C ( 1 f ) f =. 37 o q (5.27) and for U. major: C = fc + ( 1 f ) C o. 41C = fc +. 3C ( 1 f ) f =. 16 o q (5.28) The model also requires the length of irrigation period (T i) which can be estimated from f, the burrow volume (V b) and the irrigation rate (Q b): T i = fv Q b b (5.29) Making these assumptions irrigation periods of 2 minutes for C. japonica and 18 minutes for U. major were estimated. A further assumption was made that the shrimp only actively pumped water through their burrow for 5% of the irrigation period, based on irrigation behaviour measured in Forster and Graf (1995). The respiration rates for both species were measured at full oxygen saturation, yet it is likely that the respiration rate is a function of the ambient oxygen concentration. This effect was studied in detail by Paterson and Thorne (1995) for Trypaea australiensis. They found that respiration reduced with lower oxygen concentrations, and shut down completely below a critical oxygen concentration. Their findings were applied to the model here, assuming the same shape for the respiration vs oxygen concentration relationship. Koike and Mukai (1983) also commented that they observed a decrease in respiration rate at lower oxygen concentrations

146 Koike and Mukai (1983) conducted their experiment 4 to 5 days after burrow construction, and so the model was run for 5 days. The model was also run for 2 days, with negligible differences in the results. A series of model runs were conducted testing changes to burrow radius, irrigation behaviour (f) and porosity to assess the model parameters that would bring the model into reasonable agreement with measurements. Each run spanned a range of P DOC values (2 12 µmol L -1 d -1 ). The results given in Figure 5.19 used the following parameter values: P DOC = 7 µmol L -1 d -1 ; r 1 =.4 cm; φ =.4; f =.2 (C. japonica) and f =.16 (U. major). In both cases oxygen is well matched and NO 3 levels are overestimated (by up to a factor of 4.5). The extremes of the NH 4 + range are poorly predicted, although the modelled values fall within the measured range. Oxygen saturation (%) 5% 45% 4% 35% 3% 25% 2% 15% 1% 5% % µmol N /L Oxygen saturation (%) 5% 45% 4% 35% 3% 25% 2% 15% 1% 5% % µmol N /L Measured oxygen Modelled oxygen Measured nitrate Modelled nitrate Measured ammonium Modelled ammonium Measured oxygen Modelled oxygen Measured nitrate Modelled nitrate Measured ammonium Modelled ammonium Figure 5.19 Measured and modelled burrow O 2, NO 3 and NH 4 + concentration ranges for Callianassa japonica (left) and Upogebia major (right). Measured values are from Koike and Mukai (1983). The difficulty in finding a better match lies with the fact that measured NO 3 levels were low, despite the presence of significant levels of NH 4 + in an oxygenated environment. The difficulties faced in attempting to model this scenario are a direct contrast to those experienced when attempting to model the Callianassa truncata data, where measured NH 4 + levels were much lower than could be explained by the model. A possible model modification that may account for the very low NO 3 levels is a substantial increase in the rate constant for denitrification, k 4. Following Blackburn and Blackburn (1993), this value has been set at 3 day -1, however other papers by the same authors have used significantly different values. Blackburn (1996) used a value of k 6 = 5 day -1 and Blackburn et al. (1994) employed k 4 = 5 day -1. An extraordinary range of values (spanning two orders of magnitude) has been used with the justification of

147 creating a better fit to measurements. The higher rates imply a very dense and large biomass of denitrifying bacteria (Blackburn et al. (1994)). I tested the effect of a more extreme value. Increasing k 4 to a value of 1 day -1 brought the modelled NO 3 levels closer to the measured range (at most a factor of 2.6 too high, Figure 5.2). Oxygen saturation (%) 5% 45% 4% 35% 3% 25% 2% 15% 1% 5% % µmol N /L Oxygen saturation (%) 5% 45% 4% 35% 3% 25% 2% 15% 1% 5% % µmol N /L Measured oxygen Modelled oxygen Measured nitrate Modelled nitrate Measured ammonium Modelled ammonium Measured oxygen Modelled oxygen Measured nitrate Modelled nitrate Measured ammonium Modelled ammonium Figure 5.2 Measured and modelled burrow O 2, NO 3 and NH 4 + concentration ranges for Callianassa japonica (left) and Upogebia major (right). Same model parameters as Figure 5.19, except k 4 = 1 day -1. Measured values are from Koike and Mukai (1983). The model results were quite insensitive to the massive increase in k 4, with the only real effect being a reduction in NO 3 levels (to roughly half their previous values). The nitrification and denitrification rates for both scenarios are shown in Table 5.4. As expected, the denitrification rates are higher in the second scenario (where k 4 = 1 day -1 ), although the rates (and efficiencies) have increased by no more than 5% despite the large increase in k 4. Table 5.4 Model nitrification and denitrification behaviour under two k 4 values. Nitrification rate (nmol/cm/day) Denitrification rate (nmol/cm/day) Denitrification :nitrification Denitrification efficiency (%) C. japonica (k 4 = 3 day -1 ) % 29% C. japonica (k 4 = 1 day -1 ) % 43% U. major (k 4 = 3 day -1 ) % 39% U. major (k 4 = 1 day -1 ) % 54% These rates are expressed as rates per unit burrow length, and so the length of burrow beneath each m 2 of surface sediment is needed to convert these rates to an areal estimate. At first glance, this appears to be a simple matter of multiplying the rates by the average burrow length and the areal density of burrows. At this point, however, it

148 becomes clear that the experimental configuration may have created burrow conditions which are not representative of natural conditions. As mentioned earlier, the natural density of these shrimp in situ is approximately 2 individuals m -2, which corresponds to a value of r 2 = 12.6 cm. The laboratory tank was only 3 cm wide, so limiting r 2 to a smaller value. A value of r 2 = 1.5 cm was used in the model, which represents a lower bound on this distance. The distance determines the potential accumulation of NH + 4 in the sediment. For large values of r 2, relatively little of the NH 4 + being produced will diffuse to oxygenated burrow walls and so it will accumulate within the sediment. For small values of r 2, there is a lot of oxygenated burrow area beneath the sediment and so more of the NH 4 + will diffuse into these areas, where it can be transformed through nitrification. The eventual NH 4 + concentrations, and the amount of NH 4 + oxidised to NO 3, is most critically determined by the burrow spacing. The model was run again for 2 days, setting r 2 = 12 cm. As expected, this simple alteration produced a significant difference in the results (Figure 5.21). Most notably, the predicted NO 3 and NH + 4 levels were larger than measured values. This effect would be further increased if the model run time had been extended, as the system had still not reached steady state after 2 days. Oxygen saturation (%) 5% 45% 4% 35% 3% 25% 2% 15% 1% 5% % µmol N /L Oxygen saturation (%) 45% 4% 35% 3% 25% 2% 15% 1% 5% % µmol N /L Measured oxygen Modelled oxygen Measured nitrate Modelled nitrate Measured ammonium Modelled ammonium Measured oxygen Modelled oxygen Measured nitrate Modelled nitrate Measured ammonium Modelled ammonium Figure 5.21 Measured and modelled burrow O 2, NO 3 and NH 4 + concentration ranges for Callianassa japonica (left) and Upogebia major (right). The model has been modified to set r 2 = 12 cm. Measured values are from Koike and Mukai (1983). The predicted nitrification and denitrification rates are roughly double the rates predicted in the smaller-radius cases (Table 5.5). The proportion of NO 3 being

149 denitrified reduced by a small amount (from ~8% to ~6%), but the denitrification efficiency has dropped substantially (from ~45% to 1%). Table 5.5 Model nitrification and denitrification results when r 2 = 12 cm. Nitrification rate (nmol/cm/day) Denitrification rate (nmol/cm/day) Denitrification :nitrification Denitrification efficiency C. japonica (k 4 = 1 day -1 ) % 1% U. major (k 4 = 1 day -1 ) % 1% In the absence of information on the burrow locations and density within the tank, the choice of an appropriate r 2 value is difficult. Certainly the correct value will be somewhere between 1.5 cm and 12 cm. We can infer from the results of these two extremes that NO 3 levels will be generally over-estimated by the model. NH 4 + levels fall within the measured range, however the extent of the range was not predicted. The importance of r 2 makes interpretation of the laboratory experiments difficult. I found a model configuration that provided a reasonable approximation to the laboratory measurements. It may be valid simply to increase the value of r 2 to the field value and infer the new burrow concentrations and rates. This would assume, however, that the shrimp behaves in the same manner, regardless of the surrounding chemistry. A more likely scenario is that a shrimp will respond to its surroundings and will alter its irrigation pattern to create a burrow environment best suited to its needs. These considerations make it difficult to estimate the likely nitrification and denitrification rates Low tide During the low tide simulation, Koike and Mukai (1983) noted a rapid drop in Upogebia major burrow oxygen levels (54% to 16% saturation within one hour) followed by a relatively constant oxygen concentration for the remaining 4 hours (Figure 5.22). These observations proved difficult to reproduce in my model. In the absence of a respiring shrimp, the burrow walls would still have been consuming oxygen at a considerable rate, making it difficult to explain a constant oxygen concentration persisting (and even increasing slightly at one point) for a period of several hours. The authors suggested that oxygen continued to be supplied to the burrow during low tide: they observed the shrimp showing frequent irrigation motions while occupying the upper portions of the burrow. It is not clear that such activity would markedly increase the supply of oxygen

150 to the burrow water. There is no capacity for the shrimp to produce a continuous flow through its burrow because the water level is reduced to the sediment level, so oxygen would have to diffuse across the air-water interface at the burrow opening. There are two possible explanations for the relatively constant oxygen levels. Shrimp respiration may shut down completely at a relatively high value; in previous runs it was assumed that shrimp respiration ceases at 6 µmol L -1 O 2, based on the results published by Paterson and Thorne (1995). The second possibility is that oxygen was injected into the burrow during measurements. Each oxygen measurement required the withdrawal of approximately 12-16mL of water from the burrow which was circulated past an oxygen sensor and re-injected into a different section of the burrow. The volume of the external system was 3.5 ml, which represented approximately 8% of the burrow volume. It is possible that the water residing in the external system became oxygenated between measurements (taken every 3 minutes), and this partially oxygenated water would then have been injected into the burrow at the start of each measurement. Three model scenarios were run: Case 1: Model parameters the same as in previous section (with r 2 = 1.5 cm, k 4 = 1 day -1 ); Case 2: Same as Case1, with shrimp respiration altered to shut down at 35 µmol L -1. Case 3: Same as Case 2, with oxygen injection at each measurement (assumed that 8% of the burrow is replaced with oxygen at 6% saturation). The results, plotted in Figure 5.22, show that Case 3 gives the closest match to measured data. The Case 2 results demonstrate clearly that in the absence of shrimp respiration the oxygen levels are still expected to decline significantly as a result of oxygen consumption by the surrounding sediment. Only a continuing supply of oxygen can ensure the maintenance of a steady oxygen concentration over a period of hours. Case 3 demonstrates that if the external volume of 3.5 ml were oxygenated to 6% and injected into the burrow, this would be sufficient to keep oxygen measurements reasonably constant. Note, however, that Case 3 still requires respiration to be shut down at 35 µmol L -1 O

151 Oxygen concentration µmol/l time (h) Figure 5.22 Oxygen concentration in the burrows of Upogebia major under simulated low tide condition. Circular markers are data points taken from Koike and Mukai (1983). The lines are the three model scenarios (Case 1, dotted; Case 2, dashed; Case 3, solid) Summary of thalassinidean shrimp cases Despite the availability of good experimental and field data on thalassinidean burrow chemistry, and the existence of well-accepted models of sediment nitrogen dynamics, the preliminary attempt here to combine this knowledge created as many problems as it solved. This experience reinforced the oft-repeated advice that experimentalists and modellers should work together. Given that the experiments considered here were conducted in the absence of any prior knowledge about within-burrow concentrations, they provided an excellent starting point by providing a snapshot of concentration distributions in burrows. These snapshots, however, were insufficient to resolve modelling questions. More will be gleaned from future experiments if they are conducted with an explicit theoretical framework underlying the design, with the aim of integration with modelling work. Methodological considerations were highlighted, particularly in regard to laboratory measurements. For example, it may be insufficient to have the shrimp inhabiting their native sediment if some attempt is not made to recreate the sediment geometry and organic matter distribution if meaningful estimates of in situ burrow chemistry are sought. In particular, if an animal usually has a 1cm wide annulus of sediment surrounding its burrow, this should be recreated in the laboratory if the aim is to measure burrow chemistry

152 Attempts to model Callianassa japonica and Upogebia major measurements were more successful than attempts to model Callianassa truncata and Callianassa subterranea measurements. This was largely attributable to the fact that data on three of the chemical species of interest, oxygen, NO 3 and NH 4 +, were available, and the authors had measured individual shrimp respiration rates. This work raised several questions in need of answers, including: 1. What are the solute transport processes within the burrow and within the surrounding sediment? Diffusive transport assumptions were not able to explain the sediment oxygen data by Forster and Graf (1995). Advective flows through the sediment may be important in the system studied by Ziebis et al. (1996). 2. What is the sediment respiration rate, and how is it spatially distributed within the sediment? Experiments could include measuring respiration rates from cores of differing length, and repeating the experiments with uniformly-mixed sediment cores of differing length. Work along similar lines by Aller and Aller (1998) yielded very surprising results. In particular, these authors found that quantities such as NH + 4 production rates were highly sensitive to the distance to the nearest no-flux boundary. 3. How does population density (burrow separation distance) affect concentrations and flushing rates? Is there any evidence of altered irrigation behaviour in response to altered burrow separation? Experiments could measure burrow concentrations and flushing rates with a range of distances to no-flux boundary conditions. Such experiments would help determine whether animal behaviour in narrow tanks is likely to differ from field conditions. 5.6 Port Phillip Bay Extensive measurements were made in the Port Phillip Bay Environmental Study (PPBES), with the aim of calculating nutrient budgets and inferring nutrient pathways in the system. The final analysis suggested that 7-9% of the loss of nitrogen from the Bay is through the denitrification pathway in the Bay sediments. These rates of denitrification are vital for preventing eutrophication in the Bay

153 The single most important discovery of the PPBES is that denitrification in the sediments prevents soluble and available N (ammonia and nitrate) from accumulating in the water column because it is removed from the main Bay environment as nitrogen gas (N 2). (Harris et al. (1996), p.91) Much has been made of the potential role of burrowing invertebrates in these processes. For example: A key feature disclosed by PPBES is bioirrigation of the sediments with oxygen rich water through the activities of benthic invertebrates One of the important results of bioirrigation is to carry dissolved oxygen to sites deep in the sediments (at least to 5 cm) thus facilitating aerobic degradation of organic material carried into the sediments by burrowing and grazing animals. (Harris et al. (1996), p.91) and One can think of the burrow system as a great expansion of the sediment-water interface, analagous to a lung. In fact, there is evidence from other sediment studies that the large burrows, such as those of the callianassids in Port Phillip Bay, are just the trunks of an extensive system of finer burrows, so that the lung analogy may be a good one. While some 1-D models have tried to represent the effect of bioirrigation as a non-local transport process, it is not clear that this is an appropriate solution. (Harris et al. (1996), p.174) Sediment profiles of excess 21 Pb and 228 Ra deficiency suggested animal activities to a depth of 5 cm (Hancock and Hunter (1999)). Tracer losses from benthic chamber studies in the bay have also indicated that sediment bioirrigation could be occurring to depths of up to 5 cm (Berelson et al. (1999)), and the authors wondered at the cause: The biological aspects of the various sites should be integrated with the physical data reported here. What are the benthic organisms which produce such high pumping rates? (Berelson et al. (1996))

154 A major outcome of the study was the development of a nutrient cycling model which integrated physical and ecological processes (Murray and Parslow (1997)). Sediment denitrification was incorporated into the model using an empirical relationship which suggested high denitrification efficiencies were possible at low sediment respiration rates, with efficiencies dropping at high sediment respiration rates. However the authors stressed the importance of sediment biogeochemistry, and suggested further investigation into the underlying processes: There is a great deal of scope for further development of process models of sediment biogeochemistry. More use can be made of the existing sediment profile and flux data (Murray and Parslow (1997), p.186) Although some process modelling of nitrification and denitrification was conducted as part of the PPBES, it was limited to a vertical one-dimensional non-local model, and the authors suggested the need to look further at the effects of three-dimensional structure in the sediment. Unfortunately, very little is actually known about this threedimensional structure in PPB. Certainly, there are thalassinidean shrimp in the bay (Neocallichirus limosus, Biffarius arenosus and Callianassa ceramica) and their burrow structures are well described (Bird et al. (1997)), however there is considerable uncertainty in the density of their population. Early surveys counted Neocallichirus limosus as the second most numerous species recorded, yet the more recent surveys have found a substantially reduced population density (Wilson et al. (1998)). It should be noted, however, that surveys were conducted by remote grab samples, which captured only the upper 1cm of sediment. If shrimp were lurking at more distant depths they would not have been recorded, so we can assume that reported population densities were an under-estimate. Consequently, there is insufficient information to determine the nature of the burrow structures across the bay, particularly at depth. The PPBES field program provided details of porosity, concentration profiles and solute fluxes. Unknown details are the burrow geometry, irrigation behaviour and the spatial distribution of organic matter mineralisation within the sediment. The aim in this work was to use models to determine the range of burrow radii, inter-burrow distances and flushing rates that could explain the measured porewater concentrations and nutrient fluxes. The denitrification efficiency was of particular interest. The onedimensional process modelling conducted by Murray and Parslow (1997) was unable to explain the very high efficiencies (>8%) recorded in the bay

155 The previous case studies considered in this chapter involved contrived circumstances in that in all cases the sediment was mixed vertically prior to shrimp colonisation. For this reason I felt justified in assuming vertically uniform P DOC distributions, an essential assumption if the 1-D radial model is to be valid. I employed this same 1-D approach when modelling PPB sediments, however its validity is questionable given the likely vertical distribution of organic matter within these sediments. In particular, highly labile organic matter will not be uniformly mixed to great depths by the shrimp, as evidenced by the distribution of the short half-life 228 Th data from PPB in Chapters 3 and 4. A further difficulty in using the one-dimensional radial approach is that this model provides only the burrow fluxes, yet total sediment fluxes comprise both sediment surface and burrow components. These concerns prompted me to conduct further modelling exercises using a modified version of the model, implemented with full cylinder geometry One-dimensional radial model If a large lung-like structure in the sediment is responsible for the observed denitrification behaviour in PPB, how deep and how dense must these burrows be to achieve this effect? A first attempt to answer this question was made using the onedimensional radial model. Embedded in the implementation of this model were three main assumptions: 1) Uniform organic matter distribution in the sediment; 2) Nitrification and denitrification zones surrounding burrows are the main sites for these reactions (ie. Nitrification and denitrification occurring just below the sediment surface forms a negligible contribution to the total rates); 3) Steady state conditions with constant partial flushing in permanent burrows. The conceptual model of this case is a volume of well-mixed sediment irrigated to such a depth that irrigated burrow surface area far exceeds planar sediment-water surface area. The chief drawback with this model is that, although sediment mixing is substantial in the bay, labile organic matter mineralises at such a rate that it is likely to remain concentrated at the very surface of the sediment only. Nevertheless, this approach provides a tangible starting point

156 A series of model runs were run spanning a range of DOC production rates, burrow flushing rates, porosity and burrow geometry (see parameter values in Table 5.6). Table 5.6 Parameter values used in model runs (144 runs total). Parameter Values modelled φ.9,.65 r 1 r 2.15,.2,.3,.4,.5 cm 1,2, 3, 4, 5, 6 cm P DOC 2, 5, 1, 2, 5, 7, 9, 12 nmol/cm 3 /d F l 48, 24, 24/5 d -1 Given that each run specifies a burrow geometry, porosity and DOC production rate, we can find a burrow length, L (cm), that is required to produce a given areal respiration rate, R c (nmol C /cm 2 /d). The volume of sediment surrounding a burrow is 2 2 π 2 1, and the surface area implicit in the areal respiration rate is A V = r r L = πr 2 2. At steady state the DOC production rate must balance the measured areal respiration rate, so φp DOC V = R c A. Substituting in the expressions for V and A and re-arranging we get L = Rcr2 P r 2 2 φ DOC 2 r1 2 (5.3) By prescribing the porosity and the areal respiration rate, the distribution of denitrification efficiencies can be plotted as a function of burrow length, radius and separation distance. In other words, the theoretical relationship between denitrification efficiency and the depth and density of burrows can be explored. It is assumed that all organic matter decomposition takes place within the burrowed layer of the sediment. This assumption is reasonable, as there is no mechanism for significant amounts of organic matter to be transported below the burrowed zone, as burial rates have been shown to be negligible in PPB. The most detailed denitrification studies were conducted at Site 37 and Site 16 in Port Phillip Bay. These sites were considered typical of two extremes in the Bay: Site 37 had a low areal respiration rate (~2 mmol/m 2 /day CO 2) and high denitrification efficiency (>7%), and Site 16 a high respiration rate (>5 mmol/m 2 /day CO 2) and lower denitrification efficiency (<5%) (Figure 5.23). These results were found by direct measurement of nitrogen gas accumulation in benthic chambers deployed at the two sites (Heggie et al. (1999))

157 Figure 5.23 Percentage denitrification, measured ( N2 N2 + DIN) v. CO 2 flux at Site 16 and Site 37, January (Taken from Heggie et al. (1999)). The measured field values for porosity and areal respiration rate were used to produce maps of denitrification efficiency as a function of burrow spacing (r 2) and burrow length (calculated from equation (5.3)) from the collection of model runs. This procedure produced 15 maps for each site, as there was a map for each combination of burrow radius (5 values) and flushing rate (3 values). Interpolation between model data points was calculated with Matlab s interp2 function, using a bicubic interpolation method. Four of the maps for each site are shown in Figure 5.24 and Figure 5.25, and these represent the four extreme cases: (a) minimum r 1 value and maximum F l value; (b) minimum r 1 value and minimum F l value; (c) maximum r 1 value and maximum F l value; (d) maximum r 1 value and minimum F l value

158 Figure 5.24 Denitrification efficiencies interpolated from model runs (efficiencies represented as fractions rather than percentages). Data points from model runs represented by black circles (some points not shown, as L > 1cm). Sediment porosity and areal respiration rate have been set at values typical for Site 37 in Port Phillip Bay (φ =.9, R c = 2 mmol/m2/day), burrow length has been calculated from equation (5.3). The purple line marks the 7% denitrification efficiency contour (efficiencies typically exceed this value at Site 37). (a) r 1 =.15cm, F l = 48 day -1 ; (b) r 1 =.15cm, F l = 24/5 day -1 ; (c) r 1 =.5cm, F l = 48 day -1 ; (d) r 1 =.5cm, F l = 24/5 day

159 Figure 5.25 Denitrification efficiencies interpolated from model runs (efficiencies represented as fractions rather than percentages). Data points from model runs represented by black circles (some points not shown, as L > 1cm). Sediment porosity and areal respiration rate have been set at values typical for Site 16 in Port Phillip Bay (φ =.65, R c = 5 mmol/m2/day), burrow length has been calculated from equation (5.3). Purple lines mark the 5% denitrification efficiency contour, which is a typical value measured at this site. (a) r 1 =.15cm, F l = 48 day -1 ; (b) r 1 =.15cm, F l = 24/5 day -1 ; (c) r 1 =.5cm, F l = 48 day -1 ; (d) r 1 =.5cm, F l = 24/5 day -1. There are three factors to consider in interpreting these diagrams: burrow geometry, flushing rate and areal respiration rate. Generally speaking, the more densely populated an area (ie the lower the r 2 value), the higher the denitrification efficiency. This generalisation breaks down as soon as a critical r 2 (referred to as r peak in the following discussion) is reached, and denitrification efficiency drops off rapidly below this point. The explanation is that at very low r 2 values there are insufficient anoxic regions separating burrows to yield significant denitrification rates, so increasing r 2 increases denitrification until r 2 = r peak, beyond which the volume of anoxic sediment producing NH + 4 begins to overwhelm the volume of oxic sediment able to nitrify this NH + 4. The value of r peak is dependent on burrow radius, flushing rate and burrow length (or respiration rate per unit volume of sediment). Lowering r 1 has the effect of

160 lowering r peak, but the extent to which r 1 changes r peak is highly dependent on flushing rate. For example, the difference between Figure 5.24(a) and Figure 5.24(c) is much smaller than the difference between Figure 5.24(b) and Figure 5.24(c). Here there are two effects: decreasing the flushing rate lowers the value of r peak, but it can slightly increase the peak denitrification rate (compare Figure 5.24(c) and Figure 5.24(d)). Reducing the flushing rate reduces the supply of oxygen to the sediment, and so reduces volume of nitrifying sediment and hence r peak. However, the reduced flushing rate also effectively traps NO 3 into the sediment as it is not being flushed out of the burrow as rapidly, and so provides more opportunity for the NO 3 to be converted to nitrogen gas. In terms of specific predictions for PPB, the model suggests that at Site 37, denitrifying efficiencies exceeding 7% could occur from a range of burrow density, burrow length and flushing rate combinations. For example, burrows as shallow as 1cm can allow sufficiently high denitrification rates, if they are separated by approximately 4cm (r 2 = 2cm). If burrows are as deep as 5cm (as widely speculated in the final report), burrow separation distances of 4cm to 11cm can yield sufficiently high denitrification rates. Are these numbers physically realistic? Firstly, it should be noted that these specific burrow lengths and separation distances are based on a burrow radius of.5cm, which is typical of thalassinidean burrows. The population density of shrimp measured in the Bay in 1995 were 21 m -2 for Biffarius arenosus (sandy sediments) and 1 m -2 for Neocallichirus limosus (muddy sediments) (Bird et al. (1997)). The relationship between burrow density (N m -2 2 ) and burrow separation distance (2r 2) is given by Nπr 2 = 1, so r 2 = 2cm and r 2 = 5.5cm correspond to burrow densities of approximately 8 burrows/m 2 and 1 burrows/m 2. Even allowing for the fact that there are multiple burrow shafts per individual (and that Bay populations are likely to be underestimated by grab sampling techniques), clearly the measured population density falls well short of the density required by the model to produce the very high denitrification rates. It is certainly unrealistic to contemplate sediment filled with 1cm diameter burrows at a density of 8 per m 2. It is more possible to consider sediment containing 1 burrows/m 2, as these densities are not uncommon for thalassinidean shrimp. It is not clear, however, that such a burrow density would be maintained to a depth of 5cm (it is more typical that animals maintain multiple openings to the sediment surface, and a sparser burrow network at depth)

161 Turning to the model runs based on Site 16 (Figure 5.25), we see that there is a far narrower range of burrow geometry and flushing combinations that can yield high denitrification efficiencies. Thus, qualitatively, it matches the field observations that a higher areal respiration rate is correlated with lower denitrification efficiencies. When the respiration rate is reduced even further to 12 mmol/m 2 /day (measured in March 1995 at Site 16, see Figure 5.23), there are even fewer regions of high denitrification efficiency (Figure 5.26). Figure 5.26 Denitrification efficiencies interpolated from model runs (efficiencies represented as fractions rather than percentages). Data points from model runs represented by black circles (some points not shown, as L > 1cm). Sediment porosity and areal respiration rate have been set at values typical for Site 16 in Port Phillip Bay (φ =.65, R c = 12 mmol/m2/day), burrow length has been calculated from equation (5.3). Purple lines mark the 1% denitrification efficiency contour the measured field denitrification efficiency was under 1%. (a) r 1 =.15cm, F l = 48 day -1 ; (b) r 1 =.15cm, F l = 24/5 day -1 ; (c) r 1 =.5cm, F l = 48 day -1 ; (d) r 1 =.5cm, F l = 24/5 day -1. The seasonal changes in denitrification efficiency at Site 16 provided some of the most compelling evidence for the empirical function reported in the PPBES relating denitrification efficiency to sediment respiration rate. Site 16 experienced dramatic changes in sediment respiration rate and denitrification efficiency, and the two were

162 seen to be inversely related. Just over a doubling in sediment respiration rate (from 5 mmol/m 2 /day to 12 mmol/m 2 /day) corresponded to a reduction in denitrification efficiency from around 5% to below 5% (Figure 5.23). Note that while the model produces the same qualitative picture (reduced denitrification efficiency with increased respiration rate, as seen by comparing Figure 5.25 and Figure 5.26), the effect is less pronounced than the field measurements. For example, geometry configurations that yielded ~5% efficiency in Figure 5.25 typically produce efficiencies of 25%-3% in Figure 5.26 a reduction by a factor of two, as compared to a factor of ten measured in the field. However this analysis is not particularly helpful, as such a dramatic change in sediment respiration rate is likely to have an impact on the animals living in the sediment. Indeed, it is likely that population density would fall (r 2 increase) in response to increasing levels of anoxia in the sediment. Under these circumstances, further drops in denitrification efficiency can be explained by the model results. Due to the extensive nature of the PPB investigations, there are other checks we can make against the model predictions. In particular, the field studies measured pore water concentrations in sediment cores and benthic chamber fluxes at each site (Nicholson et al. (1996)). Figure 5.27 shows predicted NH + 4 and NO 3 concentrations and fluxes for r 1 =.3cm and F l = 24 day -1 (middle of the range of r 1 and F l values modelled) with porosity and respiration rate set to typical Site 37 values (φ =.9, R = 2 mmol m -2 day -1 ). The 7% denitrification efficiency contour has been superimposed on these maps, to show the region of r 2 and L values which match the field denitrification rates. These maps were used to produce the data shown in Table 5.7, where field and modelled concentrations and fluxes are compared

163 Figure 5.27 Plots of denitrification efficiency, average NH 4 +, average NO 3, NH 4 + flux, NO 3 flux and O 2 flux against r 2 and burrow length (calculated from equation (5.3)). Respiration rate and porosity were fixed at typical values for Site 37 (φ =.9, R = 2 mmol/m 2 /day), r 1 =.3 cm and F l = 24 day -1. The purple line shows the 7% denitrification efficiency contour (efficiencies typically exceed this value at Site 37). Units are µmol/l for concentration and mmol m -2 day 1 for fluxes. Denitrification efficiencies are represented as a fraction rather than a percentage

164 Table 5.7 Comparison between measured and modelled concentrations and fluxes for Site 37 in PPB. Numbers are approximate only, having been read by eye from Figure 5.27 (above) and bar graphs from Figure 31 in Nicholson et al. (1996). Modelled range within 7% denitrification efficiency contour Measured range Average NH 4 + <1 to 5 µmol/l 1 to 3 µmol/l Average NO x 2 to 6 µmol/l 2 to 5 µmol/l NH 4 + flux.1 mmol m -2 day 1 to.3 mmol m -2 day 1 NO x flux.5 to 1 mmol m -2 day to.3 mmol m -2 day 1 O 2 flux -2 to -8 mmol m -2 day 1-28 to -15 mmol m -2 day 1 In general, the comparison between field and model concentrations and fluxes are good (Table 5.7), although the model over-predicts the NO x flux (at least.5 mmol m -2 day 1 modelled compared to at most.3 mmol m -2 day 1 measured). Tracer studies were also used to estimate in situ sediment irrigation rates in PPB (Berelson et al. (1999); Hancock et al. (1997)). Irrigation rates can be calculated for each model run, providing another point of comparison with field data. For each model run, we have the burrow geometry (radius r 1 and length L calculated from equation (5.3) for a specified respiration rate) and a burrow flushing rate constant, F l. The volumetric flushing rate per unit area (L/m 2 /day) is: 2 Q = πr 1 LF l N (5.31) where N is the number of burrows per unit area. Again, choosing r 1 =.3 cm and F l = 24 day -1, the volumetric flushing rate per unit area have been plotted against r 2 and L (Figure 5.28). Superimposed on this graph are the 7% and 5% denitrification contours for Site 37 and Site 5 respiration rates respectively. Based on this graph, flushing rates of between 6 and 3 L/m 2 /day are predicted for Site 37, and over 1 L/m 2 /day for Site 16. These rates can be directly compared to irrigation rates inferred in Berelson et al. (1999) and Hancock et al. (1997). Berelson et al. (1999) measured the loss of deuterium from benthic chambers, and used a diffusion and non-local exchange model to infer the rate at which chamber water is pumped into the sediment. Their rates were in units of ml/h, which I have converted to L/m 2 /day, using the chamber area of 73cm 2. They deduced that flushing rates were 52 to 64 L/m 2 /day to a depth of 2-25cm at Site 37, which sits across the lower boundary of the 6 to 18 L/m 2 /day range inferred from Figure 5.28 (found by

165 .5 intersecting the region of >7% denitrification efficiency with the burrow length region of 2-25 cm). Berelson et al. (1999) deduced a much higher flushing rate to a much greater depth at Site 16, although the result was inconclusive as there was the possibility of chamber leakage at this site (based on a comparison between Cs and deuterium losses from the chamber). Hancock et al. (1997) analysed radium balances to derive a flushing rate of 59 L/m 2 /day, which is the same as the rate calculated by Berelson et al. (1999). length of burrow (cm) r (cm) 2 Figure 5.28 The black contours show flushing rates (L/m 2 /day) for burrows with r 1 =.3cm and F l = 24 day -1. The red contour is the 7% denitrification contour for a typical Site 37 respiration rate and porosity, and the blue contour is the 5% denitrification contour for a typical Site 16 respiration rate and porosity (calculated from model runs using r 1 =.3cm and F l = 24 day -1 ; eg. the red contour comes from Figure 5.27 (a)) Berelson et al. (1999) stated that their estimated flushing rates were in agreement with measurements of Callianassa truncata flushing rates made by Forster and Graf (1995), yet I find it hard to draw the same conclusion. Forster and Graf (1995) measured areal flushing rates of 5.3 L/m 2 /day (2.8 L/m 2 /day regular irrigation L/m 2 /day irregular irrigation) for Callianassa truncate, which are well below the flushing rates discussed in the previous paragraph. Berelson et al. (1999) chose to convert all literature flushing rates to a flushing rate across their benthic chamber area (73 cm 2 ), so a rate of 5.3 L/m 2 /day converts to 16.2 ml/h (across 73cm 2 ). Yet Berelson et al. (1999) cited a converted rate of 48 ml/h for Callianassa truncata, which is roughly a factor of 3 too high. I can only assume Berelson et al. (1999) made a mistake in their calculation. While Berelson et al. (1999) concluded that their irrigation rates compared well with measurements of thalassindean pumping rates, I can only conclude that the

166 .5 rates are extremely high, and are certainly not physically realistic given our understanding of typical thalassinidean population densities and flushing rates. Forster and Graf (1995) measured a population density of 21 ind/m 2, and estimated that 1.6 m 2 of burrow surface lie beneath each m 2 of surface sediment. Calculations of burrow surface area for the range of model runs are shown in Figure The estimate from Forster and Graf (1995) falls at the low end of model burrow surface area. length of burrow (cm) r (cm) 2 Figure 5.29 The black contours show burrow surface area (m 2 /m 2 ) for burrows with r 1 =.3cm. The red contour is the 7% denitrification contour for a typical Site 37 respiration rate and porosity, and the blue contour is the 5% denitrification contour for a typical Site 16 respiration rate and porosity (calculated from model runs using r 1 =.3cm and F l = 24 day -1 ; eg. the red contour comes from Figure 5.27 (a)) The conclusions from these results are unclear. It seems that the one-dimensional radial burrow model can explain the measured denitrification rates, porewater concentrations, porewater fluxes and to some extent flushing rates. However, the implied burrow density and flushing rates do not appear physically realistic. Certainly, they are not realistic for thalassinidean shrimp populations: measured flushing rates and population densities are far lower than those predicted by the model results. While PPB shrimp densities are not high enough, the total benthic fauna density exceeds 1 ind/m 2 (Wilson et al. (1998)), and so it is conceivable that other animals may be important. The problem with this conclusion is that only the thalassinidean shrimp build permanent, irrigated burrows to the required depths of greater than 2 cm. Another possibility is that smaller animals build burrows extending out from the main burrows of thalassinidean shrimp. Thus a higher density of burrows could be maintained to considerable depths. Very little is known about the burrow structures in

167 Port Phillip Bay sediments; useful future work would be to investigate the sediment burrow networks more thoroughly, perhaps by resin-casting species cylinder model There are many drawbacks of the one-dimensional burrow model used so far. Two issues of particular importance are: 1. It calculates only the burrow contribution to total fluxes, where in reality the total flux between the sediment and overlying water is the sum of the flux across burrow surfaces and the planar sediment-water interface. 2. The burrow flux estimates will be inaccurate, particularly if shallow burrows are being modelled, as they fail to account for interactions with surface reactions which will alter concentrations at depth. For these reasons, it is desirable to take a look at a full cylinder model in order to more completely estimate the relative roles of surface and burrow processes. I ve implemented such a model to make a comparison with the 1-D radial model results. The nitrogen model outlined in equations (5.3) to (5.1) was modified in three ways: an extra dimension (vertical distance z) was included in the diffusion equation, a zero-flux boundary condition was prescribed at the maximum z value, and SO 4 2- and HS concentrations were excluded (reducing the number of chemical species from 7 to 5). Model parameters and solution method remained the same. A 5-species version of the one dimensional model was also constructed, and comparisons with the full 7-species version were very good (eg. Figure 5.3)

168 O NO 3 2 DOC N NH r (cm) Figure 5.3 Comparison between 5-species and 7-species 1-D radial models. Solid line: 7-species model; Dotted line: 5-species model Comparisons between 1-D and 2-D models As a first step, the results were compared between 1-D and 2-D models, where in both cases a uniform P DOC = 15 µmol/l/day had been prescribed. The length of the burrow was set to 1cm for the 2-D model runs. Four parameter combinations were tested: 1. r 2 = 3 cm, F l = 24 day -1 ; 2. r 2 = 3 cm, F l = 24/5 day -1 ; 3. r 2 = 5 cm, F l = 24 day -1 ; 4. r 2 = 5 cm, F l = 24/5 day -1. Using a porosity typical of site 37 at PPB (φ =.9), and assuming P DOC = below z = 1cm, the areal respiration rate is 13.5 mmol/m 2 /day

169 At the base of the 2-D model (i.e. at a depth of 1cm), the radial concentration distribution was found to be essentially the same as that produced by the 1-D model (Figure 5.31), which suggests that below this depth there is negligible surface influence, and so the 1-D and 2-D models would produce the same results in the L>1cm region. O NO 3 2 DOC N NH r (cm) Figure 5.31 Comparison between 1-D and 2-D 5-species radial models. Solid line: 1-D model; Dotted line: 2-D model at depth of 1cm. Units are %saturation for O 2 and µmol/l for all other species

170 Within this top 1cm, however, the planar sediment surface can make significant contribution to the sediment-water exchange rates. The cylinder model flux formulae are (modified from Aller (198) to represent flux per unit of planar sediment surface): J = J + J J J z z where r_ 2D φds 2π = 2π r_ 2D A = A r z r 2 r1 r 2 φd C z r d r r dr L s L C d z r d z (5.32) where A z is the surface area of the cylinder top at z = and A r is the surface area of the burrow wall. Alternatively, because the system is at steady state, the total flux for a species can be found by integrating that species reaction term: 2 L r2 φrr dr d z r1 J = 2 r 2 (5.33) The spatial resolution of the 2-D model was coarse (1mm), with the exception of a 3mm band at the burrow wall, which had a grid size of.1 mm in the r-direction (and 1mm in the vertical). For this reason, fluxes calculated from the gradient at the burrow wall will be more reliable than fluxes calculated from the gradient at the sediment surface. Hence the expression for J r_2d in equation (5.32) was used to calculate the burrow component of the flux, and equation (5.33) was used to calculate the total flux, J. The difference between the two is the surface component of the flux, J x. The planar sediment surface has an influence on flux calculations in two ways: directly, as material diffuses across this surface (J z); and indirectly, as the surface reactions affect the concentrations at depth, and so influence the value of J r_2d. The significance of both these effects can be assessed by comparing flux estimates from equations (5.32) and (5.33) with fluxes calculated from equation (5.15). The ratio, J z/j for the four cases shows that the surface component of the total flux is substantial, particularly for a low population density of poorly flushed burrows (Table

171 5.8). This presents a serious problem for the 1-D model, and this limitation needs to be borne in mind when interpreting its results. Table 5.8 J z/j calculations from 2-D model runs, demonstrating the relative importance of the surface contribution to the total flux. O 2 NO 3 DOC N 2 NH 4 + r 2 = 3 cm, F l = 24 day r 2 = 3 cm, F l = 24/5 day r 2 = 5 cm, F l = 24 day r 2 = 5 cm, F l = 24/5 day Knowing that the model can only hope to represent the burrow component of the fluxes between the sediment and overlying water, how well does the 1-D model predict these fluxes? Results in Table 5.9 show that the surface has a significant effect on the burrow flux values. Most notable is the difference in NH 4 + fluxes predicted by the two models. The presence of the sediment surface provides more sites for nitrification reactions, hence lowering NH 4 + concentrations within the sediment and depth. Consequently, the NH 4 + gradient between the burrow wall and surrounding sediments is less, and so the burrow flux is less. These interactions are not picked up by the 1-D model, so it seriously over-estimates the burrow component of the NH 4 + flux. Table 5.9 J r/j r_2d calculations. J r was calculated from the 1-D model data using equation (5.14), and J r_2d was calculated from the 2-D model data using the expression in (5.32). O 2 NO 3 DOC N 2 NH 4 + r 2 = 3 cm, F l = 24 day r 2 = 3 cm, F l = 24/5 day r 2 = 5 cm, F l = 24 day r 2 = 5 cm, F l = 24/5 day The combined effects of the direct and indirect surface influences indicate problems for the 1-D model. J r/j calculations (Table 5.1) show that the 1-D model can produce some flux estimates that are 1% of the total flux (eg. NO 3 ), and other flux estimates which are more than treble the total flux (eg. NH + 4 ). These effects are most pronounced for lower flushing rates and burrow densities

172 Table 5.1 J r/j calculations. J r was calculated from the 1-D model data using equation (5.14), and J was calculated from the 2-D model data using the expression in (5.32). O 2 NO 3 DOC N 2 NH 4 + r 2 = 3 cm, F l = 24 day r 2 = 3 cm, F l = 24/5 day r 2 = 5 cm, F l = 24 day r 2 = 5 cm, F l = 24/5 day So is it all bad news? It must be remembered that away from the surface (certainly 1cm away), the 1-D and 2-D models are identical. This means that the 1-D model will accurately predict the 2-D concentrations at depth, and it will also generate the correct burrow fluxes at these depths. Difficulties only arise when the bulk of the burrow lies within the sediment layer which is influenced by the sediment surface. Given the level of speculation within the PPB reports that very deep (~5 cm) burrows were responsible for high denitrification rates, it seemed reasonable to neglect the sediment surface and analyse the extent to which burrows could be contributing to these processes. The 1-D burrow model was an ideal tool for this purpose, as many runs could be executed to cover a broad range of parameter combinations. Such an extensive exploration would not have been possible with the slower cylinder model. The extent to which the surface is important, in terms of predicted nitrification and denitrification efficiencies, strongly depends on the burrow geometry and flushing rate. The 1-D model performs best under high burrow density and flushing rate conditions, worsening substantially as both of these parameters decrease in value (Table 5.11). While the extent of the difference varies, the direction is always the same: the 1-D radial model always underestimates nitrification and denitrification efficiencies. Given that I was using the 1-D model to see if particularly high values were possible, the model results represented a cautious analysis: any predicted denitrification rate represents a minimum estimate for that particular model geometry and flushing rate

173 Table 5.11 Nitrification and denitrification efficiencies produced from 1-D and 2-D models. In all cases the P DOC is uniform, with a value of 15 µmol/l/day. r 2 = 3 cm f l = 24 day -1 r 2 = 3 cm f l = 24/5 day -1 r 2 = 5 cm f l = 24 day -1 r 2 = 5 cm f l = 24/5 day -1 1-D nitrification efficiency 2-D nitrification efficiency 1-D denitrification efficiency 2-D denitrification efficiency 88% 94% 6% 65% 3% 74% 24% 51% 42% 82% 23% 5% 11% 73% 8.5% 44% Summary The emphasis in the PPBES reports was on the physical changes brought about by benthic fauna: redistribution of organic matter and the irrigation of sediments (use of the lung analogy). Chemical changes such as burrow lining, animal excretion and microbial stimulation were also mentioned. This prompts the question of the relative roles of physics and chemistry. Here a modelling approach has been used to assess whether physical changes alone, in the form of altered geometry of the sediment-water interface due to burrows, could explain the observations. A one-dimensional analysis demonstrated that a high areal density of deep well-flushed burrows could explain the observed porewater concentrations and fluxes in PPB, however the required burrow density and flushing rates were too high to be physically realistic. A two-dimensional analysis illustrated the drawbacks of the one-dimensional approach, showing that the inclusion of the sediment surface makes a substantial difference to model predictions. Nitrification and denitrification rates were always under-estimated in the 1-D model, although by differing amounts. Thus the usefulness of the 1-D model is limited to the fact that thousands of runs can be run with relative ease, which is not possible for the present version of the 2-D model. A practical compromise is to use the 1-D model to conduct a thorough search of the parameter space, and then employ the 2-D model to concentrate on specific cases of interest. Away from the surface influence, the 1-D and 2-D models are identical, and so the 1-D model is a good tool for calculating concentrations and burrow fluxes at depth. A question that deserves further attention is the role of heterogeneous organic matter distribution in the sediment. In these model runs a uniform P DOC distribution has been

174 assumed, which is highly unlikely in real sediments. It is more realistic to expect high organic matter content in a thin surface layer, and perhaps lining the burrow wall (which I partially addressed in previously sections). A related question is: what is the source of NH + 4 at depth? Is it produced at depth from organic matter or does it diffuse down from a thin layer of labile organic matter at the surface? These questions could be tackled relatively easily using this model. 5.7 Conclusions In this chapter I integrated information from a range of sources to model experimental and field observations from three thalassinidean shrimp studies. The work, while identifying divergences between observations and model results, also pointed to some important methodological considerations when conducting experiments on these creatures. In particular, it underlines the need for transport assumptions and conclusions to be made more explicit in experimental studies. In addition, a greater emphasis on budgets and flows between pools of nutrients would be beneficial. At a larger scale, the emphasis in the PPBES on quantifying not just the pools, but also the exchanges between pools in the system proved to be invaluable in interpreting Baywide processes. Burrow ventilation has been invoked as an explanation for high denitrification efficiencies in PPB. The modelling in this chapter represents an attempt to yield the expected changes from simple assumptions about burrow irrigation. While burrows certainly make a large difference to nitrification and dentrification rates, it was found in PPB at least they are unlikely to be the sole contributor. This would suggest that other hypothesised factors, such as the high concentrations of microphytobenthos are likely to be significant contributors to these rates. Another possibility not explored is that of anoxic nitrate production during manganese reduction, which would feed higher denitrification rates (Hulth et al. (1999)). The work in this chapter is by no means comprehensive, and it has opened up many avenues for future research. In particular, more work should be done on the role of heterogeneous organic matter distribution. I conducted some model runs in Sections and 5.5.2, where heterogeneous mineralisation rates were applied to the burrow irrigation model. These limited examples suggested that organic matter heterogeneity coupled with heterogeneous oxygen injection to the sediment can yield very different outcomes, especially in terms of total nitrification and denitrification rates. More

175 investigations along these lines are required. In particular, heterogeneous organic matter distributions should be incorporated into the full cylinder model. Recent work by Furukawa et al. (21) has done this and found that the cylinder (or more complicated microenvironment) model is an excellent tool for exploring standard assumptions about organic matter and C/N ratio distributions in sediments. Throughout this chapter I ve limited my approach to steady state model solutions, yet dynamic considerations are likely to be important. In particular, the start-up effects associated with new burrow formation should be investigated

176 6 Conclusions and future work In this thesis my emphasis has been to integrate measurements and theoretical understanding into a quantitative modelling framework. The aim has been to better understand the interaction between burrowing animals and the sediments they inhabit, and the macroscale biogeochemical consequences of this interaction. A further aim was to explore the ability of current theoretical understanding (as captured in models) to predict the outcome of these animal-sediment interactions. The bioturbation and bioirrigation models used in this work serve useful diagnostic functions in at least four ways: 1. they provide a method for testing whether current theoretical understanding matches experimental observations (for example, the nitrogen modelling in Chapter 5 highlighted significant differences in theoretical understanding and measurements in three thalassinidean shrimp experimental studies); 2. they allow the exploration of system sensitivity to underlying processes (for example, the sensitivity analyses conducted in Chapter 3 using maximum likelihood methods generated confidence bounds on sediment mixing parameters); 3. they allow specific hypotheses to be tested (for example, the nitrogen modelling in Chapter 5 was used to explore the possibility that irrigated burrows may be responsible for high denitrification rates in Port Phillip Bay); and 4. they provide a method for identifying important knowledge gaps that need to be addressed in future work. A further model use is to aid experiment design, both by drawing attention to implications of design decisions (such as the use of narrow tanks) and by clarifying measurement priorities (for example, the importance of determining mass budgets and quantifying flows between pools of nutrients). Conclusions specific to each section of work are reviewed here, followed by more general suggestions for the future

177 6.1 Laser scanner experiments A laser scanner was used to create a time series of three-dimensional maps of sediment mounds created by burrowing shrimp, Trypaea australiensis. The use of a laser scanner is a novel approach to this problem, and a particular advantage is its accuracy in measuring volumes expelled to and subducted from the sediment surface. The laser scanner experiments of Trypaea australiensis burrow mounds represented a more detailed study of mound dynamics than is usually possible by sediment trap or tracer core methods. Results from these experiments pointed to two clear conclusions: the rate of material falling back down into a burrow can be just as significant as the rate at which sediment is being expelled through the burrow opening; and the exchange of material between surface and depth represents a rapid oscillation that will not be detected if data is collected at low sampling frequencies. I have found no discussion on either of these points in the literature. These results raise questions about appropriate measurement approaches in the field, and also suggest that rapid two-way non-local transport needs to be included in sediment mixing models for sediments inhabited by thalassinidean shrimp D EIC model The laser scanner results were the motivation for the development of the Excavate, Infill and Collapse (EIC) model. This simple non-local model for sediment mixing by biota is a combination of the upward-conveyor belt and burrow-and-fill models; the model includes excavation of material to the surface, the collapse of burrows and the burrow infill with surface material. Maximum likelihood estimation and model comparison techniques demonstrated that the EIC model performed better than the diffusion model in modelling radionuclide cores from Port Philip Bay (PPB). In all PPB cores, the best EIC model parameter estimates required some level of burrow infill, which offers further evidence of the importance of subsurface injection of surface material. A significant advantage of this model over the diffusion model is its ability to produce a broader range of profiles while capturing the important non-local exchanges between the sediment surface and depth. Further model testing involving porosity variation and non-uniform burrowing with depth suggested that these alterations make significant differences to model results

178 6.3 Higher dimensional EIC model The EIC model was expanded to two and three dimensions to demonstrate its usefulness as a simple mechanism for introducing lateral heterogeneity to model sediments. Significant heterogeneity occurs in real sediments and is thought to have first order impacts on biogeochemical processes. The 2-D EIC model was used to generate a synthetic data set in order to perform a critical assessment of the scope for extracting actual mixing parameters from one-dimensional cores. This work demonstrated that even when the underlying mixing process is known exactly, heterogeneity and sampling procedures can confound reliable recovery of the original mixing parameters. A valuable extension to the higher dimensional EIC model would be to couple it to the automatic burrow generation routine developed by Koretsky et al. (22). Detailed resin casting of burrow networks could provide good data for creating more realistic burrow maps. The equations of the 2-D or 3-D EIC model could be applied equally well to porewater species, so allowing the possibility of multi-dimensional solid and porewater modelling. Such a model could be used to investigate how sediment diagenesis is altered by the simultaneous introduction of pockets of organic matter and oxygen to depth in the sediment. 6.4 Burrow irrigation modelling The burrow irrigation modelling work combined two well-accepted modelling approaches (radially symmetrical burrow geometry and a nitrogen diagenesis model) to investigate the impacts of burrow irrigation on sediment nitrogen chemistry. Comparison with data from three published thalassinidean experimental studies served to illustrate the model s use in interpreting experimental data, and its ability to explore the extent to which current theoretical understanding, as encapsulated in the model, can explain the measurements. Resolving the divergences between theory and observation are potentially fruitful areas for further work. For example, the model was unable to explain both low ammonium concentrations and relatively small oxygen penetration distances measured in sediment surrounding Callianassa truncata burrows. Conversely, the model predicted higher nitrate and lower ammonium levels than measured within Callianassa japonica and Upogebia major burrows. The model demonstrated that assumptions about diffusive transport through sediment are unable

179 to explain the rapid changes in oxygen measured in the sediment surrounding a Callianassa subterranea burrow. The model predicts that concentration measurements made in thin tanks will not represent in situ burrow concentrations, even if the animal has been established in sediment taken from its native environment. In particular, tank geometry determines the extent to which species such as ammonium can accumulate within the sediment. There are robust theoretical grounds for drawing this conclusion, and experiments aimed at testing and exploring these effects further are needed to ensure these artefacts do not confound future experimental results. The irrigation model was used to estimate the depth and density of irrigated burrows required to explain high denitrification rates measured in Port Phillip Bay sediments. Model geometry could be configured to match a comprehensive range of data from the Bay, but the geometry and the flushing rates were not consistent with typical thalassinidean shrimp population densities and burrow flushing rates. I conclude that while irrigated burrows can contribute significantly to high denitrification rates, the relatively sparse deep burrows of thalassinidean shrimp are unlikely to be sufficient to explain the full extent of these rates. Further work is needed to more fully understand the nitrogen dynamics within Bay sediments. Resin casting of Bay sediments could be used to reveal the nature of the burrow networks. Dense burrows of other species in the surface layer, and commensals burrows extending out from thalassinidean shrimp burrows would provide further explanation for the observed denitrification rates. Another possibility includes further investigation into the role of microphytobenthos residing on the sediment surface. Limitations of one-dimensional representations of nitrogen diagenesis were explored via comparisons between the 1-D models and the full cylinder model. Nitrogen chemistry surrounding a burrow is a genuinely three dimensional problem and any one-dimensional approach will suffer significant drawbacks. For example, the onedimensional radial model of sediment surrounding a burrow is only appropriate at depths from the sediment surface exceeding approximately 1 cm. Within the upper 1 cm, the direct and indirect effects of the sediment surface have profound impacts on the fluxes between the sediment and overlying water. More work is needed to ascertain the circumstances under which lower dimensional models are appropriate

180 6.5 Future work The importance of sediment biogeochemical processes in coastal systems is well recognised, and there are continuing efforts to measure, model and better understand their role in our waterways. The development pressure on these systems is substantial (most people in Australia live near estuaries) and models are increasingly called upon to make specific predictions to guide decision-making in the real world. For example, the Port Phillip Bay Environmental Study developed a water quality model for the Bay to test and assess management scenarios. Yet it is clear that the modelling capability falls well short of that required for robust predictions; in Port Phillip Bay our capacity to explain and model the measured denitrification rates remains limited, yet the ability of Port Phillip Bay sediments to denitrify incoming nutrient loads holds an important key to its ecological health. Around the world there are continuing improvements to the transport assumptions underlying sediment diagenesis models. In particular, there is growing adoption of higher-dimensional models that allow the representation of lateral heterogeneity. Conclusions in the previous section point to some specific future developments that would make welcome contributions. In addition, further work in the following more general areas would be valuable. 1. Clear procedures for identifying and propagating uncertainty in models are needed quite urgently as a means of assessing model reliability. The already considerable work on parameter estimation techniques needs to be expanded to encompass more aspects of the modelling process. These tools are needed to better understand what the predictive capability of a model really is, and to determine when a new model represents a genuine advance over existing models. 2. The most comprehensive biogeochemical models are complicated, and are often impractical to link to full transport models that resolve a water body spatially. Modellers face inevitable trade-offs between complexity, computability and reliability. More work is needed to ascertain the circumstances under which we can safely model spatial averages only, and when higher-dimensional descriptions are necessary. More broadly, this is a question of determining the level of model detail required to capture the system behaviour of interest. There is always a need to find the simplest model that will perform the required task well

181 3. Questions of feedback between the animal behaviour and sediment environment were not addressed in this work, yet are extremely important. Animal-induced alterations to the sediment can serve to improve the animals circumstances (Aller s cylinder model demonstrated that by crowding together burrowing animals make it easier to detoxify their environment). The reverse could also apply, where once population levels fall below a critical density it becomes harder for the population to remain viable, as the extra effort required to modify their environment through burrow irrigation becomes prohibitively expensive. Another important interaction is that deposit feeders do not merely passively transport organic matter but actively mine it from the sediments. Another example is that in high sedimentation environments some benthic animals react by increasing their burrowing activity, yet in my EIC model I assumed burrowing rates were independent of sedimentation rate. These points need to be considered in bioturbation models. 4. I would like to see future collaborations between experimentalists and modellers nurtured and encouraged for the benefit of both parties, and so that knowledge advances more effectively. Experimentalists contribution to my work has been immeasurable. In return, insights gained from modelling can provide valuable feedback on experimental procedures, the design of future experiments and the interpretation of results. The complexity of biogeochemical interactions is such that modelling approaches are required to disentangle them

182 7 Appendix: A note about non-local irrigation models Boudreau (1984) demonstrated that a one dimensional diffusion model with a nonlocal source/sink term can be mathematically equivalent to radially integrating Aller s cylinder model, which provided a very useful method for including bioirrigation into one-dimensional diagenesis models. Limitations to the technique have been known for some time. For example: + This type of model is extremely useful in describing, for example, average NH 4 distributions in bioturbated deposits, but does not readily describe the effects of boundary reactions such as burrow wall nitrification, and interactions such as denitrification/nitrification balances resulting from specific geometries. (Aller (1988) p. 329) Aller (21) discussed these limitations also, concluding that a non-local model is only a good replacement for a full cylinder model (or a more general microenvironment model) if there is no radial dependence of reaction or transport properties (ie. reaction zonation around burrows cannot be important) and if the reaction kinetics are linear. Clearly these requirements are violated in multi-component diagenetic models, which consist of multiple species interacting in non-linear ways. In gauging whether to use a non-local model for the irrigation modelling in Chapter 5, I explored the difference between the cylinder and non-local models for some specific examples. 7.1 Background Aller s cylinder model equation is: 2 C = D C Ds + t z r r r C s r + R C, z, r 2 (7.1) where z is vertical distance from the sediment surface, r is the horizontal distance from the burrow centre, C is the concentration, D s is the molecular diffusion coefficient corrected for tortuosity and R is a reaction term

183 Typical boundary conditions are C = C z = C = C r = r 1 C r r r = = 2 C B z L z = = (7.2) where r 1 is the radius of the burrow, r 2 is the outer radius of the solid annulus surrounding the burrow and L is the depth of the burrow. B represents a prescribed flux at the base of the burrowed zone. Following Boudreau (1984), equation (5.1) can be multiplied by r r r integrated laterally to give, and 2 C = D C s 2 t z 2Ds r r r C r 2 1 r= r1 + R C, z, r (7.3) where the laterally averaged concentration is defined as r r 2 2 2π rcdr 2 rcdr r1 r1 C = = r2 2π rdr r 2 2 r 2 1 r1 (7.4) and the laterally averaged reaction term is R C, z, r = r2 21 r rr C, z, r dr r2 r (7.5) Boudreau (1984) made the assumption that C can be approximated by r r = r1 C r r= r 1 C C r r 1 (7.6) where r is the (unknown) point at which C occurs (r 1 < r < r 2). Figure 7.1 illustrates the approximation

184 C C r 1 r r 2 r Figure 7.1 The approximation made in equation (7.6). The dotted line represents the approximation to the concentration gradient at r 1. With the further assumption that R C, z, r = R C, z, equation (7.3) can be converted to a one-dimensional diffusion equation with a non-local term: 2 α C = D C s C C + R C z 2, t z (7.7) where α z, t = 2Dsr1 r 2 r 2 r r (7.8) Boudreau s analysis proved to be very useful, as it demonstrated sound physical reasons for employing equation (7.7) as a representation of bioirrigation in one dimensional sediment diagenesis models. The questions addressed in this chapter are: 1) To what extent is the approximation in equation (7.6) valid? 2) Equation (7.8) suggests that the coefficient, α, is a function of diffusivity and r, both of which are functions of the chemical species, and can both be functions of depth. Yet it is common practice simply to use a constant value across many species and at all depths in a model. Does this simplification matter? 3) Integrating the reaction term is only possible if reaction kinetics are linear and independent of the concentration of other species. How well does this approach work for a set of coupled non-linear differential equations, as is typical for multiple species diagenesis models?

185 Each of these questions is answered in the following three sections by making direct comparisons between solutions to the full cylinder model and non-local model. The next two sections draw on cylinder model results from Aller (198). Aller (198) employed the cylinder model with the following reaction term eq 1 R = k C C + R exp η z + R (7.9) to model the sediment-water exchange of SO 4 2-, NH 4 + and Si, where k, R, η and R 1 are empirical constants and C eq is an equilibrium concentration. The analytical steady state solution to the cylinder model is (Aller (198)): s n= n 2 G U r n µ n C z, r = C + Bz + 1 sin λ nz 2 LD µ U µ r n 2B 1 U µ nr sin nz 2 L n nu nr λ = λ µ 1 n 1 (7.1) where n =, 1, 2,... 1 λ n = n + 2 µ = k D + λ G n s n n π L n k C Ceq R kb n = + 1 ηe 1 λ R 2 2 λ λ λ η + λ n U µ r = K µ r I µ r + I µ r K µ r n 1 n 2 o n 1 n 2 n n n ηl n n and I ν (z) and K ν (z) are the modified Bessel functions of the first and second kind respectively of order ν. Aller (198) set parameters to the values shown in Table 7.1 (with only r 2 used as a fitting parameter) and found excellent agreement between modelled and field profiles. This configuration of the model is used in the next two sections

186 Table 7.1 The cylinder model parameters used in Aller (198). Variable 2 SO 4 + NH 4 Si C 14.7 mm.2 mm.74 mm B -.1 mm/cm.11 mm/cm.6 mm/cm D s.717 cm 2 /day 1.33 cm 2 /day.687 cm 2 /day η.36 /cm.61 /cm R mm/day.267 mm/day R mm/day.81 mm/day k.2 /day C eq mm r 1.5 cm.5 cm.5 cm r cm 2.1 cm 2.1 cm L 15 cm 15 cm 15 cm 7.2 To what extent is equation (7.6) valid? The solutions to the specific examples used in Aller (198) are shown in Figure 7.2. The dotted lines in the figures plot r (the location of C at each depth). For all three chemical species, r is located roughly midway between r 1 and r 2. The radial concentration gradients are steepest at r = r 1, and a visual inspection of the data suggests that the concentration gradient between r 1 and r is not linear. These observations call into question the validity of the assumption made in equation (7.6). SO 4 2 (mm) Si (mm) NH 4 + (mm) z (cm) r (cm) r (cm) r (cm) Figure 7.2 Concentration distributions of SO 4 2, Si and NH 4 + calculated from equation (7.1) using the model parameters in Table 7.1. The dotted line plots r. A comparison between the analytical radial derivative at r = r 1 and the approximation given in equation (7.6) confirms that in this particular case the approximation is a poor one (Figure 7.3).

187 5 5 5 z (cm) SO gradient (mm/cm) 4 Si gradient (mm/cm) NH 4 + gradient (mm/cm) Figure 7.3 Comparison between the analytical radial derivative at r 1 (solid line) and the approximation given in equation (7.6) (dotted line). Does this poor match matter? Equation (7.8) was used to calculate α, z and Matlab s ode15s function was used to find a steady state numerical solution to equation (7.7). The solution was compared to the average concentration profiles calculated from the full cylinder model (Figure 7.4) z (cm) SO 4 2 (mm) Si (mm) 15.5 NH 4 + (mm) Figure 7.4 Comparison between the average concentration profile calculated from equation (7.1) (dotted line) and the steady state numerical solution to equation (7.7) (solid line). The value of α has been calculated from equation (7.6)

188 This particular example demonstrates that the approximation given in equation (7.6) is not necessarily a good one. Further, the one dimensional non-local model derived from this approximation can produce significantly different solutions to the radiallyaveraged concentration profiles generated from the full cylinder model. 7.3 Validity of using a constant value for α The most common implementation of equation (7.7) assumes that α is a constant across all chemical species and at all depths. Values of α are found by calibration against measured profiles, and the expression given in equation (7.8) is rarely used in practice (although it has been used by Furukawa et al. (2) and Koretsky et al. (22)). Using this approach for the above example, I found that there are values of α that can be used to generate a very good match to the radially-averaged concentration profile for each species. Values of α SO4 =.11, α Si =.1 and α NH4 =.2 produce profiles which are visually indistinguishable from the radially-averaged analytical solutions. It should be noted that the value of α differs for each species, as predicted by Boudreau s analysis. If the same value of α (α =.14) is used for all three species, the match between the two models is still good, but less convincing (Figure 7.5). The implication is that the use of a single value of α is not necessarily valid for multiple species models z (cm) SO 4 2 (mm) 15.5 Si (mm) NH 4 + (mm) Figure 7.5 Comparison between the average concentration profile calculated from equation (7.1) (dotted line) and the steady state numerical solution to equation (7.7) (solid line). A process of trial and error was used to find a single value of α that provided a good match across all three species (α =.14)

189 Given that a useful value of α does exist, I sought to modify Boudreau s formulation slightly by substituting r for a value r' such that the approximation in equation (7.6) is true: = 1 + r' z, t r C C C (7.11) r r = r 1 The definition of α then becomes: α z, t = 2Dsr1 r 2 r 2 r' r Equations (7.11) and (7.12) were used to calculate r' and α for NH 4 +, SO 4 2 (7.12) and Si (Figure 7.6). The resulting α profiles match the constant values found earlier (with the exception of the very top and base of the profiles). The values of r' are roughly the same for each species (to one significant figure accuracy), and so the values of α simply scale with D s. Other authors have suggested assuming that α values scale with D s, so that values inferred from one species can be used to calculate the relevant values for other species (eg. Aller and Yingst (1985)). Unfortunately, there is no guarantee that r' will be the same for all species, and so this approach needs to be employed with caution (as noted by Boudreau (1997), p. 143). These results suggest that cylinder and non-local models could be equivalent if equation (7.11) is used instead of equation (7.6) in the derivation of α. Note that just as equation (7.8) is rarely used by practitioners, equation (7.12) is equally impractical. The replacement of r with r' merely represents a minor modification to Boudreau s original derivation, and direct field measurement of r' would not be feasible. It is possible, however, that equation (7.12) could be used in a modelling context. For example, a one-dimensional radial model could be used to estimate = C r r r1 and C at depth. These values could be substituted in to equations (7.11) and (7.12) to derive a value for α which could, in turn, be used to calculate vertical profiles and fluxes. I had hoped to use this approach in my work. Complications involved in multi-species modelling (discussed in the next section) ruled out this possibility in my work; however the approach would be appropriate in many other contexts

190 5 5 z (cm) z (cm) r α Figure 7.6 Values of r' and a calculated from equations (7.11) and (7.12) (NH 4 + solid line, SO 4 2 dashed line, Si dotted line). 7.4 Can a non-local model describe a multiple-species system well? The analysis so far has been limited to single-species models with reaction terms that are easily integrated radially. Sediment diagenesis models are typically coupled multiple-species models. In such models the reaction terms are more complicated, and because they are dependent on other species concentrations, they are likely to vary with r. I ve used the multiple-species cylinder model from Chapter 5 (see Section 5.6.2) to demonstrate the potential problems of using a radially-averaged approach for such systems. The model was compared with the one-dimensional models: (1) a diffusion-only radial model (which models a transect radiating out from the burrow wall); (2) a diffusiononly vertical model (which models a vertical profile in sediment with no irrigated burrows); and (3) a non-local vertical model. The cylinder model was run with the same burrow geometry as earlier examples. A 2- day run was sufficient to bring the cylinder model solution to steady state. P DOC was uniformly distributed with a value of 2 nmol/cm 3 /d (which corresponds to a carbon mineralisation rate of 25.5 mmol/m 2 /d for this burrow geometry, assuming P DOC is zero below the burrowed zone of sediment). Figure 7.7 shows the steady state concentration distributions from this configuration. Comparison with the diffusiononly vertical model confirms that irrigation has a strong influence on the radially

191 averaged concentration profiles (Figure 7.8). For example, in the cylinder model NH 4 + escapes through burrow walls as well as the surface and so there is less flux of NH 4 + to the sediment surface. Consequently, NH 4 + does not build up to such high concentrations near the sediment surface, and so oxygen can penetrate further. These kinds of interactions between species are not obvious when looking only at tracers with first order decay. Figure 7.7 Two-dimensional steady state concentration distributions from the 5-species nitrogen cycling model. The two-dimensional concentration distributions were used to estimate r' using equation (7.11), and these values were used to calculate α from equation (7.12). Unlike in previous examples, both r' and α profiles varied substantially with depth (Figure 7.9 and Figure 7.1). Further, the equations produced negative r' and α values for NO 3, DOC and NH + 4. Negative values make little physical sense, and so these results suggest that the use of equations (7.11) and (7.12) is inappropriate for this model