A morphodynamic model of atoll-islet evolution

Transcription

1 University of Wollongong Thesis Collections University of Wollongong Thesis Collection University of Wollongong Year 2008 A morphodynamic model of atoll-islet evolution Stephen J. Barry University of Wollongong Barry, Stephen J, A morphodynamic model of atoll-islet evolution, PhD thesis, School of Earth and Environmental Sciences, University of Wollongong, This paper is posted at Research Online.

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3 A Morphodynamic Model of Atoll-islet Evolution A thesis submitted in fulfilment of the requirements for the award of the degree Doctor of Philosophy from The University of Wollongong by Stephen J. Barry BSc (Hons) School of Earth and Environmental Sciences Faculty of Science 2008

4 Certification I, Stephen J. Barry, declare that this thesis, submitted in fulfillment of the requirements for the award of Doctor of Philosophy, in the School of Earth and Environmental Sciences, University of Wollongong, is wholly my own work unless otherwise referenced or acknowledged. The document has not been submitted for qualifications at any other academic institution. Stephen J. Barry Date: ii

5 "There are two kinds of truths: those of reasoning and those of fact" Gottfried Wilhelm Leibniz "All models are wrong, but some are useful" George E. P. Box iii

6 Acknowledgements This work would not exist at all without the contributions and support of many friends, colleagues and family. I owe many thanks to Dr Peter Cowell, whose knowledge, guidance, and enthusiasm, has been fundamental to the progression of this research and to my education in both academic and coastal behavior. Thanks are also due to my supervisor Colin Woodroffe for providing the initial funding and motivation for this research. I am indebted to the staff of the School of Earth and Environmental Sciences for their support during this period. Thanks also to Paul Carr, Laurie Chisholm, and Marji Puotinen for providing advice and perspective; John Reid, Mike Robinson, Heidi Brown, and John Marthick for technical support. I am pleased to acknowledge and thank Dr Peter Eklund, Dr Mark Sifer, and Dr Reza Zamani from the Faculty of Commerce for providing advice, employment, and encouragement during my time at the University of Wollongong. Support from fellow students, past and present, has been essential during this project; thanks to Paul Grevenitz, Ava Simms, Paolo Aballe, Gareth Davies, Kevin Pucillo, and George Susino. Thanks also to Peredur Jones who was a reliable companion for extramural amusements. Most of all I would like to thank my family for their support throughout these years of study. I am especially pleased to be able to thank my wife, Lynne; without her unwavering perseverance and commitment this thesis would not have been completed. iv

7 Table of Contents Certification...ii Acknowledgements...iv Table of Contents...v List of Figures...viii List of Tables...x List of Notation...xi List of Abbreviations...xiv List of Publications...xv Abstract...xvi 1 Introduction General Introduction Atoll-islet Morphology Modeling Coastal Behavior The Sediment Allocation Model Thesis Organization A Morphodynamic Model of Atoll-Islet Evolution Abstract Introduction Purpose of the Model The Sediment Allocation Model Model Parameters A Hypothetical Atoll-islet Parameter Estimation Using Inverse Techniques Optimization Constraints Parameter Estimation Results Discussion...39 v

8 2.8 Conclusion Growth-Limiting Size of Atoll-Islets: Morphodynamics in Nature Abstract Introduction Methods The Sediment Allocation Model Field Data Boundary Conditions and Cell Definition for Transect Data-model Choice of Isochron Pattern for Transect Data-model Progradational data-model Hybrid Data-model Results Data-model Comparison Parameter Estimation A Simulation of Islet Transect Development Discussion Simulating Islet Growth Sampling Strategy Conclusion Atoll-Islet Morphology and Energy Exposure: Comparative Metrics Abstract Introduction Models of Atoll Development Rationale for a Comparative Metric Measuring Islet Location Materials and Methods Field Setting Energy Exposure Exposure Regions Antecedent Morphology Measurement of Islet Location Results vi

9 4.5 Discussion Energy Exposure and Islet Location Cocos_E and Marakei_W Marakei_W and Tarawa_SW Tarawa_N and Marakei_E Conclusion Islet Response to SLR Acknowledgements Conclusions Introduction Directions for Future Research Conclusion References Appendix A Appendix B Appendix C vii

10 List of Figures Figure 1.1 Idealized representation of an atoll-islet environment...4 Figure 1.2 Response of sediment sequestration to atoll-islet evolution...7 Figure 2.1 The Sediment Allocation Model...18 Figure 2.2 The forward model of islet development...19 Figure 2.3 Evolution of the islet sediment sequestration ratio...23 Figure 2.4 An example of the physical setting of an atoll islet...25 Figure 2.5 The sediment allocation function generated using the parameter values b = 200, r = 10, and V max = 600, Figure 2.6 Output from the forward model shows the growth of an atoll isle...26 Figure 2.7 The sediment allocation function when the r parameter is altered...27 Figure 2.8 The effect on the SAM for different values for the parameter r...28 Figure 2.9 The sediment allocation function when the b parameter is altered...29 Figure 2.10 The effect on the SAM for different values for the parameter b...30 Figure 2.11 Graphs of the averaged estimated parameters...38 Figure 2.12 The output produced by the forward model...39 Figure 3.1 Schematic scenarios of island formation...47 Figure 3.2 A conceptual model of aggradational reef-island growth...48 Figure 3.3 An example of a sediment allocation function...52 Figure 3.4 Location of the field sites...54 Figure 3.5 Transect data for Cocos (Keeling) Island...54 Figure 3.6 Transect data for Makin Island...55 Figure 3.7 A conceptual model of the processes contributing to reef-island...56 Figure 3.8 The modified transect images...60 viii

11 Figure 3.9 The elliptical isochrons used to calculate the transect volumes...62 Figure 3.10 Graphs of transect development based on evolution by lateral accretion evolution by hybrid accretion...64 Figure 3.11 Sediment allocation functions for the study sites...66 Figure 3.12 Output from the SAM using estimated parameters...66 Figure 3.13 Simulation of reef-island transect development...70 Figure 3.14 An interpretation of the effect that the SAM parameters...72 Figure 4.1 Representation of an atoll-islet as morphological sub-units...88 Figure 4.3 The location of the sites showing the regions of the islets sampled...95 Figure 4.4 The Tarawa_N region contains significant conglomerate deposits Figure 4.5 The sampling method used to measure the distance from the reef rim to the atoll islet Figure 4.6 Boxplot of reef-rim to islet-beach distance ix

12 List of Tables Table 2.1 Parameters used to represent an atoll-islet environment...25 Table 2.2 Constraint values for the inversion algorithm...32 Table 2.3 Selected Sink 1 volumes from the forward model...35 Table 2.4 The average values of estimated parameters...37 Table 3.1 Estimated parameters for the SAM...65 Table 3.2 Measurements used to model the reef-island profile Table 4.1 Energy exposure variables considered for the four study sites...97 Table 4.2 Ranking of off-reef energy-exposure by region...99 Table 4.3 Summary measurements of islet location for sampled islet regions x

13 List of Notation a b height at the transect origin. sediment transport impedance term. The linear parameter in the logistic function representing transport gradients relevant to sediment sequestration by the island. c D i 1-f(v/V max ) ln(h/a)l. a Sloss variable; the dispersal factor. increment counter. island sediment sequestration ratio. The proportion of available sediment sequestered by the reef-island. f(v/v max ) f1_error the proportion of available sediment that bypasses the reef-island. 1-f(v/V max ) when v/v max = 1. When the accommodation volume is filled no sediment is sequestered by the island. This means that, ideally, the value of 1- f(1) = 0, however, since 0 < f(v/v max ) < 1, this will never be the case. Instead, the f1_error can be used as an error term in the nonlinear optimization routine. h L m p ridge height. length measurement. meter. the perturbation amount calculated as p = v (i) p max x. This value was used to perturb the exact value, v (i), during testing of the inversion procedure. For example, if v (i) = 100, p max = 0.1, and x = -0.5, then p = = -5, then the value of v (i) used in the parameter recovery process is v (i) = v (i) - -5 = = 105. xi

14 p max the maximum perturbation ratio. If the measured value was 100 and the maximum perturbation ratio was 0.1, then, the maximum absolute value of the perturbation is = 10. Q r a Sloss variable; the quantity of sediment available for deposition. transport-impedance sensitivity to morphology. The nonlinear parameter in the logistic function that represents morphodynamic feedback parameter, i.e. the response of sediment transport processes to island growth. R s s i t a Sloss variable; the accommodation available for sediment deposition. rate of sediment production; seconds. sediment produced during step i of the forward model. time. t 0 t = 0. v v 0 vector of island volume measurements. initial volume of the island at t = 0. In the solution to the logistic equation b = (1/t 0 ) 1, therefore t 0 0, and v 0 0. This implies that the island forms as the result of a pre-existing perturbation of the sediment transport field. For the purposes of testing it was assumed that the initial island volume was 0 m 3. v V max vector of island volumes calculated using the forward model. accommodation volume, the maximum volume available for island development. w() weighting function. This is used to magnify the f1_error value so that it is significant when compared to the calculated error term in the optimization process. xii

15 x unit of length; a random number in the range [-1, 1]. y yr unit of length. year. xiii

16 List of Abbreviations ECD ENSO GIS IQR LD OD SAM SLR VBA Effective conglomerate distance El Niño Southern Oscillation Geographic Information System Interquartile range Lagoonside deposition Oceanside deposition Sediment Allocation Model Sea-level rise Visual BASIC for Applications xiv

17 List of Publications Barry, S.J., Cowell, P.J. and Woodroffe, C.D., A morphodynamic model of reefislet development on atolls. Sedimentary Geology, 197(1-2): Barry, S.J., Cowell, P.J. and Woodroffe, C.D., Growth-limiting size of atollislets: Morphodynamics in nature. Marine Geology, 247(3-4): Barry, S.J., Cowell, P.J. and Woodroffe, C.D., Atoll-islet morphology and energy exposure: comparative metrics, Manuscript submitted. xv

18 Abstract A morphodynamic model of atoll-islet evolution, the Sediment Allocation Model (SAM), was developed based on the assumption that islets are equilibrium landforms. The assumption that islets are equilibrium landforms implies that the volume of sediment sequestered in an islet reaches stability when considered over the time-scale of islet evolution. Stability of the islet implies that the maximum volume of an islet (accommodation) can be quantified. Morphodynamic feedback between the islet and sediment transport processes is manifest in the rate of sediment sequestration by the islet. Initial increase of islet volume is rapid, sediment sequestration slows as islet volume increases, and stops when accommodation is full. The rate of islet development is constrained by antecedent morphology, sediment supply, and sediment transport. The constraints and accommodation for islet evolution are difficult to measure directly. Estimates of the accommodation, sediment supply, the effect of antecedent morphology, and the morphodynamic feedback between islet volume and sediment sequestration are generated using measurements of islet volume in an inversion algorithm. Experimentation carried out using a hypothetical test-case indicated that errors in the measurements of islet volume used in the inversion algorithm would result in a damping of the morphodynamic feedback and overestimation of both sediment supply and accommodation. The SAM was implemented using published radiocarbon-dated samples for surveyed islets transects. A data-model template was developed to incorporate the published data into the SAM. In the data-model template an atoll-islet was represented by a xvi

19 combination of morphological sub-units with sub-unit sequestration dominated by either lagoon processes or ocean processes. The input to the SAM was a series of sediment volumes representing islet evolution. Sediment volumes were calculated using isochron patterns that represent a hybrid of lateral accretion and vertical accretion, recognized modes of sediment accumulation on atoll-islets. Sediment volume measurements calculated using the hybrid-accretion patterns were consistent with the assumption of morphodynamic feedback between islet morphology and sediment accretion processes. Comparative metrics are required to test and refine models of islet evolution, however, suitable comparative data are not currently available. The hybrid-accretion pattern of islet evolution formed the basis for the development of a metric to determine islet location on the reef platform. Islet location was measured on four atolls as the distance from the reef crest to the oceanward beach-toe. The results of the measurement procedure were evaluated in the context of three factors that influence islet evolution: energy-exposure, sediment supply, and antecedent morphology. The results indicate that the distance from the reef crest to the oceanward beach-toe is a measurable response to the influence of the three factors. xvii

20 1 Introduction Preface This thesis has been prepared as a series of manuscripts written for publication. As a consequence, each chapter stands alone and, therefore, their introductions are detailed, citing relevant literature and setting the context for each of the aims addressed. However, the reader will find that this results in the inevitable overlap of some material between the Introduction sections of each individual chapter and that contained in the overview presented in this chapter. The material in this chapter, therefore, only briefly touches on certain subject matters while attempting to give a general background to concepts arising in the subsequent chapters.

21 1.1 General Introduction Islets on atolls are landforms that exist as a result of a balance between the constructive effort of reef-produced calcium carbonate sediment and destructive forces (waves, currents, bioerosion, and chemical dissolution) (Richmond, 1992). There is a concern that sea-level rise (SLR), an anticipated impact of global warming, may threaten the existence of the islet landforms (Roy and Connell, 1991; Leatherman, 1997; Cowell and Kench, 2001; Dickinson, 2004; Kench and Brander, 2006a; Woodroffe, 2008). Erosion of the atoll-islets is expected to be the initial response to raised sea-levels, however, a number of recent reviews have stated that further research into atoll-islet evolution is required to understand islet response to SLR (Stephenson and Brander, 2003; Masselink and Hughes, 2003; Kench and Cowell, 2001; Roy and Connell, 1991; Woodroffe, 2008). The aim of this thesis is to construct a model of atoll-islet evolution that can be used as an experimental testbed to investigate modes of islet formation over the regime timescale, 10 3 years, relevant to islet development. The contributing factors to islet formation are thought to be the hydrodynamic processes (wind waves, ocean swell, tides), antecedent morphology (pre-existing islet and reef structure) and sediment supply. While the contributing factors have been identified, data suitable for modeling, measurements of the contributing factors at the relevant scale, are not readily available. In addition, because the hydrodynamic processes responsible for sediment accumulation are nonlinear, forward models of the hydrodynamic processes are inherently unstable over long time-periods (de Vriend et al., 1993; Cowell and Thom, 1994). More specific aims of this thesis were: 1) to determine the feasibility of the development of a model of 2

22 islet evolution based on morphodynamic principles; and 2) to determine whether such a model could be applied to the problem of islet response to SLR. 1.2 Atoll-islet Morphology Atoll-islets are accumulations of sediment on atoll-reef platforms. These islets are composed predominantly of unconsolidated Holocene sands and gravels locally derived from reef organisms such as coral, coralline algae, molluscs and foraminifera (Stoddart and Steers, 1977; Yamano et al., 2000; Woodroffe, 2002; Kench et al., 2005). Islet evolution occurs over a time scale (10 3 years) that cannot be observed directly; instead, conceptual models of islet evolution have been proposed as a means of understanding the processes contributing to islet evolution (e.g. Cloud, 1952; Stoddart, 1965; Bayliss- Smith, 1988; Richmond, 1992; Woodroffe et al., 1999). There is scope, therefore, for the interpretation of the conceptual models into a numerical framework that can be used to examine the conceptual models in greater detail. The model of islet evolution developed in this thesis is based on three assumptions regarding islet behavior: (a) sediment supply for islet development is locally generated; (b) islets accumulate sediment to the limit of available accommodation space (the space available for sediment accumulation); and (c) islets are equilibrium landforms in adjustment with the controlling processes of the reef environment (Figure 1.1a). Sediment supply and accommodation space are both difficult to measure. Estimates of sediment production for reef environments are available but have not been related to availability for sequestration by atoll-islets. Sequestration of sediment by atoll islets 3

23 isolates the sediment from the sediment transport field. Similarly, accommodation space for islet growth is not directly quantifiable as the limitations to islet development have not been identified. Nonetheless, the recognition that sediment accumulates to a limit implies that the accommodation is quantifiable. In addition, the assumption that islets are equilibrium landforms indicates that sediment sequestration is controlled by a morphodynamic interaction; i.e. feedback between islet morphology and the hydrodynamic processes responsible for sediment transport (Figure 1.1b). a Ocean Wind Waves Lagoon Circulation Lagoon Atoll reef Ocean Ocean swell b sediment supply bypassing islet volume 100% 0% sequestration islet volume accommodation sequestration Figure 1.1 Idealized representation of an atoll-islet environment. (a) Sediment transport and sequestration is driven by hydrodynamic processes (e.g. wind waves, swell, tidal fluctuations, and lagoon circulation). (b) The morphodynamic response of sediment sequestration to islet volume. As islet volume increases, the proportion of available sediment sequestered by the islet decreases. 4

24 Atoll-islets are the tangible byproduct of hydrodynamic processes that transport sediment on the reef flat. From a modeling viewpoint, the morphology of an islet represents the integral of the process function residuals over the time-period required for islet evolution. For example, if an islet transects contains 1000 m 3 sediment accumulated over a 3000 yr period then the sediment accumulation rate is approximately 0.33 m 3 /year. If the transect is on a reef 1000 m wide with a tidal range of 1 m then, assuming a semidiurnal tidal regime, an annual flux of approximately 700,000 m 3 of water results in the accumulation of only 0.33 m 3 of sediment. Islets, then, represent the integration of the residual, the difference between deposition and erosion, relentlessly aggregated over the time-span of islet evolution. The difference in physical scale of the fluids (10 5 ) required to mobilize sediment (10-1 ) mean that it would be difficult to develop a model of sediment transport based on hydrodynamic processes. Islet morphology therefore contains some information about the sediment-transport processes, but, not enough information to identify the contribution of individual processes. Nonetheless, an improved understanding of islet morphology may provide insight into the processes contributing to islet evolution. The scope for this research is limited to the study of the relationship between islet morphology and the processes responsible for islet evolution. 1.3 Modeling Coastal Behavior The temporal scale of processes on atoll reefs contributing to sediment production and sediment transport ranges from seconds to thousands of years (Gourlay, 1988; Hatcher et al., 1987). The time taken for islet evolution (10 3 years) means that the complete range of conditions and processes that contribute to islet morphology cannot be directly observed over the human life-span. Geomorphologists have always used models to 5

25 progress their understanding of natural geomorphic systems. Bras et al. (2003) proposed that numerical models are a useful extension to the conceptual modeling process as they can; (a) serve as deductive tools that help guide research; (b) force rigor in the development of hypotheses and interpretations; (c) highlight errors of understanding and concept; (d) lead to unpredictable insights; and (e) serve as virtual laboratories. Prior to the development of the Sediment Allocation Model (SAM) implemented in Chapter 2, models of atoll-islet evolution were descriptive (Stoddart, 1965; Flood, 1986; Richmond, 1992) or conceptual (Bayliss-Smith, 1988; Woodroffe et al., 1999; Kench et al., 2005). Numerical models have been used to investigate islet response to SLR (Cowell and Kench, 2001; Kench and Cowell, 2001) but the study of islet evolution using numerical models has not been attempted previously. This study is the first attempt to codify knowledge of islet-evolutionary behavior in a numerical framework. Process-based models of sediment transport and accumulation are unstable over the time-scale of islet evolution (10 3 years). The models are unstable because the equations used to describe hydrodynamic processes are highly nonlinear. Instead, this study implements a behavior-oriented methodology and examines the atoll-islets produced: the outcome of the sediment-transport processes as described by their accommodation space, fluid dynamic regimes, and sediment budget, the Sloss variables: (Sloss, 1962; Swift et al., 2003b; Thorne and Swift, 1991). Werner (2003, p 143) classifies behaviororiented models, such as the SAM, universalist models that are not meant for direct study of natural systems, but rather as tools with which to assess measurements and 6

26 develop more realistic models. Universalist models are usually implemented with few variables and simplified dynamics. The approach to this study has been to view atoll-islets as sediment sinks (Figure 1.1b) with accommodation for sediment sequestration determined by three controlling factors: hydrodynamic processes, antecedent morphology, and sediment supply, however, the effect of these factors have not been quantified by a method suitable for inclusion in simulation models. Instead the SAM is formulated on the basis of the behavior of the islet morphology as a response to the controlling factors. Some previous studies of atoll islets (e.g. Richmond, 1992; Stoddart and Steers, 1977) have suggested that atoll-islets are equilibrium landforms with morphodynamic feedback between the islet development and the processes responsible for sediment sequestration: as islet volume increases, sediment sequestration decreases (Figure 1.2). The rate of sediment sequestration is the physical manifestation of morphodynamic feedback between islet morphology and hydrodynamic processes. 100% 100% Islet Volume Islet volume Sediment sequestration Sediment sequestration 0% 0% Time Figure 1.2 Morphodynamic response of sediment sequestration to atoll-islet evolution. Islet volume is limited by accommodation. The rate of sediment sequestration by the islet decreases as islet volume increases. 7

27 1.4 The Sediment Allocation Model The goal of this study is to improve the understanding of atoll-islet evolution by the implementation of conceptual models in a numerical framework. The SAM has been developed in order to achieve this goal. The formulated model is necessarily universalist (Werner, 2003) since this is the initial foray into the study of islet evolution using numerical models. There are a number of reasons why the behavior-oriented modeling approach is a productive area for research: (a) it is a novel approach to the problem of modeling atoll-islets; (b) the SAM can be applied at all scales since it has no scales inherent in its formulation; (c) it can incorporate estimates of sediment production; (d) estimates of islet volume can be made using appropriate sampling techniques; (e) it estimates maximum islet volume and therefore provides information about the limits to islet expansion; (f) the inversion method implemented in the SAM can be applied to other models based on Sloss variables (e.g. Sloss, 1962; Paola, 2000); and (g) experimentation using the model can provide insight into conceptual models of atoll-islet evolution. Beyond the purely research-specific goals of this study, there is also scope for this research to benefit a wider range of societal targets. The effect of SLR on atoll-islets is, as yet, unknown but expected to be largely negative (Mimura et al., 2007; Buddemeier et al., 2004). Mimura et al. (2007) found that the uncertainties associated with islet response to SLR needed to be reduced so that national and local-scale adaptation strategies for small islands could be better defined, but that available observational data 8

28 was inadequate to reduce the uncertainty. Indeed, the uncertainty surrounding the effect of SLR on atoll-islet formations marginalizes the populations of island inhabitants (Farbotko, 2005) and restricts the development of appropriate policies to address issues associated with SLR (Barnett, 2001; Barnett and Adger, 2003; Mimura et al., 2007). Based on the outcomes of the SAM implementation, this study proposes that the it may be possible to reduce the uncertainty associated with islet response to SLR by generating comparative datasets that measure islet location. Prior to the development of the SAM there were no suitable models of islet evolution available to identify suitable comparative data. Inhabitants of islet landforms could therefore benefit from this study through reduced uncertainty regarding islet response to SLR leading to improved policies that deal with the associated issues. 1.5 Thesis Organization This thesis describes the development and application of the SAM. The theoretical formulation of the model is presented in Chapter 2 and the model performance is tested using a hypothetical atoll islet. Testing of the model was carried out to determine the effect that errors in the measurement of islet volume have on the parameter values estimated by the inversion algorithm. In Chapter 3, radiocarbon-dated samples from islet transects on the Cocos (Keeling) Islands and Little Makin Island were used to estimate parameter values for the model. Implementation of the model required the development of a conceptual representation of islet transect evolution. In Chapter 4, the conceptual model of islet evolution is discussed in the context of atoll-reef processes. Placing islet evolution in the context of reef processes provided the theoretical basis for a procedure to investigate the processes 9

29 responsible for sediment sequestration and to monitor islet response to SLR. In Chapter 5, the principal findings in this study are summarized. 10

30 2 A Morphodynamic Model of Atoll-Islet Evolution Preface This chapter describes the rationale for the formulation of a numerical model of atollislet evolution as a morphodynamic landform. The model was implemented in a computer program. An inversion algorithm was implemented to recover model parameters using measurements of islet volume. The inversion algorithm was tested using volume measurements generated by the forward model. The physical interpretation of SAM parameters (Section 2.4 and Figure 2.3) was contributed by Peter Cowell. Publication: Barry, S.J., Cowell, P.J. and Woodroffe, C.D., A morphodynamic model of reefisland development on atolls. Sedimentary Geology, 197(1-2):

31 2.1 Abstract Atoll-islets are deposits of reef sediment that have accumulated through sediment transportation processes. The vulnerability of atoll-islets has been a focus of discussion regarding the impact of sea-level rise related to global warming. The morphodynamic processes involved in the development of these landforms are poorly understood and difficult to observe over long time-periods. Several conceptual models of atoll-islet development have been proposed and implementation of the models in a computational framework can assist in the further development and refinement of these concepts by taking into account physical limitations such as sediment availability. Aggregated-scale and lumped-parameter models of coastal environments are being developed based on the observed morphology of the coastal region rather than the physical processes. This paper presents an aggregated-scale, behavior-oriented model based on a logistic function. The model simulates sediment accumulation in an atoll environment. A requirement of the behavior-based modeling approach is that the model parameters need to be determined for each application of the model. An inversion algorithm was used to estimate values for model parameters, including the maximum volume of the islet and the sediment production rate. The inversion technique is able to successfully recover the parameter values using selected measurements of the islet volume. This approach demonstrates that the implementation of suitable models can aid researchers in their understanding of atoll-islet processes. 12

32 2.2 Introduction Atolls are reefs with an annular reef rim, enclosing a lagoon; they occur in mid ocean. Atoll-islets are the small islets that form on the reef rim; they are composed predominantly of unconsolidated Holocene sands and gravels, usually bioclastic skeletal sands derived from reef organisms such as coral, coralline algae, molluscs and foraminifera (Stoddart and Steers, 1977; Woodroffe et al., 1999; Yamano et al., 2000; Woodroffe and Morrison, 2001; Woodroffe, 2002; Kench et al., 2005). Such islets are geologically very young, having formed primarily during the late Holocene. Although superficially similar to each other, atolls differ significantly in terms of the number, size, continuity and morphology of atoll-islets. Reef islets on atolls appear to be particularly vulnerable to the impact of rising sealevels due to global warming (Roy and Connell, 1991; Leatherman, 1997; Kayanne, 2000; Kench and Cowell, 2000; Kench and Brander, 2006a); however, the genesis and development of these landforms is poorly understood (Cowell and Kench, 2001; Stephenson and Brander, 2003). The islets are accretionary landforms formed during the past few thousand years particularly as a result of a relative fall of sea-level (Schofield, 1977; Dickinson, 1999). A widely recognized distinction has been made between sand and shingle islets, often termed motu, and found especially on high-energy windward reef margins, and sand cays, found on a wider range of lower energy or leeward margins (Stoddart and Steers, 1977). However, even this differentiation can prove difficult, and mixed shingle and sand islets complicate this dichotomy. 13

33 The morphology of individual reef rims, and the morphodynamics of islets formed upon them, appears to be a function of the climate setting in which the reefs occur (Shimazaki et al., 2004). Several conceptual models of how atoll-islets might have formed were postulated by Richmond (1992), and Woodroffe et al. (1999); the shoreline translation model of barrier-islet or sand barrier formation has been modified to apply to atoll-islets (Cowell and Kench, 2001; Kench and Cowell, 2001). More recently, a new model involving initial mid-holocene building and subsequent late Holocene resilience has been proposed for islets on an atoll in the Maldives (Kench et al., 2005). Each of these models implies a different pattern of atoll-islet formation and behavior, from oceanward progradation, to roll-over, to lagoonward progradation. Radiometric dating of West Islet in the Cocos (Keeling) Islets (Indian Ocean) indicated that the atoll-islet had built predominantly by gradual oceanward progradation over the past 3500 years (Woodroffe et al., 1999). A similar pattern of oceanward progradation was indicated by radiocarbon dating of sediments from Makin islet at the northern end of the Gilbert chain in western Kiribati (Woodroffe and Morrison, 2001). Episodic tropical storms can also have profound erosional and depositional effects on reef-islet geomorphology (McKee, 1959; Bayliss-Smith, 1988; McLean and Woodroffe, 1994; Woodroffe, 2002); Hurricane Bebe added about 10% to the land area on Funafuti when it hit in 1972 (Maragos et al., 1973). Even so, there may be variation in accretionary history between islets or around the margin of the atoll, and observations on one reef platform may not apply to other islets. Recent coral reef studies have identified the need to incorporate sediment dynamics into sediment budgets (Yamano et al., 2000; Kench and Cowell, 2001; Harney and Fletcher, 2003; Kench and McLean, 2004), however, sediment transportation processes are inherently nonlinear and unstable to model over long time-periods (de Vriend, 1993; 14

34 Cowell and Thom, 1994). Here we apply a behavior-oriented approach to the investigation of atoll-islet development on atolls using a numerical model of sediment allocation based on the logistic function. This model translates components of the conceptual models introduced by Bayliss-smith (1988), Richmond (1992), Woodroffe et al. (1999) and Woodroffe (2000) into a computational framework enabling further experimentation with these models. In particular, this model assumes that atoll-islets on atolls approach a state of dynamic equilibrium with the environmental climate regime (Kench and Brander, 2006b; Woodroffe, 2002). The logistic, or population, function has many applications in the biological, medical, and statistical sciences and has characteristics that make it suitable for application to the atoll-islet environment. The equation has been thoroughly studied since its original publication in 1847 (Weisstein, 2003) and is a classic example of a feedback mechanism. The equation is typically used to model growth of a population that has an upper limit called the carrying capacity. This concept is readily applicable to the situation of atoll-islets where islet growth is limited by the accommodation volume, the maximum steady-state volume available for atoll-islet growth, available on the reef flat. The accommodation volume is a simple concept to understand but difficult to explicitly quantify due to our limited understanding of the morphodynamics of atoll-islets on atolls. Physically, the maximum surface area available for islet development is the area on the atoll reef rim itself. The atoll reef rim, however, only represents a physical limit. The true limits to the horizontal growth of the atoll-islet are imposed by the physical processes in the atoll environment such as wind, ocean currents, storm frequency, and waves (Kench and Brander, 2006b). This concept is well known, for example, coastal 15

35 lagoons connected to a tidal inlet (Kragtwijk et al., 2004) and atoll lagoons (Purdy and Gischler, 2005) can effectively be full of sediment, even though they contain a water body, because of equilibrium considerations. For the same reason that it is difficult to assess the limits to an islets physical growth in the horizontal direction, it is also difficult to quantify the limits to vertical growth. Nevertheless, the assumption that islets attain a long-term, steady-state equilibrium (Richmond, 1992) with the climatic regime infers that the accommodation volume can be quantified. There are no studies of atollislets on atolls available that address the issue of whether or not the islet has filled the available accommodation volume. The model presented here incorporates morphodynamic feedback between the atoll-islet and the processes responsible for the sediment deposition. The model does not directly address the spatial issue of assessing the physical limitations to growth due to environmental processes, nor does it explicitly identify the processes involved, these are implicit in the model formulation. Instead, it looks at the outcome of these processes, the volume of sediment that has been accumulated on the atoll-islets: i.e., it uses a behavior-oriented approach (Cowell et al., 2003a). This study uses the measurements of the islet volume at stages in its development to determine the parameters for the model using an inversion algorithm Purpose of the Model Conceptual models of islet growth on atolls have been proposed by Bayliss-smith (1988), Richmond (1992), Woodroffe (2002) based on the assumption that atoll-islets exist in an dynamic equilibrium with the environmental climate. Such an equilibrium has also been proposed for tidal deltas (Kragtwijk et al., 2004) and atoll lagoons (Purdy and Gischler, 2005). Other studies have proposed that atoll-islet building has slowed 16

36 over time (Woodroffe et al., 1999) or stopped (Kench et al., 2005). The model presented here is an implementation in a computational framework that represents atoll-islet development based upon the assumption that islets grow to an equilibrium volume. The model can be used to simulate the volumetric growth of islets on atolls. These islets go through phases of both growth and decay (Bayliss-Smith, 1988; Woodroffe et al., 1999) and this implementation of the model only applies to phases where the islet volume is increasing. It is a lumped-parameter model that averages the growth of the islet with respect to time. This means that the effects of individual events, e.g. cyclones, that can deposit large amounts of sediment onto the reef flat in short time periods, are not examined as individual events, instead, the effects of these events are aggregated over the time-step period represented in the simulation. 2.3 The Sediment Allocation Model The Sediment Allocation Model (SAM) was formulated to simulate the development of atoll-islets using an aggregated-scale, behavior-oriented approach. The model focuses on the accumulation of the materials that form these islets (boulders, gravel, sand and coral grit (Stoddart and Steers, 1977) and assumes that this sediment was produced on the reef (Yamano et al., 2000; Kench and Cowell, 2001; Woodroffe and Morrison, 2001) and can, therefore, be calculated as a function of the reef properties. The SAM is an implementation of a regime morphodynamics model in that it is guided by physical insights that are generally incomplete and rely heavily upon empirical tuning (Cowell and Thom, 1994, p 46). It partitions the sediment into two sinks: Sink 1 is the atoll-islet and Sink 2 is the combined volume of all other sinks such as the reef flat, atoll lagoon and off-reef deposition. The maximum volume of sediment that can be sequestered by the islet is the accommodation volume. The model assumes that the 17

37 amount of sediment sequestered by the islet is a function of morphodynamic feedback between the atoll-islet and sediment transportation processes. Sequestration of sediment by the atoll-islet depends upon how full the accommodation volume is at any given time and, when the volume of the islet equals the accommodation volume, the accommodation volume is full and no further sediment can be sequestered by the islet. The amount of available sediment that can be sequestered by the islet is a function of islet volume, v, and the accommodation volume, V max. A suitably parameterized logistic function, f(v/v max ), was used to allocate sediment produced on the reef to the sediment sinks (Figure 2.1). This sediment allocation model was then implemented in a time-step framework, the forward model of atoll-islet development, and used to simulate the volumetric growth of the atoll-islet (Figure 2.2). Sediment production model Apportion sediment using f(v/v max ) Sink 1: island Sink 2: lagoon/reef flat/ocean Figure 2.1 The Sediment Allocation Model. The model is used to allocate available sediment into two sinks. The amount of available sediment sequestered by the islet is a function, f(v/v max ) with v being the current volume and V max the accommodation volume. Thus, sediment sequestration is a function of how much of the accommodation volume is currently filled with sediment. 18

38 Initialize parameters s, v, V max, steplength, numsteps, i Determine total sediment available for sequestration, s i Allocate sediment to sinks using the SAM Update volumes of Sink1 and Sink2 Increment i i < numsteps True False Finish Figure 2.2 The forward model of islet development on atolls is an implementation of the SAM in a time-step framework. The parameters specific to the forward model are: s, the sediment production rate; v, the current islet volume; V max, the maximum islet volume; numsteps, the number of time steps that the model runs for; steplength, the time period covered by each step; i, the step counter; and s i, the amount of sediment produced at time-step i. The sediment allocation function determines the proportion of the total sediment produced that is deposited in each sediment sink. This function has been implemented using the continuous logistic equation: 1 f ( x) = (2.1) 1 rx + be where x is the independent variable, b=(1/x 0 ) 1, and r is the Malthusian parameter representing the maximum rate of change of the function (Hutchinson, 1978; Weisstein, 2003). 19

39 For this application, the value v/v max was then substituted for x to give f = 1 1+ be (2.2) ( v / Vmax ) rv / Vmax where v is the volume of the islet, V max is the potential accommodation volume available for islet development, b is the sediment transport impedance term representing transport gradients relevant to sequestration of sediments by the islet from the sediment transport field, and r is the transport impedance sensitivity to morphology representing the effect of islet size on the rate of change to islet sequestration of sediments. The values for v, V max, b, and r are all constrained to be positive real numbers. The islet volume v is limited to values in the range 0 v V max and, as a result, v/v max lies in the range 0 v/v max 1. The value v/v max is a measure of how much of the accommodation volume is filled with sediment. In this implementation, f(v/v max ) takes on values in the range 0< f(v/v max ) < 1 and represents the proportion of available sediment that bypasses the islet. The proportion of available sediment that is sequestered by the islet (Sink 1) is, therefore, given by the islet sediment sequestration ratio1-f(v/v max ). The value of the logistic function when no islet is present, i.e. v = 0, is explicitly related to the sediment transport impedance term, b, through the relationship b = (1/x 0 ) 1 (Weisstein, 2003) where x 0 = v 0 /V max. This value, therefore, determines the minimum proportion of the sediment produced by the reef that bypasses the islet sink, Sink 1, and is thus lost to Sink 2 (Figure 2.1). This parameter also acts as a linear feedback coefficient in the sediment allocation function, Equation (2.2). Changes to the value of the parameter b, then, have the effect of altering both the value and the slope of the 20

40 function when v = 0. The physical interpretation of this parameter is that it represents the capacity of the atoll-islet to sequester available sediment. The transport impedance sensitivity to morphology parameter, r, is the nonlinear feedback coefficient in the equation. This parameter reflects the sensitivity of the sediment transport processes to the growth of the atoll islet. A relatively large value for r reflects a high degree of sensitivity to islet development and a low r reflects low sensitivity. The r parameter is, therefore, a measure of the sediment transport system s response to islet morphology. The parameters v, the islet volume, and V max, the accommodation volume, are physical measurements directly related to the atoll islet. The independent variable, v/v max, represents the proportion of the available accommodation volume that is occupied by the islet. Note that this model is only intended to represent the sequestration of sediment by the islet when it is in the growth phase. In this model, then, the islet cannot grow beyond the maximum accommodation available. The islet volume variable, v, lies in the range (0 v V max ) and v/v max lies within the range (0 v/v max 1). We assume that as an islet grows it sequesters less sediment, the residual along-platform transport processes increase in importance compared with cross-platform transport (Kench and Brander, 2006b) and more of the sediment is deposited into the lagoon. Thus, what the model identifies, through the ratio v/v max, is the proportion of the sediment sequestered by the atoll islet. This, in turn, influences the growth rate of the islet and provides a feedback into the sediment transport system. 21

41 The inputs to the simulation model are the sediment production and the accommodation volume of the atoll platform. The atoll-islet is designated Sink 1 and Sink 2 represents all sediment that bypasses the atoll islet. At each timestep, i, a proportion of the total sediment production, s i, bypasses Sink 1, the islet, and is accumulated in Sink 2. The volume of sediment that bypasses Sink 1 is calculated as Sink 2 f / ( v Vmax ) s i V = (2.3) The remaining sediment is accumulated in the islet itself, Sink 1. The volume of sediment that is allocated to the islet at each timestep, i, is calculated as [ 1 f ( v Vmax )] s i V = (2.4) Sink 1 / where 1 f(v/v max ) is the islet sediment sequestration ratio. 2.4 Model Parameters More specifically, 1 - f(0) is the initial ratio of sediment sequestration by islets or protoislets. Initial islet sequestration ratios of 1 - f(0) 1 could be induced, for example, by the presence of a conglomerate platform in the early period of islet formation or rubble deposits after severe storms (Figure 2.3a-d). The antecedent morphology under these conditions creates a topographic barrier to sediment throughput causing high initial sequestration of sediments. Similarly, the antecedent morphology could also be defined by the state of an islet at an arbitrary time in its evolution if the purpose of the model is to simulate subsequent evolution of islet. For high values, 1 - f(0) 1, negative morphodynamic feedback can be expected to involve a decrease in 1 - f(v/v max ) with islet evolution as the topographic adjustment leads to greater sediment bypassing until islet shorelines become steady state transport pathways (Figure 2.3a-d). 22

42 Island sediment sequestration ratio, 1-f (v/v max) b = 0 r > b = 1 r = b = 1 r = b = 10 r = 6 a b c d morphologic time b = 0 r > 0 b = 1 r = 0 b = 1 r = 1 b = 10 r = 4 e f g h Figure 2.3 Evolution of the islet sediment sequestration ratio for different values of b, r and initial conditions: (a-d) initial full interruption of transport field, islet sediment sequestration ratio = 1, (fh) initial fully unimpeded transport field, islet sediment sequestration ratio = 0. The horizontal axis, morphologic time, is in units of time required for full response of the transport system in the absence of morphodynamic feedback; i.e. when r = 0 (e.g. a, b, e, f). When the sequestration ratio is constant there is no morphodynamic feedback. In cases where initial sediment bypassing is high (e.g. due to planar reef surface and uniform flow field associated with tides and wind-generated currents, and negligible ocean wave energy impinging on the reef), initial sequestration ratios are low: 1 - f(0) 0 (Figure 2.3e-h). Islets must form from a topographic kernel, such as rubble deposited by a severe storm, that disturb the uniform transport field (Richmond, 1992; Woodroffe et al., 1999). Morphodynamic feedback involves increasing perturbation in the transport field through time, stronger transport convergence, and a greater rate of islet sequestration of available sediment production. Negative morphodynamic feedback 23

43 and islet stabilization in this case is governed by the accommodation volume and the effect it has on reducing sediment production as inter- and sub-tidal areas are eliminated. Zero transport gradients (b = 0) can be due to lack of transport, no sediment production, or unabated transport across and along reef (Figure 2.3a, 2.3e): e.g., if ocean waves are weak compared to tidal and wind generated currents prevailing on the reef surface. Weak negative gradients in sediment transport (small b) cross and along reefs means that throughput of sediments occurs with little islet accumulation for a given spatially averaged rate of sediment production (Figure 2.3c-d). The model does not deal with positive transport gradients related to islets, and hence islet evolution cannot involve a reduction in volume: i.e., erosion is not possible. Where islet morphology serves to intercept sediments moving on the reef platform, islet sequestration grows through time until attaining a constant proportion with the sediments added to sand aprons (Figure 2.3f-h). When r = 0, islet morphology has no effect on sequestration (Figure 2.3a-b). As r increases, the effect of the increasing size of evolving islets becomes stronger in changes to f(v/v max ) through time, either inducing negative morphodynamic feedback in transport gradients (Figure 2.3b-d), or a stabilization in the sequestration ratio (Figure 2.3f-h) A Hypothetical Atoll-islet The purpose of atoll-islet environment devised for this study was to illustrate the behavior of the SAM and then to test the performance of the parameter estimation 24