THE ROLE OF BODY SIZE IN GENERATING PLANKTON PATCHINESS

Transcription

1 THE ROLE OF BODY SIZE IN GENERATING PLANKTON PATCHINESS Simone McCallum Supervisors Dr Anas Ghadouani Prof Gregory N Ivey Submitted in part completion of Bachelor of Engineering, Environmental, November 2005

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5 Abstract ABSTRACT Plankton communities constitute an essential component of aquatic ecosystems. Plankton influence ecosystem processes such as food web dynamics, the potential yield of fisheries and have a large role in global carbon cycles. The spatial distribution of plankton was historically considered to be homogeneous for convenience in modelling and because heterogeneity had not been considered important. It has subsequently been discovered that the spatially heterogeneous distribution of plankton, or patchiness, is an important part of plankton survival and production. There are many forces that determine the characteristics of the spatial distribution of plankton and create patchiness. Patchiness arises from interacting physical, biological and chemical forces, such as the interaction of ambient flows, birth and death rates, and available nutrients. This project focuses on understanding the role of particle size in the distribution of plankton. We investigate whether the size of plankton compared to turbulent length scales has an effect on the spatial distribution of the plankton. These effects are investigated by observing the behaviour of small plastic particles that are smaller than the largest turbulence length scale to represent plankton in two regimes when particles are either larger or smaller than the smallest turbulent length scale in the velocity field. Particle behaviour is studied in a 61 x 61 x 91cm Plexiglas tank enclosing an oscillating grid that produces homogeneous turbulence with zero mean flow. The turbulent conditions created in the tank can be calculated from the grid frequency and stroke length. Particles are placed in the tank in filtered tap water and photographed at high resolution after they have been stirred sufficiently to capture their horizontal spatial distribution. To make a measure of particle patchiness, each image is processed to determine the distance between each pair of particles. Processing involves reducing noise, then finding the centroid of each particle to determine their relative positions from which the appropriate distances can be calculated. These distances form a distribution, which for completely random particles would be a Normal distribution. Particle separation distributions from different combinations of particle size and turbulent length scale are compared to determine whether there is any difference between regimes defined by relative sizes of particles and Kolmogorov scale. The spatial distribution of particles is found to be independent of Kolmogorov scale when the particle diameter is smaller than the Kolmogorov scale. However, when the particle diameter is similar to or larger than the Kolmogorov scale, the spatial distribution of particles is dependent on the Kolmogorov and Taylor length scales. i

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7 Acknowledgements ACKNOWLEDGEMENTS Any project such as this requires varied support from many others for their expertise, experience, equipment or encouragement. I take this opportunity to thank those who have helped me in the beginning, during and eventually in finishing this project. Firstly Anas Ghadouani and Greg Ivey for working together on an unusual project like this one, their enthusiasm, technical insight, and confidence in my abilities. Peter Kovesi for some very quick, short notice help with some code for processing and analysing the photos. Frank Tan, for some upgrades to the grid stirred tank, happy to do some as soon as possible jobs and demonstrating that an experiment can be held together with G-clamps. Leon Boegman, Geoff Wake, and Kenny Lim for help and instruction for the work with cameras, tanks and the lab in general. Ross for feeding me biscuits and cups of tea while this was being written. Andy and Jim for putting up with me at home or not at home. All the kids in my class in the CWR for their empathy at all stages of this project, and all the fun times that are had in a small class like this these days will be missed. iii

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9 Contents CONTENTS Abstract... i Acknowledgements... iii Contents...v List of Figures... vi List of Tables... vi Chapter Introduction...1 Chapter Literature Review Patchiness Theory Measuring Patchiness...9 Chapter Methods Experimental Apparatus Analysis...17 Chapter Results...23 Chapter Discussion Characterising Spatial Distributions Interpretation of Results Plankton Distributions Limitations of the Study...31 Chapter Conclusions Summary Future work...34 Appendix A...35 References...37 v

10 List of Figures LIST OF FIGURES Figure 1.1 Temperature, chlorophyll a concentration, and zooplankton abundance along a latitudinal transect in the North Sea in spring (redrawn from Levin 1992)... 3 Figure 3.1 Schematic diagram of the experimental apparatus Figure 3.2 Sketch of oscillating grid in plan view Figure 3.3 Section of tank viewed from the right hand side of that in Figure 3.1, showing the image capture arrangement Figure 3.4 Sample from experiment 1 to demonstrate image processing Figure 3.5 Finding the centres of particles using nonmaxsuppts Figure 3.6 Normal fit to separation distance histogram Figure 4.1 Mean particle separation vs. Kolmogorov microscale Figure 4.2 Mean particle separation vs. Taylor microscale Figure 4.3 Mean particle separation vs. Integral length scale Figure 4.4 Correlation of mean particle separation and standard deviation Figure 5.1 Sketch of energy spectrum of turbulence LIST OF TABLES Table 3.1 Tank and grid specifications Table 3.2 Specifications of fourteen experiments performed Table 5.1 Characteristic, and typical plankton sizes and turbulent conditions in experiments and lakes and ocean surface mixing layers vi

11 Chapter 1 Introduction CHAPTER 1 1Introduction The word plankton is derived from the Greek planktos, meaning wandering. It is used to describe the small, usually immotile, freely floating organisms living in aquatic habitats (Powell et al. 1975). Plankton drive energy cycling in aquatic ecosystems as they are the productive base of food webs, converting basic forms of energy into forms usable by higher trophic levels (Vilar et al. 2003). Phytoplankton are small forms of plant life, their photosynthetic activity converts the energy in solar radiation and carbon dioxide into forms of carbon other organisms can utilise. The animal plankton, zooplankton, are primary consumers of phytoplankton along with some small fish, before they are preyed upon by higher order organisms. Due to their fundamental role in aquatic ecosystems, and their consumption of carbon dioxide, plankton determine the survival of all other aquatic organisms, and along with terrestrial primary producers, are drivers of global carbon cycles (Daly & Smith Jr 1993; Vilar et al. 2003). Plankton, comprising 30% of global primary production (Daly & Smith Jr 1993) are of great interest, and an active area of study as they determine the success of fisheries and affect global biogeochemical cycles and climate. Plankton research aims to determine the role and relative importance of various environmental parameters that affect plankton to be able to model and predict plankton productivity, behaviour and dynamics. Plankton growth and dynamics depends on the characteristics of their environment light and nutrient availability, temperature, salinity, ph, currents, turbulence, and predation intensity. The survival of higher order organisms depends on whether plankton have suitable environmental conditions for growth as well as on the behaviour and dynamics of plankton individuals and populations. Ecosystem models take an approach that simplifies ecosystem components and interactions, attempting to incorporate enough information to produce results similar to observations without including the vast and complex detail present in ecosystems in nature (Levin 1992). The heterogenous distribution of terrestrial organisms has been a topic of interest for some time, particularly in those plants and animals where spatial structure can be easily identified. Plankton distributions however, have historically been modelled as being homogeneous. There are two main reasons for this, firstly the small size, lack of motility, and apparent lack of social behaviour of individual planktonic organisms do not intuitively suggest the formation of structure (Pinel-Alloul et al. 1988). Secondly, for convenience or simplicity in developing models heterogeneity has been omitted, considered an unnecessary complication in equations or calculations (Pickett & Cadenasso 1995). The heterogeneous nature of plankton distributions had been observed as early as the 1930 s in 1

12 Chapter 1 Introduction the different collections of zooplankton found in net tows on opposite sides of a ship. This heterogeneity has subsequently been found to exhibit non-random structure and found to be significant in plankton survival and productivity (Avois-Jacquet & Legendre submitted; Denman & Dower 2001; Pickett & Cadenasso 1995). The existence and significance of heterogeneous plankton distributions is emphasised in laboratory studies that show that a greater abundance of homogeneously distributed prey is necessary to support predators than average concentrations measured in lakes and oceans (Davis et al. 1992; Denman & Dower 2001). An understanding of the generation, characteristics, and consequences of the spatial distribution of plankton is required to understand aquatic ecosystems. Spatially heterogeneous plankton distributions are often referred to as patchy distributions, and due to the effects of patchiness on plankton productivity, plankton patchiness has been an active field of study in aquatic ecology (Goldberg et al. 1997; Martin 2003). A patchy distribution is most broadly defined as any distribution of organisms that is neither uniform or random (Elliott 1977; Vandermeer 1981). Characteristic length scales of patchiness can range from millimetres to kilometres, and patches may persist on time scales from seconds to months. The spatial distribution of plankton is affected by a wide range of biological, chemical, and physical forces, it is crucial to ecosystem modelling to determine which particular forces dominate at particular time and length scales and when transitions occur (Daly & Smith Jr 1993; Denman & Dower 2001). An example of chemical forcing is the iron fertilisation used by both Abraham et al. (2000) and Boyd et al. (2000) to stimulate phytoplankton growth for the study of large scale structures. Biologically, zooplankton diel vertical migration, predator avoidance, finding food, and mating control zooplankton patchiness, and grazing and growth are the main biological controls on phytoplankton patchiness (Folt & Burns 1999; Pinel-Alloul 1995; Price 1989). Physically, plankton are observed to be closely coupled to the movement (as a result of their relative immotility) and temperature of the water (Daly & Smith Jr 1993). There are many more specific parameters that generate or affect the generation of particular spatial distributions. To understand and manage aquatic ecosystems, plankton patchiness and the forces that drive patch formation and characteristics on the various time and length scales on which it exists need to be understood. There have been many studies relating to plankton patchiness, yet due to its complex nature, a clear understanding of the forces that drive plankton spatial distributions has not been developed (Vilar et al. 2003). The main difficulty in characterising the causes and controls on plankton patchiness is the wide range of scales on which patches occur and the different dominant forces on different scales. In studying patchiness, the scales at which observations are made will affect the results obtained (Folt & Burns 1999; Levasseur et al. 1983; Levin 1992). This again complicates the understanding of patchiness since the observer may not resolve forces or spatial structure outside the observational 2

13 Chapter 1 Introduction scales. The extent of current understanding of plankton patchiness has been limited by observational capabilities. The development of further knowledge of patchiness requires observations of the processes that describe plankton behaviour on a population level, with resolution down to the individual (Denman & Dower 2001). Such observations may soon be possible due to recent advances in technology that allow for more detailed observations of plankton and their dynamics (Martin 2003). Figure 1.1 Temperature, chlorophyll a concentration, and zooplankton abundance along a latitudinal transect in the North Sea in spring (redrawn from Levin 1992). Chlorophyll a concentration is a measure of phytoplankton abundance. The solid line box indicates an area where phytoplankton abundance is relatively high while zooplankton abundance is relatively low. The dashed line box indicates an area where phytoplankton abundance is relatively high and coincides with a relative peak in zooplankton abundance. An interesting feature of the spatial distribution of plankton is that zooplankton distributions are observed to have a finer scale structure than those of phytoplankton in the same area (Denman & Dower 2001; Martin 2003). Observed phytoplankton distributions have a similar structure to physical quantities such as sea surface temperature, both of which have little variability at short length scales, in the same area where zooplankton abundance varies at both short and long length scales (Abraham 1998). A commonly cited example is the data from Mackas (1977) dissertation which highlights the difference between phytoplankton and zooplankton spatial structure along the same transect, reproduced in Figure 1.1 (Levin 1992; Mackas & Boyd 1979; Vilar et al. 2003). The zooplankton abundance has features on a similar length scale as the phytoplankton, as well as on a finer scale, 3

14 Chapter 1 Introduction giving higher frequency variability the zooplankton distribution. It is also apparent that zooplankton abundance does not necessarily coincide with phytoplankton abundance zooplankton abundance may be high where phytoplankton abundance is low (Figure 1.1). Pinel-Alloul (1988) finds that the spatial distribution of zooplankton depends on their body size while being weakly dependent on species, this indicates that physical body size dominates the behavioural differences that may be associated with body size in controlling the level of heterogeneity exhibited in zooplankton distributions. The observation of different spatial distributions of phytoplankton and zooplankton and the idea that body size may affect spatial distributions formed by organisms has motivated this project. It is the thesis of this project that the body size of plankton has a role in generating plankton patchiness and may account for the difference between zooplankton and phytoplankton distributions given that in general, phytoplankton body size is significantly smaller than that of zooplankton. The central aim of this project is to determine the role of body size in determining the characteristics of plankton spatial distributions. This specific topic is motivated by the differences in the spatial distributions of phytoplankton and zooplankton observed in nature, which suggests a fundamental difference in the way the two types of plankton form spatial patterns. A laboratory experiment is developed using turbulently mixed water to simulate turbulent conditions in the upper layers of lakes or oceans, and small particles to represent plankton. Representative particles are used to isolate body size from biological or chemical influences that would otherwise affect plankton in nature. A zero mean flow, near isotropic, steady turbulent field is used to investigate the interaction between particles and turbulent eddies uncomplicated by temporal or spatial variation in the flow. The experiment is intended to generate particle distributions that reflect the effects of particle size and particle interactions with turbulent eddies only. The spatial distributions of particles in different turbulent conditions are compared to determine the way spatial patterns formed change. The methods used for capturing and measuring plankton distributions are considered for their applicability for use in field observations of plankton. 4

15 Chapter 2 Literature Review CHAPTER 2 2Literature Review The spatial and temporal patterning of organisms and the associated length and time scales is a central problem in ecology (Avois-Jacquet & Legendre submitted; Levin 1992; Pickett & Cadenasso 1995). There is a wide range of literature regarding the spatial distribution of both terrestrial and aquatic organisms. The way patchiness is defined in ecology is largely based on the spatial distributions exhibited by larger organisms that were initially observed to form patterns. Plankton patchiness has become an area of growing interest due to the significance of plankton to aquatic ecosystems and energy cycling. Plankton spatial distributions are complex; they involve contributions from many different forcing mechanisms in biological, chemical and physical forms. Complexity is also found in the length and time scales on which patchiness is found, and the different time and length scales on which the forces that affect plankton spatial distributions act. The majority of plakton patchiness theories involve modelling the interaction between some select forces to determine whether their interaction will generate patchy distributions. Many studies are based on describing plankton simply as a diffusive tracer in a fluid including a mathematical description of physical conditions, such as temperature, or plankton growth. Some studies include zooplankton grazing on phytoplankton populations, later studies explicitly examine the distribution of zooplankton and phytoplankton (Abraham 1998). There are few mechanistic field or laboratory studies that examine the generation of patchiness, one being that of Price (1989) that examines the interaction of krill with algal patches. 2.1 Patchiness Theory Patchy plankton distributions were initially considered as collections of coherent circular patches. This early view of patchiness lead to the development of models such as those of Kierstead and Slobodkin (1953) and Skellam (1951). Kierstead and Slobodkin frame their model in the idea that a phytoplankton population will exist in a water mass that has suitable temperature, light, and nutrient conditions for phytoplankton growth. If water masses such as this are surrounded by water unsuitable for phytoplankton growth or highly dispersive flow conditions, there should be a minimum size of water mass that can support continued phytoplankton growth. The model represents phytoplankton as a reactive tracer. The phytoplankton concentration dynamics are defined by a simple differential equation that states the change in phytoplankton concentration with time is controlled by two opposing forces; a phytoplankton growth term to increase the population and dispersion by physical currents to move the phytoplankton from the suitable water mass (Kierstead & Slobodkin 1953). Phytoplankton growth is incorporated as a net or effective growth rate that accounts for growth, respiration, and 5

16 Chapter 2 Literature Review predation. Dispersion by physical currents and turbulence are incorporated as an effective diffusivity resulting in the reaction-diffusion differential equation, P = μ P + κ 2 P (2.1) t where P is the phytoplankton concentration, t time, μ effective growth rate, and κ effective diffusivity. The equation states that the change in a phytoplankton concentration at a point in space with respect to time is equal to the sum of the effects of population growth and the diffusion of phytoplankton away from a region of high concentration. By finding the solutions of (2.1), the simple result is that for a circular patch to survive, it must have a radius greater than the critical value, r = 2.4 κ μ. This model depends on specific environmental conditions and does not represent the range of variability observed in plankton distributions in nature, but the work of Kierstead and Slobodkin (1953) and Skellam (1951) are some of the first on plankton patchiness and have provided a basis for further research. The plankton distributions described by a reaction diffusion model like this represent the averaged behaviour of plankton. This is a direct result of using averaged parameters the effective diffusion and growth. Representing plankton as a reactive tracer restricts the interpretation of results to large scale plankton distributions. This contrasts with the use of a diffusive term used to describe the physical forcing of the water since diffusion does not well represent larger scale fluid motions of advection and stirring. Both the growth and diffusion terms have been altered in subsequent research, the growth being explicitly augmented by zooplankton grazing and the diffusion term altered to account for the change in dispersive effects on large length scales. All these patchiness theories are highly sensitive to the choice of model used to represent growth and the parameters used to represent the physical forcing (Martin 2003). These models represent an averaged behaviour that can be used to approximate plankton distribution characteristics if growth and flow conditions can be approximated by those used in the model. In the theory of Kierstead and Slobodkin (1953) and Skellam (1951), patchiness is generated by correlation with pattern in environmental conditions. Levin and Segel (1976) developed a theory based on the idea that the interaction between zooplankton and phytoplankton can generate patchiness without requiring environmental heterogeneity. The idea originates in the theory of Turing (1952) to explain morphogenesis, the formation of patterns and structures in organisms. Levin and Segel (1976) develop a model in which zooplanktonic herbivores having a higher rate of dispersion than phytoplankton combined with lower grazing efficiency in increased densities of phytoplankton generates patchy distributions. The model responds to an infinitesimal positive perturbation in phytoplankton population density and propagates that perturbation to form a patch by the action of the zooplankton diffusing away from the phytoplankton patch due to their high motility and low within 6

17 Chapter 2 Literature Review patch grazing efficiency. This mechanism of patch generation tends to limit the spread of phytoplankton patches due to relatively high zooplankton numbers outside the patch (Brentnall et al. 2003), which results in the phytoplankton distribution being more highly variable than that of the zooplankton, the opposite of which is observed in nature. Brentnall et al. (2003) use a reaction diffusion equation similar to (2.1), modified to simultaneously describe zooplankton and phytoplankton concentrations. Small increases in phytoplankton density are allowed to propagate due to a delayed response of grazing zooplankton. The studies of Levin and Segel (1976) and Brentnall et al. (2003) indicate that interactions between zooplankton and phytoplankton populations can act to generate patchiness in their distributions Young, Roberts and Stuhne (2001) explore another combination of processes and the generation of patchy distributions caused by the interactions between them. This model is different from those already discussed as it retains resolution of planktonic individuals by representing them as separate particles rather than as a concentration of a reactive tracer. A numerical simulation of a field of individuals with periodic boundary conditions to avoid edge effects, and an initial Poisson distribution of individuals is generated. A number of operations are performed at each time step. The movement of each individual is that of a random walk, and are appropriately termed brownian bugs, so the movement at each time step is in a random direction. Randomly distributed births and deaths occur, and advective stirring is imposed. Patchy distributions form after a number of time steps, becoming more clustered with increasing time. The patchiness arises due to the condition that the birth of an individual must occur adjacent to an existing individual. The use of discrete points to represent plankton individuals rather than using reactive tracer models allows for a smaller scale study of plankton behaviour. More recently, interest in the difference between phytoplankton and zooplankton distributions has developed. (Abraham 1998), uses a numerical description of plankton concentration, zooplankton abundance, a carrying capacity (a description of the capacity of physical and nutrient conditions to support phytoplankton growth) and advective stirring. Phytoplankton and zooplankton are assigned growth rate, zooplankton maturation time, grazing rate, and mortality rates. This results in a phytoplankton distribution that follows closely the carrying capacity and a zooplankton distribution that exhibits a much finer grained variability. The resulting distributions of phytoplankton and zooplankton give a good agreement with the spectrum of variability in observed distributions. The wider range of length scales of variability in the zooplankton distribution is found to be caused by the maturation time of the zooplankton, and phytoplankton structure produced by zooplankton grazing. This model describes phytoplankton and zooplankton dynamics well on kilometre length scales. 7

18 Chapter 2 Literature Review The complexity of plankton patchiness is highlighted by the number of different models using different combinations of forces which all give rise to patchy formations. The descriptions of plankton that use concentrations to describe plankton can be used as a predictive tool for large scale plankton distributions, remembering that the assumptions of averaged plankton and fluid flows must be representative of real flows and the results must be interpreted as representing average behaviour. It has been observed that numerical modelling results seem to indicate that even very minor changes in parameter values lead to vastly different spatial distributions (Martin 2003; Powell & Okubo 1994). As may be expected, as general descriptors of plankton dynamics, newer models are generally better as they have been improved to include more accurate descriptions of physical flows, describing stirring and advection rather than diffusive behaviour. The applicability of a model depends on the time and length scales of interest, most studies of plankton dynamics are on long time and length scales to enable the use of analytical methods and averaged behaviour of plankton and physical flows. Small scale models are less common since they are difficult to describe analytically, but patchiness on these scales need to be understood since aggregates of plankton individuals interact with forces on larger length scales to form large scale patterning. A smaller number of studies have focused on the forces that affect the generation of plankton patchiness on short time and length scales. One such study is that of Price (1989) that examines the generation of patchiness in krill populations in the presence of patches of algae. The study was initiated to determine the relative importance of physical and behavioural controls on the distribution of zooplankton. Determining the controls on zooplankton behaviour with respect to phytoplankton distributions will allow for more accurate modelling of energy cycling than the current understanding that assumes mean field phytoplankton distributions. Price finds that the krill density within a particular area increases after a patch of algae is introduced due to alteration of swimming behaviour. Swimming paths become more horizontal, speed increases, and sinking bouts mostly eliminated. Laboratory studies such as this can be used to determine the forces that dominate in the generation of small scale plankton patchiness, and develop sampling techniques that allow for the observation of the formation and dynamics of plankton patches in nature. Observations of plankton in controlled conditions or detailed observations of plankton in the field are necessary to determine those forces that dominate the formation of patchiness on different time and length scales. These observations will dictate the time and length scales on which particular predictive models of plankton behaviour are appropriate, and indicate the boundaries of regimes within which new models of plankton dynamics should be focused. Mathematical models of forces that generate patchy distributions are useful to suggest mechanisms that might generate plankton patchiness, but these need testing in the field or laboratory to determine whether the forces suggested are dominant in 8

19 Chapter 2 Literature Review plankton dynamics. There is a lack of laboratory based mechanistic studies of the forces involved in generating plankton patchiness. A comprehensive series of laboratory studies that isolate different features of plankton patchiness and the forces that are involved in the generation of patches would be instrumental in determining the relative importance of various forces in the generation of patchiness and determine the time and length scales on which those forces are important. With this information more powerful predictive models of plankton dynamics and a better understanding of energy cycling in aquatic ecosystems would be possible. 2.2 Measuring Patchiness A number of methods have been developed to measure patchy distributions of organisms in a two dimensional field. Most measures of patchiness have been developed for application to quadrat counts, symptomatic of their development for use in terrestrial environments, particularly for examining the spatial distribution of plants. These methods are transferable to plankton distributions if their abundance is measured as counts (not concentrations) and if quadrat counts are taken on a short time scale relative to those characteristic of plankton movements. The following methods for measuring the patchiness of the spatial distribution of organisms are presented in order to establish an idea of the type of measurements taken to characterise spatial distributions and to present the strengths and weaknesses of each. In addition, these measures are used to develop a measure of spatial distribution to be used in this project Variance to mean ratio The variance to mean ratio as a measure of patchiness is based on a comparison to a random distribution. A distribution that can be described by the Poisson distribution will have a variance equal to the mean and this indicates that the distribution is random. The variance to mean ratio (I) is given by 2 I = s / x, (2.2) where x is the mean of counts x 1 to x n, of individuals in each sample, and s 2 the variance between those samples. Three critical values of the variance to mean ratio indicate the type of distribution present. I = 1 indicates a Poisson distribution, I < 0 indicates a more uniform distribution, and I > 1, a patchy distribution. The method is simple but I is dependent on the number of individuals in a sample, so it is only useful for comparison between samples if the number of individuals in each sample is the same (Elliott 1977). In addition, the variance to mean ratio is dependent on the quadrat size, the choice of quadrat size may determine whether a distribution is found to be patchy or not (Elliott 1977). 9

20 Chapter Lloyd s mean crowding Literature Review Lloyd s index measures the mean number of individuals lying within some radius of each individual. Sampling uses a grid will a cell size that is characteristic of the individual s ambit to represent the range over which the individual will be influenced by other individuals and the surrounding environment. Lloyd s mean crowding, individuals in the same quadrat, m * = N N X i * m is measured as the mean number, per individual, of other, (2.3) where X i is the number of other individuals in the same quadrat as individual i (Lloyd 1967). Mean crowding, * m is compared to the mean density of the sample and can give the result that the * m distribution is patchy if is greater then the mean density. Lloyd s index, like the variance to mean ratio is dependent on the grid cell size chosen Morisita s index Morisita s index, I d, is calculated as the probability that the individuals in a pair drawn from the total sample are from the same quadrat divided by that same probability for a randomly distributed population (Vandermeer 1981). The quadrats are arranged in a grid formation. Q X ( X i i 1) I d = Q, (2.4) N( N 1) where Q is the number of quadrats used to sample the population, and N is the total number of individuals sampled. The index has critical values where I d = 1 [(n 1) / (Σx 1)] indicates a uniform distribution, I d = 1, a random distribution, and I d = N, a patchy distribution. This form of the index is dependent on the sample size, and usually gives similar results to Lloyd s mean crowding (Vandermeer 1981). A modified version of the index, the standardised Morisita s index is more powerful and independent of sample size. The index is calculated as usual, and then two significance points are calculated using a chi-squared distribution as 2 n M = χ u 1 ( x) x, (2.5) which indicates a uniform distribution of individuals, and 10

21 Chapter 2 2 n M = χ c 1 ( x) Literature Review x, (2.6) which indicates a patchy, or clumped distribution of individuals, and χ j 2 is the value of chi squared with n 1 degrees of freedom with proportion j of area to the right. The standardised index is then calculated from one of the following: I M >1.0, d c M I 1.0, c > d I d M c I = p (2.7) M c 1 I d 1 I = p 0.5 (2.8) M c > I d > M u, 1.0 > M u > I d, I d 1 I = p 0.5 (2.9) M u 1 I d M u I = p (2.10) M u This standardised index indicates a random distribution when I p =0, a more patchy distribution when I p >0, and a more uniform when I p <0. The index is more powerful because it indicates patchiness or uniformity with 95% confidence for I p =±0.5 (Krembs et al. 1998; Vandermeer 1981) Areal scale The three indices discussed so far can only effectively characterise a distribution as either uniform, random or patchy. A more useful measure is one that can distinguish between different degrees of patchiness. Using one of the measures already discussed the areal scale on which a population is most patchy can be determined. Patchy distributions of individuals are indicated by large index values. For the variance to mean ratio and Lloyds mean crowding, if the index value is plotted against quadrat size, a maximum in the plot will indicate the areal scale at which quadrat size approximates patch (Elliott 1977) Nearest Neighbour Measures Measurement of nearest neighbour distances is useful for when the exact location of each individual in a sample is known. Nearest neighbour distance information can give a more powerful description of a spatial distribution since it uses the maximum amount of spatial information available. A commonly used technique is to calculate the average nearest neighbour distance of a distribution, 11

22 Chapter 2 Literature Review ri N r = N, (2.11) along with the value of this average if the individuals were distributed randomly 1 r random =, (2.12) 2 N / A where N is the number of individuals, i is the ith individual, and A is the sample area. Given these, the ratio r rrandom can be used as a measure of patchiness. A ratio of one, indicates a random distribution, a patchy distribution will give a ratio of less than one, a lower ratio indicating a more patchy distribution (Vandermeer 1981). This ratio as a measure of patchiness is not as strongly dependent on sample size. Additionally, it will detect patchiness within a sample regardless of the characteristic length scale of that patchiness. The techniques for measuring patchiness are all indices that reveal whether a distribution is random, patchy or uniform. Generally, they are not formulated to measure a characteristic of the spatial distribution further than to make a distinction between those three types. Also, in general the indices cannot be used for comparison between populations or different distributions because they are dependent on the sampling size used. Comparisons made must be of samples of similar size and they may only be a qualitative statement of relative patchiness, uniformity or randomness. Using the technique of the areal scale, an approximation of patch size can be found, a one dimensional analogy of which is the average nearest neighbour distance. The areal scale measure is useful for comparing the spatial distribution of highly clumped patterns; distributions that are less clumped, but still have spatial structure will not yield a useful areal scale of patchiness. The measures of patchiness discussed are useful as a qualitative comparison of different spatial distributions of individuals. Existing studies of plankton patchiness are generally focused on large scale analytical or numerical models of spatial patterns. The small scale structure of plankton distributions is a topic that requires understanding for the implications this has on plankton interactions with larger scale forces and the resulting population structures. A distinct difference between the spectral form of phytoplankton and zooplankton distributions is observed in nature, zooplankton populations exhibiting a much finer scale structure. This difference motivates the study of the mechanisms that might generate such a difference. It is suggested that the general difference in body size between zooplankton and phytoplankton will contribute to the formation of decoupled small scale patterns in plankton populations. A method to characterise the spatial distribution of individuals that does not require a clumped distribution as such is required to study this disparity. The nearest neighbour measure is an appropriate one, since a sample grid is not required, the grid size of which for the other methods must reflect a clump size. 12

23 Chapter 3 Methods CHAPTER 3 3Methods An experiment is run in controlled conditions in the laboratory to examine the role of plankton body size relative to characteristic length scales of turbulence in generating plankton spatial distributions. To isolate the effect of body size from biological and chemical interactions, inanimate representative particles are used in the place of live plankton. Turbulence is isolated from other physical forces and generated in a simple form, using a technique that generates steady, horizontally isotropic turbulence and zero mean flow. Temporal variability is not considered here, the experiment is isolated from time dependence by the use of short time scale of sampling and steady turbulent fields. Images of the horizontal distribution of the turbulently mixed particles are captured for analysis of the characteristics of the spatial distributions formed. This chapter explains the experimental apparatus, analysis techniques, and the experiments performed to explore the effect of body size relative to the length scales that characterise the turbulent field. 3.1 Experimental Apparatus drive shaft Perspex box to prevent surfaces wave interfering with light light plane oscillating grid Figure 3.1 Schematic diagram of the experimental apparatus. Tank dimensions: 61 61cm square base, height 91cm, grid positioned 17.5cm from tank base, light plane 20.5cm from grid position. 13

24 Chapter Particles Methods Pliolite particles are used to represent plankton. These particles are used for their reflectivity and neutral buoyancy. The high reflectivity makes it possible to obtain clear images of particle distributions when illuminated in a plane. Their neutral buoyancy means sinking or floating will not contribute to their dynamics. The pliolite particles are a polymer manufactured for use in tyres by Goodyear and are used to trace velocity fields in turbulent flows (Coates & Ivey 1997). The pliolite particles are obtained in a mixture of particles that range in size from microns to millimetres. The pliolite particles are sieved to select for those in the range 212 to 300 microns as a representative plankton body size, also a suitable size for imaging. This particle size will also be suitable to represent particles smaller and larger than the smallest length scales of turbulence obtainable in the tank. Pliolite particles are immersed in a tank of water with the aid of a surfactant; Photo-Flo, usually used in developing photographic film, is used in this experiment Turbulence Turbulent mixing is generated in a square based Plexiglas tank by a vertically oscillating, horizontal grid (Figure 3.1). Steady turbulence is produced by the oscillating grid which is isotropic in the horizontal plane and generates zero shear. Parameters that characterise the turbulent field can be calculated using known grid parameters (De Silva & Fernando 1994). The tank has dimensions cm and is filled with 281 litres of filtered tap water. The grid, positioned at a mean distance of 17.5 cm from the bottom of the tank consists of 1 cm square bars each 5 cm apart. M Figure 3.2 Sketch of oscillating grid in plan view. M is the mesh size. Open end conditions for the grid reduce stress gradients at the grid-wall interface (Fernando & De Silva 1993). To produce isotropic turbulence the oscillating grid is required to have a solidity of less than 40% (the area of the bars should be at less than 40% of the total grid area), the frequency of oscillation should be less than 7Hz, and the edges of the grid should follow the design in Figure 3.2 (Fernando & De 14

25 Chapter 3 Methods Silva 1993). The tank and grid used by O Brien et al. (2004), which satisfies the conditions on the grid dimensions is used in this experiment. The characteristics of turbulence can be calculated using measurements of mesh size (M) (Figure 3.2) and stroke (S), the total vertical excursion of the grid, and frequency (f) of oscillation. The grid stroke length and frequency of oscillation are varied between experiments to vary the turbulent conditions and hence the relative magnitude of Kolmogorov scale and particle size. Table 3.1 Tank and grid specifications. C 1, C 2, and C 3 from O Brien (2004). Mesh size 6 cm Stroke range mm Frequency range 1 7 Hz C ± 0.04 C ± 0.05 C Solidity 33% z 205 mm Through extensive study of oscillating grid induced turbulence, it has been found that the variations of the horizontal and vertical root mean square velocities at a distance z from the mean position of the oscillating grid can be expressed as (De Silva & Fernando 1994): 3/ 2 1/ 2 1 u 0 = C1S M fz, (3.1) 3/ 2 1/ 2 1 w 0 = C2S M fz, (3.2) and the integral length scale, a length scale which represents the size of the largest eddies in turbulent flow as: l 3 0 = C z. (3.3) Furthermore, the energy dissipation rate the rate of dissipation of kinetic energy from the turbulent eddies: 3/ / 2 9 / 2 3 u 1 2C1 + C 2 M S f ε = =. (3.4) 4 l0 C 3 3 z The smallest length scale of turbulent eddies is formulated by considering that the turbulent kinetic energy is eventually lost at a small length scale where viscous forces dominate. This length scale is known as the Kolmogorov microscale, 1/ 4 3 ν η =. (3.5) ε 15

26 Chapter 3 Methods An intermediate length scale, the Taylor scale (λ T ) represents the cascade of turbulent eddies (Tennekes & Lumley 1972), and is defined by ν λ = 15 T u ε. (3.6) To ensure the turbulence is fully established, measurements are taken more than three mesh sizes away from the grid, and after 10 minutes of stirring (De Silva & Fernando 1994) Imaging planar mirror camera halogen lights Figure 3.3 Section of tank viewed from the right hand side of that in Figure 3.1, showing the image capture arrangement. Light plane is situated 20.5cm from the grid, the mirror is 105cm above the light plane, and camera a total of 153cm from the light plane. A thin plane of light to illuminate the particles is generated by twelve 50W halogen globes positioned behind a slit inside a metal casing (Figure 3.3). The light plane is positioned 38cm from the bottom of the tank, and 20.5cm from the mean position of the oscillating grid (Figure 3.1). Light reflected upwards from the particles is then reflected by a mirror to a high resolution digital still camera to record the particle distribution in the tank (Figure 3.3). Before mixing in particles, a Perspex sheet marked with a 2cm grid is used to focus the camera lens in the light plane, and to generate a calibration image to scale from pixels to metric units. Images are taken in a 2000 by 3008 pixel area, which after calibration converts to a 133.5mm by 200.8mm field of view. A small Perspex box of dimensions cm is positioned to prevent surface waves from interfering with light from 16

27 Chapter 3 Methods particles in the field of view of the camera (Figure 3.1, Figure 3.3). Particles are stirred in before starting the grid oscillating. Before taking any images, ten minutes is allowed after turning on the grid to allow the turbulence to fully establish (De Silva & Fernando 1994). Ten images are then taken, to produce ten samples of the particle distribution in each experiment. The experiment is repeated for fourteen combinations of oscillation stroke and frequency (Table 3.2). Table 3.2 Specifications of fourteen experiments performed. Experiments using stroke S, frequency f, to produce energy dissipation rate ε, Kolmogorov scale η, and Taylor length scale λ T. experiment S (cm) f (Hz) ε (m 2 s -3 ) η (mm) λ T (mm) Analysis Patchiness Measure A new method of characterising the spatial distribution of particles is used to compare the spatial distribution of particles between separate runs of the experiment. This method is based on the separation distance between each pair of particles in the field of view. These separation distances are binned to form a histogram that will reveal length scales that characterise the spatial distribution of particles. For example, a clumpy distribution of particles will exhibit a peak in the frequency of separation distances at a length scale that characterises the patch size and other peaks that characterise inter-patch distances. Distinctly clumpy distributions are not expected to be generated in the turbulent field generated in these experiments due to its uniformity. Rather than identifying a clump size, particle distributions will be characterised by fitting the distribution of distances to a normal distribution, and the mean and standard deviation used as parameters to characterise each particle distribution. 17

28 Chapter 3 Methods The mean of the separation distances distribution indicates the average separation distance between each pair of particles in the distribution. The standard deviation of the separation distances distribution indicates the spread of the separation distances from the mean. A small standard deviation would indicate that most pairs of particles are separated by a distance close to the mean; a large standard deviation indicates that the separation distance between pairs of particles varies over distances significantly smaller and larger than the mean. The length scales of pattern this method will be able to resolve is limited by the length scale proportional to the square root of the sample area. The mean and standard deviation are limited at the lower and upper by zero and the size of the sample area respectively. a) b) c) d) e) Figure 3.4 Sample from experiment 1 to demonstrate image processing. Circles indicate a point where a spec has been eliminated by the procedure. a) raw image; b) greyscale image; c) black and white image after threshold application d) black and white eroded image; e) image used to assign position vectors to each particle. 18

29 Chapter 3 Methods This method requires a significant number of particles to be in the sample space to permit the use of a fit to the normal distribution. It is more appropriate for this experiment than the other methods of making measurements of spatial distributions discussed as it does not require the prior identification of clumps or knowledge of an expected clump size. Rather than indicating the length scale of clumps in the distribution, this method describes the distribution by the distances between individuals, and the variation of those distances from their mean. To find the particle separation distributions, the digital images of particle distributions are processed using some simple software to determine the relative position of each particle in each image Image Processing Images are first processed to reduce noise, that is light reflected from the grid stirrer, the bottom and sides of the tank and particles that are only partially in the light plane. A function executed using Matlab is used to process the images and generate the measure of patchiness outlined above (Appendix A). The particles are the brightest objects in the field of view, taking the threshold will eliminate some of the partially lit particles and any reflections from the tank and grid. The images are first converted from colour to a grey scale (Figure 3.4a, b). These are then converted to black and white by taking pixels above and below a threshold light level, setting the bright pixels to white, and the dark ones to black (Figure 3.4c). A function called imerode is then used to remove light patches smaller than the pliolite particles. The function will eliminate noisy particles that are smaller than the structuring a) brightness b) scanning window pixels brightness pixels Figure 3.5 Finding the centres of particles using nonmaxsuppts. a) Dashed lines indicate the action of the scanning window; b)circles indicate points determined to be the centre of particles. 19