the role of body size in generating plankton patchiness · plankton communities constitute an...

THE ROLE OF BODY SIZE IN GENERATING

PLANKTON PATCHINESS

Simone McCallum

10013129

Supervisors

Dr Anas Ghadouani

Prof Gregory N Ivey

Submitted in part completion of Bachelor of Engineering, Environmental, November 2005

Abstract

i

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.

Acknowledgements

iii

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.

Contents

v

CONTENTS

Abstract .................................................................................................................................................... i Acknowledgements ................................................................................................................................ iii Contents....................................................................................................................................................v List of Figures ........................................................................................................................................ vi List of Tables.......................................................................................................................................... vi Chapter 1 ..................................................................................................................................................1

1 Introduction .......................................................................................................................................1 Chapter 2 ..................................................................................................................................................5

2 Literature Review..............................................................................................................................5 2.1 Patchiness Theory ...................................................................................................................5 2.2 Measuring Patchiness .............................................................................................................9

Chapter 3 ................................................................................................................................................13 3 Methods...........................................................................................................................................13

3.1 Experimental Apparatus .......................................................................................................13 3.2 Analysis ................................................................................................................................17

Chapter 4 ................................................................................................................................................23 4 Results .............................................................................................................................................23

Chapter 5 ................................................................................................................................................27 5 Discussion .......................................................................................................................................27

5.1 Characterising Spatial Distributions .....................................................................................27 5.2 Interpretation of Results........................................................................................................28 5.3 Plankton Distributions ..........................................................................................................30 5.4 Limitations of the Study .......................................................................................................31

Chapter 6 ................................................................................................................................................33 6 Conclusions .....................................................................................................................................33

6.1 Summary...............................................................................................................................33 6.2 Future work...........................................................................................................................34

Appendix A ............................................................................................................................................35 References ..............................................................................................................................................37

List of Figures

vi

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............................................................... 13

Figure 3.2 Sketch of oscillating grid in plan view................................................................................. 14

Figure 3.3 Section of tank viewed from the right hand side of that in Figure 3.1, showing the image

capture arrangement. ..................................................................................................................... 16

Figure 3.4 Sample from experiment 1 to demonstrate image processing.............................................. 18

Figure 3.5 Finding the centres of particles using nonmaxsuppts........................................................... 19

Figure 3.6 Normal fit to separation distance histogram. ....................................................................... 21

Figure 4.1 Mean particle separation vs. Kolmogorov microscale ......................................................... 23

Figure 4.2 Mean particle separation vs. Taylor microscale................................................................... 24

Figure 4.3 Mean particle separation vs. Integral length scale ............................................................... 25

Figure 4.4 Correlation of mean particle separation and standard deviation .......................................... 25

Figure 5.1 Sketch of energy spectrum of turbulence............................................................................. 29

LIST OF TABLES

Table 3.1 Tank and grid specifications.................................................................................................. 15

Table 3.2 Specifications of fourteen experiments performed................................................................ 17

Table 5.1 Characteristic, and typical plankton sizes and turbulent conditions in experiments and lakes

and ocean surface mixing layers.................................................................................................... 31

Chapter 1 Introduction

1

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


2

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


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


4

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.

Chapter 2 Literature Review

5

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


predation. Dispersion by physical currents and turbulence are incorporated as an effective diffusivity

resulting in the reaction-diffusion differential equation,

PPtP 2∇+=∂∂ κμ (2.1)

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, μκ4.2=r . 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


7

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.


8

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


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.

2.2.1 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

xsI /2= , (2.2)

where x is the mean of counts x1 to xn, of individuals in each sample, and s2 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


2.2.2 Lloyd’s mean crowding

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, is measured as the mean number, per individual, of other

individuals in the same quadrat,

*m

N

Xm N

i∑=* , (2.3)

where Xi is the number of other individuals in the same quadrat as individual i (Lloyd 1967). Mean

crowding, is compared to the mean density of the sample and can give the result that the

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.

*m*m

2.2.3 Morisita’s index

Morisita’s index, Id, 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.

)1(

)1(

−

−=

∑NN

XXQI Q

ii

d , (2.4)

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 Id = 1 – [(n–1) / (Σx–1)] indicates a uniform

distribution, Id = 1, a random distribution, and Id = 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

( )Mn x

xu =− +

−

∑∑

χ0 9752

1.

, (2.5)

which indicates a uniform distribution of individuals, and

10


( )Mn x

xc =− +

−

∑∑

χ0 0 252

1. . , (2.6)

which indicates a patchy, or clumped distribution of individuals, and χj2 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:

0.1>≥ cd MI , ⎟⎟⎠

⎞⎜⎜⎝

⎛−

−+=

15.05.0

c

cdp M

MII (2.7)

0.1≥> dc IM , ⎟⎟⎠

⎞⎜⎜⎝

⎛−−

=115.0

c

dp M

II (2.8)

ud MI >>0.1 , ⎟⎟⎠

⎞⎜⎜⎝

⎛−−

−=115.0

u

dp M

II (2.9)

du IM >>0.1 , ⎟⎟⎠

⎞⎜⎜⎝

⎛ −−−=

u

udp M

MII 5.05.0 (2.10)

This standardised index indicates a random distribution when Ip=0, a more patchy distribution when

Ip>0, and a more uniform when Ip<0. The index is more powerful because it indicates patchiness or

uniformity with 95% confidence for Ip=±0.5 (Krembs et al. 1998; Vandermeer 1981).

2.2.4 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).

2.2.5 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


N

rir N∑

= , (2.11)

along with the value of this average if the individuals were distributed randomly

ANrrandom /2

1= , (2.12)

where N is the number of individuals, i is the ith individual, and A is the sample area. Given these, the

ratio randomrr 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

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

oscillating grid

light plane

drive shaft

Perspex box to prevent surfaces wave interfering with light

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

Chapter 3 Methods

3.1.1 Particles

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.

3.1.2 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 61 ×

61 × 91 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

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. C1, C2, and C3 from O’Brien (2004).

Mesh size 6 cm Stroke range 10 – 130 mm Frequency range 1 – 7 Hz C1 0.18 ± 0.04 C2 0.22 ± 0.05 C3 0.1 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):

12/12/310

−= fzMSCu , (3.1)

12/12/320

−= fzMSCw , (3.2)

and the integral length scale, a length scale which represents the size of the largest eddies in turbulent

flow as:

zCl 30 = . (3.3)

Furthermore, the energy dissipation rate – the rate of dissipation of kinetic energy from the turbulent

eddies:

4

32/92/12/322

21

30

3

321

zfSMCC

Clu

⎟⎟⎠

⎞⎜⎜⎝

⎛ +==ε . (3.4)

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,

15

4/13

⎟⎟⎠

⎞⎜⎜⎝

⎛=

ενη . (3.5)

Chapter 3 Methods

An intermediate length scale, the Taylor scale (λT) represents the cascade of turbulent eddies

(Tennekes & Lumley 1972), and is defined by

ενλ 15uT = . (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).

3.1.3 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 30 × 30 × 25 cm is positioned to prevent surface waves from interfering with light from

16

Chapter 3 Methods

17

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 λT.

experiment S (cm) f (Hz) ε (m2s-3) η (mm) λT (mm)1 13 5 7.85×10-3 0.11 0.54 2 13 4 4.02×10-3 0.13 0.60 3 13 3 1.70×10-3 0.16 0.70 4 13 2 5.02×10-4 0.21 0.85 5 4.8 6 1.53×10-4 0.28 1.04 6 4.8 5 8.87×10-5 0.33 1.14 7 4.8 4 4.54×10-5 0.39 1.27 8 4.8 3 1.92×10-5 0.48 1.47 9 4.8 2 5.68×10-6 0.65 1.80

10 4.8 1 7.09×10-7 1.09 2.55 11 1.3 4 1.27×10-7 1.67 3.40 12 1.3 3 5.36×10-8 2.08 3.91 13 1.3 2 1.59×10-8 2.82 4.80 14 1.3 1 1.99×10-9 4.74 6.79

3.2 Analysis

3.2.1 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.

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

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.

3.2.2 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

bri

ghtn

ess

scanning window pixels

pixels

bri

ghtn

ess

b)

a)

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

Chapter 3 Methods

element, and those that are sufficiently large to represent the desired particles will be retained (Figure

3.4d). This eroded image is then multiplied by the grey scale image to generate an image that has

eliminated noise and retains the variation in brightness in the image of each particle (Figure 3.4e). The

brightness variability within each particle is useful in identifying a central point for each particle, to

define the relative position of each particle.

The central point of each particle is found using a Matlab function called nonmaxsuppts (Kovesi

2000). This function finds the brightest point on each particle in a way that is best illustrated by a one

dimensional example. Note that this example is only representative of the function, refer to Kovesi

(2000) for an exact description. The particle images can be represented on a plot of the numerical

value given to the brightness of each pixel (the shaded area in Figure 3.5). Nonmaxsuppts uses a

scanning window based on a radius that represents the particle size specified by the user to scan across

the image (Figure 3.5a). The function begins by taking the brightest pixel within the scanning window

and assigns this value to all the pixels within the window in a second image that it generates. The

window is then moved across one pixel to do the same thing again, until the full image has been

processed (Figure 3.5b). The function then finds the points at which the brightness values in the

original image equal those in the generated image (circled in Figure 3.5b); these points define the

approximate centre of each particle.

The separation distances, D are then found by simple trigonometry:

221

221 )()( yyxxD −+−= (3.7)

These distances are then fit to a normal distribution to find the mean and standard deviation as

characteristic parameters of the spatial distribution of the particles (Figure 3.6). This process is

performed for each of the fourteen experiments run (Table 3.2) to examine the changes in spatial

patterning caused by varying the relative magnitude of Kolmogorov scale and particle size.

20

Chapter 3 Methods

0 50 100 150 200 2500

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

occu

rren

ces

of s

epar

atio

n d

ista

nce

D

separation distance, D (mm) μμ–σ μ+σ

Figure 3.6 Normal fit to separation distance histogram. On the horizontal axis, μ is the mean separation distance and σ the standard deviation from that mean, representing the width of the distribution. This histogram was generated using an image from experiment 1. This particular histogram has been produced to illustrate the methods, and uses less bins than those used to calculate results.

21

Chapter 4 Results

CHAPTER 4

4Results

Parameters characterising the spatial distribution of particles for fourteen experiments are presented

here. The mean separation distance is presented as the main result. The standard deviations are not

explicitly presented as they exhibit very similar trends to the means. The method used was based on

varying the Kolmogorov microscale between experiments. The first relationship examined is that

between the mean particle separation and the Kolmogorov microscale.

23

Dependence of particle separation on the Kolmogorov microscale

r2 = 0.6835slope: 20

75

77

79

81

83

85

87

89

0 1 2 3 4

Kolmogorov microscale (mm)

Mea

n p

articl

e s

epara

tion

(m

m)

5 2d d

Figure 4.1 Mean particle separation vs. Kolmogorov microscale The d and 2d on the horizontal axis represent the mean, and twice the mean particle diameter. A linear function is fitted to the first eight points where the Kolmogorov scale is smaller than twice the particle diameter. The last six points are best represented by their mean value of 86.0 mm.

Two regions that exhibit significantly different behaviour are found. In the region where η>2d the

variation in the mean separation of particles, D is best represented by the mean of those means. In the

region where η<2d the variation in D is linearly dependent on η where D ~ 20 η.

24

Dependence of particle separation on the Taylor microscale

r2 = 0.7071slope: 8.2

75

77

79

81

83

85

87

89

0 1 2 3 4 5 6 7

Taylor microscale (mm)

Mean

par

ticl

e se

par

atio

n (

mm

)

2d d

Figure 4.2 Mean particle separation vs. Taylor microscale There is a linear relationship between D and Taylor microscale, λT in the region where η<2d.

The variation in D in the region in which η<2d can be described by a linear relationship with the

Taylor microscale, λT where D ~ 8.2 λT.

The integral length scale, l is constant in all experiments performed (Figure 4.3). In the region where

η>2d the mean separation of particles is independent of both η and λT. Here, D is approximately 4.2

times the integral length scale, D ~ 4.2 l.

The behaviour of the standard distribution with respect to each length scale of turbulence is similar to

that of the mean. It is therefore not presented explicitly here, the correlation between the two

parameters is shown in Figure 4.4.

Chapter 4 Results

Dependence of mean particle separation on the integral length scale

75

77

79

81

83

85

87

89

0 5 10 15 20 25

Integral length scale (mm)

Mea

n p

articl

e s

epara

tion

(m

m)

Figure 4.3 Mean particle separation vs. Integral length scale The integral length scale remains constant through all experiments, since it is dependent only on the grid dimensions and the distance from the grid that a measurement is taken.

Correlation between mean and standard deviation of the particle separations

r2 = 0.6442

40

41

42

43

44

75 77 79 81 83 85 87 89

Mean particle separation (mm)

Sta

ndard

Dev

iation (

mm

)

Figure 4.4 Correlation of mean particle separation and standard deviation The standard deviation varies in the same way as the mean separation, as the mean separation of the particles becomes larger, the number of pairs of particles that have a significantly larger or smaller separation distance increases.

25

Chapter 5 Discussion

27

CHAPTER 5

5Discussion

Body size has a role in generating spatial distributions of particles. The characteristics of the spatial

distribution of particles in these experiments change in different turbulent conditions. The focus of

discussion is the different spatial distributions of particles exhibited in the two regions where particle

diameter is smaller and larger than the smallest length scales of a turbulent flow. These results are

considered for indications of the forces that dominate the characteristics of spatial distributions formed

in different situations and the changes that occur to elicit a change in the spatial behaviour of the

particles. The observations made about the particles in the experiments will be considered for the

implications for zooplankton and phytoplankton spatial heterogeneity in nature.

The method used to characterise spatial distributions will be discussed, examining its usefulness in

characterising the spatial distribution of particles, how it compares to other measures, and how it can

be improved. The characteristics of particle distributions generated in a variety of different turbulent

conditions are the main focus of the project. The changes in the character of particle distributions is

discussed, with some ideas about what it could be that governs those changes in behaviour. The

expected behaviour of particle distributions outside of the range used in the experiments is considered,

and further related study of a similar nature that would provide insight into plankton spatial dynamics.

5.1 Characterising Spatial Distributions

The parameters of the distribution of separations distances used to quantify the spatial distribution of

particles allows for the characterisation of plankton spatial arrangement. This characterisation uses the

mean separation distance between particles, and the standard deviation from that mean. For use

outside of these experiments, the measure can be used when the relative position of each individual is

known at an instant in time and when there are enough individuals in a sample to obtain a fit to the

normal distribution. The mean separation distance indicates the how close each individual is to the

other individuals. An example of an interpretation of this measure is that a particle distribution with

particles that are all close together would yield a low mean and low standard deviation, whereas one

that has a random distribution of particles would yield a larger mean, and wider standard deviation.

Exact interpretations of this measure of spatial distribution are not considered in the context of this

project, but rather used as a measure to distinguish between different distributions; a relative measure.


28

This measure characterises the spatial distribution of particles in the fourteen experiments with

different turbulent conditions. A glance at the initial results (Figure 4.1, Figure 4.2) suggests the

separation scale varies logarithmically relative to the Kolmogorov and Taylor microscales. The exact

functional relationship between the turbulent length scales and mean separation is not discussed since

it is relating a characteristic descriptor of the spatial distribution to a characteristic descriptor of the

turbulence; an exact functional form relating the two inexact quantities is not a meaningful exercise.

The purpose of this study is to determine whether there are any differences in the formation of spatial

pattern in different turbulent conditions, where those changes occur, and the dominant drivers of

spatial pattern in the different regions.

5.2 Interpretation of Results

The results reveal two regimes in the behaviour of the mean separation of particles with respect to the

Kolmogorov microscale (η) (Figure 4.1). The transition between the two regimes occurs around the

point where the Kolmogorov microscale is equal to twice the particle diameter, η=2d. For those

experiments where η>2d, the particle size is smaller than the Kolmogorov microscale and the mean

separation distance between particles in the distribution is independent of changes in the turbulent

conditions. For η<2d, the experiments are in a region where particle size is similar to or larger than the

Kolmogorov microscale but always smaller than the integral length scale (Figure 4.3). In this region,

the mean separation can be described as being linearly dependent on either the Kolmogorov or Taylor

microscale (Figure 4.1, Figure 4.2). To make the purpose of the experiments clear, the region in which

η>2d, will be referred to as ‘small particles’ and conversely, where 0<η<2d, as ‘large particles’

herein.

5.2.1 Turbulent Energy Spectrum

The generated turbulence can be described as a cascade of eddies from large eddies characterised by

the integral length scale which, by energy losses generate smaller eddies of length scales down to

those characterised by the Kolmogorov microscale, where the turbulent kinetic energy is dissipated by

viscous action. This cascade of eddies can be visualised as an energy spectrum, that describes the large

amount of energy contained in the largest eddies, and the cascade of energy down to the smallest

eddies (Figure 5.1). The transition from the small particles region to the large particles region can be

described by a transition between two regions in the turbulent energy spectrum. The proportion of the

energy spectrum of turbulence that a particle may sample changes when moving between the two

regions. When the particle size is smaller than the Kolmogorov scale, it samples the entire spectrum of

turbulent energy and the full cascade of turbulent eddies. When the particle size is larger than the


Kolmogorov scale, it is within the range of eddy length scales, and does not sample the energy in the

higher wave numbers (Figure 5.1). The effect of changing the size of particle relative to the turbulent

energy spectrum can be illustrated by analogy with a leaf and a large ship on wavy water. The leaf

would be significantly affected by the waves, whereas the ship would not know the waves were there.

The difference in character of the spatial distribution of particles thus appears to be controlled by the

range of turbulent eddies it experiences.

E(k)

1 l

1 λ

1_ dlarge

1 η

1_ dsmall

k

Figure 5.1 Sketch of energy spectrum of turbulence The spectrum demonstrates the amount of energy in turbulent eddies of various length scales. The parameter k is wavenumber, with units of the inverse of length.

5.2.2 Small Particles

In the small particles region, the particles are smaller than the Kolmogorov length scale and

experience the entire spectrum of turbulent energy. This exposure to the wider range of turbulent

eddies may explain the wider distribution of separation distances (larger standard deviation, Figure

4.4) seen for the smaller particles, since eddies of a range of different sizes have input into the motion

of each particle. The characteristics of spatial distributions formed in the region where η>2d are

similar in this region which indicates that the spatial distribution of particles in this regime is

determined by a forcing parameter that is also constant over this region. There are two length scales

that satisfy this condition, the integral length scale and the sampling area. Although the mean

separation is a multiple of the integral length scale, the mechanism by which eddies characterised by

the integral length scale would control the spatial distribution is not clear. The effect of changing the

integral length scale, and the mechanism by which it affects the spatial distribution of particles could

be examined by performing experiments vary the integral length scale by varying the distance of the

measurement point from the oscillating grid.

29


30

Randomly generated simulated particle distributions of a similar number of particles to those used in

the experiments resulted in a mean particle separation of 88mm and a standard deviation of 44mm.

The parameters for the measured distribution are close to those for the random ones. A larger number

of experiments in a larger range of turbulent conditions would produce more definitive results to

determine whether the spatial distribution of particles asymptotes to a random distribution for low

energy turbulent flows, as expected.

5.2.3 Large Particles

In the large particles region, there is a strong linear correlation between the mean separation distance

and the Taylor and Kolmogorov microscales (Figure 4.1, Figure 4.2). Turbulent eddies that are smaller

than the particle diameter are not expected to have a significant effect on the spatial distribution of

those particles because the perturbation of the path of the particle will be smaller than the particle itself

(a small scale example of the leaf and ship analogy). This leads to the inference that the spatial

distribution of particles in region two can be described by its correlation to the Taylor microscale,

which is a characteristic of the turbulent energy spectrum. The mean separation distance is expected to

continue to decrease as the particles become larger compared to the Kolmogorov microscale. Another

transition in the spatial distribution of particles would be expected as the particle size approaches, and

exceeds the integral length scale, since particles would then be of a size such that the motions

generated by all turbulent eddies are small compared to the particle size. In this region where particle

size is greater than the integral length scale, is one where in nature, organisms tend to be more motile

and their behaviours would have a larger role in their spatial distribution.

5.3 Plankton Distributions

The particles in this experiment can be interpreted as being representative of individual phytoplankton

in the region where particle size is smaller than the Kolmogorov microscale, and zooplankton in the

region where particle size is in between the Kolmogorov microscale and the integral length scale of

turbulence. Table 5.1 provides a comparison of the relevant length scales, and shows similar

conditions to those used in the tank are present in nature. Using a direct relation between the

distribution of particles in the experiments and phytoplankton and zooplankton in nature, the

difference in ‘large’ and ‘small’ particle distributions supports the idea that body size has a role in the

formation of spatial distributions of plankton and the difference in body size between phytoplankton

and zooplankton has a role in generating their different distributions. At close observation of the

distribution characteristics formed, the small particles have a distribution close to a random

distribution. The large particles are distributed with a smaller separation distance, and less deviation


31

from that distance, which indicates a more heterogeneous distribution. This seems to contradict the

findings of Pinel Alloul et al. (1988) that the distribution of large zooplankton is less heterogeneous

than that of small zooplankton. This contrast in results may be due to different methods for measuring

the distributions (Pinel Alloul et al. use samples of plankton number density). Additionally, the

sampling scale used in their study is on the order of meters, at least ten times that used in this

experiment. This contrast highlights the need to consider that there may be a force (that is not physical

body size), correlated with body size, such as behavioural traits that dominates the generation of

spatial distribution, that would change the correlations observed in this experiment.

Table 5.1 Characteristic, and typical plankton sizes and turbulent conditions in experiments and lakes and ocean surface mixing layers

Parameter Typical Value Reference Phytoplankton size, d 10 μm (Kalff 2002) Zooplankton size, d 1 mm (Kalff 2002) Kolmogorov scale, η 1 mm (O'Brien et al. 2004)

Surface mixed layer

Range of d/ η ~10–2 to ~1 Particle size, d 256 μm Kolmogorov scale (min) 106 μm Kolmogorov scale (max) 4.74 mm

Experiment

Range of d/η ~10–2 to ~2

The distributions formed in the experiment appear to result in a finer scale structure in the larger

particles as indicated by a shorter separation distance and lesser deviation from that distance. This

result agrees with the observations of Mackas (1977) presented in the introduction (Figure 1.1) and the

plankton distributions resulting from the model used by Abraham (1998). It is however important to

note that body size cannot be deduced to be a determining factor in these distributions since the

variability is on the order of kilometres, a far larger length scale than those examined here.

Furthermore, it is difficult to postulate the small scale structure has on large scale variability. The

connections between spatial distributions on different length scales, and the larger scale effects of

small scale heterogeneity is an important aspect of patchiness theory and is under explored (Brentnall

et al. 2003; Levin 1992).

5.4 Limitations of the Study

The study is based on the assumption that the particles used are a suitable representation of plankton

outside of biological, chemical and physical forces (other than turbulent mixing). It is partly due to this

assumption and the nature of the measure of spatial structure that the spatial patterns formed are

discussed in terms of their general behaviour rather than their functional form.


32

5.4.1 Imaging

A correlation is found between the separation distance and the number of particles found after

processing each image is examined. This correlation suggests that the number of particles found by the

software may have an effect on the distribution characteristics calculated. Although the sample area

remains constant, the separation distance decreases along with the number of particles. This excludes

the possibility that the lower numbers of particles gives a lower density, as this would behave in the

opposite manner to increase the mean particle separation. Since this is the case, it is difficult to

quantify exactly any error that might be introduced by this phenomenon. The mechanism by which the

change in numbers of particles found occurs is that the light conditions for photographing the particles

become worse as the turbulent energy increases. The effect of this is difficult to quantify, and on

inspection, the image quality does not appear to significantly affect the ability of the software to find

particles. This problem may be avoided by using a brighter and more coherent light source, such as

laser light to reduce noise and allow short shutter speeds to be used without compromising image

quality.

References

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CHAPTER 6

6Conclusions

6.1 Summary

Plankton populations are the main primary producers in aquatic environments and the world,

contributing 30% of total global primary production (Daly & Smith Jr 1993). The spatial distribution

of plankton, in particular the spatial patterns formed significantly affects plankton productivity.

Understanding and predicting plankton spatial distribution and dynamics is an essential to the study of

aquatic ecosystems, fisheries, energy budgets, biogeochemical cycles and climate. Determining the

forces that generate, sustain or affect the formation of spatial structure in plankton distributions is

central to understanding their spatial patterns. Research in this area is conducted to determine the

relative importance of the different forces that act on plankton populations at different spatial scales. A

significant feature of plankton distributions is the difference observed between phytoplankton and

zooplankton distributions.

The experiments performed in this study demonstrate that changing from a regime where particle size

is smaller than the Kolmogorov length scale to one where it is larger changes the spatial distribution

from one that is near random to one that is more structured with a smaller separation distance between

particles. The particles used in this experiment are intended to represent plankton minus forces other

than those under consideration. Assuming the particles used are a reasonable representation of

plankton, the results imply that differences in the distributions formed by the particles in these

experiments represent the effect plankton body size has on their distribution in nature. In particular,

the body size of individual phytoplankton is expected to contribute to generating more random

phytoplankton distributions, and the generally larger body size of zooplankton is expected to generate

distributions with smaller scale variability. The results of these experiments appear to correlate with

studies of phytoplankton and zooplankton distributions on kilometre length scales in a qualitative

sense This agreement cannot be interpreted as validation of either study since the action of the

turbulent eddies does not scale up to the mesoscale fluid dynamics, although it is unknown what

implications this small scale variability has for larger scales.

References

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6.2 Future work

This project has developed a framework to examine the purely physical behaviour of planktonic

organisms, independent of their biological changes and behaviours. The laboratory and analysis

methods used can be adjusted for further work in the same vein, or for use in field observations of

plankton. Areas for future work are to further develop the measure of patchiness, obtain more

descriptive results for the experiment in this project, extend this work to include different turbulent

conditions, and to make observations in the field to determine whether there is a correlation with the

distributions observed in the laboratory.

The accuracy of the results from this project, and the scope for interpretation of correlations between

particle size, turbulent conditions and particle dynamics would be improved by performing more

experiments in a wider range of turbulent conditions. Along with this, the measurement of spatial

distributions requires further development to identify which characteristics it is meaningful to

measure. The measure of spatial distribution used in this project requires further development to

establish more definite interpretation of the results of measurements.

The idea of work such as this is to take phenomena that occur in nature and isolate them to examine

how they work, and what effects they have. To further develop the work in this project, other turbulent

conditions generated in nature could be simulated using similar grid stirring devices. For example, the

daily variations in wind forced turbulence could be represented by varying the turbulent conditions in

a tank, simply by varying the oscillation frequency. The results would reveal whether different spatial

patterns form than those in steady turbulence, and the response time of the plankton formation to

changes in turbulent conditions. The experiment could be altered again to generate heterogeneity in the

turbulence, perhaps by using two different oscillating grids. This could be used to investigate the

behaviour of plankton populations at the interface between areas of different turbulent conditions, to

simulate, for example the interface between an area that is exposed to wind and one that is sheltered.

In a wider scope, the relative contribution of body size to the characteristics of plankton distributions

requires observations of plankton in their natural environment and mechanistic studies of other forces

on the same length scales as this study. The general methods used in this study could be carried over to

new studies of different forcing mechanisms, that is to isolate and study particular forces in the

laboratory. The imaging and analysis techniques may be augmented for observation and analysis in

field studies. Finally, currently there is little understanding of the links between spatial and temporal

scale of plankton population structure. An important aspect of understanding the spatial distributions

of plankton is to discover the role of small scale patchiness in the formation of larger scale variability.

Appendix A

35

APPENDIX A function patchiness(imagename) %enter imagename as 'imagename.jpg' or similar function calculates the %distance between the particles in the field of view, then generates a %histogram, writes the distances to a data file, and generates the normal %parameters for the distribution. %% threshold and radius to use to process image threshold = 60; radius = 6; %% PROCESS IMAGE %% Read image in, convert to grey, and cast to double im = double(rgb2gray(imread(imagename))); % Simple thresholding that should be the same for each image compared bw = im > threshold; % Erode to eliminate noisy particles using a 2x2 square structuring element se = ones(2); bw = double(imerode(bw,se)); % multiply oringinal image with eroded black and white image to mask out % insignifiant particles and brighten bright points maskedim = im.*bw; %% FIND PARTICLE POSITIONS %% Find local maxima in masked image, returned %in r and c [r,c] = nonmaxsuppts(maskedim, radius, threshold, im); % FIND DISTANCES BETWEEN PARTICLES % find the distance between each point and every other point N = length(r); dist = zeros(1,(N-1)/2*N); ind = 1; for pt = 1:(N-1) for pt2 = (pt+1):N dist(ind) = sqrt((r(pt) - r(pt2))^2 + (c(pt) - c(pt2))^2); ind = ind+1; end end %% CONVERT PIXELS TO MM %% % for october 12th 14.98pixels = 1mm, diagonal distance, 3612pix = 241mm dist = dist./(14.98); % DISTRIBUTION OF DISTANCES % bins distances into histogram form and draws a bar graph bins = 0:241; h = histc(dist, bins); %% GAUSSIAN PARAMETERS %% [muhat,sigmahat,muci,sigmaci] = normfit(dist, 0.05); %% WRITE DATA TO DATA FILES %% % writes the distance data to distances.dat fid = fopen('distances.dat','a'); fprintf(fid,'%5.2f ', dist); fclose(fid); % writes the number and normal parameters for each photo to %distribution.dat gauss = [N; muhat; muci; sigmahat; sigmaci]; fid = fopen('distribution.dat','a'); fprintf(fid,'%5.2f ', gauss); fprintf(fid,'\r'); fclose(fid)

References

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