influence of coagulation, sedimentation, and grazing by...

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 103, NO. C8, PAGES 15,601-15,612, JULY 15, 1998

Influence of coagulation, sedimentation, and grazing by zooplankton on phytoplankton aggregate distributions in aquatic systems

Alexandria B. Boehm and Stanley B. Grant Department of Civil and Environmental Engineering, University of California, Irvine

Abstract. Phytoplankton growing at the surface of lakes and oceans are removed from the water column by gravitational sedimentation and/or zooplankton grazing. Both of these processes are influenced by the aggregation state of the phytoplankton, which, in turn, may be altered through particle-particle coagulation. In this study, we present a mathematical analysis of these phenomena in an attempt to better understand the physical and biological factors that control phytoplankton concentrations in aquatic systems. During phytoplankton blooms, grazer concentrations are relatively low, the concentration of phytoplankton in the mixed layer is high, and phytoplankton production at the surface is countered by coagulation and sedimentation. In this case, dynamic scaling theory indicates that the concentration of total phytoplankton aggregates No and the volume fraction of phytoplankton N• should decay as power laws of depth z: No • z- • and N• • z *:. The values of the power law exponents 3' and e are determined by the physical and chemical processes responsible for coagulation and sedimentation in a given system. Under nonbloom conditions, the concentration of grazers is relatively high, the phytoplankton concentrations are relatively low, and phytoplankton generated at the surface are quickly transferred to higher-trophic levels by grazing. In this case, N o and N• decay with depth in an approximately exponential fashion. These results suggest that the principle mechanism by which phytoplankton are removed from the water column in natural aquatic systems may be differentiated by the depth evolution of No and N•.

1. Introduction

On a global scale, phytoplankton are responsible for fixing -40 x 109 t of carbon per year, which is 40% of Earth's total primary production [Falkowski, 1994]. The fate of these organisms ultimately determines the fate of the carbon which they fix. Phytoplankton may aggregate with other phytoplankton, or detritus, and settle under the influence of gravity JAildredge and Jackson, 1995]. The flux of these aggregates through the water column controls transport of carbon from the surface layers to the benthos, where the carbon may be remineralized, incorporated into sediments, or consumed by benthic organisms [Billerr et al., 1983; Smetacak, 1985]. Alternatively, phytoplankton may be consumed by herbivorous grazers and enter the pelagic food web. In this case, the carbon and nutrients contained in the phytoplankton are passed on to higher- trophic levels.

There are many different processes that regulate the concentration of phytoplankton in natural aquatic systems, including growth conditions (light and nutrients) [Raymont, 1963a], zooplankton grazing [Banse, 1992], transport phenomena (ocean currents, wave action, upwelling, and gravitational sedimentation) [Weilenmann et al., 1989], and physicochemical phenomena (coagulation and uptake of trace metals) [Burd and Jackson, 1997; Fowler and Knauer, 1986]. These processes have an unequal impact on the fate of phytoplankton in different systems. For example, during the initial phase of a phytoplankton bloom, coagulation and gravitational sedimenta-

Copyright 1998 by the American Geophysical Union.

Paper number 98JC01307. 0148-0227/98/98JC-01307509.00

tion appear to control phytoplankton concentrations, while grazing is less important [Bttrd and .lacksott, 1997]. During the well-stratified periods of summer, however, phytoplankton growth in the open ocean is nearly completely balanced by grazing [Banse, 1992]. The goal of this paper is to develop a theoretical framework for understanding how coagulation, gravitational sedimentation, and grazing by zooplankton control the vertical distribution of phytoplankton in natural aquatic systems. We accomplish this by solving the gcncral dynamic equation (GDE) that tiescribes these phenomena for a special case where the distribution of phytoplankton bclow the mixed layer achieves a steady state. Our results are inter- preted in the context of dynamic scaling theory for the coagulation and sedimentation of particles.

2. Model Development Consider the water column illustrated in Figure 1. At the

surface there is a mixed layer in which most of the primary production takes place [Falkowski, 1994]. The depth of the mixed layer varies from a few meters to a few hundred meters [Falkowski, 1994]. In this paper, we focus on the region below the mixed layer for the following reasons: (1) The mixed layer has been the subject of numerous theoretical investigations [Ackleh and Fitzpatrick, 1997; Burd and Jackson, 1997; Jackson and Lochmann, 1992], while the region below the mixed layer has received less attention, and (2) our results indicate that the manner in which the aggregate size distribution evolves with depth below the mixed layer may provide fundamental infor- mation regarding the physical, chemical, and biological processes that control phytoplankton levels there. For simplicity,

15,601

15,602 BOEHM AND GRANT: PHYTOPLANKTON AGGREGATES

Exponential Grazer Distribution

Unimodal Grazer Distribution

Mixed Layer

Figure 1. The physical system analyzed in this study. Single phytoplankton leave the mixed layer at a constant rate, entering the water column where they may coagulate, settle under the influence of gravity, or be grazed upon by zooplankton. Exponen- tial and unimodal vertical zooplankton distributions may be representative of nighttime and daytime conditions, respectively.

we assume that single phytoplankton particles settle out of the mixed layer at a continuous rate. Once they enter the region below the mixed layer, they coagulate with other phytoplankton, settle under the influence of gravity, and are consumed by grazers. Although our analysis is premised on the assumption that the phytoplankton leaving the mixed layer are not aggregated, the results we present later in the paper suggest that this assumption is not as restrictive as it first appears.

The GDE for phytoplankton undergoing coagulation, grazing, and gravitational sedimentation in one dimension may be represented as follows (see notations for definition of symbols):

dnj(z, t) 1 • dt = 5 K,,j_,n,(z, t)n,_,(z, t) t=l

-- E gt,jFli(Z' t)FIj(Z, t) -- R s -- Rg t=l

The particle distribution function, ns(z, t), represents the fluid concentration of phytoplankton aggregates composed of j primary particles (j-mers); Ki, s is the rate constant for coagulation between i-mers and j-mers; and R s and R a are the sedimentation and grazing removal terms, respectively. In this equation, the temporal change in the concentration of j-mers (term on left-hand side) depends on their formation by coagulation of particles of size i and j - i (first term on right-hand side), loss by coagulation with all other sized aggregates (second term on right-hand side), removal from the fluid phase by gravitational sedimentation (third term on right-hand side), and grazing by zooplankton (last term on right-hand side). Our goal is to derive from this GDE mathematical expressions for the phytoplankton aggregate size distribution, n s , the concentration of total phytoplankton aggregates, No = Z• n s, and the volume fraction of phytoplankton which is proportional to N• = • jn s. In sections 2.1, 2.2, and 2.3 we develop mathematical expressions for Ki,j, Rs, and R a.

2.1. Coagulation Kernel

The coagulation kernel K,, s appearing in (1) represents the rate constant for the coagulation between i-mers with j-mers. Its magnitude reflects the transport processes that bring aggregates into proximity and the physical and surface-chemical interactions that occur between aggregates upon close ap- proach. To account for these processes mathematically, the coagulation kernel is often expressed as the product of a collision frequency function/3i, s and an attachment efficiency a/,s:

Ki, J = O•l,jj•l, J (2)

where /3i,s represents the collision frequency between i-mers and j-mers under one or more transport mechanisms and ai,s represents the fraction of these collisions that result in aggregation. Commonly employed collision frequency functions are listed in Table 1.

Kernels may also be described in terms of more general mathematical properties. For example, van Dongert and Ernst [1985] (hereafter referred to as VDE) classify kernels on the basis of two parameters, A and/•,

Table 1. Expressions for Several Collision Frequencies

Mechanism Collision Frequency, /3,j

Fluid shear

Rectilinear

Curvilinear

Differential settling Rectilinear

Curvilinear

BrownJan diffusion, DLA RLA

Product kernel

1/3(s/v)ø'S(r, + r,) 3 3/D 10(s/v)ø'S(r, + r,)2r,, where r, << r, 3/D

•r(r, + r•.)2w, - w• 1 + 1/D 0 0.5rr(r, 2) w,- wj 1 + 1/D 2/D 2kbr/31•,,(1/r, + 1/r•)(r, + r•) 0 -1/D ß '. 1 0

K• • (ij) 'ø 2to •o

Values A and/• were calculated for each kernel by assuming that both the hydrodynamic radius and the collision cross-section radius scale with aggregate size as r, • i •/ø. Values X and • describe the homogeneity and the small i/j limit of the collision frequency, respectively. DLA and RLA are diffusion and rate limited aggregation, respectively. Value e is the turbulent energy dissipation rate, v is the kinematic viscosity of the fluid, T is the absolute temperature, kb is the Boltzman constant, r• is the radius of a monomer, •, is the dynamic viscosity of the fluid, pf is the density of the fluid, and to is an exponent characterizing the surface area of an aggregate.

BOEHM AND GRANT: PHYTOPLANKTON AGGREGATES 15,603

Ka,.aj = aXgi,j X -< 2 (3a)

Ki,j '" (i)•(j) x-• j >> i, X - p• -< 1 (3b)

where a is a positive constant. Here X represents the kernel's degree of homogeneity with respect to arguments i and j, and /x is an exponent governing the small i/j limit of Ki, j. The restrictions noted above for the values of X and/x arise from physical constraints on aggregate-aggregate interactions. These two parameters reflect the underlying physics and surface chemistry of the coagulation process. If/x > 0, then the coagulation takes place predominately between clusters of the same size; if/x < 0, reactions between small and large clusters predominate. The value of X indicates how the reactivity between clusters of the same size grows with size. If X > 0, for example, the rate constant for coagulation between 10-mers will be 10 x times greater than the rate constant between mono- mers.

One important implication of the VDE classification is that the kinetics of coagulation are governed primarily by the mathematical characteristics of the kernel (i.e., X and/x) and not by the exact analytical structure of Ki.i. Hence relatively little is lost by choosing simple mathematical forms of the kernel. In this study, we adopt a coagulation kernel of the following form:

gi, j = K•(ij) • (4)

This kernel can be derived from physical considerations by assuming that the reactivity of a k-mer is proportional to its effective surface area sk so that Ki, j • $isj. The above expression is then obtained if one assumes that s k • k •ø for large clusters, where w is the exponent characterizing the aggregate surface area [Ernst and Hendriks, 1984]. In terms of the VDE classification,

X = 2w (5a)

and

for this kernel.

p• = to (5b)

2.2. Gravitational Settling

The removal of an aggregate from a parcel of water due to settling may be described mathematically by

Rs = wj O z (6)

where w• is the settling velocity of a j-mer. A mathematical expression for w• can be derived by considering the two major forces exerted on an aggregate: gravity and drag. The gravitational force experienced by the aggregate is

F% = p•j(pp- pf)g (7)

where v• is the volume of a single particle, pp is the density of the individual phytoplankton particles, and p7 is the density of the fluid in which the aggregate is settling.

The gravitational force acting on an aggregate is balanced by a drag force that can be estimated from Stokes' Law:

Fz>j: 6•'l•vRhWj (8)

where/x v is the dynamic viscosity of the fluid and the hydrodynamic radius, Rh, represents the radius of a sphere that experiences the same drag force as the aggregate at the same

translational velocity. In general, the hydrodynamic radius of an aggregate grows with aggregate size j according to the following power law relationship [Miyazima et al., 1987; Rogak and Flagan, 1990]:

Rh '•r•j • (9)

where r• is the primary particle radius and the magnitude of exponent a depends on the degree to which primary particles in the aggregate occupy the embedded space, or the aggregate's fractal dimension, D. The following relationship between a and D is often assumed [Rogak and Flagan, 1990]:

a = 1/D (10)

Combining (8) and (9), we obtain the following expression for the drag force:

Fv, • 6,rtx•r•j•wj (11)

An expression for the terminal fall velocity of a j-mer can be obtained by assuming an equality between the left and right- hand sides of (11) and setting (11) equal to (7):

where

Wj = Wlj 1-a (12)

- p) g 6,r/xvrl

2.3. Grazing

The removal of phytoplankton aggregates by zooplankton in the water column can be represented mathematically as

Ra = k;(z)n• (13)

where k•(z) is defined as the rate at which a population of zooplankton removes j-mers per unit time at a particular depth, z. The grazing rate constant depends on the clearance rate for j-mers, ½• (with units of volume/time/grazer), and the concentration of grazers, G(z), present at depth z:

k•(z) = ½•G(z) (14)

A number of researchers have studied the relationship between the size of a food particle and the rate it is cleared from suspension by zooplankton, ½ [e.g., Frost, 1972; Gaudy, 1974; Gonzalez et al., 1990; Monger and Landry, 1991; Mullin, 1963; PaffenhOfer, 1971; Richman and Rogers, 1969; Wimpenny, 1973].Monger and Landry [1990] used a "force-balance model" to theoretically examine this relationship for passive feeding microzooplankton, like zooflagellates, and determined that the clearance rate of a grazer on a particle increases with the particle's radius: ½t oc radius. Because phytoplankton aggregates are fractal in nature [Logan and Wilkinson, 1990], we adopted the hydrodynamic radius to relate the clearance rate to the size of a phytoplankton aggregate: ½ oc Rh oc j•. Accordingly, the following expression applies for the clearance rate of j-mers:

15)

where ½• is the clearance rate of a single phytoplankton particle and the exponent a determines how the hydrodynamic radius varies with the number of constituent particles (see (9) and (10)). The magnitude of ½• depends on the physical feeding mechanisms employed by the zooplankton under consideration. The size, swimming speed, and number of cilia, fla-


gella, and pseudopodia will determine the volume of water processed by the zooplankton [Kiefer and Berwald, 1992]. Ex- amples of clearance rates of zooplankton for different size phytoplankton can be found in, or extrapolated from, the stud- ies by Frost [1972], Monger and Landry [1991], and Gonzalez et al. [1990]. In formulating a mathematical expression for grazing, we have assumed that (1) the zooplankton present are passive feeders that encounter food by direct interception, (2) the nutritional value attributed to the phytoplankton is high, and (3) the clearance rate increases linearly with the aggregate's hydrodynamic radius. With respect to the last assumption, it has been found that for any individual zooplankton species there is a limited range of food sizes for which the clearance rate increases with prey size [Kiefer and Berwald, 1992]. For food particles that are too small or too large, it is difficult for an individual zooplankton to handle the prey because of physical constraints. Hence the relationship employed here should be regarded as a first approximation of a complex phenomenon that is only partially understood at the present time [Hansen et al., 1997].

The concentration of grazers G in the water column varies seasonally, daily, regionally, and with depth [Raymont, 1963b]. This spatiotemporal variability in zooplankton concentration is controlled by many factors, including primary production rates, presence of predators, nutrient availability, and sunlight inten- sity [Raymont, 1963c]. Diel migration of zooplankton affects the vertical distribution of most zooplankton [Raymont, 1963d]. After sunset, some species of zooplankton migrate to the surface layers to feed when there is little danger of being seen by a predator. During daylight hours the zooplankton retreat to greater depths. In this study, we do not explicitly take

sider two "static" grazer profiles that may be representative of night and day conditions (see Figure 1). For the exponential distribution the concentration of grazers declines exponentially from a maximum of Go directly below the mixed layer:

G(z) = Goe :/• (16)

The parameter (5 represents the characteristic length scale over which the grazer concentration drops a factor 1/e. For the unimodal distribution the grazer concentration increases with depth from G -- 0 at z = 0, reaches a maximum of 2Go/e 2 at z = 2(5, then declines exponentially for z > 2(5:

G(z) = Gd2(z/8)2e -•/• (•7)

3. Mathematical Solutions

After substituting previously derived expressions for Ki, s, R,, and R, into (1) and letting the unsteady term go to zero, we obtain the following GDE for thc system illustrated in Figure 1'

/- I x K• dna

0:-2- • [i(j -i)]•n,n,__,- K• 5'• (ij)'øn,n,- w d'-" dz

- ½,j"G(z)% (18)

To minimize the number of separate variables that require consideration, the above equation can be recast in a nondi- mensional form by multiplying all terms by 2/K• •N•(O)'

J-1 oc

0-- • [i(j- i)]øø•i•j_ t -2 Z (iJ)øø•i•j d• _j•_,dh• t=l t=l

- Gof(7/a)j% (•9)

The reduced variables in this equation are defined as follows: 1. Value n s = ns/No(O ) represents the concentration of

j-mers at some depth normalized by the concentration of total aggregates at z = 0.

2. Value • = z/w•r• is the time required for a single particle to settle a distance z under the influence of gravity, relative to the characteristic timescale for the coagulation of single particles to form dimers, r• = 2/[K• •No(O)].

3. Value 0o = r•./% represents the characteristic coagulation timescale r•., relative to the characteristic timescale % = 1/G o • for the scavenging of single particles by grazers at z = 0.

4. Value f(•/8) is a function that determines the shape of the grazer distribution with depth. The parameter 8 = 8/w • r• represents the characteristic time for a single phytoplankton particle to settle through the portion of the water column where grazers are most abundant divided by the characteristic timescale for coagulation rc. The function f takes on the following forms for the exponential and unimodal distribution, respectively: f(g) = e - t and f(g) = g2e - g/2.

Equation (19) represents an infinite set of coupled differential equations for which no solutions are presently known. Attempts to solve this equation numerically are complicated by the so-called "finite domain" problem, in which an infinite number of aggregate size classes must be integrated on ma- chines with finite memory [Kobraei and Duncan, 1986]. Al- though several numerical schemes have been proposed to address the finite domain problem [e.g., Lawlet et al., 1980; •iiiams, 1981], each is subject to the limitations and errors inherent in such efforts.

An exact solution to (19) can be obtained for the choice of • = to = 1/2. To a first approximation, • is related to the fractal dimension of aggregates through (10), and therefore a choice of • = 1/2 corresponds to a fractal dimension of D = 2. Logan and Wilkinson [1990] have measured the fractal dimension of marine aggregates and found that D falls in the range of 1.26 < D < 2.14; hence, our choice of D = 2 is within natural limits. The choice to = 1/2 implies that our coagulation kernel has a homogeneity • = I and/z = 1/2. Ball et al. [1987] have shown that • = 1 and/z = 0 for reaction-limited aggregation (RLA) in which many particle-particle collisions occur for every sticking event. Our kernel fulfills the homogeneity requirement for RLA when to = 1/2 but fails to match the condition on/z. Because/z > 0 our solution will underestimate the reactivity between small and large clusters relative to RLA. It has been postulated that the collision efficiency favors the coagulation between equal-sized clusters in natural aquatic systems [McCave, 1984] which would be consistent with/z > 0, as assumed here. The choice of • = 1 is also not too different

from the homogeneity of the kernel for orthokinetic coagulation, for which • = 1.5 and /z = 1 if D = 2 (see Table 1). Although the values of the parameters • and/z for orthokinetic coagulation do not exactly match the values for the kernel employed in this study, the basic mathematical properties of both kernels are the same; that is, interactions between clusters of the same size increase with size, and coagulation occurs predominately between clusters of the same size. The choice of the constant K• would ultimately determine whether the ker-


nel employed in this study approximates RLA or orthokinetic coagulation (see Table 1).

The mathematical solutions for the particle size distribution, total particle size, and volume fraction are derived in the appendix and given as follows:

x J-1 I(• = O) -1/2

hj(•) = (1 + x) J+lj I(D (20a)

N0(D [ x ](I(•:0))( 1 ) (20b) No(O• = Lil/2 1 + x 1(2) x(1 + x)

No(O----• = Li_•/2 1 + x 1(2) x(1 + x)' (20c) In these expressions, Lib.[y] is the polylogarithmic function,

Lie[y] = • F (21) t=l

1(2) is an integrating factor determined by the grazer distribution,

If 1(2): exp Oof(•/•) d•

and x is a scaled depth,

(22)

these two moments of the aggregate size distribution are plotted in a double logarithmic format against depth, they should appear as straight lines with slopes that reflect the underlying coagulation and sedimentation mechanisms operative in a particular system.

When Grant et al. [1996] tested this idea on phytoplankton volume distributions published by Weilenmann et al. [1989] for Lake Zurich, Switzerland, they found that No and N• both decayed like power laws with depth. However, interpretation of the power law exponents was complicated by the fact that it was not clear what effect grazers might have on the similarity predictions for N o and N l. For the model solution derived in the present paper (see (20a), (20b), and (20c)), we have assumed that a = 1/2 and ,• = 1. Therefore, if grazers were not present in the water column, we would expect that No and N l would decay like power laws of depth with exponents given by e = 0.5 and y = 1.5 (see equations (25a) and (25b)).

In Figure 2, we present model predictions for N o and N• corresponding to three different grazer distributions: no grazers (first column), exponentially distributed grazers (second column), and unimodally distributed grazers (third column). For the case where there are no grazers, our solutions for N o and N• are given by (20b) and (20c) with

1(2) = 1 (26a)

x: • (26b)

' 0) x = dg (23)

These solutions are valid for any choice of the spatial distribution of grazers G(z), so long as it is a continuous function of depth.

4. Model Predictions

4.1. Predictions for No and N•

Grant et al. [1996] showed that when particles undergo steady state coagulation and settling, the aggregate size distribution predicted by the GDE develops a shape that is invariant with depth (i.e., "self-similar") and independent of the particle concentration at z - 0 (i.e., "universal"). In this case, the aggregate size distribution can be written in the following factored form:

na= s(z)•(rl) (24)

where r/is a reduced form of the aggregate size, r/ = j/r(z), s(z) and r(z) are functions of depth, and {• determines the "shape" of the aggregate size spectrum. Substituting this factored form of the aggregate size distribution back into the GDE for steady state particle coagulation and sedimentation, a similarity analysis reveals that N o and N• should both decay like power laws of depth, where the power law exponents are determined by the homogeneity of the governing coagulation kernel ,• and the way the hydrodynamic radius varies with aggregate size a:

No • z -• y > 0 (25a)

N1 • z -• e > 0 (25b)

where 3' = (2 - a)/(3 - 2a - ,•) and e = (1 - a)/(3 - 2a - ,•). From a practical point of view, these results suggest that when

The resulting expressions are plotted against reduced depth 2 in Figure 2d. Consistent with power law expressions for No and N•, these two moments appear linear when plotted in the double logarithmic format employed in the figure. The power law exponents, as represented by the slope of the curves in Figure 2d, can be computed directly from our solutions, and these values are plotted in Figure 2g. As predicted by similarity theory, the power law exponents for N o and N• (3' and e, respectively) achieve constant values of 1.5 and 0.5, respectively, provided that 2 > 100.

When zooplankton are present in the water column, the variables x and 1(2) appearing in (20a)-(20c) take on the following expressions for the exponential distribution,

1(2): exp [-•0•e -•/•] (26c)

x = •e-Oø•{Ei[Oo•] - Ei[Oo•e-•/•]} (26d)

and for the unimodal distribution,

1(2): exp [-(22 + 2•2 + 232)00e-S/•/•] (26e)

x = exp -G08 + o •g 2• ( + 2•s• + 2•g) ds• (26f)

where Ei[y] = J'Y-o• et/t dt. Plots of N o and N• for these two different grazer distribu-

tions are shown in the second and third columns of Figure 2, for the choice of 0o = 2 x 10 -s and • = l0 s. The same total number of grazers are present in the water column in these two example grazer distributions. The fact that there appears to be more grazers associated with the exponential distribution is an artifact of the logarithmic presentation. Interestingly, the presence of grazers in the water column only affects the depth evolution of No and N• at intermediate depths (10 4 < • < 107). The reason grazers do not affect No and N• for • < 10 4


No Grazers

lO

8

•,o 6 ,• 4

2 A o

I I I

lO 0 lO 3 lO 6 lO 9

Exponential Distribution

10 •

8

•, 6

'•'4 2

0 I I I

10 0 10 3 10 6 10 9

Unimodal Distribution

lO

8

•,o 6 ,• 4

o I I I

lO 0 lO 3 lO 6 lO 9

lO-5

o_1O

I I I

lO 0 lO 3 lO 6 lO 9

2.0 i 1.5 1.o 0.5

0.0

I I

100 103 106 10 o

10 -5

10 -10

I I I I

10 0 10 3 10 6 10 9

2.0

1.5

1.0

0.5

0.0

I I I

10 o 10 3 10 6 10 9

10 -5

1ø"ø / F x,, I I I I I

10 0 10 3 10 6 10 9

2.0

1.5 --'s 1.0

0.5

0.0

I I I I

10 o 10 3 10 6 10 9

Reduced Depth, •

Figure 2. The effect that (a)-(c) three grazer distributions G have on the (_d)-(f) evolution of No (solid line) and N• (dashed line) and on the (g)-(i) power law exponents for No and N•.

follows from our mathematical formulation of the grazing process: We assume that the clearance rate increases in propor- tion to the hydrodynamic radius of the phytoplankton aggregates. Hence scavenging of the phytoplankton occurs most efficiently at intermediate depths where the phytoplankton are present in large clusters. At intermediate depths (104 < 2 < 107), the moments deviate significantly from the power law behavior predicted by similarity theory, as is manifested by the "bumps" in the curves for No and N] (Figures 2e and 2f) and the excursions in the calculated power law exponent values (Figures 2h and 2i). Presumably, if the phytoplankton leaving the mixed layer were aggregated, grazers would impact the values of No and N• at a shallower depth.

For 2 > 107 the moments appear to recover their power law behavior, and the power law exponents reestablish the values predicted by similarity theory. At these large depths, the grazer concentrations have dropped to near zero and therefore grazing rates are low, although this observation alone cannot account for the reestablishment of the power laws. The power laws for No and N• are reestablished for 2 > 107 because of the "universality" of steady state coagulation and sedimentation. As the phytoplankton aggregates settle out of the region where grazing has the most impact, coagulation and sedimentation are the dominant processes, and No and N• have no "memory" of grazing that occurred at shallower depths.

The plots presented in Figure 2 are for a particular choice of the parameters Go and &. The extent to which zooplankton grazing p_erturb_s the power laws for No and N• depends on the product Go x &. Also, the reduced depth at which the grazers impact No and N• is determined by the magnitude of Go. Figures 3 and 4 illustrate these points. In Figure 3 we have

•_ ISimilaritv Theory ~ 1 ß - .

• 2.0 I ' •Jø =tv - • 1.5

•_! 1.0

-• 0.5

o 0.0 ,oX3=5

I I I

•0 ø •0 3 •0 • •0 •

t- Q)

0

X

0

•0 = 10-2

1.5- •,0x3=5

1.0 - 1.0 0.5

0.0

I• Similarity Theory I I

10 o 10 3 10 6 10 9

Reduced Depth,

Figyre 3. _The effect •;o X • has on the power law behavior of No_ and N• when the grazers are distributed exponentially and Go = 10 -2.


plotted the power law expon•ents calculated for the exponentially distributed grazers for G o = 10- 2 and for three different choices of the parameter Go x •i. This last parameter represents the characteristic time •5/w• that single phytoplankton spend settling through the portion of the water column dominated by grazers, divided by the characteristic time r a for the scavenging of single phytoplankton particles by grazers at z = 0. Grazing has little influence on the power law behavior of No and N• when the settling time is short and the grazing time is large (i.e., (7o X • -< 0.1 in Figure 3). As (7o X • is increased above this value, the magnitude of the perturbation becomes larger, and it extends over a larger range of depths. When G O x •i is very large (> 10), most of the phytoplankton are lost to grazers and both No and N• decay with depth according to the following expressions for exponentially distributed grazers:

No, N• • exp [Go•5(e -•/• 1)] (27a)

and for unimodally distributed grazers

No, N• • exp [GoS(e-S/•Oo(•2• + 2• 2- 2•3)/2• 2- 1)]

(27b)

If 0 o x • is sufficiently large to perturb the power law behavior of No and N1, the reduced depth at which the perturbation occurs increases with decreasing Go, as illustrated for two choices of this parameter in Figure 4.

To summarize, No and N 1 evolve with depth differently depending on whether sedimentation or grazing is the dominate sink for phytoplankton. In the former case, No and N1 are predicted to decay like power laws of depth with exponents that can be mathematically related to the underlying physical and chemical processes responsible for coagulation and sedi-

rSimilarity Theory

1.5

1.0

0.5

= 10- 10-6

o

x

o. 0.0

I I I

10 0 10 3 10 6 10 9

1.5-

1.0-

0.5-

00 =.•0 -2 •0x3=5

N• 10 6.•-•

o 0.0

a. Similarity Theory I I I

10 0 10 3 10 6 10 9

Reduced Depth, •

Figu[e 4. The effect •;o has on the power law behavior o•f•o and N1 when grazers are distributed exponentially and Go x

Table 2. Hypothetical Ocean Parameters

Variable Value

100 m a

•1,1 10-19 m3/S P articlesb 1,1 1'0c • 2.0 txm d

A iø = iøp- Of 0.2 g/cm 3e ½1 1 X 10 -•s m3/grazer S f w• = d•2(Ap)g/181x,, 10 -7 m/s Kll = o/11•311 10 -i9 m3/s particles

Important Nonbloom Bloom Parameters Conditions Conditions

No(0) 10 3 cells/ml g 10 6 cells/ml h _G O 10 4 grazers/m 3 10 3 grazers/m 3i Go 0.06 3 X 10 -6 15 0.17 350

0o X • 0.01 0.001 % 6 X 109 S 3 X 106 S % 1 x 10•s 1 X 10•2s •/w• i X 109s i X 109s

aEstimate of depth below mixed layer where grazer numbers fall to 1/e of their maximum value.

bFollowing Burd and Jackson [1997], we use a shear rate of s/v - 1.0 s-• for the rectilinear formulation of the shear kernel (see Table 1).

CWe assume every collision between phytoplankton results in a sticking event.

dDiameter of A. anophagefferens reported by Smayda and Villareal [1989] for a bloom event in Narragansett Bay.

eBurd and Jackson [1997] estimate 9p - 1.2 g/cm 3 is typical for algal cells. We take 9f = 1.0 g/cm 3.

fObtained by extrapolating data of Monger and Landry [1991]. gConcentration of algal cells during nonbloom conditions approxi-

mated to be 3 orders of magnitude lower than that during bloom. hAlgal cell concentration of A. anophagefferens during a bloom in

Narragansett Bay taken from Smayda and Villareal [1989]. iprior to a spring bloom, the concentration of zooplankton is rela-

tively low. Here we choose a concentration i order of magnitude smaller than that used for the nonbloom conditions.

mentation. In the latter case, No and N• decline with depth in an approximately exponential fashion. For the mixed case, where both removal processes are important, the depth evolution of N O and N1 depends in detail on the relative magnitude of grazers and phytoplankton at the surface, the spatial distribution of grazers, and clearance rates of the dominate zooplankton species considered.

What does the model tell us about the likelihood that power laws for No and N1 will occur in ocean or lakes systems? To address this question we, (1) compiled estimates for each of the model parameters assuming either bloom or nonbloom conditions (see Table 2), (2) assumed that the collision frequency function can be estimated from the rectilinear formulation of the shear kernel (see Table 1), (3) assumed that the zooplankton are distributed exponentially with depth with •i = 100 m, and (4) assumed that the maximum number of grazers found during bloom conditions is an order of magnitude less than that for nonbloom conditions. The last assumption is based on the hypothesis that decreased grazer numbers gives rise to a phytoplankton bloom [Turner and Tester, 1997].

Given the above assumptions, values for the power law exponents 7 and e were calculated using (20b) and (20c) for both bloom and nonbloom conditions and plotted against depth in Figure 5. For the bloom condition (Figure 5b), the model predicts that No and N1 achieve power law status within <100


o o_ x

iii

o

1.5• T Non-Bloom Conditions 1.0--

0.5--

0.0

10 o

œ

i i i iiiii I i i i iilil I i i i iiiii I i i i iiiii

101 10 2 10 3 10 4

1.5--

1.0-

0.5-

0.0-

Bloom Conditions -

10 o 101 10 2 10 3 10 4

Depth (meters)

Figure 5. Behavior of the power law exponents under hypothetical oceanic conditions. When bloom conditions exist, the power law behavior of •o and • is established between 10 and 100 m below the mixed layer.•When nonbloom conditions exist, the power law behavior of No and N• is not observed.

m below the bottom of the mixed layer, and the similarity predictions for 7 and s are sustained over the rest of the water column. In contrast, for the nonbloom condition (Figure 5a), No and N] do not achieve power law status except for physi- cally unreasonable depths in > 10 km below the bottom of the mixed layer. These predictions are consistent with previous suggestions that coagulation and sedimentation will dominate phytoplankton dynamics only during bloom events [Billet et al., 1983; Riebesell, 1991a, b; Wassmann et al., 1990]. Importantly, our results suggest that the power law behavior of N O and N] predicted for phytoplankton blooms may reveal much about the underlying physical and chemical processes at work.

4.2. Aggregate Size Distribution

In this section, we examine how coagulation, sedimentation, and grazing influence the phytoplankton aggregate size distributions that evolve with depth. As mentioned earlier, when no zooplankton are present in the water column, the aggregate size distribution •i adopts a self-similar form below some crit- ical depth and the size distribution can be written in the factored form given by (24). For the case where there are no grazers in the water column, the functions s(2), •(r•), and r• take on the following forms for our solutions:

S(2) •'[5/2{2) (28a) -- .t • 0.5 •,

r• = j•0.s(2) (28b)

•(r/) = r/-•/2e-'• (28c)

/•/0.5 = E jO.5•j. j->l

In Figure 6a, we have plotted the aggregate size distribution predicted by our solutions when there are no grazers present in the water column and for several different depths below the mixed layer. If the factored form (24) of the particle size distribution is satisfied, then the size distributions corresponding to each depth should collapse onto the curve given by •(r•) when ni/s (2) is plotted against r•. When the size distributions in Figure 6a are rescaled according to similarity theory (Figure 6b), they all collapse onto (28c) (line in Figure 6b), provided that 2 -> 100. Hence the primary assumption underlying the similarity theory discussed in the previous section; namely, the aggregate size distributions adopt a self-similar form, is satisfied in this case. In Figure 6c we have plotted the aggregate size distribution predicted by our solutions for the case where grazers are exponentially distributed in the water column and •o = 10-2 and • = 5 x 102. Comparing Figures 6a and 6c, we find that grazers influence the aggregate size distributions in two ways, by (1) depleting the large end of the phytoplankton size spectra for depths below 2 = 100, and (2) elevating the concentration of phytoplankton aggregates at the small end of the size spectrum for 2 - 1000. This last result is somewhat counterintuitive but follows from the fact that grazers reduce the overall number of phytoplankton in the size distribution and therefore the transfer of small aggregates to the large end of the size spectrum by coagulation occurs at a slower rate. When these aggregate size distributions are rescaled according to similarity theory (Figure 6d), they also collapse onto (28c) (line in Figure 6d). Although the shape of the distribution is the same, the definition of the scaling function •(2) and the reduced aggregate size • are now modified by the function /(•)'

S(2) -- [I(2)/I(0) -13/2•'[5/2[ *• J • • 0.5 •,/" ! (29a)

il = j[I(2)/I(O)]No.s(2) (29b)

When there are no grazers present in the water column, I(2) = 1, and the expressions for • (2) and • reduce to those given by (28a) and (28b). These results indicate that the aggregate size distribution that develops in the presence of grazers can still be described as self-similar and that the exact

nature of the functions •(2) and • depends in detail on the vertical distribution of zooplankton, as represented through the functionality of I(2).

4.3. Phytoplankton Partitioning

What factors determine whether carbon fixed at the surface

enters the pelagic food web or settles to the benthos? Our mathematical solutions provide a first-order answer to this question. The fraction of phytoplankton leaving the mixed layer that is lost to zooplankton is given mathematically as follows:

F = 0of(2/•) • j•j d2 (30) j->l

If we substitute our solutions for fii into (30), we obtain the following simple result, which is valid for both exponentially and unimodally distributed grazers:

F= 1-exp(-00x•) where (31)


10 o

10 -5

fiJ 10 '1ø 10 -15

I I I

10 0 10 2 10 4 10 6

1øø 1•+• c fl. 10 '5 +

J

10-•o

10-15 I I I

10 0 10 2 10 4 10 6

10 0 _

10 -5 _

10 -•o _

10 -15 _

10 -1

++

I I

10 0 10 1 77

10 2

+ 1

ß lOO

o lOOO

i• lOOOO

10 ø_

10-5 + ++

10-•o

10 -15 I I

10 '1 10 0 10 1 10 2

Figure 6. The phytoplankton size distributions predicted by the model (a) when no zooplankton are present in the water column, or (c) when the grazers are exponentially distributed with •;o = 10-2 and • = 5 x 102. (Figures 6b and 6d) When these aggregate size distributions are rescaled according to similarity theory, they collapse onto a single curve.

This equation predicts that the flux of phytoplankton to higher trophic levels is controlled exclusively by the product Go x •i (see Figure 7). When this parameter is unity, the phytoplankton mass is roughly equally partitioned between the benthos and zooplankton. When Go x •i > 1, the majority of phytoplankton leaving the mixed layer are recycled to higher-trophic levels via zooplankton, and when •o X • < 1, most of the phytoplankton aggregates fall to the bottom to be incorporated into sediments or consumed by benthie organisms. It is important to acknowledge that our model does not incorporate the

x

0.8--

0.6-

0.4-

0.2-

To Sediment \ \

To Higher \

Trophic L•/// \ \ ._ _ 0,0-- I I I I I

10 -3 10 -2 10 -1 10 0 101

Figure 7. The fraction of phytoplankton leaving the mixed layer that enters the pelagic food web by zooplankton con- sumption (solid line) or falls to the benthos (dashed line) as a function of the product G o x •i.

mineralization of phytoplankton by bacterial degradation; this process may play a major role in determining how the planktonic carbon is partitioned in the aquatic community [Landry, 1992].

5. Conclusions

Much progress has been made over the years in elucidating the individual processes that affect phytoplankton concentrations in natural aquatic systems. In this study, we investigated the combined effect these processes have on phytoplankton distributions by directly solving the GDE that describes their coagulation, sedimentation, and removal by zooplankton grazing.

In solving this GDE, it was necessary to make several sim- plifying assumptions, which limit the solution's applicability to natural systems. One of the potentially restrictive assumptions is that the particle size distribution maintains a steady state throughout the water column. In practice, both the phytoplankton distributions and the zooplankton profiles will fluc- tuate on diel and seasonal timescales. However, our solutions may still apply if the timescale associated with the relaxation of the particle size distributions to their steady state condition is sufficiently short, and the total concentration of grazers at the surface (No(z = 0)) and grazer distribution (G(z)) vary slowly with time. Further research is needed to determine if these "quasi-steady state" conditions are likely to prevail in natural systems. We also assume that single phytoplankton particles flux from the mixed layer at a constant rate. In reality, aggregation occurs in the mixed layer and aggregates may flux


from this region as well. However, because of the universality of coagulation and sedimentation, the particle size distributions that develop at depth should have no memory of the aggregation state of phytoplankton near the surface.

When zooplankton concentrations are low, our solution to the GDE can be described using the language of dynamic scaling theory. Namely, the aggregate size distribution develops a shape that is self-similar and universal, and the total particle concentration No and the particle volume fraction N• decay like power laws of depth. The power law exponents can be directly related to the physical and chemical processes responsible for coagulation and sedimentation. When grazers are present in the water column at high concentrations, the phytoplankton size distributions are still self-similar, but No and N• decay exponentially with depth. From a practical point of view these results suggest a simple methodology for distin- guishing between systems that are dominated by coagulation and sedimentation on the one hand and zooplankton grazing on the other hand. In the former case, No and N• should decay like power laws of depth with exponents that are within the bounds dictated by similarity theory (see (25a) and (25b)), while in the latter case, both moments should decay exponentially with depth.

Our model also provides a conceptual framework for determining whether phytoplankton settling out of the mixed layer are consumed by zooplankton or escape to the benthos. The partitioning of phytoplankton between these two pathways is determined •by the magnitude of a single dimensionless number 0o x 8, which represents the ratio of timescales for sedimentation and for grazing. When Go x 8 is computed for oceans, we find that its magnitude is in the range where small changes in the values of key system variables (like the concen- traticm eft phytoplankton at the surface) have a significant influence on the fraction of phytoplankton partitioned between these two pathways.

Appendix In this appendix, we derive the expressions for h•, No, and

N• presented in the text. We begin by moving the third term on the right-hand side of (19) to the left-hand side and rewriting the equation in terms of a new dependent variable p• =

-- ot• . t/j.

dpj j-1 • d2- • [i(j- i)]ø•+•-•pipj_,- 2 • (ij)ø•+•-lpipj

/=1 ,---1

- •Jof(•/•)j2•-•pj (A1)

The explicit dependence of the right-hand side on i and j can be eliminated by choosing w = a = 1/2'

pj j- 1 • d2 = •;• P'PJ-'- 2 • PiPj- •aof(2'/•)pj (A2)

t=l t=l

This equation can be further simplified by applying an integrating factor 1(2)/1(2 = 0), where

1(2) = exp Gof(US) d• (A3)

and defining a new dependent variable b i = py 1(2)/1(2 = 0)'

I(Z•) (dbj'• j-1 1(2, = O) dO, ] = •;• K;'j-ibibj-i- 2 •;• i=1 i=1

(A4)

A change of variables is made such that d/dx = 1(2)/1(2 = O) d/d2:

db• , , dx = • Ki,•-ibib•-i- 2 • Ki,•bibj

i=1 i=1

(A5)

which corresponds to

fo s I(s r = 0) x: i(sr • d• (A6)

Equation (A5) is to be solved subject to the boundary •on- ditions

bl(• = 0) = 1 (A7a)

b•>,(•. = 0) = 0 (A7b)

These boundary conditions imply that all particles entering the system at z = 0 are present as single particles.

Equation (A5) is mathematically identical, or "isomorphic," to yon Smoluchowski's [1917] equation for the coagulation of particles for a constant kernel for which the solution is given as follows:

XJ - 1

b• = (1 + x) j+l (A8)

From the definition of by and py we recover the solution presented in the text (see (20a)). This solution collapses to the solution derived pr?iously for the case where there is no biological removal, Go - 0 [Grant et al., 1996]. Expressions for N O and N• (see (20b) and (20c)) follow from their respective definitions.

Notation

d l diameter of a single phytoplankton [L].

D fractal dimension of aggregates. f(•/•) function that determines the shape

of the grazer distribution. F fraction of phytoplankton leaving

the mixed layer that is lost to zooplankton.

FD• drag force experienced by a j-mer [ML/T2].

F% gravitational force experienced by a /-mer [ML/T2].

# gravitational acceleration [L/T2]. G O determines the maximum number

of zooplankton in the water column [l/L3].

Go = •'c/•'e reduced form of the maximum zooplankton concentration Go.

G(z) concentration of zooplankton [l/L3].

I(z) integrating factor determined by the grazer distribution.


i, j, k number of constituent particles in an aggregate.

k i rate at which a population of zooplankton removes j-mers [ 1 / T].

Ki, i kernel for the coagulation between phytoplankton aggregates of containing j and i constituent particles [L 3/T].

n• concentration of phytoplankton j- mers [l/L3].

• = nj/No(O ) reduced concentration ofj-mers. No = 5• na the total concentration of

phytoplankton aggregates [ 1/L 3]. No(z) = No(z)/No(O ) reduced form of phytoplankton

concentration.

N• = 5• jv•na the volume fraction of phytoplankton [ 1/L 3 ].

N•(z) = N•(z)/N•(O) reduced form of the volume fraction.

= jo - reduced form of 1/2th moment of l•o. s 5•>_ • 'S n• the aggregate size distribution.

r• radius of single phytoplankton [L]. R h hydrodynamic radius of an

aggregate [L ]. R e removal rate of phytoplankton

aggregates by zooplankton [ 1 / TL 3 ]. R s removal rate of aggregates via

sedimentation [ 1 / TL 3]. s i the effective surface area of an

aggregate containing i primary particles [L 2].

s (z) function that relates n i to •(r•) [l/L3].

wi gravitational settling velocity of a j-mer [L/T].

x scaled depth [L]. z depth below the mixed layer [L].

2 = rs/r c reduced depth.

Greek letters

a exponent relating cluster size to hydrodynamic radius.

ai, i attachment efficiency for the coagulation between aggregates containing j and i phytoplankton.

/3•,• collision frequency function for coagulation between aggregates containing j and i phytoplankton [L3/T].

8 length scale for grazer distribution [L].

8 = 8/w•r c reduced form of the length scale $. e exponent describing power law

behavior of N•. •/ exponent describing power law

behavior of N O . F shearrate[l/T]. r• reduced form of the aggregate size. X homogeneity of coagulation kernel. /• an exponent governing the small i/j

limit of K•,•. /•v dynamic viscosity of the water

[M/LT].

pp density of a phytoplankton particle [M/L3].

pf density of the fluid [M/L3]. rc = 2/[K• •No(0)] characteristic timescale for

coagulation [ T]. rg - 1/Go• characteristic timescale for grazing

[r]. r s - z/w • characteristic timescale for settling

[r]. •,• volume of a single phytoplankton

[L3]. •o exponent characterizing aggregate

surface area.

•(r/) "shape" of the normalized aggregate size distribution.

½j clearance rate of aj-mer [L3/T].

Acknowledgments. The authors would like to thank Bruce Mon- ger, Hans Dam, and an anonymous reviewer for their thoughtful com- ments on the manuscript. This work was supported by a NSF Career Award (BES-9502493) to SBG. Matching funds from the National Water Research Institute (NWRI-21851) are gratefully acknowledged.

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(Received January 19, 1998, revised April 9, 1998; accepted April 9, 1998.)

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