The theory of advective effects is extended to include: localization effects due both to the attenuation of light with depth in the ocean, and to nutrient transport into the euphotic zone of finite duration in time, and/or over a limited horizontal domain. Also nutrient uptake is generalized to nonlinear Michaelis-Menten kinetics. The characteristic curves are solved for explicitly and a symbolic general solution is obtained for arbitrary biological dynamics. Some exemplary results are presented for the effect of light, nutrient, and grazing limitations on primary productivity in an NPZ-model. The theory is now applicable to further studies of more realistic oceanic processes.

Interactive physical-biological dynamical processes in the sea are of central importance to fundamental and applied research in many areas of interdisciplinary ocean science today, including e.g., the dynamics of ecosystems and biogeochemical cycles. Three major interactive processes are physiological thermodynamics, 1 diffusion and mixing, and advection. This study is focussed on the advective process and attempts to provide an idealized general theoretical framework for the exploration of effects of various phenomenological flow fields which occur in the ocean over a broad range of time and space scales. It is intended to complement related process research based upon experimentation and simulation. Additionally this theoretical approach may be of some general interest for applications to analogous problems of the reactive dynamics of advected tracers in other fluids, e.g., in chemical and engineering problems.

The first part of this study (Robinson, 1997) introduced a biological dynamical model consisting of growth, self-interaction and bilinear interactions among n state variables occurring in the presence of a stretching deformation flow field. A general theoretical approach via the theory of characteristics was formulated and some solutions were obtained for dynamical processes homogeneous in space and impulsive in time. In this second part, the nonlinear dynamics is extended to include the Michaelis-Menten nonlinear formulation for the uptake of nutrients by phytoplankton. The ocean is divided into an upper ocean where sunlight is available for photosynthesis (euphotic zone) and a deeper ocean which is not illuminated (aphotic zone). Kinematical flow fields are introduced which allow for the localization of advective effects both horizontally and in time. A general solution is obtained for these kinematics and dynamics and some idealized illustrative examples presented. These developments should serve as the basis for theoretical studies of realistic oceanic processes.

Section 2 presents the model, section 3 solves for the characteristics and section 4 solves for the dynamics. Section 5 derives the general solution, section 6 provides examples and section 7 summarizes and concludes.

The general model to be studied is that of equation (2.3) of part I (which we will refer to as equation (I.2.3)) for n biological state variables f_i in two spatial dimensions with diffusion neglected

For the study of light limitation and to illustrate the effects of flow kinematics on biological dynamics we adopt the general NPZ-model of Part I section (4d), but with Michaelis-Menten kinetics of nutrient uptake and light attenuation (Parsons et al., 1984; Kirk, 1994). Then for phytoplankton (f₁ = P), zooplankton (f₂ = Z) and nutrient (f₃ = N), the governing equations are (with reference to (I.2.10) and (I.4.11)

The dependent variables N,P,Z  ([ml^-3]) are all dimensionalized by a biomass density M, where M is to be chosen as a characteristic value of N advected into the euphotic zone during any injection event of interest. The independent variables (x,y,t) are scaled respectively by (x₀,y₀ = y_e,t₀ = K(Ma₁₃)^-1) where x₀ is an horizontal length characterizing a localized upwelling event. The horizontal and vertical velocities (u,v) are scaled respectively by (u₀, v₀ with u₀ = v₀x₀y₀^-1), which yields a nondimensional continuity equation u_x+v_y = 0. This implies a scaling of y₀ = u₀y₀ = v₀x₀.

The kinematical flow field is chosen such that the horizontal velocity is independent of y and the product of a function of x times a function of t. As in part I both dimensional and nondimensional variables are represented by the same symbols. Thus nondimensionally

For the flow system given by the stream function of equation (2.4a) the characteristic equations (I.2.12a) take the form

The function f(t) is chosen so as to represent either i) a steady-state flow (S) or ii) a time-periodic flow with nondimensional frequency w, which over a half-period may be taken to represent an advective event (E). Thus

Two initial value problems in s are necessary to carry out the dynamical studies of the next sections. We assume that light penetrates, and biological activity occurs, only to the base of the euphotic zone located at y = 1. The first problem is the time initial value problem (T) as in part I. The second problem is the boundary value problem (B) which specifies the values of the state variables as they are advected across the base of the euphotic zone from the deeper aphotic zone where l = 0. As will be seen below, the T problem is relevant to the deep pure advection (TA), and to the biological dynamics in the water located initially in the upper euphotic zone (TE). Most interestingly, the B problem describes the biological activity, in the euphotic zone, of water located initially below the euphotic zone. Under equations (I.2.13) and (3.2)

Combining the space-time kinematics, i.e., solving equations (3.3) with the structures of (3.4) yields six flow fields summarized in Table 1. The twelve sets of characteristic curves corresponding to the T and B problems for each flow are presented in Table 2. To study light limitation effects, it is necessary to evaluate the light attenuation coefficient and its integral in s, i.e., l(y) of equation (2.3d) in terms of y(s; p,r,q) which is given in Table 3. The F(s,p) is obtained from equations (3.4a,b) evaluated with t = s+p for the S and E flows; p = 0 and q = 1 respectively for the T and B problems.

The s-domain biological dynamics for the NPZ system to be studied here satisfies the nondimensional equations

The presence of the light attenuation (l) and nutrient saturation (d) parameters in addition to the grazing parameter (b) generalize the NPZ system of equations (I.4.11). To elucidate basic processes we shall consider the dynamics of systems of one, two, and three state variables.

The Malthusian growth of phytoplankton with unlimited nutrient and no predation (Z = 0) satisfies (4.1a) with d >> 1

We retain Z = 0, insert P from (4.1d) into (4.1c) and integrate, whence

To continue analytically following the procedure of equations (I.4.12a,b,c) it is necessary to restrict consideration to a uniformly illuminated euphotic zone overlying a deep completely aphotic ocean, i.e.,

For the case of b = 1 we can generalize the results of (I.4.12d,e) for dN £ 1. Keeping three terms in the Taylor series expansion of the exponential, the denominator of (4.5b) becomes

For the n state variables of equation (2.1), there are n-dynamical equations

The general solution to (5.1a) is expressed as

Consider now the case of pure advection everywhere, i.e., set B_i = 0 in (5.1a). Then the advective solution (5.2b) is valid also in

^ y

< y < 1

. Also, however, for pure advection the state variables are uncoupled and the general solution (5.3a) reduces to

Thus the general solution (5.3a) with biological dynamics has the final form

The general results derived here are intended to provide the basis for a number of theoretical studies relevant to real ocean processes. Here we will simply illustrate effects in terms of a few idealized simple examples.

We will first consider the localization effects arising from biological activity being restricted to the euphotic zone, in terms of the time initial value problem and deformation field flow of part I., i.e., the DSB and DST flows of Table 2. We assume initially no nutrient in the euphotic zone but a reservoir of nutrient and seed populations of plankton below the euphotic zone. The TE problem is trivially advective. For y ³ 1 (5.2b) takes the form, e.g.,

The simplest example is the Malthusian growth of equation (4.2). Without loss of generality we can set d = 1, i.e., by replacing a₁₃ in (2.4g) by d^-1a₁₃. We consider two cases: i) uniform light; 3 ii) linearly decreasing light. Then

i)

l = l0 ,   s = l0 s =

l0 a

ln1/y ,      es = (1/y)[(l0)/(a)] ,

      

and

ii)

l = 1-y ,      s =

1 a

[ln1/y+(y-1)] ,      es = [1/y e(y-1)][1/(a)]

( i) and ii) comprise 6.2a)

For the case of no x-dependence, the solutions (4.2) are

i)

P(y,t) = P0(yeat)(1/y)[(l0)/(a)] ,      P(1,t) = P0(eat) ,

P(

^ y

,t) = P0(1)el0t

      

and

ii)

P(y,t) = P0(yeat)[ 1/y e(y-1)][1/(a)] ,   P(1,t,) = P0(eat) ,

P(

^ y

, t) = P0(1)et+[((e-at-1))/(a)].

( i) and ii) comprise 6.2b)

Consider first the case of a deep reservoir of seed plankton which is independent of depth, P₀ = 1. Then for

(

^ y

< y < 1)

there is a steady state solution, since water parcels reaching a given depth at any time have spent the same amount of time in the euphotic zone. The shallowest parcels have spent the longest time under illumination, resulting in the simple exponential growth at

^ y

, independent of a, for case (i). For short times and depths near unity the two solutions are dominated by advection. If the time interval of interest is

t <

^ t

and

a

^ t

>> 1

, then the choice of an effective uniform illumination

l0 = 1-(a

^ t

)-1

equates

P(

^ y

,

^ t

)

for the two cases. Any criteria to determine an effective l₀, e.g., integrated net production, must take into account the flow (a) and duration

(

^ t

)

.

We extend the case of uniform deep reservoirs with DS flow to the two (2) and three (3) state variables biological systems and consider essentially a unit source of nutrient together with very small amount(s) of background plankton(s). The solutions retain the character of establishing a steady state below a shoaling front. The analytical solutions to equations (4.3a) and (4.5e) are particularly simple with (2) B = 1, and (3) D = 1 respectively, which is achieved by choosing

For the N-P model (4.3a) now becomes

For the N-P-Z model the solutions (4.5e) reduce to

Figure 1. Profiles of N (dot-dash), P (solid), Z (dash) versus depth. Dependencies upon: (a) advection for fixed e = 0.1, (a = 1: upper P_m, a = 0.1: middle P_m, a = 0.01: lower P_m); (b) seed plankton for fixed a = 0.1 (e = 0.1: larger P_m, e = 0.01: smaller P_m). N_m=Z_m at P_m identifies the associated curves. The

^ t

scale to the right of (1a) is for a = 1; for a = 0.1 (0.01) multiply by 10 (10²).

Now consider the case that at t = 0 nutrient increases linearly with depth from zero at the base of the euphotic zone to unity at a nondimensional depth (H) in the aphotic zone. We retain the assumptions of no x-dependence, no nutrient in the euphotic zone initially in time, and a uniform deep reservoir of seed phytoplankton. Thus

Figure 2. Isolines of N (dashed), P (solid) in the y-t plane as a function of a,d,e. The dynamically inert upper euphotic zone is shaded.

Finally spatial localization is illustrated by the solution of this problem with idealized coastal upwelling kinematics (LS flow). Figure 3 shows sections (xy plots) of phytoplankton concentration for as a function of a,e,d also with l = 1-y and H = 2. These plots are for the last time shown on the corresponding plots of Figure 2. The P(y) at x = 0 on Figure 3 are thus identical to the final profiles of Figure 2. The front is advancing as

Figure 3. As in Figure 2 for P but in the y-x plane for fixed t as indicated.

A general theoretical solution has been obtained for a model ocean in which a dynamically active near-surface euphotic zone overlies a deeper region in which biological material is passively advected by the physical flow field. Illustrative dynamical solutions have been presented in one-to-three state variables for an NPZ-model in which nutrient uptake is nonlinearly modeled by Michaelis-Menten kinematics. Parametric dependencies are represented in terms of four nondimensional parameters: i) the ratio of the nutrient uptake rate to the advection rate (a); ii) the ratio of the zooplankton grazing rate to the uptake rate (b); iii) the ratio of biomass to the saturation constant (d); and iv) the ratio of the seed plankton biomass to nutrient mass in the aphotic zone (e). A sensitivity analysis has been initiated. Interesting results are indicated for the location, shape and magnitude of phytoplankton maximum and associated nutricline in the euphotic zone, and for the dynamical mechanism by which phytoplankton mediate the conversion of nutrient to zooplankton biomass. For general biological dynamics kinematical flow fields have been introduced representative of coastal upwelling, isolated open ocean eddies and wave fields; and upwelling events which set-up in time over a finite time interval. Explicit solutions for the associated family of characteristic curves have been obtained.

These results provide a theoretical framework for further studies of more realistic oceanic processes. Weak background mixing in the lower euphotic zone will merely provide some smoothing of the solutions. For the upper euphotic zone, a mixed layer model has been added to the model. Work is in progress extending the model to include zooplankton mortality (Steele and Henderson, 1990). Interesting application areas include mesoscale eddy nutrient injection events (McGillicuddy et al., 1998), wind-driven upwelling events (Franks and Walstad, 1997) equatorial upwelling (Murray et al., 1995) and spring blooms (Fasham, 1995).

Fasham, M.J.R. 1995 Variations in the seasonal cycle of biological production in subarctic oceans: A model sensitivity analysis. Deep-Sea Res., 42, 1111-1149.

Franks, P.J.S. & Walstad, L.J. 1997 Phytoplankton patches at fronts: a model of formation and response to wind events. J. Mar. Res., 55, 1-29.

Kirk, J.T.O. 1994 Light & photosynthesis in aquatic ecosystems, Cambridge: Cambridge University Press.

McGillicuddy, D.J., Robinson, A.R., Siegel, D.A., Jannasch, H.W., Johnson, R., Dickey, T.D., McNeil, J., Michaels, A.F. & Knap, A.H. 1998 Influence of mesoscale eddies on new production in the Sargasso Sea. Nature, 394, 263-265.

Murray, J.W., Johnson, E. & Garside, C. 1995 A U.S. JGOFS Process Study in the Equatorial Pacific (eqPac): Introduction. Deep-Sea Research, 42 (2-3), 275-293.

Parsons, T.R., Takahashi, M. & Hargrave, B. 1984 Biological oceanographic processes, Oxford and New York: Pergamon Press.

Robinson, A.R. 1997 On the theory of advective effects on biological dynamics in the sea. Proc. R. Soc. Lond., A, 453, 2295-2324.

Steele, J.H. & Henderson, E.W. 1992 The role of predation in plankton models. J. Plankton Res., 14, 157-172.

Steele, J.H. & Henderson, E.W. 1995 Predation control of plankton demography. J. Mar. Sci., 52, 565-573.

Tritton, D.J. 1988 Physical fluid dynamics, Oxford and New York: Oxford University Press.

Designation       Structure DS       Steady Upwelling DE       Upwelling Event LS       Steady Coastal Upwelling LE       Coastal Upwelling Event WS       Steady Wave or Eddy Field WE       Wave or Eddy Event

Flow       y(s;p,r,q) D       qe-aF       F = F(s,p) L       q[e-aFe-r+(1-e-r)] W        q sinr 2 [ e-aFtanr/2+e aFtanr/2]

Footnotes:

1 the effects of environmental conditions upon biological rates.

2 This notation is equivalent to equation (I.2.13) but simpler.

3 These idealized dependencies yield simpler analytical solutions than the more accurate exponential decay which can be treated later.