Models of Harmful Algal BloomsAuthor(s): Peter J. S. FranksSource: Limnology and Oceanography, Vol. 42, No. 5, Part 2: The Ecology and Oceanographyof Harmful Algal Blooms (Jul., 1997), pp. 1273-1282Published by: American Society of Limnology and OceanographyStable URL: http://www.jstor.org/stable/2839018Accessed: 17/08/2010 14:04

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Limnol Oceanogr, 42(5, part 2), 1997, 1273-1282 ? 1997, by the Amerncan Society of Limnology and Oceanography, Inc

Models of harmful algal blooms

Peter J. S. Franks Scripps Institution of Oceanography, UCSD, La Jolla, California 92093-0218

Abstract Models used to study harmful algal blooms are a subset of those used to examine more general planktonic

processes. Most models have been heuristic, examining the likelihood of certain processes generating a harmful algal bloom. Several models have been more closely coupled to field data and have been used to gain insights into the dynamics underlying the observations. As better physical and biological models are developed, models may play an increasingly important role in harmful algal bloom research. Techniques such as data assimilation may increase the predictive power of models, suggest strategies for improving field sampling, and better constrain poorly known parameters and processes.

Beginning with Kierstead and Slobodkin's (1953) model of the opposing effects of growth and diffusion in the for- mation of a bloom, models have served useful purposes in the study of bloom dynamics. These range from the testing of theoretical hypotheses to the identification and quantifi- cation of processes contributing to bloom formation. More recent advances in modeling techniques may allow models to play an increasingly important role in understanding and predicting blooms.

Models of harmful algal blooms (HABs) are a small sub- set of models used in biological oceanography or plankton ecology in general. However, HABs present a particular set of problems that makes them interesting, socially relevant, and possibly excellent models for more general plankton dy- namics. The most obvious aspect of HABs is that they often consist of a single species that grows toxic or unusually dense. To describe the dynamics of this species we must be able to adequately describe the dynamics of the ecosystem in which the species is embedded.

In the sections below, I present many of the models used to explore HABs (Table 1). I have arranged the models by degree of complexity, beginning with highly aggregated bi- ological models with no physical dynamics. These models segue into biological models with more resolution of taxa or species-less aggregation, more complexity, and more un- known parameters and processes. I then explore coupled physical-biological models used to study HABs, beginning with those using only simple physics (e.g. diffusion or swim- ming) and then examining those with more detailed physical dynamics. Throughout the presentation of these models, I emphasize the similarities of model structure, the way in which the models have been used, and the aspects of the models that might help us understand the dynamics under- lying harmful algal blooms. Finally, I discuss some of the directions that I feel might be most fruitful for models of HABs, including a closer coupling with field programs and the use of data assimilation to better dissect the dynamics of the blooms.

Acknowledgments I thank Sharon Franks for constructive comments on this manu-

script. This work was supported by the Gulf of Maine Regional Marine

Research Board.

Model review

Aggregated models-One of the earliest models of a red tide is that of Wyatt and Horwood (1973). In this simple model, the reduction of grazing pressure as the concentration of cells increases due to swimming allows the cells to grow relatively unchecked. The equations governing their model are basically

dPn, I'Y dt rn- C - -Pn, (1)

dPt dt y. (2)

Pn and Ps are the nonswimming and swimming phytoplank- ton concentrations, rn and rs the growth rates of the two types of phytoplankton, and y the grazing coefficient (units, concn time-') of grazers on the motile phytoplankton. The constant C is the sum of the grazing rates (units, time-') on the two types of phytoplankton. The critical assumption here is that as the concentration of the swimming species of phytoplank- ton increases, the amount grazed remains constant (i.e. the grazers do not respond to the increase in phytoplankton con- centration). It is argued that this could occur in calm con- ditions if the motile phytoplankton were to accumulate in a thin layer while the grazers remained distributed throughout the water column. If turbulence were to increase, for instance during a wind event, the motile phytoplankton would again be homogenized throughout the water column and subjected to their original grazing pressure.

Ignoring the time lag that Wyatt and Horwood included in their model, we can solve Eq. 1 and 2. The solution to Eq. 2, the motile (presumably red tide) phytoplankton con- centration is

Ps(t) = [Ps(o) -2exp(rst) +-7 (3)

It is easy to see that this will be exponentially increasing if rs > 0 (the growth rate is positive) and if the initial phyto- plankton concentration, Ps(O), is greater than ylr5, the ratio of the amount grazed to the intrinsic growth rate of the mo- tile phytoplankton. Thus a red tide will occur if the grazing pressure, y is reduced to the point that the growth rate allows

1273

1274 Franks

Table 1. Types of models used to study harmful algal blooms and some examples discussed in this paper.

Aggregated models Kishi and Ikeda 1986 Truscott 1995 Wyatt and Horwood 1973

Multispecies models Ebenhoh 1987 Kemp and Mitsch 1979 Kishimoto 1990 Montealegre et al. 1995

Models with simple phys- Franks 1997 ics Kamykowski 1974, 1979, 1981a,b

Kierstead and Slobodkin 1953 Kishi and Ikeda 1986 Kishi and Suzuki 1978 Levandowsky 1979 Stommel 1949 Tett 1987 Wroblewski et al. 1975

Models with detailed Kishi and Ikeda 1986 physics Yanagi et al. 1995

net growth of the cells present. Alternatively, an increase in the growth rate, without a consequent increase in the grazing pressure, will also allow the phytoplankton to bloom.

A similar, though vertically resolved model of red tide formation is given by Kishi and Ikeda (1986). In this model the phytoplankton (P) and zooplankton (Z) are coupled through the equations

dP = Vm N

and

dZ GmP (S) dt k + P

Vm and ks are the maximal uptake rate (=growth rate) and half-saturation constant for phytoplankton growth on nutri- ent N while Gm and kz are the maximal grazing rate and half- saturation constant for grazing of zooplankton on phytoplankton. There is probably a misprint in the original manuscript, because the grazers in Eq. 5 have no impact on the phytoplankton population (Eq. 4). There should be an additional term, - GmPZI(kz + P), on the right-hand side of Eq. 4. The formulation presented in Eq. 4 and 5 has phy- toplankton growth dependent only on nutrient concentration. These equations imply Monod-like responses of phytoplank- ton to changes in nutrient concentrations and of zooplankton to changes in phytoplankton concentrations. Although details of the light field, response to irradiance, and vertical diffu- sion are not given in the paper, it is intuitively satisfying that a red tide forms when the phytoplankton migrate vertically out of phase with the zooplankton. That is, the zooplankton graze at the surface during the night when the phytoplankton are in the deeper, nutrient-rich layer and vice versa during the day. In this model, a thin, dense migrating layer of phy- toplankton forms due to the lack of grazing control and net growth of the phytoplankton.

These simple models involving primarily nutrient uptake

and grazing have been more generally formulated and ex- plored by Truscott (1995). The equations he explores are

dP ( P__p2 -P=rP 1 --i- GmZ P2(6) dt K m

k2 + p2'

and

dZ p2_

yGmZ 2 + - EZ. (7) dt k1

The new variables introduced in this model are y, the pro- portion of grazed phytoplankton that becomes new zoo- plankton biomass; K, the carrying capacity (units, concn); and e, the death rate of zooplankton. Some interesting fea- tures of the model are its lack of explicit tracking of the nutrient concentration, its inclusion of a carrying capacity for the phytoplankton (effectively a nutrient limitation), and the inclusion of a Holling type 3 grazing function (which predicts near-zero grazing rates at low, but finite, phyto- plankton concentrations).

Truscott explored Eq. 6 and 7 for the case of a time- dependent change in the phytoplankton growth rate (driven, for example, by a change in irradiance). He analyzed the dynamics in light of theory on "excitable media" (see also Othmer et al. 1990; Truscott and Brindley 1994). In an ex- citable system, the dynamics can be strikingly different for small changes in forcings or initial conditions. An everyday example of such a system is a flush toilet: if the lever is pushed lightly, nothing happens. If the lever is pushed more vigorously, the toilet flushes, generating a large response to a relatively small change in forcing. Further pushing of the lever during the flush will not generate another flush until the tank has refilled. The analogy of this system to the plank- ton dynamics of Eq. 6 and 7 arises through the forcings which lead to sudden blooms of phytoplankton. Truscott found that the phytoplankton were excitable in these equa- tions and quantified the conditions which could lead to a sudden blooming of the phytoplankton.

Under "normal" (unexcited) circumstances, the phyto- plankton are constrained near an equilibrium concentration by the zooplankton grazing. An increase in the growth rate of the phytoplankton can allow the phytoplankton to tem- porarily escape this zooplankton control; the phytoplankton grow unchecked until they reach their carrying capacity (K). At this point, the phytoplankton population growth rate slows, while the slowly increasing zooplankton population exerts an increasingly strong grazing impact. Eventually the phytoplankton are grazed down, and the system returns to an equilibrium condition. The situation studied by Truscott (1995) was that of an increase in the growth rate of the phytoplankton. He showed that the system would be excited (i.e. a bloom would occur) if

1 dr --d=s>O0. (8) r dt

This condition suggests that a situation leading to a bloom would be

r(t) = rminexp(st), (9)

HAB models 1275

which is the solution to Eq. 8 with the initial growth rate being rm,n. If we allow some period T, during which this increase in growth rate takes place, then

T 1 In rma'x\0 T (n j) (10) s rmin

rmax is the final growth rate-the rate at which the phyto- plankton grow during and after the bloom. The condition for a phytoplankton bloom can be more generally defined (for this system) as

r-ax > yGmK(l - PO)PO (11) rnun ~2kze

Here P0 is the phytoplankton concentration at equilibrium when the growth rate r = rn. This inequality implies that the ratio of the final to the initial growth rates must be great- er than the rate of tracking by the increasing zooplankton population; that is, the phytoplankton are temporarily re- leased from grazer control and not yet under nutrient (carrying capacity) control.

In this very simple model with its insightful analysis, Truscott (1995) has generalized the Wyatt and Horwood (1976) and Kishi and Ikeda (1986) models and given more complete solutions for the conditions under which phyto- plankton might form a red tide. His analysis was limited to the case of changing phytoplankton growth rate; however, the analysis could be easily and fruitfully extended to ana- lyze the model behavior under changing nutrient conditions (varying K) or grazing stress (varying Gm and kz).

The analysis performed by Truscott (1995) may give us some insight into the mechanisms preceding a red tide bloom. The critical conditions are not just a relaxation of grazing pressure (as in the Wyatt and Horwood and Kishi and Ikeda models), but the rate of change of the growth rate relative to the grazing control of the zooplankton. Also, the magnitude of the change of growth rate is important-an instantaneous change of growth rate will not excite the sys- tem if the ratio rmax: rmin is too small. Thus the proportional increase in net phytoplankton growth rate could be a more stringent constraint than the rate of increase of the growth rate. In ecological terms, a system that has a low specific mortality rate of phytoplankton (allowing a high net growth rate) will be less likely to create a red tide under improved conditions than one which has a high mortality rate. Truscott (1995) extended these ideas to changes in growth rate driven by changes in mixed-layer depth-an idea similar to that embodied in the Wyatt and Horwood (1976) model.

These ideas are not new to HAB research. However, these simple models serve to identify and quantify the processes that might be most fruitfully explored in the field in order to understand the factors leading to red tides. One important point that becomes apparent in the Truscott (1995) model echoes Franks (1995): to understand the factors controlling a red tide, we must sample appropriate variables before the red tide occurs.

Multispecies models-Red tides, and harmful algal blooms in particular, are a phenomenon of the plankton com-

munity, not just a single species. This point was clearly ar- ticulated by Watanabe (1983, p. 423):

A red tide phenomenon cannot be understood as the simple concept of a bloom of a primary producer; rather it must be explained as a special event in natural plankton succession. Therefore, in order to understand the mecha- nism, the environmental factors controlling the type of species and population number in time and space should be found, and then the interaction between these factors and the organisms causing red tides should be investigat- ed. Models of red tide blooms, therefore, should be able to answer the more common question of what environ- mental factors lead to unialgal patches of red tide organ- isms, and what kind of mechanism selects the causative organism.

There has been almost no work examining marine HABs as a community phenomenon. However, there has been some interesting theoretical and heuristic work on multispecies complexes and species succession in the plankton.

The problems in understanding red tides in which one or several species of phytoplankton grow to exceptional con- centrations, and HABs in which a species of phytoplankton becomes toxic but not necessarily dominant, are somewhat distinct. Red tides are conceptually more tractable, because the response (the dense bloom) is obvious, quantifiable, and unusual (though the causes may be more obscure). HABs present a more difficult problem: one species of phytoplank- ton within a community becomes a threat to human or ani- mal health, without necessarily dominating the community. Harmful algal blooms, in particular, must be viewed in a community context.

Modeling of planktonic communities is still at a fairly nascent stage. Models are just beginning to be developed which allow the coexistence of multiple species of phyto- plankton. Such models are exploring the issue first articu- lated by Hutchinson (1961), which he termed "the paradox of the plankton." How can so many species of competing phytoplankton coexist in an apparently stable and homoge- neous environment? Much of the work since then has con- centrated on showing that the environment is not stable and homogeneous and that this fluctuating medium can allow the disequilibrium coexistence of many competing species. The- oretical models such as that of Ebenhuh (1987) have shown that an unlimited number of phytoplankton with slightly dif- ferent nutrient uptake characteristics can coexist if the nu- trient supply is fluctuating. Changing mortality rates allowed both periodic and chaotic fluctuations of species abundances; occasional dominance by a single species occurred and could be interpreted as a red tide. This model was not spatially resolved, and similar results might be expected for spatial patchiness of nutrients (e.g. Kemp and Mitsch 1979; Kishi- moto 1990).

Further exploration of the factors allowing the coexistence of multiple competing species may help us understand what factors we should be considering when studying HABs. For example, much of the high-frequency environmental fluctu- ation is driven by turbulence. While turbulent mixing can control time and space scales of nutrient, irradiance, and phytoplankton patchiness, it is also known to affect phyto-

1276 Franks

plankton growth rates. Laboratory experiments have shown dinoflagellate growth rates to be more strongly depressed than diatom growth rates in response to turbulence (White, 1976; Thomas and Gibson 1990a,b, 1992; Thomas et al. 1995; Gibson and Thomas 1995), and Berdalet (1992) has shown that this occurs through inhibition of cell division. Tynan (1993) has shown relative abundances of diatoms and dinoflagellates in the field to be related to the surface wave height-dinoflagellate populations were depressed if wave activity was high. These observations suggest that more complete models of HABs must consider not only grazing pressure and competition for nutrients and light, but also the details of the physical environment as an agent controlling competition. This is true at all scales of motion, not just the turbulent scales.

Most oceanographic and aquatic models involving more than one type of phytoplankton lump species into groups, assuming that species in such groups behave quite similarly. Although these models are more aggregated than those con- taining many species, they also tend to be more explicit in their categorization of groups. Thus dinoflagellates are typ- ically distinguished by their phototactic swimming, while diatoms tend to have more efficient nutrient uptake charac- teristics (e.g. Kishi and Ikeda 1986).

Species-specific models have been used somewhat suc- cessfully in lacustrine systems, for example the blooming of Oscillatoria agardhii (Montealegre et al. 1995). This model was constructed to simulate dense blooms in a monomictic lake in response to nutrient, light, and temperature limitation. The model had about 20 biological parameters, many of which were poorly known. Using 1 d of data (out of 19 days) to calibrate the model, the model was run, adjusting the pa- rameters until a good fit was obtained. If the data and model are accurate, then the new parameter values may be more accurate than the initial guesses based on laboratory exper- iment. Still, a stronger test of the model will be to compare it to an independent data set using the same parameters. The model achieved good agreement with the data, partly be- cause the model was calibrated to that data set, but also because the system they chose to model was relatively self- contained. The lake, unlike the coastal ocean, did not have significant inputs and outputs to the system, reducing the degrees of freedom that had to be accounted for in the mod- el.

One of the main problems with applying multispecies or less-aggregated models to field data is model initialization. Few field studies make the effort to count many different taxa of plankton, and even less is known about physiological and behavioral parameterizations of the various taxa. Inad- equate knowledge of initial conditions makes interpretation of ensuing dynamics difficult if not impossible. Transient oscillations due to imbalances between compartments in a model can overwhelm the real signal. It is imperative to separate these mathematical artifacts from true, observable oscillations. The extra factor of environmental forcing can both stimulate and damp fluctuations between compartments.

We are a long way from being able to predict patterns of succession of individual species in the oceanic plankton, and harmful algal blooms are just one manifestation of these dy- namics. Research into this aspect of plankton dynamics will

prove fruitful and will be facilitated by new molecular tools to rapidly distinguish species (Adachi et al. 1994, 1996; Scholin et al. 1994; Scholin and Anderson 1994; Miller and Scholin 1996). To make models more useful, such molecular research must be accompanied by physiological research to allow more accurate parameterizations of models.

Models with simple physics-Models coupling biological and physical dynamics are all versions of the flux divergence equation

aP a a a + -Pu(x, y, z, t) + Pv(x, y, z, t) + -Pw(x, y, z, t) At ax ay az = biological dynamics. (12)

This equation states that the local rate of change of P is driven by motion of existing gradients (left-hand terms) and local growth and death (right-hand terms). The horizontal velocities u and v, and vertical velocity w can be both space- and time-dependent and include motions due to swimming, sinking, or floating of the organisms. The values for the physically driven velocities u, v, and w are determined from measurements or versions of the momentum equations. These equations are well presented and thoroughly discussed elsewhere (Donaghay and Osborn 1997). I have not made any assumptions about the scales in presenting Eq. 12-it includes everything from the smallest to the largest scales of motion. Often the small-scale motions (those motions which are unresolved at the scale being studied) are lumped into diffusion terms:

+P a a a -+ Pu + -Pv + -Pw at ax ay az

/a2p a2p\ a2p Kh a2 + -2) + Kv- + biological dynamics. (13)

Here Kh and Kv are the horizontal and vertical eddy diffusion coefficients (units, length2 time-' and assumed constant in space) (see Donaghay and Osborn 1997). The difficulty in applying this equation is obtaining values for u, v, w, Kh, Kv,

and the biological dynamics. Generally simplifications are made to make the equations more tractable or to restrict them to a set of motions that is well understood and well con- strained (e.g. idealized internal waves or Langmuir circula- tions, see below).

One of the first and simplest coupled physical-biological models used to explore HABs was the process-oriented mod- el of Kierstead and Slobodkin (1953). This simple model explored the opposing effects of algal growth and biological diffusion (i.e. swimming-induced diffusion) in maintaining patches of phytoplankton. The equations governing these dy- namics are

ap a2p K-+ rP. (14) at ax2

Here the phytoplankton are being diffused along the x co- ordinate with coefficient K and growing at rate r. The size (L) of the smallest patch of suitable water that can maintain a bloom under these conditions is given by

HAB models 1277

L 7T i(cIr)?2, (15)

the ratio of the diffusion to the growth rate of the population. Smaller patches will be diffused faster than they can grow.

The dynamics embodied in Eq. 14 are very limited: only diffusion and exponential growth. Kishi and Suzuki (1978) looked for conditions of patch persistence in a more general equation containing advection, diffusion, exponential growth, and an arbitrary function, f which could be a Mi- chaelis-Menten source or sink term or a logistic-type density dependence of growth:

ap ap ap 182p 82p\ a+u +V =K - + d +rP+ f(x,y,t,P). at ax ay ax2 ay 2/

(16)

Here u and v are horizontal velocities that may be spatially dependent but do not vary in time-an important assump- tion. Under these conditions, patch growth is guaranteed if fJ C (a2 aI2 + + rP2l dx dy d 0. (17)

ax2 ay 2/ This is a more general formulation of Eq. 15 for a slightly wider range of conditions.

Wroblewski et al. (1975) and Levandowsky (1979) con- sidered other particular modifications to Eq. 14. These were the inclusion of grazing:

ap a2p -= K 2+ rP - y{l - exp[-A(P - Po)]} (18) at ax2

where y is the maximal grazing rate, A the grazing efficiency and, PO a threshold concentration of phytoplankton below which grazing is zero. To this, Levandowsky (1979) added logistic growth:

ap 2p P -= - K- + rPl- - -y{l - exp[-A(P -Poff

(19) and looked at the conditions controlling the maximal phy- toplankton concentrations. He found that each of the modi- fications changed the minimal patch size that would support phytoplankton growth and altered the initial conditions re- quired for an increasing population. The relationship of min- imal patch size with additional biological dynamics was not simple, and no generalizations such as Eq. 15 could be made. In most cases, however, the minimal patch size was larger than that given by Eq. 15, suggesting that biological dynam- ics (grazing, nutrient limitation) could be an important de- terminant of bloom initiation in this simple physical system.

The ideas embodied in Eq. 14, 16, 18, and 19 are probably too idealized to apply to the analysis of field data. Because of order-of-magnitude uncertainties in the diffusivity, the calculated minimum patch size should be considered at best an order-of-magnitude estimate. Thus the model tells us at most that the smallest patches must be, say, 10 km in di- ameter, rather than 1 km or less. Applications of this model (Slobodkin 1953; Olsson and Edler 1991) support the hy- pothesis that red tides can develop in fairly isolated patches of water (as indicated by horizontal salinity gradients), but

the model does not tell us much about the actual dynamics within the patch of water. For this, application of the bio- logical models such as Eq. 6 and 7 would be more useful (though more difficult). Care must also be exercised when trying to interpret a model such as Eq. 14. Any water mo- tions can be considered diffusion if motions at smaller scales are not resolved. Thus the scale of the measurements deter- mines the diffusion scale that must be used in the model. Patches smaller than the measurement scale cannot be prop- erly resolved in any case. A single measurement in a patch gives no indication of its actual scale, except that it is smaller than the distance between stations. This fact has been over- looked in several applications of Eq. 14.

The issue of small-scale (10 m-10 km) patch formation in dense algal blooms is both important and somewhat trac- table with simple models. This is because of the time scales of patches associated with these spatial scales. The patches are small enough and transient enough that local growth pro- cesses are generally less important than the interaction of behavioral responses (swimming, sinking, and floating) with the physical dynamics at these scales. Examples of such models include the streaks formed by Langmuir circulations (Stommel 1949) and the bands formed by internal waves (Franks, 1997; Kamykowski, 1974, 1979, 1981a).

The models of Kamykowski (1974, 1981a,b) are excellent examples of the marriage of laboratory, field, and modeling studies in the exploration of phytoplankton patch formation. Testing the hypothesis that the interaction of temperature- mediated vertical migration with internal waves led to the dense patches of phytoplankton observed off the Scripps pier, Kamykowski first developed a process-oriented model including these dynamics (Kamykowski 1974). The physical velocities were obtained from theoretical values for an in- ternal wave propagating in a two-layered fluid. The swim- ming velocities were obtained from laboratory studies of the organisms responding to temperature, light, and nutrient stresses.

Although the model was fairly simple, the application of it proved quite revealing. Kamykowski used two techniques that exploited the model's capabilities. The first was "dy- namic interpolation": using the model to interpolate between data gathered in the field. The second technique was sensi- tivity analysis. Once the model had been sufficiently cali- brated with the field data, it was run removing or altering various terms in the equations. In this manner, the contri- bution of different processes to the final result became clear. Kamykowski (1974) explored several cases with different swimming behaviors of phytoplankton. Dense patches de- veloped only in the case in which the phytoplankton crossed the pycnocline. This hypothesis was then tested with labo- ratory experiments examining the vertical migration behav- ior of several dinoflagellates in stratified water columns (Ka- mykowski 1981a). These results were then combined in a dynamic interpolation model coupling field measurements of the internal wave field with models of temperature-mediated vertical migration (Kamykowski 1981b). This model was thoroughly explored, including tests of the importance of different migration behaviors, different combinations of wave frequencies, and different basin geometries.

This work of Kamykowski is particularly notable in its

1278 Franks

combination of theoretical models with field and laboratory data. The models were used to understand the observations, both by simulating them (dynamic interpolation) and by not simulating them (sensitivity analysis). To some extent the models determined the field program: without adequate data, the models could not be properly initialized or run. The dy- namics embodied in the models give strong constraints on the types, frequency, and spatial distribution of data that must be gathered to understand the dynamics underlying the observations. This is one of the most useful features of mod- els for HAB research.

Tett (1987) used a vertical form of Eq. 14 coupled to a fairly detailed physiological model of phytoplankton to try to understand the dynamics behind the formation of red tides in the approaches to the English Channel. The vertical dif- fusivity was variable vertically and driven primarily by tidal friction over the bottom. For a spatially dependent diffusiv- ity, the equation to be solved is

aP a aP a= a K(Z) a* (20) at az az

Note that the gradients of the diffusion coefficient K must now be known. This necessitates a different numerical so- lution scheme than for a spatially homogeneous diffusivity- a point often overlooked.

The biological model of Tett (1987) included the cell-quo- ta nutrient model of Droop (1968), self-shading in the re- sponse to subsurface irradiance, grazing (linear function of phytoplankton biomass), and an internal nutrient-dependent swimming behavior. The physical and biological models were calibrated with data from the field. The model was run to equilibrium; the predicted spatial gradients of phytoplank- ton and nutrients were similar to those measured, but the biomasses were considerably different. Although there must be some question as to whether the model was an adequate representation of the dynamics of the system, the equilibrium solution could be examined to determine the relative contri- bution of various dynamics to the final state of the model. The analysis performed by Tett was quite enlightening: as predicted by the Truscott (1995) model, grazing exerted a strong control on the phytoplankton biomass. Decreased grazing rates allowed peak biomasses to increase by a factor of 10 or more; these new levels were still below those of a dense red tide. Experiments with vertical migration showed that nutrient-dependent vertical migration followed by a floating phase could lead to blooms of more than 100 mg Chl a m-3. This result supports the earlier model of Kishi and Ikeda (1986).

The sensitivity analyses performed by Kamykowski and Tett are the sort of model analyses that can lead to new and useful insights into the dynamics of red tides and HABs. The fundamental requirement for such analyses, however, is a complete set of field data. This is such a rare commodity as to make much of the past modeling of HABs relatively useless as a heuristic or predictive tool.

Models with detailed physics-To make models more use- ful as heuristic or predictive tools they must be forced with realistic boundary conditions such as solar insolation, wind, freshwater inputs, tides, and topography. To accommodate

such forcings, models must have more complete descriptions of the physics than those presented above. As with the mod- els with simple physics, there are degrees of complexity in models with more detailed physical dynamics. These might range from simple two-layered models forced by steady winds to models with continuous stratification and detailed turbulence submodels to accommodate transient wind and tidal forcing.

The models with more detailed physics are still based on Eq. 12. The addition to these models is the detail with which u, v, and w are specified. As models become more and more complex (and hopefully accurate), there are fewer and small- er assumptions made about the physical dynamics.

Kishi and Ikeda (1986) presented both an idealized two- dimensional and a realistic-topography three-dimensional model of red tides in the Seto Inland Sea of Japan. Their two-dimensional model was a cross-shore section with steady wind forcing, which they used to examine the effects of coastal wind-driven upwelling on red tide formation. Their manuscript does not indicate how many layers were used in the model or how the equations were solved. The model was run for only 2 or 3 d, which brings up the ques- tion of transient oscillations due to relaxation of the initial condition. The two experiments presented are a case with a day-night cycle and a case with no night. It was found, by examination of balances of terms in the equations, that a red tide would not form in the day-night case because of night- time advection of phytoplankton out to sea. In the constant daylight case, however, a red tide formed inshore as the phy- toplankton were able to use up all the upwelled nutrients before being advected offshore in the surface waters. Had Kishi and Ikeda used the cell-quota nutrient model that Tett (1987) prepared, possibly they would not have had to invoke constant daylight to induce a bloom because the cells could have taken up and stored nutrients before turning them into biomass.

One important result of the two-dimensional model of Kishi and Ikeda (1986) echoes that of Kierstead and Slo- bodkin (1953): there is a balance between physical dispersal and biological growth that must be maintained for a red tide to form.

The three-dimensional model constructed by Kishi and Ikeda (1986) had a horizontal resolution of 2.5 km, but only two layers in the vertical. Depth-averaged equations were used within these layers, and the biological model was solved at only these two vertical levels. The biological model contained seven compartments: Chattonella antiqua (the red-tide forming phytoplankter), diatoms, zooplankton, phosphorus detritus, nitrogen detritus, dissolved inorganic phosphorus, and dissolved inorganic nitrogen. The model was forced with diel light variations, spatially variable steady wind, and tides. Although high concentrations of Chattonella (and diatoms) developed, it is difficult to inter- pret these given the short (3 d) model run and the few cases presented. The severe constraints of computer power at that time precluded more detailed examination of the solutions or more extensive sensitivity analyses. The lack of a detailed data set to compare the model to and the few analyses of the model, however, limit its usefulness.

The three-dimensional model of Yanagi et al. (1995) was

HAB models 1279

not very different from that of Kishi and Ikeda (1986), ex- cept that it had three levels in the vertical. It was designed to simulate blooms of the red tide forming dinoflagellate Gymnodinium mikimotoi in the Seto Inland Sea. The dino- flagellate migrated instantaneously between the surface and middle layers around dawn and dusk, taking up nutrients following the Droop cell-quota model. Both nitrate and am- monium were offered as nutrient sources, with only am- monium being taken up at night in the middle layer. No zooplankton dynamics were included.

Several interesting aspects of the Yanagi et al. (1995) model have to do with the initialization, forcing, and running of the model. Yanagi et al. had access to temperature, salin- ity, nutrient, and cell concentration data gathered throughout the Seto Inland Sea on four days during a 2?/2-week period in July. There were also more continuous measurements of river discharge, insolation, and wind. The approach taken by Yanagi et al. was to use the physical model as an interpo- lation tool for the data. The model was initialized with the observed temperature and salinity fields from the first cruise. The model was then run with the observed forcing until the next cruise date, 3 d later. At this time, the model temper- ature and salinity fields were readjusted to the observed val- ues with a procedure called "successive correction" (Moore et al. 1987). With this technique, the model values are "dragged" toward the observed values, using a weighting function in space. The model was then run until the next cruise date and the procedure repeated. The model thus uti- lized the field data in improving its simulation of the obser- vations. This technique of "data assimilation" is discussed below.

The second interesting feature of the Yanagi et al. model is the way in which they simulated the phytoplankton. Rath- er than simulate the biological dynamics as an Eulerian field, they used a Lagrangian approach and followed patches of phytoplankton. So in contrast to the Kishi and Ikeda (1986) model (and most others like it), Yanagi et al. did not simulate the biological dynamics at every grid point in the model. Instead they initialized the model with observed patches of G. mikimotoi and allowed them to advect and diffuse in the model's physical milieu. An important assumption behind this approach is that all the initial patches of cells were mea- sured in the field and that no new patches arose during the time of the bloom. If this assumption is not met, there is little support for comparison of the model results to later field observations. This problem is not so strong in an Eu- lerian framework because of the continuous nature of the system.

Yanagi et al. (1995) achieved fair qualitative agreement of the predicted and observed cell concentrations. However, the more interesting aspect of their study was an examination of the changes in the model predictions when nocturnal nu- trient uptake, vertical migration, or advection were turned off in the model. Through this process sensitivity analysis, the dependence of the final solution (and hopefully the ob- servations) on various dynamics can be made explicit. It was found that lack of nocturnal nutrient uptake led to slightly decreased cell concentrations, lack of vertical migration caused a more north-westward migration of cells due to their permanent residence in the surface layer, and lack of advec-

tion led to a clumping of patches in the area of the initial patch observations. However, even with the generous amount of data available during this bloom, it is still not adequate to allow us to draw strong conclusions about the bloom dynamics.

Discussion

In presenting the models above, I have tried to show how they all share a common framework of physical and biolog- ical equations. Differences between models are in the choice of biological compartments, processes linking compart- ments, parameterization of processes, and choice of physical dynamics. Most models can be considered to be specific cases of a more general multicompartment physical-biolog- ical model. To my knowledge, the most general form of this model-including detailed boundary conditions, a turbu- lence-closure submodel, and a detailed biological model- has not been formulated for a particular harmful algal bloom.

What have we learned from models?-The models which have contributed most to our understanding of HAB dynam- ics are those that are heuristic and those that do dynamic interpolation. Heuristic models, which make up the bulk of extant models, have been used to test hypotheses concerning processes governing harmful or exceptional algal blooms and have widened our perception of the scope of important dynamics. To be most useful, however, heuristic models must eventually be tested in the field, otherwise they remain a theoretical curiosity.

Models used in a dynamic interpolation mode have proved useful in hindcasting the dynamics underlying observations. They can be used to test hypotheses concerning important processes and as a tool to quantify the sensitivity of the system to changes in parameterizations and dynamics. To be useful, models such as these must be calibrated and tested with thorough data sets.

When testing a model against field data, the limitations of the data quickly become apparent. Often there is not enough spatial or temporal resolution, or the correct variables were not measured. This inadequate data set usually leads to poor agreement between model and observation-a common ten- dency usually worked around by weak, qualitative compar- isons of the model to the data.

A better approach, unfortunately seldom used in ecolog- ical modeling, would be to define quantitative criteria by which the model's skill is judged. These criteria would be different for every model, indeed for every new application of a given model. This quantitative testing of the model serves two purposes, even when the model does not achieve the desired level of skill: the choosing of the judgment cri- teria can reveal a great deal about the system under inves- tigation, and the reasons for poor agreement between model and data become more apparent. Thus the poor agreement between model and observation may help to define a better observation program or refine the model dynamics to be a better representation of the system.

Models have not been used a great deal in designing HAB field studies. It has often been suggested that models should be constructed before the field program to aid in sampling

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design. This process could ensure that no obvious compart- ments are missed when the field program is initiated. Al- though this may work in some circumstances, the more like- ly reality is that unexpected dynamics will become apparent during the field program. The model will make such dynam- ics apparent by a poor agreement of the model and field data. It is at this point that the model can make a strong contri- bution to the design of the observation program. If the boundary conditions are found to be inadequate to resolve the physical dynamics, then the field program might add an appropriate type and number of moorings. If the field ob- servations are found to be completely uncorrelated, then the model will indicate that more frequent samples must be taken or that the scale of sampling must be adjusted.

This interactive sampling-modeling program will not work well for unexpected blooms that do not have a dedi- cated field program. For recurrent blooms, however, such a process can be extremely fruitful and lead to significant new insights into bloom dynamics. It is likely that in most cases this refining process would take several bloom cycles to reach the point of some predictive capability. This period is, unfortunately, generally longer than most program managers are willing to fund a project, making it less likely that a useful marriage of field and modeling work will occur. Al- though monitoring is not flashy science, it is through the regular gathering of appropriate data that the dynamics un- derlying HABs-and many other planktonic processes-will be quantified.

The future-Like meteorological models, one of the goals of HAB models is prediction. But, like weather models, the predictive ability of a coupled physical-biological model of a harmful algal bloom will be limited. Chaos theory has shown that the predictive ability of a model is a function of the accuracy of the initial conditions: two slightly different initial conditions can soon lead to widely different solutions as the model equations are integrated. The predictive ability of weather forecasts is limited to -5 d, largely because of the effect of unresolved processes in the initial condition. We cannot expect models of HABs to do any better than weather forecasts, particularly since many HABs are strong- ly influenced by the weather (wind and insolation).

Given this inherent limitation on the predictive ability of models, what can we do to improve our confidence in model output? One technique that can help is "data assimilation." Data assimilation is the occasional incorporation of obser- vations to readjust the model output, thus obviating the ef- fects of inaccurate initial conditions and the inescapable cha- otic divergence of model and reality. There are several techniques used for inserting data into models (Cressman 1959; Gandin 1963; Kalman 1960; Lewis and Derber 1985), usually involving a processes of gently nudging the model solution toward the data without introducing unrealistic os- cillations in the model. These techniques have been used extensively with oceanographic physical data (Haidvogel and Robinson 1989; Ghil and Malanotte-Rizzoli 1991; Ben- nett 1992; Brasseur and Nihoul 1994); however, assimilation of biological data presents unique challenges. The evolution of the physical fields is well described by the equations of motion for the fluid: momentum, vorticity, mass, and energy

are usually conserved, giving strong constraints on possible dynamics. Assimilating data into a biological system is not so obvious, where communities of different species com- position, different physiological states, and with different forcings may behave quite differently and unexpectedly. We can never accurately sample all the species and their appro- priate rates (growth, grazing, nutrient uptake, etc.), and any model will have a high degree of abstraction and lumping of species and trophic interactions.

One of the more likely candidates for data assimilation is sea-surface pigment concentration from satellite observa- tions. New techniques are being developed to rapidly iden- tify and enumerate particular algal species, and models should be developed which are capable of usefully incor- porating such data. There are several issues to consider when assimilating biological data into a model. One is the con- version of the assimilated data into model units. Biological models are typically constructed with nitrogen as the mea- sure of biomass. Conversion between the nitrogen content of phytoplankton and the values of pigment concentration inferred through remote sensing is not constant or obvious. Another is conservation of mass. In a multicompartment bi- ological model, how should the nonassimilated compart- ments be readjusted to accommodate the changes in mass driven by assimilation of data into one compartment? Final- ly, how does the observed field relate to the modeled field? Remotely sensed pigment concentrations (for example) are a weighted average of some variable depth in the water col- umn. How does this depth correspond to the model's vertical grid and resolution? One of the first coupled physical-bio- logical models to use data assimilation is that of Ishizaka (1990). He simply replaced the phytoplankton fields in the model with CZCS estimates of sea-surface pigment. The re- maining biological fields (dissolved nutrient N, zooplankton Z, and detritus D) were updated in three different ways: con- stant (low) values for N and Z while D was calculated as- suming temperature-dependent conservation of mass, model- derived values of N and Z while D was calculated as above, and assuming a constant ratio of N: Z: D and conservation of mass. The three techniques gave quite similar results. Nu- merical techniques minimized possible oscillations created by the data assimilation.

Armstrong et al. (1995) used a similar technique to in- corporate satellite-derived pigment concentrations into a physical-biological model. The phytoplankton concentra- tions in the top model level were adjusted toward the satellite values by adding a fraction, E, of the difference between the model (Pmodel) and observed (Psat) phytoplankton concentra- tions at each time step At:

Ep = EAt(Psat - Pmodel)* (21) The parameter e determines the rate at which the model is dragged toward the data; slow rates of change are termed "nudging." The other model components were allowed to readjust without any special treatment; nitrogen was not con- served. These techniques greatly improved the agreement between the model and observation and improved the pre- dictive ability of the model.

Data assimilation can be useful not only for improving model predictive ability, but also for recovering poorly

HAB models 1281

known model parameters from data. Lawson et al. (1995) used a model to generate a time series of biological data which was then subsampled at intervals as a real system might have been. These "field" data were then assimilated into the same model, using the adjoint method to estimate the unknown parameter values which generated the original data. Their results are quite revealing: many of the original model parameters could be accurately recovered with suffi- cient data assimilated into appropriate state variables. How- ever, there were many cases in which the data assimilation did a rather poor job of predicting the model parameters, even though the data were obtained from the same model structure into which they were assimilated. This suggests that it may be some time before we can use data assimilation accurately for real-world ecosystems with unknown struc- tures. However, HABs, which often behave quite indepen- dent of the larger ecosystem, may be excellent vehicles to explore the utility of data assimilation techniques and their ability to generate accurate predictions of HAB dynamics.

Conclusions

Although models have been important in adding to our understanding of harmful algal bloom dynamics, they can play a much more useful and important role. Before models can fill this role, we need:

* thorough data sets around which to formulate models * more quantitative and critical comparison of models to

data * development of data assimilation techniques to improve

the predictive power of models * development of data assimilation techniques to allow bet-

ter dynamic interpolation of data by models and the re- covery of poorly known parameters

* more interaction between model development and field sampling design.

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