Surface-water quality and flow Modeling Interest Group
1USGS, 345 Middlefield Rd, MS 496, Menlo Park, CA 94025
2Environmental Fluid Mechanics Laboratory, Stanford University,
Stanford, CA 94305-4020
Please direct correspondence to:
Lisa Vidergar Lucas
Postdoctoral Research Associate, U.S. Geological Survey
345 Middlefield Road, MS 496, Menlo Park, CA 94025
Internet: llucas@usgs.gov
Phone: (650) 329-4588 or (650) 723-1825
FAX: (650) 329-4327
Citation:
Lucas, L.V., Cloern, J.E., Koseff, J.R., Monismith, S.G., and Thompson, J.K.,
1998, Does the Sverdrup critical depth model explain bloom dynamics
in estuaries?: J. Marine Research, 56(2), 375-415.
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We present results of simulation experiments which assume that vertical transport and net phytoplankton growth rates are horizontally homogeneous. In the present approach the temporally and spatially varying turbulent diffusivities for various stratification scenarios are calculated using a hydrodynamic code that includes the Mellor-Yamada 2.5 turbulence closure model. These diffusivities are then used in a time- and depth-dependent advection-diffusion equation, incorporating sources and sinks, for the phytoplankton biomass.
Our modeling results show that, whereas persistent stratification greatly increases the probability of a bloom, semidiurnal periodic stratification does not increase the likelihood of a phytoplankton bloom over that of a constantly unstratified water column. Thus, for phytoplankton blooms, the physical regime of periodic stratification is closer to complete mixing than to persistent stratification. Furthermore, the details of persistent stratification are important: surface layer depth, thickness of the pycnocline, vertical density difference, and tidal current speed all weigh heavily in producing conditions which promote the onset of phytoplankton blooms.
Our model results for shallow tidal systems do not conform to the classical concepts of stratification and blooms in deep pelagic systems. First, earlier studies (Riley, 1942, for example) suggest a monotonic increase in surface layer production as the surface layer shallows. Our model results suggest, however, a non-monotonic relationship between phytoplankton population growth and surface layer depth, which results from a balance between several "competing" processes, including the interaction of sinking with turbulent mixing and average net growth occurring within the surface layer. Second, we show that the traditional SCDM must be refined for application to energetic shallow systems or for systems in which surface layer mixing is not strong enough to counteract the sinking loss of phytoplankton. This need for refinement arises because of the leakage of phytoplankton from the surface layer by turbulent diffusion and sinking, processes not considered in the classical SCDM. Our model shows that, even for low sinking rates and small turbulent diffusivities, a significant percentage of the phytoplankton biomass produced in the surface layer can be lost by these processes.
Sverdrup's original problem was the seasonal development of phytoplankton biomass in the open ocean as a response to seasonal stratification by heat input. In shallow coastal waters, other mechanisms of physical variability can overwhelm the annual cycle of thermal stratification. For example, in estuaries and shallow shelf waters (regions of freshwater influence, Simpson et al., 1991), salinity stratification can be a stronger stabilizing force than thermal stratification. However, even the stabilizing influence of freshwater inputs can be offset by the strong turbulent mixing of shallow waters by tidal currents and wind stress. As a result, shallow coastal systems have more complex stratification dynamics than the open ocean, with components of variability associated with seasonal and event-scale fluctuations of river flow as well as the semidiurnal and weekly fluctuations in tidal energy (Simpson et al., 1990). These shallow coastal systems also have more complex population dynamics of phytoplankton, with episodic and high-amplitude fluctuations of biomass superimposed onto seasonal cycles (Cloern, 1996). Although much of this biomass variability is correlated with fluctuations in stratification driven by the seasonal variability of river flow and hourly-daily variability of tidal stirring (Sinclair et al., 1981), we have not yet developed a general theory to define the physical conditions under which phytoplankton blooms can develop in shallow coastal waters. We ask here if the critical depth concept can be used to explain the association between stratification dynamics and bloom dynamics in shallow coastal systems such as estuaries.
This paper is the third in a series to explore the linkages between bloom dynamics and physical dynamics of shallow coastal waters. Our approach is to use an evolving numerical model of coupled biological-hydrodynamic processes to search for general principles of bloom regulation in estuarine waters. In the first paper (Cloern, 1991) we showed how phytoplankton population growth can change in response to daily fluctuations in vertical mixing over the neap-spring cycle. In the second paper (Koseff et al., 1993) we showed the importance of hourly-scale fluctuations in mixing over the semidiurnal tide cycle, and that stratification is a necessary condition for bloom inception in shallow (~10 m deep) waters where algal production is constrained by light availability and where losses can include rapid consumption by benthic invertebrates. In this paper, we extend this theoretical foundation to show how the details of salinity stratification influence the development of blooms. Blooms can arise from many different mechanisms. For example, in a certain class of systems (e.g Georges Bank, see Franks and Chen, 1996) the presence of fronts is very important for the occurrence of phytoplankton blooms, whereas for other systems, such as Puget Sound (Winter et al., 1975), the York River estuary (Haas et al., 1981), and the lower St. Lawrence estuary (Sinclair, 1978), blooms develop when the local balance between production and consumption processes is changed by the establishment of vertical density stratification. We therefore consider here only the importance of local processes of algal production-consumption-transport that can be included in the framework of a vertical one-dimensional model. In the next phase of our analysis we will consider the additional importance of advective processes by extending the model to include horizontal variability and transports.
The marine domains considered here are very different physical systems from the deep pelagic domain originally considered by Sverdrup. In his exploration of the spring bloom in the Norwegian Sea, Sverdrup followed the weekly development of thermal stratification that forms surface layers tens to hundreds of meters deep. Here, we consider the hourly fluctuations of salinity stratification that forms surface layers shallower than ten meters in thickness. In Sverdrup's pelagic system, the primary mechanisms of phytoplankton loss were conceived to be respiration and zooplankton grazing. Here, we consider shallow pelagic systems strongly connected to the benthos, which is an additional (sometimes dominant) sink for phytoplankton production. Additionally, the portion of the water column which Sverdrup studied was much less turbid than the shallow systems we are considering. Thus, the values of Zcr associated with the system we are studying are generally much smaller than those of Sverdrup. Therefore, our search for principles of bloom regulation in estuaries must consider additional processes as well as physical dynamics operating at different spatial and temporal scales from those originally conceived by Sverdrup. With this framework in mind, we designed model experiments to address the following questions:
Simpson et al. (1990) have described and analyzed an important stratification mechanism, known as Strain Induced Periodic Stratification (SIPS), for partially mixed/stratified estuaries where a longitudinal salinity gradient exists between the ocean and the freshwater source. Their description is as follows and is sketched in Figure 1. Assume we start with a homogeneous water column at the start of the flooding tide (Figure 1a). During the flood tide (Figure 1b), salty water is carried over fresher water by the vertically sheared tidal current, producing unstable stratification and thus inducing vertical mixing. In this case, the vertical shear in the horizontal current is due only to the presence of the bottom boundary layer. However, on the ebb (Figure 1c), stable stratification develops when the sheared tidal current carries fresher water over salty water. This stratification reduces the vertical mixing of momentum and increases the velocity shear (Monismith and Fong, 1996), further increasing the rate of stratification production (Jay and Musiak, 1996; Nepf and Geyer, 1996).
Depending on the strength of the tidal currents, turbulent mixing can eliminate the stratification before the end of the ebb, or the water column can remain stratified into the next flood tide. This process can be periodic on longer timescales, with the stratification strengthening during neap tides and weakening during spring tides (Simpson et al., 1990; Nunes Vaz and Simpson, 1994). A sample of this process is seen in Figure 2, a plot of Spring 1995 data for South San Francisco Bay (Friebel et al., 1996). In this figure, the salinity difference between sensors located at mid-depth and near-bottom of the 15 m deep water column is plotted as a function of time for a station near the San Mateo Bridge. The predicted daily maximum tidal current speed for a station near the San Mateo Bridge is plotted as well (Cheng and Gartner, 1985). The vertical salinity difference displays a semidiurnal oscillation throughout the record, indicating the presence of tidal straining (SIPS). In addition, during the neap tides (when the daily maximum current speed is relatively low), the stratification does not break down completely on the semidiurnal timescale but, rather, persists for a number of days. Similar observations have been made in other estuaries, notably the York (Sharples et al., 1994), Columbia (Jay and Smith, 1990b), Hudson (Nepf and Geyer, 1996), Spencer Gulf (Nunes Vaz et al., 1989), and the Tamar (Uncles and Stephens, 1990). Evidently, SIPS is a common feature of a large class of estuaries.
Monismith et al. (1996) argued that the SIPS condition only occurs in a one-dimensional channel when
(1)
where Rix is a stability parameter; g is gravitational
acceleration; 
is the longitudinal
density gradient [kg/(m3-m)] (assumed to be constant over the flow
depth and in time); H is the depth of the channel; Umax is the
maximum tidal velocity on the surface; CD is the bottom drag
coefficient; and 
is the reference
density. This condition is based upon the assumption of a local
one-dimensional balance of salinity and momentum (i.e., replacing the estuary
with a fictional one-dimensional channel), and is supported by modeling
(using the methods and code described below) and salinity data from Northern
San Francisco Bay.
When Rix is greater than about 1, the stratification strengthens each tidal cycle, a condition referred to as "persistent" or "runaway" stratification. While the fundamental structure assumed by the model may be oversimplified, the general picture inferred by Monismith et al. (1996) appears consistent with field measurements. Model runs conducted in the course of the present study (to be reported elsewhere) suggest that the critical condition is also a function of the tidal excursion, behavior that most likely reflects the temporal element of flow evolution. Runaway stratification appears to be attributable to the strong nonlinearity that comes from the longitudinal salinity gradient; it provides the baroclinic pressure gradient that drives a flow which always acts to stratify, as well as providing the "source" term for the stratification itself. As the water column stratifies, the baroclinic flow strengthens (Jay and Musiak, 1996; Monismith et al., 1996), thus intensifying the stratification and further reducing the mixing rates. Since this sheared flow is superimposed on the tides, it ultimately can overcome the destratification that takes place on the floods.
The hydrodynamic model we use follows the approach of Hamblin (1989) and Simpson and Sharples (1991): we solve momentum, salt and turbulence balance equations representing turbulent flow in a hypothetical one-dimensional estuary; i.e., the governing momentum and salt conservation equations are simplified to retain only vertical variability in the velocities and salinities. For the momentum balance, this is accomplished by neglecting advective accelerations and by specifying tidal and non-tidal barotropic pressure gradients as well as a non-tidal baroclinic pressure gradient. The salinity we compute only varies in the vertical; however, since horizontal advection of salt plays a key role in the formation of stratification, we follow Simpson and Sharples (1991) and introduce a source term that represents the effects of horizontal advection. That is, the total salinity at a point is assumed to be of the form:
(2)
where 
, the mean longitudinal
salinity gradient, is a constant. Under these conditions, the salinity, S',
evolves according to a balance between unsteady change, diffusion, and
horizontal advection, i.e.:
(3)
where Kz(z,t) is the eddy diffusivity for salt, and U(z,t) is the
computed horizontal velocity. Note that it is vertical variability in U that
gives rise to stratification through the SIPS mechanism outlined above. In
order for this approximation to remain valid, we must assume that
does not vary with x.
Variations with time as well as with depth can be included without altering
the model structure (Simpson and Sharples, 1991).
The momentum balance is that used by Simpson and Sharples (1991) as implemented by Monismith et al. (1996) and summarized here. The barotropic tide is represented by a known time varying surface slope
(4)
where 
is the hypothetical
variation in water surface elevation necessary to drive an inviscid tidal
flow with a maximum velocity of Umax (see discussion in Monismith
and Fong, 1996) and period T (we use T=12 h, i.e., roughly an M2
tide). For simplicity, we consider only a single tidal constituent; however,
multiple constituents can also be used. The horizontal salinity gradient,
, which we specify, provides a
baroclinic pressure gradient that is independent of x and increases linearly
with depth. To account for the surface slope associated with the baroclinic
flow, an extra constant surface pressure gradient equal to a dimensionless
constant, 
, multiplied by the
depth-averaged baroclinic pressure gradient is imposed. The constant determines the tidally averaged flow
that results. In our application, we iteratively found (typically in the range -0.1 to -0.5)
to minimize the net depth-averaged flow over a tidal cycle. Note that the
flow depth is not allowed to change through the tide; the barotropic pressure
gradients we impose are only expressed in terms of surface slopes for the
sake of convenience.
With these assumptions and this structure, the momentum balance is:
(5)
where 
is the eddy viscosity, 
is the coefficient of saline
expansivity (such that = (1/)(), and z is the
depth measured negative downwards from the surface.
The turbulent diffusivities are found using Galperin et al.'s (1988) version of the Mellor-Yamada level 2.5 (MY2.5) closure commonly used to model turbulence in geophysical flows (Mellor and Yamada, 1982; Blumberg and Goodrich, 1990). MY2.5 defines eddy mixing coefficients as the products of a turbulence velocity scale (q), a turbulence lengthscale (l), and stability functions, Sm and SH, designed to represent the effects of stratification on turbulent mixing:
(6)
(7)
(8)
0.53 q/N
(9)Since the tide is dominant in inducing vertical mixing in many shallow systems, especially during non-storm conditions, our use of this model thus far has been confined to cases for which the interaction of the current with the bottom roughness is the primary source of turbulence; this study does not consider wind-induced mixing. Bottom friction is parametrized by specifying a bottom roughness (we use 1 cm) and assuming that U at the point closest to the bottom conforms to the law of the wall written in terms of the bottom roughness (Blumberg and Mellor, 1987).
The transport equations for salinity (Eq. 3), momentum (Eq. 5),
q2, and q2l are solved on a staggered finite difference
grid using the code described by Blumberg, Galperin, and O'Connor (1992),
hereinafter referred to as BGO. Turbulence quantities are used to update the
momentum and salt fields (via and
Kz), which in turn are used to update the turbulence fields. In
our application the grid typically has 5 cm vertical resolution (much finer
than what might be used with a full 3D circulation model), while the time
step is usually 100 s or less. The resulting eddy diffusivities, used in the
phytoplankton model runs discussed below, were calculated for SIPS flows and
runaway stratification cases, as well as for constant stratification cases
where the vertical density distribution is specified and the salinity
evolution equation is omitted. The version of the model used for the first
two types of flows we refer to as "BGO-SIPS," while the latter we
refer to as "BGO-SPEC." Apropos to the discussion above, the use
of the two different forms of the model allows us to represent a wide range
of estuaries, albeit in a simplified fashion.
S) versus time; and Figures 3d and 4d plot depth-averaged
tidal velocity (Uave) versus time for each case.
In the SIPS case (Figure 3), attenuation of mixing in the upper water column
and an overall increase in S is seen
in each ebb/early flood period. By mid-flood, however, mixing and reverse
straining are strong enough to begin to erode the stratification, producing
the observed decrease in vertical salinity difference. During mid-ebb, tidal
mixing is enough to partially homogenize the salinity profile, resulting in a
temporary decrease in S. For the
runaway stratification case (Figure 4), mixing in the upper water column is
constantly attenuated by the pycnocline (once the permanent stratification
forms), and the vertical salinity difference grows with time. Nonetheless, a
semidiurnal signal can still be seen in the vertical salinity difference,
indicating that gravitationally induced runaway stratification and SIPS can
act concurrently.
The SIPS case (Figure 3) is typical of lagoonal systems like South San
Francisco Bay where, except in very wet years (e.g. 1995), the influence of
freshwater is relatively weak and tidal mixing is usually able to destroy any
temporary stratification which may form. On the other hand, the runaway
stratification case (Figure 4) is typical of strongly stratified estuaries
such as North San Francisco Bay during the spring, where freshwater flow
through the Sacramento-San Joaquin Delta may generate a strong longitudinal
density gradient which dominates tidal mixing (Monismith et al., 1996).
Actual data (Figure 2) show S
ultimately declining after a runaway stratification event due to the increase
in Umax (tidal forcing) as a spring tide is approached, resulting
in a breakdown of the runaway stratification. In the simulations discussed
in this paper, Umax is constant; therefore, variation over a
spring-neap cycle is not explicitly modeled.
In our approach we have neglected the vertical velocity w, which we justify
as follows. In our simulations, the rising and falling of the water suface
throughout the tidal cycle are not modeled: instead the grid is fixed. In a
barotropic tide, w can be scaled as w 
(
H/t)(1 + z/H), where z is the local depth
and H is the total depth. If we assume that the depth varies periodically as
H = Ho + Asin[2
* t/T], where T is the period of motions (~12.4
hours) and A is the amplitude of tidal motions, then w is at most [2* A/T]. If A is about 1 m
(e.g. for San Francisco Bay), then vertical velocities are about 0.15 mm/s,
which is quite small. Therefore, we argue that the vertical velocities
induced by the tide are not significant in light of other approximations we
have made in constructing the model. This picture certainly will change for
flows in several dimensions where features like fronts can induce signficant
vertical velocities.
Most importantly, this idealization allows us to generate turbulent mixing results like those described above that reflect different forms of stratification, i.e., persistent or intermittent. In either case, it is important to bear in mind that vertical mixing in stratified estuaries results primarily from two sources of turbulence production if wind is neglected: the bottom boundary layer and shear that is internal to the water column (Abraham, 1988; Monismith and Fong, 1996). As we will demonstrate below, this is important to phytoplankton dynamics because bottom-generated turbulence produces a bottom mixed layer that entrains fluid from above as it grows. This results in phytoplankton cells being mixed from the photic zone (if it is shallow) and circulated over the deeper (and hence darker) part of the water column. This is different from the case found in lakes or in the ocean where mixed layer deepening involves entraining fluid from below, hence retaining cells in the upper mixed layer, though reducing the average light exposure of those cells. In later sections, we will show that this difference is important to understanding why the details of the stratification matter to estuarine phytoplankton dynamics.
The phytoplankton model is not calibrated or "tuned"; rather, it is based upon standard forms of equations for scalar transport and phytoplankton growth and employs parameter values representative of field measurements. Because South San Francisco Bay (SSFB) has been the source of detailed biological records over the last two decades (Cloern, 1996), this system serves as our "laboratory" for investigating phytoplankton dynamics in shallow estuaries. Thus, the parameter ranges used in our model (e.g. depths, benthic grazing rates, light attenuation coefficients, and rates of sinking, zooplankton grazing, and respiration) are typical of SSFB.
(10)
is the benthic grazing rate
[m3/(m2-d)], which is nonzero only at the bottom
boundary. The phytoplankton growth rate is calculated using Jassby and
Platt's (1976) hyperbolic tangent function for productivity, a constant ratio
of phytoplankton cellular carbon to chlorophyll, and an exponentially
decaying irradiance with depth. Values typical of SSFB are used for maximum
growth rate, respiration rate, zooplankton grazing rate, average daily
surface irradiance, sediment-related light attenuation, benthic grazing rate,
and sinking speed, which are all taken to be constant in time (see Table 1,
Koseff et al. (1993), and Appendix for details).Table 1. Variables and constants relevant to models and results.
The current version of V1D is based upon the model developed by Cloern (1991) and later refined by Koseff et al. (1993). However, instead of a finite difference formulation, this version uses a finite volume approach (MacCormack and Paullay, 1972), which is mass-conservative and greatly simplifies implementation of flux boundary conditions. The model employs a staggered grid which is divided into control volumes, or cells (Fig. 5). On this grid, B and µnet (biomass and sources/sinks) are defined at cell centers, while Kz and Ws (all flux-related quantities) are defined at cell faces. In this manner, we can enforce mass conservation, i.e. that for a given control volume:
) and tidal forcing
(Umax) conditions. Each case depicts a particular balance between
tidal mixing and river flow. A depth of 15 m was chosen because it is
typical of the channel in South San Francisco Bay. These hydrodynamic
simulations were used to provide the turbulent mixing information
(Kz's) for the phytoplankton model (V1D), in which we varied the
light attenuation and benthic grazing strength. Overall, we note the
following:
). Each curve represents a unique combination of
hydrodynamic conditions which are described in Table 2. For each point on
each curve, several V1D simulations were performed (each with a different
kt) to iteratively find a kt value which just allows a
bloom. For all curves, the phytoplankton sinking rate is Ws=0.5
m/d. Abiotic light attenuation and benthic grazing rate were chosen as the
independent variables because they each have the potential to control net
phytoplankton population growth in the water column and may vary widely over
time. At first glance, the combination of kt and on the same plot may seem peculiar since,
in the field, kt may vary on timescales of minutes to hours, while
may vary on timescales of weeks.
However, in the phytoplankton simulations performed with V1D, kt
is taken to be an average value of light attenuation over timescales of
days.
Table 2. Describes cases associated with curves in Figure 6. "Hydrodynamic Code" describes version of BGO used to calculate turbulent diffusivities for that case.
In Figure 6, the x-axis is proportional to the sink term (benthic grazing),
and the y-axis increases inversely with the source term (light-driven
production). Thus, as a point departs from the origin, grazing losses
increase and mean light exposure decreases, thus diminishing the likelihood
of a bloom. The curves bound the conditions under which blooms will likely
occur: for kt-
conditions producing a point above a particular curve, a bloom will not occur
for the hydrodynamic case represented by the curve; whereas conditions
producing points below the curve indicate that a bloom will occur for that
hydrodynamic case. These threshold curves thus demonstrate how physical
processes influence the balance between light-driven production and grazing
losses. For example, Curve 'd', which corresponds to = 0.065 psu/km and Umax=0.9
m/s (an intermittent stratification case) is indicated by the "- -
-" line. If the benthic grazing rate is 5
m3/(m2-d), then in order for a bloom to occur the light
attenuation must be less than 0.5 m-1 (a condition extremely rare
in SSFB). However, Curve 'i' shows that if = 0.261 psu/km and Umax=0.5 m/s (a runaway
stratification case), for similar benthic grazing conditions (5
m3/(m2-d)) the light attenuation can be as high as 2.1
m-1 (a condition common in SSFB) and a bloom will still occur.
The following points emerge from examination of Fig. 6. First, the
phytoplankton model predicts that for a 15 m deep water column, extremely
clear water is necessary to produce a bloom for tidally intermittent
stratification (SIPS), as well as for the constantly unstratified case ( = 0), even with zero benthic grazing.
In fact, the bloom threshold curves for the tidally intermittent
stratification case essentially overlay those for the constantly unstratified
case, suggesting that the SIPS mechanism does not increase the likelihood of
a bloom beyond that of a constantly unstratified water column. This trend is
attributable to the fact that in the unstratified and SIPS cases mixing of
the phytoplankton down through the water column is faster, on average, than
their growth. Second, it is evident that runaway stratification allows a
bloom to occur under much more turbid conditions than in the unstratified and
SIPS cases. Runaway stratification lengthens the timescale for vertical
transport of the phytoplankton relative to the timescale for growth, allowing
the phytoplankton to remain in the upper water column (photic zone) long
enough to multiply. Third, the intermittent stratification and unstratified
threshold curves exhibit steeper overall slopes than the runaway
stratification cases, indicating, as expected (Cloern, 1991), that the
effects of benthic grazing on an unstratified or intermittently stratified
water column are more marked than on a persistently stratified water
column.
Under runaway stratification conditions, a vertical density structure similar to that sketched in Figure 7a may develop. Different runaway stratification cases may exhibit different values of surface layer depth, or Zm. As evidence of the general structure shown in Figure 7a, Figure 7b shows field measurements of salinity and chlorophyll concentrations corresponding to the large runaway stratification event in SSFB depicted in Figure 2. The surface layer and a pycnocline at ~5.5 m depth are obvious in the field data, as is the effect of the pycnocline on the phytoplankton (i.e. inhibition of downward transport).
For each runaway stratification case plotted in Figure 6, the estimated value
of Zm is listed in Table 2 alongside the corresponding , Umax, and H values. For
the unstratified and periodically stratified cases, the pycnocline is either
absent or not persistent and so is represented as "---" in the
Zm column. Notice that the curves (e.g. 'h','i') in Figure 6
appear to be grouped by values of Zm, with a significant distance
between groupings. In these cases, a shallower mixed layer is more likely to
produce a bloom. This indicates that, in addition to the issue of
intermittency versus persistence of stratification, other details of the
stratification are important to the phytoplankton dynamics as well. We
explore these details of the stratification below.
ii. Approach.
A modified BGO model was used to generate vertical turbulent diffusivity
fields associated with different scenarios of persistent stratification. In
persistent stratification cases simulated by BGO-SIPS (which allows the
stratification to evolve from a balance between the oscillating tidal
pressure gradient and a longitudinal density gradient), we can neither
predict nor directly control the stratification characteristics such as
Zm, Tpyc (the thickness of the pycnocline), or S (the vertical salinity difference).
Therefore, we developed BGO-SPEC, which allows us to specify the
stratification parameters for each run, hold them constant, and subject the
water column to an oscillating tidal current. This enables us to explore the
relationship between Zm (and Tpyc, S) and bloom initiation in a controlled
fashion.
Holding the stratification parameters constant (to emulate a constant source
of buoyancy) is not completely realistic; however, as is shown in Figure 8,
this method produces a reasonable approximation to the effects of the
physics, as modeled by BGO-SIPS, on the phytoplankton. In Fig. 8,
kt- bloom threshold
curves are shown for a range of runaway stratification conditions generated
by BGO-SIPS and by BGO-SPEC. The hydrodynamic parameters for all cases are
summarized in Table 3. For each BGO-SIPS case, we averaged the resultant
stratification parameters over the run and then used those average values for
Zm, Tpyc, and S in the associated BGO-SPEC run. The kt- threshold curves generated by V1D for
each pair of cases are very close. For example, Case 'a', for which = 0.131 psu/km and Umax=0.47
m/s, resulted in an average Zm of 0.7 m, Tpyc=0.8 m,
and S=1.7 psu. For this
particular case, the maximum light attenuation allowing a bloom (for =0) is about 3.4 m-1. A
separate BGO-SPEC simulation using the average stratification parameters and
holding them fixed (Case 'b') resulted in a maximum light attenuation for a
bloom of about 3.7 m-1. Even closer correspondence is evident in
the other cases. For example, the maximum light attenuation for the BGO-SIPS
Case 'e' (kt=1.66 m-1) is exceptionally close to that
for the associated BGO-SPEC scenario, Case 'f' (kt=1.69
m-1). This type of comparison assures us that BGO-SPEC provides a
sound approach for exploring the effects of Zm on bloom
initiation.
Table 3. Describes cases associated with curves in Figure 8.
"Hydrodynamic Code" describes version of BGO used to calculate
turbulent diffusivities for that case. For BGO-SIPS cases, Zm,
Tpyc, S are time-averaged
values.
iii. Relationship of Surface Layer Depth to Bloom Inception
and Magnitude.
V1D and BGO-SPEC enabled us to explore the effects of surface layer depth on
bloom initiation and, in particular, the applicability of the SCDM to shallow
estuarine systems. We did this by calculating phytoplankton population
growth for different ratios of surface layer depth to critical depth and then
comparing model results with the SCDM criterion that growth is positive
(blooms occur) whenever Zm/Zcr <1. To this end, we
show calculated 
, the depth-averaged
phytoplankton biomass at t=5 days, normalized by 
, the initial average concentration (3 mg chl
a/m3), plotted versus Zm/Zcr for a range of
light attenuation values (Figure 9). For all these cases, H=15 m,
Umax=0.75 m/s, Tpyc=1 m, S=5 psu, Ws=0.5 m/d, =0 m3/(m2-d), and the maximum growth
rate µmax=2 d-1. The simulation length of five
days was chosen because it is representative of the typical duration of
persistent stratification in a system for which spring/neap mixing effects
are significant (see Figure 2). Zcr is calculated as the depth at
which the integral net growth rate (see Appendix, Jassby and Platt, 1976),
including the effects of respiration, zooplankton grazing, and depth-variable
irradiance (neglecting self-shading), is zero. Each curve is the result of
several five-day phytoplankton simulations with V1D, which used turbulent
diffusivities generated by BGO-SPEC. For each curve, abiotic light
attenuation (kt) is held constant and only Zm varies.
We produced curves for different light attenuation values because
kt essentially sets Zcr (each curve is therefore
associated with a particular Zcr). If the SCDM captures the
processes controlling bloom development and collapses them into one universal
relationship, then the applicability of the SCDM should be independent of
Zcr (i.e. all the curves in Fig. 9, which differ only by
Zcr, should display the same behavior with respect to bloom
initiation).
S=5 psu, Ws =0.5 m/d,and =0 m3/(m2-d).From the perspective of maximizing phytoplankton production, it is evident that for each kt there is an "optimal" pycnocline depth, at which the peak of each curve is located. For points on either side of the peak, conditions are less than optimal such that the biomass produced in the water column is less than the maximum. Also plotted in Figure 9 is a horizontal line representing zero growth in depth-averaged phytoplankton biomass. It is evident that there are numerous cases for which Zm/Zcr < 1 but population growth is negative. Because the traditional critical depth model applies to surface layer averaged biomass rather than depth-averaged biomass, we checked the cases with Zm/Zcr < 1 that fell below the zero-growth level and confirmed that in the majority of those cases surface layer averaged biomass did not increase either. Model results are therefore inconsistent with the traditional SCDM. Specifically, our results indicate the following for the systems considered here:
i. Turbulent Mixing.
The surface layer depth, Zm, is closely associated with the
intensity of turbulent mixing in the surface layer and thus with the balance
between mixing and sinking in the upper water column. The turbulent
diffusivities (Kz's) calculated in BGO are proportional to ql,
where q is the square root of the turbulent kinetic energy (TKE) and l is the
turbulence macroscale (i.e. a typical eddy lengthscale, or scale over which
phytoplankton cells are mixed by the turbulence). Kz profiles
were calculated by BGO-SPEC for three values of Zm but the same
Umax, Tpyc, and S (Figure 10). Notice that for smaller Zm, the
maximum turbulent diffusivity in the surface layer (
) is much smaller than for larger Zm. As
Zm increases, the typical turbulent lengthscale for the surface
layer increases, thus increasing Kz, which is proportional to l.
An increase in l also indirectly (and nonlinearly) influences Kz
by increasing shear production and turbulent transport of TKE up through the
surface layer. These sources of TKE enhancement in the surface layer, as
well as decreased dissipation of TKE (which scales as q3/l), all
contribute to higher Kz's in the surface layer as it is deepened.
Furthermore, enhanced surface layer mixing is associated with greater
turbulent diffusivities at the surface layer/pycnocline interface. Thus, for
thicker surface layers, there is more turbulent leakage of phytoplankton out
of the surface layer.
S=5 psu.ii. Sinking in the Presence of Turbulent Mixing. Mixing in the surface layer can be especially important to bloom dynamics when phytoplankton sink. If mixing is strong enough, it partially counteracts the sinking loss of phytoplankton from the surface layer. The balance between turbulent mixing and sinking can be represented by the turbulent Peclet number, the ratio of the mixing timescale to the sinking timescale:
(11)
If we consider the turbulent diffusivity profiles shown in Figure 10 for a
particular combination of Umax, S, and Tpyc, the typical turbulent diffusivities
in the surface layer for Zm =1, 3, and 5 m are O(1), O(10), and
O(100) m2/d, respectively. For a sinking velocity of
Ws=0.5 m/d, the turbulent Peclet numbers for ascending
Zm range from O(1), for which sinking is important, to O(0.01),
for which sinking is relatively unimportant. Thus, larger Zm
values are associated with more intense surface layer mixing and, in turn,
with lower turbulent Peclet numbers and, therefore, reduced sinking losses of
phytoplankton from the surface layer. Platt et al. (1991) also suggested an
inverse relationship between sinking-related losses and Zm.
iii. Average Net Growth Rate: Surface Layer and Below
Pycnocline.
Surface layer depth controls bloom intensity in other ways. This is
demonstrated in Figure 11, which shows two scenarios associated with
different surface layer depths. Figure 11a is for a shallow surface layer,
while Figure 11b is for a deeper surface layer. The shallow surface layer
has less intense turbulent mixing, higher turbulent Peclet numbers, and,
therefore, greater sinking losses than the deeper surface layer. However,
the shallow surface layer has less turbulent leakage of phytoplankton than
the deeper surface layer. The shaded area is the region of positive local
net growth (where local growth rate exceeds respiration and zooplankton
grazing losses). On the right, we show schematic vertical profiles of net
growth rate and water density associated with each case. A smaller
Zm is associated with higher 
, the average net growth rate over the surface layer
(because mean light exposure of the surface layer phytoplankton increases as
Zm decreases).
Finally, depth-averaged biomass is affected by the net production that occurs below the surface layer. This subsurface phytoplankton is easily transported down by sinking while it is in the pycnocline region (since mixing there may be very weak) and by turbulent mixing below the pycnocline. In other words, if the depth at which local growth is zero extends below the surface layer (i.e. in the pycnocline or lower), phytoplankton will be produced that may be easily lost to the lower aphotic water column, as opposed to remaining longer in the surface layer, where it experiences positive net growth rates and is relatively isolated from benthic grazers. The significance of retaining cells in the region of maximal growth was emphasized by Smetacek and Passow (1990) as a key to bloom initiation.
The relationship of these four Zm-related processes to the
non-monotonic behavior of the curves in Figure 9 is illustrated further in
Figure 12. This schematic of population growth (/) versus
Zm/Zcr shows that for Zm less than optimal
(to the left of the peak), is large
and turbulent leakage from the surface layer is minimized, but sinking losses
may be significant and a large fraction of the production may occur below the
surface layer. Opposite trends apply to Zm values greater than
optimal (to the right of the peak). The peak of the curve represents an
optimal balance of these four conditions.
In Figure 13, we illustrate the diversity of functional relationships that
can exist between phytoplankton population growth and mixed layer depth. The
plot of , depth-averaged biomass at
t=5 days, versus Zm is shown for both zero and nonzero sinking
rates (Fig. 13a). Both relationships are non-monotonic. This is because
depth-averaged phytoplankton population growth is smaller when a large
fraction of the production occurs below the surface layer, rather than solely
in the surface layer. The Ws=0.5 m/d curve shows that sinking
augments the non-monotonic behavior. The plot in Figure 13b, on the other
hand, shows 
, surface layer averaged
phytoplankton biomass at t=5 days, versus Zm for zero and nonzero
sinking rates. In the absence of sinking, decreases exponentially (i.e. monotonically) as
Zm increases. Thus, if sinking is zero, the -Zm relationship is consistent
with that proposed by Riley (1942). The reason for this is that surface
layer averaged biomass is not affected by the production occurring below the
surface layer. Therefore, if sinking is zero, then surface layer averaged
biomass will decrease as Zm increases; whereas, if depth-averaged
biomass is considered, or if sinking is nonzero, then there will be a
non-monotonic relationship between surface layer depth and bloom intensity.
S=5 psu, Umax=0.75 m/s, H=15 m,
kt=.5 m-1, =0.0.Note that the previous discussion is applicable to a purely tidally driven system. If wind is a significant source of TKE, the system will likely be less sensitive to sinking out of the surface layer because enhanced surface layer mixing would better counteract sinking (i.e. the turbulent Peclet number would be lower). Such a system would therefore tend to have a more monotonic relationship between depth-averaged phytoplankton biomass and surface layer depth. This relationship also depends on the function chosen to describe phytoplankton growth rate. For example, photoinhibition of algal growth (not included here) would suppress bloom development in very shallow surface layers, shifting the initial rise of the curve in Figure 12 toward the right.
For a system in which the primary source of turbulence is the interaction of
the current with the bottom roughness, the interfacial turbulent diffusivity,
or 
, generally increases as
Umax, the maximum tidal current velocity, increases. Furthermore,
since stratification inhibits the transport of turbulence up through the
water column, decreases with
increasing S and Tpyc.
These relationships are demonstrated in Table 4, which shows values of
time-averaged interfacial turbulent diffusivity, 
, predicted by BGO-SPEC for different combinations of
Zm, Umax, Tpyc, and S. Also shown is the ratio of to 
, the
time-averaged maximum Kz over the depth. For each hydrodynamic
case, the V1D phytoplankton model was used to simulate two scenarios, each
for a different light attenuation coefficient. For each phytoplankton
simulation, two quantities were calculated: 1) Fturb, the
turbulent leakage flux of phytoplankton from the surface layer over one
timestep; and 2) Prod, the surface layer production over one timestep.
Included in Table 4 is the time-averaged ratio of Fturb to Prod,
or the average fraction of phytoplankton biomass produced in the surface
layer that is lost via turbulent mixing.
Table 4. (a) , turbulent
diffusivity at surface layer/pycnocline interface averaged over five days,
for various combinations of Zm, Umax, Tpyc,
and S; for each such hydrodynamic
case, two values of light attenuation, kt, were used; (b) normalized by , the five-day average maximum turbulent diffusivity over
the total water column depth; (c) the five-day average ratio of
Fturb, the turbulent leakage flux from the surface layer, to Prod,
the surface layer production, for the various hydrodynamic conditions as well
as different values of kt. For all values in (c),
Ws=0.0 and =0.0.
The results in Table 4 show that, even for values of which are 103-104 times smaller than
the average maximum diffusivity, it is possible for the surface layer to lose
50% or more of its integral production via turbulent mixing. In fact, in
more turbid water, phytoplankton growth rates may be so slow that turbulent
mixing may remove more than 100% of what the surface layer produces (i.e.
remove all of what was produced plus a portion of what existed previously).
Thus, even "small" turbulent diffusivities can be responsible for
significant losses of phytoplankton out of the surface layer. In the next
section, we elaborate on the direct effects of surface layer leakage--both
advective and turbulent--on phytoplankton dynamics.
:
(12)
= total net growth in surface layer over one timestep [mg chl a]
= total advective plus turbulent diffusive flux out of surface layer over one
timestep [mg chl a]
(the superscript "int" refers to "surface layer/pycnocline interface")
represents the net
accumulation of phytoplankton biomass in the surface layer normalized by the
amount produced in the surface layer over one timestep. This quantity
reflects the balance between production in the surface layer and leakage out
of the surface layer. If = 1,
there are no leakage losses out of the surface layer; if 0 < < 1, there is positive net
accumulation in the surface layer, despite the occurrence of leakage; if < 0, then leakage dominates
production, and the surface layer experiences net loss. Sustained positive
values of are associated with
a bloom in the surface layer; whereas, sustained negative values of are associated with a decline in
surface layer biomass.
We have plotted versus time
for two cases which vary only by maximum tidal current speed (Figure 14).
BGO-SPEC was used to calculate turbulent diffusivities for both cases, with
H=15 m, Zm=1 m, Tpyc=1 m, and S=5 psu. In V1D, Ws=0.5 m/d and kt=1
m-1 for both cases. For reference, the lower Umax case
(represented by the solid line) resulted in =29.3 mg chl a/m3 (a large bloom); whereas, the
higher Umax case (represented by the dashed line) yielded =6.6 mg chl a/m3 (a much
smaller bloom). The plot shows that is always less than 1 for both cases, indicating that
leakage fluxes are constantly occurring. For Umax=0.95 m/s, a
strong quarterdiurnal signal is evident, with oscillating between positive and negative values. This
indicates that the leakage contains a strong tidal mixing component which
dominates the production during those portions of the semidiurnal tidal cycle
when mixing is most intense. In between such leakage-dominated episodes,
returns to positive values,
indicating that tidal mixing is weak enough such that production is
temporarily able to exceed leakage. This result underscores the importance
of semidiurnal tidal variability, as hypothesized by Koseff et al. (1993).
For Umax=0.75 m/s, only a faint tidal signal is evident,
indicating that sinking is the most dominant of the two leakage processes for
this case. Furthermore, for this low Umax case, is always positive, indicating that
the surface layer biomass must be increasing in time, resulting in a much
higher than for the higher
Umax case. The comparison of these two Umax cases may
explain why blooms in SSFB always occur during periods of low tidal energy
(Cloern, 1991; Cloern, 1996).
S=5 psu, and Ws=0.5 m/d.
Earlier, we asserted that leakage from the surface layer is the reason for
which our bloom predictions do not adhere to the traditional Sverdrup
Critical Depth Model. We then explained the leakage mechanisms and
demonstrated their effect on phytoplankton blooms. We now finally show that
if all leakage is completely removed, our results are consistent with the
SCDM. We have plotted 
, the
phytoplankton biomass averaged over the surface layer, versus time for four
different cases (Figure 15). Common to all cases are the following: H=15 m,
Umax=0.75 m/s, Tpyc=1 m, S=5 psu, and kt=4 m-1. For this light
attenuation coefficient, Zcr=5.5 m. According to the critical
depth model, Zm=5 m (< Zcr) should result in a
surface layer bloom for these irradiance conditions; whereas, Zm=6
m (> Zcr) should not. We see that for Zm=5 m, the
model predicts no bloom if leakage of any sort occurs; whereas, if sinking
and interfacial mixing are turned off, there is a surface layer bloom, as
predicted by the SCDM. If Zm=6 m and all leakage is removed,
there is no bloom, demonstrating further consistency with the SCDM under
these ideal conditions.
S=5 psu, and =0.0.
Zm and Ws are in [m] and [m/d], respectively.
Although the persistence of stratification is important, the details of the persistent stratification are important as well. Specifically, surface layer depth, thickness of the pycnocline, vertical density difference, and tidal current speed all weigh heavily in producing conditions which promote the onset of phytoplankton blooms. Our investigation of the effects of such hydrodynamic details leads to an explanation of why we might expect a range of functional relationships between phytoplankton growth and surface layer depth. First, there may be a non-monotonic relationship between phytoplankton population growth and surface layer depth. Thus, a shallower surface layer is not necessarily "better," from the perspective of maximizing phytoplankton production. This non-monotonic behavior is the result of the influence of Zm, the surface layer depth, on several "competing" processes:
What are the differences between shallow tidally driven systems and deeper pelagic systems for which the traditional SCDM appears applicable? First, the primary source of turbulence in the ocean is the wind; whereas, for the system we have studied, turbulence is generated primarily at the bottom of the water column due to the interaction of the tidal current with the bottom roughness. In the ocean, therefore, as the wind blows, the surface layer is deepened and the euphotic zone may remain within the surface mixed layer. On the other hand, in shallow tidally driven systems, the "real" mixed layer is the bottom layer, which, as turbulence continues to be generated, expands upward possibly into the euphotic zone, entraining phytoplankton cells and mixing them downward into aphotic conditions. Where enhanced mixing may deepen the surface layer in the ocean, somewhat decreasing the average surface layer net growth rates, enhanced tidally driven mixing in the shallower system may have stronger negative effects on phytoplankton population growth by shallowing the surface layer and removing phytoplankton biomass from the euphotic zone.
A second difference is that the deeper wind-mixed surface layer may not incur as severe sinking losses as the shallower system, because greater turbulent diffusion in the deeper surface layer (due to both the wind source and the larger turbulent lengthscale) results in smaller turbulent Peclet numbers in the surface layer. Thus, a deep wind-driven system may not have as much advective leakage as the shallower wind-free system. For this reason, a deeper system may not have as strong of a non-monotonic relationship between phytoplankton growth and surface layer depth as the shallower tidally driven system has. Inclusion of wind effects on the shallow system would most likely diminish advective surface layer losses and cause the relationship between phytoplankton concentration and surface layer depth to be more monotonic. However, it is important to note that wind-induced mixing would likely enhance turbulent leakage of phytoplankton from the surface layer and augment the deviation from the SCDM. Similarly, we might expect that deeper systems could also experience enough turbulent surface layer leakage to deviate substantially from the SCDM.
In summary, the surface and bottom layers of a persistently stratified water column are not truly "decoupled" as has often been believed. This was pointed out by Sharples and Tett (1994) who, in order to match model results with observations of a mid-water chlorophyll maximum, had to allow some small degree of transport between the two layers. Thus, it may be best to conceptualize a pycnocline as a physical feature that merely detains phytoplankton cells in the surface layer (slowing their downward transport) as opposed to retaining them in the surface layer (completely preventing their downward transport). This vertical transport can occur by sinking or turbulent diffusion and, even at low levels, can severely reduce the likelihood of a bloom and lead to substantial departure from the traditional Sverdrup Critical Depth Model.
(13)
, the ratio of phytoplankton
cellular carbon to chlorophyll a, is a constant equal to 50 (Cloern, 1991).
Furthermore, we assume that ZP, losses to zooplankton grazing, are constant
in depth and time. Thus, the relationship for the net phytoplankton growth
rate is as follows:
(14)
(15)
(16)
A finite volume discretization originates with the "conservation law form" of the continuous equation. The defining characteristic of an equation cast in conservation law form is that the flux terms are combined into one term, which is the divergence of the total flux. The general phytoplankton transport equation in conservation law form is:
(17)
Uk = velocity in k-direction
ik = unit vector in k-direction
Q = source term
, and applying Gauss' Divergence Theorem, we get:
(18). S is the surface enclosing , and dS is a surface element of S
with the direction of the outward normal to S. Discretizing Eq. 18 for our
1D case (Fx/x = Fy/y = 0), we get the following:
(19)
, realizing that for our 1D water column S=1 and = z, and making use of the fact that the benthic grazing term
is zero everywhere but at the bottom boundary, Eq. 19 becomes the following
for all interior cells:
(20)The turbulent diffusivities (Kz's) are obtained from the BGO code, and are located exactly where they are needed--at the cell faces. Furthermore, the staggered grid conveniently locates biomass concentrations between faces such that centered differences are easily implemented for the spatial derivatives in the diffusion terms. The implicit treatment of diffusion produces a tridiagonal system of equations, which is solved using the Thomas Algorithm.
The advection terms are slightly more challenging. In order to calculate B*, our estimate of B at the cell face, we use Leonard's (1979) QUICK (Quadratic Upstream Interpolation for Convective Kinematics) method. QUICK uses a three-point upstream-weighted quadratic interpolation for the concentration at a cell face. This method helps to minimize instabilities associated with central differencing of convection-dominated problems, and it does not produce the significant artificial diffusion introduced by classical upwind methods. The QUICK estimate of B* is of the following form:
(21)
(22)
(23)
(24)
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