Journal of Plankton Research Vol.19 no.12 pp.1859-1880, 1997
Convection in ice-covered lakes: effects on algal suspension
Dan E.Kelley
Department of Oceanography, Dalhousie University, Halifax, Canada
Abstract. Convection occurs in ice-covered lakes if solar radiation warms near-surface water from
the freezing point towards the temperature of maximal density. One effect of convective mixing may
be to suspend non-motile phytoplankton in the upper water column, providing cells with enough light
for growth during ice-covered periods. Observations of the diatom Aulacoseira baicalensis under the
ice cover of Lake Baikal, Siberia, support the hypothesis that convective mixing causes net suspension of cells. This paper presents a theoretical examination of the conditions under which convective
flow fields can suspend algae in the photic zone of the upper water column. It is shown that the
efficiency of algal suspension depends on the ratio of the still-water algal sinking rate, Wp, to convective updraft speed, Wu. The suspension efficiency is also shown to be affected by asymmetries in
the flow field and night-time cessation of convection, but only if Wp and Wu are comparable in value.
It is concluded that convection in Lake Baikal should be vigorous enough to increase the mixed-layer
residence time of A.baicalensis from a few days to over a month, at least during years with thin snow
Introduction
The common existence of under-ice algae in lakes and in the ocean is becoming
increasingly apparent (Vincent, 1981; Mtiller-Haeckel, 1985; Homer et al., 1992;
Ackley and Sullivan, 1994; Spaulding et al, 1994; Kirst and Wiencke, 1995). It is
also becoming clear that these algal populations may be quite important to highlatitude ecology (Michel et al., 1996; Melnikov, 1997). For example, Legendre et
al. (1992) have estimated that ice-bottom algae account for 25% of the annual
primary production in seasonally ice-covered Arctic waters.
Recent studies reveal that the physical and biological systems are closely interlinked (e.g. Ackley and Sullivan, 1994). One implication of the linkage is that
future field programs should avoid sampling the physical and biological systems
in isolation from one another. A second implication (and a principal motivation
for the present study) is that improved understanding of the physical environment may shed light on biological questions. If it can be established that particular aspects of the physical regime are likely to have a large influence on the
biology, then it may be possible to optimize future monitoring of physics and
biology together.
To the extent that lake systems and ocean systems are analogous, improved
understanding of physical/biological coupling in fresh water may also elucidate
the marine state. This may be relevant to studies of global warming, since global
climate models predict that high-latitude regions will be affected more than midlatitude regions by climate change (Houghton et al, 1995). Insight into physical/biological coupling in lakes, for example, could help to answer important
questions about ocean carbon sequestration in a wanned climate with reduced
ice cover.
Unfortunately, field studies in high-latitude ice-covered marine systems are
expensive and logistically difficult. Even though logistical constraints may be
© Oxford University Press
1859
D.E.Kelley
fewer in lakes, intensive coupled physical and biological sampling in fresh water
has also been quite rare. It makes sense to minimize wasted effort in future field
programs by determining in advance the most relevant parameters and the
magnitude of expected signals. This provides compelling motivation for laboratory and theoretical work, as an adjunct to field studies.
The present contribution falls into the latter category, being a theoretical study
motivated by observations in Lake Baikal, Siberia. This is the deepest lake in the
world, reaching -1600 m, and one of the largest, spanning an arc that is -80 km
wide and -600 km long (Kozhov, 1963). It experiences under-ice diatom blooms
at somewhat irregular intervals of 2-5 years (Kozhov, 1963; Bondarenko et ai,
1996). The cause of the interannual variability in the bloom is not certain, but one
possibility is that it is related to interannual variability in the physical regime, and
this is a prime motivation for the present study.
The goal here is to estimate the magnitude of convective effects on algal suspension in Lake Baikal and similar ice-covered lakes, bearing in mind the possible importance of physical/biological coupling. This may help to guide future
sampling in such systems and may aid retrospective analyses of physical and biological time series. (Much of the analysis has analogy to the ocean as well, with
shear-driven motions taking the part of convective motions.) I begin with a somewhat general discussion of the physical and biological constraints on phytoplankton in moving fluids, and the requirements for aflowfield to suspend heavy
phytoplankton cells. Then the particular case of convective water motion is discussed in detail, including effects offlowasymmetry and diurnal variation in solar
forcing. Conclusions and suggestions for future work are given at the end.
Physical and biological constraints on under-ice algal blooms
A major requirement for the maintenance of algae under ice is that enough light
be able to pass through the ice to fuel algal growth in the water below. For this
to be the case, snow cover must be thin, since snow shades much more effectively
than ice. (A mere 0.05 m of snow atop 1 m of clear ice reduces light transmission
by a factor of 50; see Appendix.) Therefore, growth can proceed readily only if
snow cover is minimal. This is often the case in large lakes that are swept clear of
snow by winds extending over long fetches. Lake Baikal provides a good
example. Often it has no snow over the ~1 m of ice cover during thefirst2 months
of the calendar year. By late March to early April, however, warming converts
the ice surface to a snow-ice amalgam (Kozhov, 1963, Table 31). The amalgam
provides increasingly effective shading over time. Competition between the
increasing shading and the increasing solar radiation through these early months
of the year results in a maximum in through-ice radiation in early March.
March is also the time when blooms of the diatom Aulacoseira baicalensis have
been observed in Lake Baikal (Kozhov, 1963, Figure 95). (In referring to
Kozhov's work, note that this diatom was called Melosira baicalensis at that time.)
During this period, the algal cells are distributed with approximately uniform
concentration in a surface mixed layer some tens of meters thick. They are also
1860
Conrection in ice-covered lakes and algal suspension
found in similar concentrations within the ice, and in much larger concentrations
in a thin aggregation layer just under the ice (S.I.Heaney, unpublished data).
As is the case with most diatoms, A.baicalensis cells are heavier than water, so
they can be expected to sink out of the mixed layer in a few days, given a mixedlayer depth of -20 m (Kozhov, 1963) and a sinking rate of ~5 m day-1. [This stillwater sinking rate, from unpublished data of T.L.Richardson, is within the range
of values reported for marine algae (Smayda, 1970; Smayda and Bienfang, 1983)
and freshwater algae (Reynolds, 1984).] The fact that high concentrations are
observed throughout the mixed layer for periods exceeding this time scale, with
much lower concentrations in deeper water, implies the presence of a mechanism of active suspension of the cells.
How might this suspension be achieved? In other regimes, notably wind-driven
coastal upwelling systems, steady vertical motion can suspend algae. This cannot
occur in Lake Baikal during March, since the land-fast ice insulates the water
from wind stress. By the same token, turbulent motions driven by wind stress
(Gregg, 1987) can be eliminated from consideration, as can Langmuir circulation
arising from the interaction of waves and shear. In the absence of turbulence
caused by strong horizontal currents, therefore, this leaves just one main possibility for suspension: convective motion. Such motion is the focus of this paper.
In ice-covered lakes, convection can be caused by solar heating. The mechanism is unique to freshwater systems in which surface temperatures are cooler than
~4°C, the temperature at which fresh water has maximum density. If cold surface
water is warmed by solar radiation, its density will increase, and the resultant
gravitational instability will cause convection (Woodcock and Riley, 1947; Fanner,
1975; Matthews, 1988; Bengtsson, 1996; Kenney, 1996). Although the convective
mechanism has been invoked previously to explain suspension of algae in lakes
(Baker, 1967; Hawes, 1983; Matthews and Heaney, 1987), the details have not
been worked out fully. This is partly because the mechanism is described by nonlinear equations whose solution is difficult, and partly because direct measurements of convection, required to calibrate theory, have not been carried out.
Direct measurements are needed because the indirect evidence is somewhat
ambiguous. The main indication is the existence of spoke-like patterns in the ice
surface and snow cover. Such patterns are commonly observed in lakes and ponds
(Katsaros, 1981). The first explanation proposed for the patterns was that they
resulted from differential melting of ice under which flowed warm and cold convective plumes [see Woodcock and Riley (1947) for an early introduction and
Matthews (1988) for a much more extensive recent treatment]. More recently, the
alternative hypothesis has been suggested that the patterns result from convection in melt water above the ice, rather than convection below the ice (Woodcock
and Lukas, 1983). Resolution of these conflicting hypotheses will be achieved
only with direct measurements of convective flows. One goal of this paper is to
indicate the magnitude of the physical signals (e.g. currents) in such systems.
Related biophysical effects, such as those deriving from the varying light and
nutrient levels encountered by phytoplankton cells moving with and through the
fluid (Denman and Gargett, 1983; Lewis et al, 1984; Lande and Lewis, 1989;
Mann and Lazier, 19%), will not be discussed here.
1861
D.E.KeUey
The convection being discussed falls into the category of mixed-layer deepening driven by destabilizing buoyancy flux. Thus, somewhat paradoxically, the
response of cold fresh water to surface warming is analogous to the response of
salt water (and warm fresh water) to surface cooling. The result is growth of the
surface mixed layer through entrainment of the denser waters below. The initial
conditions for springtime convection are as follows (Figure 1). Surface temperatures are ~0°C, owing to thermodynamic equilibrium between ice and water.
Deeper temperatures are higher, with bottom temperatures approaching 3-4°C.
This is the result of autumn overturning, during which surface waters cooled to
~4°C sink towards the lake bottom (Killworth et al., 19%). The details of the temperature profile depend on such things as plume dynamics during autumn overturning, ice formation, etc. Such details are not of concern in this general and
preliminary calculation. I rely only on the basic pattern of surface water near 0°C,
separated by a thermocline from warmer waters below (Figure 1). This pattern
0°C
4°C
Fig. L Definition sketch illustrating a cross-section of ice cover and the convective mixed layer below.
The temperature profile at the left indicates the mixed layer with a temperature of -0°C and denser
deep waters with a temperature of ~4°C. Just below the ice is a thin region of warm water that provides the driving force for convection. The convective motions, indicated schematically by circular
streamlines with an updraft between, are the proposed mechanism of suspension of algal cells (indicated by dots) within the mixed layer.
1862
Convection in ice-covered lakes and algal suspension
holds in most ice-covered lakes, and certainly in Lake Baikal (Kozhov, 1963,
Figures 19-21).
The cold surface temperatures ensure that the base state is susceptible to convection driven by surface heating. As discussed above, the convection should be
strongest in Lake Baikal during March, coinciding with the A.baicalensis bloom.
It will be argued here that convective mixing is important to this bloom since it
can facilitate growth by suspending the diatoms near the surface. One may speculate as to the adaptive strategy involved in such a physical/biological coupling, but
that would be well beyond the reach of the present discussion. My goals are more
modest, starting with a demonstration that convective motions are swift enough
to suspend the algae.
Requirements for algal suspension
The essence of the suspension of heavy particles (e.g. algal cells) by fluid motion
is simple: updrafts can capture cells that sink at a speed equal to or less than the
updraft speed. In a bounded domain, the updrafts feed horizontalflowsand recirculating downdrafts from which sinking cells may escape, perhaps to be recaptured in another updraft in a circulation pattern below. Flow systems may be
made up of steady eddies with simple geometry or of more complicated unsteady
flows. In each case, updrafts can hold cells aloft provided that their still-water
sinking rate, denoted Wp, is less than the typical updraft speed, Wu. A field of
eddies, spread horizontally or vertically, can suspend cells that sink slower than
updrafts rise, i.e. cells with Wp < Wu. The mechanism was first studied in the
context of algal suspension by Stommel (1949), who was considering Langmuir
circulations. The mechanism also applies to other geophysical systems, e.g. transport of snow and sediment (Nielsen, 1992).
For a givenflowfield,the efficiency of cell suspension may be measured by the
fractional area of theflowfieldthat can suspend a cell of a given still-water sinking
rate. This measure of suspension efficiency, denoted E, is central to this paper.
There are two possible modes of suspension. First, a cell may be suspended
motionlessly, if the water flow is equal and opposite to the cell sinking rate.
Second, a cell may be suspended if it moves along a closed trajectory, alternately
rising and falling, analogous to a juggled ball.
To understand the functional dependence of E on flow properties, one must
first note that not all cells with Wp < Wu will be suspended with equal efficiency.
Consider a simple circulation pattern, or eddy, in the form of a spinning cylinder
of fluid. A neutrally buoyant cell inserted anywhere in such a flow will circle
forever, tracing a closed streamline, but the fate of a heavy cell depends on the
location of insertion. Inserted in a downdraft, it could escape from the eddy by
sinking past the streamline on which it had been inserted. Inserted in a sufficiently
strong updraft, it could be suspended indefinitely, either motionless or tracing a
closed trajectory. In other words, only certain portions of a circulation pattern
can suspend cells of a given still-water sinking rate.
Two limiting cases illustrate the dependence of E on Wp and Wu. Neutrally
buoyant cells can be suspended everywhere, regardless of the flow direction or
1863
D.E.KeUey
strength, so the efficiency must approach 100% for Wp/Wu —> 0. Cells sinking
faster than the swiftest updraft cannot be suspended anywhere in theflowfield,
so E must fall to zero as Wp approaches or exceeds Wu. This suggests two end
points in the functional dependence of E = E(Wp/Wu,...).
Most of the following analysis will concentrate on E, which is easily calculated
for a givenflowfield.For some purposes, it is also useful to calculate the expected
residence time of cells in the surface region. One way to estimate this is with an
area-averaged effective sinking rate W.. This accounts for both suspension by the
flow and sinking through theflow.It is difficult to formulate W, in closed mathematical terms, because to do so requires Lagrangian integration, tracing an
ensemble of cells through possibly complex spiraling trajectories. For this reason,
the present analysis centers on £ as a measure of suspension. However, a rough
estimate of W, can be made quite simply. Consider cells inserted randomly within
a domain offluiddemarking a closed eddy. Let the area of this domain be denoted
A. Within a subdomain whose area is EA, cells will be suspended indefinitely.
Therefore, the effective sinking rate is zero in that subdomain. However, within
the rest of the domain, of area (1 - E)A, cells will be flushed on a time scale proportional to the ratio of subdomain height to still-water sinking rate. Taking the
two subdomains together yields an estimate of the area-averaged sinking rate:
W. ~ (1 - E)Wp
(1)
Much of the remainder of this paper aims to show how E and W, depend on
Wp and Wu, as well as on the geometry and time variation of theflowfield.First,
I must illustrate how circulation patterns, and particularly the updraft speed Wu,
depend on the solar forcing and ambient conditions.
Scales of convective motion
Buoyancy flux
The rate of solar heating as a function of depth depends on the non-reflected solar
insolation and the optical properties of the medium through which the radiation
passes. Here, the focus is on water motions driven by solar radiation passing
through ice, so the starting point is the downward solar radiation at the bottom
of the ice. The Appendix describes how this was calculated, using a so-called 'twostream' radiation model. As to the distribution of this flux through the water
column, matters are more uncertain because the species of phytoplankton below
the ice and their concentration will affect the optical properties of the water
column.
I will start with the premise that the algal aggregation near the ice-water interface is highly absorbent of solar radiation, compared to the water column below.
The assumption is, therefore, that the solar heating occurs in a thin region at the
top of the mixed layer. This contrasts with the assumption of Matthews and
Heaney (1987) that heating is distributed broadly through the water. The present
limit may be more appropriate in the case of Lake Baikal, given recent observations that the concentration of cells in a very thin layer near the ice-water
1864
Convection in ice-covered lakes and algal suspension
interface, possibly in the form of an algal mat attached to the ice, exceeds that in
the water column by 1-2 orders of magnitude (S.I.Heaney, unpublished data).
This pattern of localized heating in a thin surface layer corresponds, in the limit,
to the well-studied case of mixed-layer convection driven by a boundary flux of
destabilizing buoyancy. Therefore, it is possible to base the present analysis on
scaling laws that have been developed and tested in other contexts [see, for
example, Bo Pedersen (1980) or Turner (1973)].
Because the increase in water density (or equivalently the decrease in buoyancy) is the root cause of convection, the forcing is best measured by the buoyancy flux, not by the more directly observed radiation, or heat flux. If the heat
flux is denoted F (units W nr 2 ), then the buoyancy flux is:
^
(2)
pc p
where / is in W kg"1. Here g = 10 m s~2 is the gravitational acceleration, p = 1000
kg nr 3 is the water density and Cp = 4200 J kg"1 "C"1 is the specific heat of water
at constant pressure. These quantities are nominally constant for the present
application. However, the thermal expansion coefficient of water, a, varies
greatly with temperature. It ranges from -7 X 10~5 °C"1 at 0°C to zero at 4°C. The
dependence is essentially linear, with
a = (1.69 X 10"5 °C" 2 )(r- 3.95°C)
(3)
holding to within 2% for 0 < T < 3.5°C. It is because a < 0 for fresh water cooler
than 4°C that solar warming can cause convection in ice-covered lakes.
Length and velocity scales
The length scale of convective motions forced at a level boundary is related to
the mixed-layer depth, H, and the aspect ratio of convection streamlines. In most
convective systems, the latter is of order one (Turner, 1973), so the length scale
is approximately H. Because erosion of the thermocline by convective plumes
proceeds much more slowly than the convection itself (Bo Pedersen, 1980; Kelley,
1987), H may be taken as a constant for the present purpose.
The scale of convection speed is related to the mixed-layer depth and the buoyance forcing. Dimensional analysis and empirical studies reveal that the root
mean square convective speed is:
Wnns-^/fl)!*
(4)
where H is the mixed-layer depth and C\ is a non-dimensional number of value
»0.6 [see Farmer (1975) and also the review by Bo Pedersen (1980)]. In the following, it will be assumed that W^ provides a scale for convective updraft speed Wa.
For a solar insolation F ~ 10 W m"2 (see Appendix), an ambient temperature
of 2°C (midway between surface and bottom temperatures), and a mixed-layer
1865
D.E.KeUcy
thickness H ~ 20 m, one may estimate J ~ 10~9 W kg-1. This implies a convective
updraft velocity Wu ~ 10"3 rar'.A sensitivity test of the dependence of Wu on
through-ice solar flux and mixed-layer depth reveals suspension speed to range
over a factor of ~3 for mid-latitude lakes such as Lake Baikal (Figure 2). This
range is comparable to that resulting from uncertainty in c\ and the variation in
a over a reasonable range of ambient temperature. The range also encompasses
a value calculated by Matthews and Heaney (1987) using a different technique
applied to Heywood Lake, Antarctica. This suggests a robustness of the scale
value, since, as noted above, the Matthews and Heaney (1987) technique assumed
continuous buoyancy forcing through the water column, instead of the localized
(point) source assumed here.
Diurnal variation
The buoyancy forcing associated with solar heating ceases at night, but convective motion will continue, unforced, until friction damps it out. During this time,
cells will remain suspended, according to whether their still-water sinking rate
continues to exceed the gradually decreasing convective updraft speed. After
convective motion ceases, cells will fall at their still-water sinking rate, provided
that other motions do not suspend them.
The e-folding time for frictional damping, T, is roughly the ratio of the turbulent kinetic energy to the rate of dissipation of this kinetic energy by friction.
[Energy may also be lost through propagation of internal waves (Peters et ai,
1995), but so little is known about either the convection or the internal wave field
in the present scenario that this effect is best ignored for now.]
The kinetic energy per unit mass (units J kg"1) is proportional to the square of
the convective speed (Tennekes and Lumley, 1972):
0
10
20
30 40 50 60 70
Mixed-layer depth, m
80
90
100
Fig. 2. Variation of scale for suspension speed, Wu, as a function of solar insolation through the ice
and mixed-layer thickness, calculated for a nominal ambient temperature of 2°C. The gray region
indicates the range of parameters likely to apply in ice-covered Lake Baikal.
1866
Conrection to Ice-covered lakes and algal suspension
KE ~ cliJH)™
(5)
In convective turbulence, the dissipation rate per unit mass, denoted e (units
W kg-1), is proportional to the destabilizing buoyancy flux (Farmer, 1975):
e = c2/
(6)
Observations indicate that c2 = 0.6 to within a factor of -1.5 (Lombardo and
Gregg, 1989). Therefore the e-folding time scale for frictional damping is:
T = 0.6(//2//)1/3
(7)
For Lake Baikal, this is ~1 h, which is long enough to average over variations in
solar intensity caused by scattered clouds, but short enough to prevent convective motion from continuing through the whole night.
Therefore, during most of the night algal cells should sink with their still-water
velocity, in the absence of other fluid motions. Some cells, given their sinking rate
and depth, will sink past the daytime zone of suspension, and will thus be lost.
This is not a factor for neutrally buoyant cells, since they do not sink, nor for cells
that sink very rapidly, since they cannot be suspended in any case. Therefore, cells
of intermediate still-water sinking rate will be most affected by night-time loss.
This is one of several effects on algal suspension that will be examined more
quantitatively in the next section.
Retention in various eddy circulation fields
Symmetrical eddies
The essence of the suspension mechanism may be illustrated with the example of
a steady circulationflowfield(eddy) with simple geometry. Such simpleflowsare
amenable to laboratory simulation (e.g. Tooby et al, 1977) and to theory. An
extension of the analysis to unsteady, geometrically complicated circulation patterns could, in principle, be accomplished by numerical simulation using the
Navier-Stokes equations of motion for fluid flow (e.g. Squires and Eaton, 1991),
but for manyflowsof interest this is beyond the capabilities of present-day computers (Sundaram and Collins, 1996).
The model two-dimensional eddyflowfieldto be used here was first studied in
an algal suspension context by Stommel (1949), as a model of Langmuir circulation. It has stream function:
v|i = »J(Osin(jtx/Lx)sin(7iz/Lz)
(8)
where x and z are coordinates in the horizontal and vertical directions, respectively, Lx and Lz are the corresponding dimensions of the eddy, and 4»0 is a scale
factor. The model domain is 0 < x < Lx and -Lz < z < 0; outside this domain, the
circulation is mirrored and repeated. For concreteness, it will be assumed that
i|»o < 0, which yields an upwelling plume at the right boundary and fluid motion
1867
D.E.Keiey
towards x = 0 at the top boundary. Velocity components, given by spatial
derivatives of iji, are:
u=
) Wusin(jijc/Lx)cos(jtz/Lj)
w = Wucos(7tj:/L_c)sin(7tz/Lz)
(9)
where Wu is the maximal updraft speed. (For this flow field, Wu = <\i^Lz.) This
stream function is symmetrical about midpoints x = L/l and z - -LJl, with a
stagnation point (location of zero velocity) at each corner and at the center
(Figure 3).
A cell with density equal to that of the water will trace along the velocity field
with a trajectory that is nominally square along the edges, becoming circular
towards the stagnation point at the center.
A simple modification of the velocity will allow description of the trajectory of
cells that are heavier than water. For cells that sink at speed Wp in quiescent
water, the result is:
u = -(
w - Wucos(7tJ:/LJ.)sin(7tz/Lz) - Wp
(10)
0.0
,
/
/
•
, *
^
\ t
N
^
\ t
-0.5 -
!jjj!i;;::::.M.t
!! 111 \»
;; j ; ;
j V \ V \ v v v -
'
\ \ v \ \ \ >» ^»
"
, ,
,
SS
-1.0
0.5
xlL,
Fig. 3. Velocity field for a symmetrical eddy with a velocityfieldgiven by equation (9). Arrow length
indicates displacement of a neutrally buoyant cell in 1/30 of a time unit defined by the ratio of eddy
height, Lz, to peak updraft speed, Wa.
1868
Convection in ice-covered lakes and algal suspension
Here, the cell sinking rate and the water velocity are assumed to combine linearly,
which is true for cells experiencing small inertial forces, i.e. with low Reynolds
number (Tooby et al, 1977), such as relatively slowly sinking algal cells of interest here.
In this velocityfield,the interior stagnation point is shifted towards the updraft
plume (Figure 4). In addition, two stagnation points appear along the boundary
of the updraft. In the case where Wp = WJ2, for example, these stagnation points
are z ° -0.2Lz and z « -0.8Lz. The stagnation points mark the corners of a retention zone for cells with the indicated still-water sinking rate. Tracing the trajectory using dx/dt = u and dz/dt = w, starting from the neighborhood of the top
stagnation point, yields a boundary between the non-retention zone and the
retention zone. Cells inserted into the non-retention zone will be swept from the
domain in an interval that is roughly proportional to L^Wp. On the other hand,
cells within the retention zone will be held aloft, tracing closed orbits, forever.
As the still-water sinking rate Wp approaches the maximal updraft speed Wu,
the retention zones progressively shrink (Figure 5). They also become more localized to the interior of the domain. This localization towards the middle of the
mixed layer could have biological consequences. Consider two algal cells moving
-0.5
-1.0
x/U
Fig. 4. As Figure 3, but for a cell with still-water sinking speed equal to half the maximal updraft
speed (W_ = WJ2). Arrows indicate cell velocities in theflow.The thin line traces theflowfieldfrom
the neighborhood of a stagnation point at z « -0.2Lz to another at z~ -O.SLZ. To the right of this boundary, cells are retained, tracing deformed circles counterclockwise. In the rest of the domain, cells are
flushed out of the domain by the combination of the fluid flow and their own sinking.
1869
D.E.KeDey
-0.5 -
-1.0
x/L,
Fig. 5. Boundaries of retention zones for cells with various still-water sinking rates. The flow is
defined by equation (10) and the ratio of fall speed to maximal updraft speed, Wj/Wu, is indicated
along curves. (Compare the Wp/Wu = 0-5 case with Figure 4.)
along two distinct trajectories with equal time-averaged depths, but different
ranges in depth. The time-averaged light level experienced by the cell moving
nearer the surface will not equal that of the cell with smaller trajectory, for a nonlinear light profile. This, together with non-linear photosynthetic response to a
given light intensity (Lande and Lewis, 1989; Cullen, 1990), will result in different growth rates for the two cells, even though their trajectories may have identical average depth. Thus, the shape of the retention zone may have biological
consequences in addition to the physical consequence of suspension.
In the hypothetical case of a source of algal cells with still-water sinking rates
uniformly distributed across values from 0 up to Wp, the suspension efficiency
function E = E(Wp/Wu,...) would determine the histogram of cell concentration.
(This presumes that no sink terms depend on Wp/Wu.) By definition, E is the relative area of retention. This index decreases from unity at Wp/Wu = 0 to zero at
Wp/Wu = 1 (Figure 6). The end points are in agreement with the intuition outlined
in the section on the requirements for algal suspension. However, intuition is of
little help in predicting how E might vary for intermediate values of Wp/Wu. The
present calculation indicates that the dependence is quite non-linear for the
velocity distribution under consideration. Specifically, efficiency falls off faster
than it would in a linear law. For example, 50% efficiency occurs when
Wp/Wu = 0.3, not at a ratio of 0.5 as it would for a linear law (Figure 6).
1870
Convection in ice-covered lakes and algal suspension
Relative fall speed, Wp/Wu
Fig. 6. Relative retention area as a function of algal still-water sinking rate relative to plume speed,
i.e. suspension efficiency function E = E(Wp/Wu), for flow field of equation (10).
So far, the discussion has employed analytical mathematics, since theflowfields
are simple enough to be integrated. For more complicated flow fields, especially
unsteady ones, this is not necessarily possible. Then numerical techniques are
useful. This is true of the biology as well as the physics. Increasingly, biological-physical models of plankton employ the strategy of following individual
algal cells as they move through space, encountering different light and nutrient
levels. This may be done on a concentration basis (e.g. Richardson et al., 1997)
or on a cell-by-cell basis. Although the second method is much more expensive
computationally, it allows greater generality. This generality is useful for modeling factors such as the non-linear effects of nutrient and light history on algal cells.
(Non-linear effects are necessarily smeared in the averaging that is inherent in
concentration models.) In a larger ecological context, tracking individual cells
also allows the study of effects such as predator-prey interactions. As such
models grow in complexity, it may become feasible to shed new light on longstanding questions regarding phytoplankton in moving fluids (Lund, 1959;
Hawes, 1983; Spigel and Imberger, 1987).
In the present context, a Monte Carlo numerical simulation was carried out to
measure the rate of winnowing of algal cells in various flow fields. (Numerical
techniques are not required for this simpleflowfield;the computer program was
developed for a broader application with non-linear biological parameters.) To
begin, 106 imaginary cells were randomly inserted in the flow field described by
equation (10). These cells were distributed throughout the eddy (i.e. 0 < x < Lx
and -Lz < z < 0). Each cell was assigned a still-water sinking rate selected randomly between 0 and the updraft speed (i.e. 0 < Wp < Wu). A Euler integration
scheme was used to trace the trajectories.
The numerical results verify the theoretical predictions in detail. In particular,
the predicted efficiency function E - E(Wp/Wu) was recovered well (compare
Figures 7 and 6). The numerical calculation reveals extra information that the
1871
D.E.KeUey
0
0.5
Relative fall speed, WpIWu
FTg. 7. Temporal development of a sinking rate histogram in Monte Carlo numerical simulation using
a symmetrical eddy flow field denned by equation (10), for 106 imaginary algal cells initially distributed randomly through the domain, and with the sinking rate, Wp, randomly selected in the range
O-Wu. The darkening histograms indicate distributions of cells still in the domain -hz < z < 0, sampled
at the initial time (light gray histogram near the top axis, indicating statistically uniform concentration
as a function of Wp/W,,), and thereafter at time intervals of LJ(2Wa). For Lake Baikal, this time interval is roughly 3 h. Compare the converged distribution (black histogram) to Figure 6.
steady-state theory cannot, such as the rate of adjustment to the predicted
histogram shape (Figure 7).
Roughly speaking, the suspension efficiency depends on the proportion of the
flow field with vertical velocity exceeding the still-water cell sinking rate. This
leads to the question of retention in asymmetrical eddy fields. A particular case
of interest is the sort of asymmetry to be found in fully developed top-forced convection, in which updrafts are broader, and more homogeneous in velocity, than
downdrafts.
Asymmetrical eddies
The velocity field given by equation (9) corresponds to the solution of the classical Rayleigh instability problem of the onset of convective motion between two
stress-free parallel plates, the lower held at a relatively warm temperature and
the upper held at a relatively cool temperature [see Rayleigh (1916) or modern
reviews such as those given by Chandrasekhar (1961) or Turner (1973)]. As such,
it provides a reasonable description of a convective eddy with relatively weak
forcing, i.e. with Rayleigh number barely exceeding the critical value.
Several differences arise in cases of more intense forcing. One is that eddies
become unsteady. Eventually, the fluid may become turbulent if the forcing is
intense enough. The detailed prediction of cell motion is not possible in this limit,
although cells may be traced over the time scale of eddy rotation, suggesting that
the present approach may continue to produce meaningful results.
1872
Convection in ice-covered lakes and algal suspension
-0.5 -
x/Lr
Fig. 8. As Figure 3, except for the stream function [equation (11)] with asymmetry parameter n = 2.
The scale factor «fo is adjusted to give maximal updraft speed matching that in Figure 3.
Another result of increased forcing is that eddies become more asymmetrical,
with thin boundary layers along the walls and thin plumes sweeping fluid from
these boundary layers into the interior. The effect of asymmetry may be simulated crudely by modifying the convectiveflowfield.A simple approach is to write
the stream function as:
-x/L x ) n (l + z/Lz)nsin(nx/Lx)sin(TizlH)
(11)
where n is a factor controlling flow asymmetry. With n = 0, the previous velocity
field [equation (9)] is obtained. Higher values of n yield progressively thinner
boundary layers. They also yield thinner downward plumes, relative to upward
plumes (Figure 8), so this equation may illustrate the effects of convective asymmetry of interest here.
Monte Carlo numerical experiments were carried out with n = 1 and n = 2, and
compared with the n = 0 case described in the previous subsection. In each case,
ijjo was adjusted to give identical maximal updraft speeds.
As asymmetry increases, the suspension efficiency function progressively
departs from a linear dependence on Wp/Wu. The departure is greatest for stillwater sinking rates of about half the updraft speed. With Wp/Wu ~ 0.5, the suspension efficiency is nearly halved as the asymmetry factor is increased from n = 0
to n = 2.
1873
D.E.KeDcy
Relative fall speed, WpIWu
Fig. 9. Steady-state distribution of captured cells in numerical calculations with aflowfield[equation
(11)] with n = 0 (top curve), n = 1 (middle curve) and n = 2 (lower curve). The n = 0 results are identical to those in Figure 7 and the other curves represent the effect of increasing asymmetry in the flow
field.
In the limit, Wp/Wu —* 0; however, the results converge to E = 1, regardless of
the degree of asymmetry of theflowfield.This means that algal cells will not be
greatly affected byflowassymmetries if convection speeds greatly exceed sinking
rates. Such is the case in Lake Baikal, where the ratio Wp/Wu is of order 10~2.
Simply stated, the flow is more than able to suspend the cells, regardless of the
precise geometrical pattern of the circulation, so that the details of flow asymmetry become irrelevant.
Effect of night-time loss
During the non-convective hours of the night, assuming as above that no other fluid
motion is present, algal cells will fall at their still-water sinking rate. This was simulated in the Monte Carlo model by cutting convection instantaneously for a designated period. Day and night lengths were assumed to be equal for Lake Baikal in
March, given the 11 h day (Kozhov, 1963, Table 31) and an hour of continued convection after dusk (see the subsection on diurnal variation in the section on scales
of convective motion). The symmetrical flow field [equation (10)] was used.
As expected, the simulation indicates that night-time cessation of convection
reduces suspension efficiency (Figure 10). There is no effect on neutrally buoyant
cells, so that E is unaltered in the limit as Wp/Wu —> 0. Night-time sinking also has
no effect for swiftly falling cells, because they cannot be suspended in any case;
thus, E is unaltered as Wp -» Wa. For intermediate values of Wp/Wu, however, E
is reduced by 10-20% due to sinking over the course of a single night. The reduction is greatest for Wp = 0.2Wu to 0.6Wu, i.e. for cases in which Wp and Wu are of
comparable value.
The modification of E is similar to that introduced byflowasymmetry (see the
previous subsection). In each case, the modification reduces the efficiency
1874
Convection in ice-covered lakes and algal suspension
0
0.5
Relative fall speed, Wp/Wu
1
Fig. 10. Temporal development of the sinking rate histogram in a Monte Carlo simulation in which
convection was stopped at night. Only the night-time evolution is indicated: the top curve represents
the histogram at dusk (as in Figure 7), with darker regions indicating the state at time intervals
LJ(2Wa), or roughly 3 h.
function, most strongly at intermediate values of Wp/Wu. This decrease in suspension efficiency yields a corresponding decrease in the area-averaged effective
sinking rate W,.
For the present application to the suspension of A. baicalensis in Lake Baikal,
Wp/Wu is of the order 10~2. Given such a low ratio, the suspension efficiency E
will exceed -95% regardless of diurnal variation (orflowasymmetry, as discussed
in the previous subsection). Correspondingly, the area-average effective sinking
rate W*, proportional to (1 - E), will be -20 times smaller than the still-water
sinking rate. Therefore, A.baicalensis cells will be suspended within the surface
mixed layer for a time scale of a month, instead of the few days that would apply
in the absence of convection. This may have important biological consequences
which might be addressed profitably in other studies.
Summary and conclusions
The efficiency of the suspension of heavy cells in a turbulent fluid is primarily a
function of the ratio of still-water sinking rate, Wp, to the typical speed of updrafts
in the fluid, Wu. For solar-induced convection in lakes, the updraft speed is determined by the mixed-layer thickness and the solar insolation passing through the
ice. Diurnal variation in solar insolation reduces the suspension efficiency
because cells can sink freely after convection ceases. The reduction is most pronounced for Wp/JVu in the range 0.2-0.6, and vanishes for cells with sinking rates
much smaller than updraft speeds. A qualitatively similar reduction in suspension
efficiency results for the asymmetricalflowfieldsof vigorous convection, as compared to weakly forced flows.
1S75
D.E.KeUey
In the present application, a study of suspension of the diatom A.baicalensis in
Lake Baikal, Wp/Wu is of the order 10~2. For such a low ratio, the effects of diurnal
cycling and flow asymmetry are slight, because convective velocities are more
than sufficient to suspend the cells. In fact, the suspension efficiency is calculated
to be 95% or higher, which implies that the area-averaged effective sinking rate
will be only ~l/20th of the still-water sinking rate. As a result, the convection
increases the mixed-layer residence time of these cells from a few days to over a
month. Given an algal doubling time of order days, then, convection is probably
important to the growth of under-ice algae, provided that the ice is clear enough
to transmit light efficiently.
This suggests that interannual variations in light transmission through the ice
might explain observed variations in the strength of algal blooms in Lake Baikal.
Since even a thin layer of snow cover will effectively inhibit under-ice convection,
the most relevant physical parameter may be the opacity of the snow cover. This
motivates the hypothesis that springtime blooms of A.baicalensis in Lake Baikal
will be most intense following winters with minimal snowfall.
Acknowledgements
This work was motivated by a question posed by Tammi Richardson, and it gives
me great pleasure to thank her not just for that, but also for encouragement and
advice throughout the course of the analysis. This paper would not have been
written otherwise. In addition, I thank both Chris Taggart and Brian May for
reading an earlier version of the manuscript and providing many helpful comments. Much of the material discussed here is outside my usual scope, and I have
Jim Christian, John Cullen, Richard Davis and Mary-Lynn Dickson to thank for
providing background on under-ice algae in marine systems and the units used
by biologists to measure light. Finally, I thank the two anonymous reviewers for
their advice on improving the manuscript, and especially for their leads into a
Lake Baikal literature with which I am only slowly growing familiar. Financial
support for this work was provided by a research grant from the Natural Sciences
and Engineering Council of Canada.
Appendix: Calculation of light penetration through ice
The passage of light through a substance (in this case, ice, snow or water) is
affected by both absorption and scattering. These processes are measured by
coefficients k and r, respectively (units nr 1 ); each coefficient depends on the
wavelength of radiation. (In some applications, the scattering coefficient is split
into components for forward scattering and backward scattering. Here it is not
split, because such data are unavailable for ice.)
The combined results of absorption and scattering are commonly represented
in the oceanographic literature by a single measure of light attenuation named
the 'extinction' coefficient [see Strickland (1958) for a comprehensive introduction]. The use of a single attenuation parameter is partly a matter of convenience,
given historical measurement techniques such as the use of Secchi disks. Separate
1876
Convection in ice-covered lakes and algal suspension
consideration of absorption and scattering is common in the meteorological
literature, but less so in the oceanographic literature. For this reason, and to
explain values of light transmission used in the text, a brief review of the method
is presented here.
A two-stream calculation of irradiance will be used here. This means that two
conservation equations must be solved simultaneously: one for downwelling
irradiance, Ed, the other for upwelling irradiance, £ u . Both Ed and Eu are integrated across all angles of incidence and are measured in units W nr 2 . (This
energy-based unit is useful for the present calculations of flux of heat and buoyancy; biological applications often employ the unit of moles of photons per unit
area and unit time.) Downwelling energy is lost by absorption and by scattering,
and gained by scattering of upwelling energy. The three effects are described by
the three terms on the right-hand side of the conservation equation:
d
dz
=-kEd-rEd
+ rEu
(Al)
where z is distance through the medium. The origin z = 0 will be taken at the
surface (air-ice interface) and z increasing downwards. (This coordinate system
matches optical convention, but differs from that used in the text for convective
motions.) The upwelling 'stream' is described by:
^
k Eu + rEu - rEd
(A2)
oZ
The sign change in these equations results from formulating Ed positive when flux
is downward and Eu positive when flux is upward. The coefficients k and r are
functions of the wavelength (X) of the radiation, and are different for different
substances. Values will be taken from the Perovich (1990) summary of published
values for various snow, ice and water types.
Equations (Al) and (A2) are complete only when appropriate boundary conditions are stated. Many applications consider layered systems, and in this case
boundary conditions are required at the top and bottom of the domain, in
addition to continuity conditions at each interior interface. Given the present
application to solar insolation, the surface boundary condition relates Ed to the
incoming flux from the sun and the sky, and the bottom boundary condition is
that the upwelling flux vanishes. The latter condition is only met if the domain
extends far enough that the light is effectively extinguished.
In a homogeneous domain, i.e. a layer of constant (k, r) properties, the surface
boundary condition is:
Ed = (1 - R)S + REU, at z = 0
(A3)
where 5 is the incoming solar irradiation and R is the specular reflection
coefficient. The first term on the right-hand side of this equation accounts for loss
1877
D.E.Kelley
of incoming radiation by reflection and the second accounts for the translation of
upwelling flux into downwelling by reflection at the surface. For an interface
between ice and air, the reflection coefficient is R = 0.05 (Perovich, 1990). The
bottom boundary condition is:
Eu = 0, at z = D
(A4)
where D is the thickness of the layer.
Solution of (A1)-(A4) is straightforward. Differentiating (A2) with respect to
z, applying (Al) and denning
K = (k2 + 2kr)m
(A5)
Ed = Aexp(-Kz) + Bexp(Kz)
(A6)
Eu = Aaexp(-xz) + Bfiexp(i<z)
(A7)
yields
where
a =
-K+k+r
(A8)
(A9)
and A and B are constants of integration, determined by the boundary conditions
to be:
-(p/q)exp(2KP)(l-fl)S
O/a)(l tfa)exp(2/cD) + (1 R$)
K
'
For illustration, these equations were solved for the case of Lake Baikal. Ice
optical properties were taken as those for 'cold blue ice' (r = 1.8 nr 1 and k = 1.5
m"1) and 'melting blue ice' (r = 1.2 m"1 and k = 1.2 nr 1 ) reported by Perovich
(1990). In each case, k represents an average over the relevant wavelengths of
solar radiation. Given these values, and assuming D to lie in the range of
0.8-1.1 m appropriate through January-March (Kozhov, 1963, Table 31), one
finds that between 4 and 11% of the solarfluxwill penetrate through to the water.
This number is reduced by a factor of 50 by the addition of 0.05 m of snow cover
{k = 12 m-1, r = 160 nr 1 ; Perovich, 1990), as occurs by early April (Kozhov, 1963,
1878
Convection in ice-covered lakes and algal suspension
Table 31). This leads to the conclusion, in the text, that March is the time of
maximal radiation flux to the water beneath the ice. During this month, the noontime incident flux is S = 150 W nr 2 (Kozhov, 1963, Table 31). (Kozhov's kilolux
unit was converted to the SI unit W irr2 by multiplying by 4.1; Strickland, 1958).
Thus, it is estimated that the water receives an irradiance of 5-15 W nr 2 at noon
during March. This irradiance is quenched almost entirely by the appearance of
only a thin snow layer in April. These calculations support the heatflux,10 W nr 2 ,
used in the text for the calculation of buoyancy forcing for convection.
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Received on April 24, 1997; accepted on August I, 1997
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