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Limnol. Oceanogr., 38(4), 1993, 798-817
0 1993, by the American Society of Limnology and Oceanography, Inc.
Vertical mixing in a shallow, eutrophic lake: Possible
consequences for the light climate of phytoplankton
Sally Maclntyre
Marine Science Institute, UCSB, Santa Barbara, California 93 106
.
Abstract
Profiles of temperature-gradient microstructure are used to define the size and location of mixing regions,
the intensity of turbulence, and the potential exposure of phytoplankton to fluctuating irradiance in a
shallow, turbid, productive lake. The part of the water column which was mixing tended to be subdivided
into two regions with different dynamics, one in which the turbulence was active and one in which it was
constrained by buoyancy. Generally the upper layer, which ranged from 0.3 to 1.5 m deep, was actively
mixing. Energy dissipation rates were on the order of lo- 7 m2 s-~, vertical eddy diffusivities ranged from
1O-3 to 1Op5 m2 s- I, and overturns mixed on a time scale of minutes. Phytoplankton could become well
mixed before turbulent transport within overturns ceased and, while the wind persisted, were likely to
experience continuous fluctuations in irradiance. In one of the largest overturns, phytoplankton could
circulate between the 90% light level and the 5% light level in 3-4 min. Where buoyancy affected turbulence,
energy dissipation rates ranged from 1O-9 to 1O-7 m2 ss3 and vertical eddy diffusivities from 1O-7 to
1O-4 m2 s- ’ . Mixing times based on these diffusivities exceeded r/N, the time scale for turbulent transport,
indicating overturns would mix only partially. Phytoplankton could still experience large fluctuations in
irradiance, but the fluctuations probably were not continuous.
Evidence linking vertical mixing in lakes and water oxygen change and integration of the
oceans to phytoplankton photosynthesis (Mar- results from static bottles (Melack 1982; Me-
ra 1978a,b; Mallin and Paerl 1992) is limited, lack and Kilham 1974) provide further sup-
but the rapid photoadaptation of a number of portive evidence for the importance of vertical
physiological processes (Ferris and Christian mixing.
199 1; Prezelin et al. 199 1; Vincent 1990) sug- Several approaches have been taken when
gests there is a coupling between the fluctua- modeling photosynthesis to include the fluc-
tions in irradiance caused by vertical move- tuations in irradiance caused by vertical water
ment of phytoplankton and photosynthesis. motions. Woods and Onken (1982) predicted
Differences in estimates of primary production trajectories of individual cells assuming that
for cells circulating in an idealized way in a vertical motions were caused by Langmuir cells
mixed layer are obtained between models in- and small-scale turbulent eddies and that the
corporating photoadaptive responses and those vertical water motions only occurred in a mixed
that do not (Patterson 1991). Discrepancies layer whose depth varied diurnally. Lande and
between photosynthetic estimates based on free Lewis (1989) contrasted photosynthetic rates
based on models in which irradiances received
by phytoplankton were based either on track-
Acknowledgments ing individual cells or average photoadaptive
This research was conducted when I was at the Center properties of cells at a given depth. These ap-
for Water Research ofthe University of Western Australia.
J&-g Imberger made available the use of his facilities, in-
proaches assumed constant mixing rates in the
cluding the microstructure profiler and software for data mixed layer as did Denman and Gargett (1983),
analysis. I thank him for assistance in all aspects of this who estimated time and length scales for phy-
project. I also thank Carolyn Oldham, David Pullin, Cathy toplankton to circulate within the mixed layer.
Thomson, and Dottie McLaren for help with computa- Analyses by Imberger ( 1985) and by Shay
tions and graphics; Terry Smith and Roger Fletcher for
help with fieldwork; Tom Shay, John Melack, and John and Gregg (1986) indicated that the mixed lay-
Patterson for useful discussions; and Jenny Davis and John er does not mix uniformly but is subdivided
Melack for unpublished data on North Lake. into different regions, sometimes demarcated
This work was supported by the Center for Environ- by temperature differences as small as O.O2”C,
mental Fluid Dynamics of the University of Western Aus-
tralia, the Australian Research Council, NSF grant ST1
in which the intensity of the turbulence differs,
89-96 152 to R. K. Zimmer-Faust and S. MacIntyre, and with the uppermost layer termed the surface
ONR grant NOOO14-89-J-3206 to A. Alldredge. mixing layer. Because of these subdivisions,
798
Mixing in a shallow lake 799
vertical movement of phytoplankton in the field is the Kolmogorov microscale, Lk = (y3/
upper mixed layer should not be modeled as- p, where v is kinematic viscosity and E the
suming uniform mixing rates. Biological evi- energy dissipation rate. Itsweire and Helland
dence supporting this view includes the dif- (1989) found that eddies 3-100 times larger
ferences in the photoadaptive properties of than the Komogorov microscale were respon-
phytoplankton near the surface and base of the sible for most of the turbulent transport of
mixed layer (Falkowski 1983). In conse- solutes in a fluid. The Ozmidov (1965) length
quence, Yamazaki and Kamykowski (199 1) scale, L, = (c/W)“, where Nis the Brunt-V&a-
modeled trajectories of phytoplankton based la frequency, is often considered the maximum
on vertical eddy diffusivities that varied with size of an overturn in a stratified fluid. In fact,
windspeed and depth in the water column. it was derived theoretically as the largest eddy
However, these eddy diffusivities are not based possible before buoyancy begins to affect tur-
on observations but on empirical relations be- bulence, i.e. overturning scales >L, will be
tween windspeeds and energy dissipation rates affected by the buoyancy forces in the fluid and
in the water column (Oakey and Elliot 1982; overturning scales CL, will not. In profiles col-
but see Gargett 1989). In addition, Yamazaki lected from oceans and lakes, Dillon (1982)
and Kamykowski assumed that the largest ed- found that L, ranged from 0.4 to 20 L, and
dies were equivalent to the Ozmidov length Imberger and Ivey (199 1) found an even wider
scale, whereas in actively turbulent waters the range indicating that trajectories of phyto-
Ozmidov length scale is frequently 5-30 times plankton should not be based upon this the-
larger than overturning scales (Imberger and oretically based scale.
Ivey 199 1). In situ measurements indicating In addition, data from the profilers allows
the portions of the water column that are mix- estimation of time scales of vertical circula-
ing and the length and time scales of mixing tion. Denman and Gargett (1983) hypothe-
within these regions are essential for establish- sized that the time scale for the large eddies to
ing estimates of trajectories of phytoplankton. circulate phytoplankton is Z/u, where I is the
Temperature-gradient microstructure pro- scale of the eddy and u the vertical velocity
filers provide the in situ data required to obtain scale. u can be obtained within a turbulent
the turbulent length and velocity scales and patch from the energy dissipation rate, E = u3/
mixing rates that are needed for this problem. 1. Strictly speaking, this latter expression ap-
The vertical extent of overturning regions in plies when buoyancy does not greatly affect
the water column (Thorpe 1977; Dillon 1982) turbulence (Taylor 1935), which is the case in
can be obtained from analysis of density pro- a turbulent patch. A time scale for turbulent
files. Energy dissipation rates (Dillon and diffusion, based on the variance in the rate of
Caldwell 1980) and vertical eddy diffusivities spreading of diffusing solutes, is P/2&; it es-
(Ivey and Imberger 199 l), from which mixing timates an average time for a patch of phy-
rates are computed, can be obtained from spec- toplankton or solute centered at depth z to
tral analysis of the temperature-gradient pro- spread distance I (Bowden 1983). K, is the co-
file. efficient of vertical eddy diffusivity (Osborne
Analysis of microstructure data allows es- and Cox 1972; Osborne 1980; Ivey and Im-
timation of several length scales important for berger 1991). Contrasting the two time scales
describing the vertical circulation of phyto- indicates when movement of phytoplankton
plankton. The first is the actual size of over- can be assessed based on scaling arguments
turning regions, the maximum displacement and when a model invoking turbulent diffusion
length (Q, where displacements (d’) are the is required.
vertical distances parcels of water would need The length and time scales of turbulence, by
to be moved in order for the measured density themselves, are insufficient to predict vertical
profile to become monotonic (Thorpe 1977). movement of phytoplankton. Two other prob-
AS not all phytoplankton will circulate over lems must be considered. One, phytoplankton
the full extent of an eddy, another length scale do not always follow the flow, either because
can be defined, the root-mean-square (rms) their speeds of rising or sinking exceed or are
displacement scale (L,) or Thorpe scale. The comparable to the turbulent velocities or be-
smallest length scale of a turbulent velocity cause shear is induced when their size is greater
800 MacIntyre
than that of the eddies in which they are em- olds number diagram, and compare the time
bedded. The second problem, and the one that scales for circulation within the large eddies,
is considered in more detail here, is that the the time scale for turbulent diffusion, and the
water column where phytoplankton live tends time scale for buoyancy flux. I also determine
to be either stably or unstably stratified and when the motility of the dominant phyto-
stable stratification tends to damp out turbu- plankton or the turbulent motions contribute
lent mixing. more to the cells’ movement.
The Froude-Reynolds number diagram (Im-
berger and Ivey 199 1) indicates when buoy- Study site
ancy forces affect turbulence. Plotting turbu- North Lake (Davis and Rolls 1987; Bayley
lent Froude numbers, Fr, = u/NZ, and turbulent et al. 1989) is a turbid, productive shallow lake
Reynolds numbers, Re, = Z&J, on the diagram 14 km south of Perth, Western Australia, and
indicates whether the motions in the fluid are 7 km from the Indian Ocean in the Cockburn
actively turbulent, are purely viscous motions, chain of wetlands. The lake is oval and had a
are internal waves alone, or are a combination surface area of 29 ha and a maximum depth
of turbulence and wavelike motions or of tur- of 2.6 m in December 1987. Malaleuca wood-
bulence and motions driven by free convec- lands fringe the shores. During spring, blooms
tion. The vertical movements of phytoplank- of Microcystis aeroginosa frequently develop
ton will differ in all these cases. with chlorophyll a concentrations exceeding
After a stratified flow becomes turbulent, 100 pg liter-’ (J. A. Davis pers. comm.). In
numerical models (Riley et al. 198 1) and lab- December 1987, the attenuation coefficient of
oratory investigations (Itsweire et al. 1986; photosynthetically available irradiance was 2.1
Barrett and Van Atta 199 1) show a buoyancy m- *, Chl a concentrations ranged from 10 to
flux that is initially positive; heavier fluid is 140 pg liter-l with an average of - 30 pg li-
mixed upward. However, the buoyancy flux is ter - l, and M. aeroginosa and a species of An-
positive for only a short period of time, r/N abaena were the dominant phytoplankters in
in the experiments of Itsweire et al., n/2N in the lake (J. M. Melack pers. comm.).
the model of Riley et al., and over a similar
range for Barrett and Van Atta’s three exper- Methods
iments. Subsequently the direction of the Temperature, temperature gradients, and
buoyancy flux changes, indicating restratifi- pressure were measured at a frequency of 100
cation and wavelike motions, especially at the Hz with a portable rising microstructure pro-
larger scales. This pattern, mixing-restratifi- filer modified from Caldwell and Dillon (198 1)
cation-mixing-restratification, persists, but the and Carter and Imberger (1986). The profiler
positive buoyancy flux in the first period is is -0.3 m long and uses a Kevlar link to trans-
substantially greater than either the positive or mit the data to shipboard electronics. When
negative buoyancy flux in the subsequent pe- deployed, it sinks until a weight is jettisoned
riods. The large initial flux suggests that the at a desired depth and then rises to the surface
initial period n/N or n/2N can be considered at a speed of -0.1 m s-l. The profiler has a
as a time scale for turbulent transport, 7BF. If pair of thermistors 0.027 m apart and a depth
either Z/u or 12/2K, are larger than TBF, the fluid transducer 0.120 m below them. All data were
may not become well mixed unless overturns digitally enhanced to give a frequency response
continue to form. One consequence of strati- of 40 Hz and to smooth the signals (Fozdar et
fication on the flow may be incomplete mixing al. 1985).
in overturns and subsequent reduction of the A meteorological station and thermistor
frequency and magnitude of fluctuations of ir- chain were located in the north-central portion
radiance that phytoplankton in an overturn of the lake. Thermal stratification was moni-
could experience. tored with seven thermistors each -39 cm
In this paper, I analyze a time series of pro- apart; they were calibrated against a Hewlett-
files of temperature-gradient microstructure, Packard quartz crystal thermometer and in-
determine the length and velocity scales of tercomparable to O.Ol”C. Two propeller ane-
overturning eddies, construct a Froude-Reyn- mometers located 2.2 m above the surface of
Mixing in a. shallow lake , 801
the water measured the two horizontal com- both the theoretical one-dimensional spec-
ponents of windspeed; the sensor array also trum for homogeneous isotropic turbulence
included shielded thermistors to measure air (the Batchelor spectrum) to obtain an estimate
temperature and a pyranometer to measure of 6(Gibson and Schwarz 1963) and to an auto-
diffuse short-wave radiation. Meteorological regressive model (Imberger and Ivey 199 1).
and thermal stratification measurements were Values of vertical eddy diffusivity K, were
performed 14-l 7 December 1987, and micro- calculated with Osborne’s (1980) model, K, =
structure profiles were obtained during the last RY/( 1 - Rf) cIN2, with values of the flux Rich-
-’ 2 cl of this period at locations - 100-200 m ardson number Rf calculated from values of
from the meteorological station. Fr, and Re, (Ivey and Imberger 199 1). Al-
though the value of Fr, at which buoyancy first
Calculations begins to affect the turbulence (FrtCRIT) is un-
Displacement lengths and displacement known, Ivey and Imberger (199 1) suggest
scales (Thorpe 1977; Dillon 1982) were ob- FrtCRIT equal to 1.8 for turbulence in air and
tained by reordering the density profile so that to 1.2 for turbulence in water. FrtCRIT equal to
the profile increased monotonically with depth. 1.2 may be too low, for values of R+alculated
The depth to which any water particle is dis- assuming FrtCRIT = 1.2 remain close to the
placed in the reordering is a Thorpe displace-. maximum expected well below Fr, = 1, a result
ment (d’) and the root-mean-square sum (rms) that is unlikely given the rapid decline in buoy-
of the displacements L, = (d’2) %, where angle ancy flux once buoyancy forces begin to effect
brackets indicate averaging, is the Thorpe scale turbulence (Itsweire and Helland 1989). Val-
or displacement scale. The displacements were ues of RYobtained by examination of Linden’s
filtered and centered at the center of the over- (1979) experiments are intermediate to those
turn (Imberger and Boashash 1986). L, is the obtained from these two sets of calculations.
largest displacement. Differences in temperature between the
The rate of energy dissipation (E) was com- thermistor chain and the two thermistors of
puted from the maximum frequency (fjJ of the the microstructure profiler were observed dur-
Wigner-Ville distribution of the temperature- ing data analysis. The temperature profiles from
gradient profile (Imberger and Boashash 1986) the two thermistors had similar features, i.e.
where fm is proportional to the smallest scale comparable mixed-layer depths and locations
of the flow, the Batchelor scale LB = (JvD*)” of overturns, indicating the errors in the mag-
where D is thermal diffusivity (Batchelor 1959). nitude of the temperature will not affect esti-
The determination is done for record lengths mates of displacement lengths, of energy dis-
of 20 samples so that the turbulence is effec- sipation rates, or of segment lengths but may
tively stationary. For purposes of averaging, cause errors in estimates of N and in subse-
regions of the water column where the tem- quent estimates of Fr,. During the period be-
perature-gradient signal was stationary, and by fore the sea breeze, the maximum discrepancy
implication any turbulence present was statis- between the two sensors in the temperature
tically stationary, were determined by a seg- difference between the surface and base of the
mentation algorithm (Imberger and Ivey 199 1). two profiles was 1“C, with the larger differences
Within each, values of E were averaged and observed with thermistor 1. This error will
average values of the Brunt-V&ala frequency, cause errors as high as 20% in estimates of Fr,.
N = (-g/p dp/dz)” where g is gravity, p is den- After the sea breeze, discrepancies in the tem-
sity, and z is depth, were computed for the perature difference did not exceed 0. 1°C and
original and the monotonic density profiles. were as little as O.Ol”C; again differences were
Average values of Fr,, Re,, u, and L, were cal- larger with thermistor 1. Errors in N and Fr,
culated following Luketina and Imberger were on the order of 5% for the larger tem-
(1989). perature difference. Because few of the seg-
Following Dillon and Caldwell (1980), I ap- ments were found to be turbulent before the
plied a Gaussian window to the segments of sea breeze commenced and the errors in N and
the temperature-gradient profile; the power Fr, are small after the sea breeze and because
spectral densities obtained were matched to data from both thermistors are used to define
802 MacIntyre
Fig. 1. A. Air temperatures (“C), windspecd (m s-l) and direction (degrees), solar radiation (W mb2), and water
temperatures (“C) for 16 December 1987 at depths (cm) shown. Arrows indicate the times when microstructure casts
were taken. B. As panel A, but for 17 December 19‘87. ’
the attributes of the flow field, the description the sea breeze. For instance on 16 December
of turbulence that follows is expected to be (Fig. 1A) the sea breeze commenced in late
accurate. morning when only a 2°C temperature gradient
occurred in the upper meter, whereas on 17
Results December (Fig. 1B) the wind came up at 1220
Thermal stratification and meteorological hours when the temperature gradient in the
forcing-Each day of the study had cloud-free upper meter ivas 4°C. Despite persistent wind-
skies, high insolation, and the sudden onset of speeds ~7 m s-l, the water column at the
a sea breeze with moderate to high winds (Fig. meteorological station became isothermal only
1). The diel cycle consisted of a period when for brief periods in the afternoon (data are not
the water stratified due to low windspeeds and shown for 17 December). The subsequent re-
high insolation, followed by a wind-mixing pe- stratification as well as differences in mixed-
riod that began with the onset of the sea breeze, layer depth between the thermistor chain data
generally by late morning or early afternoon. and microstructure data are evidence of hor-
At night the lake mixed to a depth of at least izontal heterogeneity. This topic will be ex-
2 m due to combined forced and free convec- panded upon elsewhere.
tion. Day-to-day differences in stratification Microstructure -The microstructure pro-
arose because of differences in the timing of files form a progression, with those from 17
Fig. 2. A. Temperature profiles scaled to show the thermal structure at 1149, 1157, 1208, 1224, 13 14, and 1341
hours on 17 December. The temperature scale differs for each profile. B. Filtered, centered displacement lengths for
the same times. C. Profiles of the t.:te of energy dissipation obtained from the maximum frequency of the Wigner-
Ville transform. Segments where the signal was below the instrument threshold are marked BIT. Data are from
thermistor 1. Stippled and open bars left of the axes indicate segments where the temperature-gradient signal was
statistically stationary; numbers in panel A are ,the segment numbers referred to in the tables and text; numbers in
panel C are the values of Fr, for the segment.
Mixing in a shallow lake 803
Temperature (OC)
25 26 27 28 29 26 28 30 25 26 27 28 26 28 26.5 27 26.8 27
A 0.0 1
0.3
0.6
z 0.9
z
‘i 1.2
a
* 1.5
t I
1.8
t
2.0
1149 1157
2.4 t
Lt0-N
0.0 0.6 0.0 0.8 0.0 0.3 0.0 0.6 0.0 0.6
I I
r:i
::
::
::
:a
2.1
1149 1157 1208
2.4 t
tz (m2sB3)
0.12
::%
0.60
0.30
0.70
0.1
BIT BIT
BIT
0.6
1.8
2.1
2.4 E
1149
i
1208
L 1224
804 MacIntyre
December taken just before and within 2 h ciated with Fr, > 1; when E < 1O-7 m2 s-3,
after the initiation of the sea breeze and those values of I+, were < 1. At 13 14 hours, Fr, was
from 16 December taken 3-5 h after the sea < 1 except in the uppermost segment. At 134 1
breeze started. For this reason, these data will hours Fr, was 2 1 in the upper 1.5 m and de-
be considered as a series, with the 17 Decem- creased from a maximum of 4 in the upper-
ber data examined first. most segment to a minimum of 0.6 at 1.6-m
The microstructure casts on 17 December depth. Such decreases in Fr, with depth are
document the changes that occurred with the expected for wind-mixed layers (Imberger and
onset of the sea breeze (Fig. 2). Profiling started Ivey 199 1).
at 1149 hours, just before the onset of the sea The four temperature profiles taken on 16
breeze at 1220, and continued until 134 1 hours. December, more than 3 h after the sea breeze
The first four profiles (Fig. 2A) are highly strat- began, had temperature differences ranging
ified with at least a 3°C difference between the from 0.6 to 0. 1°C from the surface to 1.8 m
surface and 1.8 m. The displacement profiles (Fig. 3A). Temperature inversions and insta-
indicate instabilities occurred in the upper 0.9 bilities occurred throughout the water column
m except at 1157 (Fig. 2B). Instabilities are in all profiles (Fig. 3B) with the largest over-
also associated with the warmer water below turning, 1.5 m, associated with near isothermy
1.5 m. The segmentation algorithm indicates in the upper 1.5 m at 1546 hours. Energy dis-
that each profile had several distinct segments sipation rates were measurable in all segments
where the temperature-gradient signal was sta- and ranged from 2 x lo+ to 9 x 1O-7 m2 s-~.
tionary. Energy dissipation rates were ob- The temperature inversions below the sur-
tained within these segments in the upper 0.9 face segments at 13 14 hours on 17 December
m (Fig. 2C). Although dissipation rates were and 1506 hours on 16 December are imbedded
measurable near 1.8 m, data are not shown in stably stratified regions and therefore likely
because errors are likely, due to acceleration caused by shear instabilities. Inversions at 16 11
of the profiler as it was beginning to rise. Val- hours on 16 December, when the temperature
ues of Fr, tended to be < 1. difference in the upper 1.2 m was <O.O5”C,
In contrast to the profiles taken earlier, the may have been caused by intrusions.
temperature difference from the surface to 1.8 Batchelor spectra -Dissipation spectra from
m was only 0.65”C at 13 14 hours and only all segments were examined for a Batchelor
~0.25OC at 1341 hours. Unstable regions oc- spectrum to ensure that the fluctuations in the
curred throughout the water column. At 13 14 temperature-gradient signal resulted from tur-
hours a displacement of 0.6 m (Fig. 2B, seg- bulence as opposed to noise (J. Imberger pers.
ment 25) occurred in the upper segment. This comm.). After the sea breeze, the dissipation
layer was separated from the water below by spectra of all the segments except those near
a temperature gradient of -0.25”C and by the the base of the profile fit a Batchelor spectrum
displacements dropping to zero, which may (e.g. Fig. 4A), although some were noisy (e.g.
indicate zones where the water is not mixing Fig. 4B). Prior to the sea breeze, spectra from
and which form at least a temporary barrier only 4 of the 19 segments fit a Batchelor spec-
to vertical exchange. trum. An example of a dissipation spectrum
At 1341 hours the upper 0.6 m was still a that does not match the Batchelor spectrum is
distinct segment; however, the overturns with- presented in Fig. 4C. Because of noise in the
in it were <0.2 m in vertical extent. The over- dissipation spectra, estimates of E obtained by
turns may have been isolated from each other matching the dissipation and Batchelor spectra
and from ones below. If so, a <O.O2”C tem- were often overestimated. When the dissipa-
perature gradient at 0.3 m demarcated the sub- tion spectra were fitted by hand to the Batch-
division in the upper 0.6 m. At 13 14 hours, elor spectra, results were in concordance with
there were overturns centered at 0.9 and at 1.2 estimates from the Wigner-Ville transform.
m; the deeper one was now 0.5 m in vertical Table 1 provides the Fr,, Re,, and turbulent
extent. length and time scales for all segments where
By 13 14 hours, turbulence extended to a the dissipation spectrum matched the Batch-
depth of 1.7 m with E exceeding lop8 rn” s-3 elor spectrum. Only data from these segments
(Fig. 2C). In all the profiles after the sea breeze are used in subsequent calculations.
began, values of E > 10m7 m2 sm3 were asso- Eddy dicffusivities- Values of K, computed
Temperature(OC)
25.7 25.8 25.9 25.2 25.4 25.6 25.8 26.0 26.2 25.4 25.6 25.8
0.0 -: I I
ii
I
45E
0.3 ii- 46
; 47
:. 1
0.6 ii- 48C
3:42
0.9 ij-
::
>
1.2 :::-
‘J-
.
1424 1506 1546 1611
2.4
t I
L t (m)
0.0 0.5 0.0 1.5
B 0.0
0.3
0.6
.
;r
1 E
1.8
2.4 L 1424 1506 1546
t
1611
dm2s -3)
10-lo lo-” 1O”O 1o-6
3.0
1.2
0.6
0.6 0.5
0.8
2.2
1
1.8
2.4
Fig. 3.
1506
E
As Fig. 2, but at 1424, 1506, 1546, and 1611 hours on 16 December.
1546
t
1611
806 Maclntyre
0,
10-5
1
l- A
lo-"r ' '111111' ' '111111' ' ' JuJl
100 10' lo2 lo3 loo IO' 102 lo3 100 10' lo* 10'
Wave Number (cpm)
Fig. 4. Dissipation spectra (solid lines) from (A) 16 11 hours on 16 December (segment 58), (B) 13 14 hours on 17
December (segment 27), and (C) 1224 hours on 17 December (segment 20). Dashed lines- the fitted Batchelor spectra;
dotted lines- the auto-regressive model.
for both values of FrlcKIT are plotted as a func- phytoplankton. However, as stratification per-
tion of Fr, in Fig. 5. Values of K, are approx- sisted long after the sea breeze commenced,
imately a factor of two higher below FrlCKrT buoyancy forces were important and may have
and a factor of two lower above FrlCKrT when affected the circulation of phytoplankton by
K, is calculated assuming Fr,,,,, = 1.2. Results the turbulent flow field. Consequently, before
are presented assuming Fr,,,,, = 1.2, as trends discussing the circulation of phytoplankton, it
were comparable to those obtained with Kz is necessary to determine where and when the
values computed assuming FrlCRIT = 1.8. Val- flow was actively turbulent, where and when
ues of K, decrease by up to three orders of it was turbulent but affected by buoyancy.
magnitude below the maximum at Fr, = 2 be- The Froude-Reynolds number diagram ana
cause of the inefficiency of mixing due to brloy- attributes of the mixing layers-The data plot-
ancy forces and because of the reduction in E. ted in the Froude-Reynolds diagram (Fig. 6)
clarify the times and depths where buoyancy
Discussion affected the turbulence. Prior to the sea breeze,
Dynamics of the mixing layers-The water most of the water column was not turbulent!
column in North Lake constitutes a surface but where it was, buoyancy tended to affect the
layer (Imberger and Patterson 1990)-the lay- turbulence. One segment with a Batchelor
er affected by exchanges of heat and momen- spectrum fell into the region of the diagram
tum across the air-water interface. In deeper indicating the flow not to be turbulent. How-
bodies of water, it is only a fraction of the ever, as the boundaries of this region have beer
upper mixed layer. Microstructure profiles determined based on laboratory experiments
showed large variations in thermal structure they may shift somewhat as more data frorr
and in energy dissipation rates and ubiquitous lakes and oceans become available.
overturning regions whose vertical extents var- After the sea breeze began, the Fr, and Re,
ied from a few centimeters to 1.5 m. Different from the segments nearest the surface indicat-
patterns were found on different casts. These ed the flow to be actively turbulent, or tur-
data permit calculation of the length and time bulent but somewhat influenced by buoyancy.
scales of the turbulence and the possible range The values of Fr, and Re, (Table 1) correspond-
of fluctuations in irradiance experienced by ed to those Imberger and Ivey (199 1) found
Mixing in a shallow lake 807
Table 1. Shown for each segment where the dissipation spectrum fit a Batchelor spectrum and the Fr,-Re, diagram
indicated turbulence was present is the depth of the top of the segment (Z,), average values of the turbulent Froude
number (Fr,), turbulent Reynolds number (Re,), Thorpe displacement scale (I,,), segment thickness (L,), turnover time
of the displacement scale (I/U), diffusional time scales P/8K, where 1 is L, and L,y, and the expected duration of turbulent
transport (r/IV). Data are presented for both thermistors on the microstructure profiler, and sequence of time is ordered
with respect to the onset of the sea breeze.
l/U L,2/8K,* L,2/8Kzt L,2/8K,* L,2/8K,t T/N
Therm- Time z, L L,
istor (hours) SC& (cm) Fr, Re, (cm) (cm) (s)
17 Dee
1 1208 6 0 0.2 500 20.9 90 87 6,479 12,402 148,482 284,219 60
2 11 10 0.3 550 19.3 75 68 1,159 2,334 17,500 35,249 64
1 1224 16 16 0.7 80 3.9 15 19 27 62 401 915 42
2 24 70 1.5 6 9 15 12 4 - 1,406 - 57
1 1314 25 0 1 500 17.6 61 62 27 52 324 620 195
26 60 0.4 150 10.3 17 71 1,007 2,049 2,742 5,583 89
27 78 0.6 300 13.5 66.5 60 99 220 2,40 1 5,329 114
2 30 0 1.4 500 16.2 53 53 20 21 211 224 232
31 65 0.7 250 11.8 37 56 58 129 569 1,269 123
32 110 0.8 160 9.1 41 52 39 86 787 1,746 130
33 160 0.8 80 6.0 17 45 40 91 310 709 112
1 1341 34 0 4 100 4.1 57 17 6 4 1,218 690 212
35 65 2 220 6.8 38 21 8 4 246 139 132
36 103 1 500 13.1 46.5 35 15 29 188 360 108
2 38 0 4 120 5.0 59.2 21 8 4 1,095 621 260
39 65 5 70 3.1 34 14 5 3 619 351 222
40 100 0.5 70 12.7 13 232 4,456 9,302 4,669 9,747 364
41 115 0.7 180 11.8 36 77 85 190 789 1,769 170
16 Dee
1424 42 0 2 700 15.8 184 36 13 8 1,814 1,028 224
2 43 0 3.5 210 7.3 116 26 10 5 2,403 1,362 282
44 120 2 900 18.7 69 39 15 8 198 112 243
1506 45 0 3 220 7.4 17 25 9 5 49 28 233
46 24 1.2 80 4.5 8 26 9 15 30 47 97
47 32 0.6 100 6.5 21 42 103 228 1,075 2,385 80
48 55 0.5 200 11.0 6 60 215 463 64 138 95
49 67 0.8 300 9.7 43 32 22 48 434 948 79
50 110 2.2 250 6.5 18 17 6 4 49 28 115
2 51 0 2.8 100 4.9 20 24 9 5 150 85 207
52 20 1 180 8.3 30 39 17 34 222 438 122
53 55 0.8 400 14.4 55 52 36 78 523 1,136 130
1546 54 0 2 2,000 40.1 148 80 30 17 411 233 505
2 55 0 1.8 2,000 45.9 130 105 40 22 317 180 596
57 130 0.6 250 3.0 25 20 6 14 423 938 201
1611 58 0 1 650 19.5 39.3 58 25 48 103 196 184
59 40 0.6 2,500 60.1 8 144 194 430 3 8 272
60 53 0.4 900 38.9 7 168 847 1,802 27 58 211
61 60 0.5 450 20.7 104 95 261 569 6,588 14,354 149
2 63 0 1 350 13.7 35 54 23 45 153 295 170
65 70 0.4 600 27.1 52 122 675 1,429 2,487 5,260 154
67 130 2.5 70 3.2 36 15 5 3 694 393 116
* Fr,,,,, = 1.2.
j’ Fr,,.,,, = 1.8.
for wind-mixing layers. Values of Re, are large December; at this time the upper 0.5 m was
in actively mixing layers when penetrative cooler than waters below, indicating that free
convection occurs (Imberger and Ivey 199 1). convection was likely to have contributed to
Re, only exceeded 1,000 at 1546 hours on 16 the mixing. The data from segments below the
808 Madntyre
-2
10 4
-3
10 j
0
l 8
YN : 0
0
a
-6
10 :
-7
10 - I I I
0 1 2 3 4 5 6
Turbulent Froude Number
Fig. 5. K, vs. Fr, for Fr,,,,, = 1.2 (0) and 1.8 (0).
surface segment primarily fell into the region dissipation (Imberger 1985). Discriminating
where turbulence is affected by buoyancy, the between actively mixing regions and regions
exceptions being a few segments found below where buoyancy affects turbulence further in-
the thermocline or just below the surface seg- dicates the problems inherent in defining an
ment. Values of Fr, and Re, corresponded to upper mixed layer based on temperature pro-
those found when strong shear at the base of files and also shows that the uppermost mixing
the thermocline, intrusions, or boundary-layer layer may not be actively turbulent. If the up-
mixing were the cause of the turbulence (Im- per mixed layer is considered a discrete, nearly
berger and Ivey 199 1). The different causes for isothermal mixing region, such regions oc-
mixing suggest differences in the frequency of curred at 13 14 hours on 17 December (seg-
overturns. In the wind-mixing layers, over- ments 25, 30), and 1506 hours (segments 45,
turns will be continuous as long as the wind 5 l), 1546 hours (segments 54, 55), and 1611
persists. Where overturns are caused by shear hours (segments 58, 63) on 16 December. At
working against a stable density gradient, over- these times the vertical extent of the near-iso-
turns will not necessarily be continuous. thermal water column or unstable water col-
Previous studies have distinguished be- umn is the same size as the largest displace-
tween the conventionally defined mixed layer ment length and in close correspondence with
discriminated by temperature profiles and the the length of the segment. Circulation can be
mixing layer as determined by rates of energy described by one large eddy. However, if the
Mixing in a shallow lake 809
Re t
Fig. 6. Froude-Reynolds number activity diagram. The critical value of Fr, that separates the zone of active
turbulence (A) from the zone of buoyancy-affected turbulence (B) lies between Fr, = 1 and Fr, = 2; horizontal lines are
drawn at both values. The slanted line separates the zone where buoyancy dominates (C) from the zone of buoyancy-
affected turbulence; the equation for the line is E = 15vW. Data are from uppermost segments before winds (o), deeper
segments before winds (A), uppermost segments after winds (0), and deeper segments after winds (0).
dynamics of the segments are considered in- of several large eddies. For instance, the upper
stead of just the stratification, and if the as- 1.8 m at 1424 hours on 16 December was one
sumption is made that an upper mixed layer segment. Within the upper 0.6 m, displace-
is one that is actively mixing (Imberger 1985), ment lengths were ~0.2 m whereas deeper in
some of the segments that appear to constitute the water column were three large overturning
a mixed layer will be discounted (segments 25, regions, the largest of which was 0.6 m (Fig.
5 8, 63) for they have Fr, = 1 and are likely to 3B). Because the displacement lengths did not
be affected by buoyancy, whereas others that go to zero, there may have been exchange be-
are thermally stratified (segments 34, 35, 38, tween eddies. The upper meter at 134 1 hours
39, 42, 43, and 44) have values of Fr, and Re, was also weakly stratified and actively mixing,
that indicate active mixing. In this paper, the but it was subdivided into two different seg-
upper mixing layer is considered the part of ments, each with different dynamical charac-
the water column above the thermocline that teristics. Within the upper 0.6 m, which com-
is mixing; it may be stably stratified. prised one segment, were two discrete eddies,
The upper mixing layer is often comprised each of which was ~0.2 m in vertical extent.
810 Madntyre
These eddies (segments 34 and 38) are smaller 54) did mixing occur within the entire segment
than those in the previous profile, suggesting in less than the time scale r/N. In particular,
that as mixing progresses the upper mixed lay- - 1800 s would be required to mix segment
er does not necessarily continue as one over- 42, but r/N is only 224 s. In contrast, mixing
turning layer even if its size remains un- times (Table 2) for the three large overturns in
changed. Consequently, the length scale for segment 42 are all less than r/N. This obser-
calculating circulation times of phytoplankton vation supports the contention that circulation
in actively mixing regions is not necessarily of phytoplankton cells be computed for over-
the vertical extent of the actively mixing re- turning regions, not the vertical extent of ac-
gion. When wind mixing and penetrative con- tively mixing regions.
vection from heat loss both occur, as at 1546 The differences in the three time scales is
hours on 16 December, circulation can be de- systematically explored through the use of ra-
scribed by one large eddy; however, when heat tios in Figs. 7, 8, and 9. When the time scale
is not being lost at the surface, several distinctZ/u is divided by 12/2K,, the relation K, = a! UZ
overturning regions may occur within a mixing is obtained. In fact, K, - a UZwhere cx is a
layer. coefficient of order 1 is a basic definition in
Time and length scales of vertical mixing- phenomenological theories of turbulence
Tables 1 and 2 include estimates of the over- (Tenuekes and Lumley 1972). cyis nearly con-
turning time scale Z/u, the time scale of tur- stant with a value of -0.7 abo; e FrtCRrT when
bulent diffusion P/8.&, and the time scale for I is L, (Fig. 7). Although the near constancy
turbulent transport, n/N. In Table 1, Z/u is cal- arises in part from observations and assump-
culated with L,, the rms displacement scale, tions used to calculate the flux Richardson
whereas in Table 2, Z/u, is calculated with L,, number (Ivey and Imberger 199 l), it does sug-
the maximum displacement length. L,, the gest that L, is a good representation of the
vertical extent of regions where the turbulence mixing length in turbulent diffusion models.
is stationary, is also used in the computation Circulation times by the large eddies, as rep-
of F/8K,. The time for a patch to spread from resented by L,, are nearly equivalent to mixing
the center of a turbulent region with length times. The decrease in a! below FrlCRTTand the
scale Z to the edges is (Z/2)2/2Kz (i.e. P/M,). resulting discrepancy between the two time
The time for particles to spread from one edge scales arises because of the inefficiency of mix-
of a patch to the other is P/2& ing, that is, the correlation between the fluc-
As mentioned previously, the time scale for tuations in velocity and concentration that in-
turbulent transport estimates the length of time dicate turbulent transport decreases as
in which mixing is expected to persist in an buoyancy affects the turbulence (Itsweire et al.
eddy. Consequently, if either Z/u or Z2/8Kzex- 1986). I propose that when Fr, < FrICRIT, the
ceeds r/N, phytoplankton cells will not be well time scale Z/u is statistically correct for the
mixed. Using the overturning time scale im- phytoplankton cells transported by the ener-
plies that transport by the large eddies is a getic eddies but that fewer phytoplankton cells
coherent process, whereas using the time scale will be transported resulting in longer mixing
for turbulent diffusion implies that transport times of the population. The value of cy may
is by a wide range of eddy sizes. reflect the probability of turbulent transport
Overturning time scales are always rapid, by energetic eddies at the time of sampling.
ranging from 12 to 232 s (Table 1). Z/u is always When Z/u and 12/2K, are computed based on
less than or comparable to a/N and is less than the largest displacement lengths, as opposed
LC2/8Kzwhen Fr, < 1. LC2/8Kzis less than r/N to L,, a smaller value of (x is obtained above
except when Fr, is 5 0.6. Because the K, values FkRIT (Fig. 7B). This result implies that cir-
are based on the average Fr,, Re,, E, and N for culation within the largest eddies would have
each segment, Ls2/8Kz should provide good es-’ been much more rapid than mixing of the pop-
timates of the time to mix a segment even ulation over a comparable distance. If phy-
though these are not necessarily overturning toplankton are transported distances d’, the
entities. However, only at 1506 hours (seg- profile of displacement lengths (Fig. 10) at 1546
ments 45 and 46) and 1546 hours (segment hours on 16 December illustrates the many
Mixing in a shallow lake 811
Table 2. Location of the center (2,) and length (L,.) of pronounced overturns within segments. Irradiance at the
top (Ito,,) and bottom (I&, circulation times of single cells (I/U), mixing times of assemblages (P/8K,), and the time
scale for turbulent transport (r/N) are given for each of the pronounced overturns.
Time
Thermistor (hours)
17 Dee
1 1314 25 0.31 0.57 95 29 135 283 541 195
27 0.79 0.26 25 15 93 367 815 114
27 1.15 0.38 .13 6 119 764 1,695 114
2 3W 0.30 0.59 100 29 124 261 277 232
31 0.80 0.22 24 15 84 201 449 123
32 1.18 0.35 12 6 126 573 1,273 130
1 1341 34$ 0.18 0.15 81 59 39 82 47 212
35$ 0.83 0.14 20 15 34 33 19 132
1.24 0.50 13 4 84 218 416 108
2 zi+ 0.20 0.18 79 55 48 96 54 260
3% 0.87 0.08 18 15 26 34 19 222
41 1.33 0.35 9 4 159 746 1,672 170
16 Dee
1 1424 42$ 0.65 0.30 35 19 54 48 27 224
42$ 1.20 0.48 13 5 74 123 70 224
42$ 1.50 0.60 8 2 86 193 109 224
43* 0.22 0.13 72 55 37 30 17 282
43$ 0.99 0.33 18 9 69 194 110 282
44$ 1.52 0.54 7 2 78 121 69 243
1506 45$ 0.14 0.20 92 61 47 65 37 233
48 0.55 0.34 45 22 128 2,100 4,532 95
49 0.70 0.33 33 16 71 256 559 79
50$ 1.20 0.17 10 7 32 44 25 115
51$ 0.12 0.11 69 40 45 26 207
53 0.71 0.47 i:: 14 113 382 830 292
53 0.84 0.58 32 9 130 582 1,263 292
1546 54$ 0.75 1.39 89 183 362 205 505
55$ 0.88 1.60 85 : 241 480 272 596
1611 58 0.13 0.22 96 61 63 32 62 184
58 0.38 0.38 67 30 91 96 183 184
60 0.55 0.48 52 19 191 1,263 2,686 211
61 0.87 0.93 43 6 257 5,268 11,478 149
63 0.15 0.19 90 60 68 47 90 165
65 0.79 0.80 44 8 251 5,886 12,451 157
67 1.41 0.19 6 4 48 193 110 116
* FT,(.R,+ = 1.2.
t FTrCRIT = 1.8.
# Actively mixing segment.
different trajectories of phytoplankton follow- (Lazaro and Lasheras 1992). In contrast, a
ing the flow. Knowing how many phytoplank- model of mixing based on K, (Fisher et al.
ton are circulated by the large eddies is an 1979, equation 2.37) indicated the concentra-
important issue in assessing the irradiance to tion of particles transported a distance L, would
which phytoplankton are exposed. be - 1% of the initial concentration in a time
Recent research indicates that coherent ed- L,lu. If circulation of phytoplankton in over-
dies are important for particle dispersion in turns is a coherent process, more are likely to
several types of flow (Lazaro and Lasheras experience large fluctuations in irradiance at
1992). In a shear layer, concentrations of par- the time of initial overturning than would be
ticulates in overturns at maximum displace- predicted based on eddy diffusivity models.
ment reached 25% of initial concentrations The time scale for vertical mixing is con-
812 MacIntyre
.
0 1 2 3 4 5 6 I
10-2 1
Turbulent Froude Number 0 1 2 3 4 5 6
Turbulent Froude Number
1.0 Fig. 8. Ratio of time scale for turbulent mixing, P/SK=,
6 to time scale for turbulent transport (i.e. buoyancy flux,
T& vs. Fr, when K, is calculated assuming Fr,,,,, = 1.2
0.8 and I is the displacement scale; O-T~~ = T/N, l -raF =
1r/2N.
x
’ 2s 0.6
. segments that were much smaller than the
2 l overturns in which they were embedded, mix-
..
ing of a segment (Fig. 9A) would only be com-
s 0.4
\ plete for Fr, between 1 and 3, emphasizing the
2 importance of computing mixing rates for
0.2 overturns.
Because complete mixing in the lifetime of
an overturn is more likely above FrlCRIT, ac-
tively mixing overturns, typically near the sur-
0 1 2 3 4 5 6 face, are likely to become well mixed in a short
period of time (i.e. the time scale of the over-
Turbulent Froude Number
turn). However, where buoyancy has an effect,
Fig. 7. Ratio of the time scale l/u to P/2K, vs. Fr, where only partial mixing occurs. Complete mixing
K, is computed assuming Fr,,,,, = 1.2. In panel A, 1 is the will depend on continued formation of over-
displacement scale; in panel B, I is the largest displace-
ment. turning regions which are likely to be formed
intermittently when induced by shear insta-
bility (Gregg 1987).
trasted with the time scale for turbulent trans- When overturns are caused by shear, Fr, is
port rBF and plotted against Fr, (Figs. 8 and 9). initially near 1 where complete mixing is ex-
The length scale is Z/2 implying diffusion from pected. The observations of incomplete mix-
the center of a region of length 1 to the edges, ing at Fr, < 1 can be explained by the growth
i.e. 12/8K,. The transition from incomplete of overturns after initiation (Barrett and Van
mixing to complete mixing occurs for Fr, be- Atta 199 1); it is the larger overturns that have
tween 0.6 and 0.8 for TBF between n/N and evolved which are sampled and which do not
a/2N when L, is the turbulent length scale (Fig. have time to fully mix before energy is dissi-
8). For segments and overturns with dimen- pated in the overturn (J. Imberger pers. comm.).
sion L,, incomplete mixing occurred even for Alternatively, shear from the bottom could
Fr, > 1 (Fig. 9). With the exception of a few cause restratification (J. Imberger pers. comm.).
Mixing in a shallow lake 813
d’ (ml
0.00
0.6
Turbulent Froude Number
1.8
2.1
2.4
0 I I Fig. 10. Profile of displacement lengths at 1546 hours
10-l ’
0 1 2 3 4 5 6 on 16 December, thermistor 1. Every tenth point is plot-
Turbulent Froude Number
ted. -
Fig. 9. Ratio of time scale for turbulent mixing, 12/8Kz,
to time scale for turbulent transport 7BFfor 7gF = T/N and comparable time scales for overturning and
FrrCRIT = 1.2. In panel A, 1is the vertical extent of a segment mixing by diffusion when L, is used as the
(L,); in panel B, 1 is the largest displacement (L,).
turbulent length scale, but not when L, is the
turbulent length scale, indicate that a wide range
In summary, where buoyancy affects the tur- of eddy sizes contributes to vertical transport.
bulence, Z/u is -K P/2& indicating that some In a nonlinear problem, such as assessing the
phytoplankton cells will be rapidly transported exposure of phytoplankton to irradiance, mod-
a distance I but the time for a population to els based on vertical eddy diffusivities may be
spread the same distance will be considerably inaccurate.
longer; because P/8& > rBF, mixing will be Fluctuations in irradiance induced by cir-
incomplete. In the actively mixing overturns, culation of phytoplankton-The previous re-
P/8K, is 5 ~BF, mixing will be completed. The sults have indicated that circulation of phy-
814 MacIntyre
toplankton should be determined for time for particles to be transported a distance
overturning regions, not larger dynamical L, is 4 times this. Consequently, the time for
zones. Because phytoplankton cells are being the population to experience the extremes in
circulated by eddies ranging from L,, to -,$ irradiance in the overturn in segment 42 would
whose distribution in homogeneous flow is have been 192 s if FrtCRIT = 1.2. However, only
typically assumed to be Gaussian, the range of a few percent of the initial concentration of
fluctuations that individual cells could expe- particles would spread a distance L, in this
rience is large. The largest fluctuations possible time according to the turbulent diffusion mod-
at the time of overturning and the overturning el above. Similarly, a small percent of popu-
times are presented in Table 2 (see Figs. 2B lation would have experienced the extremes in
and 3B). The time scale Z/u is computed with insolation in segment 54 in 24 min if FrlCRrT
I= L, and u estimated from E = u3/Z with E the = 1.2. However, if the cells are transported by
average value within the segment. The ap- coherent large eddies, as suggested by studies
proach of computing u based on the overturn- of particle dispersion within overturns (Lazaro
ing scales was verified in Ivey and Imberger and Lasheras 1992), more particles are likely
(199 l), but it must be remembered that u is a to experience the large fluctuations.
rms velocity, and thus approximates a statis- Within segments where buoyancy affected
tical average for the velocity of large eddies. the turbulence (Table 2, unmarked segments),
The length of time for individual cells to cir- cells will not become well mixed but, due to
culate the length of an overturn, Z/u, was al- the large overturning scales, could still expe-
ways fast, ranging from 30 to 257 s. For ex- rience large variations in irradiance. For ex-
ample, in the overturn centered at 0.65 m at ample, at 13 14 hours on 17 December (Fig.
1424 hours on 16 December (segment 42), a 3A, segment 25) the upper 0.6 m was over-
cell could go from the 35% light level to the turning. There, phytoplankton cells could go
19% light level in a minimum time of 54 s. At from the 95% light level - 2 cm below the
1546 hours on 16 December (segment 54), a surface to the 29% light level in 135 s. The
cell could go from the 89% to the 5% light level rms displacement length was 0.18 m (Table 1).
in a minimum time of 183 s. Below the upper layer there were two- smaller
Not all cells will circulate a distance L,. billows. In the overturn centered at 0.79 m
However, by assuming L, represents the stan- (segment 27, 1st overturn), a phytoplankton
dard deviation in the vertical of the displace- cell could have circulated between the 25 and
ments of particles following the flow, particle 15% light levels in 93 s. The rms displacement
movement on time scales of turbulent diffu- length was 0.14 m. Because turbulent transport
sion can be computed with a random walk would have ceased in these overturns before
model or diffusion equation (Fisher et al. 1979, they would have become well mixed, estimat-
equation 2.28). For instance, for the overturn ing the time for the population to experience
in segment 42,68% of the particles would have the extremes in irradiance based on the dif-
been within +O. 16 m of their initial position fusional time scales for each overturn does not
within the diffusional time scale, 8-l 3 s (Table appear to be relevant.
1); 27% of the particles would be between 0.16 Eflects of turbulence on buoyant phytoplank-
and 0.32 m of their initial position. Similarly, ton -Realistic models of phytoplankton tra-
68% of the particles in the overturning region jectories must include the effects of the motil-
at 1546 hours on 16 December (segment 54) ity or buoyancy of the phytoplankton. The
would have been within kO.4 m of their initial cyanobacteria in North Lake are buoyant, with
position within 30 s assuming FrtCRIT = 1.2; a observed rising speeds for Microcystis ranging
further 27Oh would have been between 0.4 and from 10 to 250 m d-l (Humphries and Lyne
0.8 m of their initial position. 1988; Reynolds et al. 1987); recorded sinking
The time scales for turbulent diffusion in speeds are lower, 30 m d-l (Reynolds et al.
Table 2 represent the time for particles at the 1987). If the motion of turbulent eddies is
center of the overturn to reach the edges when dominant, then the time scale for vertical mix-
they are transported by the range of displace- ing (T,?) will be less than the time scale for the
ments characteristic of the turbulent flow. The cells to rise (T,) (i.e. T,,.,: T, K 1). This ratio
Mixing in a shallow lake 815
10’ _ I I I 3
can be computed for individual cells embed-
ded in and circulated by the large eddies using A -
the velocity scale u of the large eddies to com-
pute the overturning time scale T,, as Z/u where
I is the scale of an overturn. In the second i3
approach, the time scale for vertical mixing, ‘5 100 - :+ : . .
- . .
T,,, is computed as P/2K,. T, is UV where v .i3 .* .
(. l . . ’
is the speed of the cell. The ratios T,,/ : T, and .
- $0 i
.i’ 0 o
0
T : T, for the overturns of Table 2 are plotted ‘c
8
oe o E”
00
o o 0
Cl
agznst Fr, in Fig. 11. When cells rise slowly 0
.z Ol
(Fig. 11 A), the velocity scale of the large eddies 2 10-l z
is between 10 and 100 times faster than the 8
cells’ velocity; the large eddies will dominate + +
- +++++ + + +
cell movement. When cells rise at a rate of 100 - t- + ++ + + + +
or 250 m d-l, the two time scales are com- ++ + +++ +
parable and the cells’ positions will depend on I
10-J
both the large eddies and their own motility. 0 1 2 3 4 5 6
Different results are observed with the ratio Turbulent Froude Number
T,, : T,. At low cell speeds, the movements of
the cells are dominated by the turbulent mo-
tions only when Fr, is between 1 and 2 and
only in some cases. When cells rise rapidly and
Fr, < 1, the cells’ rising speeds determine their
position. Otherwise, the cells’ buoyancy and
the turbulent motions both affect the cells’ tra-
jectories.
For rapidly rising cells, cell speed is nearly
equivalent to the turbulent velocity scale. Be-
cause turbulent velocities decrease as eddy sizes
decrease, it is likely that only eddies of a certain
size and larger will have sufficient energy to
transport these buoyant phytoplankton. Its-
weire and Helland’s (1989) power spectra of
vertical velocity is flat from - 1,000 Lk to 20
Lk where turbulence is active, indicating a wide
range of scales have sufficient energy to trans-
port cells. However, as buoyancy begins to af-
fect flow, the vertical velocity power spectra
drop rapidly, and in fact by 100 L,, have
Turbulent Froude Number
dropped an order of magnitude. Based on Its-
weire and Helland’s results, I hypothesize that Fig. 11. A. Overturning time scale divided by rise time
(i.e. T,,,lT,,) vs. Fr, (T,,,, is l/u and T, is NV where v is cell
as buoyancy affects the flow, only the largest vertical speed). B. Mixing time scale divided by rise time
eddies will have sufficient energy to transport (i.e. T,,,/T,.) vs. Fr, (T,,, is P/2K, and FrlCRIT = 1.2). v =
motile phytoplankton, and, as the range of 10 m d-l (+); v = 100 m d -l (0); v = 250 m d-l (0). In
scales that can transport phytoplankton de- all cases, the length scale is L,.
creases, the proportion of phytoplankton
transported by turbulent eddies at each instant
in time is also likely to decrease. Support for Conclusions
this hypothesis is the greater importance of the The analysis of temperature-gradient micro-
cell’s buoyancy in determining its position at structure data from this shallow, turbid lake
low Fr, when T,, is compared with T, as op- illustrates the complicated thermal structure
posed to comparing Tmrwith T,. and differences in mixing dynamics in the sur-
816 MacIntyre
face layer. Within this shallow layer, the water and laser-induced fluorescence. Phys. Fluids A 3: 132 l-
column was subdivided into regions with dif- 1332.
ferent dynamics. In particular, during a windy BATCHELOR, G. K. 1959. Small-scale variations of con-
vected quantities like temperature in turbulent fluids.
period the upper portion was actively turbu- J. Fluid Mech. 5: 113-133.
lent and the deeper portion was turbulent, but BAYLEY,P.,D.M. DEELEY, R. HUMPHRIES,ANDG.BOTT.
the turbulence was affected by buoyancy and 1989. Nutrient loading and eutrophication of North
mixed less efficiently. These regions with dif- Lake, Western Australia. Environ. Prot. Auth. Perth,
W. Aust. Tech. Ser. 33.
ferent dynamics were often comprised of sev- BOWDEN, K. F. 1983. Physical oceanography of coastal
eral overturning regions with different length waters. Wiley.
and time scales. It is within the overturns that CALDWELL, D. R., AND T. M. DILLON. 198 1. An oceanic
mixing times and circulation times of phyto- microstructure measuring system. Oregon State Univ.
Tech. Rep. 8 1- 10.
plankton can be assessed. In overturns with CARTER, G. D., AND J. IMBERGER. 1986. Vertically rising
active turbulence, mixing took <5 min. In microstructure profiler. J. Atmos. Ocean. Technol. 3:
contrast, mixing in the deeper regions was un- 462-471.
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