MACINTYRE_ SALLY. Vertical mixing in a shallow_ eutrophic lake by gdf57j


									 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
             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,
                                         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


z    0.9

‘i   1.2
*    1.5

           t                      I
               1149                            1157
     2.4   t
                            0.0                0.6         0.0     0.8             0.0    0.3           0.0      0.6        0.0    0.6
                                      I    I


               1149                            1157                1208
     2.4   t
                                                                    tz   (m2sB3)


                                                                         BIT                     BIT


     2.4   E
                                                                                    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
     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
  0.3 ii-                             46
      ;                               47
      :.                                 1
  0.6 ii-                             48C
  0.9 ij-
  1.2 :::-


                  1424                              1506                                  1546                  1611
        t                                                                  I
                                                                 L t (m)
                                              0.0      0.5                         0.0         1.5
B 0.0

                                         1 E

  2.4   L         1424                              1506                                  1546

                                                             dm2s      -3)
                                        10-lo         lo-”              1O”O               1o-6
  0.6                                                      0.5




            Fig. 3.
                       As Fig. 2, but at 1424, 1506, 1546, and 1611 hours on 16 December.
 806                                                Maclntyre

            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

                10            4

                10 j

                                                                              l      8
           YN                                                                        :          0

                10        :

               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.

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 =
’ 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


                      Turbulent Froude Number



                                       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

 T : T, for the overturns of Table 2 are plotted          ‘c
                                                                                   oe           o     E”
                                                                                                                 o          o       0

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
 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-
      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.
likely to be completed on times scales of over-             DAVIS, J. A., AND S. W. ROLLS. 1987. A baseline bio-
turns. Phytoplankton     cells in the 0.2-1.5-m-                logical monitoring programme for the urban wetlands
                                                                of the Swan coastal plain, Western Australia. Environ.
deep wind-mixed        upper layers could *be                   Prot. Auth. W. Aust. Bull. 265.
circulated within a broad range of light inten-             DENMAN, K. L., AND A. E. GARGETT. 1983. Time and
sities on time scales of minutes. Fluctuations                  space scales of vertical mixing and advection of phy-
would be experienced by these cells as long as                  toplankton in the upper ocean. Limnol. Oceanogr. 28:
wind mixing persisted. Cells within the mixing              DILLON, T. M. 1982. Vertical overturns: A comparison
zones where buoyancy affected turbulence also                   of Thorpe and Ozmidov length scales. J. Geophys.
experienced rapidly fluctuating light levels, but               Res. 87: 9601-96 13.
because the overturns were likely to have been                        AND D. R. CALDWELL. 1980. The Batchelor spec-
induced by shear working against a stable den-                   trum and dissipation in the upper ocean. J. Geophys.
                                                                 Res. 85: 1910-1916.
sity gradient, they were likely to have been                FALKOWSKI, P. G. 1983. Light-shade adaptation and ver-
formed intermittently.     Consequently, fluctu-                 tical mixing of marine phytoplankton:      A comparative
ations may have been experienced discontin-                      field study. J. Mar. Res. 41: 215-237.
uously. The length scales of eddies circulating             FERRIS, J. M., AND R. CHRISTIAN. 199 1. Aquatic primary
                                                                 production      in relation to microalgal responses to
phytoplankton     were likely to be constrained                  changing light: A review. Aquat. Sci. 53: 187-217.
relative to those in actively turbulent flow.               FISHER, H.B.,E.J.       LIST, R.C.Y. KoH,J. IMBERGER,AND
    Trajectories and mixing rates of phytoplank-                 N. H. BROOKS. 1979. Mixing in inland and coastal
ton cells are required to design experiments                     waters. Academic.
                                                            FOZDAR, F. M., G. J. PARKER, AND J. IMBERGER. 1985.
and to model primary production in fluctu-                       Matching temperature and conductivity           sensor re-
ating light environments.      The variability   of              sponse characteristics. J. Phys. Oceanogr. 15: 1557-
length and time scales within the mixing layers                   1569.
in North Lake indicates the inadequacy, in                  GARGETT. A. E. 1989. Ocean turbulence. Annu. Rev.
many cases, of parameterizing the upper mixed                    Fluid Mech. 21: 419-451.
                                                            GIBSON, C. H., AND W. H. SCHWARZ. 1963. The universal
layer by one coefficient of eddy diffusivity as                  equilibrium      spectra of turbulent velocity and scalar
is commonly done in models of phytoplankton                      fields. J. Fluid Mech. 16: 365-384.
circulation. Using estimates of time and length             GREGG, M. C. 1987. Diapycnal mixing in the thermo-
scales of vertical motions from microstructure                   cline: A review. J. Geophys. Res. 92: 5249-5286.
                                                            HUMPHRIES, S. E., AND V. D. LYNE. 1988. Cyanophyte
data in conjunction      with models of upper                    blooms: The role of cell buoyancy. Limnol. Oceanogr.
mixed-layer dynamics would permit more re-                       33: 79-91.
alistic models of trajectories of phytoplankton             IMBERGER, J. 1985. The diurnal mixed layer. Limnol.
to be developed than the random walk models                       Oceanog. 30: 737-770.
 or turbulent diffusion models currently in-                -,        AND B. BOASHASH. 1986. Application of the Wig-
                                                                  ner-Ville distribution to temperature gradient micro-
voked.                                                            structure: A new technique to study small-scale vari-
                                                                  ations. J. Phys. Oceanogr. 16: 1997-20 12.
                                                            -,        AND G. IVEY. 1991. On the nature of turbulence
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