Limnol.
Oceanogr., 34(2), 1989, 420-434
0 1989, by the American Society of Limnology
and Oceanography,
Inc.
Dynamics of horizontal turbulent mixing in a nearshore
zone of Lake Geneva
U. Lemmin
Laboratoire de recherches hydrauliques,
CH- 10 15 Lausanne, Switzerland
Ecole Polytechnique
Fed&ale de Lausanne,
Abstract
Time series of currents measured during the fall/winter
period in a nearshore zone (near a
headland and in an adjacent bay) of Lake Geneva, Switzerland, were used to determine horizontal
turbulent mixing coefficients. The data were filtered to obtain components in the time range 0.512 h, which are suitable for mixing studies because spectra showed no energy source or system
response there. From the filtered data, eddy diffusion coefficients were calculated by the integral
time-scale method and eddy viscosity coefficients by the Ertel mixing length method. Viscosity
coeficients were systematically higher than diffusion coefficients by a factor of at least two. Longterm horizontal turbulent mixing coefficients calculated over periods of several months were of
order lo3 cm2 s I, a factor of 10 lower than coefficients obtained for coastal zones of larger lakes.
Short-term (3 d) mixing coefficients varied between 102 cm2 s-l for quiet background situations
and 10“ cm2 s I for periods of wind-induced,
large-scale advection. This variability was traced to
the history of wind events, indicating that energy input is not sufficiently homogcncous in time to
maintain mixing at a constant level. Sheltering by local topography caused the sensitivity of the
mixing coefficients to wind direction. Variability
in space, with larger values at the headland and
smaller ones inside the bay, underlined the modifying influence of shoreline geometry. Inside the
bay, diffusion diagrams show that near the surface mixing is generated by shear, while near the
bottom inertial subrange diffusion is found. The importance of shear is also evidenced by a
correlation between wind forcing and diffusion coefficients for events of long wind fetch. Near the
headland none of these simple concepts hold because local topography influences mixing.
The intensive use of the water resources
in lake systems has caused great concern
about the spreading of biochemically active
materials due to their potential influence on
the balance of lake ecosystems. Most of these
materials, acting as nutrients or (toxic) pollutants, enter the lake via rivers and outfalls.
They are thus initially introduced into the
nearshore zone which is known as the coastal boundary layer. The incoming water
masses quickly lost their excess momentum
and their further distribution
in the lake
mainly depends on water movement within
the coastal boundary layer, which serves
then as a buffer zone receiving materials
from rivers and outfalls on the one side and
dispersing them into the offshore part of the
lake on the other. A study of the structure
of the current field in the coastal boundary
layer is of direct importance for determining
the trophic state of the lake as a whole.
Advective displacement due to the mean
Acknowledgments
C. Perrinjaquet supervised the fieldwork and carried
out the data preprocessing and programming with care
and efficiency. Comments by two anonymous reviewcrs and W. H. Graf greatly improved the manuscript.
(larg,e scale) motion and diffusion due to the
random component of the current field are
among the processes acting simultaneously
on water parcels entering the coastal boundary layer. The structure of the advective
component has been studied for several
cases. It is generally assumed that advective
flow in the coastal boundary layer is driven
by large-scale water movement and follows
the depth contours (Murthy et al. 1986).
Approaching the shoreline, bottom friction
becomes increasingly important in the balance of forces due to reduced water depth.
In a case study of Lake Huron, Murthy and
Dunbar (198 1) showed the presence of a
well-defined
coastal boundary layer with
smooth offshore gradients of the alongshore
current component. A well-established profile :may be locally modified due to shorelines forming bays and headlands.
The effect of onshore winds-one
component of the diurnal wind field-on
the
resultant alongshore currents was investigateld by Murray (1975). Less is known about
the diffusive component of the current field
in the coastal boundary layer of lakes. Murthy and Filatov (198 1) derived turbulent
420
421
Horizontal mixing in a nearshore zone
mixing coefficients from spectra of currents
measured in the coastal boundary layer of
large lakes. Over the range of time scale
from synoptic to semidiurnal (each identified by a peak in the spectra), the coefficients
dropped from 1Ohto 1O4cm2 s-l. Diffusion
coefficients calculated from the spreading of
dye, heat, and radioactive
materials released from power station sites in the coastal
boundary layer of the Great Lakes have been
discussed by Lam et al. (1984). In these lakes,
the width of the coastal boundary layer may
be up to 10 km, which is the total width of
Lake Geneva. In all studies, only long-term
mean values were obtained. The dynamics
of turbulent diffusion have not been treated.
No data seem to exist for coastal boundary
layers of smaller lakes. It is not a priori evident that the coefficients calculated for large
lakes also apply to smaller lakes. The present study investigates the diffusive aspect of
the current field and makes only brief reference to the advective component.
Horizontal eddy diffusion coefficients in
lakes which are frequently used in predictive transport models have been derived
from Lagrangian studies of spreading heat
and matter (Lam et al. 1984). On the other
hand, by using moored automatically
recording current meters, long time series of
Eulerian horizontal current fluctuations at
fixed points that can also be interpreted in
terms of mixing parameters can be obtained
with relative ease. From the Eulerian measurements, horizontal momentum transfer
coefficients can be calculated as the ratio of
current covariances
and mean current
shears. Unfortunately
current shear is not
easily determined. This method of calculation becomes even less suitable for the
coastal boundary layer where local topographical effects may generate mesoscale
current structures such as dead waters or
recirculating gyres. In addition, momentum
transfer coefficients usually appear to be
larger than diffusion coefficients (Pond and
Pickard 1986) and may not be good substitutes.
As an alternative, the method used to determine Lagrangian diffision coefficients can
be adapted to Eulerian measurements. The
Lagrangian eddy diffusion coefficient is defined as (Hinze 1975)
1
Kf, = -9
IAt2 R(t)
0
dt
(1)
where R(t) is the autocorrelation
function
of the velocity fluctuation u’. The autocorrelation function drops off from 1 to small
values beyond some time t = r if the data
contain no periodic components. For times
t > r the integral
in Eq. 1 will approach a
constant value T, the integral time scale.
Eulerian velocity fluctuations can be substituted for Lagrangian ones for homogeneous and stationary turbulence. Based on
simultaneous
Lagrangian
and Eulerian
measurements, it was found that due to advection the Eulerian measurements may
overestimate the frequency of the fluctuating component. Therefore Hay and Pasquill
(1959) proposed adjusting the time scale for
the effect of advection by introducing a constant ,& assuming that the Lagrangian and
Eulerian autocorrelation
functions have
about the same shape. The Lagrangian autocorrelation function R,(t) can then be related to the Eulerian R,(t) by
R,(t) = &(Pt)
(2)
where /3 > 1. The appropriate value of @
can be determined if simultaneous Lagrangian and Eulerian measurements exist. Observations range from p = 1.4 (Schott and
Quadfasel 1979) for the Baltic Sea to p >
10 for atmospheric diffusion, with a most
likely value of ,0 x 4. This method implies
that the shapes of the Lagrangian and Eulerian autocorrelation
functions are identical after appropriate scaling with & which
is not necessarily true. However as Csanady
(1973, p. 79) pointed out, “. . . the prediction of spreading is not very sensitive to the
exact shape of the autocorrelation
coefficient and therefore it is not surprising to
discover that the simulation technique based
on Eulerian velocities gives in practice good
diffusion predictions.”
The eddy diffusion coefficient then becomes
KII = ,&i-T.
(3)
For the Lagrangian case 6 = 1; for the Eulerian case p > 1. The assumption that the
turbulence is stationary is only a gross ap-
422
Lemmin
proximation
in geophysical flows. The influence of nonstationarity
may be reduced
if data are selected so that the mean over a
range of averaging times is no longer a random variable indicating a wide separation
of the advective from the diffusive components of the flow field. The averaging
times should be long compared to the integral time scale T. This method of calculation is used here.
Another parameter frequently used to
characterize diffusion is the turbulent length
scale L defined as (Hinze 1975)
L = UT
(5)
Mm-thy (1976) showed that for b = 1 diffusion is controlled by shear, while for b =
4/3 inertial subrange diffusion is found.
Combining oceanic and large-lake dye diffusion data, Lam et al. (1984) found b = 1.1
in the upper layers and b = 1.4 for the hypolimnion. Thus horizontal eddy diffusivity
increases markedly with the length scale. In
the upper layers, lateral diffusion coefficients were found to be an order of magnitude smaller than the longitudinal
ones;
this was taken to indicate anisotropic turbulence. Smaller values ofthe diffusion coefficient found in the hypolimnion
are explained by a decrease of available turbulent
energy with depth.
Diffusion coefficients determined from
dye experiments in Swedish lakes were found
to be much smaller than those reported for
the Great Lakes. This led Ottensen Hansen
(1978) to postulate that diffusion coefficients are fetch-dependent. He proposed the
following expression:
KH = const 6~~
where u* is the friction
velocity
A,, = (ut2)“1,
(4
where U is the time mean value of the Eulerian velocity and T the time scale as delined above. One of the commonly used relationships initially introduced as “diffusion
diagrams” by Okubo (197 1) for oceanic
conditions is the horizontal diffusion coefficient as a function of length scale. A theoretical explanation is provided by Bowden
et al. (1974). A log-log plot of the diffusion
coefficient against the length scale determines a straight line which is given by
KH = aLb.
(vertical) length scale which he took as equal
to the water depth for fully developed flows
or equal to the epilimnion depth in stratified
flows. His data indicated a scatter of more
than a decade for the constant, which he
attributed to the wrong choice of length scale
6. Eider (1959) had found const = 0.23 for
fully developed shear flow in laboratory
flumes.
Eddy viscosity coefficients can also be calculated by the Ertel method (Shtokman
1970):
--
(6)
and 6 is a
(7)
t+h
where IX = 0.5
S
u’ dt
is the mixing length and X is the time interval in which the fluctuations of the current velocity maintain the same sign. For
oceanic conditions, eddy viscosity coefficients given in the literature are larger than
eddy diffusion coefficients. For comparison,
calculations with this method are also presen ted here.
The Lake Geneva environment
Lake Geneva (local name, Lac Leman) is
located at the border of Switzerland and
France. It has a surface area of 582 km2, a
maximum depth of 3 10 m, and a mean depth
of :I57 m. River-induced throughflow is not
important since the mean residence time of
the water is about 11 yr. The lake is strongly
stratified during summer and fall. After the
end of December a weak stratification
of
about 0.1 K in the upper 100 m remains
during the mixing period. Strong winds
which result from large-scale atmospheric
pressure differences are channelized by the
Alps to the south and the Jura mountains
to the north (Fig. 1) and are predominantly
oriented along a SW-NE axis. Local topography strongly modifies the distribution
of
wind stress above the lake. The eastern part
of the lake is surrounded by steep mountains on both the north and south shores.
Winds from the southwest, locally known
as “vent,” enter the lake basin from the flat
surroundings. They have the longest fetch
over the lake. Above the mountainous eastern part they are observed with greatly reduced speeds. Winds from the opposite di-
Horizontal mixing in a nearshore zone
423
a
Fig. 1. Location of current meter mooring stations at the northern shore of Lake Geneva. Depth contours
in meters. The 75-m contour roughly indicates the width of the coastal boundary layer calculated from typical
parameters for the lake. Instruments were placed at 10-m depth at all stations (SSl and SS7 at 8 m due to
limited depth). At SS3, SS5, and SS6 additional current meters were placed -5 m from the bottom. a. Lake
Geneva. The study area (in black) is at the north shore. The directions and names of the prevailing winds are
indicated; they are forced by the mountainous terrain shown in panel b. b. Location of Lake Geneva (in black)
in Switzerland. The outline of the Jura (J) mountains and the western parts of the Alps (A) indicate regions with
elevations up to several thousand meters above lake level.
rection (local name “bise”) occur most
frequently during winter. Due to the mountainous topography along the eastern and
central part of the northern shore, the northern nearshore sections in which the present
study was carried out and the eastern part
of the lake are sheltered from these winds.
The overall effect of bise winds on the lake
hydrodynamics is less pronounced than that
of vent winds. Wind speeds remain ~5 m
s-l most of the time and local winds due to
differential heating may at times become
important
in the force balance (Ganter
1978).
Field program
Self-contained
automatically
recording
current
meters (Aanderaa)
have been
moored in a section of the northern coastal
boundary layer in Lake Geneva to study the
nearshore water movement. The recording
interval of the instruments was set at 30
min. Recording started on 24 October 1984
and continued until 11 March 1985.
The stations are arranged in two clusters
(Fig. l), each of which has a set of moorings
in comparable water depth. In both clusters
the stations are spread out in alongshore and
offshore directions between water depths of
13 and 77 m. A depth of 75 m in this area
corresponds roughly to the width of the lateral friction-induced
boundary layer (order
1 km) for typical conditions in Lake Geneva. At all stations, one current meter was
placed at - 1O-m depth, and at stations SS3,
SS5, and SS6 additional instruments were
placed close to the bottom. At the two nearshore stations, SS 1 and SS7, the instruments
were placed at 8-m depth because of limited
water depth. A depth of - 10 m would be
424
Lemmin
Fig. 2. Energy density spectra for stations SS 1, SS3,
SS6, SS7, and SS9 for the entire observation period,
24 October 1984-12 March 1985. Alongshore component-solid
lines; cross-shore component at station
SS 1 -dash-dotted
line, and SS7 -dashed line. 95% C.I.
as indicated. The frequency range (R) is considered in
the present calculation of mixing coefficients.
near the bottom ofthe Ekman layer for winds
> 5 m s-l. However, as winds over Lake
Geneva are ~3 m s-l > 80% of the time,
the direct influence of the wind on the recorded currents is small. A net buoyancy of
at least 25 kN on each line is sufficient to
keep the line straight under the weather conditions of this lake. Wave effects were reduced by placing the upper float of the
mooring about 5 m below the surface.
Stations SSl-SS4 in the first cluster are
located along the headland of St. Sulpice.
In the second cluster, stations SS5-SS9 are
found in the Bay of Vidy (Lausanne). This
configuration
of stations and instrument
depths was chosen to determine whether
nearshore currents will follow depth contours (Murthy et al. 1986) under the complex topography of a headland with an adjacent bay.
Results
To calculate mixing coefficients, we require that the advective component of the
current field, lake responses such as internal
waves, and energy inputs such as diurnal
winds all be eliminated from the data. This
definition
is different from that used by
Murthy and Filatov (198 1) who included all
motions up to the synoptic scale. To determine which frequency range of the data
would be free from these “perturbations,”
it was necessary to calculate spectra for all
stations. Spectra of the longshore components for stations SSl, SS3, SS6, SS7, and
SS9 in Fig. 2 have no significant peaks that
could be related to energy input. There is a
small peak at the diurnal frequency. The
difference in the low frequency end of the
spectra between the stations reflects the
weaker currents inside the bay area. At low
frequencies cross-shore components, also
shown in Fig. 2, generally have lower energy
than alongshore components. For frequencies > 1 cpd, all spectra become smooth.
Therefore this range was taken as representativc for movements that could be described by a diffusion coefficient. Energy
levels in this frequency range inside the bay
(SS6, SS7) are significantly lower than those
at the headland (SSl, SS3).
On the basis of the above observations,
the data are low-pass filtered with a cutoff
period of 15 h. This eliminates all movements with periods < 12 h. The high frequency component (HF) is then obtained as
the difference between the original and the
low-pass filtered data. The removal of the
slowly varying advective component results
in a zero mean current for the HF data (Fig.
3), and short period fluctuations are the
dominant characteristic of the HF data. This
example demonstrates that the amplitudes
of the HF component at SS7 inside the bay
are much smaller than those at the headland
during periods of weak currents (the first
half of the record segment). Strong winds
from the southeast (with long fetch) after 10
November 1984, which lead to a general
increase in the low-pass current amplitudes,
also generate more energetic HF fluctuations everywhere.
In order to test the time dependence of
the HF fluctuations and to establish a measure of the stationarity of the movements
investigated, I calculated cumulative averages of the kinetic energies of the mean motion and the fluctuating component of the
Horizontal mixing in a nearshore zone
b
425
0
0
0
0
5
10
15
Nov 1984
20
Fig. 3. a. Sample trace of the measured (solid lines) and the low-pass filtered (cutoff period 15 h; dashed
lines) current components at stations SSl (nearshore) and SS3 (offshore) at the headland and SS7 (nearshore)
inside the bay for 5-20 November 1984. b. The difference between the measured and filtered signals (termed
HF) for the three stations. Alongshore component-a;
cross-shore component-c.
HF series. Accumulation commenced at the
beginning of the records. Means were determined for the respective record length
and fluctuations calculated as deviation from
that mean value. For a record length of several days the mean energy value drops to
zero and remains there, indicating stationary conditions. The energy of the fluctuating
component shows variability
in time and
space (Fig. 4). Highest energies are found
nearshore at the headland- station SS2 (not
shown) being almost identical to SSl . Energies drop with distance from shore (SS3).
Lower energies are found inside the bay area
with lowest levels at SS5 and SS6 (not shown
here) where comparable values were found.
Even though SS3, SS5, and SS9 are in the
same water depth (75 m), energy levels drop
from the headland (SS3) to the open water
(SS9) to the bay area (SS5). There is also a
drop in energy between the two nearshore
stations, SS 1 at the headland and SS7 inside
the bay. An increase in energy in the fluctuating component was observed after about
30 d. This increase which had already been
noted in the structure of the advective and
fluctuating components (Fig. 3) may be the
result of a strong wind-pulse from the SW
(vent), which was unusually long and stable
for this lake. It lasted for more than 5 d at
12-h mean speeds > 5 m s-l. Winds from
that direction have the longest fetch for the
study area. The increase in energy is less
pronounced inside the bay area (SS5 and
SS7). Another increase is seen after about
70 d at the beginning of January. This was
again linked with a vent wind-pulse.
In
spring, input of wind energy is low and en-
426
Lemmin
Table 1. Turbulent mixing coefficients derived from the HF current components for all stations. Calculated
for the total length of the record and for the periods 24 October-3 1 December 1984 (destratifying-84)
and 1
January-l 1 March 1985 (dcstratificd-85).
Currents are split into alongshore (C,, positive toward east) and
SSI
k,
A,,
Total
84
85
Total
84
85
ss2
_~
ss3
SS3B
ss4
c,
C,
C,
Cl
C,
C
C,
Cl
C,
C,
3.27
3.62
3.24
9.82
12.4
8.63
2.05
1.77
2.35
3.91
3.65
4.19
2.65
3.49
2.04
6.88
11.2
4.38
2.34
2.58
2.19
3.91
4.36
3.54
1.53
1.64
1.53
3.44
4.08
3.09
1.71
1.75
1.75
2.88
3.39
2.66
1.02
1.33
0.72
3.71
5.44
2.31
0.57
0.66
0.51
1.53
1.99
1.21
1.03
1.66
0.46
2.85
5.49
1.04
0.75
0.93
0.59
1.22
1.69
0.88
-_
ergy levels drop continuously
toward the
end of the records. The structure indicates
that individual events are important in the
energy balance.
Turbulent mixing coeficients - Turbulent
mixing coefficients were calculated with both
the integral time-scale method and the mixing length method. The availability
of long
time series permits study of the effect of
record length on the magnitude of the resulting mixing coefficients. In order to determine the minimum length for which stationary conditions are achieved, I calculated
integral time scales for record subsections
of different length. Each time the beginning
of the subsection was shifted by one data
point in a manner used to calculate running
means. It was observed that for > 100 lags
(55 h) the integral time scale converges to
a constant value of around 1 h for all records. For the mixing length method, histograms were established of the time during
which the sign of the fluctuation remains
constant. For all records they have maxima
around 1.5-2 h. The coefficients (Table 1)
obtained by the two methods of calculation
over the total record length are of the same
order of magnitude. The difference is a factor of at least two. The difference between
the alongshore and the on-/offshore com-
* SSI
b ss3
0 ss9
ss7
x ss5
l
Fig. 4. Cumulative averages (12 h) of the kinetic energy of the HF current componets at stations SSl, SS3
at the headland and SS5, SS7 in the bay. Station SS9 is in front of the bay in the same water depth as SS3 and
ss5.
427
Horizontal mixing in a nearshore zone
cross-short components (C,,, positive toward north). Eddy diffusion coefficients derived from the integral timescale method-&,;
eddy viscosity coefficients derived from the mixing length method-A,,.
Near-bottom
instruments are indicated by “B” following station identification.
Units are lo? cm2 s--l.
ss5
SSSB
ss9
SSI
SS6B
SS6
C,
C,
C,
Cl
C,
Ct.
C,
C,
C,
C,
C,
c,
0.74
1.03
0.5 1
1.78
2.79
1.05
0.53
0.67
0.43
1.06
1.43
0.76
0.28
0.21
0.35
0.62
0.55
0.69
0.35
0.27
0.39
0.85
0.75
0.93
0.52
0.67
0.37
1.36
2.05
0.89
0.3 1
0.36
0.29
0.62
0.74
0.55
0.68
1.11
0.31
1.71
2.99
0.74
0.53
0.87
0.22
1.21
2.09
0.49
0.83
1.21
0.47
2.37
3.81
1.32
0.46
0.68
0.29
0.91
1.39
0.59
0.82
1.11
0.54
1.93
3.07
1.11
1.12
1.58
0.68
2.06
2.94
1.24
ponents is small in all cases. Mixing coefficients calculated separately for the first half
(fall 1984, destratifying) and the second half
(winter 1985, destratified) of the record
show small seasonal variations; the shift to
lower values in winter (1985) is slightly more
pronounced in the bay area. This shift is
mainly the result of a change in meteorological forcing as is shown below. If we consider these calculation periods “long term,”
typical values for the long-term horizontal
diffusion coefficients are of order lo3 cm2
s-l in the study area.
Calculations of mixing coefficients were
also carried out for shorter subsections of
the records in order to study the effect of
variable meteorological forcing which has
already been documented for the energy distribution (Fig. 4). As integral time scales
were constant (- 1 h) for periods > 55 h,
eddy diffusion coefficients were calculated for
3-d periods (=72 h or 144 data lags). For
the mixing length method, this period corresponds to integration over -40 cycles. As
can be seen from a peak in the spectra (Fig.
2), 3-4 d are a typical duration for the passage of large-scale atmospheric pressure systems. Calculations were carried out in a running mean manner with a shift of 1 d between
consecutive estimates. For all stations the
diffusion coefficients calculated in this manner by the integral time-scale method show
a dependence of the magnitude of the coefficients on meteorological forcing (Fig. 5).
From the equal magnitude of the two
components north and east of the wind field
(Fig. 5) measured at Cointrin airport near
Geneva at the SW end of the lake, it is ob-
vious that forcing is along an axis of about
45” as discussed above. The wind is organized in events of vent and bise with relative
calm periods between. Highest wind energy
is found during bise events. For all stations
the alongshore and cross-shore components
of the diffusion coefficients are of comparable magnitude, indicating homogeneity of
the flow field (Fig. 5). The variation in magnitude of the diffusion coefficients is unsymmetrically correlated to the wind structure.
The highest diffusion coefficient values result from a strong vent event after about 30
d in November. Each of the following vent
events, even though they may be of relatively low amplitude due to the 3-d averaging, leads again to augmentation of the
values. During such events the magnitude
of the coefficients may rise by a factor > 10
over the long-term background values found
between events (SSl and SS2; Fig. 5). Inside
the bay area, the vent-related peaks of the
coefficients are generally less pronounced
than those found near the headland. This
difference becomes most evident when
comparing the two nearshore stations, SSl
and SS7. After each of the events the magnitude of the coefficients drops within a few
days toward low background values. Bise
events on the other hand do not show a welldefined correlation. The strong bise event
at the end of the year has a limited effect
most evident at the nearshore headland. At
other stations inside the bay it appears that
it was the short vent pulse at the end of the
year between the two large bise events that
created an increase in diffusion coefficients.
Toward the end of the observations several
428
Lemmin
1984
,
Nov
1984
1985
,
40.
Dee
,
Jan
,
1985
Feb
Colntrlrl
I
::
,I
_-
104
-.
7
3
0) 10
yz
‘o’,02
;
’
’
1984
[
10’
I
100 i
;,
Dec’--
Jan
’
III
I
Feb’
1985
Fig. 6. Time history of eddy diffusion coefficients
calculated by the integral time-scale method for 3-d
periods with a l-d shift. Shown are the coefficients for
the near-bottom instruments at stations SS3 (75-m water
depth; 70-m instrument depth), SS5 (75 m; 70 m), and
SS6 (35 m; 29 m). Further details given in legend of
Fig. 5.
11
t-1
f
Nov
1984
1985
Fig. 5. Time history of eddy diffusion coefficients
for the near-surface instruments calculated by the integral time-scale method over consecutive 3-d periods
with a 1-d shift. Given at the top arc the wind-squared
components, calculated as u x (uz + v’)“? and v x (u2 -t
vz)” and averaged over consecutive 3-d periods shifted
by 1 d, from Cointrin airport near Geneva. Directions
of bise and vent winds shown in Fig. 1. Alongshorc
components positive to the east-solid
lines; crossshore components positive to the north-dashed
lines.
bise events occurred, but currents and therefore also diffusion coefftcients inside the bay
Sal1off.
The coefficients from the near-bottom instruments at SS3, SS5, and SS6 are shown
in Fig. 6. Values there are generally lower
than those at the same stations at 1O-m depth
but may reach the same order of magnitude.
The same tendencies with respect to forcing
and station location that were observed near
the surface are found again. Near the bottom, periods of currents below the instrument threshold (2 cm s-l) are more frequent
and more extensive, particularly inside the
bay and toward the end of the recording
period when bise events dominated wind
history. No mixing coefficients could be obtained for those periods.
For a comparison of the two methods, the
same type of calculation (3-d periods with
1-d shift) was also carried out with the mixmg length method. The dynamics (Fig. 7)
resemble those calculated with the integral
time-scale method above. As was already
seen for the long-term coefficients, turbulent
mixing coefficients derived from the mixing
length method are higher. Although the difference in general is a factor of about two,
it rnay reach a factor of four for the peak
values at station SS3 shown here. Near the
shore at the headland (SSl and SS2) the
difference in peak values may be even higher.
Both components of the coefficients are of
comparable magnitude.
429
Horizontal mixing in a nearshore zone
30-i
'
Nov
' g,84
Dee
1
Ian
1g,85
Feb
,
KH=u’*-T
0
50
days
100
Fig. 7. Time history of eddy viscosity coefficient (a) calculated by the mixing length method and eddy
diffusion coeffkient (b) calculated by the integral time-scale method for station SS3. Alongshore components
positive to the east-solid
lines; cross-shore components positive to the north-dashed
lines.
Turbulent length scale -Length
scales
were calculated according to Eq. 4, following the same averaging procedure as used
for the diffusion coefficients. Data have been
assembled for each of the station clusters
along the headland (SSl-SS4) and the bay
(SSS-SS9) as well as the three near-bottom
instruments at SS3, SS5, and SS6. In each
set the analysis was carried out separately
for the alongshore and the cross-shore component. In general, an increase of the diffusion coefficient with length scale is observed (Fig. 8). However the scatter is
significant, as can be seen from the subsequent regression analysis carried out on these
data to verify Eq. 5 with results given in
Fig. 8. The horizontal eddy diffusion coefficient as a function of the turbulent length scale. Data for the
longitudinal and lateral component of the station cluster inside the bay (SSS-SS9) have been combined.
430
Lemmin
Table 2. Relationship between dispersion coeffkient K,, and length scale L. Regression analysis was carried
out for the equation K,, = aL” (Eq. 5). For the calculations, data were grouped together for the station clusters
SSI-SS4-headland,
SS5-SS9-bay,
and near-bottom instruments at SS3, SS5, SS6-bottom.
In each cluster,
alongshore and cross-shore directions were treated separately. Given are the correlation coefficient c between
K,, and L, the constant a, the exponent b, and the standard error SE(b) of the exponent.
CllMCT
Headland
Bay
Bottom
Direction
c
Alongshore
Cross-shore
Alongshore
Cross-shore
Alongshore
Cross-shore
0.47
0.51
0.64
0.69
0.8 1
0.77
‘Table 2. The product-moment
correlation
coefficient is smallest for the headland where
data scatter is largest. The best correlation
is found in the near-bottom layers. The exponent of Eq. 5 for the headland data is
-0.5, which does not correspond to any of
the diffusion regimes. Inside the bay the exponents are close to 1, which indicates dominance of shear-induced diffusion. Near the
bottom, exponents around 1.3 represent inertial subrange diffusion. The constants in
Eq. 5 for the alongshore and cross-shore
components are of the same order of magnitude.
Atmospheric forcing-The data were analyzed for a relationship between the diffusion coefficients and the atmospheric forcing according to Eq. 6. Data were separated
in subsets at the headland (SSl-SS4) and
inside the bay (SSS-SS9) and near the bottom (SS3, SS5, SS6). When all data were
included scatter of the “constant” for each
station cluster was over four decades, clearly
indicating that a general correlation does
not exist. As a correlation between vent
winds and diffusion coefficients was already
apparent from Fig. 5, only vent situations
were treated subsequently. This reduced the
overall scatter to less than two decades.
Generally constants for us -C 0.25 cm s-l
wcrc about twice as large as those for u* >
0.25 cm s-l (Fig. 9). When only data for u*
> 0.25 cm s-l, which corresponds to a wind
speed of - 2 m s- l, were retained, the scatter
dropped to less than one decade. With 6 =
75 m, mean values of the constant and the
standard deviations as given in Table 3 were
obtained for the different clusters. The respective values for each station were close
to those of the whole clusters. In the bottom
--
a
b
SE(b)
323.1
264.4
53.53
46.18
18.21
18.55
0.46
0.55
0.91
0.96
1.32
1.28
0.036
0.039
0.04 1
0.032
0.043
0.046
cluster, station SS5 was omitted because
much smaller values were found there. The
mean value of the constant for the bay cluster and the bottom cluster is close to the
value given by Elder ( 1959) for lateral diffusion in shear flow. For the headland cluster the value of the constant is about three
times that of Elder.
Discussion
The above calculations have shown that
the integral time-scale T and the cycle time
X remain constant for periods > 55 h. In
both methods of calculation the turbulent
mixing coefficients become a function only
of the fluctuating velocity component. In
this lake the amplitude and energy of the
fluctuating component are not constant over
long periods of time (Figs. 3 and 4). It was
documented that the variability of this component and thus of the diffusion coefficient
is linked to meteorological
forcing. Even
though the differences in time are sometimes smaller than a factor of 10 and by the
methods used may not be significant, some
insight into the diffusion dynamic may be
gained by looking at systematic differences
in time and between stations. Support for
this approach comes from the spectra (Fig.
2) .where a significant difference in energy
levels between stations at the headland and
stations inside the bay is apparent. Winds
with long fetch (vent) generate elevated energy levels and diffusion coefficients. Since
the amplitude of the HF component is linked
to the advective component of the flow field
(Fig. 3), some of the dynamics of the turbulence and diffusion coefficients may be
explained by the advective current pattern.
A first analysis of the currents in the study
431
Horizontal mixing in a nearshore zone
,
10-q
0
1
0.2
,
,
,
,
,
0.6
u,
,
,
,
1.0
,
,
,
,
1.4
,
,
,
T
1.8
(cmd)
Fig. 9. Nondimensional
eddy diffusion coefficient as a function of shear velocity. Data were taken from the
station cluster inside the bay (SSS-SS9) and limited to vent situations. The dashed line indicates the value found
in laboratorv shear flow studies (Elder 1959). The solid line represents the mean value of the present data for
shear velocities >0.25 cm s I.
area (Lemmin et al. 1987) has shown that
during vent events a strong alongshore current from the west passes along the headland
and, with somewhat reduced speed, also
through the bay. In these events, currents
follow depth contours. The bay is part of
the large-scale current pattern and a significant exchange of water mass takes place,
leading to “flushing of the bay.” During bise
events when the wind blows from the opposite direction, the current offshore and at
the headland is oriented east to west, is less
steady in time, and is of a smaller amplitude
than during vent events. Frequently a slowly recirculating gyre is set up inside the bay
Table 3. Determination
of the constant in Eq. 6 for
three data subsets: cluster headland comprising stations SS I-SS4, cluster bay (SS5-SS9), and cluster bottom (SS3 and SS6). Calculated are the mean values
(CM) and standard deviations (SD) for vent situations
for U. > 0.25 cm s ’ with 6 = 75 m. Alongshore and
cross-short components are combined.
Cluster
CM
SD
Headland
Bay
Bottom
0.62
0.29
0.25
0.49
0.27
0.17
area during bise events. In this case the bay
is cut off from the large-scale circulation,
and currents no longer follow depth contours. This flow pattern explains why even
during strong bise events smaller amplitude
elevations of the diffusion coefficient are observed at the headland and why little effect
is seen inside the bay area. Thus large-scale
current patterns, which are already different
on the basis of their forcing by winds from
opposite directions, are further modified locally by shoreline topography.
Within this pattern, station SS5 is inside
the bay area. It has the same water depth
as SS3 and SS9, but is has lower energy
levels (Fig. 4) than these two stations. This
effect is also evident when comparing nearbottom measurements. Inside the bay, coefficients can no longer be calculated for extended periods in February and March because of low currents, whereas values at SS3
are still usable.
Stratification seems to influence the mixing coefficients only to a limited extent. Each
vent event is noted in the hypolimnion
during fall when the temperature difference
between the epilimnion (depth, 20 m) and
432
Lemmin
hypolimnion
is between 5 K (mid-November) and 3 K (mid-December). It is not quite
clear to what extent the next bay to the west
affects flow at the headland. Some influence
may be envisioned by comparing stations
SS2 and SS4. Even though they are both in
intermediate water depth and only 1 km
apart, at SS4 (to the west) amplitudes are
lower and the response behavior is not always clear (Fig. 5). The damping effect of
the bay becomes quite evident when comparing SSl and SS2 at the headland with
SS7 and SS6 in the bay-each pair being in
comparable water depth.
The turbulent mixing coefficients in this
coastal zone of Lake Geneva vary between
about 1O4cm2 s-l for wind-induced current
events and 1O2cm2 s-l for quiet background
situations with long-term means of - lo3
cm2 s-l. The eddy diffusion coefficients resulting from the integral time-scale method
are smaller than the eddy viscosity value
resulting from the mixing length method.
This difference compares favorably with
findings from oceanic mixing studies. Mixing coefficients are about a factor of 10 below the coefficients given by Murthy and
Filatov (198 1) for frequencies above inertial
in large lakes. They are also smaller by an
order of magnitude than the Great Lakes
coastal zone edcly diffusion coefficients derived from Lagrangian studies cited by Lam
et al. (1984). For lack of simultaneous Lagrangian measurements, the @value in Eq.
3 cannot be determined. By using the value
of @= 1, the present results can be taken as
the lower limit of the probable diffusion
coefficients. Taking the value of @ = 1.4
given by Schott and Quadfasel (1979) for
the Baltic Sea would not bring my diffusion
coefficients into the range of the large lakes.
Even using the most probable value of p =
4 cited by Hay and Pasquill (1959) would
only bring some peak values during windinduced current events into the large lake
range, leaving mean value and background
coefficients below those from the large lakes.
Thus this coastal boundary layer of Lake
Geneva seems less energetic than those of
larger lakes. Further evidence of this point
is found by comparing the spectra of current
fluctuations given by Murthy and Filatov
(198 1) with those from Lake Geneva. In
Lake Geneva, the spectral energy in that
frequency range is also smaller by a factor
lo., Turbulent mixing coefficients for Lake
Geneva calculated by the method of Murthy and Filatov (Hesselberg formula) remain below those from the large lakes, which
indicates that the differences in the mixing
coefficients are linked to the current structure and are not the result of the different
methods of calculations used.
As had been observed in dye experiments
in the ocean and in large lakes before (Lam
et al. 1984), the diffusion coefficients generally increase with increasing length scale.
Lam et al. concluded (p. 27) that “A linear
increase of the diffusion coefficient with
length scale would be a reasonable approximation for the upper layers.” The scatter
observed in the diffusion diagrams (Fig. 8)
stresses again the variability
of the mixing
field over the 6 months of observation included in the present analysis and the probability of different processes contributing to
mixing simultaneously.
Nevertheless the
exponents found in the bay area and near
the bottom indicate that diffusion in Lake
Geneva is controlled by the same processes
of shear and inertial cascading as in the large
1ak:esand the oceans.
Murthy (1976) found that shear produced
diffusion in the epilimnion and inertial subrange diffusion in the hypolimnion
of Lake
Ontario. In Lake Geneva these exponents
hold also for the time of weak to no stratification, indicating that probably due to the
limited input of wind energy the shear effect
is limited to the upper layers of the lake. In
contrast to Murthy’s results from Lake Ontario, alongshore and cross-shore regression
constants were always of the same order in
Lake Geneva. This can be explained by the
fact that, in contrast to Murthy, larger scales
linked with the advective current field had
been filtered out in the present analysis. The
alongshore coefficients from Lake Ontario
may therefore be dispersion rather than diffusion coefficients. In part the filtering of
the data may also be the reason why length
scales in Lake Geneva are much smaller than
those reported by Mm-thy. On the other
hand, the shorter fetch length due to the
smaller size of Lake Geneva and the topographic constraints on the wind field over
Horizontal mixing in a nearshore zone
this lake generally make for smaller mean
currents which enter into the length scale
calculation. This difference becomes evident from the constant in the equation, also
given in Table 2. For small values of the
diffusion coefficient, the points deviate from
the regression line. In this range the current
meters approach their threshold limit and
spinup and spindown of the rotor may influence results. Omission of these data has
no influence in the regression analysis. Near
the headland of St. Sulpice the regression
analysis did not provide exponents that
would allow identification
of the mixing regime within the diffusion diagram framework. Headlands are known to be areas
where local topography modifies the current
field and processes such as vortex shedding
may play a role. The complexity of this region was documented above by difference
in diffusion coefficients at SS2 and SS4. It
was previously shown that in the area of the
headland of St. Sulpice nonlinear effects are
important (Bohle-Carbonell
and Lemmin
1988).
It would have been desirable to develop
a universal functional relationship between
eddy diffusion coefficients and forcing by
the wind. However as the mixing coefficients are found to be sensitive to wind direction and are influenced by the the largescale current pattern generated by local
shoreline geometry, this seems not to be
meaningful for this study. The parameterization for a vent situation according to Eq.
6 resulted in scatter equal to that observed
by Ottcnsen Hansen (1978) who argued that
this was due to the wrong choice of length
scale. However, particularly for smaller wind
stress, mesoscale flow structures can be expected which may be in different states of
development, resulting in a more complex
Aow field than is specified by the assumptions underlying the theory of fully developed flow applied in the above analysis. In
this case, scatter decreased for larger friction
velocities, indicating that more stable flow
fields became established. A length scale of
6 = 75 m resulted in constants in the clusters
inside the bay and near the bottom that were
close to those found by Elder (1959; const
= 0.23) for diffusion in shear flow. Inside
the bay the correlation between the diffu-
433
sion coefficient and the turbulent length scale
discussed above is explained by shear. In
the study area a (vertical) length scale of 75
m corresponds to the depth of a lateral
boundary layer in which friction is important in the force balance for winds >3 m
s-l. At the headland the constant is almost
three times the value reported by Elder.
Thus, in line with the above observed lack
of correlation between diffusion coefficients
and turbulent length scales, simple theories
again fail to explain mixing in this area.
Conclusion
The present study has been concerned
with horizontal mixing in a nearshore region composed of a headland and a bay. For
time scales < 12 h, the spectra of the currents measured at fixed points were smooth.
The high frequency component with periods < 12 h showed variability in time and
space. An alongshore spatial variability that
was observed over distances of 1 or 2 km
can be attributed to local topography. Deviations of the shoreline contour from a
straight line have a modifying effect on the
amplitude and response dynamics of the
current field. Near the headland, current
fluctuations are more energetic than inside
the bay. This can be linked to the structure
of the advective component which at times
cuts off the bay area from the mean largescale circulation and may form a recirculating gyre there. The variability in time is
due to the sheltering effect of the surrounding topography, which significantly alters the
fetch length and thus the energy input for
winds from opposite directions. The time
history of the cumulative kinetic energy of
the HF component was found to be sensitive to this variation in input of wind energy. This pattern shows that events of wind
energy input do not follow each other in
sufficiently rapid succession to provide the
lake system with enough steady energy to
allow a continuous, stationary cascading of
energy from large-scale to small-scale motion. Consequently individual
events become important in the study of the mixing
dynamics.
Long-term mean diffusion coefficients, 0
( lo3 cm2 s-l), are an order of magnitude
smaller than those given by Murthy and Fi-
434
Lemmin
latov (198 1) for large lakes for periods < 12
h and those obtained from tracer studies by
Lam et al. (1984). However, as was shown
by the calculation for subsections of 3-d duration, short-term mixing coefficients may
vary over a range of a decade around the
long-term mean values, and long-term mean
coefficients may not characterize the actual
situation. In addition, there are spatial variations resulting from bottom and shoreline
topography
Despite this variability,
regression analysis of the correlation between the diffusion
coefficient and the turbulent length scale
showed that mechanisms for generating
mixing inside the bay are the same as observed in large lakes and oceans. Near the
surface, diffusion is caused by shear; near
the bottom, inertial
subrange diffusion
dominates. The situation near the headland
is too complex to be described by this relatively simple theory. The importance of
shear in generating mixing was further evidenced by an analysis between wind forcing
and the diffusion coefficient. For events of
long fetch and strong winds the correlation
coefficient for stations inside the bay was
relatively close to that expected for shear
flow. Again, at the headland no correlation
was found. This suggests that at the headland local topographic effects are important
in generating mixing.
In such a nonstationary
(long term) situation, selection of a particular length of
record may influence the results. Realizing
this sensitivity of turbulent mixing coefficients to local eflects, it becomes evident
that in assessing the mixing potential of a
shore zone, care must be taken that the mixing coefficients applied are representative of
the situation and site in question.
References
BOHLE-CARBONELL, M., AND U. LEMMIN. 1988. Observations on non-linear current fields in the Lake
of Gcncva. Ann. Geophys. 6: 89-100.
BOWDEN, K.F.,D.P.
KRAUEL,AND R.S. LEWIS. 1974.
Some features of turbulent diffusion from a continuous source at sea. Adv. Geophys. MA: 315329.
CSANADY, G. T. 1973. Turbulent diffusion in the environment. Reidel.
ELIJER, J. 1959. The dispersion of marked fluid in
turbulent shear how. J. Fluid Mech. 5: 544-560.
GANTER, Y. 1978. Contribution
a l’etude des brises
du Lac Leman. Rapp. Inst. Suisse Meteorol. 83.
43 p.
HAY, J. S., AND F. PASQUILL. 1959. Diffusion from a
continuous source in relation to the spectrum and
scale of turbulence. Adv. Geophys. 6: 345-365.
HINZE, J. 0. 1975. Turbulence. McGraw-Hill.
LAM, D. C. L., C. R. MURTHY, AND R. B. SIMPSON.
1984. Effluent transport and diffusion models for
the coastal zone. Springer.
LE~,IMIN, U., W. H. GRAF, AND C. PERRINJAQUET.
1987. Les courants dans une couche littorale du
LKman: la zone de Vidy. Ing.-Arch. Suissc 113:
272-280.
MURRAY, S. P. 1975. Trajectories and speeds of winddriven currents near the coast. J. Phys. Oceanogr.
5: 347-360.
MURTHY, C. R. 1976. Horizontal
diffusion characteristics in Lake Ontario. J. Phys. Oceanogr. 6: 7684.
-, AND D. S. DIJNBAR. 1981. Structure of the
flow in the coastal boundary layer of the Great
Lakes. J. Phys. Oceanogr. 11: 1567-1577.
-AND N. N. FILATOV. 198 1. Variability of currents and horizontal
turbulent exchange coefficients in Lakes Ladoga, Huron, Ontario. Occanology 21: 322-325.
--,
T. J. SIMONS, AND D. C. L. LAM. 1986. Simulation of pollutant transport in homogcncous
coastal zones with application to Lake Ontario. J.
Geophys. Res. 91: 9771-9779.
OKUBO, A. 197 1. Oceanic diffusion diagrams. DeepSea Res. 18: 789-802.
OT~ENSEN HANSEN, N. E. 1978. Mixing processes in
lakes. Nord. Hydrol. 9: 57-74.
POND, S., AND G. PICKARD. 1986. Introductory
dynamical oceanography. Pergamon.
SC~IOTT, F., AND D. QUADFASEL. 1979. Lagrangian
and Eulerian measurements of horizontal mixing
in the Baltic. Tellus 31: 138-144.
SHTOKMAN, V. B. 1970. Selected works on physics
of the sea. Gidrometeoizdat,
Leningrad.
Submitted: 23 October 1987
Accepted: I5 September 1988
Revised: 10 January 1989