Content uploaded by Youyu Lu
Author content
All content in this area was uploaded by Youyu Lu on Jul 18, 2014
Content may be subject to copyright.
M
AY
2000 855LU ET AL.
q 2000 American Meteorological Society
Turbulence Characteristics in a Tidal Channel
Y
OUYU
L
U
,* R
OLF
G. L
UECK
,
AND
D
AIYAN
H
UANG
School of Earth and Ocean Sciences, University of Victoria, Victoria, British Columbia, Canada
(Manuscript received 28 January 1999, in final form 3 May 1999)
ABSTRACT
A broadband ADCP and a moored microstructure instrument (TAMI) were deployed in a tidal channel of
30-m depth and with peak speeds of 1 m s
21
. The measurements enable us to derive profiles of stress, turbulent
kinetic energy (TKE), the rate of production and dissipation of TKE, eddy viscosity, diffusivity, as well as
mixing length, and to test the parameterization of dissipation rate in the model of Mellor and Yamada. At
middepth in the channel where the influence of stratification was present, the Ellison length agrees with the
Ozmidov length. The measured mixing length is smaller than the simple z-dependence formulation proposed
for unstratified turbulence. The diffusivity of density and heat, and the viscosity for momentum, are correlated
and comparable in magnitudes. The 20-min averaged production rate deduced from the ADCP agrees with the
dissipation rate estimated from microstructure measurements. The dissipation rate calculated with the Mellor–
Yamada model agrees with the measured values with TAMI, but the empirical constant B
1
derived from the data
is larger than that conventionally used in the model. In the near-bottom layer, there is a tight correlation between
the production rate and the closure-based dissipation rate. The Reynolds stress at 3.6 m above the bottom is
consistently 2.5 times smaller than the shear velocity squared ( ), which is inferred from fitting the velocity
2
u
*
profiles to a logarithmic form. A logarithmic velocity profile almost always exists and reaches heights of 5.6
to 20 m, but the Reynolds stress is seldom constant in any part of the logarithmic layer.
1. Introduction
Our ability to predict the behavior of coastal envi-
ronments depends largely on our understanding of the
flow and mixing processes. Deriving the flow and tur-
bulence characteristics from measurements is important
for understanding coastal dynamics and the develop-
ment of numerical models. Turbulence measurements
are few compared to the vast pool of mean flow data.
The important turbulent quantities of practical interest
are the frictional force on the flow, turbulence intensity,
and various coefficients describing the mixing of mo-
mentum and scalars. Turbulent quantities undergo com-
plicated variations in space and time. A turbulent bound-
ary layer is formed above the seabed by bottom friction.
Within the boundary layer the flow is attenuated, the
shear and frictional force are enhanced, and theturbulent
kinetic energy (TKE) production is intensified. The
height of the boundary layer is proportional to the scale
of turbulent velocity (e.g., Bowden 1978), and can ex-
* Current affiliation: Department of Oceanography, Dalhousie Uni-
versity, Halifax, Nova Scotia, Canada.
Corresponding author address: Rolf Lueck, School of Earth and
Ocean Sciences, University of Victoria, P.O. Box 1700, 3800 Finnerty
Road, Victoria, BC V8W 2Y2, Canada.
E-mail: rlueck@uvic.ca
tend over the whole water depth in shallow seas (Souls-
by 1983). It is generally believed that the structure of
the oceanic boundary layer bears many similarities to
that in atmospheric and laboratory flows. More evi-
dence, particularly from the oceanic boundary layer, is
required to convincingly establish this analogy.
In general, numerical models need to parameterize
turbulence, partly due to the constraint of computers
and partly due to the classical problem that the equations
for turbulent moments are not closed. Turbulent closure
schemes are commonly based upon scaling arguments
and contain constants that must be determined from
measurements. Mellor and Yamada (1974, 1982) pro-
posed an hierarchy of turbulent closure models for geo-
physical boundary layer flows, and their level-2.5 ver-
sion has been implemented in practical modeling of
coastal water circulation (e.g., Blumberg and Mellor
1987; Lynch et al. 1996). The feasibility of this closure
scheme and the values of the empirical constants need
to be tested by oceanic measurements. Whereas tur-
bulent parameterization can be indirectly tested by the
ability of a model to reproduce the mean flow field, a
more critical test is the ability to describe the depth
dependence and time evolution of turbulence (Simpson
et al. 1996).
In this paper, we present an analysis of the turbulent
quantities from measurements in a tidal channel with
flow magnitudes that are typical for coastal waters. The
856 V
OLUME
30JOURNAL OF PHYSICAL OCEANOGRAPHY
F
IG
. 1. Duration of measurements made in Cordova Channel with
various instruments.
F
IG
. 2. Stick diagrams of the 20-min mean velocity measured by
the ADCP and averaged over the profiling range. Open circles in
panel (c) mark the time of the 17 density profiles shown in Fig. 5.
data describe the vertical variation and temporal evo-
lution of turbulence in a tidal boundary layer and are
used to test a major parameterization in the Mellor–
Yamada model, the closure for dissipation rate. In sec-
tion 2, we describe the experiment and background mea-
surements made in the study area. In section 3, we in-
troduce the turbulence measurements made with a
broadband acoustic Doppler current profiler (ADCP)
and a moored microstructure instrument. Section 4 pro-
vides a brief account of turbulence parameterization. In
section 5, we describe the variations of turbulence
throughout the water column based on measurements
with the ADCP. Sections 6 and 7 present a quantitative
analysis of turbulent characteristics at middepth and in
the near-bottom layer, respectively. Section 8 is a sum-
mary of the results of this study.
2. Study area and experiment
Cordova Channel is a side channel among a series of
narrow passages that link Juan de Fuca Strait to the
Strait of Georgia (between Vancouver Island and the
mainland of North America). There is a substantial es-
tuarine circulation in this area due to the runoff from
several rivers, of which the Fraser River is the largest.
The tidal flow is strong in this area and the mixing that
it generates in channels and narrow passages influences
the estuarine circulation (e.g., Thompson 1981; Fore-
man et al. 1995).
A multi-investigator experiment in Cordova Channel
was conducted in the early fall of 1994 from 19 to 30
September. Most of the data analyzed in this paper are
from instruments deployed in the narrowest part of the
channel, where it is about 1 km wide and 30 m deep
(Lu and Lueck 1999a, Fig. 1). The eastern boundary
(James Island) is smoothly curved. The western side is
smooth and straight south of the measurement site but
the channel broadens northward due to Cordova Spit
and Saanichton Bay. The influence of coastline curva-
ture has been noted previously and will be discussed
further in this paper.
The two instruments deployed by our group are a
600-kHz broadband ADCP andthe tethered autonomous
microstructure instrument (TAMI) (Lueck et al. 1997;
Lueck and Huang 1999). Figure 1 summarizes the du-
ration of all measurements.
Throughout the experiment, the wind speed was typ-
ically less than3ms
21
and reached 5 m s
21
only oc-
casionally. The wind stress on the sea surface was neg-
ligible compared to the frictional stress at the bottom.
The water surface was calm and the wave heights did
not exceed 0.2 m during the experiment, according to
our visual observation.
A multibeam survey indicates a smooth bottom along
the axis of the channel with random undulations of less
than 0.1 m peak to peak. The ADCP was at a ‘‘high’’
point in the channel and the bottom slope was 0.015
and 0.01 to the south and north, respectively, for a dis-
tance of 200 m. There are small ripples of amplitude
0.1 m at 200 m on either side of the ADCP. The cross-
channel slope on the west side of the channel (between
isobaths of 15 and 35 m) has many crevices that are
irregularly spaced, about 1 m deep and 20 m wide.
Divers reported that the bed was composed mainly of
fine gravel with diameters ranging from 2 to 8 (310
23
m), and the bed contained neither mud nor silt.
The flow in the channel was mainly tidal, directed
northward during the flood and southward during the
ebb (Lu and Lueck 1999a). During the experiment, the
tide changed from spring to neap, and the diurnal con-
stituents became increasingly dominant. Figure 2 pro-
vides the time variations of the depth-mean flow for the
3.8 days of ADCP measurements. During these three
intervals, the ADCP collected velocity profiles every 3
seconds. An asymmetry between ebb and flood was ob-
served (Lu and Lueck 1999a). The ebb tide was com-
plicated due to the influence of Cordova Spit and the
shallows of Saanichton Bay. During the ebb, there were
M
AY
2000 857LU ET AL.
F
IG
. 3. Consecutive profiles of (
s
t
, in units of kg m
23
) collected
at the south end of Cordova Channel during the interval indicated
by open circles in Fig. 2. Successive profiles are shifted by 0.3 kg
m
23
to the right.
frequent reversals of streamwise shear, fluctuations of
flow direction, strong secondary circulation, and trans-
verse shear. During the flood, the flow above the ADCP
was nearly aligned with the channel axis and it was only
weakly affected by the curvature of the western shore
and the shallows.
CTD profiles were taken nominally every 20 min dur-
ing the period shown in Fig. 1, from the CSS Vector
anchored near the south entrance of the channel (about
1.5 km to the south of the ADCP). Figure 3 shows 17
consecutive density (
s
t
) profiles over one-half semidi-
urnal tidal period from day 29.1 to 29.4 The stratifi-
cation varied with time and it correlated with the var-
iation of flow strength (cf. Fig. 2c). During strong flows,
the density gradient was larger above than below mid-
depth, and unstable overturns were observed during the
ebb. During slack current the whole water column was
stratified. The effects of stratification on turbulence will
be examined in section 6, using the density gradient
measured by CTDs mounted on TAMI.
3. Turbulence measurements
a. ADCP
The ADCP was mounted in a quadripod on the sea-
floor and the steady readings from its tilt sensors indicate
that the instrument remained motionless during the ex-
periment. The ADCP measured velocities along its four
inclined beams, and the data were transferred via a cable
to a computer on shore. About 3.8 days of rapidly sam-
pled velocity data were collected with the standard
working mode (mode 4). The data span a range of 25
m [3.6 to 27.6 mab (meters above bottom)] with a ver-
tical resolution of 1 m.
The along-beam velocities, denoted by (b
1
, b
2
, b
3
,
b
4
), are low-pass filtered at a cutoff period of 20 min
to separate the mean (tidal) and turbulence components.
The difference and sum of the turbulent components,
namely ( , , , ), provide estimates of two-com-
b9 b9 b9 b9
1234
ponents of the Reynolds stress and a quantity Q. The
formulas used for the calculation are
1
2222
(u9w9,
y
9w9) 5 (b92b9 , b92b9 ), (1)
1234
2 sin
u
1
2222
Q 5 (b91b91b91b9
1234
2
4 sin
u
2 4D), (2)
where
u
5 308 is the beam inclination angle and D is
the bias in the variances of the along-beam velocities
due to Doppler noise. The combination of the Reynolds
stress and mean shear provides estimates of the TKE
production rate
]u ]
y
P 52 u9w91
y
9w9 . (3)
12
]z ]z
The quantity Q is related to the TKE density q
2
/2 5
(u9
2
1
y
9
2
1 w9
2
)/2 by
Q 5
g
q
2
/2, (4)
where the factor
g
5 (1 1 2
a
tan
22
u
)/(1 1
a
)isde-
termined by the anisotropy,
a
5 w9
2
/(u9
2
1
y
9
2
). The
value of
g
ranges from 1 to 2.7, corresponding to
a
5
0 (extremely anisotropic turbulence) to
a
5 0.5 (iso-
tropic turbulence). In this analysis we use
g
5 1.8, hence
a
5 0.2, which is the value estimated by Stacey (1996)
from measurements in an unstratified tidal channel.
This technique of estimating turbulent quantities with
the variances of ADCP velocities has been reported by
Lohrmann et al. (1990), Stacey (1996), Lu (1997), and
Lu and Lueck (1999b). Stacey et al. (1999) pointed out
that the profiling resolution must be smaller than the
sizes of the energy-containing eddies. Estimates of the
turbulence length scales are required to determine if this
is true.
b. The moored instrument
The moored microstructure instrument TAMI was de-
ployed twice during the experiment at a nominal depth
of 15 m. The turbulent velocity and temperature fluc-
tuations were measured, respectively, by shear probes
and fast thermistors mounted on TAMI. The TKE dis-
sipation rate,
e
, were estimated by fitting the velocity
spectra to the theoretical spectra in the inertial subrange
(Huang 1996; Lueck and Huang 1999). The temperature
spectra
c
(k) provide estimates of the weighted-mean
temperature spectral level
5/3 1/3
z
5
c
(k)k
e
, (5)
where the overbar denotes an average over the inertial-
convection subrange. The dissipation rate of tempera-
ture fluctuation variance, 2
x
, is related to
z
by
x
5
z
/
b
, (6)
where
b
is a constant. This constant ranges between
0.35 to 1.15 (e.g., Gargett 1985). Following Edson et
al. (1991), we choose
b
5 0.79 in this study.
858 V
OLUME
30JOURNAL OF PHYSICAL OCEANOGRAPHY
4. Turbulence parameterization
a. Mixing coefficients and length scales
The effects of turbulence in transferring momentum
and in mixing scalers are usually parameterized by in-
troducing viscosity and diffusivity coefficients. In this
study, the measurements with the ADCP and TAMI en-
able us to get three estimates of the vertical viscosity
and diffusivity coefficients, namely
P G
e
2
x
r
T
A 5 , K 5 , K 5 , (7)
yy y
22 2
SN(]T/]z)
where A
y
is the vertical eddy viscosity and and
r
T
KK
yy
are, respectively, the vertical diffusivity for density and
heat. The profiles of A
y
are derived from the production
rate and mean shear (S) measured with the ADCP. The
time series of and are derived from the quantities
r
T
KK
yy
measured with TAMI, where N is the buoyancy fre-
quency, G is the mixing efficiency, and ]T/]z is the mean
vertical gradient of temperature. In oceanic environ-
ments, G varies between 0.04 and 0.4 (Peters et al. 1995)
and is typically #0.2 (Osborn 1980). The estimates of
are based on assuming a local balance between the
T
K
y
rates of dissipation of temperature fluctuation variance,
2
x
, and its production (Osborn and Cox 1972).
The turbulent length scales derived from the mea-
surements are the mixing length
1/2
P
l 5 (8)
m
3
12
S
and the Ozmidov scale
1/2
e
l 5 . (9)
O
3
12
N
According to Stacey et al. (1999), the mixing length l
m
is related to the Ellison length l
E
(a scalar analogy to
l
m
)byl
E
5 3l
m
, and l
E
is argued to be the characteristic
length of turbulent eddies. In stratified flow, the Oz-
midov length, l
O
, characterizes the largest possible over-
turn that turbulence can accomplish (Turner 1973);
hence l
O
sets an upper limit on l
E
. Determining the char-
acteristic length scale of the TKE-containing eddies is
also important because the ADCP averages the velocity
vertically over 1 m. The variances of the measured ve-
locity fluctuations may be reduced if the size of the
TKE-containing eddies is smaller than O(1 m).
In unstratified wall-bounded turbulent flows the mix-
ing length is proportional to the distance from the wall.
In shallow waters, the growth of the eddies is con-
strained by the presence of both the seabed and surface.
A simple z-dependent mixing length, l
z
, is sometimes
(e.g., Simpson et al. 1996) proposed as
1/2
z
l 5
k
z 1 2 , (10)
z
12
h
where
k
5 0.4 is von Ka´rma´n’s constant, h is the total
water depth and z is the height above the seabed.
b. The Mellor–Yamada closure model
In an hierarchy of models proposed by Mellor and
Yamada (1974, 1982), the turbulent viscosity and dif-
fusivity coefficients are parameterized in terms of the
turbulent intensity (q) and a master length scale (l), that
is,
(A
y
, K
y
,)5 (S
m
, S
h
, S
q
)lq,
q
K
y
(11)
where is the diffusivity for TKE, K
y
the diffusivity
q
K
y
for tracers, and (S
m
, S
h
, S
q
) are three stability functions.
The Mellor–Yamada level-2.5 model carries the gov-
erning equation for TKE density, namely
22 2
] qq]]q
q
1 u · = 2 K
y
12 12 12
[]
]t 22]z ]z 2
5 P 2
e
2 B. (12)
Besides the rates of production (P) and dissipation (
e
),
the additional term on the right-hand side of (12) is the
rate of loss of TKE to buoyancy (B). Conventionally,
P and B are parameterized by the local vertical shear
and density gradient (P 5 A
y
S
2
, B 5 N
2
). The rate
r
K
y
of dissipation is parameterized in the Mellor–Yamada
model by
3
q
e
5 , (13)
MY
Bl
1
where B
1
is an empirical parameter.
The stability functions were formulated by Galperin
et al. (1988) as
(g 2 gG) g
23h 6
S 5 , S 5 ,
mh
(1 2 gG)(1 2 gG)(12 gG)
4 h 5 h 4 h
S 5 0.2,
q
(14)
where
2
l
2
G 52 N , (15)
h
2
q
and g
2
,...,g
6
are empirical constants. The values of
these empirical constants were determined by appealing
to data from the laboratory and the atmosphere under
neutral conditions (Mellor and Yamada 1982). The val-
ues cited by Galperin et al. are g
2
5 0.393 27, g
3
5
3.0858, g
4
5 34.676, g
5
5 6.1272, g
6
5 0.493 93, and
B
1
5 16.6.
The Mellor–Yamada closure of the TKE dissipation
rate,
e
MY
, and also the stability functions S
m
and S
h
, can
be calculated using q
2
measured with the ADCP, pro-
vided that the master length (l) is known. Let us first
examine how l is linked to the mixing length in a special
case. Defining L5q
2
/|u9w9|, we can rewrite (11) as
M
AY
2000 859LU ET AL.
F
IG
. 4. Depth–time sections of the 20-min mean local friction velocities (a) u
*
s
, (b) u
*
n
(m s
21
), (c) log
10
P (m
2
s
23
), (d) q (m s
21
), and (e) log
10
A
y
(m
2
s
21
). The blank areas in (c), (d)
and (e) represent negative values. The black (white) curves in panel (a) indicate the height of the log-layer during flood (ebb).
860 V
OLUME
30JOURNAL OF PHYSICAL OCEANOGRAPHY
1 q
l 5 (16)
S L S
m
and (13) as
S
m
2
e
5LqS. (17)
MY
B
1
If we choose l 5 l
m
, then from (16) S
m
is simply
S
m
5L
21/2
. (18)
In the case that the TKE budget (12) is reduced to a
balance between the three terms on the right-hand side
of (12), that is,
P 5
e
1 B 5 (1 1G)
e
, (19)
then combining (17) and (18) gives
B
1
5 (1 1G).
23
S
m
(20)
In the absence of stratification G
h
5G50; hence S
m
5 g
2
5 0.393 27 according to (14) and B
1
5 16.4 ac-
cording to (20). Hence, the values for g
2
and B
1
cited
by Galperin et al. (1988) are consistent with assuming
l 5 l
m
and that dissipation balances production in un-
stratified flow. Stacey et al. (1999) argued that the equiv-
alence of l and l
m
may also apply in stratified turbulent
boundary layers. Following them, we shall set l 5 l
m
to calculate
e
MY
and S
m
in this analysis. Itis worth noting
that according to (20), B
1
is not an universal constant,
but rather it varies with changes in S
m
and G.
5. Depth–time variations of turbulence in the
channel
Figure 4 shows the depth-time sections of the 20-min
mean estimates of turbulent quantities from measure-
ments taken with the ADCP. Plotted in panels (a) and
(b) are the ‘‘local friction velocities’’
(2u9w9)(2
y
9w9)
sn
u* 5 , u* 5 , (21)
sn
1/2 1/2
|(2u9w9)| |(2
y
9w9)|
sn
where (2u9w9)
s
and (2
y
9w9 )
n
are the along- and cross-
channel components of the Reynolds stress (the along-
and cross-channel directions are defined as parallel and
normal to the depth-mean flow, respectively). During
the flood, the alongchannel stress is positive (warm
shading) and decreases with increasing height. During
the ebb, the alongchannel stress is negative (cold shad-
ing) and its magnitude also decreases with increasing
height, but only in the lower half of the water column.
Above middepth, the alongchannel stress frequently re-
verses sign, corresponding to the sign reversals of the
streamwise shear. The cross-channel stress is small dur-
ing the flood and large during the ebb. This ebb–flood
asymmetry of the cross-channel stress reflects the asym-
metry of the transverse shear, which is related to vari-
ations in the strength of the transverse flow (Lu and
Lueck 1999a). The extremely large stress estimates, ob-
tained during the turning of the tide, are unreliable be-
cause these estimates are dominated by single events
possibly associated with the large horizontal eddiesshed
from Cordova Spit (Lu and Lueck 1999b).
The rate of TKE production [panel(c)] intensifies to-
ward the seabed, which is a characteristicof wall-bound-
ed turbulence. However, during the ebb, events of large
production occur at heights above the log-layer. These
events correspond to the sign reversal of stress (and
shear) above middepth and are caused by the entrain-
ment of slower water from the shallows of Saanichton
Bay (Lu and Lueck 1999a,b). Negative estimates of P
(blank areas) are either due to round-off (and are usually
small), or they are caused by unreliable stress estimates
obtained during the turning of the tide. The magnitude
of P spans about three decades, ranging from 10
24
m
2
s
23
(W kg
21
) near the bottom to 10
27
m
2
s
23
during
weak flows.
Panel (d) shows the turbulent intensity q, calculated
by (4) from the estimates of Q. The Doppler noise level
(D) is assumed to be uniform and equal to 1.25 3 10
24
m
2
s
22
, as determined from tests in an inlet with very
weak flows (Lu and Lueck 1999b). The Doppler noise
should to some extent depend upon the abundance of
sound scatters, which may vary with sites and flow con-
ditions. In this analysis, some negative estimates of q
[the blank areas in panel (d)], obtained during weak
flows, indicate that the Doppler noise may not have
always been as large as 1.25 3 10
24
m
2
s
22
. The TKE
decreases with increasing height, similar to the stream-
wise friction velocity. The largest estimates of q are
obtained at the beginning and end of the ebb and the
smallest ones are obtained during weak flows.
The eddy viscosity coefficient A
y
[panel (e)] is cal-
culated by dividing P with the squares of the shear (7).
The variations of A
y
range from about 10
23
m
2
s
21
dur-
ing weak flows to 0.3 m
2
s
21
during strong flows. The
eddy viscosity increases with increasing height in the
lower half of the water column and reaches a maximum
near middepth.
The white curve in panel (a) depicts the height of the
log-layer obtained by fitting the streamwise velocity
profiles to a logarithmic form with 1% accuracy (Lueck
and Lu 1997). During strong flows, the top of the log-
layer reaches more than half way to the surface. The
height is predicted well by 0.04u*/
v
, where u* is the
friction velocity derived from log-layer fitting, and
v
is
the angular frequency of the dominant tidal constituent,
M
2
.
6. Turbulence characteristics at middepth
Both the ADCP and TAMI took measurements at
middepth in the channel. The two instruments wereapart
by about 50 and 100 m during the first and second
deployments of TAMI, respectively. The vertical dis-
placement of TAMI was less than 61 m (Huang 1996).
For the analyses in this section, the measurements from
TAMI are averaged into 20-min ensembles, and the
M
AY
2000 861LU ET AL.
F
IG
. 5. Estimates of the gradient Richardson number (Ri) at mid-
depth using N
2
measured with TAMI and shear measured with the
ADCP. Each open circle represents a 20-min average.
F
IG
. 6. Time variations of the Ellison length (open circles) and the
Ozmidov length (solid lines with crosses) for the two deployments
of TAMI. The dashed line depicts 3l
z
, where l
z
is the z-dependent
mixing length calculated with (10). The quantities are estimated at
middepth.
F
IG
. 7. The rates of TKE dissipation,
e
, measured with TAMI (solid
lines with crosses) vs production, P, (open circles) measured with
the ADCP at middepth.
quantities estimated with the ADCP are averaged over
three levels near middepth over the same 20-min inter-
vals.
According to CTD profiling near the south entrance
of the channel (Fig. 3), sharp density gradients occurred
mostly near and above middepth during strong flows.
At middepth, the moored instrument TAMIcarried three
CT sensors which the outer pair spaced vertically by 3
m. No shear estimates were available at the site of
TAMI; however, we estimate the gradient Richardson
number (Ri) by dividing N
2
at TAMI by the shear
squared above the ADCP (averaged over 3 m at mid-
depth). A total of 3.2 days of Ri, each value representing
a 20-min ensemble mean, are shown in Fig. 5. During
the flood, 66% (31%) of the Ri values were greater (less)
than ¼, and 3% were negative (indication of overturns).
During the ebb, 56% (34%) of the Ri values were greater
(less) than ¼, and 10% were negative. Note that a neg-
ative Ri does not mean that the stratification was un-
stable for the entire 20 minutes represented by a datum.
Hence, at middepth, the water column was stable more
often than it was unstable during the flood, whereas the
chances of stability and instability were roughly equal
during the ebb. These statistics of Ri indicates that the
influence of stratification cannot be excluded at the mid-
depth.
Figure 6 shows the time series of the Ellison length
(l
E
5 3l
m
) and the Ozmidov length (l
O
). Both length
scales vary significantly with time. The magnitudes of
l
E
and l
O
are comparable. For the total of 3.8 days of
data, 34% of the l
E
values are negative (due to negative
production rate P), the remaining 66% are all greater
than 1 m, while 51% are greater than 3 m. Following
Stacey et al. (1999), if we take l
E
as the characteristic
length scale of TKE containing eddies, then all the data
points shown in Fig. 8 correspond to eddies with scales
larger than 1 m, the vertical resolution of the ADCP.
Hence, at middepth, there should not be any reduction
of the beam velocity variances due to the vertical av-
eraging of the ADCP. Consequently, we anticipate that
the turbulence products are not underestimated at mid-
depth.
At middepth, l
E
is generally smaller than 3l
z
. Hence,
l
m
is smaller than the mixing length l
z
5 4.3 m (10)
based on geometric considerations alone. The reduction
of the mixing length from the simple z-dependent for-
mulation can be explained by the influence of stratifi-
cation, which tends to inhibit the growth of turbulent
eddies.
Figure 7 compares the estimates of
e
from TAMI
against the estimates of P from the ADCP. The time
862 V
OLUME
30JOURNAL OF PHYSICAL OCEANOGRAPHY
F
IG
. 8. Scatter diagram of the rates of TKE dissipation vs production at middepth. Panels (a)
and (b) present the data from the first and second deployments of TAMI, respectively. Solid line
denotes a ratio of 1 and dashed lines represent ratios of 5 and 1/5 between the two quantities.
F
IG
. 10. (a) The TKE density q
2
/2 (heavy solid lines) vs stress
magnitude u9w9 (thin solid lines with crosses); (b) The stability func-
tion S
m
calculated using (18) (solid line) and (14) (crosses) vs S
m
5
0.393 27 (dashed line); (c) the stick diagram of the flow. All quantities
are estimated at middepth.
F
IG
. 9. Time variations of the diffusivity for (a) density ( ) and
r
K
y
(b) temperature ( ) (both denoted by solid lines with crosses) com-
T
K
y
pared against the vertical eddy viscosity A
y
(open circles, bothpanels).
Panel (c) shows the 20-min flow. All quantities are estimated at mid-
depth.
variations of
e
correlate well with those of P. The peak
values of both rates are about 2 3 10
25
m
2
s
23
(W kg
21
),
and the minimum values are about 2 3 10
28
m
2
s
23
.
Note that during the ebb from day 24.3 to 24.6, both
e
and P increased by a factor of 30 when the flow direction
fluctuated compared to their values between the inter-
vals of fluctuation when the direction was steady (see
the stick diagram in Fig. 1). The agreement between the
two rates is slightly better during the first deployment
of TAMI, probably because the two instruments were
closer together. Figure 8 shows a scatter plot of P versus
e
. During the first deployment of TAMI [panel (a)],
almost all of the
e
values agree with P within a factor
of 5, and this is also true for the majority of the
e
values
from the second deployment [panel (b)]. The difference
between the estimates of P and
e
reflects statistical var-
iations more than it does the separation between the two
instruments (e.g., Moum et al. 1995). The agreement is
remarkable, considering the two rates are obtained with
two completely different instruments using very differ-
ent sensors.
Figure 9 compares the vertical eddy viscosity coef-
ficient A
y
measured with the ADCP against the two
diffusivities obtained with TAMI: (a) for density and
r
K
y
(b) for heat. The values of are calculated with
T
r
KK
yy
(7) and using G50.2. The agreement between A
y
and
is very good and both ranged between 10
23
and 1
r
K
y
m
2
s
21
. The correlation between A
y
and is generally
T
K
y
good except that occasionally has spikes of up to 10
T
K
y
m
2
s
21
, due to small mean temperature gradients.
Variations of the TKE density, q
2
/2, and the magni-
tude of the Reynolds stress |u9w9| at middepth are shown
in Fig. 10a. There is a clear correlation between the
variations of the two quantities. The mean q
2
to |u9w9|
ratio, L, is 12.1 6 0.9. Two estimates of the stability
M
AY
2000 863LU ET AL.
F
IG
. 11. Scatter diagram of the TKE dissipation rate measured with
TAMI against q
3
/l
m
measured with the ADCP. The factor of propor-
tionality is B
1
5 46.6.
F
IG
. 12. The rates of TKE dissipation,
e
, measured with TAMI
(solid lines with crosses) vs the dissipation,
e
MY
, calculated using (13)
with l 5 l
m
and B
1
5 46.6 (crosses).
function S
m
, one calculated with Eq. (14) (assuming l
5 l
m
) and the other with (18), are shown in Fig. 10b.
Due to stratification, both estimates of S
m
are less than
0.393 27, the value in unstratified flow. The mean value
of S
m
, calculated using (18), is 0.287. The extremely
large and small values near day 24.6 and 24.75 are
unreliable because they occur just after the start and just
before the end of the weak flood when the turbulence
was not stationary.
Figure 11 shows a scatter diagram of
e
measured with
TAMI against q
3
/l
m
measured with the ADCP. A straight
line is fitted to the points with the least squares method.
The constant of proportionality is B
1
(13), the value
required to match the closure-based rate of dissipation
e
MY
to the rate
e
derived from TAMI. The mean value
of B
1
is 46.6 and it has a 95% confidence interval of
64.6, which was obtained using a bootstrap method.
The estimate of B
1
may be biased either high or low
depending on the degree of isotropy (4). If the anisot-
ropy is even greater than
a
5 0.2, then B
1
is larger than
46.6 and hypothetically as large as 112 for the extreme
case of total anisotropy,
a
5 0. Under total anisotropy
the flow is no longer turbulent and there is no stress
due to the absence of vertical velocity fluctuations. If
the isotropy is greater than
a
5 0.2, then B
1
is less than
46.6 and is as small as 23 in the other extreme of com-
plete isotropy,
a
5 0.5. Isotropic turbulence is impos-
sible because there is no stress and, hence, no production
of TKE. The smallest possible value of B
1
is still greater
than the value of 16.6 cited by Galperin et al. (1988).
Correction of the TKE estimates for Doppler noise does
not produce a substantial bias because all estimates of
TKE are much larger than the noise variance. The re-
maining possible explanation for this elevated estimate
of B
1
is the influence of stratification, which is unmis-
takably present at middepth. By assuming l 5 l
m
and a
local balance of the TKE budget, (20) predicts an in-
crease of B
1
with a decrease of S
m
in a stratified envi-
ronment, and this is consistent with our estimate of B
1
.
Figure 12 compares
e
from TAMI and
e
MY
calculated
with B
1
5 46.6 and l 5 l
m
. The values track each other
well over a range of 2.5 decades and the agreement
between
e
and
e
MY
is slightly but not significantly better
than that between
e
and P (Fig. 7).
At middepth, there are no clear tidal signals in the
estimates of the turbulent quantities. Variations of the
turbulent parameters appear to be only correlated with
changes in TKE density. From Fig. 4c, the region of
bottom-enhanced TKE production, which does display
a tidal signal, is generally below middepth during the
flood. During the ebb, the region of enhanced production
protrudes above middepth, but this increase of height
is due to the entrainment of water from the shallows of
Saanichton Bay into the main stream.
7. Turbulence structure in the near-bottom layer
During strong flows, stratification is weaker in the
lower half of the channel than at middepth (Fig. 3). This
decrease of N
2
is accompanied by an increase in shear
toward the bottom. Hence, the gradient Richardson
number should be mostly less than its critical value in
the near-bottom layer, and we anticipate that stratifi-
cation plays a less significant role in suppressing tur-
bulence than at middepth.
Figure 13a shows the time variations of the mixing
length l
m
at 3.6 mab (the lowest bin of the ADCP). For
all 3.8 days of data, 73% of the l
m
values are between
0.3 and 1 m, 19% are greater than 1 m, and the remaining
864 V
OLUME
30JOURNAL OF PHYSICAL OCEANOGRAPHY
F
IG
. 13. (a) Mixing length l
m
(circles) vs l
z
(dashed lines) (in meters)
and (b) vertical eddy viscosity A
y
(m
2
s
21
). Panel (c) shows a stick
diagram of the 20-min flow (m s
21
). All quantities are estimates at
z 5 3.6 m.
F
IG
. 14. (a) TKE density, q
2
/2, (heavy solid lines) vs stress mag-
nitude u9w9 (thinner lines with crosses) (both in m
2
s
22
). (b) Values
of the stability function S
m
calculated with (18) (solid line) vs S
m
5
0.393 27 (dashed line); (c) the stick diagram of the flow. All quantities
are estimated at z 5 3.6 m.
F
IG
. 15. Scatter diagram of the rate of TKE production against
q
3
/l
m
. Open circles and crosses are for speeds greater than and less
than 0.35 m s
21
, respectively. The constant of proportionality is B
1
5 26.3. All quantities are estimates at z 5 3.6 m.
8% are either less than 0.3 m or negative (corresponding
to negative production rate). Unlike at middepth, the
characteristic length of turbulent eddies (l
E
5 3l
m
)is
not significantly greater than the vertical averaging
length (1 m) of the ADCP.
Is the Reynolds stress in the near-bottom layer un-
derestimated because of vertical averaging by the
ADCP? In the appendix, we analyze additional velocity
data that has a vertical resolution of 0.1 m and was
collected using the coherent mode of the ADCP. The
analysis indicates that spatial averaging reduces the es-
timated stress by no more than 5%.
Figure 13(b) shows the time variations of the vertical
viscosity coefficient A
y
. Except during the turning of
the tide and during the weak flood between day 24.55
and 24.75, the eddy viscosity is almost independent of
flow magnitude at logarithmic scales, ranging between
0.02 and 0.04 m
2
s
21
. During the weak flood, A
y
drops
to 5 3 10
23
m
2
s
21
and lower.
Variations of the TKE density and the Reynolds stress
are well correlated (Fig. 14a). Both q
2
/2 and |u9w9| con-
tain clear tidal signals, but they are frequently elevated
during the beginning and the end of the ebb when the
flow is turning. The mean value of L5q
2
/|u9w| is 9.84,
with a 95% confidence interval of 60.61. The stability
function S
m
, shown in Fig. 14b, is calculated from the
q
2
-to-stress ratio with (18). During strong flows when
the magnitudes of the stresses are large, the values of
S
m
are close to g
2
5 0.393 27, as predicted by (14) when
N 5 0. By comparing the near-bottom estimates of S
m
against those from middepth (Fig. 10b), it is evidentthat
the effects of stratification are weaker near the bottom.
The correlation between q
3
/l
m
and P at z 5 3.6 m
(Fig. 15) is tight and much closer than that between
q
3
/l
m
and
e
at middepth (Fig. 11). The constant of pro-
portionality is B
1
5 26.3 6 1.2 for speeds exceeding
0.35 m s
21
. Matching the rate of production to the rate
of dissipation predicted by the Mellor–Yamada closure
requires an adjustment of B
1
from 16.6 to 26.3. For B
1
5 16.6, the predicted rate of dissipation exceeds the
measured rate of production. Given that, on average, (i)
the measured rate of production agrees with the mea-
sured rate of dissipation at middepth, (ii) the Richardson
number is smaller near the bottom than at middepth,
(iii) vertical averaging does not significantly reduce the
M
AY
2000 865LU ET AL.
F
IG
. 16. Time series of local friction velocities (a) u
*
s
and (b) u
*
n
(open circles) at z 5 3.6 mab, and u
*
(crosses) obtained by fitting
the streamwise velocity profiles to a log-layer.
estimated stress, and (iv) the rate of production of buoy-
ancy is at most about 20% of the rate of dissipation,
then it is unlikely that the rate of dissipation is much
less than the rate of production. Uncertainty about the
actual degree of isotropy cannot explain the larger than
predicted value of B
1
. The estimate of B
1
is biased high
if the actual isotropy is larger than
a
5 0.2 because a
larger isotropy would increase our estimate of TKE us-
ing (4). A larger isotropy is possible, but it would have
to be
a
5 0.41 to make our estimate B
1
equal to 16.6
and such a high degree of isotropy is not plausible. The
remaining possible explanation for this elevated esti-
mate of B
1
is, like at middepth, the influence of strati-
fication according to (20). Some estimates of S
m
are
smaller than g
2
5 0.39 (Fig. 14b) even when the flow
is stronger than 0.35 m s
21
. Thus, stratification is not
always negligible at 3.6 m.
The along- and cross-channel local friction velocities
u
*
s
and u
*
n
at 3.6-m height are shown in Fig. 16. The
cross-channel local friction velocity is large during the
ebb, with peak values of 0.025 m s
21
, but small during
the flood. The alongchannel friction velocity reaches
0.04 m s
21
during peak flow, corresponding to a stream-
wise stress magnitude of 1.6 3 10
23
m
2
s
22
. The mag-
nitude of u
*
s
is, however, consistently smaller than the
shear velocity u
*
obtained by fitting the streamwise ve-
locity profiles to a log-layer (Lueck and Lu 1997). The
mean ratio of to is 0.41; that is, the log-layer-
22
uu
s
**
fitted bottom stress is on average larger than the mea-
sured alongchannel Reynolds stress by a factor of 2.5.
Vertical profiles of the Reynolds stress (not shown) are
neither constant nor linear, even within the log-layer.
One possible explanation for the discrepancy |u9w9|
and is the influence of horizontal inhomogeneity
2
u
*
caused by bedforms. Theoretical analyses (e.g., Belcher
et al. 1993) have shown that the turbulent boundary
layer over small-scale topographic features can be sig-
nificantly distorted from the classical boundary layer
over smooth walls. However, a factor 2.5 increase of
bottom stress requires sand waves of amplitude 1 m and
wavelength 10 m, and there is no evidence for such
bedforms in Cordova Channel. The existence of bed-
forms causes a form drag that influences the flow field
farther away from the bottom than does skin friction.
In oceanic bottom boundary layers, measurements have
revealed the existence of multiple log-layers (Chriss and
Caldwell 1982; Sanford and Lien 1999). The bottom
stress inferred from fitting the velocity profile in the
outer log-layer (extending more than a few meters from
bottom, as in this study) likely contains a contribution
from form drag, in additional to the local Reynolds
stress.
8. Conclusions
A bottom-mounted ADCP and the microstructure in-
strument TAMI, moored at middepth, measured tur-
bulence, flow, and density stratification in Cordova
Channel. The flow in the channel is mainly tidal and
with peak speeds of 1 m s
21
. The data enable us to
derive estimates of Reynolds stress, TKE density, the
rates of TKE production and dissipation, eddy viscosity
and diffusivity, and turbulence length scales.
Depth–time variations of turbulence in the channel
are revealed by measurements with the ADCP. The var-
iation of the Reynolds stress with depth corresponds to
the vertical structure of the mean shear. The along-
channel component of the stress contains clear tidal var-
iations, but the cross-channel component is only sig-
nificant during the ebb when the secondary circulation
is strong. The production of turbulent kinetic energy is
generally enhanced near the bottom, bearing the char-
acter of wall-bounded turbulence, but events of large
production rate can occur at heights above the log-layer.
The TKE density changes more strongly with time than
with depth. The eddy viscosity has a maxima at mid-
depth.
The two instruments provided simultaneous mea-
surements at middepth. Statistics of the gradient Rich-
ardson number indicate that stratification was important
at this depth. Estimates of the Ellison length and Oz-
midov length are comparable in magnitude. The mixing
length is smaller than that predicted by the simple z-de-
pendent formulation (10), which does not take stratifi-
cation into account. Two estimates of eddy diffusivity
and one estimate of viscosity are obtained. The three
estimates are comparable and they correlate over the
range of 10
23
to1m
2
s
21
. The q
2
to |u9w9| ratio is
estimated to be 12.1. Independent estimates of the TKE
production and dissipation rates agree within a factor
of 5 for 20-min ensembles. Both rates ranged between
2 3 10
28
and 2 3 10
25
m
2
s
23
.
The stability function S
m
in the Mellor–Yamada mod-
el, calculated using q
2
and mixing length measured with
the ADCP, has a mean value of 0.287, smaller than
0.393 27 for unstratified flow. The measured dissipation
rate is proportional to q
3
/l
m
. By taking the mixing length
866 V
OLUME
30JOURNAL OF PHYSICAL OCEANOGRAPHY
l
m
as the master length, the dissipation rate calculated
with the Mellor–Yamada model is proportional to the
rate of dissipation measured with TAMI, and both rates
agree if B
1
5 46.6. A possible explanation for this large
estimate of B
1
(compared to the value of 16.6 in the
literature) is the influence of stratification. By assuming
l 5 l
m
and a local TKE balance, the relationshipbetween
B
1
and S
m
based on the Mellor–Yamada closure actually
predicts an increase of B
1
with decreasing S
m
in a strat-
ified environment.
Close to the bottom, the measured turbulent quantities
contain stronger tidal variations than at middepth. The
influence of stratification is expected to be small because
of the strong shear and weak density gradient.Estimates
of the eddy viscosity range between 0.02 and 0.04 m
2
s
21
and are fairly steady except during the turning of
the tide and during very weak flows. The q
2
to 2|u9w9|
ratio is 9.84 6 0.61. The stability function S
m
is smaller
but closer than at middepth to the value used in the
Mellor–Yamada model in unstratified flow. There is a
tight correlation between the dissipation rate calculated
with the Mellor–Yamada model and the rate of produc-
tion estimated from the shear and stress. The two rates
match for the choice B
1
5 26.3.
Although the mean velocity profiles are fitted accu-
rately to a log-layer, the Reynolds stress is not constant
within the log-layer. At 3.6-m height, the magnitude of
the along-channel stress is smaller than the log-layer
fitted bottom stress by a factor of 2.5. Interestingly,
Johnson et al. (1994) found a factor of 3 discrepancy
between the bottom stress obtained from log-layerfitting
and that derived from dissipation estimates using data
collected in the Mediterranean outflow. We speculate
that form drag causes a discrepancy between the mag-
nitude of the near-bottom Reynolds stress and bottom
stress obtained from a fit of velocity to a logarithmic
profile.
Acknowledgments. We would like to thank D. New-
man and J. Box for their technical support to the field
program, and D. Farmer who provided the CTD data.
Comments from Steven Monismith and the anonymous
reviewers led to significant improvement to the original
manuscript. This work was supported by the U.S. Office
of Naval Research under Grant N00014-93-1-0362.
APPENDIX
The Influence of Profiling Resolution to Stress
Estimates
The profiling range of the ADCP is broken into equal-
ly spaced segments called depth cells. The along-beam
velocity at each cell is the average of the velocity in a
volume that has a cross section equal to that of the
transducer and a length equivalent to the cell size. The
actual vertical weighting is that of a triangle with 50%
overlap between adjacent cells. The concern for tur-
bulence measurement is that the averaging volume
should not exceed the size of the energy- and stress-
containing eddies. For the results presented above, we
used data collected using the standard mode (mode 4)
of the ADCP and with 1-m cell size. At 3.6 m above
the seabed, the Ellison length is not significantly larger
than the cell size and the Reynolds stress may be un-
derestimated due to spatial averaging.
During the experiment, we also collected data using
the coherent mode (mode 5) of the ADCP. This mode
provides much finer profiling resolution (0.1 m) and
lower noise, but at a cost of very limited range O(1 m).
We obtained useful data in eight bins and used these to
compare two estimates of the alongstream Reynolds
stress. The first estimate,
t
1
, is the average of the stresses
at the eight levels, while the second estimate,
t
2
, is the
stress obtained from the vertically averaged velocity.
The second estimate is a proxy for the results obtained
with mode 4. Estimates are obtained for two segments
of data that are 34 and 44 minutes long, respectively.
For segment 1
t
1
5 8.82 and
t
2
5 8.52, while for
segment 2,
t
1
5 9.19 and
t
2
5 8.59, all in units of 10
24
m
2
s
22
. The ratio of the two estimates is 0.96 and 0.93,
and we conclude that the stress derived with 1-m cells
is underestimated by only 5% near the bottom.
REFERENCES
Belcher, S. E., T. M. J. Newley, and J. C. R. Hunt, 1993: The drag
on an undulating surface induced by the flow of a turbulent
boundary layer. J. Fluid Mech., 249, 557–596.
Blumberg, A. F., and G. L. Mellor, 1987: A description of a three-
dimensional coastal ocean circulationmodel. Three-Dimensional
Coastal Ocean Models, N. S. Heaps, Ed., Coastal and Estuarine
Sciences, Vol. 4, Amer. Geophys. Union, 1–16.
Bowden, K. F., 1978:Physical problemsofthebenthicboundary layer.
Geophys. Surv., 3, 255–296.
Chriss, T. M., and D. R. Caldwell, 1982: Evidence for the influence
of form drag on bottom boundary layer flow. J. Geophys. Res.,
87, 4148–4154.
Edson, J. B., C. W. Fairall, P. G. Mestayer, and S. E. Larsen, 1991:
A study of the inertial-dissipation method for computing air–sea
fluxes. J. Geophys. Res., 96, 10 689–10 711.
Foreman, M. G. G., R. A. Walters, R. F. Henry, C. P. Keller, and A.
G. Dolling, 1995: A tidal model for eastern Juan de Fuca Strait
and the southern Strait of Georgia. J. Geophys. Res., 100, 721–
740.
Galperin, B., L. H. Kantha, S. Hassid, and A. Rosati, 1988: A quasi-
equilibrium turbulent energy model for geophysical flows. J.
Atmos. Sci., 45, 55–62.
Gargett, A. E., 1985: Evolution of scalar spectra with the decay of
turbulence in a stratified fluid. J. Fluid Mech., 159, 379–407.
Huang, D., 1996: Turbulent mixing in a tidal channel. M.S. thesis,
School of Earth and Ocean Sciences, University of Victoria, 125
pp.
Johnson, G. C., R. G. Lueck, and T. B. Sanford, 1994: Stress on the
Mediterranean outflow plume. Part II: Turbulent dissipation and
shear measurements. J. Phys. Oceanogr., 24, 2084–2092.
Lohrmann, A., B. Hacket, and L. P. Roed, 1990: High resolution
measurements of turbulence, velocity and stress using a pulse-
to-pulse coherent sonar. J. Atmos. Oceanic Technol., 7, 19–37.
Lu, Y., 1997: Flow and turbulence in a tidal channel. Ph.D. thesis,
University of Victoria, 140 pp.
, and R. G. Lueck, 1999a: Using a broadband ADCP in a tidal
M
AY
2000 867LU ET AL.
channel. Part I: Mean flow andshear. J.Atmos.OceanicTechnol.,
16, 1556–1567.
, and , 1999b: Using a broadband ADCP in a tidal channel.
Part II: Turbulence. J. Atmos. Oceanic Technol., 16, 1568–1579.
Lueck, R. G., and Y. Lu, 1997: The logarithmic layer in a tidal
channel. Contin. Shelf Res., 17, 1785–1801.
, and D. Huang, 1999: Dissipation measurement with a moored
instrument in a swift tidal channel. J. Atmos. Oceanic Technol.,
16, 1499–1505.
, , D. Newman, and J. Box, 1997: Turbulence measurements
with a moored instrument. J. Atmos. Oceanic Technol., 14, 143–
161.
Lynch D. R., J. T. C. Ip, C. E. Naimie, and F. E. Werner, 1996:
Comprehensive coastal circulation model with application to the
Gulf of Maine. Contin. Shelf Res., 16, 875–906.
Mellor, G. L., and T. Yamada, 1974: A hierarchy of turbulence closure
models for planetary boundary layers. J. Atmos. Sci., 31, 1791–
1806.
, and , 1982: Development of a turbulence closure model
for geophysical fluid problems. Rev. Geophys. Space Phys., 20,
851–875.
Moum, J. N., M. C. Gregg, R. C. Lien, and M. E. Carr, 1995: A
comparison of turbulent kinetic energy dissipation rate estimates
from two ocean microstructure profilers. J. Atmos.Oceanic Tech-
nol., 12, 346–366.
Osborn, T. R., 1980: Estimates of the local rate of vertical dissipation
from dissipation measurements. J. Phys. Oceanogr., 10, 83–89.
, and C. S. Cox, 1972: Oceanic fine structure. Geophys. Fluid
Dyn., 3, 321–345.
Peters, H., M. C. Gregg, and T. B. Sanford, 1995: Detail and scaling
of turbulent overturning in the Pacific Equatorial Undercurrent.
J. Geophys. Res., 100, 18 333–18 348.
Sanford, T. B., and R.-C. Lien, 1999: Turbulent properties in a ho-
mogeneous tidal bottom boundary layer. J. Geophys. Res., 104,
1245–1257.
Simpson, J. H., W. R. Crawford, T. P. Rippeth, A. R. Campbell, and
J. V. S. Cheok, 1996: The vertical structure of turbulent dissi-
pation in shelf seas. J. Phys. Oceanogr., 26, 1579–1590.
Soulsby, R. L., 1983: The bottom boundary layer of shelf seas. Phys-
ical Oceanography of Coastal and Shelf Seas, B. Johns, Ed.,
Elsevier Oceanogr. Ser., Vol. 35, Elsevier, 189–266.
Stacey, M. T., 1996: Turbulent mixing and residual circulation in a
partially stratified estuary. Ph.D. thesis, Stanford University, 209
pp.
, S. G. Monismith, and J. R. Burau, 1999: Observations of tur-
bulence in a partially stratified estuary. J. Phys. Oceanogr., 29,
1950–1970.
Thompson, R. E., 1981: Oceanographyof the British Columbia Coast.
Canadian Special Publication of Fisheries and Aquatic Science,
Series 56, 291 pp.
Turner, J. S., 1973: Buoyancy Effects in Fluids. Cambridge University
Press, 367 pp.
