WIND-DRIVEN
CIRCULATION
IN
UNSTRATIFIED
LAKES
John A. T. Bye
Scripps Institution
of Oceanography,
University
of California,
San Diego, La Jolla, California
ABsTRAcr
Observations
of the wind-driven
circulation
in the vertical plane of unstratified
lakes,
including
some new results indicating
a logarithmic
current profile near the lake surface,
are prescntcd.
A thcorctical
velocity profile that is in fair agreement with a stress-driven
laboratory
flow, and that may approximate
to the steady circulation
in the central section of lakes of
nearly uniform
depth, is also dcvcloped.
From this profile,
an estimate is made of the
effect on the wind setup of a gently sloping lake bottom.
also found that the surface current was
independent of wave action.
Further results were obtained by the
present author on Lough Neagh in Northern Ireland in 1962. Two cxpcrimcnts
(E221 and E223) were carried out in 10 m
of water, about % mile ( 0.8 km) from the
east (downwind)
short of the lough. The
experiments consisted in determining
simultaneously the current profile to a depth
of 1 m below and the wind profile to a
height of 10 m above the water surface;
the respective measurements were made by
timing the drift of a series of floats and
from a mast of sensitive cup anemometers.
Prior to the observations, a wind of 78 m/set, approximately constant in dirccOBSERVATIONS
tion, had been blowing for several hours
The surface current. The relation bewith a fetch of 14.5 km, and conditions in
tween the surface current ( U, ) and the
the air-water interface region throughout
surface wind ( U,) has been investigated
the
measurements were very close to isoby many workers. It appears to be adethermal.
quatcly known up to wind speeds of 15
The drift floats were made of 2.5 x 2.5
m/set.
cm cross-section wood, varying in length
U,h
For Reynolds
Numbers,
2 10S, between 2.5 cm and 1 m, and wcrc
V
weighted at one end to float with their
where v is the dynamic viscosity of water,
tops
just breaking the water surface2 (Fig.
and h is the depth of the lake, Keulegan
1).
( 1951) in wind tunnel experiments found
In a shearing flow, each float moves at
that
a velocity, ( U,), such that it experiences
zero resultant drag force. We can express
u, - 0.033.
(1) this condition analytically as follows:
INTHODUCTION
The purpose of this article is to review
existing observations of wind-driven
fully
turbulent circulation in the vertical plane
of unstratified lakes of constant depth and
to develop a theoretical velocity pro&
based on the similarity theory of turbulencc, This profile is shown to be in fair
agreement with a laboratory stress-driven
flow and may approximate to conditions
in the central section of lakes of nearly uniform depth.
For such lakes, an estimate is made of
the effect on the wind setup of a gently
sloping lake bottom.
us
Thcsc results were extended by Van
Dorn ( 1953) in a shallow basin *( 2 m
deep ) up to wind spcedsl of 15 m/set.
He obtained the same proportionality,
and
l U, was mcasurcd
at 1 m.
;JWC&&-U)2dx
=;JWC,QJIiT-U)2dx,
(2)
2 The original idea for the floats was by Profcssor J. R. D. Francis, and the experiments were pcrformed from Professor I?. A. Sheppard’s cxpcrimental lough platform.
451
452
JOHN
A. T. BYE
where k,,; is von Karman’s constant in the
water, xOlu is the water roughness parameter, and U,‘” is the water friction velocity
defined by
U,“= ( F,>1
Pw
in which F, is the surface stress, and pLcis
the density of water; it may be deduced
from equation (3) that, provided xozuQ L,
each float will travel at the velocity it experiences at a depth 0.2725 of its length
below the surface, that is, A = 0.2725L.
This simple result implies that the float
profile corresponding to equation (4) is
also logarithmic in form, and given by
FLOAT
FIG. 1.
The drift floats.
Up= u,-- U,"
where x is measured downwards from the
surface and L and W arc, respectively, the
length and width of the float. U is the
water velocity, CD is the drag coefficient of
the float at depth x, and A is the depth
at which the float moves at the water velocity (U = U,).
For the range of Reynolds Number
(UP - u)w ] (where 2) is the dynamic
[
u
viscosity of water), encountered during
our drift, the drag coefficient CD for a
square cross-section is approximately constant, and ignoring a thin plate which was
fitted around the longer floats to prevent
excessive bobbing under wave action (Fig.
1)) W is also constant, so that equation
( 2 ) can be simplified to:
,“J(U,-U)2dx=;J(UF~-U)2dx.
(3)
Hence, if the velocities of a series of floats
of different lengths are known, the current profile in which they drift may be
deduced. Strictly, equation (3) is a vector
equation, but in the present experiments
all the floats moved in the direction of the
wind.
In particular,
if this profile is logarithmic, of form
u=u,-gln2f-,
20
x(J7”
k W In
-$
+ 1.30-
&”
kw *
(5)
In both our experiments we observed
that the float velocities, computed by timing the floats over a distance of 25 m,
were logarithmically
distributed.
We induccd that the current profiles are also
logarithmic.
They are plotted log-linearly
in Fig. 2, the slopes of the graphs were
estimated by eye.
The wind profiles during each float release were also found to be logarithmic
and can be represented in a similar manner
to the current profiles;
the subscript a in equation ( 6) refers to
air quantities, and x is measured upwards
from the surface.
The parameters of equations (4) and
( 6)) determined from the observations, are
summarized in Table 1, and have been
computed as follows :
First, assuming a value for von Karman’s
constant in air of k, = 0.41, and that pw =
1, and pn = 0.0012, an estimate of von
Karman’s constant in water can be obtaincd by comparing the logarithmic slopes
of the air and water profiles. The mean
estimate of our two experiments (k, = 0.4)
is identical to that found by Charnock
( 1959) in tidal currents near the sea bed.
CIRCULATION
100
IN
I
I
UNSTRATIFIED
453
LAKES
noted at the beginning of the paper, for
on substituting for U, from equation ( 1 ),
using the drag relationship
I
50
F, = C,looPJJ a32
and assuming the drag coefficient at 1 m,
Cdl00 = 0.0021, which is the mean of our
two observations; and also consistent with
a consensus of measurements under a variety of conditions
(Sibul 1955; Deacon
1962), we have,
20
IO
5
E
0
N
U,=0.033(--&J=21UP.
2
I
\
0.5
(7)
It thus appears that in fully turbulent
flows, at least for wind speeds < 15 m/set,
the relationship between the friction velocity and the surface current is
u, =20 to 25 U,“.
(8)
0.2
5
FIG. 2.
u, cm/s:
Surface current profiles.
Second, the surface current has been estimated by superposing the wind and current profiles. The intersection in the ( U,
x ) plane gives 27, and x0. This procedure
is mathematically
equivalent to extrapolating the logarithmic profiles to the roughness height, which is assumed to have the
same value in air and water,
xow = x0” = x0 .
A detailed investigation of the nature of
the current and wind profiles near the interface is clearly necessary before its validity may be judged.
The calculated surface currents, however, are in close agreement with the observations of Keulegan and Van Dorn
TABLE 1.
Winclspccd
at 2 meters
(m/see)
E 221
E 223
8.4
7.5
t7.G
(cm/see)
34.1
30.3
Further investigations under a variety of
conditions are needed before a more precise dependence is established.
The velocity profile,
Velocity profile.
except for the observations in the first
meter described above, has not yet been
Francis
measured in lake conditions.
( 1953) however, measured the profile in
a tank having boundary conditions that
were essentially similar. The natural wind
stress was replaced by a much more powerful stress produced by jets of water in the
surface layer; the experimental profile is
shown in Fig. 3, The features of the profiIe are, 1) the velocities and velocity
gradients are large near the surface, and
2) near the bottom there is a thin boundary layer of approximately O.lh thickness,
where the return current is reduced to
zero.
The bottom stress, The bottom stress has
been measured by Francis ( 1953) and Van
Dorn ( 1953). Van Dorn found that in a
Surface current parameters
uzw
kw
(cm/see)
2.75
2.32
us
(cm/see)
24.8 U,”
24.6 U,”
k 1”
0.39
0.42
ZO
(cm)
Cd00
0.015
0.012
0.0022
0.0020,
454
JOHN
I
A. T. BYE
where P is the pressure, I?,, is the horizontal steady stress, and g is the acceleration
of gravity.
Taking the x-axis vertically downwards,
the x-axis along the line of action of the
surface stress, and considering the solution
of the equations with the boundary conditions,
atx=O
and
Fm = Fs
F,,=O
at x =h.
I
&y-
8-2
FRANCIS'
EXPERIMENTAL
PROFILE
0.4
N 0.6
ILARITY
PROFILE (B=I)
0.0
5
FIG.
3. Total velocity profiles.
shallow basin the bottom stress, Fz, 5 0.1
F,. Francis observed, in two determinations with the experimental arrangements
described above, that
Fb - O.O14F, .
IIence, the measured bottom stress is a
very small percentage of the surface stress.
A theoretical velocity profile
velocity
In this section, a theoretical
profile for steady l-dimensional
flow in a
fluid of constant density and depth (h) is
derived, It is assumed that the circulation
is in a vertical plane parallel to the surface stress and that there is zero net transport,
;JUdx=O.
The model is applicable
The bottom boundary conditions have
been approximated here so that the similarity principle can be used to obtain the
solution. Physically the approximation implies that the thin bottom boundary layer
in which the return current is reduced to
zero is omitted in the theoretical solution.
First, the integration of equation (9a)
yields
F,, is now expressed in terms of a mixing
length for momentum exchange by the
Prandtl relation,
and the mixing length (1) is estimated
from von Karman’s similarity
principle
which states :
to the measure-
l=kwg/$
ments by Francis described above, and the
observed and calculated profiles are compared in Fig. 3.
The model is not claimed to be an adequate description of flow in many natural
lakes, but a comprehensive theory cannot
be put forward at present because the effects on the circulation of such factors as
bottom topography and the variability
of
the wind stress have not been adequately
observed.
The equations of motion for the flow
are as follows :
and
,=-kg,,,
(9a)
(10)
F
where U is the velocity at depth x.
Substituting in equation ( 10) from these
relations gives
(11)
and integrating
u=-
equation
F (l-z!)’
21)
(
+Bln
W
( 11) twice,
I II ,
B-u-;)”
B
+C
CIRCULATION
IN UNSTRATIFIED
where B and C arc integration constants.
This result is analogous to that derived by
Hunt (1954) for flow in a channel.
After a further integration,
and eliminating C by the zero1 transport condition
:I‘ U dx = 0, we find the general velocity
profile,
U=+Y
‘IV
-%+
f
B(B+%)
(1-g)*+Bln
+B(B2-1)ln-B-
B-l
u,F=:23u~w
B(B+‘h)
which the depth varies gently along the
direction of the applied wind stress.
The analysis may approximate to conditions in the central sections of lakes having
a fairly uniform depth over a large area.
In these cases it provides the correction to
the wind setup formula due to the inertia
terms in the equations of motion discussed
by Urscll ( 1956).
First, for lakes of constant depth, there
is the well-known
expression for the surface slope ( at/ax).
B-(1-;)+
1.
(12)
[ 1
B
f
The constant B is estimated empirically
by considering equation (12) at the surfact. Substituting
the observed surface
current for U at x = 0 gives the equation
+B3]n
B-l
-+?h
1
,
which is satisfied by B = 1.00002.” Thus,
a very close approximation to the solution,
except for very near to the surface, is given
by putting B = 1 in equation ( 12)) namely,
455
LAKES
1.01 F,
a,$ F,- Fb
-z-wax
(13)
P&
P&
’
where LJis the mean elevation above the
undisturbed level.
In lakes of variable depth, however, the
inertial terms in the equations of motion
cannot bc ignored, and to a first approximation of small bottom gradient that neglccts vertical accelerations, the setup rclationship is modified as Follows :
a,$
-=
dx
F,- Fb
pg(h-t-t) -g(
1
h+t)
?- “J U2 diz;. (14)
ax -E
For a small bottom gradient, we may
appro’ximate J U2 dx from the constant
depth solution ( 12), namely,
This profile is plotted in Fig. 3 together
with Francis’ experimental
profile.
The
agreement is fair, except near the bottom.
Other similarity profiles (Ellison 1960)
can be derived using different hypotheses
to estimate the mixing length, but they are
not significantly
different from equation
( 12), and the present observations arc not
precise enough to distinguish
between
them.’
The wind setup over gently sloping bottoms
The above results will now bc used to
determine the wind setup in a lake in
3 In channel flow Vanoni (IIunt
1954) shows
in
that B = 1.02. If B = 1, the surface velocity
lake flow is infinite.
4 I thank a refcrec for drawing my attention to
this paper.
Substituting
in equation
g-
810&h
[ la-
( 14)) we obtain
+ t) ] pgt:+
~> , (15)
where
6 = 1 if F, is +ve,
= -1 if F, is -ve.
This is an implicit equation for the modified setup. Two special cases will be discussed below:
Lake with a level bottom. For a lake initially of constant depth, dh/dx = 0,
at 1.01 F,
-?=5
8% pg(h+t)
alOF,
[ ‘-p&+6)
1
’
This equation gives the modification
to
the constant depth setup relation produced
by the setup itself. It is a systematic ‘setdown,’ but in all realistic observational
456
JOHN
A. T. BYE
conditions, it is entirely negligible (<.l%
for [at/k]
< 1O-4) .
An analogous effect is produced by the
inertia of the surface waves, It has been
discussed by Longuet-Higgins
and Stewart
( 1964)) and is usually called the radiation
stress of the waves. In our case, it leads
to a further modification of the setup relation.
It is assumed throughout that the water
velocities are separable into mean, turbulent, and wave components; the turbulent
components are incorporated,
from the
start, as Reynolds Stresses in the total stress
terms, The departure from the mean elevation (t) at a given time is expressed as the
sum of the setup ( & ), and the wave elevation ( y), thus
First, considering
of motion,
the vertical
(16)
in which t is time, w is the vertical velocity
component, and the components of total
stress, F,, and F,,, are
aw
F,, =-2pvaz+pw.
%-<
au
F,, = - 2pud3; + Pi?
between the surface and the bottom, to obtain
lh
=-FOE
S
APdz-dx
pl_:s!!f+-[
!$]“t
an d substituting
the above relationship
(equation 18) for the pressure. Forming
the mean over time of the resulting equation, and denoting the time mean quantities by an overbar, gives
-S
U2dx
=-- 1”
p-5
S-%pg(z+()
Ax -5
S
wdx + ~2 - w -E2+ ’ zdz
-5
+g(x+t)
in which the total stress, F,,, is given by
a 7L
2
Integrating this equation with respect to x
between the surface (-t), and a depth x,
gives
ax
S
au
at
equation
+g,
and
Equation
( 18) defines the pressuredepth relationship.
The required setup relation is sought by integrating the equation of horizontal motion,
-
+p(wet2-w”)
where the terms, which are identically
,
>dx
ax
zero,
(17)
where .P, is the atmospheric
pressure.
I
?.!&&
If now the stress is uniform,
-5
is negligible, so that we may write equation ( 17) as
S
have been omitted,
and three other terms,
CIRCULATION
because the stress is uniform
J&J%F
dx>dx,
IN UNSTRATIFIED
and
because
the
waves are assumed to be irrotational, have
also been neglected.
If we evaluate the integrals in equation
(ZO), and represent the left-hand side of
the equation as a sum of the mean ( U,)
and wave (u) components of the velocity,
we obtain,
~
~
d gc2 aw “h
=-&gh’---+
ax 2
+ &-;$w2dz
-9.
(21)
Finally, assuming that the waves are
deep water waves, so that J&dz
=
Jz12dz, and that the mean-square wave
elevation is much greater than the meansquare setup, to” < t7
after substituting the results of the text, WC
h ave
1 dWwt2
-2
1 a[2
ax -3x’
(22)
The surface waves therefore, give rise
to two extra set-down terms that are significant if there is an appreciable growth in
wave amplitude along the direction of the
wind stress.
Lake with a sloping bottom. If the bottom slope, ah/ax is much greater than
@/ax, which is usually the case in natural
basins, equation (15) shows that an error
of 10% is induced in the simple setup relation (13) by a bottom slope of 1 in 100.
Here, however, the cffcct of the inertial
terms is not systematic, and can be eliminated (or calculated L bY comparing resuits with a wind of given speed blowing
from opposite directions.
In addition it may be noted that, differ-
457
LAKES
entiating equation (18) with rcspcct to X,
gives an expression for the pressure gradient. Between two pressure recorders situated at a depth, II, below the level of wave
action, the time-mean gradient is given by
CJP,
-=gjg
dX
Substituting
ac$ aw-(2
+Pyy.
into equation
( 22))
%I
Zy5(12?!!!$)~&~,
dX
The mean pressure gradient, therefore,
also has a set-down term, [l/( 2pgh) ] X
( aF/ax), due to the downwind growth of
waves. However, as the pressure recorders
measure the total pressure, that is, hydrostatic and dynamic, pressure gradient measurements cannot resolve the additional
set-down, ( l/g) ( d~-~~/ax),
in equation
(22), due to the wave dynamic pressure
gradient.
CONCLUSION
In this article various aspects of the simplest wind-driven circulation in an unstratificd lake have been discussed. There are
good observations relating the surface current and bottom stress to the wind stress,
but knowledge of the velocity profile is
confined to two determinations in the surface 1 m. The velocity profile of an analogous laboratory circulation,
however, is
in fair agrccmcnt with a profile derived
from the similarity theory of turbulence.
It is clear that much additional research
is needed before a satisfactory undcrstanding can be obtained of this important scale
of hydrodynamical
phenomena.
REFERENCES
CIIAHNOCK, M.
1959. Tidal friction
from currents near the seabed. Geophys. J., 2: 215-221.
DEACON, E:. L. 1962. Aerodynamic
roughness of
the SCR. J. Gcophys. Res., 67: 3167-3172.
ELLISON,
T. H. 1960. A note on the velocity profile and longitudinal
mixing in a broad open
channel.
J. Fluid MC&,
8: 3340.
FRANCIS,
J, R. D.
1953. A note on the velocity
distribution
and bottom stress in a wind-driven
;;a;;
current system. J. Marine Rcs., 12:
458
JOHN A.T.
&NT,
J. N. 1954. The turbulent
transport
of
suspended sediment in open channels.
Proc.
Roy. Sot. (London),
Ser. A, 224: 322-335.
KEULEGAN, G. H. 1951. Wind tides in small
closed channels. J. Rcs. Natl. Bur. Std., 46:
358-381.
LONGUET-IIIGGINS, M. S., AND R. W. STEWART.
1964. Radiation
stresses in water-waves;
a
physical discussion, with applications.
DeepSea Res. 11: 529-562.
BYE
SIBUL, 0.
1955. Water surface roughness and
wind shear stress in a laboratory
wind-wave
channel, p. 1-42. Tech. Mem. 74. U. S. Beach
Erosion Board, Corps of Engineers.
URSELL, F. 1956. Wave generation by wind. p.
216-219. In G. K. Batchclor and R. M. Davies
[ eds.], Surveys in mechanics. Cambridge University Press.
VAN DORN, W. G. 1953. Wind stress on an artificial pond.
J. Marine Res., 12: 249L276.