Coastal Engineering, 11 (1987) 513-526
Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands
513
A L a g r a n g i a n Model for Wind- and Wave-Induced
N e a r - S u r f a c e Currents
ALASTAIR D. JENKINS*
Oceanographic Center, P.O. Box 1883 Jarlesletta, N- 7001 Trondheim, Norway
(Received October 9, 1986; revised and accepted January 19, 1987)
ABSTRACT
Jenkins, A.D., 1987. A Lagrangian model for wind- and wave-induced near-surface currents. In:
P.P.G. Dyke (Editor), J O N S M O D
86 Coastal Eng., 11: 513-526.
A theory is outlined for time-dependent currents induced near the sea surface in deep water,
away from coastal boundaries, by a variable wind stress and deep-water wave field. It is based on
the theory of Weber (1983) which uses a second-order perturbation expansion of the NavierStokes equations in Lagrangian coordinates and includes the Coriolis effect. It uses an eddy viscosity formulation for both wave dissipation and momentum transfer within the current field: the
eddy viscosity v may be allowed to vary with depth. The wind stress may be time-varying and the
wave field may vary in both space and time.
For the case of a constant ~, the results agree with those of Ursell (1950), Hasselmann (1970)
and Pollard (1970) in the limit v-.0, and the steady-state resultsagree with those of Weber. For
a particular case of depth-varying v, results (obtained from numerical simulations) are in better
general agreement with observations of wind-induced surface driftthan when a constant v isused.
An outline is given of the application of the theory to the case of a random sea state.There are
good prospects for using output data from numerical wave prediction models to drive the equations
of this near-surface current model.
INTRODUCTION
T h e c u r r e n t n e a r t h e sea s u r f a c e is a n i m p o r t a n t e n v i r o n m e n t a l p a r a m e t e r
for o f f s h o r e o p e r a t i o n s a n d design, a n d for c a l c u l a t i o n o f drift, s p r e a d i n g a n d
d i s p e r s i o n o f p o l l u t a n t s . I t is, h o w e v e r , s u b j e c t to c o n s i d e r a b l e u n c e r t a i n t i e s
as r e g a r d s b o t h m e a s u r e m e n t a n d modelling, m a i n l y as a r e s u l t o f t h e effect o f
s u r f a c e waves.
S t o k e s (1847) p r e d i c t e d t h a t o c e a n w a v e s w o u l d g e n e r a t e a d r i f t c u r r e n t
( t h e Stokes drift) in t h e d i r e c t i o n o f w a v e p r o p a g a t i o n , w h i c h is p r o p o r t i o n a l
to t h e s q u a r e o f t h e w a v e a m p l i t u d e , a n d w h i c h d e c r e a s e s e x p o n e n t i a l l y w i t h
*Present address: Bergen Scientific Centre IBM, All~gaten 36, N-5000 Bergen, Norway.
0378-3839/87/$03.50
© 1987 Elsevier Science Publishers B.V.
514
depth. Ursell (1950) demonstrated, however, that if the Coriolis effect is taken
into account, waves cannot induce a steady-state drift current in an inviscid
ocean. This paradox was resolved by Hasselmann (1970) and Pollard (1970),
who both dem6nstrated that a wave train which propagates into a non-viscous
ocean region initially at rest induces inertial oscillations, but that such oscillations are superimposed on a zero mean drift current and decrease exponentially with depth in the same way as the Stokes drift. Hasselmann used a
perturbation expansion in an Eulerian coordinate system to obtain his results,
whereas Pollard obtained an exact solution to the equations of motion using
Lagrangian coordinates.
The Coriolis effect also has an important effect on wind-induced currents in
a viscous ocean in the absence of waves. This was first demonstrated by Ekman
(1902, 1905), who showed that the wind-induced current became confined to
a layer near the surface. (If there is no Coriolis effect, momentum from the
wind diffuses indefinitely into the interior.) If the viscosity is constant, the
steady-state surface current is turned 45 ° to the right of the wind (in the
northern hemisphere), and reaches this state after executing some oscillations. Ekman also pointed out that the appropriate value for viscosity that
should be used to calculate wind-induced currents is not the Newtonian (molecular) viscosity of the fluid, but an eddy viscosity, much larger than the molecular viscosity, whose value is chosen to account empirically for the vertical
momentum transfer due to turbulent motions. Such an eddy viscosity can vary
with depth: wind induced currents in the presence of a depth varying eddy
viscosity have been calculated analytically by Dokbroklonskiy (1969), Lai and
Rao (1976), Madsen (1977) and Weber (1981), and numerically by Heaps
(1984) and Davies (1985a,b). The nature of the vertical profile of the eddy
viscosity was found to have a strong influence on the behaviour of the wind
induced current.
Madsen, in particular, used a vertical profile of eddy viscosity which gave
results in general agreement with observations of surface drift - a steady-state
current of about 3% of the wind speed, about 10 o to the right of the wind ( N.E.
Huang, 1979; J.C. Huang, 1983). He assumed that the eddy viscosity v was
proportional to depth, and given by:
v=Ku.~
(1)
where ~c~ 0.4 is yon K~irm~in's constant, u . = v/p is a friction velocity ( ~ is the
wind stress and p is the water density), and ~ = - z is the distance from the sea
surface. Equation (1) represents an extension of the concept of a constantstress turbulent boundary layer to the case of the near-surface layer of the
ocean.
Longuet-Higgins (1953) investigated the effect of viscosity on the waveinduced current (in the absence of the Earth's rotation). In his analysis he
solved the equations of motion in a curvilinear Eulerian coordinate system
515
which was attached to the moving water surface. He found that the presence
of even a small viscosity not only modifies the motion in thin boundary layers
(vorticity layers) near the surface and the bottom, but also produces significant changes, of the same order as the Stokes drift, in the wave-induced current
in the interior. These changes are intimately related to the viscous damping of
the waves: the wave momentum which disappears as a result of damping is
converted into momentum associated with the current field (Liu and Davis,
1977).
The relationship between the conservation of wave momentum and wave
induced currents was used explicitly by Bowen (1969) and Longuet-Higgins
(1970a,b) in the study of longshore currents: the rate of wave momentum loss
as a result of wave breaking provides a stress which drives the longshore current.
Chang (1969) and Unliiata and Mei (1970) reproduced the results of Longuet-Higgins using Pierson's (1962) perturbation expansion of the NavierStokes equations in Lagrangian coordinates. The same method was used by
Liu (1977) to determine mean currents induced by standing waves as well as
progressive waves. Madsen (1978) used the method to extend the theory of
near-surface wave-induced currents in deep water to include the Coriolis effect, but an error in his treatment of the surface boundary condition was pointed
out by Weber (1983a,b). Weber determined wind- and wave-induced currents
as influenced by the Coriolis effect for (a) the case of a horizontally homogeneous field of decaying swell, the rate of wave decay being determined by the
viscosity, and (b) the case of a steady-state horizontally homogeneous current
induced by both wind and waves. He obtained, in the steady-state case, using
plausible values for the (constant) eddy viscosity, a Lagrangian mean surface
current about 3% of the wind speed, directed 20-30 ° to the right of the wind
direction.
COMBINATIONOF WIND AND WAVEEFFECTS
Care must be taken when performing a theoretical investigation of the combined effect of wind, waves and the Earth's rotation on the near-surface current, for three reasons. The first reason is that the wind causes waves to grow
as well as feeding momentum into the current. The second is that if an eddy
viscosity is used to transfer momentum, from the wind stress for example,
between the currents at different depths, it will also act to cause wave damping.
The third reason is the difference between the Lagrangian mean and Eulerian
mean current (the Stokes drift) due to the wave motion - we need to know
which of these two types of mean currents is the one which gives rise to the
Coriolis force and which is the one whose gradient gives rise to shear stresses.
Weber's (1983a,b) theory provides a promising first step in resolving the
above problems, since his use of eddy viscosity to damp the waves and transfer
momentum within the current field is self-consistent, and his mathematical
516
development makes explicit how the Coriolis force is to be treated. He considered only two special cases, however: a decaying swell without wind forcing,
and a steady-state case of a wind- and wave-induced current. If we wish to use
octhe theory to model wind and wave induced currents under realistic field
conditions, it will be necessary to extend it to take account of arbitrary time
variations of the wind stress and wave field.
In his two papers, Weber derived partial differential equations and surface
boundary conditions for the current which appear at first sight to be inconsistent. For the case of decaying swell (Weber, 1983a), he obtained the equation:
4~
vWcc - Wt -ifW=P~2oe-2pt ak318e2kC ---~-e ~. t c o. s ~. c - -. s m y c j )
(2)
and the surface boundary condition:
wc =o
(3)
(c=O)
where W=u+iv is the complex representation of the Lagrangian mean current (u and v are the x- and y-components of the current), c is the Lagrangian
vertical coordinate which corresponds to the Cartesian coordinate z for an undisturbed sea, so that c--0 always corresponds to the sea surface, t is time, f is
the Coriolis parameter, ~oe -pt is the wave amplitude, a is the wave angular
frequency, k is the wavenumber and ~ = (a/2v) ½is the reciprocal of the surface
vorticity layer thickness. The c and t subscripts denote partial differentiation
with respect to the appropriate variable. For the steady-state case (Weber,
1983b), the differential equation is:
4~
VWcc-ifW=~2oakaI4e2kC---~e~(cos~c-sin~c) ]
(4)
and the surface boundary condition is
W~ = - 2~o2ak 2 + # , 2 / v
(c=0)
(5)
If we consider the time-dependent equation (2) and attempt to derive the
time-independent equation by setting fl = 0 and Wt--0, we produce an incorrect coefficient for the term proportional to e 2h¢ on the right-hand side of eqn.
(4). If we consider the boundary condition [ eqn. ( 5 ) ] and attempt to remove
the effect of wind stress by setting u. = 0, we again come to an incorrect result.
Weber explains the difference in the boundary conditions [ eqns. (3) and ( 5 ) ]
in terms of the wind-induced air pressure variation, in phase with the surface
slope, which causes wave growth according to the theory of Miles (1957). He
explains that the difference between the differential equations (2) and (4) is
directly due to the damping of the waves by viscosity. Jenkins (1986) made an
explicit derivation of the differences in the equations and boundary conditions,
and was able to extend the theory so that it can be applied to cases where the
517
wind varies with time and where the wave field has an almost arbitrary space
and time dependence.
Weber also only considered the case of a constant eddy viscosity: eddy viscosity can often only be applied successfully to problems in hydrodynamics if
it can be allowed to vary spatially, and the case of ocean currents would appear
to be no exception. The theory of Madsen (1977), which uses an eddy viscosity
which increases with depth, gives a steady-state current which is generally
closer to observations of wind-induced drift than Weber's (1983b) results, even
though it does not consider wave-induced currents directly. It would be valuable to have a theoretical model, which not only combines wind and wave effects in a self-consistent manner, but which also can allow for a depth-varying
eddy viscosity. Such a model has been developed: it is described in detail by
Jenkins (1987), and a summary of it is given later in this paper.
Time-varying wind and wave forcing
The extension to Weber's theory developed by Jenkins (1986) allowed the
wave amplitude to vary both in space and in time. The rate of wave amplitude
variation is required to be slow with respect to the wave period (2~ a) -1, but
can be rapid with respect to the inertial period (2nf) - 1. It turns out, when the
differential equation and boundary conditions for the current are derived using
the perturbation analysis in the Lagrangian coordinate system, that Weber's
equations (2) - (5) are reproduced as special cases. Equations (2) and (3) are
reproduced when there is no energy input from the atmosphere, and the waves
are damped by the action of the eddy viscosity p. Equations (4) and (5) are
reproduced when both the spatial and temporal rates of change of the wave
amplitude are zero, the wave amplitude is maintained against viscous damping
by cyclic vertical stress variations at the sea surface, and the appropriate wind
stress is applied.
If Weber's (1983b) values for eddy viscosity, sinusoidal wave amplitude and
wind stress are applied to a water mass initially at rest, the Lagrangian mean
current (drift current) initially jumps from zero to a value equal to the Stokes
drift due to the wave field. The tip of the current vector then moves along a
spiral path [similar to the Fredholm spiral described by E k m a n (1905)] and
eventually reaches an equilibrium value identical to Weber's steady-state value
for the appropriate depth.
It should be noted that for engineering applications, although the Lagrangian mean current is the relevant current for studies of the drift of pollutants,
it is not the correct current to use for calculating forces on offshore structures.
This is because it includes the Stokes drift, which is already taken into account
when wave forces are calculated. The relevant current for force calculations is
the Eulerian mean current, equal to the vector average water particle velocity
at a fixed spatial position. It turns out that the equations derived by Jenkins
518
for the current take a particularly simple form when what he called the "quasiEulerian current" is used. This current is equal to the Lagrangian mean current minus the Stokes drift, and can be thought of as being equal to the EulerJan mean current referred to a Lagrangian coordinate system.
If the eddy viscosity is set to a very small value, and a wave field is made to
propagate into a region of the ocean previously at rest, inertial oscillations are
generated. These inertial oscillations decrease in amplitude with increasing
depth, in proportion to the rate of decrease of the Stokes drift with depth. The
behaviour of the wave-induced inertial oscillations is thus in agreement with
the predictions of Hasselmann (1970) and Pollard (1970) for the case of zero
viscosity.
Vertically varying eddy viscosity
The effect of a vertically varying eddy viscosity on wave-induced currents
within the boundary layer at the sea bottom was investigated by Longuet-Higgins (1958), who demonstrated that the wave-induced mean current just outside the boundary layer is independent of the form of the variation of the
viscosity within the boundary layer. This result requires the wave damping to
be slow compared to the wave period (as in the corresponding case for a constant viscosity). It also requires the eddy viscosity acting on any individual
fluid particle to be constant (i.e. that v is a function of the Lagrangian vertical
coordinate). The boundary layer in this case has thickness ~ (2VW/a) ~, where
vw i8 a typical value for v within the boundary layer. Longuet-Higgins' theory
was developed further by Johns (1970, 1975, 1977) and by Trowbridge and
Madsen (1984a,b), for more general cases of space and time variation of v
within the bottom boundary layer.
In the present case, however, we are considering the effect of wind and waves
on the current near the sea surface in deep water, and shall thus ignore the
bottom boundary layer. We expect the boundary layer near the surface, i.e. the
surface vorticity layer, to be thin in comparison with the region of wave influence, and we also expect v to vary with depth within the water column, outside
the vorticity layer. We assume, in fact, that v is constant within the vorticity
layer. It may be possible (though it has not been attempted here) to use arguments similar to those of Longuet-Higgins (1958) to demonstrate that the
precise nature of the eddy viscosity profile within the vorticity layer is not in
fact important.
The theory of Jenkins (1987) extends the investigation of near-surface currents induced by time-varying wind and waves to the case of where v is a function of the Lagrangian vertical coordinate c. Barstow (1981) used a similar
technique to calculate wave-induced currents in the presence of an eddy viscosity which depended on the Eulerian vertical coordinate (z), but it seems
more natural to allow it to depend on the Lagrangian coordinate, particularly
519
x-vel.
cm/s / / ~
Surface
zo:
Wind
and
wave direction
1 0 ~
y-vel.
I
10
cm/s
~ 20m
2m
t
-10
Fig. 1. Development in time (0 to 48 pendulum hours) of the Lagrangian mean currents for a
wind speed of 10 m/s and a sinusoidal wave field of amplitude 0.88 m and period 4.5 s applied
simultaneously at t = O. Wind drag coefficient is 1.8 × 10 -3. The dots mark each pendulum hour
from 0 to 15. [After Jenkins (1987).]
since p may vary over vertical distances which are smaller than the wave
amplitude.
A perturbation expansion is again performed on the Lagrangian-coordinate
equations of motion to second order in wave slope. In the formulation of the
momentum equation, it is assumed that the eddy viscosity u acts in the same
way as a hypothetical spatially varying Newtonian viscosity. Some extra terms
are introduced in the expansion, proportional to the vertical gradient of v. On
solving the equations to first order in the wave slope, it turns out that the
surface gravity wave field has a small rotational component within the water
column, in addition to the main irrotational component and the near-surface
vorticity layer solution. The rate of wave dissipation (in the absence of wave
energy input from the atmosphere, caused by cyclic stress variations at the
surface) is found to be proportional to a weighted depth integral of the eddy
viscosity: the weighting function is proportional to e 2kc,i.e. it decays with depth
at the same rate as the Stokes drift.
Equations for the time development of the current, to second order in the
wave slope, were derived and solved numerically, using Madsen's (1977) formulation of the eddy viscosity profile [ eqn. (1), with 2 replaced by - c ], except
near the surface (c > - k - 1), where p was set to a higher (constant) value to
account explicitly for the effect of breaking waves. As in Weber (1983b), a
520
"o 4.0
~
3.0
Lagrangian
2.0,
Ouasi-Eulerian
1.0.
0.0
"
10
900]
6001
15
20
Quasi-Eulerian
2'5
30 m/s
Wind at lOrn level
~ ~ /
/ / / / /
o
30o 1
°°o°
Lagrangian
~
lb
l's
2'o
2'5
3'0 m/s
Wind at lOm level
Fig. 2. Long-term mean value of surface current as a percentage of wind speed, and direction of
surface current with respect to the wind and wave direction. Sinusoidal wave fields were used.
[After Jenkins (1987).]
wind drag coefficient ranging from 1.8× 10 -~ to 2.7× 10 -3 was used, and a
sinusoidal wave field was applied. The system was started from rest at t = 0.
More detailed information on the input parameters is given in Jenkins (1987).
The results of the simulation for a wind speed of 10 m / s and corresponding
sinusoidal wind-wave field applied to a sea region initially at rest are shown in
Fig. 1. In Fig. 2 the long-term mean value of the current at the sea surface is
shown for a range of wind speeds, as a percentage of wind speed, and its deviation to the right (in the N o r t h e r n hemisphere ) of the wind and wave direction.
The Lagrangian mean current ranges from 2.2% to 2.8% of the wind speed and
is directed between 12 ° and 17 ° to the right of the wind direction. The deviation from the wind direction is generally closer to observed values t h a n when
a constant ~ was used.
APPLICATION TO THE CASE OF A RANDOM WAVEFIELD
The theory described above for wind- and wave-induced near-surface currents using a vertically varying v is straightforward to apply to the case of a
random field of travelling waves, since the equations for the current are linear
in the current and in the square of the wave amplitude. Contributions to the
forcing of the equations from individual components of the wave spectrum can
thus be added together vectorially. The space and time derivatives of the wave
amplitude must be estimated, and this is best done by considering explicitly
521
the energy and m o m e n t u m balance during the process of wave growth and
dissipation.
If we consider a random Gaussian field of travelling waves, each component
with wavenumber vector k contributing F ( k ) to the total variance of surface
displacement E, we have:
E=~F(k)
(6)
k
where we use an approximating Fourier sum instead of an exact Fourier-Stieltjes
integral in order to simplify the presentation (Hasselmann, 1962). The Stokes
drift is given by:
u (')
=
~2ake2~:F(k)
(7)
&
(The scalar wavenumber h is equal to ]k I. )
The rate of change of wave energy at wavenumber k is:
F(k) : S i n ( k ) "~-Snl ( k ) "~-Sds(k )
(8)
where Sin is the rate of wave energy input from the atmosphere, S.~ is the rate
of energy input from other wavenumber components as a result of conservative
non-linear wave-wave interactions, and - Sd~ is the rate of wave energy dissipation [see, for example, Hasselmann (1974)]. If we assume that all contributions to Sa~ can be modelled by the choice of a suitable profile for v, we
obtain:
F
0
a2
Se~ (k) = - 2 F ( k ) | 2 v ( 0 ) k 2 + f k~c,2{v(c')}e2hC'dc' ]
I-
(9)
0
ga
=-4F(h)
J
u(c')k ~ e2he' d(2kc')
--GO
In particular, if p is constant, Sa~ ( k ) oc a4F(k), and if u is proportional to
depth (apart from in the surface vorticity layer where we assume that the
required constant value of ~ is sufficiently small ), Sd~(k) ~: a2F ( k ). Interestingly, this latter a2-dependence of Sa~ is the same as that obtained by Hasselm a n n (1974) for the case of dissipation due to whitecapping, which was found
to give good results when applied in numerical simulations of fully developed
sea states (Komen et al., 1984).
We obtain the following equation for the current:
au(r)
a/
--+rxu'"-g
at
,
(10)
522
where U ~) is the vector representation of the quasi-Eulerian current (the
Lagrangian mean current minus the Stokes drift) and f is a vector of magnitude [ pointing vertically upward (in the Northern hemisphere). The surface
boundary condition is:
0U(r)
Z' 0~---- ('c/p) + ~ [-- (a/k)kSi,(k) +4vakkF(k)]
(c=O)
(11)
where v is the vector wind stress.
The first term on the right-hand-side ofeqn. (11) represents the momentum
input from the wind. Part of this momentum goes into the waves instead of the
current; this occurs via the first term in square brackets. The second term in
brackets represents the wave momentum which is transferred from the waves
to the current at the surface as a result of wave dissipation. However, this term
depends just on the eddy viscosity at the surface and not on the eddy viscosity
at greater depths - the contribution to momentum transfer from the waves to
the current as a result of ~ taking a value different from its surface value appears via the first term in parentheses on the right-hand side of eqn. (10). The
second term in parentheses on the right-hand side of eqn. (10) represents the
Coriolis force due to the Stokes drift part of the current. In eqn. (11), the
contribution from Snl (k) disappears when we sum over all k, because the relevant wave-wave interactions conserve wave momentum.
If we compare eqns. (10) - (11) with Weber's equations ( 2 ) - ( 5 ), we notice
that the terms, with factors (cos),c- sin~c), corresponding to the vorticity layer
contribution to the current, do not appear. In fact, their contribution to the
current itself turns out to be relatively small, though their contribution to the
vertical derivative of the current near the surface is significant. This latter
effect has been taken into account by amending the boundary condition (11),
so that the vorticity-layer contribution again becomes negligible. [ For further
details see Jenkins (1986, 1987). ]
For all but the simplest wave spectra and eddy viscosity profiles, it will almost certainly be necessary to use numerical methods to solve the equations:
on the other hand, the linearity of the equations means that their solution by
numerical techniques should be straightforward.
The near-surface wind- and wave-induced current can thus be simulated
numerically with a random sea state and a vertically varying eddy viscosity,
using eqns. (10)-(11), provided that we know:
(1) The wind stress T.
(2) The directional wave spectrum F ( k ) . In fact, only a coarse directional
resolution is needed, since we require effectivelyjust the total wave momentum
for each value of ]k J.
(3) The rate of wave energy input from the wind Sin(k). This can be obtained, for example, by applying appropriately the theory of Miles (1957) sup-
523
plemented by the experimental parametrizations of Snyder et al. (1981) and/or
Mitsuyasu and Honda (1982).
(4) The verticalprofile of eddy viscosity ~ (c), or, alternatively,the dissipation source function Sd~ (k): the two quantitiesare relatedthrough eqn. (9).
This parameter is the one subject to the greatest uncertainty, and it is likely
that a considerable amount of"tuning" willbe required in order for the simulation to be successful.
It m a y be possible to simulate the near-surface current profileusing actual
fieldmeasurements of directionalwave spectra. However, in order to obtain a
better understanding of the model's properties at this early stage, it will very
probably be more instructiveto drive it with the output, for some simple situations, from a numerical wave prediction model. The wave prediction model's
values of F ( k ) and S~n( k ) can be used directly, and its Sd~( k ) function can
be used to estimate v (c) within the region of wave influence.
CONCLUSIONS
The current near the sea surface caused by the action of wind and waves is
an important parameter for offshore design and operations, and for calculating
the drift, spreading and dispersion of pollutants. When considering these applications it is important to use the correct type of current. In pollution studies,
the Lagrangian mean current is the one to apply, whereas in calculating forces
on offshore structures one must consider the Eulerian mean current, since the
Stokes drift, the difference between these two types of currents, depends on
the wave field and is already taken into account when wave forces are calculated.
A perturbation expansion, to second order in the wave slope, of the equations
of motion in Lagrangian coordinates, was used by Jenkins (1987) to derive
partial differential equations and boundary conditions for the evolution of the
near-surface vertical current in deep water away from coastal boundaries. His
treatment was a natural extension of the work of Longuet-Higgins (1953),
Pierson (1962) and Weber (1983a,b). The effects of wind stress, the Earth's
rotation and a time-dependent surface gravity wave field were taken into account. Eddy viscosity (v) was used in a self-consistent way, both to cause wave
damping and to transfer m o m e n t u m between currents at different depths. The
derivation of the equations for the current profile was performed using an eddy
viscositywhich is a function of the Lagrangian verticalcoordinate c.
Solutions to the equations were obtained for a wind stress and a sinusoidal
wave field which have step-function time dependence. For constant v and a
sinusoidal wave fieldwhich is an approximation to a fullydeveloped wind sea,
the current profile tends to Weber's (1983b) steady-state solution as time t - , oo
(surface drift current ~ 3% of the wind speed, directed 20 °-30 ° to the right of
the wind and wave direction in the Northern hemisphere). In the absence of
wind, the effect of a sinusoidal swell for small values of ~ approximates the
524
wave-induced inertialoscillations,decreasing exponentially with depth, which
were predicted in the case of zero v by Hasselmann (1970) and Pollard (1970).
For a particular case of a vertically varying eddy viscosity - ~ (c) proportional
to c except for near the surface when it was set to a constant value - results
were obtained (long-term mean drift current 2.2-2.8% of the wind speed,
12 °-17 ° to the right of the wind direction) which have a deviation from the
wind direction somewhat closer to observed values than for the constant P case.
Equations and boundary conditions are derived for the case of a random sea
state [eqns. (10) - (11) ]. In order to simulate realisticallythe evolution of the
current profile,the following parameters will be required: the wind stress,the
wave spectrum (with a coarse directional resolution), the rate of wave energy
input from the wind at different wavenumbers, and the vertical eddy viscosity
profile. The eddy viscosity profile is related to the wave energy dissipation
function through eqn. (9). Although it m a y be possible to use field measurements of the wave spectrum, it is probably best at this stage to drive the model
with output data and parameters from a numerical wave prediction model, so
that a better understanding of the behaviour of the near-surface current model
under fairly simple conditions can be obtained.
ACKNOWLEDGEMENTS
This work was partly supported through the project "Analysis of Oceanographic Data", which was funded by Conoco Norway Inc., Elf Aquitaine Norge
A/S, Norsk Hydro A/S, Norske Shell, Saga Petroleum A/S, Statoil, Total Marine Norsk A/S and the Royal Norwegian Council for Scientific and Industrial
Research (NTNF). The rest of the work was supported by NTNF, SINTEF
and the Oceanographic Center.
REFERENCES
Barstow, S.F., 1981. Wave induced mass transport: theory and experiment. Ph.D. thesis,Dept. of
Offshore Engineering, Heriot-Watt University, Edinburgh, 219 pp.
Bowen, A.J.,1969. The generation of longshore currentson a plane beach. J. Mar. Res.,27" 206-215.
Chang, M.-S., 1969. Mass transport in deep-water long-crestedrandom gravity waves. J. Geophys.
Res., 74: 1515-1536.
Davies, A.M., 1985a. Application of a sigma coordinate sea model to the calculation of windinduced currents. Continental Shelf Res., 4: 389-423.
Davies, A.M., 1985b. A three dimensional modal model of wind induced flow in a sea region.Prog.
Oceanogr., 15: 71-128.
Dobroklonskiy, S.V., 1969. Drift currents in the sea with an exponentially decaying eddy viscosity
coefficient.Oceanology, 9:19-25 (in English language edition).
Ekman, V.W., 1902. O m jordrotationens inverkan pA vindstr~immar i hafvet. Nyt Magazin for
Naturvidenskab, 40 (1): 37-64.
525
Ekman, V.W., 1905. On the influence of the earth's rotation on oceancurrents. Ark. Mat. Astron.
Fys., 2 (11): 1-53.
Hasselmann, K., 1962. On the non-linear energy transfer in a gravity-wave spectrum. Part I.
General theory. J. Fluid Mech., 12: 481-500.
Hasselmann, K., 1970. Wave-driven inertialoscillations.Geophys. Fluid Dyn., 1: 463-502.
Hasselmann, K., 1974. On the spectraldissipationof ocean waves due to white-capping. Boundary
Layer Meteorol., 6: 107-127.
Heaps, N.S., 1984. Development of numerical models for the prediction of currents. J. Soc. Underwater Technol., 10 (2): 8-18.
Huang, J.C., 1983. A review of the state-of-the-artof oil spillfate/behaviour models. Proc. 1983
Oil Spill Conf., San Antonio, Texas, Feb. 28-March 3, 1983. American Petroleum Institute,
pp. 313-322.
Huang, N.E., 1979. On surface driftcurrents in the ocean. J. Fluid Mech., 91: 191-208.
Jenkins, A.D., 1986. A theory for steady and variable wind- and wave-induced currents.J. Phys.
Oceanogr., 16: 1370-1377.
Jenkins, A.D., 1987. Wind- and wave-induced currents in a rotating sea with depth-varying eddy
viscosity. J. Phys. Oceanogr., 17: 938-951.
Johns, B., 1970. On the mass transport induced by oscillatory flow in a turbulent boundary layer.
J. Fluid Mech., 43- 177-185.
Johns, B., 1975. The form of the velocity profile in a turbulent shear wave boundary layer. J.
Geophys. Res., 80: 5109-5112.
Johns, B., 1977. Residual flow and boundary shear stress in the turbulent bottom boundary layer
beneath waves. J. Phys. Oceanogr., 7: 733-738.
Komen, G.J., Hasselmann, S. and Hasselmann, K., 1984. On the existence of a fully developed
wind-sea spectrum. J. Phys. Oceanogr., 14: 1271-1285.
Lai, R.Y.S. and Rao, D.B., 1976. Wind drift currents in deep sea with variable eddy viscosity.
Arch. Met. Geoph. Biokl., Ser. A, 25: 131-140.
Liu, A.°K. and Davis, S.H., 1977. Viscous attenuation of mean drift in water waves. J. Fluid Mech.,
81: 63-84.
Liu, Ph. L.-F., 1977. Mass transport in the free-surface boundary layers. Coastal Eng., 1: 207-219.
Longuet-Higgins, M.S., 1953. Mass transport in water waves. Philos. Trans. R. Soc. London, Ser.
A, 245: 535-581.
Longuet-Higgins, M.S., 1958. The mechanics of the boundary-layer near the bottom in a progressive wave. Appendix to Russell, R.C.H. and Osorio, J.D.C. An experimental investigation of
drift profiles in a closed channel. In: Proc. 6th Conf. on Coastal Eng. Council on Wave Research, Univ. of California, Berkeley, pp. 171-193.
Longuet-Higgins, M.S., 1970a. Longshore currents generated by obliquely incident sea waves, 1.
J. Geophys. Res., 75: 6778-6789.
Longuet-Higgins, M.S., 1970b. Longshore currents generated by obliquely incident sea waves, 2.
J. Geophys. Res., 75: 6790-6801.
Madsen, O.S., 1977. A realistic model of the wind-induced Ekman boundary layer. J. Phys. Oceanogr., 7: 248-255.
Madsen, O.S., 1978. Mass transport in deep-water waves. J. Phys. Oceanogr., 8: 1009-1015.
Miles, J.W., 1957. On the generation of surface waves by shear flows. J. Fluid Mech., 3: 185-204.
Mitsuyasu, H. and Honda, T., 1982. Wind-induced growth of water waves. J. Fluid Mech., 123:
425-442.
Pierson, W.J., 1962. Perturbation analysis of the Navier-Stokes equations in Lagrangian form
with selected linear solutions. J. Geophys. IRes., 67: 3151-3160.
Pollard, R.T., 1970. Surface waves with rotation: An exact solution. J. Geophys. Res., 75: 5895-5898.
Snyder, R.L., Dobson, F.W., Elliott, J.A. and Long, R.B., 1981. Array measurements of atmospheric pressure fluctuations above surface gravity waves. J. Fluid Mech., 102: 1-59.
526
Stokes, G.G., 1847. On the theory of oscillatory waves. Trans. Cambridge Philos. Soc., 8: 441-455.
Trowbridge, J. and Madsen, O.S., 1984a. Turbulent wave boundary layers. 1. Model formation
and first-order solutions. J. Geophys. Res., 89: 7989-7997.
Trowbridge, J. and Madsen, O.S., 1984b. Turbulent wave boundary layers. 2. Second order theory
and mass transport. J. Geophys. Res., 89: 7999-8007.
Llnliiata, 0. and Mei, C.C., 1970. Mass transport in water waves. J. Geophys. Res., 75: 7611-7618.
Ursell, F., 1950. On the theoretical form of ocean swell on a rotating earth. Mon. Not. R. Astron.
Soc., 6: 1-8.
Weber, J.E., 1981. Ekman currents and mixing due to surface gravity waves. J. Phys. Oceanogr.,
11: 1431-1435.
Weber, J.E., 1983a. Attenuated wave-induced drift in a viscous rotating ocean. J. Fluid Mech.,
137: 115-129.
Weber, J.E., 1983b. Steady wind- and wave-induced currents in the open ocean. J. Phys. Oceanogr., 13: 524-530.