## 1. Introduction

It is a well-known fact that surface waves carry mean momentum (Stokes 1847). For monochromatic waves in a viscous nonrotating fluid, the pioneering paper on this subject is Longuet-Higgins (1953). He applied an Eulerian fluid description with curvilinear coordinates to solve this problem. For a direct Lagrangian approach to wave drift in a rotating ocean, earlier treatments are found in Chang (1969), Ünlüata and Mei (1970), and Weber (1983). Also, the generalized Lagrangian mean formulation of Andrews and McIntyre (1978) can be applied to this problem.

Numerical general ocean circulation models (GCMs) are widely used to predict oceanic motions caused by the wind. Such models are usually based on an Eulerian description of motion. Furthermore, they often assume hydrostatic balance in the vertical, and accordingly they do not capture wind-induced surface waves. Even nonhydrostatic models do not resolve surface waves, or they lack the forcing conditions that allow the generation of wind waves. Thus, it is not expected that such models would include the mean drift resulting from periodic wave motion. As pointed out by McWilliams and Restrepo (1999), GCMs may underestimate the currents by taking only the mean horizontal wind stress into account, yielding the traditional Ekman current. One often asks if the Eulerian approach is capable of determining the total mean wave-induced current, because the Stokes drift in this case is confined between the wave crests and the wave troughs at the surface (Phillips 1977). The aim of the present paper is to compare results for the wave-induced drift obtained by an Eulerian and a Lagrangian analysis. To determine the wave-drift current in the entire fluid column from an Eulerian approach is rather laborious (e.g., Longuet-Higgins 1953), while it is fairly simple to do so from a direct Lagrangian starting point (Weber 1983). We shall therefore be content with making this comparison for the mass or volume fluxes in the oceanic surface layer. In this case the Eulerian approach is the simplest. Here we extend Phillips’ (1977) analysis to a viscous rotating ocean. For a simplified ocean with constant (eddy) viscosity we can use earlier results (e.g., Weber and Melsom 1993a) to obtain the mean Lagrangian fluxes. This approach yields the additional bonus that an explicit expression for the form drag is obtained. For simplicity, we consider periodic waves with amplitudes that vary slowly in time and space separately. Earlier, Jenkins (1986, 1987) treated these cases simultaneously.

This paper is organized as follows: First we compare the results for the mass fluxes derived by Eulerian and Lagrangian starting points in a viscous rotating ocean when the waves are 1) spatially periodic with amplitudes that may vary slowly in time, and 2) temporally periodic with amplitudes that may vary slowly in space. Then we derive a general set of equations that includes both these cases for a single wave component along the *x* axis. This set of equations is generalized to a spectral formulation, where the wave-forcing terms are evaluated for a theoretical one-dimensional frequency spectrum for a saturated sea. Last, we formulate our wave-influenced mass flux problem for a two-dimensional wave spectrum that can be obtained from an operational wave prediction model. The wave-induced forcing terms computed from this model drive a numerical barotropic ocean surge model. Results from this model are compared with results from similar simulations using the wind stress computed from the 10-m wind speed (Large and Pond 1981).

## 2. The Eulerian approach

*x*axis in a Cartesian coordinate system, where the

*x*and

*y*axis are situated at the undisturbed sea surface (see Fig. 1). The

*z*axis is vertical and directed upward. The corresponding unit vectors are (

**i**,

**j**,

**k**), respectively. Our system rotates with constant angular velocity

*f*/2 about the

*z*axis, where

*f*is the Coriolis parameter. We first consider a traditional Eulerian description where the velocity

**v**= (

*u*,

*υ*,

*w*) and the pressure

*p*are functions of time

*t*and the spatial coordinates (

*x*,

*y*, z). The Navier–Stokes equation and the continuity equation for a viscous, rotating fluid of constant density can be written as

*ρ*is the density,

*μ*is the dynamic viscosity, and

*g*is the acceleration due to gravity. Furthermore, ∇ and ∇

^{2}are the gradient and Laplacian operators, respectively. In the real ocean the presence of turbulence, and its generation, maintenance, and interaction with the mean flow are not known in detail. There are elaborate models for some of these problems, but they are often highly speculative. However, we know that laminar waves propagating in a turbulent fluid suffer attenuation (Ölmez and Milgram 1992). To model this effect in the simplest possible way, we make the traditional Boussinesq approximation of isotropic turbulence with a constant eddy viscosity, that is, all of our variables in (1) and (2) are averages and do not contain turbulent fluctuations. Hence, the entire effect of turbulence is embedded in the value of the viscosity coefficient in (1), hereinafter referred to as the dynamic eddy viscosity.

The wind and waves in our problem are directed along the *x* axis, and we assume that the variables are independent of *y*. We follow Phillips’ (1977) approach, and integrate the horizontal components of (1) between a constant depth *z* = −*H*, where the viscous stresses are assumed to vanish, and the material surface *z* = *ζ*(*x*, *t*); see Fig. 1. The surface layer is assumed to be so deep that it encompasses the Ekman layer as well as the deep-water wave field. In practice *H* will be comparable to the Ekman depth *D _{E}* =

*π*(2

*ν*/

*f*)

^{1/2}in the open ocean, where

*ν*=

*μ*/

*ρ*is the kinematic eddy viscosity.

*Q*

^{(x)}and

*Q*

^{(y)}represent the Lagrangian mass transport.

*τ*

^{(t)}is the tangential wind stress and

*τ*

^{(n)}is the normal wind stress along the sea surface. Utilizing (6), (3) and (4) reduce to

*x*axis, the velocity component in the

*y*direction is very small, which means that the friction term in (8) can be neglected. Similarly, the friction term

*μw*(

_{x}*z*=

*ζ*) in (7) is small. It practically vanishes when averaged over the wave cycle. Hence, to

*O*(

*ζ*

^{2}),

In a recent paper Ardhuin et al. (2004) discuss the mean mass transport by integrating the Eulerian equations in the vertical from the ocean bottom to the free surface. Like Hasselmann (1971), they define their mean flow by integrating the horizontal mean velocity to the position of the mean sea level, and the wave part (the Stokes mass transport) by the averaged integral of the velocity from the mean sea level to the position of the free surface. In this way the correlation between the variable air pressure and the free surface slope becomes a part of the forcing of the Stokes mass transport. In addition, Ardhuin et al. do not apply the dynamic surface boundary conditions, which introduce the normal and tangential surface stresses into the problem. In this way the role of the form drag [averaged last term on the right-hand side of (10)] as the main source term for the total wave-induced mean mass transport is obscured. We think that Phillips’ (1977) formalism, as used here, is the most convenient method for describing the total mean mass transport resulting from wind and waves in an Eulerian context.

*x*. The amplitude is spatially homogeneous, but may grow or decay slowly in time. By averaging over one wavelength, (10) and (11) reduce to

*Q*≡

*Q*

^{(x)}+

*iQ*

^{(y)}, where

*i*is the imaginary unit. In (12) the mean tangential wind stress is essentially responsible for the traditional Ekman transport. We are particularly interested in the wave-induced part of the mean transport. This is basically related to the action of the form drag

*τ*, defined as

_{D}*τ*

^{(n)}(Phillips 1977). Also, a fluctuating tangential viscous stress in the air (skin friction) may give rise to wave growth (Lamb 1932). However, the skin friction in phase with the surface elevation appears to be a minor factor in transferring energy from the wind to the wave field under realistic conditions (Chalikov and Makin 1991). The Ekman transport forced by

*Q*=

*Q*

_{Ekman}+

*Q*

_{wave}, we may introduce wave-induced volume fluxes (

*U*,

*V*) such that

*τ*can then be written, with

_{D}*W*≡

*U*+

*iV*, as

*τ*= 0), that is, decaying waves, the mass transport is zero when averaged over one inertial cycle; see Weber (1983), who derived this result from a direct Lagrangian approach. For waves in the absence of friction, this was first shown by Hasselmann (1970). In a nonrotating viscous ocean (

_{D}*f*= 0) with no wind, we find that the total mean wave momentum must be conserved. This means that the decaying Stokes drift, which in an Eulerian description is confined between the wave crests and the wave troughs, induces a compensating mean Eulerian current in the fluid.

*U*,

_{S}*V*), as easily can be seen by inserting for a linear wave component (e.g., Longuet-Higgins 1953). The averaged procedure (16) corresponds basically to the subdivision made by Hasselmann (1971). In the present problem the Stokes transport becomes

_{S}*ω*is the angular wave frequency and

*ζ̂*is a wave amplitude that is allowed to vary slowly with time (slow relative to the wave period 2

*π*/

*ω*). Accordingly, if we write the averaged integrals in (16) as

*W*≡

_{E}*U*+

_{E}*iV*. We realize that the Stokes transport terms −∂

_{E}*U*/∂

_{S}*t*and −

*fU*appear as forcing terms in the

_{S}*x*and

*y*directions, respectively, in this equation for the wave-induced Eulerian transport. However, it is somewhat deceptive to regard the problem in this way. The important fluxes in the oceanic surface layer affecting the mass balance are the Lagrangian fluxes, that is, in general (

*U*+

_{E}*U*,

_{S}*V*+

_{E}*V*). As shown by (15), the only forcing term in the spatially periodic wave-drift problem is the form drag.

_{S}## 3. Lagrangian approach

*a*,

*b*,

*c*). The particle position at later times (

*X*,

*Y*,

*Z*) and the pressure

*P*will then be functions of

*a*,

*b*,

*c*, and time

*t*. Velocity components and acceleration are given by (

*X*,

_{t}, Y_{t}*Z*, ) and (

_{t}*X*,

_{tt}*Y*,

_{tt}*Z*), respectively. For plane waves along the

_{tt}*x*axis, the deviations (

*x*,

*y*,

*z*,

*p*) from the initial state will not depend on

*b*. We then may write

*q*≡

*x*+

_{t}*iy*. By including the effect of the earth’s rotation, the equations for the conservation of momentum and mass can be obtained from Lamb (1932). With the present notation, the momentum equations become, to second order in wave steepness,

_{t}^{2}

_{L}≡ ∂

^{2}/∂

*a*

^{2}+ ∂

^{2}/∂

*c*

^{2}is the Laplacian operator in Lagrangian coordinates, and

*J*(

*A*,

*B*) ≡

*A*−

_{a}B_{c}*A*is the Jacobian. The conservation of mass (here volume) leads to

_{c}B_{a}*c*= 0. When averaging in time or space, we then obtain

*β*is the wave growth/decay rate, and

*ζ*

_{0}is the initial wave amplitude. The real coefficient

*B*is related to the vorticity part of the primary wave field. Neglecting the effect of the tangential fluctuating wind stress in phase with the surface elevation, which is of minor influence in comparison with the fluctuating wind stress in phase with the surface slope in the wave generation process, we find that

_{r}*k*is the wavenumber, and

*γ*= (

*ω*/2

*ν*)

^{1/2}is the inverse viscous boundary layer thickness at the surface. The form drag in this problem can be written

*σ̃*is the real part of the fluctuating wind stress component normal to the sea surface, and

*is the real part of the vertical displacement of the primary wave field. Inserting for the primary wave field from Weber and Melsom (1993a), we find that*

*z̃**β*= −2

*νk*

^{2}. The form drag (28), with

*ζ̂*=

*ζ*

_{0}exp(

*βt*), can also be written

*ρU*=

_{S}*ρωζ̂*

^{2}/2 is the total horizontal wave momentum, and

*τ*is the virtual wave stress originally introduced by Longuet-Higgins (1969). It can be written [Weber 1997, his (77)] as

_{w}*ν*as the bulk turbulent eddy viscosity. We see right away from (30) that for purely decaying waves (

*τ*= 0)

_{D}*τ*transfers the lost wave momentum from all kind of dissipative processes.

_{w}*p*

_{am}over a smooth wave component can be approximated by

*ρ*is the density of the air and

_{a}*u*

_{10}is the wind speed at 10-m height. Hence, we can write the form drag (27) over a wave component as

*s*=

*ρ*/

_{a}*ρ*. Associating the wave component in question with the peak of the wind–wave spectrum, we have approximately that

*ω*/

*k*=

*u*

_{10}for this peak. Applying the deep-water dispersion relation

*ω*

^{2}=

*gk*, (36) can be written as

*ν*= 15 cm

^{2}s

^{−1}for

*u*

_{10}= 10 m s

^{−1}and

*ν*= 122 cm

^{2}s

^{−1}for

*u*

_{10}= 20 m s

^{−1}. For rougher sea, where breaking occurs, the form drag is larger than over a smooth wave (Banner 1990). Then (35) underestimates the value of the form drag. We shall return to the effect of wave breaking on the eddy viscosity in section 5.

## 4. Temporally periodic waves

_{x}= 0). Disregarding the mean tangential wind stress, and considering only the mean volume fluxes in (14) induced by temporally periodic wave motion, (10) and (11) finally reduce to

*τ*is defined by (13) and

_{D}*C*=

_{g}*ω*/(2

*k*) is the group velocity. In the derivation of (38), we have neglected the small friction terms

*R*

^{(mn)}

**i**

_{m}

**i**

_{n}] of Longuet-Higgins and Stewart (1960) can be written for a single wave component as

*l*

_{1}and

*l*

_{2}are the horizontal wavenumber components, and

*E*=

*MC*is the wave energy density (Starr 1959). For waves along the

*x*axis (

*l*

_{1}=

*k*,

*l*

_{2}= 0) we find that

*M*=

*ρU*, and hence

_{S}*E*=

*ωρU*/

_{S}*k*. By insertion into (39), it is then seen that the last term of the wave-induced stress in (38) is just the remaining nonzero component of the divergence of the radiation stress tensor per unit density.

*τ*becomes equal to the virtual wave stress

_{D}*τ*, and the relation (33) reappears.

_{w}*D*for deep-water gravity waves with an uncontaminated surface is determined by the irrotational part of the wave field (Phillips 1977), and is given by

*T*=

*λ*/

*C*in this case is thus given by

## 5. Effect of wave breaking: A simplistic approach

*ζ*

_{cr}and

*ζ*

_{0}, respectively, the energy lost per wavelength for an infinitely long wave train during the breaking event is

*T*. Then, by analogy with (44), this energy loss must be compensated by the work done by the equilibrium form drag, that is,

_{B}*β*is the growth rate of the fastest growing waves. From experimental data we find that

*U*

_{*}is the friction velocity in the air. A typical value for

*K*is 1 × 10

^{−2}(Plant 1982). Furthermore, experimental observations indicate that Δ

*e*lies between 10

^{−2}and 10

^{−1}(Melville and Rapp 1985). We then obtain approximately from (49)

*c*≈ 2 × 10

_{D}^{−3}for the drag coefficient. We note the interesting fact that this expression for the form drag has the same functional dependence of the wind velocity and the wave steepness as (35) for nonbreaking waves.

*ν*associated with a breaking wave component can now be obtained by combining (33) and (52). For the spectral peak component this leads to

_{B}*K*= 1 × 10

^{−2}and

*c*= 2 × 10

_{D}^{−3}, as suggested above, the coefficient in (53) becomes 2 × 10

^{−5}. This is about 25% higher than the corresponding value estimated for smooth waves in (37), and appears to be a reasonable result.

*U*

_{*}in the air and

*g*as

*U*

^{3}

_{*}/

*g*, or equivalently as

*u*

^{3}

_{10}/

*g*, because

*U*

^{2}

_{*}=

*c*

_{D}u^{2}

_{10}. This also follows from applying the law-of-the-wall distribution

*ν*= −

*κu*

_{*}

*c*for the eddy viscosity in the ocean (Madsen 1977). Here

*κ*is von Kármán’s constant and

*u*

_{*}is the friction velocity in the ocean (

*u*

_{*}=

*s*

^{1/2}

*U*

_{*}). When integrating this distribution in the vertical over one wavelength, we find for the vertically averaged eddy viscosity that

*ν*

*u*

^{3}

_{10}/

*g*. Based on analogy with grid-induced turbulence, Kitaigorodskii [1996, his (25)] obtained for the bulk eddy viscosity in breaking waves in our notation

^{−1}would yield

*ν*∼1600 cm

^{2}s

^{−1}. Breaking or white capping in the open sea will increase the eddy viscosity, but not that dramatically. We therefore argue that our (53) is much closer to a realistic modeling of the magnitude of the wave-induced eddy viscosity than (54). It should be noted that the viscosity discussed here is relevant for the transport of mean momentum, that is, for ocean currents. For the wave field itself the appropriate eddy viscosity should be considerably less (Jenkins 1989; Weber and Melsom 1993a).

## 6. Spectral considerations for a fully developed sea

We now apply the general form (46) of the transport equations, valid for sinusoidal waves with temporally and spatially modulated amplitudes. In generalizing, going from a single wave component to a fully developed sea, our wave amplitude must be associated with the spectral distribution of the wave energy. Because we here have considered plane waves along the *x* axis, we shall be content with considering the one-dimensional wave spectrum for illustrative purposes. However, an extension to a two-dimensional spectrum is straightforward. This is left for the next section where we apply a wave prediction model.

*ζ*

^{2}

_{0}= 2

*φ*(

*ω*)Δ

*ω*, where

*φ*is the frequency spectrum (Bye 1967), the equilibrium virtual wave stress in (52) may be written in spectral form as

*α*between 6 × l0

_{T}^{−2}and 11 × 10

^{−2}(Phillips 1985). The expressions (55) and (57) will be integrated from the peak spectral frequency

*ω*to a high-frequency limit

_{p}*ω*, representing the upper tail of the spectrum (

_{h}*ω*≫

_{h}*ω*). Utilizing that

_{p}*ω*

^{2}

_{p}=

*gk*, and

_{p}*ω*

^{2}

_{h}=

*gk*=

_{h}*rg*

^{2}/

*U*

^{2}

_{*}, where

*r*

^{1/2}is a constant of order unity (Phillips 1985), we find from (55) that

*α*= 8 × 10

_{T}^{−2},

*r*

^{1/2}= 5 × 10

^{−1},

*K*= 1 × 10

^{−2}, and

*s*= 1.25 × 10

^{−3}, we find that the ratio (61) is about 0.6. This is the order of magnitude one would expect for an ocean where the dissipation is dominated by wave breaking (Mitsuyasu 1985; Melville and Rapp 1985; Weber and Melsom 1993b).

*T*of about 12 h (

_{s}*T*= 0.5 × 10

_{s}^{5}s), while the length scale

*L*of the wind field is 500 km. The maximum wind speed

_{s}*u*

_{10}is taken to be 20 m s

^{−1}. Utilizing our adopted values for the parameters, we find that

## 7. Model results

*U*and

*V*are obtained by a numerical simulation. First we introduce the two-dimensional frequency spectrum

*F*(ϖ,

*θ*). Here ϖ =

*ω*/2

*π*, and

*θ*is the angle of the wavenumber vector

**with respect to the**

*κ**y*axis, measured positive in the clockwise direction, that is,

**= (**

*κ**κ*sin

*θ*,

*κ*cos

*θ*). For deep water, the equation for the evolution of the wave spectrum becomes (Komen et al. 1994)

**C**

_{g}= [

*C*

^{(x)}

_{g},

*C*

^{(y)}

_{g}] = (

*π*ϖ sin

*θ*/

*κ*,

*π*ϖ cos

*θ*/

*κ*). Furthermore,

*S*

_{in}is the rate of energy input from the atmosphere,

*S*

_{nl}is the contribution from components of different wavenumbers by nonlinear wave–wave interaction, and

*S*

_{ds}is the wave energy dissipation.

*h*(

*x*,

*y*,

*t*) and a variable air surface pressure

*P*

_{0}(

*x*,

*y*,

*t*) into the model. We must also include some sort of bottom friction to dampen inertial oscillations. Here we choose a linear Rayleigh friction with a constant friction coefficient

*R*. We take the depth of integration to be considerably larger than the wavelength of the most energetic surface waves, implying that we can apply deep-water wave theory. Extending our wave-forcing formulation in (56) to two dimensions, the ocean surge model for a wave-influenced surface stress can be written in vector form as

_{f}**V**= (

*U*,

*V*). Furthermore

*τ*_{wind}= [

*τ*

^{(x)}

_{wind},

*τ*

^{(y)}

_{wind}] is the horizontal wind stress. Its form will be specified later. According to (46), the wave-induced stress component is given by

**V**

_{S}= (

*U*,

_{S}*V*) is the Stokes transport. For a single wave component, we have in the present notation that

_{S}*U*= (

_{S}*ωζ*

^{2}

_{0}sin

*θ*)/2 and

*V*= (

_{S}*ωζ*

^{2}

_{0}cos

*θ*)/2. Associating now the wave amplitude for a single component in the spectrum by

*ζ*

^{2}

_{0}= 2

*F*(ϖ,

*θ*)ΔϖΔ

*θ*, we obtain for the spectral distribution

*τ*

^{(x)}

_{w}} and {

*τ*

^{(y)}

_{w}} in (65) are related to the wave energy dissipation

*S*

_{ds}in (63) caused by turbulent dissipation and breaking [e.g., (33) and (52)]. In our spectral formulation, we can write

*π*ϖ sin

*θ*ΔϖΔ

*θ*

**i**, and 2

*π*ϖ cos

*θ*ΔϖΔ

*θ*

**j**, respectively, and integrate over the spectrum, we obtain

*} is given by (67), and the spectral form drag is*

**τ**_{w}To produce forcing data for the storm surge model, the numerical ocean wave model (WAM; Komen et al. 1994) was run for a period of 2 months, starting at 1 January 2004. As input to the wave model we used analyzed winds from the European Centre for Medium-Range Weather Forecasts (ECMWF). The model domain, which is the same for both the wave model and the storm surge model, is shown in Fig. 2. The grid is rotated spherical with the equator located at 60°N. The horizontal resolution is 0.45° in both directions. Every third hour, the wave model calculates the forcing terms on the right-hand side of (64). The storm surge equations in (64) were discretized on a C grid with centered differences in both time and space. To remove any possible spurious modes, an Eulerian (forward) time step was applied every 20 time steps. At each time step, the surface elevation was updated first and then each of the two horizontal components was updated. In this procedure, the Sielecki method (Sielecki 1968) was used. This means that all the updated variables are used immediately in the subsequent equations. The bathymetry for the computational domain is in some places more than 4000 m deep. For such deep waters, the Courant–Friederichs–Lewy (CFL) criterion imposes an extremely short time step on the storm surge model. Therefore, the water depth is limited everywhere to 200 m in this study. This of course, makes the experiments rather unrealistic as storm surge simulations for the real world. However, because the main scope of this experiment is not to forecast the surge as realistically as possible but to quantify the relative effects of the wave-forcing terms on the right-hand side of (64), we believe that this simplification can be justified. Another simplification is the removal of all the open boundaries for the storm surge model. This makes it easier to handle the boundary conditions in the model. The introduction of these artificial walls may introduce spurious reflections of long barotropic waves. The justification for this is again the fact that the main purpose of this investigation is to quantify the effects of the wave-induced forcing. Reflected barotropic waves will probably be present in both the experiment and the control runs. Our assumption is that this will have little effect on the difference between the two runs.

*u*

_{10}and

*υ*

_{10}are the

*x*and

*y*components of the 10-m wind vector

**v**

_{10}. The model drag coefficient

*c*is independent of the wind when the wind speed is below the threshold value of 11 m s

_{DM}^{−1}and is linearly dependent on the wind for stronger wind speeds, that is,

**v**

_{10}| > 11 m s

^{−1}is introduced to model, in a crude way, the increasing effect of the sea state on the momentum transfer from the atmosphere to the ocean at higher wind speeds. For the control run, the stresses calculated from (72) and (73) were used instead of

*τ*_{wind}+

*τ*_{wave}on the right-hand side of (64). Second, an experiment was run for the same period, but with the wave-forcing term

*τ*_{wave}, defined by (65), calculated from the WAM. For simplicity, the turbulent wind-forcing term

*τ*_{wind}in (64) for this case was parameterized as in (72), but with a constant drag coefficient (

*c*= 1.2 × 10

_{D}^{−3}). As already mentioned, the simulation period was the first two months of 2004. January was basically used as a period for spinning up the models. Accordingly, all of the results presented here will be for February 2004, which is taken to be the experiment period.

*c*= 1.2 × 10

_{D}^{−3}from Large and Pond (1981), which we used in our runs for the wind-induced stress, contains some wave effects as well. To remedy this, we calculated the average flat drag coefficient from five selected stations depicted in Fig. 2 (black squares) such that the average stress of the control run with (72) and (73) was equal to the mean value of

*τ*_{wind}+

*τ*_{wave}at these stations. This indicated that our wind stress could be approximated by

We realize from Table 1 that on these five stations the magnitude of the average virtual wave stress alone is nearly 50% of mean stress from the 10-m wind speed. This is in good accordance with our previous estimates in section 6. All terms involving the Stokes drift in (65) are, on the average, one to two orders of magnitude smaller than the total stress. This also agrees well with our findings in section 6. If we used the alternative expression in (71) for the wave-induced stress, the magnitudes of the radiations stress terms are found to be equally small. To illustrate further the effect of the sea-state-dependent surface stresses, the time series for the wind-induced stress in (74) and the wave-induced stress in (65) is plotted in Fig. 3 together with the mean stress in (72) and (73) calculated from the 10-m wind speed. Here, we have depicted the results from stations 1 and 5. From Fig. 3 we note the interesting fact that for small-to-moderate winds, the wind-induced stress is larger than the wave-induced stress. However, for stronger winds (large peaks in the plot) the wave-induced part is the largest. At such winds our computed value of |(*τ*_{wind} + *τ*_{wave})/*ρ*| exceeds the traditional value of |*τ*_{10}/*ρ*| in the forcing terms for the surge. The quantification of the enhanced influence of the wave part of the stress for stronger winds is important, because the assessment of damage caused by the surge is particularly relevant in the case of storm events.

## 8. Summary and concluding remarks

We have demonstrated that integration of the Eulerian momentum equation from a constant depth of vanishing motion to the oscillating surface yields the same equations for the volume transport in periodic waves as that obtained from a direct Lagrangian analysis to second order in the wave steepness. It is found that the form drag associated with the action of the fluctuating wind stress over the wave slopes is the only source term in the equation for the integrated Lagrangian volume transport induced by spatially periodic waves. Accordingly, waves that decay in time in the absence of external forcing do not induce any net transport in a rotating ocean. This is valid whether the decay is due to viscous dissipation or wave breaking. In the case of temporally periodic waves, where the wave amplitude may grow or decay slowly in space, we show that the horizontal divergence of the total wave momentum flux is an additional source term in the equations for the wave-induced volume transports. Alternatively, this term can be written in terms of the radiation stress, as shown by Phillips (1977). Comparison between analytical and empirical expressions for the form drag over smooth waves in a balanced state (statistically steady waves) leads to a simple estimate for the bulk eddy viscosity in the surface layer associated with wind waves. By modeling wave breaking in a simple way, a similar formula for the eddy viscosity in a saturated sea, where breaking dominates the dissipation process, is obtained. On the basis of the results for a single wave component, we derive equations for the wave-induced volume transports in a fully developed sea where the wave spectrum may change slowly in space and time. For an idealized one-dimensional frequency spectrum (Toba 1973), and for reasonable estimates for the time and space variation of the wind, our equations for the wave-induced volume fluxes appear to have realistic forcing terms. For the precise form of the wave spectrum in a real ocean, these equations need input from an ocean wave prediction model (e.g., Komen et al. 1994).

The wave-forcing terms for a storm surge model have been calculated for a 2-month period in 2004 by running the WAM (Komen et al. 1994) over a model domain covering the northern North Atlantic and the Nordic Seas. These terms were then used to force a storm surge model for the same period. The calculated surface elevations were compared with the results from a control run where the surface stresses were obtained in the traditional way, using the 10-m wind speed. In this way it was found that the non-wave-dependent drag coefficient for the wind stress part of the forcing could be approximated as *c _{D}* = 0.95 × 10

^{−3}. Time series of each individual forcing term for selected locations revealed that the contribution from the virtual wave stress amounted to about 50% of the total forcing for moderate to strong winds. The terms involving the time rate of change of the Stokes transport and the radiation stress in the alternative formulation were at least one order of magnitude smaller than the total stress. In the case of storm events with rough seas, the wave-induced part of the stress is larger than the wind stress. This is an important finding, because it is particularly in connection with strong winds that reliable surge simulations are needed.

The order-of-magnitude estimates from the idealized spectral formulation in section 6 for the wave-induced forcing were remarkably close to those obtained from the numerical ocean simulations in section 7, using the WAM. This lends support to the robustness of the present formulation for the wave-influenced transport in the oceanic surface layer. For shallow waters, where the influence of bottom friction and wave breaking are more prominent, the effect of the radiation stresses will also be larger. This may change the balance between the forcing terms in the surge equation. A wave-influenced storm surge model for a shallow coastal region is clearly the next step in line for this type of investigation.

## Acknowledgments

This study has been performed under the Strategic University Programme “Modelling of Currents and Waves for Sea Structures” funded by The Research Council of Norway.

## REFERENCES

Andrews, D. G., and M. E. McIntyre, 1978: An exact theory of nonlinear waves on a Lagrangian-mean flow.

,*J. Fluid Mech.***89****,**607–646.Ardhuin, F., B. Chapron, and T. Elfouhaily, 2004: Waves and the air–sea momentum budget: Implications for ocean circulation modeling.

,*J. Phys. Oceanogr.***34****,**1741–1755.Banner, M. L., 1990: The influence of wave breaking on the surface pressure distribution in wind–wave interactions.

,*J. Fluid Mech.***211****,**463–495.Bye, J. A. T., 1967: The wave drift current.

,*J. Mar. Res.***25****,**85–102.Chalikov, D. V., and V. K. Makin, 1991: Models of the wave boundary layer.

,*Bound.-Layer Meteor.***56****,**83–99.Chang, M-S., 1969: Mass transport in deep-water long-crested random gravity waves.

,*J. Geophys. Res.***74****,**1515–1536.Hasselmann, K., 1970: Wave-driven inertial oscillations.

,*Geophys. Astrophys. Fluid Dyn.***1****,**463–502.Hasselmann, K., 1971: On the mass and momentum transfer between short gravity waves and larger-scale motions. Part 1.

,*J. Fluid Mech.***50****,**189–205.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.Jenkins, A. D., 1989: The use of a wave prediction model for driving a near-surface current model.

,*Dtsch. Hydrogr. Z.***42****,**133–149.Kitaigorodskii, S. A., 1996: The influence of breaking wind waves on the aerodynamic roughness of the sea surface as seen from below.

*The Air–Sea Interface: Radio and Acoustic Sensing, Turbulence and Wave Dynamics,*M. A. Donelan, W. H. Hui, and W. J. Plant., Eds., The Rosenstiel School of Marine and Atmospheric Science, 177–187.Komen, G. J., L. Cavaleri, M. Doneland, K. Hasselmann, S. Hasselman, and P. A. E. M. Janssen, 1994:

*Dynamics and Modelling of Ocean Waves*. Cambridge University Press, 533 pp.Lamb, H., 1932:

*Hydrodynamics*. 6th ed. Cambridge University Press, 738 pp.Large, W. G., and S. Pond, 1981: Open ocean momentum flux measurements in moderate to strong winds.

,*J. Phys. Oceanogr.***11****,**324–336.Longuet-Higgins, M. S., 1953: Mass transport in water waves.

,*Philos. Trans. Roy. Soc. London***245A****,**535–581.Longuet-Higgins, M. S., 1969: A nonlinear mechanism for the generation of sea waves.

,*Proc. Roy. Soc. London Soc. London***311A****,**371–389.Longuet-Higgins, M. S., and R. W. Stewart, 1960: Changes in the form of short gravity waves on long waves and tidal currents.

,*J. Fluid Mech.***8****,**565–583.Madsen, O. S., 1977: A realistic model for the wind-induced Ekman boundary layer.

,*J. Phys. Oceanogr.***7****,**248–255.McWilliams, J. M., and J. M. Restrepo, 1999: The wave-driven ocean circulation.

,*J. Phys. Oceanogr.***29****,**2523–2540.Melsom, A., 1996: Effects of wave breaking on the surface drift.

,*J. Geophys. Res.***101****,**12071–12078.Melville, W. K., and R. J. Rapp, 1985: Momentum flux in breaking waves.

,*Nature***317****,**514–516.Mitsuyasu, H., 1985: A note on the momentum transfer from wind to waves.

,*J. Geophys. Res.***90****,**3342–3345.Ölmez, H. S., and J. H. Milgram, 1992: An experimental study of attenuation of short water waves by turbulence.

,*J. Fluid Mech.***239****,**133–156.Phillips, O. M., 1977:

*The Dynamics of the Upper Ocean*. 2d ed. Cambridge University Press, 336 pp.Phillips, O. M., 1985: Spectral and statistical properties of the equilibrium range in wind-generated gravity waves.

,*J. Fluid Mech.***156****,**505–531.Plant, W. J., 1982: A relationship between wind stress and wave slope.

,*J. Geophys. Res.***87****,**1961–1967.Sielecki, A., 1968: An energy-conserving difference scheme for the storm surge equations.

,*Mon. Wea. Rev.***96****,**150–156.Starr, V. P., 1959: Hydrodynamical analogy to

*E*=*mc*2.,*Tellus***11****,**135–138.Stokes, G. G., 1847: On the theory of oscillatory waves.

,*Trans. Camb. Philos. Soc.***8****,**441–455.Toba, Y., 1973: Local balance in the air-sea boundary processes. III: On the spectrum of wind waves.

,*J. Oceanogr. Soc. Japan***29****,**209–220.Ünlüata, Ü, and C. C. Mei, 1970: Mass transport in water waves.

,*J. Geophys. Res.***75****,**7611–7618.Weber, J. E., 1983: Attenuated wave-induced drift in a viscous rotating ocean.

,*J. Fluid Mech.***137****,**115–129.Weber, J. E., 1997: Mass transport induced by surface waves in a viscous rotating fluid.

*Free Surface Flows with Viscosity,*P. A. Tyvand, Ed., Computational Mechanics Publications, 37–67.Weber, J. E., 2003: Wave-induced mass transport in the oceanic surface layer.

,*J. Phys. Oceanogr.***33****,**2527–2533.Weber, J. E., and A. Melsom, 1993a: Transient ocean currents induced by wind and growing waves.

,*J. Phys. Oceanogr.***23****,**193–206.Weber, J. E., and A. Melsom, 1993b: Volume fluxes induced by wind and waves in a saturated sea.

,*J. Geophys. Res.***98****,**4739–4745.

Model domain for the wave and storm surge model runs. The dashed line is the monthly (February 2004) mean difference in surface elevation (m) between the control run and the experiment (positive values when the experiment elevation is larger than the surface elevation of the control run). The black squares are the locations of the stations, together with the station numbers, where the modeled wind- and wave-induced forcing has been compared with the surface forcing from the 10-m wind speed (Table 1).

Citation: Journal of Physical Oceanography 36, 11; 10.1175/JPO2951.1

Model domain for the wave and storm surge model runs. The dashed line is the monthly (February 2004) mean difference in surface elevation (m) between the control run and the experiment (positive values when the experiment elevation is larger than the surface elevation of the control run). The black squares are the locations of the stations, together with the station numbers, where the modeled wind- and wave-induced forcing has been compared with the surface forcing from the 10-m wind speed (Table 1).

Citation: Journal of Physical Oceanography 36, 11; 10.1175/JPO2951.1

Model domain for the wave and storm surge model runs. The dashed line is the monthly (February 2004) mean difference in surface elevation (m) between the control run and the experiment (positive values when the experiment elevation is larger than the surface elevation of the control run). The black squares are the locations of the stations, together with the station numbers, where the modeled wind- and wave-induced forcing has been compared with the surface forcing from the 10-m wind speed (Table 1).

Citation: Journal of Physical Oceanography 36, 11; 10.1175/JPO2951.1

Time series of the surface stresses (m^{2} s^{−2}) at two selected stations (1 and 5) for February 2004. The locations of these stations are depicted in Fig. 2. The dotted lines are the stresses calculated from the 10-m wind speed. The solid thick lines are the wave-induced stress (65) and the solid thin lines are the wind-induced stress (74).

Citation: Journal of Physical Oceanography 36, 11; 10.1175/JPO2951.1

Time series of the surface stresses (m^{2} s^{−2}) at two selected stations (1 and 5) for February 2004. The locations of these stations are depicted in Fig. 2. The dotted lines are the stresses calculated from the 10-m wind speed. The solid thick lines are the wave-induced stress (65) and the solid thin lines are the wind-induced stress (74).

Citation: Journal of Physical Oceanography 36, 11; 10.1175/JPO2951.1

Time series of the surface stresses (m^{2} s^{−2}) at two selected stations (1 and 5) for February 2004. The locations of these stations are depicted in Fig. 2. The dotted lines are the stresses calculated from the 10-m wind speed. The solid thick lines are the wave-induced stress (65) and the solid thin lines are the wind-induced stress (74).

Citation: Journal of Physical Oceanography 36, 11; 10.1175/JPO2951.1

Forcing terms (m^{2} s^{−2}) on the right-hand side of (64) at five selected stations. The table shows the monthly average for February 2004.