The role of surface waves in the ocean mixed layer

Transcription

1 The role of surface waves in the ocean mixed layer STEPHEN E. BELCHER, JEFF A. POLTON, DAVID M. LEWIS & MIGUEL A.C. TEIXEIRA Department of Meteorology, University of Reading, Berkshire, United Kingdom (submitted October 23) Current affiliation: Department of Mathematics, University of Liverpool, Liverpool, United Kingdom. Current affiliation: Department of Geophysics, University of Lisbon, Portugal Corresponding author address: S. E. Belcher. Department of Meteorology, University of Reading, PO Box 243, Reading, Berkshire RG6 6BB, United Kingdom.

2 ABSTRACT We analyse the effects of Stokes drift of surface waves on the mean current and turbulent fluctuations in the ocean mixed layer. Because the Stokes drift decays with depth, vorticity that is initially vertical is tilted into the horizontal and stretched. The vorticity arises from either small scale vorticity associated with three-dimensional turbulent eddying motions, which yields the turbulent vortex force, or from the planetary vorticity associated with rotation of the Earth, which yields the Coriolis Stokes forcing. The dynamical processes are investigated using large eddy simulations (LES) that resolve the large-scale three-dimensional turbulent motions and by developing simplified models. The Coriolis Stokes forcing is shown to change significantly the Ekman balance between Coriolis force and divergence of the turbulent stress that is usually assumed to hold in the wind-driven mixed layer. The result is that the current profile is turned further southwards and has less mean shear in the windward direction than the standard Ekman model. Results from LES and observations both agree well with a simplified model that accounts for the Coriolis Stokes forcing. We demonstrate how the Coriolis Stokes forcing can be accounted for by solving the standard Ekman balance but with a modified boundary condition at the sea surface, thereby obviating the need to resolve the thin layer where the Coriolis Stokes forcing actually acts. The effects of the turbulent vortex force on the three-dimensional turbulent eddying motions are illustrated with a linearised model. In the linear model, initially isotropic turbulence is distorted by the planar sea surface and by the strain associated with the Stokes drift of the surface waves. The linear dynamics produces streamwise vortices, i.e. Langmuir circulations, whose statistics are shown to agree well with the LES. The turbulence produced is totally different from turbulence distorted by mean shear, as in the atmospheric boundary layer. Finally, these two strands are brought together. The turbulence is shown to have characteristics of Langmuir turbulence when the mean shear is smaller than the strain rate due to the Stokes drift. The Coriolis Stokes forcing reduces the mean shear in the wind-driven mixed layer, ultimately to levels below the strain rate associated with the Stokes drift. When this happens, the turbulent mixing becomes dominated by Langmuir circulations. In this sense the Coriolis-Stokes forcing preconditions the wind-driven mixed layer to Langmuir circulations. Diagnostics from the LES support this interpretation. 1

3 1. Introduction The wind blowing over the ocean surface generates surface waves, and turbulence and mean currents in the ocean mixed layer. Observations reveal that the mean current profile in the wind-driven mixed layer has strong shear near the air-sea interface and the direction of the mean current turns strongly with depth. The near-surface current vector makes an angle of between and 45 degrees with the wind direction, and then makes the much larger angle of about 75 degrees at depths between 5 and 2m (see Lewis & Belcher 23). As we shall see later, the standard Ekman model with a constant density profile, and with either a constant or depth varying eddy viscosity, cannot reproduce this variation with depth. The turbulent mixing in the wind-driven mixed layer is also observed to be markedly different from a standard pressure-driven turbulent boundary layer near a flat surface, such as in the atmospheric boundary layer. In particular, organised rolls are observed to form aligned with the prevailing wind direction (e.g. Leibovich 1983). These rolls are manifest as streaks of foam and debris at the surface along the lines of convergence between the rolls. These streamwise rolls are known as Langmuir circulations and are believed to arise through deformation of small scale vorticity by the surface waves (e.g. Leibovich 1983). These observations raise a number of questions. Firstly, do ocean surface waves change the mean flow in the ocean mixed layer? The answer to this question is important because at present numerical models of the large-scale ocean circulation do not represent surface wave processes. This lack of representation is perhaps not surprising: the motions directly associated with ocean surface waves are confined to the upper ocean, within a distance of order k 1 = λ/2π m of the surface (here k is the wavenumber and λ is the wavelength of the dominant waves). So a second question is: by what mechanisms can surface waves change the mean flow to greater depths? Finally, there is now a well-developed instability theory for how a current profile below the air-sea interface is unstable to rolls through interaction with surface waves (Leibovich 1983). But how does this instability operate in the real ocean mixed layer, when the flow is fully turbulence and the current profile turns with height? Here we describe results of recent and ongoing research that aims to answer these questions, with a focus on the effects of Stokes drift on the dynamics of the wind-driven mixed layer. In section 2 we recall theory of surface waves and Stokes drift on a rotating ocean. In section 3 we analyse the effects of a key wave process on the mean current profile in the mixed layer. In section 4 we outline a simple model that reproduces quantitative features of 2

4 the turbulence in the mixed layer. This simple model suggests a mechanism for the production of Langmuir circulations in the fully turbulent mixed layer and a way of diagnosing the presence of Langmuir circulations in the mixed layer. These considerations lead to the main new result of the present paper, namely that interaction of Stokes drift with the planetary vorticity reduces mean streamwise shear in the ocean mixed layer, which then preconditions the mixed layer to the development of Langmuir circulations. These ideas are then tested against large eddy simulations of the mixed layer that account for key wave processes in section 5. Finally, conclusions are given in section Surface waves and Stokes drift on a rotating ocean A monochromatic sinusoidal surface water wave of small amplitude, such that ak is small (a is the wave amplitude), propagating on a quiescent non-rotating ocean is well described by linear theory. At a fixed point the Eulerian velocity induced by the passing water wave then oscillates in time with zero mean. There is however a slow net Lagrangian motion of water parcels in the direction of the wave, which is explained as follows (refer to figure 1). When under the crest of the wave the Eulerian velocity is horizontal and in the direction of the wave. Hence the water parcel moves instantaneously horizontally in the direction of the wave. At a later time when the wave has propagated, the backward facing slope is above the water parcel. The Eulerian velocity is now directed downwards, and so the water parcel moves instantaneously downwards. Later still, when the wave trough is above the water parcel, the Eulerian velocity and parcel motion is backwards. Finally, when the forward facing slope is above the parcel, the Eulerian velocity and the parcel motion is upward. At leading order in wave slope, ak, this sequence leads to the water parcels exhibiting circular orbits, with one circuit per wave cycle. There is however a nonlinear correction. When under the wave crest the water parcel is at its highest point in the orbit and closest to the mean water level; when the water parcel is below the trough the water parcel with at its lowest point in the orbit and furthest from the mean water level. Since the horizontal Eulerian velocity associated with the wave decays with distance from the mean water level, the horizontal speed and hence the displacement of the water parcel is slightly greater when it lies under a wave crest than the return displacement when it lies below the trough. The result is a slow net drift, the Stokes drift, of the water parcel. According to small amplitude theory the Stokes drift due to a monochromatic surface water wave on a non-rotating ocean varies with depth as u s = U s ke 2kz, (1) 3

5 where U s = (ak) 2 c, c is the phase speed of the wave, which is given by c 2 = g/k and k is the unit vector in the direction of wave propagation. Since the Stokes drift itself decays with depth there is a Lagrangian strain: fluid elements that are initially vertical are tilted into the horizontal and stretched. The significance here of the Stokes drift and the associated Lagrangian strain is that, in an inviscid fluid, vortex lines are carried with fluid line elements. Hence any vorticity that is initially vertical is tilted into the horizontal and stretched. The stretching then intensifies the vorticity. There are two sources of vorticity in the wind-driven mixed layer. Firstly, there is the small scale vorticity associated with the three dimensional turbulent motions actually within the mixed layer. Secondly, there is the large scale planetary vorticity. The remainder of this paper is focussed on describing the effects of Stokes drift on these two components of the vorticity. But first an alternative description is given of the origin of the interaction between the Stokes drift and the planetary vorticity. a. Surface waves on a rotating ocean When the ocean rotates the properties of a monochromatic surface wave change, see figure 2. In particular, the Coriolis force leads to generation of a North-South component of the motion, ṽ (Ursell 195; Hasselmann 197; Weber 1983). The Eulerian solutions show that ṽ is in phase with the w, so that there is a net momentum transport, namely τ = ρ ṽ w, where angle brackets denote average over a wavelength (Xu & Bowen 1993). Hasselmann (197) shows that this wave-induced stress has a non-zero vertical derivative and hence exerts a force on the mean flow, which is given by d τ dz = f u s (2) where f is the Coriolis parameter. Hence the Stokes drift can be thought of as deforming the planetary vorticity just as it does local vorticity. b. Large eddy simulation of the wind-driven mixed layer Polton et al (23) have performed large eddy simulations of the wind-driven turbulent mixed layer. Large eddy simulation resolves the large-scale three-dimensional overturning motions associated with the large scales of the turbulence. The small-scale turbulent motions are parameterised. Hence the name large eddy simulation (hereafter LES). In the LES described here only the wavelength-averaged effects of the waves are represented. 4

6 Formally, the total velocity field, U, is separated into an irrotational wave part, ũ, and a residual, vortical, part, u. Hence we expand U = ũ + u, (3) where ũ is the irrotational wave motion and the residual vortical component of the motion is u = O((ak) 2 ), which is formally of the same order as the Stokes drift. This decomposition is then substituted into the equations for conservation of momentum and mass, and then averaged over a wavelength. This procedure yields two new terms due to the interactions of the Stokes drift with the resolved vorticity, u s ω (where ω = u ), and with the planetary vorticity, u s f (Craik, 1977; Craik & Leibovich 1976). Following Skillingstad & Denbo (1995) and McWilliams et al (1997), the LES solves for the residual mean flow u from D u Dt f ( u + u s ) = 1 ρ w p + u s ω + SGS, (4) where SGS represents the subgrid scale turbulence, which is parameterised here using a standard Smagorinsky model. The water density was set to be uniform with height to model the particularly clean case when the turbulent mixing and Coriolis force lead to an equilibrium layer. The LES was run for a range of wind and wave parameters. Later the results will be presented and compared with simpler models of the mean flow and turbulence in the layer. See Polton et al (23) for full details. 3. Mean flow in the Ekman Stokes layer We now identify the role of the deformation of the planetary vorticity by the Stokes drift, the term f u s in (4), on the time-mean flow profile in the wind-driven mixed layer. Following Polton et al (23) we refer to this term as the Coriolis Stokes forcing. Hence the residual vortical current, u, is decomposed into a time mean part, u, plus a turbulent fluctuation, u. The equations governing the time mean residual flow, u, is ρ w f ( u + us ) = d τ dz, (5) where τ = ρ w (u w, v w ) is the Reynolds shear stress associated with the residual motion. The boundary conditions are τ = τ on z = u as z. (6) 5

7 These are the same equations as the standard Ekman model for the wind-driven mixed layer but with the Coriolis Stokes forcing. Hence, following McWilliams et al (1997), we refer to this as the Stokes-Ekman layer. a. Depth-integrated transport in the Stokes Ekman layer Some estimate of the importance of the Coriolis Stokes forcing compared to the standard Ekman dynamics can be obtained by comparing the depth integrated transport associated with wind-driven flow, the Ekman dynamics, to the depth integrated transport associated with the Stokes drift. This ratio is given by the Ekman Stokes number: E S = wave driven transport wind driven transport = 2kU s τ /ρ w f. (7) First take wind speeds of U = 5 to ms 1 and compute the wind stress with the drag coefficient given by equation?? in Garrett (19??). Then take waves of slope ak =.5 with wavelengths of λ = 2 to m. These values yield Ekman Stokes numbers in the range E s (8) We conclude that the transport by the waves can be a significant fraction of the wind-driven transport. These values motivate studying the effects of the Coriolis Stokes forcing on the current profiles in the wind-driven Stokes Ekman layer. b. Mean current profile in the Stokes Ekman layer Consider now how the Coriolis Stokes forcing changes the current profile through the wind-driven mixed layer. To do so requires a parameterisation for the turbulent stress in (5). For mathematical simplicity we consider here a gradient transfer model with a constant viscosity, namely τ = ρ w κ m d u dz. (9) Madsen (1978) and Huang (1979) derive the solution, and Polton et al (23) show how it can be written U = U e + U es + U s, () 6

8 where U = u + iv, and the components of the solution are U e U es U s } = (1 i)u e exp {(1 + i) zδe, (11) { = (1 i)u e exp (1 + i) z }( ) 1 U s /δ s 1, (12) δ e 2 U e /δ e (1 + i 1 δe 2 ) 2 δs { } 2 U s z = (1 + i 1 δe 2 ) exp. (13) δ s 2 δ 2 s Here U e = (τ /ρ w ) 1 2 and δ e = (2κ m /f) 1 2 are the velocity and depth scale for the pure Ekman solution and δ s = 1/2k is the depth scale of the Stokes drift. The decomposition shown in () explains how the shallow processes associated with the surface wave, in this case the Coriolis Stokes forcing, can affect the current profile over the whole depth of the wind-driven mixed layer. The first term U e is the pure Ekman solution, which is the response to the boundary condition imposed by wind stress. The Coriolis Stokes forcing gives two additional terms. Firstly there is the current forced directly by the Coriolis Stokes forcing, U s, which decays on the same depth scale as the Coriolis Stokes forcing, δ s = 1/2k. Secondly, there is an Ekman Stokes component, U es, which decays on the Ekman depth scale, δ e, and so changes the current profile over the whole depth of the layer. This component of the solution arises to ensure that the total solution satisfies the boundary condition imposed by the wind stress at the air sea interface. The solution () is written such that U e satisfies the boundary condition imposed by the wind stress at the sea surface. But, in effect, the Stokes component of the solution, U s, carries some of the imposed wind stress. The Ekman Stokes component is required to remove this stress at the surface, so that the total solution does satisfy the boundary condition. In this sense the effect of the Coriolis Stokes forcing is to change the boundary condition on the Ekman current (i.e. the deep flow below the shallow layer affected by the Stokes drift). c. Representation of the Coriolis Stokes forcing by an effective boundary condition Polton et al (23) carry this concept further and show how the effect of the Coriolis Stokes forcing can be represented as a modified boundary condition to the standard Ekman layer equations, which thus obviates the need to resolve the thin layer near the surface where the Coriolis Stokes forcing acts. Specifically, when δ e δ s, as is typical in the real ocean 7

9 mixed layer, subject to an effective wind stress: ρ w f u = d τ dz, (14) τ = τ (ˆτ ẑ ke s ) = τ ρ w f us δ s on z =, (15) u as z. (16) This formula for the effective wind stress applies to wind and waves at arbitrary angles. The Coriolis Stokes forcing thus leads to an additional vector component of the effective wind stress. The magnitude of the extra component is E s times the actual wind stress, and drives the wave-driven component of the depth-integrated transport. The direction of the extra component is 9 o to the right of the wave motion. So when the wind stress and the waves are aligned and propagate to the East, the extra component is to the South. d. Comparisons with large eddy simulations and observations Polton et al (23) also derive solutions when the eddy viscosity varies linearly with depth, which better represents the variation of turbulent transport close to a planar interface. For example, in the atmospheric boundary layer it yields the logarithmic velocity profile. The general character of the solution with the Coriolis Stokes forcing is the same as when the viscosity is constant with depth. Figure 3 shows comparisons between these analytical solutions with a linearly varying eddy viscosity and large eddy simulations of the flow computed with the model described in section (2). The results are presented as hodographs, with the locus of (u, v) plotted as the depth varies. Hence a line from the origin to each point on the hodograph line shows the current vector at a particular depth. The analytical solution is shown both with and without the Coriolis Stokes forcing. The plots show how the current vector turns and reduces with increasing depth. As U s increases so the hodographs is shifted and rotated clockwise. The resulting hodographs can be quite different to the hodograph with no Coriolis Stokes forcing. The analytical model follows the trends shown by the LES. This is perhaps surprising because, as we shall see in the next section, the turbulence resolved by the LES shows quite different characteristics to turbulence in standard surface layers, whereas the representation of the turbulence in the analytical model makes no attempt to represent this effect. The results therefore suggest that correct representation of the Coriolis Stokes forcing is the key to modelling the mean current profile in the wind-driven mixed layer. 8

10 We now compare with observations. The observational data was taken from Price & Sundermeyer (1999), who produced hodographs from the EBC and LOTUS data sets. Unfortunately no wave measurements were taken during the observations. The wave parameters were estimated by assuming the waves to be in equilibrium with the measured wind stress and using the empirical relations given in Komen et al (1994), see Polton et al (23) for details. Figure 4 shows the results together with comparisons with the analytical model with the linear eddy viscosity, both with and without the Coriolis Stokes force. The model shows remarkably good agreement with the observations, and certainly accounts for the significant differences between the observations and hodograph obtained when the Coriolis Stokes forcing is neglected. These results provide perhaps the first observational evidence that surface wave processes have an impact on current profiles in the ocean. 4. Langmuir turbulence in the Stokes Ekman layer Consider now the effects of the Stokes drift on the small-scale vorticity in the turbulence within the wind-driven mixed layer. The LES show that the Stokes drift has the effect of generating elongated vortices in the x-direction of wave motion these being the Langmuir circulations in the fully turbulent flow. These Langmuir circulations occur on a whole spectrum of scales, and thus need to be considered as the large eddies of the turbulent flow. Hence, McWilliams et al (1997) coin the term Langmuir turbulence. The statistical properties of the Langmuir turbulence can be understood with a linearised rapid distortion theory model. Teixeira & Belcher (22) have analysed the distortion of initially isotropic homogeneous turbulence by a progressive surface wave in the rapid distortion limit. More recently Teixiera & Belcher (23) have shown how these calculations can be extended to account for the inhomogeneous distortion that occurs in the wind-driven mixed layer. We now review these analyses briefly and then suggest how the current profiles within the Stokes Ekman layer might precondition the flow to generation of Langmuir turbulence. a. Rapid distortion model of Langmuir turbulence The rapid-distortion model is based on the assumption that the distortion of the turbulence by the external mean flow is stronger than the distortion of the turbulence by its own nonlinear dynamics. What aspects of the external mean flow distort the turbulence? Firstly the presence of the air sea interface acts to block the turbulence, ensuring that the vertical 9

11 velocity fluctuations are zero there (Hunt & Graham 1978). Secondly, the mean sheared current profile in the wind-driven mixed layer distorts the turbulence, as in any turbulent shear flow. Thirdly, the Stokes drift acts to tilt and stretch the vorticity in the turbulence. We have seen in section 3 that the mean shear in the direction of the wave propagation is reduced by the action of the Coriolis Stokes forcing. Teixeira & Belcher (23) show that for the case presented in McWilliams et al (1997) the straining induced by the Stokes drift is larger than the straining associated with the mean shear. Hence a simplified model is developed by Teixeira & Belcher (23) that considers the distortion to turbulence by an plane air sea interface plus the distortion by Stokes drift. In the rapid-distortion model the turbulence is distorted by these external agents according to linear dynamics. This approximation is justified because two conditions are satisfied. First, the velocity associated with the Stokes drift, of magnitude U s, is much larger than the velocity associated with the turbulence, or magnitude q. If u w is the friction velocity in the water flow, then for the McWilliams et al case U s /q (ak) 2 c/u w (17) Secondly, the strain rate associated with the turbulence is of order q/l which scales as T 1 L, the integral time scale of the turbulence, which is much smaller than the strain rate associated with the Stokes drift is du s /dz. Teixeira & Belcher show that the maximum value of the ratio of these terms is du s /dz T 1 L c w (a w k w ) 2 exp(2kd T 1) 7 1 (18) max u w for the McWilliams et al simulations. Given the two conditions (17) and (18) the turbulence evolves for short times under the external conditions according to linear dynamics (see Teixeira & Belcher 22). For short times, such that t T L, the turbulence then evolves according to linear dynamics, so that the turbulent vorticity fluctuations evolves according to D ω /Dt = ω. u s, (19) with u s (z) the specified fixed Stokes drift. The model is completed on specifying the initial vorticity field and the end time for the distortion, t d. Following many other rapid distortion studies (see e.g. Hunt 1973), the initial vorticity field is taken to be isotropic and

12 homogeneous with a von Karman velocity spectrum, namely E(k ) = q 2 g 2 (k l) 4 l (g 1 + (k l) 2 ) 17 6, (2) where k is the peak wavenumber, and g 1 =.558 and g 2 = are dimensionless constants. The end time for the distortion, t d, is an empirical constant of the model, determined to give the best agreement with the LES data (see Teixeira & Belcher 23). b. Statistics of Langmuir turbulence Figure 5 shows vertical profiles of the velocity variances in Langmuir turbulence normalised on the turbulent kinetic energy. The curves are calculated from the linearised rapid distortion model with the distortion time given by t d /T L =.43. The symbols are from the LES of McWilliams et al (1997). The simplified model agrees remarkably well with the fully nonlinear solutions from the LES. This agreement suggests the following dynamical evolution in Langmuir turbulence. Turbulent fluctuations lead to vorticity fluctuations that are tilted and stretched into the x-direction by the straining of the Stokes drift. After a time of approximately one half T L (i.e. t d /T L =.43) the nonlinear processes arrest this deformation. Figure 5 shows that when the turbulence is distorted by Stokes drift u 2 v 2 w 2. This ordering is markedly different from the profiles obtained in a turbulent boundary layer, when the turbulence is distorted by mean shear (rather than by Stokes drift). In that case u 2 > v 2 > w 2, yielding the so-called streaky structures, see e.g. Lee, Kim & Moin (199). The reason for this difference is that in the presence of mean shear the linearised evolution equation for turbulent vorticity has an extra term associated with stretching of the mean vorticity, Ω, from the mean shear, and becomes D ω /Dt = ω. u s + Ω. u, (21) where Ω is the mean vorticity in the layer. The rapid-distortion model thus explains the quantitative features of mixing in Langmuir turbulence. It also explains the qualitative differences between Langmuir turbulence and shear flow turbulence and moreover suggests that the relative roles of straining by Stokes drift and mean shear is the key to diagnosing the occurrence of Langmuir circulations. The LES data of Polton et al (23) is used next to examine this possibility. 11

13 5. On the conditions for Langmuir circulations We now draw together the two themes developed above, namely the effects of the Stokes drift, via the Coriolis Stokes forcing, on the mean shear in the wind-driven mixed layer and the effects of Stokes drift, via the turbulent vortex force, on the turbulent structures. In particular, we examine the role of the environmental shear in preconditioning the wind-driven mixed layer to Langmuir turbulence. The argument is as follows. We found in section 4 that elongated streamwise vortices, the signature of Langmuir turbulence, are produced when random turbulent vorticity is deformed by the strain associated with the Stokes drift, du s /dz. When such random turbulent vorticity is deformed by mean shear, du/dz, it is the streamwise velocity fluctuations that become large producing streaky structures. Hence we suggest that Langmuir turbulence is produced when du s dz du dz = Langmuir turbulence. (22) Next the results of the LES are examined to test this idea. The type of turbulence is measured here using the ratio of the vertical fluctuations to the horizontal fluctuations, namely R t = ( w 2 u 2 + v 2 )1 2. (23) In ordinary shear driven turbulence near a boundary R t is then small because u 2 > v 2 > w 2. In Langmuir turbulence R t is near unity because u 2 u 2 w 2. Figure 6 shows vertical profiles of R t obtained from the LES of Polton et al (23) for five cases: one with no Stokes drift and four with the combinations of two values of strength of Stokes drift U s and of penetration depth of Stokes drift 1/2k. Also shown on these plots are the profiles of du s /dz and du/dz, also obtained from the LES. Figure 6 shows that when du s /dz < du/dz, as in the cases when U s is either zero or small, R t is small through all depths. When du s /dz > du/dz, as in the two cases when U s is larger, R t is approximately unity. The simulations therefore support the suggestion that the dominant strain mechanism controls the type of turbulence generated. Now in section 3 it was shown how the Coriolis Stokes forcing controls the mean flow current profile and hence the environmental shear du/dz. Indeed the profiles of du/dz 12

14 shown in figure 6 vary with u s and k. Hence we suggest that the the Coriolis Stokes forcing preconditions the wind-driven mixed layer to the occurrence of Langmuir turbulence by reducing the mean shear and thus allowing strain of turbulence by Stokes drift to be the dominant deformation. Since the Coriolis Stokes forcing depends on the planetary vorticity, this leads to the suggestion that the occurrence of Langmuir turbulence depends on latitude. Further work is necessary to examine this conjecture. 6. Conclusions We have reviewed briefly two effects of surface water waves on the dynamics of the wind-driven mixed layer, via their Stokes drift. First, the interaction of planetary vorticity with Stokes drift leads to a new mean force that acts in the wind driven mixed layer. This Coriolis Stokes forcing is surprisingly large and changes significantly the current profiles through the whole depth of the wind-driven mixed layer. Although the Coriolis Stokes forcing acts only near the surface, it changes the current profile through the whole depth of the wind-driven mixed layer because it changes the effective boundary condition on the wind-driven component of the motion. Hence this new force, which is not currently represented in ocean general circulation models, could be effectively represented through a new boundary condition at the air sea interface that accounts for the Coriolis Stokes forcing. A second effect of the Stokes drift is to distort the small scale turbulent vorticity associated with the three-dimensional overturning turbulent motions in the wind-driven mixed layer. In the fully turbulent environment of the wind-driven mixed layer, the result is vortical turbulent structures that are elongated into the streamwise direction. This is Langmuir turbulence. In strong contrast, in an environment with mean shear, streaky structures are produced. Finally, large eddy simulations of the wind-driven mixed layer support the idea that Langmuir turbulence is produced when du s /dz > du/dz. This suggests that the Coriolis Stokes forcing, which changes the environmental current and shear profiles, plays a role in preconditioning the ocean mixed layer to Langmuir turbulence. This suggestions deserves further work. Acknowledgments. The research presented in this paper was funded by the Leverhulme Trust under grant F/239/A and the US Office of Naval Research under grant N

15 7. References Craik, A. D. D., 1977: The generation of Langmuir circulations by an instability mechanism. J. Fluid Mech., 81, Craik, A. D. D. and S. Leibovich, 1976: A rational model for Langmuir circulations. J. Fluid Mech., 73, Garratt, J.R. Hasselmann, K. 197 Wave-driven inertial oscillations. Geophys. Fluid Dyn., 1, Huang, N.E On surface drift currents in the ocean. J. Fluid Mech., 91, Hunt, J. C. R., 1973: A theory of flow round two-dimensional bluff bodies. J. Fluid Mech., 61, Hunt, J. C. R. and J. M. R. Graham, 1978: Free stream turbulence near plane boundaries. J. Fluid Mech., 84, Komen, G.J., Cavaleri, L., Donelan, M.A., Hasselmann, K., Hasselmann, S. & Janssen, P.A.E.M Dynamics and Modelling of Ocean Waves, Cambridge University Press. Lee, M. J., J. Kim and P. Moin, 199: Structure of turbulence at high shear rate. J. Fluid Mech., 216, Leibovich, S., 1983 The form and dynamics of Langmuir circulations. Ann. Rev. Fluid Mech., 15, Lewis, D.M. & Belcher, S.E. 23 Time-dependent coupled Ekman boundary layer solutions incorporating Stokes drift. Submitted to Dynamics Atmos. Ocean. Madsen, O., S A realistic model of the wind-induced Ekman boundary layer. J. Phys. Oceanogr., 7, McWilliams, J. C., P. P. Sullivan and C.-H. Moeng, 1997: Langmuir turbulence in the ocean. J. Fluid Mech., 334, 1 3. Polton, J.A., Lewis, D.M. & Belcher, S.E. 23 The role of wave-induced Coriolis Stokes forcing on the wind-driven mixed layer. Submitted to J. Phys. Oceanogr. pp 36 14

16 Price, J.F. & Sundermeyer, M.A Stratified Ekman layers. J. Geophys Res. 4, Skyllingstad, E. D. and D. W. Dembo, 1995: An ocean large-eddy simulation of Langmuir circulations and convection in the surface mixed layer. J. Geophys. Res.,, Teixeira, M.A.C. & Belcher, S.E. 22 On the distortion of turbulence by a progressive surface wave. J. Fluid Mech., 458, Teixeira, M.A.C. & Belcher, S.E. 23 The large-scale structure of Langmuir turbulence. Submitted to J. Fluid Mech. pp.12. Ursell, F. 195 On the theoretical form of ocean swell on a rotating Earth. Mon. Not. Roy. Astron. Soc. Geophys. Suppl. 6, 1 8. Weber, J.E Attenuated wave-induced drift in a viscous rotating ocean. J. Fluid Mech., 137, Xu, Z. & Bowen, A.J Wave- and wind-driven flow in water of finite depth. J. Phys. Oceanogr., 24,

17 c z u = (ak) 2 s c Figure 1: Schematic showing the geometry of the flow. Fluid parcels move under the influence of the wave in near circular orbits, but with a small net drift in the direction of the waves: this is the Stokes drift. The Stokes drift profile decreases with depth. Vorticity that is initially vertical is tilted and stretched by the strain associated with the Stokes drift. This deformation changes the dynamics of the wind-driven mixed layer. 16

18 Figure 2: Schematic illustrating the orbital paths when the ocean rotates. The orbital path is tilted into the along-crest direction. The new ṽ component of the orbital motion is correlated to w, and yields a stress, τ = ρ ṽ w, when averaged over a wavelength. The divergence of the stress can be written ρ f u s. 17

19 Figure 3: Current profiles comparing the LES (heavy solid line) with the analytical models. The light solid lines are the solutions obtained using the effective boundary condition. The dashed lines are the solutions obtained using the full boundary condition and wave parameterisation. Both analytical models use a linearly varying eddy viscosity. The cross denotes the depth at which the effective boundary condition is applied. Top row: hodographs; middle row: profile of u; bottom row: profile of v. (a) (b) v/u * 6 * v/u u/u * u/u * Figure 4: Comparisons, in the form of hodographs, between the simple analytic model, assuming a fully developed sea (solid line) or no wave effects (dashed line), and observational measurements (crosses) taken from the (a) LOTUS3 and (b) EBC data sets. 18

20 z/z i u 2 /(2/3K), v 2 /(2/3K), w 2 /(2/3K) z/l Figure 5: Vertical profiles of the normalized turbulent velocity variances. Symbols: data from LES of McWilliams et al. (1997), lines: RDT model of turbulence distorted by a surface wave. Solid lines and circles: u 2 /(2/3K), dotted lines and squares: v 2 /(2/3K), dashed lines and diamonds: w 2 /(2/3K). K is the turbulent kinetic energy. 19

21 .17m/s U s (m/s).68m/s m depth (m) depth (m) U s =.m/s 2 2 depth (m) δs=1/2k (m) m depth (m) depth (m) Figure 6: Preconditioning to generation of Langmuir turbulence. Each of the five panels shows profiles of du/dz (dashed lines on log scale) and du s /dz (dotted lines on log scale), the strains that deform the turbulence, and R t = w 2 /(u 2 + v 2 ) (solid lines on linear scale), a measure of the type of turbulence. The panel on the left has no wave processes, U s =. The four panels on the right are the combinations of two values of strength of the Stokes drift, U s, and of penetration depth, δ s = 1/2k. When du s /dz > du/dz, then R t is near unity, consistent with the formation of Langmuir turbulence. When du s /dz < du/dz, then R t is small, consistent with the formation of streaky structures. 2