## 1. Introduction

The Ekman layer is the quintessential oceanic surface turbulent boundary layer. Its canonical formulation is a steady surface wind stress, *u*_{*} the oceanic “friction velocity”), on top of an ocean with uniform density *ρ _{o}* and uniform rotation rate

*f*(Coriolis frequency) aligned with the vertical direction. The analytic steady solution with constant eddy viscosity

*κ*has a surface current to the right of the stress direction (with

_{o}*f*> 0) and a further rightward spiral decay over a depth interval

*K*-profile parameterization (KPP) (Large et al. 1994; McWilliams and Huckle 2006),

*κ*(

*z*) has a convex shape and a magnitude

*u*

_{*}/

*f*. Large-eddy simulation (LES)—with an explicit calculation of the turbulent eddies, their Reynolds stress, and the mean current—provides a validation standard for parameterizations to be used in large-scale circulation models (Zikanov et al. 2003).

The same winds that cause the Ekman layer also cause surface gravity waves, either in local equilibrium with the wind or in disequilibrium due to a transient history or remote propagation. The combination of wind and waves has a significant impact on the (wavy) Ekman layer, most importantly through the generation of turbulent Langmuir circulations (LCs) and modification of the Coriolis force through the wave-averaged Stokes drift profile *u*_{st}(*z*) acting as “vortex forces” (Skyllingstad and Denbo 1995; McWilliams et al. 1997; plus many subsequent studies reviewed in Sullivan and McWilliams 2010). Furthermore, especially for high winds and waves, the momentum transmission from atmospheric winds to oceanic currents by surface drag occurs primarily through isolated impulses associated with wind-generated surface waves when they break and penetrate into the ocean, rather than through a uniform *τ* at the surface; this is represented in a stochastic breaker model (Sullivan et al. 2007).

This paper reports on LES solutions of the Ekman layer problem, which is a simpler boundary layer configuration than most prior studies that include a depth-limiting stable density stratification and surface buoyancy flux.^{1} We contrast the Ekman layer without wave effects to the wavy layer with both Stokes drift and breaker impulse forcing, in various combinations to distinguish particular influences. The primary focus is on how the coherent structures, LCs, and breaker-induced circulations relate to the turbulent Reynolds stress—hence, the mean current profile—to be able to assess the requirements for a successful parameterization of the wavy Ekman layer. Because we do not include buoyancy effects, nonstationarity, or other types of currents nor do we survey a variety of different wave wind regimes, our results only provide an idealized case study rather than a more comprehensive characterization of wave effects in the surface boundary layer. Nevertheless, for this case it demonstrates their importance and salient characteristics.

## 2. Problem setup

The LES code solves the wave-averaged dynamical equations in Sullivan and McWilliams (2010) with forcing options among a uniform mean surface stress *τ ^{x}*, fields of stochastic breaker acceleration

*A*(

**x**,

*t*) and subgrid-scale energy injection rate

*W*(

**x**,

*t*) or mean breaker vertical profiles, 〈

*A*〉(

*z*) and 〈

*W*〉(

*z*). (Mean refers to time and horizontal averages, denoted by angle brackets;

*z*is the vertical coordinate.) The forcing options are normalized to give the same mean vertically integrated force; that is,

We focus on a particular situation where the forcing is aligned with *U _{a}* = 15 m s

^{−1}[implying a surface stress of 0.35 N m

^{−2}; hence, velocity

*u*

_{*}= (|

*τ*|/

^{x}*ρ*)

_{o}^{1/2}= 1.9 × 10

^{−2}m s

^{−1}]. The wave elevation spectrum [determining the Stokes drift profile

*u*

_{st}(

*z*)] and breaker spectrum (determining

*A*and

*W*) are empirically consistent with equilibrium for this wind for a wave age of

*c*/

_{p}*u*

_{*a}= 19 (

*c*is the phase speed of the wave elevation spectrum peak and

_{p}^{2}The profiles of 〈

*A*〉(

*z*) and

*u*

_{st}(

*z*) are shown in Fig. 1. Both are surface intensified and have characteristic vertical length scales (defined, somewhat arbitrarily, as the depth where the amplitude has decreased to 10% of its surface value) of

*h*= 1.4 m and

_{b}*h*

_{st}= 13 m, respectively. Both of these are much smaller than the turbulent boundary layer depth

*h*; the ordering

_{o}*h*≪

_{b}*h*

_{st}≪

*h*is typical in the ocean. For full wave elevation and breaker spectra, as used here, there is no uniquely correct vertical scale definition, and we use these estimates only as a rough guide for the vertical profiles shown below. We will see that the flow structure and dynamical balances are distinctive in three sublayers within the overall Ekman layer, which we designate as the breaker, Stokes, and interior shear layers. We choose a midlatitude Coriolis frequency,

_{o}*f*= 10

^{−4}s

^{−1}: hence, an Ekman boundary layer dimensional depth scale of

*u*

_{*}/

*f*= 190 m. The domain size is

*L*×

_{x}*L*×

_{y}*L*= 500 m × 500 m × 300 m, large enough to encompass the Ekman layer and its energetic turbulent eddies. The horizontal grid cell size is

_{z}*dx*=

*dy*= 1.7 m, and the vertical grid is nonuniform in the vertical with a minimum cell size

*dz*= 0.42 m near the surface and maximum of

*dz*= 5 m at the bottom where the flow is nearly quiescent. Solutions are spun up from rest to a statistical equilibrium state after about one inertial period, 2

*π*/

*f*. The solution analyses are made over a subsequent interval of several inertial periods, with temporally filtering to exclude the inertial oscillation in the horizontally averaged current at each vertical level. All our analysis results are presented in nondimensional form using appropriate factors of

*u*

_{*}and

*f*.

In this paper we distinguish among different wave effects by defining six different cases, all of which have the same mean momentum forcing (i.e., the same *u*_{*}). The case without any wave effects is designated as *Nτ*, where *N* denotes the exclusion of Stokes drift influences and *τ* denotes a surface-stress boundary condition; this is the classical Ekman problem. The case with fullest wave effects is *SB*, where *S* denotes the inclusion of Stokes drift and *B* denotes stochastic breaker forcing; we view this case as the most complete representation of wave effects. Intermediate partial wave-effect cases are *Sτ* and *NB*. In addition, to understand the importance of the transient breaker forcing, we define cases *N*〈*B*〉 and *S*〈*B*〉 in which the four-dimensional (4D) fields of acceleration and energy-injection rate are replaced by their 1D mean profiles: 〈*A*〉(*z*) and 〈*W*〉(*z*).

## 3. Solution analysis

### a. Bulk statistics

*h*; the depth-integrated value of the turbulent kinetic energy

_{o}^{3}profile,

*e*as parameterized in LES; the superscript prime denotes a fluctuation about the horizontal average and the superscript

^{s}*s*refers to a (

**x**,

*t*) local variable in the subgrid-scale energy model]; and the total depth-integrated energy injection rate,

^{4}

*τ*〈

^{x}*u*〉(0)/

*ρ*, or breaker forcing,

_{o}*u*is the

*x*velocity in the direction of the wind and waves, and

*υ*,

*w*are the transverse and vertical velocities in the

*y*,

*z*directions. All quantities in the table are listed nondimensionally.

The six cases used in this study. The depth *h _{o}* is defined as the depth at which the magnitude of the turbulent stress is 10% of its surface value. The third column is the total kinetic energy per unit area. The fourth and fifth columns are the total energy input rate

*τ*and

*B*cases, respectively.

*W*is large, and this enhancement has a similar magnitude with either stochastic or mean breaker forcing and with either

*u*

_{st}present or not; the injection rate is slightly smaller in 〈

*B*〉 cases than

*B*cases because the fluctuation correlation effect, 〈

*A*′

*u*′〉, is absent. There is also a noticeable difference in

*Sτ*and

*Nτ*, respectively); we will see in section 3b that the

*Sτ*case injection rate is smaller than the

*Nτ*case because 〈

*u*〉(0) is much reduced through the effect of the Stokes–Coriolis vortex force.

^{5}The enhanced boundary layer depth is consistent with the idea that, when Stokes drift is important, the relevant turbulent velocity scale is a composite one,

*u*

_{st}(

*z*) is more than 10 times larger than

*u*

_{*}near the surface

^{6}(Fig. 1), the estimate

*u*

_{*com}/

*f*for

*h*is more than twice as deep with Stokes drift, consistent with Table 1. The relevance of

_{o}*u*

_{*com}to the TKE balance has been previously validated in stratification-limited Ekman layers (Harcourt and D’Asaro 2008; Grant and Belcher 2009; Kukulka et al. 2010) where

*h*does not satisfy the Ekman scaling relation.

_{o}^{7}Also,

*u*

_{st}present, but this does not scale well with a bulk estimate using the composite velocity scale,

*ϵ*near the surface (Terray et al. 1996). Remarkably, there is not a direct relation between

*e*greatly. Nor does

*u*

_{st}is present, as would be suggested by the increase in

*u*

_{*com}(cf. cases

*Nτ*and

*Sτ*). This demonstrates a degree of decoupling between

*e*itself and the energy cycle throughput rates,

*ϵ*, so that the conventional turbulent scaling of

*ϵ*~

*e*

^{3/2}/

*h*, for some turbulent length scale

*h*, does not hold across the various combinations with

*u*

_{st}and

*B*. Furthermore, because eddy viscosity is commonly estimated as

*κ*~

*e*

^{1/2}

*h*, this result also raises a question about how to express the idea that breaker energy injection leads to enhanced turbulent mixing near the surface. The increase of

*e*with

*u*

_{st}does support the idea of enhanced mixing by the LCs sustained by the vortex force; however, the increase in

*κ*(section 3d) increases by far more than

*e*

^{1/2}

*h*does. In summary, Stokes drift effects increase the boundary layer depth and turbulent energy, and breakers increase the energy cycle rate, but these enhancements are not collectively well represented by simple bulk scaling estimates, even with the composite velocity scale

*u*

_{*com}in (2).

### b. Mean velocity and momentum balance

*z*= 0 (i.e., it is zero in the

*B*cases). These balances contain the mean Coriolis and Stokes-Coriolis force, the mean breaker acceleration 〈

*A*〉, and the divergence of the total horizontal turbulent Reynolds stress,

*i*and

*j*for all three spatial directions, ⊥ for a horizontal vector component, and

*z*for a vertical one. The vertical integrals of (3) relate the mean transport to the Stokes transport and the mean wind stress in

*τ*cases (or its integral equivalent

*B*cases),

The mean horizontal velocity profiles, 〈**u**_{⊥}〉(*z*) = (〈*u*〉, 〈*υ*〉) (*z*), have the familiar Ekman spiral structure of decaying amplitude and rotating clockwise with increasing depth (Fig. 2). The profiles for the different cases are primarily distinguished by Stokes drift effects, with the forcing mechanism secondary. Compared to an Ekman layer without waves, Stokes drift causes the boundary layer depth *h _{o}* to be deeper (Table 1); hence, the mean velocity magnitude is diminished near the surface to satisfy the transport constraint (5). Stokes drift further diminishes the downwind velocity near the surface. This effect is a consequence of the Stokes-Coriolis force (Huang 1979): that is, the second term in the

*y*-momentum balance in (3). It adds an anti-Stokes component to the

*x*transport in (5) and makes the surface current angle

*θ*(0) more nearly southward, −

_{u}*π*/2. The reduced value of 〈

*u*〉(0) with

*u*

_{st}leads to the reduced energy injection rate

*Sτ*with stress forcing (Table 1); in the cases with breaker forcing,

*W*〉, hence is not sensitive to 〈

*u*〉(0). Near the surface 〈

*u*〉(

*z*) has downwind, down-wave shear. Without

*u*

_{st}, this extends over the whole upper half of the layer, and it is especially large within a thin layer with stress forcing (as expected from Monin–Obukhov similarity, with 〈

*u*〉 ~ 1/

*z*) controlled in the LES by the subgrid-scale mixing. Breaker forcing limits the strength of the near-surface shear over a vertical scale of

*h*. With

_{b}*u*

_{st}, the positive

*x*shear is confined within the breaker layer

*h*. Just below in the Stokes layer, the

_{b}*x*shear is up-wave over most of the Stokes depth scale

*h*

_{st}in accord with the anti-Stokes tendency in (3). Even with wave effects, ∂

_{z}〈

*υ*〉(

*z*) does not have strong features on the scales of

*h*and

_{b}*h*

_{st}. With breaker forcing its surface boundary condition of zero, shear is approached within a thin layer controlled by subgrid-scale mixing. The magnitude of ∂

_{z}〈

*υ*〉(

*z*) is diminished with

*u*

_{st}because

*h*is bigger while the

_{o}*y*transport is the same. Overall, the oscillations with depth of the velocity component profiles (i.e., Ekman spiral) are less evident with

*u*

_{st}even in the interior shear layer (cf. appendix B). In both components breaker forcing and

*u*

_{st}cause reduced mean shear near the surface compared to surface stress forcing, and more so in the transient

*B*cases than in the mean 〈

*B*〉 cases, consistent with enhanced vertical momentum mixing by wave-induced breakers and LCs and the absence of a Monin–Obukhov similarity layer.

The Reynolds stress profiles, 〈**u**_{⊥}*w*〉(*z*) (Fig. 3), are grossly similar among the different cases except within the breaker layer near the surface. As with the mean velocity in Fig. 2, we plot the Reynolds stress as its magnitude |〈**u**_{⊥}*w*〉| and angle *θ*_{−uw}. The latter is in the direction opposite to −〈**u**_{⊥}*w*〉 to facilitate comparison with the mean shear ∂_{z}〈**u**_{⊥}〉, which can be compared within the framework of an eddy viscosity assumption of proportionality between Reynolds stress and mean shear (section 3d).

*θ*

_{−uw}≈ −

*π*before the stress magnitude becomes very small. So, the main intercase difference is due to the larger vertical scale

*h*with

_{o}*u*

_{st}. hence a slower rotation rate. In all cases, the bulk rotation rate is

*dθ*

_{−uw}/

*dz*≈ −0.7

*π*/

*h*. The Ekman spiral has a simpler manifestation in Reynolds stress than in mean velocity, where the anti-Stokes tendency partly obscures the rotation. Here 〈

_{o}**u**

_{⊥}

*w*〉(0) = ∂

_{z}〈

**u**

_{⊥}〉(0) = 0 with breaker forcing, whereas the latter quantity is nonzero and equal to

*x*direction with stress forcing. The different surface boundary conditions for surface stress and breaker forcing are accommodated within the thin breaker layer

*h*without otherwise much difference in the interior; that is, −∂

_{b}_{z}〈

*uw*〉 stays positive to the surface with eastward stress forcing while

_{z}〈

*u*〉 < 0 while −〈

*uw*〉 > 0, which is inconsistent with downgradient momentum flux; this presages the invalidity of conventional eddy viscosity parameterization in the Stokes layer (section 4). Without the Stokes-Coriolis and vortex forces, the flux is downgradient in the upper ocean and even throughout the interior shear layer (section 4). In both the upper ocean in no-wave cases and in Stokes layers, 〈

*υ*〉 < 0 (mean flow to the right of the surface wind) and ∂

_{z}〈

*υ*〉 < 0; hence, −∂

_{z}〈

*υw*〉 > 0 and −〈

*υw*〉 > 0 because of zero transverse Reynolds stress at the surface. These 〈

*υ*〉 and 〈

*υw*〉 profiles are qualitatively similar in shape with or without waves, with the transverse Reynolds stress divergence in (3) balanced in the upper part of the layer by either

*f*〈

*u*〉 or

*fu*

_{st}in

*N*or

*S*cases, respectively.

### c. Velocity variance and energy balance

Many previous studies show that the Stokes drift vortex force increases *e* and alters the anisotropic partition of variance among the fluctuation velocity components by reducing the downwind component *u*′ and increasing the transverse and vertical components (*υ*′, *w*′) as expected from the idealized geometry of LCs as longitudinal roll cells. In these aspects, we also see two groupings based on whether *u*_{st} is included (*S* cases) or not (*N* cases) (Fig. 4). The cases with different forcing specifications have more complicated distinctions: *τ* forcing enhances *u*′ variance and diminishes (*υ*′, *w*′) variance near the surface without *u*_{st} (*N* cases) and vice versa with *u*_{st} (*S* cases); *e* is much larger in the surface layer with breaker forcing than with stress forcing, and it is largest with 〈*B*〉 forcing, mainly because of a subgrid-scale *e* enhancement near the surface; and the forcing-induced differences are mostly confined to a thin layer of several times *h _{b}*. The maximum for 〈

*w*′

^{2}〉(

*z*) occurs near the surface near the base of the Stokes layer but outside the primary influence of subgrid-scale mixing and breaker forcing. It is much stronger in

*S*cases as an expression of LCs that have peak intensity in the Stokes layer (section 3e). The case

*Sτ*is anomalous in having the shallowest depth for the maximum, and it also has the largest surface extremum for 〈

*υ*′

^{2}〉; vortex force acts almost singularly in generating small-scale LCs near the surface, unless limited by the extra mixing associated with breaker forcing.

We decompose the profile of kinetic energy into three pieces: the mean-current kinetic energy (MKE), *e*(*z*) in (1), which contains both large-eddy and subgrid-scale components. Energy balance relations are derived by averaging the product of the momentum equation and the velocity and adding this to an average of the subgrid-scale model that is expressed ab initio as an energy balance.

*e*. For completeness, we record the mean energy balance in appendix A, but we focus here on the balance relation for the turbulent energy

*e*(

*z*) in statistical equilibrium; namely,

*τ*〉 or mean breaker acceleration 〈

*A*〉(

*z*) is an energy source for

*e*directly; the connection to the latter is made by a conversion through the shear production

*e*(appendix A). We assume a steady wind here, which therefore does not provide a direct source for

*e*. The transient and subgrid-scale breaker work for

*e*is

*p*is the dynamic pressure.

^{8}Index summation over

*i*is implied in the next-to-last term. Finally, the viscous dissipation term occurs entirely through the subgrid-scale model,

*τ*, energy

^{s}*e*, dissipation rate

^{s}*ϵ*

^{s}, and eddy viscosity

*κ*are local fields calculated in the subgrid-scale model (section 2).

^{s}The TKE balance without wave effects (Fig. 5, right) is a familiar story of _{u} ≈ *ϵ*, with *e* downward from the more energetic upper part to the lower part of the Ekman layer; the crossover depth from negative to positive *h _{o}*. The story is quite different with wave effects (Fig. 5, left). Breaker energy injection

*h*, and this influence is so strong that the entirety of the underlying Stokes and interior shear layers are supplied by the downward energy flux from the breaker layer,

_{b}*ϵ*is much increased in the surface layer primarily to balance the large

^{9}The negative

_{st}is much larger than

_{u}but is necessarily restricted to the Stokes layer. Within the interior shear layer

_{u}is an energy source, but small compared to transport and dissipation. Within the breaker layer, injection and transport approximately balance dissipation; over the Stokes and interior shear regions of the wavy Ekman layer, Stokes production and transport balance dissipation. The differing character of the TKE balance with depth may explain why the simple scaling estimate based on Stokes production,

*u*

_{*com}in (2), is not uniformly successful in accounting for wave effects (section 3a). Nevertheless, the importance of Stokes production, rather than shear production, gives support for the Lagrangian eddy viscosity proposed in sections 3d and 4.

In summary, the TKE balance without waves has shear production as its source, passed through the MKE budget from mean surface-stress wind work. In contrast, the TKE balance with waves has primarily breaker energy injection and secondarily Stokes production as its sources, both of which are conversions from the wave field; in this case the energy conversion from MKE though _{u} is much less important. The associated MKE balances are further summarized in appendix A.

### d. Eddy viscosity profiles

**u**

_{⊥}

*w*〉 is oppositely aligned with the mean shear ∂

_{z}〈

**u**

_{⊥}〉, hence that

*θ*= 0.

_{κ}In an Ekman layer without wave effects in case *Nτ*, *κ*(*z*) has a convex profile that extends over the whole of *h _{o}* (and even somewhat beyond), and

*θ*(

_{κ}*z*) is small (Fig. 6).

These characteristics are supportive of a full-turbulence [a.k.a. Reynolds-averaged Navier–Stokes (RANS)] eddy-viscosity parameterization scheme such as KPP, and the skill of this turbulence model is assessed in section 4. In fact, *θ _{κ}*(

*z*) is slightly positive except at the boundary layer edges

^{10}, but not to such a degree that an eddy-viscous KPP solution is inaccurate (section 4).

With wave effects in case *SB*, *κ*(*z*) is much larger and extends deeper. Both features are qualitatively consistent with Ekman layer scalings of *h _{o}* ~

*u*

_{*com}/

*f*and

*u*

_{*com}in (2). However, the

*κ*enhancement is by nearly a factor of 10 in Fig. 6, while the enhancement of (

*u*

_{*com}/

*u*

_{*})

^{2}is not even half as large, so there is a quantitative discrepancy. A much bigger discrepancy is a large positive spike of

*θ*in the Stokes layer and a broader, but lesser, maximum in the interior of the Ekman layer. This presents a significant challenge to a conventional eddy viscosity RANS parameterization.

_{κ}^{11}

*u*

_{st}(e.g., in case

*Nτ*), these quantities are the same as (12). They are plotted for case

*SB*in Fig. 6. Near the surface

*κ*is smaller than

^{L}*κ*because the Lagrangian shear is larger, but it still is much larger than

*κ*without wave effects.

^{12}The

*κ*has a depth structure that is smoothly distributed over the Ekman layer

^{L}*h*as a whole, and it has an evident suppression within the Stokes and breaker layers, for example, compared to a linear interpolation between the midlayer peak and the surface, which is characteristic of surface layer similarity with

_{o}*κ*~

*u*

_{*}

*h*when there are no wave effects. Furthermore,

_{o}z*θ*with a small negative lobe through the Stokes layer.

_{κ}^{13}This suggests that an eddy viscosity parameterization based on

*u*

_{st}≈ 0, both the conventional and Lagrangian eddy viscosity quantities are the same. So, the interior behavior of

### e. Langmuir circulations

*u*

_{st}the eddy patterns are quite different from LCs. Figure 7 shows turbulent LCs in the vertical velocity field in case

*SB*. They have smaller horizontal and vertical scales near the surface, and their longitudinal axis rotates clockwise with depth as part of the Ekman spiral. The

*w*extrema are asymmetric with larger downward speeds than upward. This asymmetry is measured by the skewness profile, shown in Fig. 8,

*S*cases) is to make Sk[

*w*] ≈ −0.8 except within the breaker layer where it decreases toward zero. In contrast, the

*N*cases have generally weaker skewness, especially in the upper half of the layer. The eddy patterns are more complex than the idealized roll cells of linear instability theory (Leibovich 1983). In particular, the largest

*w*< 0 values occur more in isolated horizontal patches than along lines, although the elongated structure is evident at a lower amplitude.

To educe the typical structure of a LC, a composite average of many individual events is employed. The vertical column is divided into 14 zones with central depths *z _{c}* to aggregate LCs with similar vertical structure; the

*z*are nonuniformly spaced to capture the finer scales near the surface. To detect a LC, a trigger criterion is defined to identify its central location. A normalized vertical velocity,

_{c}*w*

^{†}=

*w*(

*x*,

*y*,

*z*)/rms[

*w*](

*z*), is used to enable detection across a broad depth range because the magnitude of

*w*varies widely (Fig. 4). The trigger criterion is that

*w*

^{†}is a local minimum with

*w*

^{†}extrema are sorted by their magnitude, largest first. When an event is detected, a 3D local volume of size

*L*(

*z*)

_{c}^{2}·

*H*(

*z*) is then used to “black out” any other nearby events to avoid redundant captures.

_{c}^{14}All detected events in a given zone are then averaged together to produce a 3D composite spatial pattern in

**u**

_{c}(

*x*,

*y*,

*z*) and a total detection number per volume

*n*(

_{c}*z*) (i.e., per unit time). The horizontal mean is subtracted before calculating the composite fields.

Pattern recognition is inherently a fuzzy analysis procedure with potentially ambiguous event detections. So we deliberately choose conservatively large values for *L _{c}*, and

*H*. This errs on the side of undercounting the LC population by including only the strongest events based on a presumption that they will have the cleanest spatial structure. We also test that the results are not highly sensitive to the detection parameter choices, except in the total event number. The results shown here are for

_{c}*z*and for blackout exclusion sizes that increase linearly with depth,

_{c}*L*=

_{c}*H*= 2.5 m − 0.3

_{c}*z*, to match the increasing LC size (Fig. 7); for example, at the deepest

_{c}*z*= −0.95

_{c}*u*

_{*}/

*f*= −177 m,

*L*= 0.29

_{c}*u*

_{*}/

*f*= 53 m. The LC detection results in Figs. 9–11 are based on 80 temporal snapshots, with a total of 11 600 detected events used in the composite averages.

An example of a composite LC is Fig. 9 for a relatively shallow *z _{c}* = −8 m. It has a clean spatial structure of an elongated downwelling center along a horizontal axis rotated clockwise from the breaker direction, with weaker peripheral upwelling centers to the sides. The horizontal flow is forward along the rotated axis, with confluence in the rear and diffluence in front. Figure 9 (left) is in the plane of the

*w*

^{†}minimum, and it shows approximate fore–aft symmetry in the horizontal flow.

^{15}In a vertical cross section perpendicular to the axis, the primary extrema

^{16}in

*w*< 0 and

_{c}*z*=

*z*(a tilde denotes a horizontally rotated quantity; see Fig. 9 caption), with approximately the same cross-axis and vertical scales that are somewhat smaller than |

_{c}*z*|. Cross-axis horizontal convergence occurs above the central depth, and divergence occurs below. These characteristics are as we expect for LCs, although the along- and cross-axis correlation lengths are not very large in a turbulent Ekman layer.

_{c}The detected LC population density *n _{c}* is shown in Fig. 10, together with the vertical distribution of zone centers

*z*and zone boundaries. The zone size expands with depth roughly matching the increase in size of the detected LCs. The

_{c}*n*decreases with depth: there are fewer, bigger LCs deeper within the Ekman layer.

_{c}*z*=

*z*to define the horizontal rotation angle

_{c}The total contribution of the detected LC population to any mean quantity is equal to the product of population density *n _{c}* times the horizontal average of the composite quantity, summed over all zones. For example, the contribution to the vertical velocity variance profile is

_{c}

*n*〈

_{c}**u**

_{⊥c}

*w*〉, and the contribution to the eddy viscosity is Σ

_{c}_{c}

*n*. Figure 11 shows both the individual composite-zone and composite-total contributions to the 〈

_{c}κ_{c}*w*

^{2}〉(

*z*) and

*κ*(

^{L}*z*) profiles. In both quantities all zones show a similar shape varied by the peak magnitude and depth scale. So, the composite-total profiles have a similar shape. Furthermore, they are essentially similar in shape to the LES total profiles but with a smaller magnitude. The relative magnitude is somewhat larger for

*κ*than for 〈

^{L}*w*

^{2}〉, indicating that LCs are more efficient agents in momentum flux than their variance fraction would imply. We conclude that the statistical structure of Ekman layer turbulence is primarily the result of its coherent LCs. Because of the conservative design of the detection procedure to avoid false detections, we interpret the magnitude discrepancy as a consequence of an undercount of the LC population (

*n*too small). We hypothesize that this discrepancy would close with a more sophisticated detection procedure.

_{c}A striking result in Fig. 6 is the positive eddy viscosity directions, *θ _{κ}* and inconsistent with simple eddy viscosity. The LC composite flux angle is very close to the total flux angle near the surface. In the interior the rate of clockwise rotation is very small for the LC flux, and over the bottom half it rotates too slowly compared to the total flux. We conclude that the detected LCs are the source of positive

*θ*. Evidently the remainder of the turbulent fluctuations (including undetected weaker LCs) have a more rotated flux angle on average, so the total flux angle value lies in between the LC flux and mean shear values. At the bottom of the Ekman layer (

_{κ}*z*< −

*h*), all four angles coincide, but, of course, there is not much mean flow, variance, or flux down there.

_{o}### f. Breakers and downwelling jets

To illustrate the 3D structure of a typical breaker, another composite average is constructed from many transient events in case *SB*. The detection criterion is that the surface *u* in the breaker direction exceeds a positive threshold value *U*_{cr}, chosen as *U*_{cr} = 10*u*_{*} = 0.2 m s^{−1}, over a connected area of _{cr} = 1.6 × 10^{−3} (*u*_{*}/*f*)^{2} = 55 m^{2}. Again, these choices are conservative ones that select the larger, stronger breakers. For composite averaging, the origin is placed at the position of maximum *u* > 0. The composite pattern in Fig. 13 has strong downwelling in the front and weaker upwelling in the rear. The horizontal velocity is stronger in *u* than in *υ*, divergent and confluent in the rear, and convergent and diffluent in front. The depth scale is slightly larger than *h _{b}* because the composite is for relatively larger, stronger breakers. Notice that the

*y*scale is wider for breakers than for upper-ocean LCs (Fig. 9). All of these characteristics are a response to the specified shape of the breaker acceleration events,

*uw*〉(

*z*) < 0 near the surface (

*z*> −2

*h*); however, it is much weaker than for the LC composites.

_{b}In the wavy Ekman layer, an interesting phenomenon emerges: namely, coherent, downward-propagating, downwelling jets. We detect them by a variant of the LC detection procedure (section 3e): for a large *w*^{†} < 0 anomaly first detected within the top 3.5 m, a search is made for another large anomaly in the local spatial neighborhood at a subsequent time 20 s later. If the new detection is successful, the process is continued in time, always searching in the local neighborhood of the latest detection. The detection sequence is terminated when no new local strong anomalies are found. This procedure yields many examples of downwelling jets that penetrate much of the way through the boundary layer (Fig. 14). They have a typical downward propagation speed of about 0.3*u*_{*}, which is a small fraction both of the rms *w* (Fig. 4) and of their own local *w* extremum and have a typical horizontal propagation speed of several times *u*_{*}, generally following the mean flow (Fig. 2). The downwelling jet extremum typically occurs along the horizontal axis of a LC; hence, it contributes to the LC structure in *w* more as an isolated extremum along the axis than as a longitudinally uniform distribution typical of roll cells (Fig. 7). Deep downwelling jets are much less frequently detected than either breakers or LCs separately, but they are much more frequent and coherent in case *SB* than any of the other cases in Table 1. In laboratory experiments on breaking waves without evident LCs, deep downwelling jets are not seen (Melville et al. 2002).

Case *SB* also has the largest negative skewness among all the cases here, with Sk[*w*] ≈ −0.85 around *z* = −0.15*u*_{*}/*f* (Fig. 8), although its distinction from other *S* cases abates into the interior. We interpret this as an incremental effect of the strong downwelling jets on top of the primary LC asymmetry in *w*. Thus, the jets arise out of an interaction between breakers and LCs through a vertical vorticity catalyzation process provided to LCs by the finite transverse scale of the breaker acceleration, in particular the opposite-signed vertical vorticity extrema on either side of the breaker center in Fig. 13 (left). A vertical vorticity seed is tilted and stretched by Stokes drift and the mean current to grow into the longitudinal vorticity of a mature LC (Leibovich 1983; Sullivan et al. 2008). This phenomenon is more pronounced with our choice of relatively young wave age with its larger breakers than with the older waves in full wind wave equilibrium (section 2). This catalyzation process is not, of course, the only way to generate a LC because many other vertical vorticity seeds are present in a turbulent boundary layer.

### g. Surface drift

A long-standing, practical oceanic question is the lateral drift of a buoyant object at the surface. Its simplest posing is as pure fluid drift, neglecting windage and other bulk forces on the object and surfactant rheological complexity. In the Ekman problem, we have defined the Lagrangian mean flow by

**X**(

*t*;

**X**

_{0},

*t*

_{0}) as the Lagrangian horizontal coordinate of a particle released at a random location

**X**

_{0}at time

*t*

_{0}. For

*t*>

*t*

_{0}it moves with the local surface Lagrangian flow:

**U**

^{L}is defined as the ensemble average of (18) over many releases at (

**X**

_{0},

*t*

_{0}) and their

**X**(

*t*) trajectories of long duration. Figure 15 (left) is a snapshot for the wavy case

*SB*of a set of

**X**(

*t*) positions with a large

*t*−

*t*

_{0}, calculated by (18) using LES

**u**

_{⊥}fields. The locations are organized into fragmented lines and apparently have lost any correlation with their original release locations by becoming trapped in convergence zones. For this case the mean drift velocities expressed in (

*u*,

*υ*) components are

**U**

^{L}= (17.1, −3.3)

*u*

_{*}, with a large downwind

*u*

_{st}(0) = 17.5

*u*

_{*}contribution. So, the short- and long-term Lagrangian drifts are relatively little different.

^{17}Similarly small differences are seen in our other LES cases.

We calculate the composite-average surface horizontal flow conditioned on strong convergence (Fig. 15, right). There is little horizontal flow through the convergence center, which is where surface trajectories will spend most of their time once they become organized into wind rows. That is, the surface-trapped particles move into the LCs but do not move through them. A similar fore–aft asymmetry for surface flow in the LCs is shown for hurricane LES simulations with nonequilibrium wind waves in Figs. 11 and 12 of Sullivan et al. (2012) though with some instances of nonzero but weak down-axis flow ahead of the downwelling center. This flow structure contradicts the roll-cell paradigm with a positive downwind velocity anomaly extending along the cell axis. However, it does partly explain why **U**^{L} and

Using the *u*_{*}–*U _{a}* relation in section 2, we can re-express the mean drift velocity

*U*rotated 10° to the right of the wind direction for case

_{a}*SB*. In the ocean an ensemble of surface drift measurements is difficult to control for varying conditions of wind, waves, and stratification, and commonly averages are made by lumping different situations together. Ardhuin et al. (2009) uses a combination of surface radar backscatter and a numerical wave model to estimate mean surface drifts (comparable to

*U*rotated 10°–40° to the right of the wind, with higher speed and greater rotation when the stratification is strong. They explain that their speed may be an underestimate because some depth averaging is implicit in the radar backscatter process near the surface where the Stokes shear is large. Given this caveat and their lumping of many situations, we do not see our answer for

_{a}**U**

^{L}as notably inconsistent. However, there is a literature of empirical estimates of substantially larger surface drift speeds in excess of 0.03

*U*(e.g., Bye 1966; Wu 1983; Kim et al. 2009), which is not supported by our LES results or by the measurements of Ardhuin et al. (2009); we will not attempt to reconcile these historical contradictions.

_{a}## 4. Parameterization implications

Oceanic general circulation models (OGCMs) require full-turbulence (RANS) parameterization of boundary layer turbulent fluxes to calculate upper ocean currents and material distributions. Because *z*) is quite different in cases *SB* and *Nτ* (Fig. 2), we conclude that presently used OGCM parameterizations are inadequate without wave effects. In particular, the parameterization influences on boundary layer depth, vertical mixing rate, and velocity profile shape need to be changed.

**u**

_{⊥}(

*z*,

*t*) with specified wind and wave forcing in the

*x*direction [

*τ*,

^{x}*u*

_{st}(

*z*), and

*A*(

*z*)]:

**F**is the parameterization of the Reynolds stress, −〈

**u**

_{⊥}

*w*〉(

*z*). Boundary conditions are

*z*= 0 and

**u**

_{⊥},

**F**→ 0 as

*z*→ −∞. The KPP model for the unstratified Ekman layer is

*c*

_{1}and

*c*

_{2}(McWilliams and Huckle 2006). Notice that there are no wave influences in this scheme for

**F**.

We test KPP for the classical Ekman layer without wave effects: that is, case *Nτ*. First, we optimally fit the values of *c*_{1} and *c*_{2} to minimize **u**_{⊥}(*z*) between LES and KPP, normalized by the rms magnitude of **u**_{⊥}(*z*) from LES. The minimum value is *c*_{1} = 0.29 and *c*_{2} = 0.72. These constants are close to the values *c*_{1} = *k* = 0.4 for the von Kármán constant *k* and *c*_{2} = 0.7 previously proposed for an Ekman layer modeled with KPP (viz., McWilliams and Huckle 2006) but with *c*_{1} somewhat smaller here. The quality of the KPP fit to **u**_{⊥}(*z*) is good by boundary layer parameterization standards (Fig. 16). There are larger discrepancies in the shape of *κ*(*z*) than in **u**_{⊥}(*z*), but eddy viscosity itself is not the important parameterization product for OGCMs except as a means to obtain **u**_{⊥}. In particular, without stable density stratification, *κ* in LES does not vanish at depth as sharply as in the KPP model, but the deep value of *κ* is evidently not very important in determining **u**_{⊥}(*z*) after it has decayed to a small magnitude. What is most important for achieving a small value of **u**_{⊥} is large. The KPP recipe (20) is consistent with a wall-bounded similarity layer (a.k.a. log layer) where *z*| → 0; thus, the strongest constraint is on matching the product of *c*_{1}*c*_{2} with the LES answer. A caution is that the similarity-layer shear is theoretically singular, ∂_{z}*u* → *ku*_{*}/*z*; hence, LES can only provide a discretely approximate standard for such a case, and LES–1D discretization differences also limit the degree of agreement in **u**_{⊥}. The modest degree of nonalignment between 〈**u**_{⊥}*w*〉 and ∂_{z}〈**u**_{⊥}〉 (*θ _{κ}* > 0 in Fig. 6 for case

*Nτ*) is evidently not a serious obstacle to a reasonably skillful fit with the KPP parameterization scheme. By practical parameterization standards for use in OGCMs, there is little motivation to try to do better in this wind-only case, apart from improving the precision of the calibration and OGCM implementation if these are important limitations.

**u**

_{⊥}(

*z*) is substantially altered by waves (section 3). At the least, the boundary layer depth needs to be deeper and the eddy viscosity

*κ*magnitude be larger with wave effects (Table 1 and Fig. 6). McWilliams and Sullivan (2000) propose an amplified

*κ*magnitude due to

*u*

_{st}based on a case with a stratification-limited depth, and Eq. (2) suggests a scaling for the amplification of the turbulent velocity scale

^{18}(but note the cautionary remark at the end of section 3a). Figure 6 shows

*θ*>

_{κ}*π*/2 for case

*SB*around the Stokes layer. In a conventional relation of aligned flux and shear,

**F**=

*κ*∂

_{z}

**u**, this implies locally negative diffusion, which is potentially ill behaved in time integration of the 1D model (19). Recognizing the existence of flux-gradient misalignment in LES with waves, Smyth et al. (2002) propose the addition of non-eddy-viscous, countergradient flux profiles to a KPP scheme for

**F**, in analogy with its successful application in a convective regime (where ∂

_{z}〈

*T*〉 and 〈

*wT*〉 have the same sign over much of the boundary layer). This proposal has the disadvantage of complexity by needing to specify a model for the vector profile shape and orientation and, unlike in the convective regime, the eddy momentum flux here is not literally countergradient (i.e.,

*θ*≠

_{κ}*π*). A potentially simpler remedy to the ill behavior of a negative-diffusion scheme is suggested by the alternative of a stress-aligned Lagrangian eddy viscosity scheme,

*κ*≥ 0. Figure 6 shows that the problematic Stokes layer structure in

^{L}*θ*is greatly diminished in

_{κ}*SB*.

*A*〉(

*z*) and

*u*

_{st}(

*z*) profiles from Fig. 1 and a replacement for

**F**with the generalized Lagrangian eddy viscosity profiles

*κ*(

^{L}*z*) and

*θ*) is a horizontal rotation matrix representing the rotation of the shear direction into the Reynolds stress direction. In this expression,

*κ*

^{L}*z*) is an eddy viscosity tensor, dependent upon two scalar functions,

*κ*(

^{L}*z*) and

**u**

_{⊥}with good accuracy (

*SB*using the LES-diagnosed profiles of

*κ*and

^{L}**F**and LES-diagnosed Eulerian viscosities.

*κ*(

*z*) in Fig. 6 for a

*κ*

^{L}^{(1)}by matching the interior shear layer shape near its peak. This match can be done better with Lagrangian

*κ*than an Eulerian

^{L}*κ*because its peak is deeper and more in the center of the layer, as in the KPP shape. The refitted coefficients are

*c*

_{1}= 0.8,

*c*

_{2}= 1.4, consistent with bigger

*κ*and

*h*with waves. The resulting

_{o}**u**

_{⊥}(

*z*) with (

*κ*=

^{L}*κ*

^{L}^{(1)},

*θ*= 0) is a very poor fit to the case

_{κ}*SB*profile (R = 0.8), mainly due to very different

*u*(

*z*) near the surface. Step 1 is thus necessary but insufficient. Step 2: Noting that the

*κ*

^{L}^{(1)}is very much larger than the LES

*κ*near the surface (because there is no similarity layer with waves), we derive a surface layer approximation to the

^{L}*x*-momentum balance in (3) by neglecting Eulerian velocity compared to Stokes velocity in the aligned Lagrangian eddy viscosity model (21):

*z*→ 0

^{−}in the Stokes layer, because the denominator is increasing while the numerator

*S*prevents divergence of

_{o}*z*→ −∞, and the small value

*S*= 0.0025 ∂

_{o}_{z}

*u*

_{st}(0) makes a smooth transition in a composite specification,

*κ*

^{L}^{(2)}=

*κ*

^{L}^{(1)}in the lower part. Figure 17 shows that

*κ*

^{L}^{(2)}are an excellent fit to the LES-derived

*κ*above the blending point at

^{L}*z*≈ −0.18

*u*

_{*}/

*f*. In the interior shear layer,

*κ*

^{L}^{(2)}is a modest misfit to the LES

*κ*, to a similar degree as in case

^{L}*Nτ*in Fig. 16. The 1D solution for

**u**

_{⊥}with (

*κ*

^{L}^{(2)},

*Nτ*, the deeper reach of

*κ*in LES is not important for the

^{L}**u**

_{⊥}skill. Step 3: To further reduce the error, we include the misalignment effect with the smoothed and depth-truncated

*κ*=

^{L}*κ*

^{L}^{(2)}gives a very good fit in

**u**

_{⊥}(

*z*) with

Appendix B is the analytic Ekman layer solution for misaligned, Lagrangian eddy viscosity with constant viscosity *κ _{o}* and rotation angle

*θ*. It provides an explanation for the primary differences in

_{o}**u**

_{⊥}(

*z*) between the two panels in Fig. 18: near the surface, where

*u*is larger and −

*υ*is smaller, that is, less clockwise rotation relative to the wind direction, and in the interior shear layer, where

The influence of breaker acceleration *A* (versus surface stress *τ ^{x}*) is only weakly evident in the shape of

**u**

_{⊥}(

*z*) in Fig. 18 as a weak positive shear in

*u*and positive veering in

*θ*(also in Fig. 2). The primary

_{u}*x*-momentum balance in the breaker layer is between

*A*and −∂

_{z}(

*uw*), not the Coriolis force ∝

*fυ*. The most important

*A*influence is a desingularization of the surface layer, compared to a surface stress boundary condition and its associated similarity layer. For

*z*< −

*h*where

_{b}*κ*,

^{L}**u**

_{⊥}, and the Reynolds stress profiles are smooth in

*z*, and the limiting case

*h*→ 0 is mathematically and computationally well behaved and physically meaningful. In contrast, a surface stress condition in combination with

_{b}*κ*→ 0 is ill-behaved and illconceived in the presence of waves.

Thus, we have demonstrated in three steps—the first: the familiar KPP model for the interior shear layer with a wave-enhanced *κ* magnitude and depth scale; and the second: a derived dynamical approximation near the surface; and the third: a qualitatively simple, albeit unfamiliar misalignment profile shape (which could easily be expressed in a formula)—that an accurate 1D model is achieved with Lagrangian eddy viscosity in the wavy Ekman regime with both *τ* and *u*_{st} important [i.e., *τ* = 0, and its derivation assumes large Stokes shear]. This cannot be done as well with Eulerian eddy viscosity because there is no derivable analog of *κ*_{sur} for the Stokes and breaker layers, which therefore would have to be yet another empirically fitted aspect of the model; the Eulerian *θ _{κ}* shape is more convoluted (Fig. 6); and the fit to a KPP shape is less apt in the interior shear layer. One might argue that the first two steps alone—leaving out the

This demonstration does not yet yield a usable parameterization scheme, of course, because the few LES cases examined here do not make up a regime scan of wind, wave, and buoyancy influences in the surface boundary layer,^{19} with the extensive calibration and testing necessary for usability. Nevertheless, it is likely that the wave influences seen in the Ekman problem will be echoed more generally.

## 5. Summary

Under conditions close to wind wave equilibrium, the influences of surface gravity waves are quite significant in the Ekman layer. The Stokes-Coriolis and vortex forces are the main influences, while the differences between breaker acceleration and surface stress are secondary and mostly localized near the surface. The Ekman layer as a whole approximately separates into three vertical sublayers: the breaker layer where *A* is large, the Stokes layer where *u*_{st} is large, and the interior shear layer underneath, with *h _{b}* ≪

*h*

_{st}≪

*h*in the cases considered here. These distinctive sublayers are evident in the mean current and Reynolds stress profiles, as well as the momentum and turbulent kinetic energy balances. The Ekman layer with waves is deeper and more energetic, and its surface current profile

_{o}**u**

_{⊥}(

*z*) is controlled by the shapes of

*A*(

*z*) and

*u*

_{st}(

*z*)—neither of which is easily measured in the ocean—acting through

*κ*

_{sur}(23) and the 1D momentum balance (3) with Stokes-Coriolis force. This is a different conception of Ekman surface layer dynamics than either Monin–Obukhov similarity or breaker energy injection (Craig and Banner 1994); breaker energy injection

*z*) does occur distributed over

*h*, but it does not directly relate to the Reynolds stress

_{b}**F**or eddy viscosity

*κ*, hence not to the momentum balance and

**u**

_{⊥}(

*z*) profile. The cases with mean acceleration and energy injection profiles, 〈

*A*〉(

*z*) and 〈

*W*〉(

*z*), give generally similar answers to those with stochastic

*A*and

*W*, and they are much simpler and more economical to compute. The energy cycle is very different with forcing by either mean stress or breaker injection so that the latter is much to be preferred as a process depiction. The partial wave formulation of Stokes drift without breaker injection (case

*Sτ*) is ill structured approaching the surface, with LCs developing very fine scales without the regularization provided by breaker-augmented mixing and dissipation.

Breaker acceleration creates transverse overturning cells near the surface, and shear instability and Stokes vortex force create longitudinal LCs whose scale expands and horizontal orientation rotates with depth. Both types of coherent motions contribute important Reynolds stress. These influences occasionally combine to create downward-propagating downwelling jets. In the surface layers, the large Stokes shear requires rapid rotation with depth of the Reynolds stress, and in the interior shear layer the LCs rotate clockwise (i.e., have substantial vertical coherence) more slowly with depth than the mean shear (Ekman spiral); these behaviors create a moderate degree of stress–shear misalignment that is inconsistent with downgradient eddy viscosity. The mean surface Lagrangian drift of buoyant particles with waves is dominated by the Stokes drift velocity and rotated slightly rightward; this drift is only slightly different for short- and long-time particle trajectories in spite of particles become trapped within LC convergence zones.

To both explore parameterization possibilities and test our comprehension of wave influences, we solve a 1D model (19) with parameterized Reynolds stress **F**. Without wave effects (case *Nτ*), a K-profile parameterization scheme is successful. With wave effects (case *SB*) several modifications are necessary for success: a KPP profile shape with greater, deeper eddy viscosity in the interior shear layer; a Lagrangian eddy viscosity scheme (23) in the breaker and Stokes layers; and a stress–shear misalignment profile with

The ocean has a wide range of wind wave conditions, as well as various buoyancy influences. Often the transient evolution is more evident than a steady-state equilibrium in the surface boundary layer. So, the wavy Ekman layer problem solved here, while central, is hardly general. A good strategy is still needed for encompassing the general behaviors of the upper ocean in measurements and models.

## Acknowledgments

The authors are grateful to the National Science Foundation (Grant DMS-785 0723757) and Office of Naval Research (Grant N00014-08-1-0597) for support. Computations were made on the supercomputers Bluefire at the National Center for Atmospheric Research and Thresher and Trestles at the San Diego Supercomputing Center. We appreciate discussions with Fabrice Ardhuin about surface drift.

## APPENDIX A

### Mean and Total Kinetic Energy Balances

The energy analyses in sections 3a and 3c focus on the total work done by stress and breaker forcings, *e* defined in (1) contains both resolved-eddy and subgrid-scale energies and its balance relation (7). To clarify the total energy context, we complement them here with the energy balance relation for the mean flow (MKE):

*z*= 0), the mean stress and breaker acceleration injection is

*A*′ work and subgrid-scale injection

*W*are assigned to the

*e*balance in (8). The Stokes-Coriolis force provides an energy conversion with the surface gravity wave field [as does Stokes production,

*e*balance (7)]:

_{u}defined in (9) is a conversion from

*e*. Finally, notice that a mean dissipation rate associated with the subgrid-scale stress can be defined as

*i*here is only horizontal because 〈

*w*〉 = 0); however, it is already part of

_{u}in (9), so it does not contribute separately to the MKE balance.

The total energy balance relation is the sum of (7) and (A1). It has depth-integrated sources from injection, _{st} and a single dissipative sink from *ϵ*. This is shown diagrammatically in Fig. A1. Notice that all three sources contain a conversion with the surface wave field. The sum of sources equals the dissipation sink in equilibrium.

We do not show a quantitative evaluation of the MKE balances, but rather summarize them qualitatively from a volume-integrated perspective. With Stokes vortex forces (*S* cases), the primary *u*〉(0) is small, both a consequence of the Stokes-Coriolis force. [The sign of *υ*〉 < 0 by the southward Ekman transport constraint in (5).] However, *S* cases, and _{st} with breakers (*B* cases), so the two wave conversions acting directly in the TKE balance are the important sources, with the _{u} conversion from MKE a minor effect. The wavy energy route is summarized as _{st} → *ϵ*. This is very different from the Ekman layer without waves (case *Nτ*), where the MKE → TKE route is essential:

## APPENDIX B

### Analytic Solution with Misaligned Lagrangian Viscosity

*U*=

*u*+

^{L}*iυ*, using

^{L}*θ*is in a range around 0 where

_{o}*k*> 0. With

*u*

_{st}=

*θ*= 0, this is the classical Ekman solution. Otherwise, compared to the classical solution,

_{o}*u*has a flow component opposite to

*u*

_{st}; the vertical decay rate

*k*is faster and the rotation rate

*l*is slower with

*θ*> 0 (and vice versa if

_{o}*θ*< 0); and some

_{o}*θ*values are inconsistent with a boundary layer solution (e.g.,

_{o}*θ*= −

_{o}*π*/4 where

*k*= 0).

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^{1}

Polton et al. (2005) and Polton and Belcher (2007) also analyze simulations of an unstratified Ekman layer with Stokes drift.

^{2}

This age is somewhat young compared to full wind–wave equilibrium, with *c _{p}*/

*u*

_{*a}= 30. Younger waves have relatively fewer, larger breakers, and we make this choice to allow better resolution for a given spatial grid. Our conclusion in section 5 is that the details of the

*A*and

*W*profiles are only important within the breaker layer, so the age choice is not determinative overall. See Sullivan et al. (2007) for details about how the elevation and breaker spectra are specified from measurements and related to

*u*

_{st},

*A*, and

*W*consistent with conservation of momentum and energy in the air–wave–water system.

^{3}

This is distinct from the mean kinetic energy profile,

^{4}

*A*′ and subgrid-scale energy injection *W*. This separation is relevant to the separate mean *e* balances. See section 3c and appendix A for the full energy balances.

^{5}

In case *Sτ* the small wind stress injection is not the dominant energy source, which rather are Stokes-Coriolis and Stokes production conversions with the surface waves (section 3c and appendix A).

^{6}

For a full wave spectrum, the choice of *u*_{st o} is somewhat delicate because *u*_{st}(*z*) near the surface is sensitive to the spectrum shape. So, we prefer to view this scaling estimate for *u*_{*com} qualitatively rather than precisely. Similarly, the turbulent Langmuir number, *u*_{st o}. From Fig. 1 we see that *La _{t}* is a bit smaller than 0.3 in our

*S*cases, close to a local wind wave equilibrium value.

^{7}

Harcourt and D’Asaro (2008) propose a modified form of *u*_{*com} with different vertical weighting of *u*_{st}(*z*) for use in scaling the variance of *w* under more general circumstances. Kukulka et al. (2010) propose another modification when 〈*u*〉(0)/*u*_{st o} is not small (unlike in our *S* cases).

^{8}

_{st}and cancel any net contribution to

^{9}

An implication of the diagnosed transport profile is that there probably is finescale structure on a scale of perhaps 10 cm or less near the surface, which is not well resolved in our present solutions. Besides the discretization accuracy limitation that could be ameliorated with finer grid resolution, we would question the physical validity of our subgrid-scale and breaker parameterization schemes in a surface microscale realm.

^{10}

The small value of *θ _{κ}*(

*z*) is robustly nonzero with respect to computational parameters and statistical averaging accuracy in case

*Nτ*. We do not have an explanation.

^{11}

This is the short-time mean velocity averaged over an ensemble of parcels that move with **x**(*t*_{0}).

^{12}

The enhancement of *κ* near the surface is expected from a model of TKE injection by wave breaking (Craig and Banner 1994). Our solutions indicate it is an ill-determined quantity because the mean shear ∂_{z}〈*u*〉 is weak near the surface. In contrast, *κ ^{L}* is well determined; see (23).

^{13}

**u**

_{⊥st}is larger than 〈

**u**

_{⊥}〉,

*υw*〉 decreases rapidly and

*θ*

_{−uw}rotates clockwise rapidly, while

*u*

_{st}is relatively large.

^{14}

More precisely, we focus on excluding LCs with excessive lateral or vertical overlap by defining the black-out volume of a candidate LC as the union of two rectangular volumes of sizes (2*L _{c}*)

^{2}·

*H*and

_{c}*w*

^{†}extremum.

^{15}

In planes above the LC center, the aftward confluent flow is much stronger than the foreward diffluent flow, especially at the surface (also see section 3.g).

^{16}

Because we base the detection on the locally normalized amplitude *w*^{†}, it is not guaranteed that the absolute amplitude of *w _{c}* will be largest at

*z*, as it is in Fig. 9. In Fig. 11 (note the dots on the profile curves), we see that the maxima in |

_{c}*w*| and

_{c}*κ*occur at shallower depths than

_{c}*z*for the deepest detection zones, although these maxima are deeper than for those for shallower detection zones. For the shallowest zones, the profile maxima occur slightly above

_{c}*z*.

_{c}^{17}

Nevertheless, their differences are statistically significant based on standard error estimates. Across the *S* cases, the long-term drift is rotated more to the right than the short-term drift [i.e.,

^{18}

A consequence of larger *κ* is increased entrainment rate at the pycnocline (McWilliams et al. 1997). This is likely to be a general behavior in stratified boundary layers with waves.

^{19}

The Coriolis force with a nonvertical rotation axis is also influential in Ekman layers, especially in the tropics. A KPP scheme is proposed in McWilliams and Huckle (2006), but as yet its interplay with wave effects is unexamined.