## 1. Introduction

The wind blows and the waves rise and roll on. This is the regime of Langmuir turbulence in the oceanic surface boundary layer (BL), so-called because Langmuir circulations (often recognized by the windrows in the surfactants they cause) are the primary turbulent eddies whose vertical momentum and buoyancy fluxes maintain the mean ageostrophic current and density stratification. Langmuir circulations arise from the instability of wind-driven boundary layer shear in the presence of Stokes drift (Craik and Leibovich 1976; McWilliams et al. 1997). Alternatively expressed, this regime is a wind-driven Ekman layer with stable interior density stratification and surface gravity waves that induce wave-averaged vortex and Coriolis forces and buoyancy advection due to the Lagrangian-mean Stokes drift velocity. Langmuir turbulence is common in nature (Belcher 2012).

The physics of this regime is now well understood, primarily on the basis of many large eddy simulation (LES) studies of the wind-wave equilibrium state [e.g., the review in Sullivan and McWilliams (2010)]. However, in nature the wind is rarely in equilibrium with the waves (Young 1999; Sullivan et al. 2008; Hanley et al. 2010). Often this is due to transient wind changes or limited fetch, when the waves have not yet evolved into a fully developed equilibrium but are headed there over an interval of hours or over an offshore distance of tens of kilometers. Disequilibrium also occurs when swell-waves with a large amplitude and long wavelength propagate thousands of kilometers away from their generation in an earlier strong storm to a place with weaker local wind whose wind-waves are weaker and shorter. During their propagation, swell waves exhibit slow decay and slow evolution through dispersion and four-wave resonant interaction, and they have weak nonlinear interactions with the wind-waves encountered en route as long as there is significant separation in wavelength and/or direction (Masson 1993; Zakharov 2005; Depley et al. 2010). Under these conditions, a local quasi-equilibrium state of linearly superimposed swell- and wind-waves can persist for a day or more if the wind is steady. These circumstances are especially common in the tropics and subtropics in both hemispheres, with swell-waves coming from higher-latitude winter storms (Alves 2006; Hanley et al. 2010). Also, they can occur anywhere in the lull after a storm.

Several idealizations of persistent disequilibrium have been examined with LES. Commonly, the wave spectrum is simplified to a monochromatic wave at the spectrum peak (McWilliams et al. 1997), the wind-wave Stokes drift is multiplied by a factor (usually less than one, implying weaker waves with a turbulent Langmuir number La ≥ 0.3, its equilibrium value; see section 2) (Harcourt and D’Asaro 2008; Grant and Belcher 2009), or the wind-wave Stokes drift direction is rotated relative to wind alignment (Van Roekel et al. 2012). All of these formulations are ad hoc compared to realistic persistent wind regimes where the wind-sea evolves toward equilibrium. On the other hand, the general formulation of wind-wave disequilibrium is daunting in the complexity of possible transient histories. In contrast, the case of equilibrium wind-waves with added steady swell-waves with well-separated spectrum peak wavenumbers (hence La < 0.3) is an apt idealization of realistic persistent situations. Because the remote swell-generating storm and the local wind are independent, the added swell component can have an arbitrary orientation and amplitude relative to the local wind-sea. This is the problem addressed here.

## 2. Formulation

The LES code solves the wave-averaged dynamical equations in Sullivan and McWilliams (2010). These are incompressible, rotating Boussinesq fluid dynamics with a prognostic turbulent kinetic energy equation for subgrid-scale fluxes. Both model components have surface wave effects that include Stokes drift Coriolis and vortex forces and scalar advection, augmented pressure head, and subgrid Stokes drift energy production and advection. The “wind” forcing is by parameterized breakers that inject a body force *A* and subgrid work *W*; for simplicity in the present paper, ensemble-mean breaker forcing profiles *z* is the vertical coordinate.) The same dynamical formulation is used in McWilliams et al. (2012) for the wavy Ekman layer with uniform density and equilibrium wind-waves. In particular, in that paper the differences between ensemble and stochastic breaker forcing are analyzed, as well as their differences compared with a conventional surface stress forcing with an equivalent integral effect (i.e., *τ* is stress, *ρ*_{o} is the mean density, and *τ* and its associated *U*_{a} with a bulk drag law;^{1} that is, *ρ*_{a} = 1 kg m^{−3} and *C*_{D} = 1.3 for the cases here with *U*_{a} ≤ 10 m s^{−1}.

The LES model is spun up from rest and approximately equilibrates over a period of a day or so, while retaining inertial oscillations that slowly decay over many days. In the present problem, an initial mean temperature stratification *T*(*z*) is included. It initially has a well-mixed layer in the top 33 m and an interior gradient of 0.1°C m^{−1} (implying a buoyancy frequency of *N* = 0.014 s^{−1}). Turbulent entrainment develops, and the boundary layer deepens. Because this *N* value is rather large, the boundary layer depth *h*(*t*) only slowly deepens while the average mixed layer temperature *f* is 10^{−4} s^{−1} (corresponding to a latitude of 45°N) and is assumed to be spatially uniform over the small spatial scale of the domain. With the traditional approximation that neglects the horizontal projection of Earth’s rotation vector, the solutions are symmetric with respect to wind direction; hence, without loss of generality, the wind direction is chosen as

*z*≤ 0 where mean sea level is

*z*= 0. Because swell-wave spectra are narrow, the swell is represented as a linear, monochromatic surface gravity wave with period

*P*, wavelength

*λ*, and amplitude

*a*. The dispersion relation in deep water is

*g*is gravitational acceleration). The associated Stokes drift has a depth scale

*D*

^{s}=

*gP*

^{2}/8

*π*

^{2}and a surface velocity

*U*

^{s}= 8

*π*

^{3}

*a*

^{2}/

*gP*

^{3}. The term

Wave measurements are routinely made from oceanic buoys, and an extensive compendium is available from the Coastal Data Information Program (CDIP; www.cdip.ucsd.edu). Data and plots are presented in terms of *P* (or *H*_{s} (*a*). In some analyses waves with a spectrum peak *P* longer than 10 s are categorized as swell-waves, although they are wind-waves in the midst of high winds; this ambiguity is rarer at subtropical and tropical sites with fewer big storms. Climatological histograms for individual sites are presented for the joint distribution of *P* and *H*_{s}. Events with *H*_{s} > 10 m or *P* > 20 s are found, although the bulk of the swell-wave events are not so extreme. These imply possible ranges of *U*^{s} up to 0.4 m s^{−1} and *D*^{s} up to 50 m, both of which are much larger than the amplitude and depth scale for Stokes drift in a moderate wind-sea. No restriction is placed on the swell-wave direction. Figure 1 shows a particular example of mixed wind- and swell-waves with large *H*_{s}.

LES solutions are calculated in a vertical domain *H* = 100 m thick and a horizontally periodic domain (of width *L* = 300 m, except where otherwise stated). The vertical grid is stretched with a smallest spacing of 0.3 m near the surface. The horizontal grid is uniform with a spacing of 2 m. These grids encompass the boundary layer turbulence down to the resolution limit where the subgrid parameterization regularizes the flow.

The influence of swell-waves is explored through its parameters *U*^{s}, *D*^{s}, and *U*^{s} and *D*^{s} tend to be anticorrelated because of their inverse dependencies on powers of the period *P* [see just below (1)]. The LES results are nondimensionalized by *h* diagnosed from the solution averaged over the statistical analysis period *t* = 0.55 − 1.18 × 10^{5} s (i.e., one inertial period, 2*π*/*f*, with a Coriolis frequency *f* = 10^{−4} s^{−1}). The term *h*(*t*) is defined as the depth at which the mean thermal gradient ^{−1} just below the well-mixed layer. Initially, *h*(0) = 33 m in all cases.

Case sets analyzed in this paper. In all cases *A*, *W*) are in equilibrium with the eastward wind. For these case sets, the La values are 0.14–0.24 (Us1), 0.13–0.29 (Us2), 0.16 ^{−1} and **u**_{st} = 0 (La = ∞).

**u**

_{st}is evaluated at the first grid level of

*z*= −0.15 m. The value of La is weakly grid dependent because there is a logarithmic singularity in

*z*→ 0 arising from the high-frequency tail in the wind-sea spectrum of Alves et al. (2003) with

*z*→ 0. Notice that a vector misalignment between swell- and wind-waves causes a reduction in |

**u**

_{st}(0)|, hence an increase in La, compared to alignment. Depending on the swell amplitude and orientation, La may or may not be increased compared to the case of only wind-waves.

## 3. Analytic Ekman–Stokes model

*ρ*

_{o}, constant eddy viscosity

*κ*

_{o}, and surface wind stress

*τ*instead of breaker impulse

**A**:with a surface boundary condition of

**u**→ 0 as

*z*→ −∞. As is commonly done for analytic convenience, this is rewritten as a scalar equation for the complex velocity

*κ*

_{o}∂

*z*

*τ*

^{x}+

*iτ*

^{y})/

*ρ*

_{o}and

*T*,

*U*

^{w}, and

*U*

^{s}are taken to be real and positive. Also, assume that

*f*> 0 (Northern Hemisphere).

^{L}=

_{st}

For the usual large swell situation of *U*^{s} > *U*^{w} and *D*^{s} > *D*^{w}, these solutions indicate that a sufficient condition for the local wind-wave influence to be relatively unimportant in the boundary layer flow profile is for *U*^{s}*D*^{s} ≫ *U*^{w}*D*^{w}. All of the cases in Table 1 with *U*^{s} ≠ 0 are in this regime.

*θ*

^{s}relative to the wind direction (here E). Define

*U*

^{s}= 0). Then for

*U*

^{s}≠ 0 and general

*θ*

^{s}, the difference fields

*θ*

^{s}; that is,The same rotational symmetry holds for −

*κ*

_{o}∂

_{z}

**R**(

*z*) in (15)].

*z*) is augmented by an inertial oscillation

_{in}(

*z*,

*t*) such that their sum is zero at

*t*= 0 at all depths. With an additional acceleration term ∂

_{t}

*k*, anticyclonic temporal rotation of the current direction at an inertial frequency

*f*, and amplitude decay at a rate

*κ*

_{o}

*k*

^{2}. The longer-lasting components have smaller

*k*, and hence a larger vertical scale. So an estimate of the initial inertial current amplitude is made with the steady velocity averaged over the boundary layer, that is, the transport magnitude divided by the Ekman depth scaleusing (9) to evaluate the integral.

For comparison with the LES results below, the Ekman–Stokes solution is normalized by *h*_{e} for depth, and *κ*_{o}∂_{z}*U* or −*κ*_{o}∂_{z}**u**). As will be shown, the matches of the Ekman–Stokes solutions to the LES results are not precise in their vertical profile structure, as is to be expected with the strong simplifications of uniform density and eddy viscosity (cf. Fig. 6, described in greater detail below) and a monochromatic wind-sea profile for *U*^{s}, *θ*^{s}, and *D*^{s} with their LES values. Second, the measured LES boundary layer depth *h* (which is limited by stable stratification rather than *h*_{e}) is used for its normalizations of *z* and Reynolds stress. Third, for the remaining Ekman–Stokes parameters—*U*^{w}, *D*^{w}, and *κ*_{o}—the simple profile assumptions for the wind-sea Stokes drift velocity and diagnosed eddy viscosity do not correspond to the more complex profiles in the LES solution. Therefore, the Ekman–Stokes parameter values are chosen—0.125 m s^{−1}, 1.25 m, and 0.025 m^{2} s^{−1}, respectively—to give the best visual comparisons below in Figs. 3–5. These parameter values are not ill matched to the more complex *κ*(*z*) profiles in Figs. 2 and 6, respectively. Furthermore, the normalized Ekman–Stokes profile shapes are not highly sensitive to these parameter choices. At best, the Ekman–Stokes model is a simple explanation for some of the

## 4. Mean current and Reynolds stress

From a larger-scale perspective, the most important outcomes for the boundary layer are the quasi-steady mean horizontal current and slowly eroding density stratification after an initial spinup period of about a day. After filtering out the inertial current that arises from the impulsive start to the boundary layer flow, the mean current profile **u**^{L} and an anti-Stokes Eulerian flow with **u**_{st}, as in the Ekman–Stokes solution (7). Both features are familiar attributes of equilibrium Langmuir turbulence (e.g., McWilliams et al. 1997); for example, if **u**_{st} = 0 (as in case NB defined in Table 1; not shown), the Eulerian *h* is a bit larger through enhanced entrainment (section 6). Although such large swell has not been examined before, these behaviors are not surprising for Langmuir turbulence solutions.

*U*

^{s}increases for wind-aligned swell (i.e.,

*θ*

^{s}= 0; Fig. 3, left), the surface maximum in

**u**

^{L}is larger, and the Ekman spiral is fatter; that is, the flow in the bulk of the boundary layer is stronger. Compared to case NB without Stokes drift (defined in Table 1; not shown), these wave-influenced cases have a stronger surface

**u**

^{L}and a fatter Ekman. The same qualitative trend with La occurs in the Ekman–Stokes solution for analogous parameter values (section 3), which is plotted in the inset. When the swell direction

*θ*

^{s}varies away from wind alignment to the east (Fig. 3, right),

**u**

^{L}at the surface weakens and rotates toward the swell, but only over a limited azimuthal range. In the interior of the boundary layer, the upwind velocity component is strongest—the Ekman spiral is fattest—in the

*θ*

^{s}range from southeast (SE) to northeast (NE) and is much weaker over the rest of the compass range. Again, these behaviors are similar to those in the Ekman–Stokes model (shown as insets in Figs. 3, 4, and 5). In Fig. 3, there is a rough left–right symmetry with a sign reversal in

*θ*

^{s}[e.g., north (N) versus south (S)]. The dependency of

**u**

^{L}(

*z*) on the swell depth scale

*D*

^{s}(i.e., in the case set Ds) is rather slight and is not shown; with a smaller

*D*

^{s},

**u**

^{L}(

*z*) is slightly more surface intensified. Of course, the anti-Stokes aspect in

*D*

^{s}, as in (7).

**A**(

*z*) > 0 in a thin layer near the surface. The velocity on the left side is the Lagrangian-mean flow

**u**

^{L}(

*z*), and one sees the constraint that an anti-Stokes Eulerian flow

**u**

_{st}and fixed momentum forcing by the wind, with the Reynolds stress profile unlikely to exceed the surface stress. The right side forcing in (14) is minus the divergence of the difference between the Reynolds stress and the upward integral of the breaker impulse:Because of the boundary condition on Reynolds stress and the normalization for

*A*, this has a surface value of

**u**

^{L}(

*z*) (Fig. 3).

The Reynolds stress hodograph for **R**(*z*) in Fig. 4 becomes much fatter when *U*^{s} is large and *θ*^{s} is in the same quadrant as the wind direction. With large enough *U*^{s} (which occurs in the case set Us2, not shown), the magnitude of the crosswind *θ*^{s} > 0), the **R**(*z*) hodograph tilts to the left and vice versa for *θ*^{s} < 0; but again, the range in directional tilt is much less than the range in *θ*^{s}. For the quadrants away from the wind [from south to north through west (W)], the hodograph becomes quite thin. These dependencies on *U*^{s} and *θ*^{s} are also present in the Ekman–Stokes model (insets), although the shapes of its hodographs are not the same, and the left–right symmetry in *θ*^{s} is not as visually apparent, in spite of the fact that it does have the particular *θ*^{s} symmetry (10). Again, the *D*^{s} dependencies of the Reynolds stress (not shown) are much smaller than those for *U*^{s} and *θ*^{s}.

*θ*) is the horizontal rotation matrix representing the rotation of the shear direction into the opposite of the Reynolds stress direction. Here,

*κ*

^{L}(

*z*) is the positive scalar magnitude, and

*θ*

^{L}(

*z*) is the rotation angle. A conventional eddy viscosity model has

*θ*

^{L}≡ 0, but this is not sufficient to characterize an Ekman layer with surface waves (McWilliams et al. 2012, section 3d). Alternatively, an eddy viscosity vector is defined in terms of components of the Reynolds stress parallel and perpendicular to the mean Lagrangian shear:In the conventional model,

In McWilliams et al. (2012) it is shown that a Lagrangian eddy viscosity representation has simpler vertical structure in *κ*(*z*) and a smaller range of *θ*(*z*) variation than a conventional Eulerian one (i.e., defined with **u**^{L} replaced by **u**_{st} is not small. The same is true in the present stratified cases with swell-waves.^{2} The hodograph of the eddy viscosity vector *κ*^{L} in a wind-sea compared to the absence of waves (case NB; defined in Table 1) and the ensuing decrease with added swell, *U*^{s} ≠ 0. This nonmonotonic dependence on La is somewhat surprising, and this is further discussed at the end of this section. The vertical profile of *κ*^{L}(*z*) has a middepth maximum and vanishes at the edges of the boundary layer in all cases (i.e., the looping curves in the hodograph that start and end at the origin). Without waves the loop is very thin, indicating that *θ*^{L} ≈ 0, but it is not thin for either a wind-sea alone or a wind-sea with added swell. The wind-sea case with *U*^{s} = 0 has *U*^{s}, the loops shrink and rotate clockwise in *θ*^{L}, so that *θ*^{L}(*z*) < 0 at almost all depths with the largest *U*^{s}. With variable *θ*^{s} (Fig. 6, right), the magnitude of *κ*^{L} increases with leftward swell rotation up to *θ*^{s} = *π*/2 (north), then begins to shrink again and *θ*^{L} rotates to the left. With rightward rotation in *θ*^{s}, the magnitude decreases to a minimum near *θ*^{s} = −*π*/2 (south), and *κ*^{L} assumes a distended loop shape for *θ*^{s} = −3*π*/2 [southwest (SW)]. This is a very substantial left–right *θ*^{s} asymmetry in the boundary layer response to swell.

The Ekman–Stokes model (section 3) is not apt for these eddy viscosity behaviors because (*κ*_{o}, 0) is just a point on the positive abscissa of these hodographs. This implies that the analytic model is not reliable for the detailed profile structure of the mean flow and Reynolds stress, even if it is useful for indicating their qualitative dependencies on the swell-wave parameters. In contrast, a “Lagrangian diffusion” Ekman–Stokes model—analogous to (5) and its boundary conditions but with the Eulerian diffusive flux *κ*_{o}∂_{z}

Finally, there is an appreciable dependency of *κ*^{L}(*z*) on *D*^{s} in the case set Ds (Fig. 6, bottom), with a larger eddy viscosity magnitude as *D*^{s} is decreased, while maintaining a roughly similar loop shape as in Fig. 6 (left).

The complexity of the *κ*^{L}(*z*) profiles and their variations with swell-wave parameters in Fig. 6 are challenging for achieving an accurate boundary layer parameterization for the shape of *κ*_{o} (section 3). The implications of this trade-off between accuracy and simplicity are further discussed in section 8.

*κ*

^{L}decreases with the addition of an increasing swell-wave amplitude to an equilibrium wind-wave field. The viscosity magnitude from (17) is the ratio of Reynolds stress and Lagrangian-mean shear magnitudeswhile the mean horizontal momentum balance (14) below the wave-breaking layer impliesThus,To explain the trend in

*κ*

^{L}with decreasing La, for example, for wind-aligned waves, one can examine the trends in the numerators and denominators of (19) or (21). Both

*κ*

^{L}has a convex shape with a maximum near the middle of the boundary layer (Fig. 6). With decreasing La, the

^{3}thus implying the reversing trends in

*κ*

^{L}. Consistent with (20), the Reynolds stress profile curvature (i.e., its second derivative with depth) first diminishes with decreasing La, then because of the tendency of the hodograph to fatten (Fig. 4) with added swell, it also increases. The Ekman–Stokes model with its fixed Eulerian

*κ*

_{o}, in contrast, has monotonically increasing numerators and denominators, and their middepth ratios as La decreases (not shown). The implication is that, in the LES solution, changes in the wave field change both the shape of the Langmuir circulations (section 7) with their significant contribution to the momentum flux profile (e.g., demonstrated in McWilliams et al. 2012) and the outcome of the anti-Stokes competition in the Lagrangian

**u**

^{L}(

*z*) profile in seemingly subtle ways that lead to the diagnosed

*κ*

^{L}(

*z*) changes with La. A more simplistic argument is that the numerator is plausibly constrained by the wind stress magnitude

_{z}

**u**

_{st}; however, this argument does not faithfully capture the decreasing |∂

_{z}

**u**

^{L}| in the larger La regime.

## 5. Inertial current

*z*over the boundary layer and weak below. For each LES case, a depth-averaged inertial amplitude

*U*

^{s}and

*θ*

^{s}are shown in Fig. 5. The inertial amplitude increases with

*U*

^{s}(decreasing La), and the three case sets with variable La (Us1, Us2, and Ua) all show a similar functional shape. The

*θ*

^{s}variations have an inertial current amplitude dependency with a roughly

*a*+

*b*sin[

*θ*

^{s}] dependency. These behaviors are qualitatively similar in the Ekman–Stokes (12), as is evident by comparison with the insets in Fig. 5. The

*D*

^{s}in the LES (Fig. 5, bottom), consistent with the prediction in (12); that is,for large

*D*

^{s}and

*U*

^{s}. This relation explains the increase in

*D*

^{s}and

*U*

^{s}increase, La decreases. It also explains the dependency on

*θ*

^{s}. When

*θ*

^{s}is in the NE quadrant, with the minus sign in (22) the magnitude of

*θ*

^{s}is in the SW quadrant, the swell effect subtracts from the contributions from stress and wind-sea, making

^{4}Furthermore, compared to case NB without Stokes drift (defined in Table 1; not shown), these wave-influenced cases have a stronger inertial current response.

## 6. Turbulent intensity, entrainment rate, and energy production

*e*is the isotropic measure of turbulent intensity, and the vertical velocity variance

*e*(

*z*) and a shallow maximum in

*e*and

*θ*

^{s}= ±

*π*/2. For northward swell,

*e*is as large as with wind alignment, whereas for southward swell

*e*is much reduced. Both

*e*(

*z*) and

*z*< −

*h*) that indicate increased internal gravity wave excitation by the boundary layer turbulence impinging on the top of the pycnocline (Polton et al. 2008).

This is another asymmetric response to the swell angle *θ*^{s}, and it is in the same sense as the stronger inertial current response for *θ*^{s} > 0 (section 5). However, there is no temporal correlation of turbulent intensity with the phase of the inertial current. This implies that the enhanced vertical shear at the base of the boundary layer when the inertial current aligns with the mean current shear is not an important source of turbulent energy production [in contrast to the situation of “inertial resonance” when the wind rotates in phase with the inertial current (Price 1981)].

To demonstrate the swell-wave parameter dependencies, Fig. 8 shows *e* and _{st} = *e*, ^{−4/3}. This comparison curve is also drawn, and it is roughly consistent with the La dependencies in the boundary layer for these cases with added swell that extend to smaller La than previously explored. The increases in internal-wave variance below the boundary layer are an even steeper function of inverse La, roughly ~La^{−2}.

The swell-wave direction dependencies show the greatest inertial current and turbulent intensity for *θ*^{s} in the NE quadrant, consistent with Figs. 5 and 7. The *θ*^{s} dependency in Fig. 8 is a roughly symmetric decrease away from wind alignment for *θ*^{s} = ±*π*/2 in Fig. 7 show that this symmetry is not precise.^{5} The boundary layer *e* measures in Figs. 7 and 8 are even farther away from *θ*^{s} symmetry than *e* for *θ*^{s} > 0. Below the boundary layer, both *e* and *θ*^{s} that their maxima occur for northward swell (similar to the inertial current in Fig. 5). The *D*^{s} dependencies in case set Ds (not shown) are rather slight, with some evident profile sensitivities in *e*(*z*) and *e* averaged over the boundary layer to increase as *D*^{s} decreases.

Langmuir turbulence is known to increase the entrainment rate in a stratification-limited boundary layer (McWilliams et al. 1997) as a consequence of Langmuir circulations penetrating into the stable stratification and scooping colder water into the boundary layer. This is the process that causes *h*(*t*) to increase in the present solutions.

*α*as the thermal expansion coefficient. Figure 9 shows that added swell further enhances the entrainment rate, which increases as La decreases with a dependency that can be fit approximately as a linear function of La

^{−2}. The entrainment rate is largest when the swell is aligned with the wind-sea, with a similar

*θ*

^{s}dependency as found for turbulent intensity

*e*(i.e., larger for

*θ*

^{s}> 0 than for

*θ*

^{s}< 0; Fig. 8). Consistent with the

*e*change with

*D*

^{s}remarked on above, there is a weak tendency for the minimum in

*D*

^{s}decreases. Both

*e*and

As expected, there is a strong correlation between entrainment rate and the increased value of *h* averaged over the analysis period. Because of the strong thermocline, the changes in *h* are modest, ranging from only a slight increase over the initial *h*(0) = 33 m in case NB up to a value of 36 m for the wind-aligned case in case set Us1 with the strongest wind and swell. Yet again, the *D*^{s} dependencies for *h*(*t*) are modest, although there are small increases with *D*^{s}.

*θ*

^{s}asymmetry in Figs. 8–10 be explained? The

**u**

_{st}profiles are symmetric with respect to the sign of

*θ*

^{s}, as part of the symmetry of the

*κ*

_{o}assumption in Ekman–Stokes). A plausible cause is the difference in Eulerian-mean velocity

*θ*

^{s}. They are composed of an Ekman spiral in the southern quadrants to the right of the wind plus an anti-Stokes component opposite to

**u**

_{st}(

*z*). For northward swell (

*θ*

^{s}> 0), the southward anti-Stokes component broadly reinforces the Ekman flow, whereas for southward swell, the anti-Stokes component is opposing. Therefore,

^{6}Support for this conjecture is provided in Table 2, which has the normalized and depth-integrated values for the rates of mean shear and Stokes turbulent kinetic energy production [(7) in McWilliams et al. (2012)]:The mean shear production rate

_{u}, in particular, is much larger for northward swell for a given value of |

*θ*

^{s}|, so that the total production rate is also larger. In almost all cases,

_{u}<

_{st}, with westward swell as the only exception (where total production is small). Thus, stronger

Turbulent kinetic energy production rates of (25), depth-integrated and normalized by *θ*^{s} magnitude for the case set

*θ*

^{L}] > 0, as it is at all depths in Fig. 6, and it is proportional to the product of the Reynolds stress and its profile curvature. As discussed at the end of section 4, both of these factors are increasing as La decreases in the swell regime (i.e., La ≤ 0.3). This is consistent with energy production and velocity variance strongly increasing (Fig. 8), even while their ratio

*κL*in (21) is decreasing (Fig. 6). Even for weak waves with La > 1, the decrease in |

*κ*increases), so that

^{L}*e*and entrainment as La decreases across its entire range.

Finally, it can be remarked that Langmuir turbulence differs from shear turbulence in that the negative feedback between the turbulent kinetic energy production and the mean shear is not as tight: _{u} is proportional to *e* and momentum mixing increase, whereas _{st} is proportional to ∂_{z}**u**_{st}, which is not as directly reduced by changes in the boundary layer turbulence, but would be so only if the feedback on the wave field, here neglected, were strong enough. This is a partial explanation for the strong inverse dependence of *e* on La in Langmuir turbulence.

## 7. Langmuir circulations

Langmuir circulations in a nonrotating, unstratified surface layer have an idealized shape of closely packed, longitudinal roll cells, orienting with the surface stress and wind-sea and extending vertically throughout the boundary layer (Leibovich 1983). In equilibrium Langmuir turbulence (McWilliams et al. 1997), they have more fragmented shapes but still retain appreciable horizontal anisotropy; they are rotated to the right of the wind-sea increasingly with depth. At sea they are most easily visualized by buoyant debris trapped in surface convergence lines, and in LES solutions the horizontal pattern of *w* (especially *w* < 0) is a comparably useful visualization.

^{7}is calculated from

*w*(

*x*,

*y*) at each height

*z*:where again the angle brackets denote an average over all (

*x*,

*y*, and

*t*). Langmuir circulations have horizontally elongated patterns, which are expressed in

*C*

_{w}as approximately elliptical contours decreasing away from the central extremum where

*C*

_{w}(0, 0,

*z*) = 1. Therefore, an ellipse is fit to an intermediate contour of

*C*

_{w}in the (

*ξ*,

*η*) plane (e.g.,

*C*

_{w}= 0.4 for Fig. 10).

^{8}The direction of its major axis is designated as

*θ*

^{LC}(

*z*) and the ratio of its major and minor axes as

*ε*

^{LC}(

*z*). Respectively, these quantities measure the mean orientation angle and degree of elongation (anisotropy) of the Langmuir circulations. The resulting profiles of these measures are shown in Fig. 10 for several cases in case sets Us1 and

*θ*

^{LC}(

*z*) is nearly wind aligned at the surface and progressively rotates clockwise with depth, while

*ε*

^{LC}(

*z*) is largest at the surface and decreases with depth down to near isotropy at the bottom of the boundary layer. Neither of these behaviors is very different due to the presence of swell. However, when the swell is misaligned with the wind-sea, the Langmuir circulations rotate in the direction of the swell by a small amount near the surface. Below the surface, the behavior of

*θ*

^{LC}(

*z*) is very different for

*θ*

^{s}= ±

*π*/2: it rotates clockwise even more rapidly for southward swell, and it is nearly independent of depth for northward swell. Additionally, for northward swell, the circulation anisotropy is very much enhanced in the middle of the layer, compared to the surface and compared to the other

*θ*

^{s}orientations.

Beyond these statistical measures, the Langmuir circulation patterns are displayed in Fig. 11 for visual assessment. Near the surface (*z* = −1 m) the patterns are only moderately different for different values of *U*^{s} and *θ*^{s}, although some variations in *w* amplitude and anisotropy direction are discernible. By the middle of the layer (*z* = −10 m), however, the pattern differences are quite large. Besides simple amplitude dependencies for *w*—larger *w* both for larger *U*^{s} and for *θ*^{s} in the range from SE to N—the nonmonotonic dependency of *θ*^{LC} on *θ*^{s} and the highly variable degrees of pattern complexity are notable. For *U*^{s} = 0, the pattern is a familiar one from previous studies of equilibrium Langmuir turbulence. For an opposing swell (*θ*^{s} = *π*), the Langmuir circulations are regular roll cells with nearly uniform spacing and a north–south orientation perpendicular to both the wind-sea and the swell-sea. But with the swell in the SE–N sector, there are multiple Langmuir circulation orientations and scales and complicated branching patterns from a dominant *w* extremum. It is clear that no simple conception of the structure of Langmuir circulations holds across the range of swell conditions.

In Van Roekel et al. (2012), considerable attention is given to the near-surface orientation angle of Langmuir circulations *θ*^{LC} as the misalignment angle between an equilibrium wind-sea and the wind stress is varied over a range from *θ*^{m} = 0 to +3*π*/4. The principal motivation is to interpret the dependency of *θ*^{LC} monotonically increases with *θ*^{m} (their Figs. 2 and 7). In that paper several statistical predictors of *θ*^{LC} are assessed, with varying degrees of skill, and mostly *θ*^{LC} is about half way between the wind and wave angles for this set of cases. In Fig. 10, the sign of *θ*^{LC} near the surface is the same as the sign of *θ*^{s}, but its magnitude is much less than *θ*^{s}/2. In the boundary layer interior, the *θ*^{LC}(*z*) profile shapes are quite different in the north and south cases. Furthermore, the Langmuir circulation pattern complexity in Fig. 11 is not always well characterized by only a single angle, even at a fixed depth. Therefore, *θ*^{LC} does not seem to be a robust property of Langmuir turbulence with different swell-wave orientations.

The Langmuir circulation pattern dependencies are not strong with *D*^{s} variations in case set Ds (again not shown), except for a moderate enhancement of the *w* magnitudes near the surface in the case with the smallest *λ* = 50 m).

A further structural oddity emerges in the limit of large swell and weak wind (i.e., in the case set Ua for the smallest ^{−1}; Fig. 12). The Langmuir circulations become much larger in scale, both in their along-axis extent and in the spacing between neighboring circulations. In this case, they are bundles of anisotropic overturning cells rather than individual isolated cells. Within the bundles are multiple spatial scales and orientation angles. A primary orientation—roughly northwest (NW)–SE at both depths shown in the figure—characterizes the long axis of the bundle, but different orientations arise on a finer scale within the bundles—more E–W near the surface and more N–S in the boundary layer interior. This is certainly a highly organized turbulent coherent structure, but it is a long way from an idealized roll cell. For the smallest La values in Fig. 9, some sort of regime transition is occurring, with a break from the ~La^{−4/3} scaling growth of *e* are less clear.

Comparison of the statistics and patterns among the cases in set Lh indicate that there is only a weak dependency on the domain width *L*, with the widest domain (*L* = 1200 m) large enough to encompass several of the large-scale Langmuir circulations, suggesting a convergence of the LES solutions with this essentially computational parameter. If solutions were sought with even smaller La values (e.g., even weaker winds), it is likely that the Langmuir circulation size would grow even larger, requiring larger *L*. Malecha et al. (2013) have proposed a multiscale asymptotic treatment in the limit of La → 0 as an aid to computational efficiency; it is an open question whether this is a viable approach to dealing with the complex bundle structures evident in Fig. 12.

## 8. Summary and discussion

The presence of remotely generated swell waves significantly alters the Langmuir turbulence of an equilibrium wind-sea. The Lagrangian-mean flow profile **u**^{L}(*z*) has a larger surface value and a fatter Ekman spiral throughout the boundary layer with increasing swell magnitude *U*^{s}. With misaligned wind and swell, **u**^{L} rotates toward the swell direction, but only within a limited angular range that is almost equal to ±30°. The Eulerian-mean flow *U*^{s} is large, but this cancellation is not so complete as to leave **u**^{L} unaltered with added swell-waves. The Reynolds stress profile **R**(*z*) also has a fatter Ekman spiral with strong swell, and with misalignment it rotates to a moderate degree toward the swell direction within the boundary layer interior. For a given swell magnitude, the orientation of opposing wind- and swell-seas (i.e., *θ*^{s} = ±*π*) yields the weakest **u**^{L} and thinnest Ekman spiral. All of these behaviors are similar to the analytic Ekman–Stokes model solutions with constant density and eddy viscosity (section 3).

Turbulent intensity, *e* and ^{−4/3}, consistent with an argument based on Stokes production as the dominant source of Langmuir turbulence, while for the inertial current, entrainment rate, and fluctuation variances below the boundary layer, the dependencies are even steeper, perhaps ~La^{−2} (as predicted for

The Ekman–Stokes model predicts a reflection symmetry in the swell direction (i.e., independence of the sign of *θ*^{s}) for the mean boundary layer, but this does not hold in the LES solutions. While **u**^{L}(*z*) is approximately symmetric in its weakening and rotation for positive and negative *θ*^{s} (Fig. 3, right), most other boundary layer properties are asymmetric. This is strongly so for inertial current amplitude, turbulent intensity, and entrainment rate, all of which are much stronger with leftward (*θ*^{s} > 0; counterclockwise, anticyclonic, and northward) swell misalignment than with rightward misalignment. The Langmuir circulation patterns are similarly asymmetric with *θ*^{s}, with stronger, more vertically aligned, and more elongated eddies when *θ*^{s} > 0.^{9} As La decreases with larger swell, the Langmuir circulation patterns become more complex and multiscale; at the smallest value of La ≈ 0.1 included here, the outermost longitudinal and transverse correlation scales become very large compared to the boundary layer depth (Fig. 12).

Even with a large added swell, however, the general characteristics of Langmuir turbulence are similar to what occurs in the equilibrium wind-sea regime, but the quantitative differences are not small. In the swell-wave parameter space of (*U*^{s}, *θ*^{s}, *D*^{s}), the most important parameter is La in (3). With aligned swell, as *U*^{s} increases and La decreases, the Lagrangian-mean flow, turbulent intensity both within the boundary layer and below, and entrainment rate all increase. As *θ*^{s} varies away from wind alignment, La in (3) increases due to swell- and wind-sea Stokes drift misalignment. This increase in La accounts partly for the general tendencies for decreases in the preceding quantities; however, the *θ*^{s} dependency is asymmetric in its sign in the LES (unlike in the Ekman–Stokes model or in the La definition), and sometimes the asymmetry is strong enough to overcome the implied change due to La (e.g., the peak value for *e* averaged over the boundary layer occurs at *θ*^{s} = *π*/2 in Fig. 8). The *D*^{s} dependencies are mostly rather weak, although inertial amplitude *κ*^{L}(*z*) are exceptions. This is consistent with the absence of any *D*^{s} influence in the definition of La in (3), which differs from the alternative definition in Harcourt and D’Asaro (2008) in this aspect.

In Van Roekel et al. (2012, Fig. 12) functional fits are presented for **u**_{st} magnitude and on its extension beyond the boundary layer when *D*^{s} > *h*] and different angular projections for the misaligned swell, wind, and Langmuir circulations. While all of the fits are somewhat skillful, the skill levels among the alternatives are not sharply distinguished. The swell influences summarized in the preceding paragraph do not accurately collapse into a universal function of any of these alternative Langmuir numbers (not shown). In particular, the orientation angle *θ*^{LC} is a complex function of *θ*^{s} and depth (Figs. 10 and 11); hence, it is not useful as an a priori projection angle for a composite La.

Oceanic circulation models require a boundary layer parameterization because their computational grids are necessarily too coarse to resolve Langmuir turbulence. With data inputs about the surface fluxes and gravity wave field, the desired outcome is accurate profiles of **u**^{L}(*z*)], *e*, *κ*(*z*) are only means to that end. In particular, the swell influences demonstrated here do need to be parameterized.

For **u**^{L}(*z*) and the inertial current amplitude, even the simple Ekman–Stokes model with constant eddy viscosity is fairly skillful (Figs. 3, 4, and 5), even though in practice the parameterization must also account for the influence of stratification in limiting the boundary layer depth and the generally convex shape of the eddy viscosity profile [both of which are parts of the KPP scheme; Large et al. (1994)]. On the other hand, the complexity of the swell influences on the eddy viscosity profile diagnosed from the LES solutions [i.e., *κ*^{L}(*z*) in Fig. 6] indicates that neither the constant *κ*_{o} in the Ekman–Stokes model nor the [*K*(*z*), 0] in the KPP scheme would yield a highly accurate **u**^{L}(*z*). In particular, the magnitude of *κ*^{L} increases in equilibrium wind-wave Langmuir turbulence compared to shear turbulence without waves (McWilliams and Sullivan 2000; McWilliams et al. 2012), but as La further decreases with swell waves, the eddy viscosity magnitude decreases (section 4); that is, |** κ**| must have a maximum at an intermediate value of La near its wind-sea equilibrium value of 0.3. These results raise a subtle design issue about the trade-off between parameterization simplicity (which often correlates with robustness across regimes) and

**| for very small La happens even while**

*κ**e*and

*κ*~

*w*

_{s}

*h*, with

*w*

_{s}as a turbulent velocity scale. Our present view is that relatively simple rules for the dependency of

*κ*(

*z*) on the wave parameters (i.e., simpler than manifested in Fig. 6) may provide a useful accuracy for

*K*

_{s}(

*z*)∂

_{z}

*z*= −

*h*penetrates into the stable stratification. The determination of

*h*is represented in the KPP scheme for the free convection regime by a critical bulk Richardson number condition where the mean velocity scale is augmented by a turbulent velocity scale

*V*

_{t}.

^{10}The rationale is that entrainment by encroachment into the thermocline can happen even when the mean shear is weak, as in convection at low wind speed or as in Langmuir turbulence with strong waves and weak winds. To choose

*V*

_{t}, one must specify the desired entrainment rate that will result from the KPP scheme when

*V*

_{t}is large enough to dominate the mean shear in determining

*h*. For free convection, the entrainment rule is the peak value of the pycnocline buoyancy flux, which is a negative fraction,

*β*

_{T}≈ −0.15, of the destabilizing surface upward buoyancy flux

**B**> 0. For Langmuir turbulence with swell, the entrainment rule is expressed by the normalized functional dependencies plotted as the ordinates in Fig. 9, which are denoted as

*F*(

*U*

^{s},

*θ*

^{s}). With these rules a combined expression can be derived for

*V*

_{t}with both convection and Langmuir turbulence, following the arguments in Large et al. (1994, p. 372); namely,The notation is as in that paper:

*N*is the buoyancy frequency just below the boundary layer;

*w*

_{s}is the turbulent velocity scale used in scaling the eddy diffusivity for material concentrations,

*K*

_{s}(

*z*) ~

*w*

_{s}

*h*; Ri

_{cr}is the critical value for the bulk Richardson number;

*Bh*)

^{1/3}is the conventional convective velocity scale determined from the surface buoyancy flux and boundary layer depth; and

*C*

_{υ}is an empirical coefficient slightly larger than the one related to the shape of

*F*~ La

^{−2}, the scaling dependency of

*V*

_{t}from (28) in Langmuir turbulence isassuming that

*w*

_{s}~

^{−2/3}(i.e., the scaling curve in Fig. 8).

The parameterization discussion in the preceding two paragraphs sketches how the KPP scheme could be modified to include the important effects of both a wind-sea and swell in Langmuir turbulence by generalizing *V*_{t} in the condition determining *h* and by choosing the turbulent velocity scale *w*_{s} to have an appropriate dependency on the wave parameters La and *θ*^{s} (and perhaps *D*^{s}). However, a complete and implementable parameterization scheme requires extensive testing, and that task is beyond the scope of this paper.

In conclusion, the two relevant idealized regimes of equilibrium Langmuir turbulence in the ocean—with both wind and waves held steady in time and the turbulence evolving into a stationary state—are for a wind-sea alone, as previously analyzed in many papers, and for the addition of a remotely generated swell-sea, as analyzed here. Beyond these, the relevant regimes are inherently transient in both the winds and waves, and often the turbulence as well. Only a few transient situations have been addressed, for example, spinup from rest, the passage of a hurricane utilizing a transient, nonlocal wave model (Sullivan et al. 2012), and combined measurements of variable waves and currents (Smith 1992, 1998). Therefore, to make further progress it is of pressing importance to measure and model across a wide range of transient situations to discover how they differ from equilibrium Langmuir turbulence.

## Acknowledgments

This research was sponsored by the National Science Foundation (Grant DMS-785 0723757) and Office of Naval Research (Grant N00014-08-1-0597). Computations were performed on the supercomputers Bluefire at NCAR and Thresher, Trestles, and Gordon at SDSC. We appreciate Dr. Leonel Romero’s guidance on the theory and observations of swell-waves and his help in preparing Fig. 1.

## REFERENCES

Alves, J. H. G. M., 2006: Numerical modeling of ocean swell contributions to the global wind-wave climate.

,*Ocean Modell.***11**, 998–112.Alves, J. H. G. M., , M. L. Banner, , and I. R. Young, 2003: Revisiting the Pierson–Moskowitz asymptotic limits for fully developed wind waves.

,*J. Phys. Oceanogr.***33**, 1301–1323.Belcher, S. E., 2012: A global perspective on Langmuir turbulence in the ocean surface boundary layer.

*Geophys. Res. Lett.,***39,**L18605, doi:10.1029/2012GL052932.Craik, A. D. D., , and S. Leibovich, 1976: Rational model for Langmuir circulations.

,*J. Fluid Mech.***73**, 401–426.Depley, M. T., , F. Ardhuin, , F. Collard, , and B. Chapron, 2010: Space-time structure of long ocean swell fields.

*J. Geophys. Res.,***115,**C12037, doi:10.1029/2009JC005885.Grant, A. L. M., , and S. E. Belcher, 2009: Characteristics of Langmuir turbulence in the ocean mixed layer.

,*J. Phys. Oceanogr.***39**, 1871–1887.Hanley, K. E., , S. E. Belcher, , and P. P. Sullivan, 2010: A global climatology of wind-wave interaction.

,*J. Phys. Oceanogr.***40**, 1263–1282.Harcourt, R. R., 2013: A second-moment closure model of Langmuir turbulence.

,*J. Phys. Oceanogr.***43**, 673–697.Harcourt, R. R., , and E. A. D’Asaro, 2008: Simulation of Langmuir turbulence in pure wind seas.

,*J. Phys. Oceanogr.***38**, 1542–1562.Large, W. G., , J. C. McWilliams, , and S. Doney, 1994: Oceanic vertical mixing: A review and a model with a nonlocal boundary layer parameterization.

,*Rev. Geophys.***32**, 363–403.Leibovich, S., 1983: The form and dynamics of Langmuir circulations.

,*Annu. Rev. Fluid Mech.***15**, 391–427.Lemarie, F., , J. Kurian, , A. F. Shchepetkin, , M. J. Molemaker, , F. Colas, , and J. C. McWilliams, 2013: Are there inescapable issues prohibiting the use of terrain-following coordinates in climate models?

,*Ocean Modell.***42**, 57–79.Malecha, Z., , G. Chini, , and K. Julien, 2013: A multiscale algorithm for simulating spatially-extended Langmuir circulation dynamics.

, in press.*J. Comput. Phys.*Masson, D., 1993: On the nonlinear coupling between swell and wind waves.

,*J. Phys. Oceanogr.***23**, 1249–1258.McWilliams, J. C., , and P. P. Sullivan, 2000: Vertical mixing by Langmuir circulations.

,*Spill Sci. Technol. Bull.***6**, 225–237.McWilliams, J. C., , P. P. Sullivan, , and C. H. Moeng, 1997: Langmuir turbulence in the ocean.

,*J. Fluid Mech.***334**, 1–30.McWilliams, J. C., , E. Huckle, , J. Liang, , and P. Sullivan, 2012: The wavy Ekman layer: Langmuir circulations, breakers, and Reynolds stress.

,*J. Phys. Oceanogr.***42**, 1793–1816.Polton, J. A., , J. A. Smith, , J. A. MacKinnon, , and A. E. Tejada-Martinez, 2008: Rapid generation of high-frequency internal waves beneath a wind and wave forced oceanic surface mixed layer.

*Geophys. Res. Lett.,***35,**L13602, doi:10.1029/2008GL033856.Price, J. F., 1981: Upper ocean response to a hurricane.

,*J. Phys. Oceanogr.***11**, 153–175.Romero, L., , and W. K. Melville, 2010: Airborne observations of fetch-limited waves in the Gulf of Tehuantepec.

,*J. Phys. Oceanogr.***40**, 441–456.Smith, J. A., 1992: Observed growth of Langmuir circulation.

,*J. Geophys. Res.***97**(C4), 5651–5664.Smith, J. A., 1998: Evolution of Langmuir circulation during a storm.

,*J. Geophys. Res.***103**(C6), 12 668–12 694.Sullivan, P. P., , and J. C. McWilliams, 2010: Dynamics of winds and currents coupled to surface waves.

,*Annu. Rev. Fluid Mech.***42**, 19–42.Sullivan, P. P., , J. C. McWilliams, , and W. K. Melville, 2007: Surface gravity wave effects in the oceanic boundary layer: Large eddy simulation with vortex force and stochastic breakers.

,*J. Fluid Mech.***593**, 405–452.Sullivan, P. P., , J. B. Edson, , T. Hristov, , and J. C. McWilliams, 2008: Large-eddy simulations and observations of atmospheric marine boundary layers above nonequilibrium surface waves.

,*J. Atmos. Sci.***65**, 1225–1245.Sullivan, P. P., , L. Romero, , J. C. McWilliams, , and W. K. Melville, 2012: Transient evolution of Langmuir turbulence in ocean boundary layers driven by hurricane winds and waves.

,*J. Phys. Oceanogr.***42**, 1959–1980.Van Roekel, L. P., , B. Fox-Kemper, , P. P. Sullivan, , P. E. Hamlington, , and S. R. Haney, 2012: The form and orientation of Langmuir cells for misaligned wind and waves.

*J. Geophys. Res.,***117,**C05001, doi:10.1029/2011JC007516.Young, I. R., 1999:

Elsevier, 287 pp.*Wind Generated Ocean Waves.*Zakharov, V. E., 2005: Theoretical interpretation of fetch-limited wind-driven sea observations.

,*Nonlinear Processes Geophys.***12**, 1011–1020.

^{1}

A more accurate drag law should include the influence of swell waves on *C*_{D} for low wind speed (Sullivan et al. 2008).

^{2}

Similarly, Harcourt (2013) proposes a local second-moment turbulence closure model based on Lagrangian-mean shear rather than Eulerian.

^{3}

This result is based on LES solutions with an equilibrium wind-wave Stokes profiles whose amplitude is artificially reduced to increase La above its equilibrium value of 0.3. These solutions are not shown here explicitly, but both Harcourt and D’Asaro (2008) and Grant and Belcher (2009) show results for this larger La regime.

^{4}

In LES cases that go beyond the largest *D*^{s} value in case set Ds, the further increases in *D*^{s} lead to little change in the turbulent intensity and fluxes. However, the anti-Stokes component in

^{5}

Van Roekel et al. (2012, their Fig. 8) shows a monotonic decrease in *θ*^{s}/2 for their misaligned cases. See section 7 for further comparative remarks.

^{6}

To be specific we have in mind a calculation for given 1D profiles of **u**_{st}*z*, and

^{7}

An alternative statistical measure of Langmuir circulation orientation is the direction determined by a peak in the distribution function for tan^{−1}(*ω*^{y}/*ω*^{x}), where ** ω** is the vorticity vector (see, e.g., Van Roekel et al. 2012). This gives similar results to the correlation function method used here, but with somewhat greater estimation noise.

^{8}

The elliptical fit parameters are only weakly dependent on the *C*_{w} value as long as *ε*^{LC} is not too close to 1.

^{9}

The nonequilibrium misalignment cases examined in Van Roekel et al. (2012) all have nonnegative wave rotation angles relative to the wind θ equal to or greater than zero.

^{10}

In present implementations of the KPP scheme in circulation models, there are two alternative formulations for the critical Richardson number condition that determines the boundary layer depth *h*. One formulation is expressed as a ratio of the bulk differences across the boundary layer of buoyancy change and velocity change squared; in this representation