## 1. Introduction

Winds flowing over the ocean excite a spectrum of surface gravity waves ranging from ripples to kilometer-long swells. Beneath persistent winds, the statistics of the shortest gravity waves saturate in a stationary balance between wind input, nonlinear interactions between wave components, and dissipation and breaking (Phillips 1985). The saturated fraction of the wave spectrum is called the “equilibrium range.” Longer waves, or “swell,” are out of equilibrium by definition and therefore grow along the fetch of the wind.

Much of the momentum input into the ocean is transferred via form stress acting on the faces of equilibrium range waves (Grare et al. 2013; Melville 1996). The shortness of equilibrium range waves and the accompanying effects of wave breaking motivate the parameterization of air–sea momentum transfer as an effective stress imposed at the air–sea interface. Yet some fraction of the total momentum transferred between the atmosphere and ocean—perhaps as small as 5% in conditions typical to the laboratory and field experiments (Melville 1996), or as large as 25% beneath hurricane-strength winds (Fan et al. 2009)—is not transferred to the ocean at or just beneath the surface, but is instead *distributed in depth* by pressure gradients associated with the growth of swell. In this paper, we address the effects of the “swell-mediated” fraction of the total momentum flux on turbulent ocean surface boundary layers.

*ρ*

_{0}is an ocean reference density,

*P*is the ocean-side kinematic pressure, and

*g*is gravitational acceleration. Over an ocean at rest, the pressure field in (1) excites an infinite, monochromatic surface wave field with wavenumber

*k*and frequency

*a*(

*t*). If

*p*

_{a}vanishes after some time

*t*=

*T*, the outcome is an infinite, steadily propagating wave field with amplitude

*a*(

*T*). This outcome was first investigated in a seminal paper by Stokes (1847).

*x*-momentum budget for a nonrotating and inviscid ocean reveals the air–sea momentum transfer generated by (1),

*h*is a depth at which momentum fluxes vanish, and the angle brackets ⟨⋅⟩ denote a simultaneous average over horizontal directions and a time average over the rapid oscillations of swell. The time derivative in (3) is the rate of change of depth-integrated momentum over time scales much longer than the oscillation of the surface wave field.

*p*

_{a}∂

_{x}

*s*⟩ on the right of (3) describes atmospheric pressure forces impacting the tilted ocean surface and is called “form stress.” With the pressure field in (1) and surface displacement in (2), we find that momentum is transferred to the ocean by form stress at the rate

*u*⟩—is zero. In other words, (4) acting on a nonrotating ocean forces a current with ⟨

*u*⟩ = 0, and

*τ*+ ∂

_{t}

*U*

^{S}, where

*τ*is a complexified effective surface stress that models the net effect of momentum transfer through swell-incoherent viscous stresses, form stress acting on equilibrium range waves, and the concomitant effects of wave breaking. The superscript “

*L*” in (6) stands for “Lagrangian-mean,” whose agency is discussed in section 2. The governing equation for the total momentum

*U*

^{L}, which may be derived either from the rotating Navier–Stokes equations beneath a free surface, or from the Craik–Leibovich wave-averaged Boussinesq equations (9)–(11) that form the basis for the simulations in this paper (Kukulka et al. 2010), is

*f*is the Coriolis parameter.

Ursell and Deacon (1950), Hasselmann (1970), Pollard (1970), and Eq. (7) show that, in the presence of rotation, the current that arises beneath a growing monochromatic wave field is not steady as in Stokes’s (1847) problem, but instead rotates inertially. A surprising result of our numerical simulations is that, due to three-dimensionality, stratification, and preexisting turbulence, swell-transmitted flows both rotate inertially and transfer energy to turbulence via shear production, thus mixing and deepening the boundary layer. This is in striking contrast to the laminar, unidirectional flow that arises in the irrotational case analyzed by Stokes (1847) and exemplified by (5). The evolution of a turbulent boundary layer in our most basic case beneath a growing surface wave field with *τ* = 0 is depicted in Fig. 1. Setting *τ* = 0 distinguishes our work from Kukulka et al. (2010), Sullivan et al. (2012), and Large et al. (2019) that investigate time-dependent mixing processes due to a combination of time-dependent *τ* and ∂_{t}*U*^{S}, and which do not separately investigate the effects of the “additional” momentum input by nonzero ∂_{t}*U*^{S}.

Section 2 introduces the wave-averaged Boussinesq equations and discusses some of their basic properties. In section 3, we continue beyond the scenario depicted in Fig. 1 to investigate the qualitative differences between the mixing and deepening of turbulent boundary layers forced either by an effective surface stress *τ* or swell growth via ∂_{t}*U*^{S}. We conclude that depth-distributed forcing by ∂_{t}*U*^{S} produces *less mixing* than forcing by *τ*, because ∂_{t}*U*^{S} drives currents with weaker shear that relinquish less energy to turbulence. In shallow boundary layers, moreover, some of the swell-transmitted atmospheric forcing acts directly on laminar near-inertial motions below the base of the boundary layer. We emphasize that these observations pertain to the *interior* effects of boundary layer forcing mechanisms associated with the shear production of turbulence, rather than the surface-concentrated effects of wave breaking, which are neglected in our large-eddy simulations.

Langmuir turbulence is not the main focus of this work. Nevertheless, the wave-catalyzed organization of turbulent motions into the coherent structures of Langmuir turbulence (Sullivan and McWilliams 2010; D’Asaro et al. 2014) features prominently in our numerical solutions. In section 3e, we observe that the coherent Langmuir turbulence structures tend to align with the Lagrangian-mean shear, consistent with Sullivan et al.’s (2012) results beneath realistic hurricane winds and waves. After the surface stress dies down, however, the orientation of the coherent structures decouples from the weakening Lagrangian-mean shear and locks onto

Section 4 investigates the importance of swell growth rate in determining initial conditions for large-eddy simulations, and subsequent deepening of turbulent boundary layers under steady surface stress and steady surface waves. Section 5 concludes by discussing how our results may corroborate, a posteriori, some assumptions that underpin the parameterization of atmosphere–ocean momentum transfer in general circulation models. Appendix A describes the large-eddy simulation software “Oceananigans.jl,” appendix B provides the vector calculus identities required to manipulate the Eulerian-mean Craik–Leibovich equations into their Lagrangian-mean form, and Table 1 lists the large-eddy simulations used in this work.

Simulations 1–8 are reported in section 3 and simulations A–E are reported in section 4. Simulations 1–5 and A–E use 128 m × 128 m × 64 m domains with 256 × 256 × 256 grid points and uniform 0.5 m × 0.5 m × 0.25 m grid spacing in *x*, *y*, *z*. Simulations 6–8 use 192 m × 192 m × 96 m domains with 256 × 256 × 384 grid points and uniform 0.75 m × 0.75 m × 0.25 m grid spacing in *x*, *y*, *z*. The effective wave-forced stress during the growth of 100-m wavelength, 1-m amplitude deep water waves on a time scale of *T*_{w} = 4 h is *τ*_{w} = 2.72 × 10^{−5} m^{2} s^{−2}. The wind stress prescribed by McWilliams et al. (1997) is *τ*_{MSM} = −3.72 × 10^{−5} m^{2} s^{−2}. Simulation 1 is initialized with *χ*_{ψ} is a Gaussian-distributed, unit standard deviation random field for each *ψ* ∈ (*u*^{L}, *υ*^{L}, *w*^{L}, *b*), Δ_{z} = 0.25 m is the vertical grid spacing, and *N*^{2} = 10^{−6} s^{−2} is the initial buoyancy gradient. Simulations 2–7 are initialized from simulation 1 at *t* = *π*/*f* with Coriolis parameter *f* = 10^{−4} s^{−1}. Simulations A–E use *k* = 0.105 m^{−1} The simulation data were generated with Oceananigans (https://github.com/CliMA/Oceananigans.jl). Instructions for reproducing the results are at https://github.com/glwagner/WaveTransmittedTurbulence.jl.

## 2. The surface-wave-averaged Boussinesq equations

**u**

^{S}is the Stokes drift. The Lagrangian-mean velocity

**u**

^{L}(

*x*,

*y*,

*z*,

*t*) advects mass, momentum, and vorticity, and its variance (1/2)|

**u**

^{L}|

^{2}is the wave-averaged kinetic energy.

*f*-plane tangent to and rotating with the ocean surface and cast in terms of

**u**

^{L}are (Craik and Leibovich 1976; Huang 1979; Leibovich 1980; Holm 1996; Suzuki and Fox-Kemper 2016; Seshasayanan and Gallet 2019)

*p*is Eulerian-mean kinematic pressure,

*b*is Eulerian-mean buoyancy,

*f*is the Coriolis parameter, and

**q**are the stress tensor and buoyancy flux due either to molecular diffusion or a subfilter turbulent diffusion model for large-eddy simulation. We show how (9)–(11) are derived from the Eulerian-mean form of the Craik–Leibovich equations, and how they are connected to the generalized Lagrangian-mean equations derived by Andrews and McIntyre (1978) in appendix B.

Equations (9)–(11) are an asymptotic approximation of the Navier–Stokes equations beneath a small amplitude and weakly modulated surface wave field. In particular, (11) neglects a term relevant for time-dependent swell and associated with divergence of the vertical component of the Stokes drift. If accounted for, this vertical divergence would lift the mean position of each fluid parcel by (1/2)*a*^{2}*ke*^{2kz} as the swell amplitude *a* increases [see Mcintyre (1981) and Eq. (3.7) in Longuet-Higgins (1986)]. In the cases considered in this paper, however, the vertical velocity associated with this mean vertical displacement is miniscule: for example, the growth of *a* = 2 m amplitude swell with wavenumber *k* = 2*π*/100 m^{−1} lifts the sea surface by just (1/2)*a*^{2}*k* = 0.12 m over a period of 4 h. Because the effect is so small, we neglect the vertical component of the Stokes drift and prescribe *w*^{L} = 0 at *z* = 0.

The momentum equation (9) is often written as a prognostic equation for the Eulerian-mean velocity **u**^{E}, both for analysis (Craik and Leibovich 1976; Suzuki and Fox-Kemper 2016) and large-eddy simulation (Skyllingstad and Denbo 1995; McWilliams et al. 1997; Noh et al. 2004; Polton and Belcher 2007; Harcourt and D’Asaro 2008; Yang et al. 2015). We use the Lagrangian-mean velocity **u**^{L} instead as our prognostic variable for both numerical simulations and analysis. Using **u**^{L} explicitly identifies the role of atmospheric momentum forcing transmitted via growing swell, and thus ∂_{t}**u**^{S}, in driving the turbulent evolution of the simulated surface boundary layers in section 3.

Prescribing **u**^{S}(**x**, *t*) in (9) determines the effects of swell on the evolution of the Lagrangian-mean momentum **u**^{L}. Vertical fluxes of horizontal momentum into **u**^{L} through saturated surface waves in the equilibrium range are prescribed through stress boundary conditions on *z* = 0. Buoyancy fluxes are prescribed through *q*_{z}|_{z=0}. The complexified *downward* surface stress in (7) is

### a. Wave-averaged effective background vorticity

**u**

^{S}is the surface wave pseudovorticity. As discussed by Bühler (2014) in their section 11.3.2, (12) means that the

*effective*background vorticity Ω

^{†}is advected by the Lagrangian-mean velocity

**u**

^{L}. In other words, the total vorticity is

*b*= 0 and

**Ω**stands for vorticity rather than

**u**

^{L}transports mass, momentum, and vorticity, the wave-averaged deviation from planetary vorticity is ∇ ×

**u**

^{E}, rather than ∇ ×

**u**

^{L}. This altered relationship between momentum and vorticity encapsulates the dynamical effect of surface waves on the evolution of

**u**

^{L}. We use the interpretation of

**Ω**

^{†}as an effective background vorticity to explain some of the phenomena observed in our large-eddy simulations in section 5.

### b. Wave-averaged kinetic energy

**u**

^{L}|

^{2}, in unstratified flow with

*b*= 0 and inviscid flow with

*V*. The total mean kinetic energy

**u**

^{L}and steady surface waves. Equation (15) shows that forced surface waves with nonzero ∂

_{t}

**u**

^{S}are a source of oceanic momentum and kinetic energy.

**u**

^{E}=

**u**

^{L}−

**u**

^{S}into Eq. (15) yields a formula for the volume-integrated Eulerian-mean kinetic energy (1/2)|

**u**

^{E}|

^{2}= (1/2)|

**u**

^{L}|

^{2}+ (1/2)|

**u**

^{S}|

^{2}−

**u**

^{L}⋅

**u**

^{S},

## 3. Large-eddy simulations beneath growing swell

*a*= 1 m and

*a*= 2 m. Both swells grow over a time scale of

*T*

_{w}= 4 h, have 100-m wavelength, 8-s period, and the horizontal Stokes drift profiles

*g*= 9.81 m s

^{−2},

*k*= 2

*π*/100 m

^{−1}, and

*a*= 1, 2 m in (17) are chosen (i) to illustrate the swell-mediated transmission of momentum below a shallow boundary layer and (ii) to illustrate how stronger and thus deeper turbulent mixing increases the effectiveness with which the swell-transmitted momentum mixes the boundary layer.

_{t}

*u*

^{S}and total stress

*u*

_{eq}(

*z*) is defined in (17), and

*τ*

_{w}scales the stress exerted on the ocean by the atmosphere via forced swell. The maximum vertically integrated momentum forcing of the boundary layer due to (17) is

^{−1}.

^{1}

### a. Numerical methods and software

**u**

^{L}is the

*resolved*Lagrangian-mean velocity filtered to remove scales smaller than the numerical grid scale. Spatial filtering motivates downgradient approximations for the subfilter stress tensor and diffusive flux,

_{i}

*b*, where the indices

*i*= (1, 2, 3) correspond to the Cartesian directions (

*x*,

*y*,

*z*). The eddy viscosity

*ν*

_{e}and eddy diffusivity

*κ*

_{e}in (20) are modeled with the anisotropic minimum dissipation (AMD) formalism introduced by Rozema et al. (2015) and Abkar et al. (2016), refined by Verstappen (2018), and validated and described in detail for ocean-relevant scenarios by Vreugdenhil and Taylor (2018). Additional details about (20) are given in appendix A.

We solve (9)–(11) numerically with Oceananigans, a software package developed by the authors in the Julia programming language that runs on graphics processing units (GPUs) (Bezanson et al. 2012; Besard et al. 2018). The simulations in this paper use second-order finite volume spatial discretization, second-order Adams–Bashforth time discretization, a pressure projection method to ensure ∇ ⋅ **u**^{L} = 0, and a fast method based on the fast Fourier transform to solve the pressure Poisson equation discretized with second-order differences on a regular grid (Schumann and Sweet 1988). Oceananigans code and documentation are hosted at https://github.com/CliMA/Oceananigans.jl.

*x*,

*y*,

*z*, grid spacings 0.5 m × 0.5 m × 0.25 m, and resolution 256

^{3}, and a second “large” domain of dimension 192 m × 192 m × 96 m with 0.75 m × 0.75 m × 0.25 m grid spacing and resolution 256 × 256 × 384. The domains are horizontally periodic in

*x*,

*y*and have rigid top and bottom boundaries, where we impose

*w*

^{L}= 0. To absorb internal waves radiated downward from the turbulent surface boundary layer, we implement bottom sponge layers of the form

*ϕ*∈ (

*u*,

*υ*,

*w*,

*b*), where

*H*is the depth of the domain,

*μ*= 1/60 s

^{−1}, ∂ = 4 m,

*b*

^{†}=

*N*

^{2}

*z*, and

*u*

^{†}=

*υ*

^{†}=

*w*

^{†}= 0. Each

*F*

_{ϕ}is added to its corresponding equation in (9) and (10).

The simulations reported in this section and the rest of this paper are listed in Table 1. Additional information, including instructions for reproducing the simulations and figures in this paper, are hosted at https://glwagner.github.io/WaveTransmittedTurbulence.

### b. Generation of a weakly turbulent initial condition

The spinup simulations are run for half an inertial period until *t*_{spin} = *π*/*f*, where *f* = 10^{−4} s^{−1} is the Coriolis parameter. The boundary layer depth at the end of the spinup is approximately 8 m. The horizontally averaged buoyancy and velocity at *t*_{spin} = *π*/*f* are shown in Fig. 2 and serve as initial conditions for subsequent simulations reported in this section, which use the same domain, resolution, and sponge layer configuration.

Horizontally averaged fields in three LES forced by growing swell with equilibrium amplitude *a* = 2 m (solid lines), a pulse of surface stress with no waves (dashed lines) a pulse of surface stress beneath steady waves with amplitude *a* = 1 m. Shown are (a) horizontally averaged buoyancy, (b) buoyancy gradient, (c) speed, and (d) vertical velocity variance normalized by

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0178.1

Horizontally averaged fields in three LES forced by growing swell with equilibrium amplitude *a* = 2 m (solid lines), a pulse of surface stress with no waves (dashed lines) a pulse of surface stress beneath steady waves with amplitude *a* = 1 m. Shown are (a) horizontally averaged buoyancy, (b) buoyancy gradient, (c) speed, and (d) vertical velocity variance normalized by

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0178.1

Horizontally averaged fields in three LES forced by growing swell with equilibrium amplitude *a* = 2 m (solid lines), a pulse of surface stress with no waves (dashed lines) a pulse of surface stress beneath steady waves with amplitude *a* = 1 m. Shown are (a) horizontally averaged buoyancy, (b) buoyancy gradient, (c) speed, and (d) vertical velocity variance normalized by

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0178.1

### c. Near-inertial waves, turbulence, and boundary layer deepening

*a*= 1 m and

*a*= 2 m are imposed on the weakly turbulent end state of the spinup simulation. The simulation time is reset to

*t*= 0. We impose free slip conditions on the horizontal velocity components and no normal flow on the vertical velocity component at top and bottom boundaries,

_{z}

*b*|

_{z=−H}=

*N*

^{2}and the bottom sponge layer in (21) restore the near-bottom buoyancy profile to

*N*

^{2}

*z*.

Figure 1 plots contours of vertical velocity after half an inertial period *w*^{L}|_{t=π/f} and depth profiles of the ⟨*u*^{L}⟩ and ⟨*υ*^{L}⟩ for the strong swell case with equilibrium amplitude *a* = 2 m. Near-inertial shear excited by the growing surface wave field drives turbulence that mixes and deepens the boundary layer. At the same time, mixing is enhanced by the organization of turbulent motions into the coherent structures of Langmuir turbulence, which manifest in the left panel of Fig. 1 as elongated rolls of alternating positive and negative vertical velocity oriented at roughly 10° angles from the *y* axis. The penetration of vertical motions through the stratified base of the boundary produces smooth downward-emanating bulbs of vertical velocity at around *z* = −25 m. Turbulent motions at the base of the boundary layer excite downward-propagating internal waves. The right panel shows the inertial rotation and downward turbulent penetration of the swell-transmitted current in time.

### d. Surface-concentrated versus depth-distributed stress

_{t}

**u**

^{S}in (18), we run three additional simulations for

*a*= 1 m and

*a*= 2 m each with the boundary condition

*τ*

_{w}is defined in (18). The boundary condition (25) prescribes a 4-h pulse of surface stress. We combine (25) with the Stokes drift profiles

*u*^{S}= 0 (surface stress, no swell), and${u}^{S}={u}_{\text{eq}}^{S}\left(z\right)$ for${u}_{\text{eq}}^{S}\left(z\right)$ in (17) (surface stress, steady swell).

For *a* = 1 m we also run a simulation forced by both the surface stress in (25) and growing swell with **u**^{S}(*z*, *t*) from (17), which we call “surface stress, growing swell.” The cases described in section 3c are called the “growing swell, no surface stress” cases.

*w*

^{L})

^{2}⟩ after one inertial period at

*t*= 2

*π*/

*f*in the four simulations that correspond to an equilibrium wave amplitude of

*a*= 1 m. The maximum value of the friction velocity,

*z*≈ −8 m at

*t*= 0 to

*z*≈ −12 m at

*t*= 2

*π*/

*f*. A substantial fraction of the total momentum is transmitted to laminar, nonturbulent near-inertial motions below

*z*≈ −12, whose shear does not contribute to turbulent mixing. The fingerprint of momentum transmitted by swell below the boundary layer is the blue exponential tail in Fig. 2c below

*z*≈ −12 m. The boundary layers driven by surface stress are deeper, therefore, because more of the mean kinetic energy transmitted to the boundary layer is converted to turbulence.

Figure 2d shows the average vertical velocity variance ⟨(*w*^{L})^{2}⟩. The case forced by surface stress and beneath steady swell has the strongest vertical velocities. We hypothesize this is due to the formation of vigorous Langmuir structures beneath strong, steady swell in the presence of surface stress. Vertical velocities are somewhat weaker in the case with both surface stress *and* growing swell, perhaps because the Stokes drift is not as strong during active momentum forcing by surface stress. Nevertheless, the surface stress, growing swell case—which experiences the strongest horizontal momentum forcing by both surface stress and swell-transmitted stress—boasts the deepest boundary layer and the strongest horizontal velocities.

*a*= 2 m. The maximum value of the friction velocity for

*a*= 2 m is

*z*≈ −32 m. This greater deepening enables greater turbulence production by shear transmitted by the growing swell, and thus mixing rates more comparable to the boundary layers driven by surface stress. The dramatic differences in vertical velocity variance between the blue lines in Figs. 2d and 3d are evidence that Langmuir turbulence is more active and effective in the growing swell simulation with

*a*= 2 m than in the growing swell simulation with

*a*= 1 m. Despite strong vertical velocities beneath growing swell (blue line, Fig. 3d), which suggest the presence of Langmuir turbulence (McWilliams et al. 1997; Sullivan and McWilliams 2010), the boundary layer is deeper in the surface stress case with no swell (blue and orange lines, Fig. 3b).

Horizontally averaged fields in three LES forced by growing swell with equilibrium amplitude *a* = 2 m (solid lines), a pulse of surface stress with no waves (dashed lines) a pulse of surface stress beneath steady waves with amplitude *a* = 2 m. Shown are (a) horizontally averaged buoyancy, (b) buoyancy gradient, (c) speed, and (d) vertical velocity variance normalized by

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0178.1

Horizontally averaged fields in three LES forced by growing swell with equilibrium amplitude *a* = 2 m (solid lines), a pulse of surface stress with no waves (dashed lines) a pulse of surface stress beneath steady waves with amplitude *a* = 2 m. Shown are (a) horizontally averaged buoyancy, (b) buoyancy gradient, (c) speed, and (d) vertical velocity variance normalized by

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0178.1

Horizontally averaged fields in three LES forced by growing swell with equilibrium amplitude *a* = 2 m (solid lines), a pulse of surface stress with no waves (dashed lines) a pulse of surface stress beneath steady waves with amplitude *a* = 2 m. Shown are (a) horizontally averaged buoyancy, (b) buoyancy gradient, (c) speed, and (d) vertical velocity variance normalized by

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0178.1

*x*-velocity ⟨

*u*

^{L}⟩(

*z*,

*t*) and the horizontally averaged turbulent kinetic energy

*t*= 8 h, while a turbulent layer steadily penetrates the stratified fluid below the surface stress over the 8-h period of significant forcing.

A depth–time plot of the horizontally averaged *x*-velocity, ⟨*u*^{L}⟩(*z*, *t*), and turbulent kinetic energy *E*(*z*, *t*) defined in (28).

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0178.1

A depth–time plot of the horizontally averaged *x*-velocity, ⟨*u*^{L}⟩(*z*, *t*), and turbulent kinetic energy *E*(*z*, *t*) defined in (28).

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0178.1

A depth–time plot of the horizontally averaged *x*-velocity, ⟨*u*^{L}⟩(*z*, *t*), and turbulent kinetic energy *E*(*z*, *t*) defined in (28).

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0178.1

### e. Alignment of freely decaying coherent Langmuir turbulence structures perpendicular to Stokes drift

Figure 5 plots a time series of normalized vertical velocity contours at *z* = −2 m and *z* = −8 m for the surface stress, steady waves case with *a* = 2 m. A black arrow in the center of each plot indicates the direction of the mean horizontal flow at each time and depth. The time series shows that the coherent Langmuir turbulence structures rotate and grow in size over time, especially after the forcing dies out after *t* ≈ (3/4) × 2*π*/*f*.

Contours of vertical velocity normalized by its maximum absolute value at *z* = −2 m and *z* = −8 m from the strong “surface stress with steady waves” simulation (simulation 8 in Table 1), showing the rotation of coherent Langmuir turbulence structures into *t* ≈ 12 h; the flow begins to decay freely after *t* ≈ (3/4) × 2*π*/*f*. At early times, the coherent structures are roughly aligned with the direction of mean shear, which is aligned with *t* ≈ *π*/*f* the rotation rate of the structures begins to slow until *t* ≈ 2*π*/*f*, at which point the structure axes are fixed to *t* = (5/4) × 2*π*/*f* and *t* = (3/2) × 2*π*/*f*, the mean shear rotates through *z* = −2 m: much of the coherency evident at *t* = (5/4) × 2*π*/*f* is destroyed at *t* = (3/2) × 2*π*/*f*. The strength and coherency of the Langmuir turbulence structures thus pulsates as the near-inertial current rotates, strengthening when the current is aligned or antialigned with

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0178.1

Contours of vertical velocity normalized by its maximum absolute value at *z* = −2 m and *z* = −8 m from the strong “surface stress with steady waves” simulation (simulation 8 in Table 1), showing the rotation of coherent Langmuir turbulence structures into *t* ≈ 12 h; the flow begins to decay freely after *t* ≈ (3/4) × 2*π*/*f*. At early times, the coherent structures are roughly aligned with the direction of mean shear, which is aligned with *t* ≈ *π*/*f* the rotation rate of the structures begins to slow until *t* ≈ 2*π*/*f*, at which point the structure axes are fixed to *t* = (5/4) × 2*π*/*f* and *t* = (3/2) × 2*π*/*f*, the mean shear rotates through *z* = −2 m: much of the coherency evident at *t* = (5/4) × 2*π*/*f* is destroyed at *t* = (3/2) × 2*π*/*f*. The strength and coherency of the Langmuir turbulence structures thus pulsates as the near-inertial current rotates, strengthening when the current is aligned or antialigned with

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0178.1

Contours of vertical velocity normalized by its maximum absolute value at *z* = −2 m and *z* = −8 m from the strong “surface stress with steady waves” simulation (simulation 8 in Table 1), showing the rotation of coherent Langmuir turbulence structures into *t* ≈ 12 h; the flow begins to decay freely after *t* ≈ (3/4) × 2*π*/*f*. At early times, the coherent structures are roughly aligned with the direction of mean shear, which is aligned with *t* ≈ *π*/*f* the rotation rate of the structures begins to slow until *t* ≈ 2*π*/*f*, at which point the structure axes are fixed to *t* = (5/4) × 2*π*/*f* and *t* = (3/2) × 2*π*/*f*, the mean shear rotates through *z* = −2 m: much of the coherency evident at *t* = (5/4) × 2*π*/*f* is destroyed at *t* = (3/2) × 2*π*/*f*. The strength and coherency of the Langmuir turbulence structures thus pulsates as the near-inertial current rotates, strengthening when the current is aligned or antialigned with

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0178.1

*t*= (1/2) × 2

*π*/

*f*, when momentum forcing is significant. After

*t*= (1/2) × 2

*π*/

*f*, and as the mean shear weakens, the cell appears to rotate into alignment with

*t*= 2

*π*/

*f*. Parameter

## 4. Turbulent mixing following rapid and gradual surface wave growth

In this section we reveal some differences between the turbulent evolution of boundary layers forced by the (i) rapid and (ii) gradual growth of a surface wave field. In short, when *f* ≠ 0, the rapid growth of a surface wave field leads to the momentum distribution **u**^{L} = **u**^{S} and thus **u**^{E} = 0, while the gradual growth of a surface wave field leads to **u**^{L} = 0. We explore the difference between these two initial conditions on turbulent mixing driven by surface stress using large-eddy simulations.

### a. Rapid and gradual surface wave growth over laminar boundary layers

*τ*is a prescribed function of time, the solution to (7) is

*τ*= 0, such that

*f*, on the other hand, then

*U*

^{L}≈ 0.

*H*(

*t*) is the Heaviside function and

*t*= 0, when the surface wave Stokes drift profile is steady, is

**u**

^{E}= 0 is stable and cannot transition to turbulence.

^{2}In stratified surface-wave-averaged flows, however, stability is guaranteed only when the Lagrangian-mean Richardson number is greater than 1/4 (Holm 1996),

### b. Large-eddy simulations of turbulent mixing driven by surface stress following rapid and gradual surface wave growth

*a*= 0.8 m and wavenumber

*k*= 0.105 m

^{−1}. The surface stress and surface buoyancy flux are

_{z}

*b*=

*N*

^{2}in (42) and

*a*= 0.8 m,

*k*= 0.105 m

^{−1}, and

*g*= 9.81 m s

^{−2}, Ri

^{L}≈ 0.09 at

*t*= 0 and

*z*= 0 such that the sufficient condition for instability in (37) is met. The simulations are performed in a rectangular domain with dimensions 128 m × 128 m × 64 m, grid spacing 0.5 m × 0.5 m × 0.25 m, resolution 256

^{3}, and with sponge layers of the form (21).

*a*= 0.8 m in (39) as “1×” cases, and run two additional cases with excited and resting initial conditions and the same boundary conditions, except with 4 times stronger Stokes drift fields

**u**

^{S}= 0. These five simulations are labeled A–E in Table 1. A visualization of vertical velocity in the

*x*–

*y*plane after two inertial periods such that

*t*= 2 × 2

*π*/

*f*, at a depth of

*z*= −4 m, and for the reference case, 1×excited case, and 1× resting case, are shown in Fig. 6. Both the 1× excited and 1× resting cases exhibit the coherent structures of well-developed Langmuir turbulence (Sullivan and McWilliams 2010).

Contours of vertical velocity in the *x*–*y* plane and at a depth *z* = −4 m in the reference, 1× excited, and 1× resting large-eddy simulations. Simulation parameters are detailed in Table 1.

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0178.1

Contours of vertical velocity in the *x*–*y* plane and at a depth *z* = −4 m in the reference, 1× excited, and 1× resting large-eddy simulations. Simulation parameters are detailed in Table 1.

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0178.1

Contours of vertical velocity in the *x*–*y* plane and at a depth *z* = −4 m in the reference, 1× excited, and 1× resting large-eddy simulations. Simulation parameters are detailed in Table 1.

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0178.1

The main result of this section is that the excited initial conditions in (33) provide a reservoir of near-inertial shear that quickly transitions to turbulence and rapidly deepens the boundary layer at early times. This rapid early deepening, compared to the more gradual evolution of boundary layers initialized with the resting state (38), is illustrated in Figs. 7 and 8, which depict the horizontally averaged buoyancy, vertical buoyancy gradient, and Lagrangian-mean velocity fields ⟨*u*^{L}⟩ and ⟨*υ*^{L}⟩ after a quarter of an inertial period [blue lines, *t* = (1/4) × 2*π*/*f*] and after two inertial periods (orange lines, *t* = 2 × 2*π*/*f*) in the 1× and 4× cases, respectively. The effect of 1× excited initial conditions is modest—the boundary layer is approximately 20% deeper than the 1× resting case at *t* = (1/4) × 2*π*/*f*—and the two boundary layer depths become similar by *t* = 2 × 2*π*/*f* as boundary layer deepening slows.

Horizontally averaged fields in 1× large-eddy simulations: (a) buoyancy, (b) buoyancy gradient, (c) speed, and (d) vertical velocity variance. Excited initial conditions (dashed lines) cause an initial mixing event to deepen the boundary layer compared to resting initial conditions (solid lines). This is most evident in the buoyancy gradient profiles at *t* = (1/4) × 2*π*/*f* [blue lines in (b)]. By *t* = 2 × 2*π*/*f*, memory of the initial condition is lost and the buoyancy, buoyancy gradient, and vertical velocity variance profiles are similar between the excited and resting case [orange dashed and solid lines in (a), (b), and (d)]. The horizontal velocities in the excited case still exhibit faster speeds [orange dashed and solid lines in (c)] due to the excitation of a strong inertial oscillation at *t* = 0. The differences between the “no waves” and “excited” cases are discussed in McWilliams et al. (1997). Figure A1 reproduces some of the plots from McWilliams et al. (1997). Simulation parameters are detailed in Table 1.

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0178.1

Horizontally averaged fields in 1× large-eddy simulations: (a) buoyancy, (b) buoyancy gradient, (c) speed, and (d) vertical velocity variance. Excited initial conditions (dashed lines) cause an initial mixing event to deepen the boundary layer compared to resting initial conditions (solid lines). This is most evident in the buoyancy gradient profiles at *t* = (1/4) × 2*π*/*f* [blue lines in (b)]. By *t* = 2 × 2*π*/*f*, memory of the initial condition is lost and the buoyancy, buoyancy gradient, and vertical velocity variance profiles are similar between the excited and resting case [orange dashed and solid lines in (a), (b), and (d)]. The horizontal velocities in the excited case still exhibit faster speeds [orange dashed and solid lines in (c)] due to the excitation of a strong inertial oscillation at *t* = 0. The differences between the “no waves” and “excited” cases are discussed in McWilliams et al. (1997). Figure A1 reproduces some of the plots from McWilliams et al. (1997). Simulation parameters are detailed in Table 1.

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0178.1

Horizontally averaged fields in 1× large-eddy simulations: (a) buoyancy, (b) buoyancy gradient, (c) speed, and (d) vertical velocity variance. Excited initial conditions (dashed lines) cause an initial mixing event to deepen the boundary layer compared to resting initial conditions (solid lines). This is most evident in the buoyancy gradient profiles at *t* = (1/4) × 2*π*/*f* [blue lines in (b)]. By *t* = 2 × 2*π*/*f*, memory of the initial condition is lost and the buoyancy, buoyancy gradient, and vertical velocity variance profiles are similar between the excited and resting case [orange dashed and solid lines in (a), (b), and (d)]. The horizontal velocities in the excited case still exhibit faster speeds [orange dashed and solid lines in (c)] due to the excitation of a strong inertial oscillation at *t* = 0. The differences between the “no waves” and “excited” cases are discussed in McWilliams et al. (1997). Figure A1 reproduces some of the plots from McWilliams et al. (1997). Simulation parameters are detailed in Table 1.

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0178.1

As in Fig. 7, but for 4× large-eddy simulations with 4× stronger wave fields and excited initial conditions. The difference between the boundary layer depth at *t* = (1/4) × 2*π*/*f* for the resting and excited cases is dramatic, as evidenced by the blue dashed and solid lines in (b). Unlike Fig. 7, memory of the initial mixing event persists at *t* = 2 × 2*π*/*f*. The resting case exhibits stronger vertical velocities than the excited case at *t* = 2 × 2*π*/*f*, especially at the base of the boundary layer. The reason for this is unknown.

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0178.1

As in Fig. 7, but for 4× large-eddy simulations with 4× stronger wave fields and excited initial conditions. The difference between the boundary layer depth at *t* = (1/4) × 2*π*/*f* for the resting and excited cases is dramatic, as evidenced by the blue dashed and solid lines in (b). Unlike Fig. 7, memory of the initial mixing event persists at *t* = 2 × 2*π*/*f*. The resting case exhibits stronger vertical velocities than the excited case at *t* = 2 × 2*π*/*f*, especially at the base of the boundary layer. The reason for this is unknown.

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0178.1

As in Fig. 7, but for 4× large-eddy simulations with 4× stronger wave fields and excited initial conditions. The difference between the boundary layer depth at *t* = (1/4) × 2*π*/*f* for the resting and excited cases is dramatic, as evidenced by the blue dashed and solid lines in (b). Unlike Fig. 7, memory of the initial mixing event persists at *t* = 2 × 2*π*/*f*. The resting case exhibits stronger vertical velocities than the excited case at *t* = 2 × 2*π*/*f*, especially at the base of the boundary layer. The reason for this is unknown.

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0178.1

Because the surface wave field is stronger in the 4× cases, the effect of initial enhanced shear in the excited case is more dramatic: after 1/4 of an inertial period, the boundary layer is almost twice as deep in the excited simulation as in the resting and reference simulations. The imprint of the initial excited state is still evident even at *t* = 2 × 2*π*/*f*, at which point the excited boundary layer is still deeper than the boundary layer spun up from a resting state. We conclude with the visualization of the evolution of vertical velocity in Fig. 9 from the 4× resting and excited experiments at the early time *t* = (1/4) × 2*π*/*f*. In the 4× excited simulation, turbulence penetrates deeper and regions of organized vertical velocity are stronger, reflecting both the greater energy available for turbulent mixing in the excited cases, and the organization of the more energetic turbulence into stronger coherent structures.

Comparison of the vertical velocity at *t* = (1/4) × 2*π*/*f* in two large-eddy simulations with resting and excited initial conditions.

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0178.1

Comparison of the vertical velocity at *t* = (1/4) × 2*π*/*f* in two large-eddy simulations with resting and excited initial conditions.

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0178.1

Comparison of the vertical velocity at *t* = (1/4) × 2*π*/*f* in two large-eddy simulations with resting and excited initial conditions.

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0178.1

## 5. Discussion

The simulation illustrated by Fig. 1 shows that the growth of surface waves can excite sheared near-inertial waves and drive turbulence and mixing that deepens the ocean surface boundary layer. The simulations in section 3 show that, more generally, the partitioning of ocean momentum forcing into a surface-concentrated component and a depth-distributed component mediated by growing swell impacts the ensuing mean currents and turbulent evolution and deepening of the ocean surface boundary layer. The simulations in section 4 show that the rate at which swell grows also affects the evolution of the boundary layer, a fact that is well appreciated in the context of surface-concentrated momentum forcing, but less so for swell-mediated momentum forcing.

The distinction between momentum transfer via equilibrium range waves and viscous stress near or at the surface, and depth-distributed momentum transfer via the growth of swell may be important when ∂_{t}*U*^{S} comprises a significant fraction of the total water-side stress *τ* + ∂_{t}*U*^{S}. For example, Fan et al. (2009) find that ∂_{t}*U*^{S} rises to 25% of the total stress in strongly forced hurricane conditions. However, depending on the quantity of interest, the differences between boundary layers driven by *τ* and boundary layers driven by ∂_{t}*U*^{S} are probably small in typical scenarios when ∂_{t}*U*^{S} is small compared to *τ*. This is especially true in boundary layers deeper than the longest swell components, where preexisting turbulence capably converts swell-deposited kinetic energy to turbulent kinetic energy.

The *unimportance* of the depth dependence of atmospheric momentum forcing under typical conditions has implications for parameterizations of atmospheric momentum forcing in general circulation models. Current general circulation models do not partition atmospheric momentum forcing into the two components in (7). Instead, general circulation models impose atmospheric momentum forcing through a surface stress. This approximation, which is sensible because 90%–95% of the stress exerted on the ocean by the atmosphere enters via equilibrium range waves with very short decay scales (Melville 1996), is *further justified* by our results, which suggest that the depth dependence of the remaining 5%–10% of the stress exerted on the ocean is relatively unimportant except in very shallow boundary layers. More realistic observations and modeling are warranted to further evolution of boundary layers forced by both surface stress and swell growth.

## Acknowledgments

This paper’s seed was planted by arguments between the lead author and Sean R. Haney about the intricacies and obscurities of Stokes drift while Sean was a postdoctoral researcher at the Scripps Institution of Oceanography. Besides being an amazing surfer and a generous friend, Sean had a unique ability to make arguments fun and productive. Sean passed away too soon due to illness on 1 January 2021 and is deeply missed. This paper is dedicated to him. Be careful, Sean.

In addition to Sean’s inspiration, this work benefited from the constructive criticism of two reviewers and amiable banter with Stephen Belcher, Keaton Burns, Navid Constantinou, Baylor Fox-Kemper, Ramsey Harcourt, Qing Li, Brodie Pearson, Nick Pizzo, Brandon Reichl, and William Young. This work was supported by the generosity of Eric and Wendy Schmidt by recommendation of the Schmidt Futures program, and by the National Science Foundation under Grant AGS-6939393.

## APPENDIX A

### Subfilter Fluxes in Wave-Averaged Large-Eddy Simulations

*ν*is the molecular viscosity,

*ν*

_{e}is an eddy viscosity that is a nonlinear function of resolved velocity field,

*δ*

_{ij}is the Kronecker delta, and

*κ*

_{e}is the eddy diffusivity of buoyancy.

#### a. The Lagrangian-mean strain tensor and kinetic energy dissipation

Typical models for large-eddy simulations of (9)–(11) (Skyllingstad and Denbo 1995; Polton and Belcher 2007; Yang et al. 2015; McWilliams et al. 1997; Noh et al. 2004; Harcourt and D’Asaro 2008) use a subfilter stress tensor

(a) Horizontally averaged Eulerian-mean *x*-velocity ⟨*u*^{E}⟩, (b) horizontally averaged *y*-velocity ⟨*υ*^{E}⟩ = ⟨*υ*^{L}⟩, (c) horizontally averaged vertical variance ⟨(*w*^{L})^{2}⟩, and (d) horizontally averaged Lagrangian-mean ⟨*u*^{L}⟩. Here, ⟨*υ*^{E}⟩ = ⟨*υ*^{L}⟩ because the Stokes drift is in *x*, such that *z*/*h* is height normalized by *h*, the depth of the maximum horizontally averaged buoyancy gradient such that ∂_{z}⟨*b*⟩(*h*) = max (∂_{z}⟨*b*⟩). Panels (a) and (b) reproduce Fig. 2 in McWilliams et al. (1997). Panel (c) reproduces Fig. 6 in McWilliams et al. (1997).

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0178.1

(a) Horizontally averaged Eulerian-mean *x*-velocity ⟨*u*^{E}⟩, (b) horizontally averaged *y*-velocity ⟨*υ*^{E}⟩ = ⟨*υ*^{L}⟩, (c) horizontally averaged vertical variance ⟨(*w*^{L})^{2}⟩, and (d) horizontally averaged Lagrangian-mean ⟨*u*^{L}⟩. Here, ⟨*υ*^{E}⟩ = ⟨*υ*^{L}⟩ because the Stokes drift is in *x*, such that *z*/*h* is height normalized by *h*, the depth of the maximum horizontally averaged buoyancy gradient such that ∂_{z}⟨*b*⟩(*h*) = max (∂_{z}⟨*b*⟩). Panels (a) and (b) reproduce Fig. 2 in McWilliams et al. (1997). Panel (c) reproduces Fig. 6 in McWilliams et al. (1997).

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0178.1

(a) Horizontally averaged Eulerian-mean *x*-velocity ⟨*u*^{E}⟩, (b) horizontally averaged *y*-velocity ⟨*υ*^{E}⟩ = ⟨*υ*^{L}⟩, (c) horizontally averaged vertical variance ⟨(*w*^{L})^{2}⟩, and (d) horizontally averaged Lagrangian-mean ⟨*u*^{L}⟩. Here, ⟨*υ*^{E}⟩ = ⟨*υ*^{L}⟩ because the Stokes drift is in *x*, such that *z*/*h* is height normalized by *h*, the depth of the maximum horizontally averaged buoyancy gradient such that ∂_{z}⟨*b*⟩(*h*) = max (∂_{z}⟨*b*⟩). Panels (a) and (b) reproduce Fig. 2 in McWilliams et al. (1997). Panel (c) reproduces Fig. 6 in McWilliams et al. (1997).

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0178.1

The first important assumption we make is that the terms proportional to molecular viscosity *ν* in (A1) are negligible. This is justified after the fact by the results of our large-eddy simulations, where *ν*_{e} is roughly 10^{2}–10^{3} times larger than *ν* within the bulk of the boundary layer, and where |∂_{z}**u**^{S}| is largely similar to |∂_{z}**u**^{L}|. Note that neglecting terms proportional to molecular viscosity also means we neglect the term *ν*∂_{z}*u*^{S} in (A1) associated with the molecular dissipation of the surface wave field. This term is crucial for describing streaming flows in viscous boundary layers and other viscous surface wave phenomena at low Reynolds number (see, for example, Longuet-Higgins 1953). In this paper, we assume that molecular dissipation of the surface wave field has a negligible effect on boundary layer evolution. This assumption is almost always justified at the high Reynolds numbers of typical ocean surface boundary layers.

The second important assumption is that the Lagrangian-mean kinetic energy, (1/2)|**u**^{L}|^{2}, undergoes a Kolmogorovian forward cascade through a spectral inertial range en route to the small, unresolved scales of molecular dissipation. The assumption of an inertial range for spectral fluxes of Lagrangian-mean kinetic energy follows from the conservation of Lagrangian-mean kinetic energy in (15) in the absence of stratification, dissipation, or forcing by nonzero ∂_{t}**u**^{S}. We note that an alternative hypothesis that the Eulerian-mean kinetic energy, (1/2)|**u**^{E}|^{2}, undergoes a forward cascade through an inertial range is more difficult to justify because (1/2)|**u**^{E}|^{2} is not conserved [for example, Eq. (36) shows that (1/2)|**u**^{E}|^{2} oscillates between 0 and 2|**u**^{L}|^{2} in an adiabatic inertial oscillation depending on whether **u**^{L} is aligned or antialigned with **u**^{S}]. The “turbulence-induced anti-Stokes” flow observed in Pearson’s (2018) large-eddy simulations is evidence that turbulent momentum fluxes tend down Lagrangian-mean gradients, and thus tend to dissipate Lagrangian-mean kinetic energy.

*b*= ∂

_{z}

**u**

^{S}= 0, we find that

*ν*

_{e}> 0 by its definition below in (A5). As a result,

#### b. Eddy viscosity and eddy diffusivity

*ν*

_{e}in (A1) and eddy diffusivity

*κ*

_{e}in (A3). The terms

*ν*

_{e}and

*κ*

_{e}are defined to be strictly nonnegative,

*x*,

*y*,

*z*grid spacings Δ

_{i}. The hats in (A6) denote the scaled quantities

*C*in (A6) is a model constant. For simulations A–E in Table 1, which are reported in section 4, we set

*C*= 1/12 following Verstappen (2018) and Vreugdenhil and Taylor (2018). For simulations 1–8 in Table 1 and reported in section 3, we implement

^{3}a model that increases

*C*from 1/12 to 2/3 at

*z*= 0 over a scale of 4Δ

_{z}= 1 m,

*C*

_{I}= 1/12,

*C*

_{0}= 2/3, and

*d*= 1 m. The model constant enhancement in (A9) is necessary for obtaining smooth buoyancy profiles near the surface during the spinup simulations 1 and 6 in Table 1. We find that without an enhancement of the kind in (A9), the eddy diffusivity

*κ*

_{e}is too small near

*z*= 0 during free convection, which prevents a smooth transition between the boundary-adjacent cells where the unresolved diffusive flux

**q**dominates, and the turbulent interior where advective fluxes

**u**

*b*in (10) control the evolution of the buoyancy distribution.

Our use of *ν*_{e} and *κ*_{e} from the Lagrangian-mean velocity gradient energy equation (Rozema et al. 2015). In this derivation, the term −(∇ × **u**^{S}) × **u**^{L} in (9) arises as a “transport term” in the velocity gradient energy equation, and thus, similar to Coriolis accelerations, does not affect the form of *ν*_{e} (Abkar et al. 2016).

#### c. Model validation

Figure A1 plots the horizontally averaged velocity and vertical variance from the 1× excited case, reproducing parts of Figs. 2 and 6 from McWilliams et al. (1997). Our results are similar despite the differences between our subfilter flux model and McWilliams et al.’s (1997).

## APPENDIX B

### Lagrangian-Mean Form of the Craik–Leibovich Equations

*p*is the Eulerian-mean pressure, and

**u**

^{S}× (∇ ×

**u**

^{E}) is the “vortex force.” Some algebraic gymnastics lead from (B1) to (9). Starting with the vector identity

**u**

^{S}× (∇ ×

**u**

^{L}) =

**u**

^{S}× (∇ ×

**u**

^{E}) +

**u**

^{S}× (∇ ×

**u**

^{S}) and

With the identity (B6), ∂_{t}**u**^{E} = ∂_{t}**u**^{L} − ∂_{t}**u**^{S}, and the assumption ∇ ⋅ **u**^{S} = 0 (valid for weakly modulated waves, as discussed in section 2), we can convert the Eulerian-mean form of the Craik–Leibovich equation (B1) and the continuity equation ∇ ⋅ **u**^{E} = 0 into their Lagrangian-mean counterparts (9) and (11).

Finally, we note that (B1) is derived by Leibovich (1980) from the generalized Lagrangian-mean momentum equation presented in Theorem I of Andrews and McIntyre (1978). Thus, Leibovich (1980) provides a link between Andrews and McIntyre (1978) and our (9)–(11). As discussed by Leibovich (1980), the pseudomomentum appearing in Theorem I of Andrews and McIntyre (1978) and the surface wave Stokes drift **u**^{S} are nearly equivalent the scenarios we consider with relatively slow background rotation and slowly modulated waves.

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^{1}

With drag coefficient *C*_{d} = 10^{−3}, air density *ρ*_{a} = 1.225 kg m^{−3}, seawater density *ρ*_{w} = 1035 kg m^{−3}, and ocean-side kinematic stress parameterized with *τ* = 1.65 × 10^{−5} m^{2} s^{−2} for *u*_{a} = 3.7 m s^{−1}.

^{2}

While the wave-averaged momentum is **u**^{L}, the wave-averaged vorticity is ∇ × **u**^{E} [see Bühler (2014) or chapter 2.4 in Wagner (2016)]. Thus, without a source of vorticity, a flow with **u**^{E} = 0 is irrotational and cannot transition to turbulence.