1. Introduction
Among various processes that occur in the oceanic surface boundary layer, Langmuir circulations (LCs) receive attention because they are believed to modulate the air–sea heat/material exchange through an enhancement of turbulent mixing (Langmuir 1938; Smith 1992; Belcher et al. 2012; D’Asaro 2014; Li et al. 2016). In the formation of LCs, the interaction between surface waves and mean flow is believed to play a central role. Craik and Leibovich (1976, hereinafter CL76) derived an expression for the wave–current interaction called the vortex force (VF) and showed that the VF induces roll structures similar to observed LCs. Today, a number of studies investigate the effects of LCs on mixed layer turbulence using large-eddy simulations (LES) in which the wave–current interaction is represented by the VF in the momentum equation (e.g., Skyllingstad and Denbo 1995; McWilliams et al. 1997). The VF representation is popular because it enables us to simulate wave effects under the rigid-lid upper boundary condition, with which the difficulty in simulating nonhydrostatic surface waves is avoided.
When wind-driven surface current is aligned with the wave propagation direction (i.e., down-wave surface current is present), the VF causes dynamic instability, and the instability forms rolls aligned with the wind-wave direction (Craik 1977; Leibovich 1977, 1983). This driving mechanism is called the CL2 mechanism and is widely accepted as a major driving mechanism of LCs.
The CL equation has been used in various contexts to describe the effects of surface waves on mean flow. As written above, LES based on the CL equation has become a common approach to study the mixing effects of LCs and to develop or verify mixed layer turbulence models that account for the impact of LCs (e.g., Skyllingstad and Denbo 1995; McWilliams et al. 1997, 2012, 2014; Noh et al. 2004; Li et al. 2005; Harcourt and D’Asaro 2008; Van Roekel et al. 2012; Tejada-Martinez et al. 2012; Harcourt 2013, 2015; Reichl et al. 2016), and it is reported that some of these parameterizations represent better the observed spatial distribution of mixed layer depth (e.g., Li et al. 2016). Uchiyama et al. (2009, 2010); Weir et al. (2011) used a generalized form of the CL equation derived by McWilliams et al. (2004, hereinafter MRL04) to simulate littoral currents and rip currents. Hamlington et al. (2014) and Suzuki et al. (2016) investigated the effect of surface waves on submesoscale phenomena using a variant of the CL equation.
However, there remains an important issue: neither observations, laboratory experiments, nor numerical simulations have yet provided direct verification of the CL equation, a formulation arrived at variously through complex mathematical approaches (e.g., CL76; Andrews and McIntyre 1978; Holm 1996), and the CL2 mechanism, upon which the prevailing understanding of LCs is based. The lack of direct laboratory or numerical validation of the CL equation has resulted in a series of discussions regarding its validity, and a consensus view on this issue has not been reached yet (Mellor 2003, 2016; Ardhuin et al. 2008a, 2017; Aiki and Greatbatch 2012, 2013, 2014). Since this argument is mostly over the theoretical aspect of the wave–current interaction, experimental approaches, such as observations, laboratory experiments, or numerical simulations, that do not invoke the approximations and assumptions behind the CL equation can provide good evidence to support or refute the CL theory and eventually deepen our understanding of the wave–current interaction, the driving mechanism of LCs, and, therefore, ocean surface mixing.
These experimental approaches have not been commonly taken because each approach has different issues. In field observations, the measurement of waves and wave-averaged flow is not sufficiently accurate for comparing the actual wave effect on the averaged flow and the wave-related terms in the CL equation. As for laboratory experiments, the resulting flow in small tanks cannot be free from the effect of the side and bottom walls of wave tanks, which leads to complications for comparing results with LCs in the open ocean. Melville et al. (1998) and Veron and Melville (2001) conducted a series of careful laboratory experiments with a large wind-wave channel to examine mechanisms of LCs with negligibly small side/bottom effects, but the velocity scale of the resulting flow was comparable to the phase speed of waves. Such a situation usually only occurs for waves with O(0.1) m wavelengths, so the direct comparison with LCs in the typical open ocean condition is again difficult. Numerical simulations that explicitly simulate wave motions, namely, wave-resolving simulations (WRS), can be a possible approach, but numerical models that can explicitly simulate three-dimensional turbulence and deep-water waves have not been developed until recently. Takagaki et al. (2015) reported that they simulated LC-like structures in a direct numerical simulation of air–water multiphase flow. Tsai et al. (2017) investigated the LC-like flow obtained from a direct numerical simulation of free-propagating surface waves and concluded that the flow was driven by the CL2 mechanism. Tsai et al.’s (2017) conclusion is based on a comparison of the simulated results with the linear stability analysis results from the streamwise-homogeneous CL equation. However, the consistency between the simulated wave force and the CL formulation (i.e., VF and Bernoulli head), which is a controversial issue in today’s community (e.g., Mellor 2016; Ardhuin et al. 2017), still remains uninvestigated.
There is another issue: Given that the CL equation is valid, is the equation always an appropriate way to describe the wave–current interaction in the upper ocean? Although it is used in a variety of situations (e.g., mixed layer turbulence, coastal circulations, submesoscale phenomena), less care is given to the assumptions under which the equation is derived. In fact, its derivations include, explicitly or implicitly, some assumptions about spatiotemporal scales of waves and mean flow. For example, CL76 assumed that the characteristic spatial scale of mean flow is comparable to that of waves (wavelength) and that the characteristic velocity scale of mean flow is comparable to Stokes drift velocity. Evidently, this scaling is not satisfied by all of the various phenomena mentioned above.
The present study intends to address these questions. First, using a nonhydrostatic free-surface numerical model developed recently, a wave-resolving simulation of surface gravity waves and mean flow is conducted to clarify if the wave effect is consistent with the CL equation. Since the mathematical argument starting from Mellor (2003) is still going on, we consider it beneficial to support or refute (whichever the result is) the validity of the CL formulation from a different perspective: an experimental approach. Detailed dynamical analyses are conducted to study the driving mechanism of simulated flow, and it is explained from the Eulerian point of view (in contrast to the CL theory, which is based on the Lagrangian point of view). Consistency between the simulation results and the CL theory is found. Then, in order to clarify the scales of waves and mean flow for which the CL equation has been derived, a systematic description of spatiotemporal scaling assumptions in previous studies is provided. It is shown that there is disagreement among the previous studies regarding the applicability of the VF formulation when the amplitude of waves is small. Using the WRS results, it is demonstrated that the VF formulation is still valid even when the wave orbital velocity is much smaller than the characteristic velocity scale of mean flow as long as the latter is much smaller than the wave phase speed.
Throughout this paper, we focus on the situations where phase speed of the waves is much greater than the current velocity scale, and hence the wave motions are approximately linear and irrotational, as most applications of the CL equation to LCs in the open ocean assume. These assumptions are common because in many cases ocean surface waves are well described with the linear irrotational theory. Furthermore, in such cases, the Stokes drift profile can be prescribed using the well-known linear analytic solution. Note that interaction of currents with rotational waves, which is relevant to waves with smaller phase speed (wavelength) and plays important roles in local air–sea interaction processes (e.g., Craik 1982; Melville et al. 1998; Veron and Melville 2001; Phillips 2005), is not the scope of the present study.
First, simulation settings are described in section 2, and the results and analyses are shown in section 3. Then, in section 4, scaling assumptions in previous studies are systematically examined to clarify the regimes (in terms of relative spatial and temporal scales of waves and mean flow) in which the VF has been derived, and the validity of one of these assumptions is examined using the WRS results. Finally, the findings of the present study are summarized in section 5.
2. Problem setup
To directly investigate the driving mechanism of LCs, we designed a wave-resolving simulation of surface waves and currents in a configuration similar to a wave tank with a wave generator in one end. For this purpose, we used a nonhydrostatic free-surface version of the so-called kinaco model (Matsumura and Hasumi 2008) in which nonhydrostatic pressure and free-surface elevation are solved simultaneously as in Casulli and Zanolli (2002). This scheme and the efficient Poisson–Helmholtz equation solver (Matsumura and Hasumi 2008) make it possible to correctly simulate the interaction of deep-water waves and currents with reasonably limited computational resources (appendix A).
To contrast the role of waves, experiments with and without waves are conducted. The control run is the basic experiment, in which surface gravity waves, as well as wind-driven currents, are explicitly simulated. The no-wave run is the same as the control run except that surface waves are not forced. In both experiments, system rotation (Earth rotation) is not considered, and an incompressible fluid of uniform density is considered in order to focus on LC dynamics. For the same reason, uniform and isotropic eddy viscosity is assumed.
Consider a fluid in a rectangular water tank of streamwise, spanwise, and vertical dimensions of 16λ10, 4λ10, and ≥2λ10, respectively. Here, λ10 indicates the wavelength of the deep-water wave with a period of 10 s, which is approximately 156 m. Cartesian coordinates (x, y, z) are assigned, where the x, y, and z axes are aligned with the streamwise, spanwise, and upward directions, respectively (Fig. 1). The reference level (z = 0) is set so that the bottom of the tank is at z = −2λ10.
(a) Sketch of the computational domain and (b) spatial variation of damping coefficients χu and χη. In all the figures in this study, the labels of the x, y, and z axes are normalized by λ10, where λ10 ≈ 156 m is one wavelength of the deep-water wave with a period of 10 s. The shaded area at 0 ≤ x ≤ 2λ10 is the forcing domain, and the shaded area at 12λ10 ≤ x ≤ 16λ10 is the damping domain. At x = 0, χu = χη = 2 × 10−2 s−1. At x = 16λ10, χu = 2 × 10−4 s−1 and χη = 2 × 10−1 s−1.
Citation: Journal of Physical Oceanography 48, 8; 10.1175/JPO-D-17-0199.1
The computational domain is divided into three subdomains, namely, the forcing domain at 0 ≤ x ≤ 2λ10, the free propagation domain at 2λ10 ≤ x ≤ 12λ10, and the damping domain at 12λ10 ≤ x ≤ 16λ10. Waves are generated in the forcing domain using external forcing terms, propagate along the x axis through the free-propagation domain, and are finally damped in the damping domain using external forcing terms.
Numerical configuration
The numbers of grid points in the (x, y, z) direction are (1024, 256, 128), respectively. The grid spacing is uniform (λ10/64) in the x and y directions. The vertical grid spacing is uniform (λ10/64) except for the top layer, where it changes according to the surface elevation (λ10/64 + η).
The simulations are started from rest (u = 0) and a flat surface (η = 0) and integrated until the rms of 
3. Results
a. Characteristic temporal scales of simulated flows
To clarify what are “mean flow” and “waves” in the WRS result, we first look into the characteristic temporal scales of the flow. To visualize the temporal scales, the normalized energy density spectrum of w, computed from a time series of w at a fixed point, is shown in the lower-right corner of Fig. 2. It can be seen that there is a clear gap between the temporal scales of low-frequency flow (continuous spectrum in 
A three-dimensional sketch of the flow field in 4λ10 ≤ x ≤ 8λ10, 0 ≤ y ≤ 4λ10, and −0.6λ10 ≤ z ≤ η(x, y, t) at t = 36 h. The lines are the streamlines of velocity field averaged over one wave period at t = 36 h, and the transparent surface is the instantaneous surface elevation η at t = 36 h. The color of the streamlines indicates the wave-averaged vertical velocity 
Citation: Journal of Physical Oceanography 48, 8; 10.1175/JPO-D-17-0199.1
b. Overview of simulated flows
A three-dimensional sketch of the flow simulated in the control run is shown in Fig. 2. The colored lines are the streamlines of the wave-averaged flow 
Figure 3 shows the horizontal section of velocity (
Horizontal distribution of (a) 
Citation: Journal of Physical Oceanography 48, 8; 10.1175/JPO-D-17-0199.1
In Fig. 3, some features of the observed LCs (see, e.g., CL76) can be identified. There is an array of downwelling jets aligned with the wave-propagating direction. Downstream surface current velocity 
The horizontal section of the no-wave run result is also shown in Fig. 3. In the absence of waves, no roll structure is seen (flow is nearly uniform in the y direction). It should also be noted that the roll structure is not found in the experiment with wind forcing with the same magnitude as and in the opposite direction to that of the control run (not shown). These results demonstrate that the presence of both waves and down-wave surface currents is crucial for the LC-like rolls in the control experiment.
Figure 4 shows the y–z sections of the flow of the control run at x = 6λ10. Including this figure, all the y–z slices shown below are smoothed in x direction over one wavelength to remove small phase-dependent noise, unless noted explicitly. The flow structure in Fig. 4 again shows the features of observed LCs: the downstream velocity 
Spanwise–vertical (y–z) section of the averaged velocity in the control run at x = 6λ10. The blue shades indicate downstream velocity 
Citation: Journal of Physical Oceanography 48, 8; 10.1175/JPO-D-17-0199.1
Hereinafter, we shall call the rolls observed in the control run “LCs”, since 1) the downwind waves and wind-driven currents are crucial for generating the rolls and 2) the rolls have multiple features that are commonly considered to be those of LCs.
c. Driving mechanism of simulated LCs
To investigate the driving mechanism of the LCs in the control run, the mean vorticity equation is examined.
Figure 5 shows 
Spanwise–vertical (y–z) section at x = 6λ10 of (a) streamwise vorticity 
Citation: Journal of Physical Oceanography 48, 8; 10.1175/JPO-D-17-0199.1
Spanwise–vertical (y–z) section at x = 6λ10 of (a) vertical vorticity 
Citation: Journal of Physical Oceanography 48, 8; 10.1175/JPO-D-17-0199.1
In the downwelling jet region, the mean flow nonlinear term 
This and the phase relation of deep-water waves together lead to the following interpretation of the driving terms, 
Conceptual sketch showing how wave motion induces mean torque when positive mean vertical vorticity 
Citation: Journal of Physical Oceanography 48, 8; 10.1175/JPO-D-17-0199.1
d. Comparison with the CL theory
Are the WRS results and their interpretation consistent with the prevailing understanding of LCs based on the CL theory?
In fact, the flow simulated using the CL equation (with VF forcing) with the wave parameters in the control run instead of using the full Eqs. (2)–(4) produced qualitatively and quantitatively similar flow to the LCs in the control run (not shown).
As for vertical vorticity 
In his criticism of VF formulation, Mellor (2016) argued that the VF is much smaller than other wave stress terms and thus negligible, because VF terms are 
However, this does not mean that the VF terms are negligible. The 
4. Scaling assumptions of the CL equation
a. Scaling assumptions by preceding studies
In this section, we first revisit the scaling assumptions made by previous studies that derived the CL equation in order to clarify situations in which the CL equation is known to be valid. As noted before, we shall restrict our attention to the cases where waves are linear and irrotational. To describe the scales of the wave and mean flow quantities, we first introduce a set of notations as in CL76, MRL04, and Aiki and Greatbatch (2014). To begin with, suppose that velocity field 
Let the characteristic wavenumber, angular frequency, and wave slope (the product of wave amplitude and wavenumber) of the surface waves be denoted by k, σ, and ε, respectively. Assuming deep-water waves, the scales of the phase speed, amplitude, and orbital velocity follow σk−1, εk−1, and εσk−1, respectively. The temporal scale of waves is σ−1, and the horizontal and vertical spatial scales of wave quantities are both k−1 (because of the characteristics of deep-water waves).
Now the characteristic scales of the wave and mean flow quantities can all be described with four nondimensional numbers (ε, γ, δH, δV), the angular frequency σ, and the wavenumber of surface waves k. Among the four nondimensional numbers, ε characterizes the nonlinearity of waves, and the other three represent the ratio of the spatiotemporal scale between mean flow and waves. The notation introduced here is listed in Table 1.
Notations introduced in this section and scales of wave/mean flow quantities written using the notations.
1) Applicability of irrotational linear wave solution
2) CL76
3) L80
Wave amplitude is small [i.e., Eq. (17d)];
Wave motion is, in the lowest order in ε, irrotational [i.e., Eqs. (17a)–(17c)];
Reynolds number relevant to wave motion is not too small; and
Wave orbital velocity is much greater than the rotational current velocity.
4) MRL04
5) ARB08
6) Scaling diagram
The spatiotemporal scaling assumptions of the previous studies are illustrated in Fig. 8. Here, γ−1 and 
Spatiotemporal scales assumed in previous studies that derived variants of the CL equation. The horizontal axis is the ratio between the temporal scale of mean flow and the temporal scale of waves. The vertical axis is the ratio between the horizontal scale of mean flow and the spatial scale of waves. Diagrams are for (a) ε = 0.1, which corresponds to a large wave amplitude, and (b) ε = 0.01, which corresponds to a small wave amplitude. In the both diagrams, δV = 1 is chosen.
Citation: Journal of Physical Oceanography 48, 8; 10.1175/JPO-D-17-0199.1
b. Vortex force in small-amplitude cases
As shown in Fig. 8, the assumed spatiotemporal scales, defined in terms of wave slope ε, change with wave amplitude. Notably, the area that meets the requirements of L80 shrinks as the amplitude becomes small, because L80 requires much greater orbital velocity than mean flow velocity for irrotational wave field to hold. In contrast, the conditions in Eq. (17) suggest that the irrotationality of wave field is not restricted by small wave amplitude, as long as the Froude number (γ/δH,V) is small. This implies that the requirements by L80 [Eqs. (19a) and (19b)] are too strict especially when wave amplitude is small. In addition, ARB08 does not require mean flow velocity to be smaller than orbital velocity. Therefore, according to ARB08, the CL equation remains applicable even when wave amplitude is small.
Here arises a disagreement regarding the applicability of the CL equation when wave orbital velocity scale is smaller than or comparable to characteristic mean flow velocity scale. It is beneficial to investigate this disagreement and clarify whether the CL equation (especially the VF formulation) is valid in such cases because such conditions can be seen in waves interacting with frontal or coastal currents, whose speed can be as large as wave orbital velocity, while current speed is much smaller than wave phase speed.
We will take an experimental approach to investigate the above disagreement since our argument leading to Eq. (17) is based on simple scale comparison of terms and thus rather crude. To achieve this, a simulation (hereinafter referred to as the small-amplitude experiment) is carried out with an orbital velocity scale smaller than the characteristic velocity scale of mean flow. The experiment is initialized with the flow field at t = 36 h of the control run in the previous section, where LCs are active. The same forcing as the control run is imposed, except that aforce = 0.005 m, which is one-hundredth of the control run. The characteristic velocity scale of mean flow is 
The wave torque obtained from the above analysis 
Spanwise–vertical (y–z) section at x = 6λ10 of (a) 
Citation: Journal of Physical Oceanography 48, 8; 10.1175/JPO-D-17-0199.1
5. Conclusions
In this study, we first performed a wave-resolving simulation of surface waves, wind-induced current, and their interactions using a nonhydrostatic free-surface numerical model. When waves and wind-induced down-wave currents were present, roll structures of mean flow very similar to observed LCs appeared, and they did not appear if either surface waves or down-wave surface currents were absent. From these results, we concluded that the roll structures observed in the present simulation were LCs. Dynamical analysis of the mean streamwise vorticity equation showed that the streamwise vorticity of simulated LCs was forced by the torque induced by the Reynolds stress associated with wave motion, and it was shown how wave motion induced mean rotational forces as waves interacted with the mean vorticity field from the Eulerian point of view. For wave torque to enhance preexisting streamwise vorticity, the spatial patterns of the streamwise and vertical mean vorticity must be aligned in the same location in the spanwise direction. Analysis of the mean vertical vorticity equation clarified that not only spanwise to vertical tilting but also vertical stretching played a central role in maintaining such vorticity patterns. It was also confirmed that the rotational component of the wave-associated Reynolds stress is well represented by the vortex force, demonstrating the validity of the CL equation (VF formulation).
Then, we reviewed the spatiotemporal scaling assumptions made by the studies that had derived the CL equation. Among them, the derivations by Leibovich (1980) and Ardhuin et al. (2008b) that utilized GLM strategy covered a wide range of scales. However, the limitation indicated by Leibovich (1980) could be relaxed (at least for the simulated condition), as demonstrated in the present WRS result. With this relaxation, it was confirmed that VF formulation was possible in conditions where the mean flow was stronger than the wave orbital motion. Such situations may include examples of wave effects on strong frontal flow.
The original derivation of the VF by Craik and Leibovich (1976) showed that, if some scaling assumptions (section 4) were satisfied, the vorticity fluctuation ω′ arises through advection/stretching/tilting of the mean vorticity, and the wave torque 
The present study investigated LCs driven by a particular set of wave parameters (period and amplitude) and wind stress. Various sets of wave parameters and wind stress need to be investigated for an extensive understanding of LCs in the real ocean. Notably, further investigation is needed for situations where the mean flow is strong (Froude number is large). Wave parameters in the present study (period of 10 s and amplitude of 0.5 m) led to the phase speed of 15.6 m s−1 and the wave slope of 0.02, which resulted in rather fast swells passing over rather weak LCs [~0.02 m s−1, i.e., α ~ O(10−3)]. In reality, the Froude number can be much greater depending on the sea states, which may lead to the inapplicability of irrotational linear wave solutions. In such cases, the mean flow effects on waves must be explicitly taken into account, and the VF formulation may have to be modified, as in Craik (1982) and Phillips (2005). Meanwhile, active LCs tend to reduce shear in the mean field, which may affect the irrotationality of waves. In addition, the present study only simulated monochromatic waves. Presence of multiple waves might affect the dynamics of LCs, for example, through the CL1 mechanism (Leibovich 1983). As in the present study, wave-resolving simulations will be a useful approach to investigate wave–current interaction in such high–Froude number or multiple-wave situations. The interaction in these situations will be investigated in our future study.
Acknowledgments
The authors appreciate the insightful comments and suggestions from the editor and the two reviewers. This research is partially supported by Grant-in-Aid for Scientific Research on Innovative Areas (MEXT KAKENHI Grants JP15H05824 and JP15H05825), Grant-in-Aid for JSPS Research Fellow (JSPS KAKENHI Grant JP17J07923), and Initiative on Promotion of Supercomputing for Young or Women Researchers, Information Technology Center, The University of Tokyo.
APPENDIX A
Deep-Water Waves and Stokes Drift Reproduced in Model
Here, we present the capability of our numerical model to properly reproduce deep-water waves and Stokes drift.
First, the dispersion relation of the simulated waves is evaluated. The two-dimensional domain in the x–z plane is assumed, with a fixed depth of H and periodic horizontal boundaries at x = 0 and λ. As an initial condition, the surface elevation of η = a cos(2π/λ)x is added to the state of rest (a is wave amplitude and much smaller than λ and H). Periods of resulting standing waves T are evaluated at the antinodes, and experiments are conducted with various λ values.
Figure A1 shows the measured wave angular frequency (circles), σ = 2π/T, and the dispersion curve of linear surface waves (solid line), 
(a) Dispersion relation of simulated small-amplitude standing waves with different depth–wavelength ratios and (b) vertical profile of the simulated Stokes drift velocity. In (a), angular frequency σ is normalized with the angular frequency of shallow water waves, 
Citation: Journal of Physical Oceanography 48, 8; 10.1175/JPO-D-17-0199.1
Figure A1 shows the vertical profile of the simulated Stokes drift velocity (horizontally averaged on the x = 6λ10 plane) and the Stokes drift velocity calculated analytically from linear theory [Eq. (12)]. The Stokes drift velocity experimentally evaluated and analytically calculated agree well with each other.
A detailed implementation of our nonhydrostatic free-surface model and further analysis of simulated deep-water waves will be discussed elsewhere.
APPENDIX B
Fourier Analysis of Fluctuation Vorticity
To investigate how the wave-associated vorticity fluctuation arises, a Fourier analysis of the control run result is performed. We let 
Spanwise–vertical (y–z) section at x = 6λ10 of 10-s-period Fourier coefficient of (a),(b) spanwise and (c),(d) vertical vorticity fluctuation. Shown in (a) and (c) are the real part of the coefficients and in (b) and (d) the imaginary part. Black lines indicate the vorticity fluctuation obtained from the WRS result, and blue lines indicate the wave-induced vorticity fluctuation defined in Eq. (B1). Contour intervals are 5 × 10−6 s−1. Black and blue lines nearly coincide.
Citation: Journal of Physical Oceanography 48, 8; 10.1175/JPO-D-17-0199.1
The distribution of 
REFERENCES
Aiki, H., and R. J. Greatbatch, 2012: Thickness-weighted mean theory for the effect of surface gravity waves on mean flows in the upper ocean. J. Phys. Oceanogr., 42, 725–747, https://doi.org/10.1175/JPO-D-11-095.1.
Aiki, H., and R. J. Greatbatch, 2013: The vertical structure of the surface wave radiation stress for circulation over a sloping bottom as given by thickness-weighted-mean theory. J. Phys. Oceanogr., 43, 149–164, https://doi.org/10.1175/JPO-D-12-059.1.
Aiki, H., and R. J. Greatbatch, 2014: A new expression for the form stress term in the vertically Lagrangian mean framework for the effect of surface waves on the upper-ocean circulation. J. Phys. Oceanogr., 44, 3–23, https://doi.org/10.1175/JPO-D-12-0228.1.
Andrews, D. G., and M. E. McIntyre, 1978: An exact theory of nonlinear waves on a Lagrangian-mean flow. J. Fluid Mech., 89, 609–646, https://doi.org/10.1017/S0022112078002773.
Ardhuin, F., A. D. Jenkins, and K. Belibassakis, 2008a: Comments on “The three-dimensional current and surface wave equations.” J. Phys. Oceanogr., 38, 1340–1350, https://doi.org/10.1175/2007JPO3670.1.
Ardhuin, F., N. Rascle, and K. A. Belibassakis, 2008b: Explicit wave-averaged primitive equations using a generalized Lagrangian mean. Ocean Modell., 20, 35–60, https://doi.org/10.1016/j.ocemod.2007.07.001.
Ardhuin, F., N. Suzuki, J. C. McWilliams, and H. Aiki, 2017: Comments on “A combined derivation of the integrated and vertically resolved, coupled wave–current equations.” J. Phys. Oceanogr., 47, 2377–2385, https://doi.org/10.1175/JPO-D-17-0065.1.
Belcher, S. E., and Coauthors, 2012: A global perspective on Langmuir turbulence in the ocean surface boundary layer. Geophys. Res. Lett., 39, L18605, https://doi.org/10.1029/2012GL052932.
Casulli, V., and P. Zanolli, 2002: Semi-implicit numerical modeling of nonhydrostatic free-surface flows for environmental problems. Math. Comput. Modell., 36, 1131–1149, https://doi.org/10.1016/S0895-7177(02)00264-9.
Craik, A. D. D., 1977: The generation of Langmuir circulations by an instability mechanism. J. Fluid Mech., 81, 209–223, https://doi.org/10.1017/S0022112077001980.
Craik, A. D. D., 1982: Wave-induced longitudinal-vortex instability in shear flows. J. Fluid Mech., 125, 37–52, https://doi.org/10.1017/S0022112082003231.
Craik, A. D. D., and S. Leibovich, 1976: A rational model for Langmuir circulations. J. Fluid Mech., 73, 401–426, https://doi.org/10.1017/S0022112076001420.
D’Asaro, E. A., 2014: Turbulence in the upper-ocean mixed layer. Annu. Rev. Mar. Sci., 6, 101–115, https://doi.org/10.1146/annurev-marine-010213-135138.
Hamlington, P. E., L. P. Van Roekel, B. Fox-Kemper, K. Julien, and G. P. Chini, 2014: Langmuir–submesoscale interactions: Descriptive analysis of multiscale frontal spindown simulations. J. Phys. Oceanogr., 44, 2249–2272, https://doi.org/10.1175/JPO-D-13-0139.1.
Harcourt, R. R., 2013: A second-moment closure model of Langmuir turbulence. J. Phys. Oceanogr., 43, 673–697, https://doi.org/10.1175/JPO-D-12-0105.1.
Harcourt, R. R., 2015: An improved second-moment closure model of Langmuir turbulence. J. Phys. Oceanogr., 45, 84–103, https://doi.org/10.1175/JPO-D-14-0046.1.
Harcourt, R. R., and E. A. D’Asaro, 2008: Large-eddy simulation of Langmuir turbulence in pure wind seas. J. Phys. Oceanogr., 38, 1542–1562, https://doi.org/10.1175/2007JPO3842.1.
Holm, D. D., 1996: The ideal Craik-Leibovich equations. Physica D, 98, 415–441, https://doi.org/10.1016/0167-2789(96)00105-4.
Langmuir, I., 1938: Surface motion of water induced by wind. Science, 87, 119–123, https://doi.org/10.1126/science.87.2250.119.
Leibovich, S., 1977: Convective instability of stably stratified water in the ocean. J. Fluid Mech., 82, 561–581, https://doi.org/10.1017/S0022112077000846.
Leibovich, S., 1980: On wave-current interaction theories of Langmuir circulations. J. Fluid Mech., 99, 715–724, https://doi.org/10.1017/S0022112080000857.
Leibovich, S., 1983: The form and dynamics of Langmuir circulations. Annu. Rev. Fluid Mech., 15, 391–427, https://doi.org/10.1146/annurev.fl.15.010183.002135.
Li, M., C. Garrett, and E. Skyllingstad, 2005: A regime diagram for classifying turbulent large eddies in the upper ocean. Deep-Sea Res. I, 52, 259–278, https://doi.org/10.1016/j.dsr.2004.09.004.
Li, Q., A. Webb, B. Fox-Kemper, A. Craig, G. Danabasoglu, W. G. Large, and M. Vertenstein, 2016: Langmuir mixing effects on global climate: WAVEWATCH III in CESM. Ocean Modell., 103, 145–160, https://doi.org/10.1016/j.ocemod.2015.07.020.
Matsumura, Y., and H. Hasumi, 2008: A non-hydrostatic ocean model with a scalable multigrid Poisson solver. Ocean Modell., 24, 15–28, https://doi.org/10.1016/j.ocemod.2008.05.001.
McWilliams, J. C., P. P. Sullivan, and C.-H. Moeng, 1997: Langmuir turbulence in the ocean. J. Fluid Mech., 334, 1–30, https://doi.org/10.1017/S0022112096004375.
McWilliams, J. C., J. M. Restrepo, and E. M. Lane, 2004: An asymptotic theory for the interaction of waves and currents in coastal waters. J. Fluid Mech., 511, 135–178, https://doi.org/10.1017/S0022112004009358.
McWilliams, J. C., E. Huckle, J.-H. Liang, and P. P. Sullivan, 2012: The wavy Ekman layer: Langmuir circulations, breaking waves, and Reynolds stress. J. Phys. Oceanogr., 42, 1793–1816, https://doi.org/10.1175/JPO-D-12-07.1.
McWilliams, J. C., E. Huckle, J.-H. Liang, and P. P. Sullivan, 2014: Langmuir turbulence in swell. J. Phys. Oceanogr., 44, 870–890, https://doi.org/10.1175/JPO-D-13-0122.1.
Mellor, G. L., 2003: The three-dimensional current and surface wave equations. J. Phys. Oceanogr., 33, 1978–1989, https://doi.org/10.1175/1520-0485(2003)033<1978:TTCASW>2.0.CO;2.
Mellor, G. L., 2008: The depth-dependent current and wave interaction equations: A revision. J. Phys. Oceanogr., 38, 2587–2596, https://doi.org/10.1175/2008JPO3971.1.
Mellor, G. L., 2016: On theories dealing with the interaction of surface waves and ocean circulation. J. Geophys. Res. Oceans, 121, 4474–4486, https://doi.org/10.1002/2016JC011768.
Melville, W. K., R. Shear, and F. Veron, 1998: Laboratory measurements of the generation and evolution of Langmuir circulations. J. Fluid Mech., 364, 31–58, https://doi.org/10.1017/S0022112098001098.
Noh, Y., H. S. Min, and S. Raasch, 2004: Large eddy simulation of the ocean mixed layer: The effects of wave breaking and Langmuir circulation. J. Phys. Oceanogr., 34, 720–735, https://doi.org/10.1175/1520-0485(2004)034<0720:LESOTO>2.0.CO;2.
Phillips, W. R., 2005: On the spacing of Langmuir circulation in strong shear. J. Fluid Mech., 525, 215–236, https://doi.org/10.1017/S0022112004002654.
Reichl, B. G., D. Wang, T. Hara, I. Ginis, and T. Kukulka, 2016: Langmuir turbulence parameterization in tropical cyclone conditions. J. Phys. Oceanogr., 46, 863–886, https://doi.org/10.1175/JPO-D-15-0106.1.
Skyllingstad, E. D., and D. W. Denbo, 1995: An ocean large-eddy simulation of Langmuir circulations and convection in the surface mixed layer. J. Geophys. Res., 100, 8501–8522, https://doi.org/10.1029/94JC03202.
Smith, J. A., 1992: Observed growth of Langmuir circulation. J. Geophys. Res., 97, 5651–5664, https://doi.org/10.1029/91JC03118.
Suzuki, N., B. Fox-Kemper, P. E. Hamlington, and L. P. Van Roekel, 2016: Surface waves affect frontogenesis. J. Geophys. Res. Oceans, 121, 3597–3624, https://doi.org/10.1002/2015JC011563.
Takagaki, N., R. Kurose, Y. Tsujimoto, S. Komori, and K. Takahashi, 2015: Effects of turbulent eddies and Langmuir circulations on scalar transfer in a sheared wind-driven liquid flow. Phys. Fluids, 27, 016603, https://doi.org/10.1063/1.4905845.
Tejada-Martinez, A., C. Grosch, N. Sinha, C. Akan, and G. Martinat, 2012: Disruption of the bottom log layer in large-eddy simulations of full-depth Langmuir circulation. J. Fluid Mech., 699, 79–93, https://doi.org/10.1017/jfm.2012.84.
Tsai, W.-T., G.-H. Lu, J.-R. Chen, A. Dai, and W. R. Phillips, 2017: On the formation of coherent vortices beneath nonbreaking free-propagating surface waves. J. Phys. Oceanogr., 47, 533–543, https://doi.org/10.1175/JPO-D-16-0242.1.
Uchiyama, Y., J. C. McWilliams, and J. M. Restrepo, 2009: Wave-current interaction in nearshore shear instability analyzed with a vortex force formalism. J. Geophys. Res., 114, C06021, https://doi.org/10.1029/2008JC005135.
Uchiyama, Y., J. C. McWilliams, and A. F. Shchepetkin, 2010: Wave–current interaction in an oceanic circulation model with a vortex-force formalism: Application to the surf zone. Ocean Modell., 34, 16–35, https://doi.org/10.1016/j.ocemod.2010.04.002.
Van Roekel, L., B. Fox-Kemper, P. Sullivan, P. Hamlington, and S. Haney, 2012: The form and orientation of Langmuir cells for misaligned winds and waves. J. Geophys. Res., 117, C05001, https://doi.org/10.1029/2011JC007516.
Veron, F., and W. K. Melville, 2001: Experiments on the stability and transition of wind-driven water surfaces. J. Fluid Mech., 446, 25–65.
Weir, B., Y. Uchiyama, E. Lane, J. Restrepo, and J. McWilliams, 2011: A vortex force analysis of the interaction of rip currents and surface gravity waves. J. Geophys. Res., 116, C05001, https://doi.org/10.1029/2010JC006232.
Other terms induce 
Mellor (2016) assumed 
The denominators are L2 because the terms in the vorticity equation is investigated here.
This decomposition is normally considered as an Eulerian temporal filtering.
