[30] Based on the LES results, we next contrast the differences between the LC and no LC cases by analyzing the energetics, vertical transport, and Lagrangian particles.

#### 4.1. Horizontally Averaged TKE Budgets

[31] Any dependent variable *q* is decomposed into horizontal mean 〈*q*〉 and its deviation from this mean *q*′, so that

For the stationary state considered here the turbulent kinetic energy (TKE) balance is [e.g., *McWilliams et al.*, 1997]

where the terms on the right-hand side are, from left to right, TKE production due to Stokes drift shear, TKE production/distruction due to Eulerian shear, vertical energy flux divergence due to turbulent TKE transport and pressure work, and the TKE dissipation rate*ϵ*. SGS represents all other unresolved subgrid scale terms (i.e., except for *ϵ* [see *Skyllingstad et al.*, 2000]) which are typically less than 5% of the largest resolved TKE budget term (thin dashed line in Figure 4). Note that our TKE budget analysis focuses on the interior flow (away from surface and bottom boundary layers) using a LES where the small-scale turbulence close to the boundaries is modeled rather than resolved. Thus our TKE budget cannot be directly compared to that of*Tejada-Martinez and Grosch* [2007] where surface and bottom viscous sublayers are resolved.

[32] Without LC the dominant TKE balance is between TKE shear production and dissipation (Figure 4, left). Only at middepth is the divergence of vertical TKE transport also important. This is because TKE is advected from energetic regions close to the boundaries (greater TKE production) and deposited at less energetic middepth regions (smaller TKE production).

[33] With LC the TKE budget differs significantly from the one without LC, as the TKE transport term plays an important role throughout the whole water column (Figure 4, right). At depths between approximately 4 and 12 m, the two leading terms in the TKE budget are dissipation and vertical TKE transport. This finding is consistent with the LES results from *Tejada-Martinez et al.* [2012], who show that the log layer of a hydrodynamically smooth flow with its dominant balance between dissipation and production is disrupted in the presence of LC.

[34] Although wave effects significantly modify the TKE budget, the total, i.e., vertically integrated, Stokes drift shear production contributes directly only a small fraction to the total energy input to the system (due to wind and waves). This can be seen by considering the total energy input into the system due to stresses acting on the Stokes drift (*I*_{s}) and Eulerian currents (*I*)

where *u*_{*} is the water friction velocity, *u*_{s0} is the surface Stokes drift and 〈*u*〉_{0} is 〈*u*〉 at *z* = 0 . Note that is the energy surface flux due to the surface stress acting on the mean surface current 〈*u*〉_{0}. The fraction of energy input of *I*_{s} relative to *I* is

This suggests that although the energy input due to Stokes drift shear production is relatively small, the mechanism of stretching and tilting vertical vorticity into the wind direction is an important physical process, significantly redistributing TKE. Furthermore, as one reviewer of this paper pointed out, much of the TKE production is dissipated in the energetic surface and bottom boundary layers and may thus not contribute to the interior mixed layer dynamics. Therefore, the surface Stokes drift and the magnitude of the Eulerian mean surface velocity (related to *u*_{*}) are generally not sufficient to scale Langmuir turbulence characteristics.

[35] One peculiar feature is the negative TKE production between depths of about 3 m and 10 m, which is due to negative gradients in the mean velocity profile (Figure 5). Note that both velocity profiles with and without LC resemble qualitatively the respective ones obtained from the simulations of *Tejada-Martinez and Grosch* [2007]. Physically, the negative TKE production can be understood by considering the nonlocal nature of the transport, which is a common, yet insufficiently explored, phenomenon in the presence of relatively large-scale coherent motions. Large-scale coherent motions provide an efficient mechanism to flux relatively high along-wind momentum close to the surface to greater depths roughly between*z* = −10 m and *z* = −14 m. Therefore, one anticipates that the along-wind mean current 〈*u*〉 has two peaks, one near the surface and the other somewhere at greater depth where the momentum is deposited (say between *z* = −10 m and *z* = −14 m, Figure 5). As a consequence 〈*u*〉 contains a local minimum (between the two peaks), which is for our simulations roughly located at *z* = −3 m with a negative current shear between the local 〈*u*〉 minimum and the deeper 〈*u*〉 peak. Thus, locally it appears as if turbulence produces mean kinetic energy, since the momentum flux is downward throughout the whole water column. Physically, however, it is important to keep in mind that large-scale, nonlocal LC motion transports momentum away from the boundaries. Also, examination of {*u*}′ [see, e.g., *Kukulka et al.*, 2011, Figure 13] reveals that downwind velocity fluctuations are intensified in downwelling regions at the surface and near the bottom, which is consistent with the results of *Tejada-Martinez and Grosch* [2007].

#### 4.2. TKE Budget of Dominant LC Scale

[36] The previous discussion prompts the question what maintains LC energetically, in particular given that the energy input due to the Stokes drift shear is only a small fraction of the total energy input to the TKE budget. To investigate this, we derive an energy balance for the coherent motion based on along-wind averages, which exploits the symmetry of the large-scale coherent features

Thus, any field *q* can be decomposed into

Note that by definition , 〈{*q*}〉 = 〈*q*〉, , and *q*″ is the deviation from the along-wind average (“small-scale incoherent motion” or*u*_{i}″ motion), and is the “large-scale coherent motion” (*ũ*_{i} motion). Note that the same decomposition was employed by *Tejada-Martinez and Grosch* [2007]. Resolved turbulent kinetic energy budgets can be derived for both motions.

[37] The kinetic energy budget of the large-scale coherent motion (*ũ*_{i}) is expressed as

The right-hand side terms, from left to right, are tilde TKE production due to Stokes drift shear, tilde TKE production due to Eulerian current shear, TKE conversion from tilde motion to double prime motion, vertical flux divergences of (1) self-advection of tilde kinetic energy, (2) turbulent double prime stresses acting on tilde motion, and (3) tilde energy flux due to pressure work. The last two terms are the tilde TKE dissipation rate and all other subgrid scale terms.

[38] The kinetic energy budget of small-scale incoherent motion*u*_{i}″ is

The right-hand side terms from left to right are double prime TKE production due to Stokes drift shear, double prime TKE production due to Eulerian current shear, TKE conversion from double prime motion to tilde motion (note opposite sign as in tilde TKE budget), vertical flux divergences of (1) self-advection of double prime TKE, (2) advection of double prime TKE by tilde motion, and (3) double prime TKE flux due to pressure work. The last two terms are the double prime TKE dissipation rate and all other subgrid scale terms, respectively. Note that smaller scales cannot advect larger scales and larger scales do not exert work by turbulent stresses on the smaller-scale motion. The sum of the tilde and double prime budgets results in the full TKE budget described above.

[39] Similar to the study of *Tejada-Martinez and Grosch* [2007], the large-scale coherent motion is relatively small without LC, while it is larger with LC (Figure 6). In the LC case the large-scale coherent tilde motion is the largest contributor to total TKE near the ocean bottom (Figure 6, bottom left) and fluxes 60% to 80% of the total momentum at middepth range between *z* = −3 m and *z* = −14 m (Figure 6, bottom middle). In both cases the sum of the residuals of the partial TKE budgets agree well with the total TKE dissipation rate, which has been directly calculated from the LES model results (Figure 6, right). The close agreement suggests that contributions of SGS terms in the partial budgets are not significant. Note that most of the TKE dissipation is due to incoherent small-scale motion. Because the tilde motion has a larger contribution in the LC simulations, we discuss TKE balances of coherent and incoherent motions only for the LC simulations.

[40] The LC tilde TKE budget is complex (Figure 7, left). The contribution of each budget term, whose significance and sign my change with depth, however, is important. One key result of the tilde TKE budget analysis is that the coherent LC structures extract most of their energy from the Eulerian mean flow rather than from the Stokes drift shear.

[41] The double prime TKE budget is somewhat less complex, as at middepth the dissipation (including SGS contributions) is to first order balanced by double prime TKE advection due to LC motion (Figure 7, right). Near the surface and bottom the dominant balance is between shear production and dissipation. Details of the budget reveal that there are depth ranges where TKE energy production due to Stokes drift shear is significant, which is characteristic of Langmuir turbulence.

#### 4.3. Vertical Scalar Transport

[42] In order to examine the vertical scalar transport we consider the evolution of a scalar passive field *S*, whose initial distribution is given by the vertical step function *S*(*x*,*y*,*z* ≥ −*H*/2) = 1 and *S*(*x*,*y*,*z* < −*H*/2) = 0 (Figures 8 and 9). Without LC the sharp gradient at *z* = −*H*/2 is gradually smoothed until the horizontally averaged vertical *S* profile is relatively well mixed after about 40 min (Figure 9, top left). The corresponding resolved turbulent fluxes −〈*Sw*〉 are largest at middepth, where variations in *S* and *w* are largest (Figure 9, bottom left). Scalar vertical transport is significantly larger with LC for the first 20 min (by a factor of about 2 for −4 m < *z* < −12 m), so that the horizontally averaged *S* profiles appear to be well mixed 20 min after the tracer release (Figure 9, right). However, rather than being well mixed *S* remains organized because high *S* patches are transported in downwelling regions, while low *S* patches are advected in upwelling regions (see Figure 8 (right) 15 min after development). As a result, the turbulent flux of *S* changes sign after about 30 min and transports deep high *S* back upward (Figure 8, right). Note that coherent along-wind vorticies are also present without LC. However, these are significantly weaker than the ones from the LC case. After 1 h of the tracer release*S* appears to be more homogenized (mixed) without LC, although the vertical transport is significantly enhanced with LC. Such nonlocal transport due to LC cannot be captured by simple eddy viscosity concepts for horizontally averaged vertical transport.

[43] Because of the nonlocal transport and horizontally inhomogeneous organization of scalar fields the local eddy viscosity concept based on horizontal averages is not physical. In both cases eddy viscosities are very large if the transport is nonzero and vertical gradients vanish. In the LC case the eddy viscosity approaches infinity when profiles appear “well mixed” (in a horizontally averaged sense) and changes sign if *S* is transported upward the horizontally averaged concentration gradient. Thus, it is challenging to parameterize the net vertical transport because the instantaneous transport depends on the relative positions of the scalar and velocity fields.

[44] The “backward” transport in the LC case is possible because fields remain organized although transport is significant. In particular, LC currents may organize and redistribute fields while potentially suppressing small-scale mixing. Therefore, it is important to distinguish organized advective transport (“stirring”) and irreversible small-scale molecular transport (“mixing”) [see, e.g.,*Müller and Garrett*, 2001]. Small-scale mixing leads to a reduction of the scalar variance. Without boundary fluxes, the volume averaged mean square balance is

where *D* is the molecular diffusion coefficient of *S* and the over bar indicates depth averages. In the previous equation *S* is not SGS averaged. In the LES, unresolved subgrid scale fluxes lead to irreversible mixing, so that changes in are not only due to molecular processes but also include SGS stirring processes. The mean square continuously decreases as *S* transitions from the organized initial state to a less organized mixed state (Figure 10). Once fields are well-mixed approaches a constant minimum value. Interestingly without LC, *S* homogenizes more quickly, indicating that LC may organize *S*.

[45] The role of turbulence in irreversible mixing is to enhance regions of high gradients so that elevated irreversible molecular exchange takes place. Our results indicate that small-scale turbulence may increase such regions relative to the organized LC motion with relatively weak small-scale motion (seeFigure 6). Note also that TKE and associated dissipation rates are larger for the case without wave forcing, as discussed in sections 4.1 and 4.2. Finally, flow structures may enhance or suppress mixing, e.g., the interior of large-scale persistent vorticies may contain kinematical barriers to transport that constrain the motion, alike those often found in ocean gyres [*Wiggins*, 2005]. Investigating in detail potential barriers and teasing apart contributions of stirring and mixing to changes of the scalar variance in a three-dimensional LES flow, where mixing is parameterized, is a complex research task that may require a dynamical systems approach [*Wiggins*, 2005]. To investigate the nonlocal transport further it is natural to take a Lagrangian approach by tracking the evolution of particles distributions.

#### 4.4. Lagrangian Particles

[46] Lagrangian particles in LC simulations have been previously tracked in two-dimensional models [*Colbo and Li*, 1999], three-dimensional models on two-dimensional planes [e.g.,*McWilliams et al.*, 1997] and in the fully three-dimensional LES space [*Skyllingstad*, 2003; *Harcourt and D'Asaro*, 2010]. We introduce 276480 particles evenly spaced at *t*= 0 s. Initially, the first vertical level is 1 m away from the bottom and the last vertical level is 0.5 m away from the surface with 28 vertical levels in between (0.5 m vertical spacing). Each vertical level contains 96 × 96 particles with a 1 m horizontal spacing. These particles are advected by the resolved time-dependent three-dimensional flow field, so that we solve for each particle*j* the ordinary differential equations

with the initial conditions at *t* = 0, as described above. The flow field (*u*, *v*, *w*) is determined from the LES for each time step and linearly interpolated in space. The ordinary differential equations (ODEs) are solved using the same time integration scheme as for the LES (low storage third-order Runge-Kutta method). Our tests and previous studies [e.g.,*Gopalakrishnan and Avissar*, 2000] indicate that SGS velocities may be neglected to first order in well-resolved LES studies, although SGS velocities may play a role very close to the boundaries, when they are also, however, the most challenging to model. As a simple test we repeated the scalar mixing experiment with Lagrangian particles by interpreting*S* as a particle density distribution and found good agreement between the Eulerian and Lagrangian approaches.

[47] To visualize particle motion we first follow 64 selected particles (from the total of 276,480) in the crosswind-depth plane (Figure 11). With LC, instantaneous particles quickly spread throughout the whole crosswind depth plane, consistent with the relatively large transport. A closer look at the particle trajectories reveals that particles take “preferred paths,” as they are either “trapped” in an LC vortex or “sucked” into the adjacent counter rotating vorticies (Figure 11). This provides some insight on why the field *S* is more organized in the presence of LC.

[49] Taylor-derived limits of the variance,*σ*^{2}, for times much smaller and much larger than an integral timescale, which is related to *H*/*u*_{*} ≈ 31 min [*Taylor*, 1922]. For *t* ≪ *H*/*u*_{*}, *σ*^{2} varies approximately as *t*^{2} consistent with Taylor's limit (Figure 12, top). Note that is related to a turbulent diffusivity, which increases nearly linearly initially. For *t* ≫ *H*/*u*_{*}, Taylor dispersion limits yield that *σ*^{2} is proportional to *t* and = const. However, in our simulations, particle paths are constrained by the limited vertical domain (the vertical motion is confined between the surface and bottom boundaries), so that all moments approach their asymptotic limit (13) for large times (Figure 12). With LC the maximum turbulent diffusivity is more than twice as large as the one without LC. Note that with LC the variance decreases again, because particles move back toward their initial location. Such a behavior is expected when particles ride in a roll vortex (compare Figure 11).

[50] Without LC values of skewness and kurtosis are initially close to the expected values for Gaussian random velocity, for which = 0 (skewness) and = 3 (kurtosis). With LC, the relatively large negative initial skewness is due to downwelling jets, so that particles move more rapidly to greater depths. These large negative values are consistent with relatively narrow downwelling and broader upwelling regions. The large kurtosis value reflects that relatively large up and downward velocities occur more often than what would have expected from a Gaussian velocity distribution. Deviations from Gaussian statistics is expected in the presence of coherent features.

[51] Travel of particles throughout the whole water column and associated distributions have potentially important implications for nutrient and light supply of phytoplankton [*Denman and Gargett*, 1995; *Li and Garrett*, 1998]. Phytoplankton critically depends on the nutrient transport from the bottom to the surface layer. Plankton is also vertically advected in an exponentially decaying light field. Thus, we examine here the typical vertical extent over which a particle travels over a timescale *T* = 2 *H*/*u*_{*}, and we compute a variance for each particle for the vertical paths

where

The distributions of standard deviations from each particle is significantly different with and without LC (Figure 13). With LC vertical particle travel distances are significantly larger, allowing particles to approach both the bottom and surface boundary layer. Without LC particles remain much more confined to a certain depth. For example, m for 48% and 18% of particles with and without LC, respectively.

#### 4.5. Buoyant Particles

[52] Strong downwelling and surface convergence regions control the distribution of buoyant material such as, air bubbles, many plankton types, marine plastic debris, or polydisperse oil droplets [*Colbo and Li*, 1999; *D'Asaro*, 2000; *Thorpe*, 2004]. As a first step to illustrate the influence of LC on buoyant material, we introduce a buoyant rise velocity *w*_{b} and replace the vertical advection equation by

We set *w*_{b} = 0.5 cm/s which is consistent with a rise speed of frequently observed dirty air bubbles with a radius of 50 *μ*m [*Thorpe*, 1982]. We then repeat the particle tracking experiment from section 4.4 (same number of particles, same initial condition). Note that all particles must eventually surface because the SGS motion has been neglected and *w* = 0 at the surface.

[53] Because of the asymmetry of LC upwelling and downwelling regions, buoyant and sinking particles with the same terminal buoyancy speeds will be differently distributed. Buoyant particles will be transported further into the ocean interior.

[54] With LC buoyant particles remain submerged much longer, accumulating in downwelling regions (Figure 14). After 2 h from the initial particle release, 4 times as many particles stay under water and many more particles are found at greater depth. The delayed resurfacing of buoyant particles is qualitatively consistent with earlier ideas of “retention regions” in which particles may be trapped [*Stommel*, 1949]. A quantitative comparison with this model is not appropriate as our cells do not conform to the idealized cell geometry assumed by Stommel and small-scale turbulence transports particles to outside regions. Patches of high particle concentrations form at middepth under LC convergence regions. Surfaced particles are less randomly distributed in the presence of LC and organize quickly in narrow subsurface convergence regions (Figure 15). This structure agrees with the results from *Skyllingstad* [2003]. Regions of high and low buoyant particle concentrations may strongly influence chemical and biological processes, which depend nonlinearly on material concentrations [e.g., *Fasham et al.*, 1990].