## 1. Introduction

Lateral density gradients associated with submesoscale fronts are ubiquitous in the ocean surface mixed layer (Samelson and Paulson 1988; Ferrari and Rudnick 2000; Munk et al. 2000). These fronts have a width on the order of one to ten kilometers and are nominally in gradient wind balance; that is, the cross-front gradient in hydrostatic pressure is balanced by Coriolis forces associated with a vertically sheared alongfront current. There has been significant recent interest in understanding frontal instabilities and their role in the consequent restratification of the mixed layer (ML) and in the loss of balance associated with a forward energy cascade. Boccaletti et al. (2007), in particular, demonstrate that a three-dimensional (3D), essentially geostrophic (i.e., balanced) but *O*(1) Rossby number baroclinic Eady mode, which they refer to as the mixed layer instability (MLI), causes frontal slumping and restratification on time scales of a few days—much shorter than that associated with mesoscale baroclinic instability. Molemaker et al. (2005) show that a second mode, the ageostrophic anticyclonic instability (AAI), exhibits a large nonhydrostatic unbalanced component, which may be important for the extraction of kinetic energy residing in ocean mesoscale flows and its ultimate dissipation via small-scale, three-dimensional turbulence.

In addition to these 3D instabilities, lateral fronts, particularly within the ML where the vertical stratification is weak and hence horizontal gradients are comparably strong, are subject to a two-dimensional (2D; down-front invariant) gravitational/centrifugal instability referred to as symmetric instability (Haine and Marshall 1998). Linear inviscid stability analysis in an unbounded domain indicates that the symmetric mode arises for 0 ≤ Ri_{υ} < 1, where the vertical Richardson number Ri_{υ} = *N*^{2}*H*^{2}/^{2}, *N*^{2} is the ML buoyancy frequency, *H* is the ML depth, and *f*. Fully nonlinear numerical simulations confirm that the mixing accomplished by the symmetric mode continues until Ri_{υ} ≈ 1 (Haine and Marshall 1998). Since the most rapidly growing perturbations are aligned with isopycnals, the net result is an efficient but minimal *vertical* restratification of the ML. [Taylor and Ferrari (2009) show that the saturation of the growing symmetric instability can be accomplished by a vertical flux of positive potential vorticity across the thermocline following a secondary Kelvin–Helmholtz instability of the symmetric mode.] Indeed, Boccaletti et al. (2007) argue that the bulk of the restratification occurs via the MLI which ensues after Ri_{υ} ≥ 1 and which effectively releases the potential energy in the horizontal stratification.

To date, less attention has been paid to the interaction between lateral density gradients within the ML and smaller-scale convective (vertical) mixing processes that can and do arise independently of the existence of fronts. One notable exception is the recent study by Taylor and Ferrari (2010) of the impact of a geostrophically balanced lateral density gradient on turbulent thermal convection in the upper ocean. Their results suggest that the presence of the density front can modify the structure of the *vertical* stratification and the development of the surface boundary layer.

More significant, perhaps, is the omission in most prior investigations (including those referenced above) of surface wave effects on the stability of ML fronts, which not only precludes the occurrence of Langmuir circulation (LC), a primary vertical mixing mechanism in the ocean surface boundary layer under wind forced seas (Leibovich 1983; McWilliams et al. 1997; Thorpe 2004), but also the modification of other instability processes by the rectified effects of the waves through the action of the Craik–Leibovich (CL) vortex force. Not surprisingly, the focus of nearly all numerical process studies of Langmuir circulation in a density stratified environment—starting with a series of two-dimensional (downwind invariant) numerical simulations by Li and Garrett (Li and Garrett 1995, 1997), and continuing with fully three-dimensional large-eddy simulations (LES) of turbulent Langmuir circulation by, for example, Skyllingstad and Denbo (1995); McWilliams et al. (1997); Li et al. (2005)—has been on the effects of an imposed strictly vertically varying density profile on the development of the vortices and the evolution of the surface mixed layer.

Motivated by the routine occurrence of submesoscale fronts in the upper ocean, we here address the role of down-front-propagating surface waves on the evolution of these fronts. Specifically, we simultaneously investigate the influence of an *imposed lateral* density gradient on Langmuir circulation and the effect of the CL vortex force on the classical symmetric instability mode. Note that we do not study the possible two-way coupling between these instability modes and the more slowly evolving processes responsible for maintaining submesoscale fronts—although we view the present investigation as a necessary first step toward addressing the two-way multiscale interaction between fronts and mixed layer turbulence. Instead, our primary aim is to elucidate the nature of the convective instability and the physical mechanisms involved. We consider an idealized scenario in which the dynamics are 2D, with no variation in the alongfront direction, but involve all three velocity components. We also imagine that the LC evolves in a preexisting mixed layer which is “slippery” to the cellular flow and that there exists a sufficiently strong pycnocline to inhibit further layer deepening by the cells over the time scales of interest. Further details of the problem formulation are given in section 2. Using a complement of linear stability theory, energy budgets, and fully nonlinear, nonhydrostatic numerical simulations of the rotating, laterally stratified 2D CL equations (section 3 and section 4, respectively), we diagnose the physics of the fastest growing primary instability mode and its subsequent secondary instability and nonlinear evolution. We conclude in section 5 with a discussion of the potential implications of our results for ML restratification.

## 2. Problem formulation

*x*axis is aligned with the front and with the presumed direction of both the wind and the vertically varying surface wave Stokes drift velocity

*z*measures distance upward from the mean position of the sea surface, and the cross-front coordinate

*y*is oriented so as to complete a right-handed coordinate system. To explore the dynamics of LC and other phenomena characterized by time scales long compared to the typical surface wave period, we employ the rotating CL equations—a surface wave–filtered version of the incompressible Navier–Stokes equations (Craik and Leibovich 1976; Leibovich 1977; Craik 1977; Huang 1979; Leibovich 1983; Holm 1996; McWilliams et al. 1997)—on an

*f*plane (where

*f*is the Coriolis parameter) and under the Boussinesq approximation, namely,Here,

*ρ*is the density field, with

*ρ*

_{0}a constant representative value;

**U**= (

*U*,

*V*,

*W*) is the incompressible velocity field;

*p*is a modified pressure;

*g*is the gravitational acceleration; and

*ν*is the presumed constant eddy viscosity that arises from the Craik–Leibovich wave-filtering procedure. The substantial derivative operator

_{e}*D*/

*Dt*= ∂/∂

*t*+

**U**·

**∇**, where

*t*is the time variable. The final term on the right-hand side of (1) consists of the Stokes–Coriolis force (McWilliams et al. 1997; Lewis and Belcher 2004) and the CL vortex force, the cross-product of the Stokes drift velocity associated with the filtered surface waves—a prescribed input in this formalism—and the (time or phase averaged) relative vorticity vector.

*x*invariant roll vortices, or “Langmuir cells,” when the wind and waves are aligned in the

*x*direction. We presume this to be the case even when lateral stratification and Coriolis forces are incorporated and henceforth restrict attention to

*x*invariant dynamics (although we allow for flow in the

*x*direction). To nondimensionalize the governing equations, we choose the mixed layer depth

*H*to be the relevant length scale. Attributing density variations entirely to temperature anomalies, and assuming a linear equation of state relating density

*ρ*to the temperature

*T*, we scale temperature byroughly the temperature difference across a typical Langmuir cell. In (2),

*T*(

_{f}*y*) is the imposed, linear lateral stratification (with constant gradient

*dT*/

_{f}*dy*),

*α*is the coefficient of thermal expansion of seawater, and

*M*is, thus, the Brunt frequency associated with the horizontal density gradient. The typical alongfront flow speed

*O*(

*M*

^{2}

*H*/

*f*) is then used to scale all velocity components except for the Stokes drift velocity, which is scaled with

*U*

_{s}_{0}=

*U*(0). The time variable is nondimensionalized by the convective or eddy-turnover time scale

_{s}*H*/

*nondimensional*and subscripts denote partial differentiation. Here, Ω is the

*x*vorticity component, and

*ψ*is the associated streamfunction, with

*V*=

*ψ*

*≡*

_{z}*υ*and

*W*= −

*ψ*≡

_{y}*w*. For analytical simplicity, we take the Stokes drift to be a linear rather than more realistic exponentially decaying function of depth,

*U*(

_{s}*z*) =

*z*+ 1, where we set the Stokes parameter

*S*≡

*U*

_{s}_{0}/

*ν*/(

_{e}*H*), which can be interpreted as an inverse Reynolds number; the Peclet number Pe ≡

*H*/

*κ*, where

_{e}*κ*is an eddy diffusivity for heat; the ML Rossby number Ro =

_{e}*fH*); and the

*horizontal*Richardson number Ri

_{h}≡

*M*

^{2}/(

*H*)

^{2}.

*z*= 0 and along the mixed layer base

*z*= −1:where the ratio of the square of the vertical to horizontal buoyancy frequency

*γ*=

*N*

^{2}/

*M*

^{2}. Following Taylor and Ferrari (2009), we have specifically chosen the momentum and heat flux boundary conditions in (7) to minimize the effects of dynamic buoyancy forcing via Ekman drift (Thomas and Ferrari 2008) and to allow for a base state with linear vertical and lateral thermal stratification. Indeed, (3)–(6) and boundary conditions (7) admit a uni-directional base flowthat is in surface-wave-modified gradient wind balance with an imposed basic-state temperature distributionfor constants

_{h}= 1 +

*S*. In dimensional terms, this last condition can be expressed as

*αg*(Δ

*T*/

*H*)

*dT*/

_{B}*dy*= −

*f*(

*H*)(

*dU*/

_{B}*dz*+

*SdU*/

_{s}*dz*); that is, the usual gradient wind balance condition modified by the Stokes–Coriolis torque. Strictly speaking, it is inappropriate to refer to this base state as a front, of course, since (8) has no horizontal structure. In this study, we nevertheless continue to use this terminology (again following, e.g., Taylor and Ferrari (2009)) but with the understanding that we are modeling a

*L*=

*O*(1) km wide section of a lateral front encompassing tens of Langmuir cells and across which the temperature field may reasonably be approximated as linearly varying. Except for the total temperature

*T*, all fields are taken to be

*L*-periodic in the

*y*coordinate.

The traction boundary condition *U _{z}* = 1 in (7) ensures that the imposed down-front wind stress

*exactly*cancels the so-called geostrophic stress (Thomas and Rhines 2002), eliminating the possibility of a residual Ekman drift. As discussed in Thomas and Ferrari (2008), down-front winds driving a dimensional frictional surface shear

*H*(=

*M*

^{2}/

*f*in the absence of the Stokes–Coriolis force) will induce an Ekman drift that tends to destratify the mixed layer by advecting heavy fluid over light; conversely, down-front winds driving a frictional surface shear less than the geostrophic shear will induce an Ekman drift that drives vertical restratification. Given our nondimensionalization, the implied water friction velocity

*u*

_{*}=

*O*(0.005) m s

^{−1}, using

^{−1}and

*H*= 50 m as representative values. When

*S*= 1, these winds and waves are appropriate for the dynamical regime identified as “Langmuir turbulence” by McWilliams et al. (1997).

In what follows, we take *γ* ≥ 0 and consider the linear and nonlinear stability of the imposed front to down-front invariant disturbances. We fix La and Pe at moderately small and large values, respectively, and treat Ri_{h} and Ri_{υ} ≡ *γ*Ri_{h}, the vertical Richardson number, as the key control parameters.

## 3. Linear stability analysis

*x*velocity component and the temperature into basic-state (denoted with “

*B*” subscripts) and perturbation contributions,

*U*=

*U*+

_{B}*u*and

*T*=

*T*+

_{B}*θ*, and linearizing the dynamics about this basic state gives the (linearized) equations governing the evolution of the perturbations:The perturbation fields are

*L*periodic in the

*y*coordinate and satisfy homogeneous boundary conditions along

*z*= 0 and

*z*= −1:Decomposing the generic perturbation fieldand substituting into (10)–(14) yields an ordinary differential eigenvalue problem in

*z*for each horizontal wavenumber

*k*, where

*σ*are the (generally complex) vertical eigenfunction and eigenvalue.

^{−1}= 0) in a vertically unbounded domain, so that

*σ*depends only on the ratio

*λ*= −

*m*/

*k*rather than on

*k*and

*m*separately. An analytical expression for

*σ*is readily found to be given bywhere in the absence of waves

*S*= 0 and

*Ro*

^{−1}= Ri

_{h}while in the presence of waves

*S*= 1 and

*Ro*

^{−1}= Ri

_{h}/2. It is instructive to consider various limiting cases of (16) representing distinct and hybridized modes of linear instability; see Table 1 for a taxonomy of these various instability modes.

Taxonomy of 2D (down-front invariant) linear instability modes of the base state (8)–(9) modeling the central region of a submesoscale lateral front in the surface ML. Note that in case (i), Coriolis accelerations are (artificially) suppressed (i.e., *Ro*^{−1} = 0 but Ri_{h} is finite), while in case (ii), lateral density gradients are suppressed (i.e., Ri_{h} = 0 but *Ro*^{−1} is finite). In scenarios (iii) and (iv), *Ro*^{−1} = Ri_{h} and in case (v), *Ro*^{−1} = Ri_{h}/2. The subscript “*f*” refers to a property of the fastest-growing linear mode. (Concise analytical forms for certain properties have been obtained by taking representative asymptotic limits.)

As suggested in Fig. 2, in the absence of surface waves, vertical stratification within the ML, and Coriolis accelerations [case (i)], counterrotating cellular disturbances inclined at 45 degrees to the vertical in the direction of the density gradient are most efficiently able to release the potential energy stored in the lateral front by moving less dense fluid above more dense fluid. In addition to this purely buoyancy-driven instability mechanism, which leads to inclined convection cells, inertial instability is possible when Coriolis accelerations are incorporated, even in the absence of density stratification [case (ii)]. Specifically, vertical gradients in the perturbation down-front velocity component *u* tilt vertical planetary vorticity filaments into the *x* direction, creating counterrotating cellular disturbances. Energy stored in the down-front basic-state shear flow is extracted by the cellular flow; in terms of momentum fluxes, the rate at which down-front momentum near the surface is advected into the convergence zones between the cells exceeds the rate at which momentum is extracted near the ML base. Interpreted in this way, this form of inertial instability is loosely analogous to the CL2 mechanism responsible for Langmuir circulation, except that 1) in LC, the imposed surface wave Stokes drift tilts vertical vorticity associated with spanwise-varying streamwise velocity perturbations into the streamwise direction, while in the inertial instability, vertical gradients in the streamwise velocity perturbations tilt the imposed vertical planetary vorticity filaments into the streamwise direction (as noted above); and 2) the fastest-growing inertial modes are inclined at 45 degrees to the vertical because the imposed planetary vorticity is of one sign unlike the perturbation vertical vorticity field in the LC case.

In pure symmetric instability (i.e., in the absence of surface waves), the buoyancy and vortex-tilting mechanisms are both operative and contribute equally to the growth rate of the disturbance, which is approximately equal to the square root of the horizontal Richardson number [case (iii)]. In what may be termed “classical symmetric instability” [case (iv)], stabilizing residual vertical stratification within the mixed layer (which often is stronger than the lateral stratification there) significantly reduces growth rates; in fact, the idealized analysis suggests that classical symmetric instability is completely inhibited when Ri_{υ} ≥ 1, a result that full numerical simulations confirm persists into the nonlinear regime. Moreover, the streamlines of the fastest growing classical symmetric modes are roughly parallel to isopycnal surfaces (since the fastest growing mode has a wavenumber ratio satisfying *λ _{f}* ≈

*M*

^{2}/

*N*

^{2}), implying that symmetric instability accomplishes little cross-isopycnal transport. All these results are consistent with previous analyses; see, for example, Hoskins (1974), Haine and Marshall (1998), and Taylor and Ferrari (2009).

Remarkably, the linear analysis suggests that in the presence of down-front-propagating surface waves [case (v)] the fastest growing modes again are inclined at 45 degrees to the vertical, implying significant cross-isopycnal transport and greatly enhanced growth rates relative to classic symmetric instability. Indeed, for Ri_{υ} = 1, a regime in which the classical symmetric mode is suppressed, the properties of the fastest growing mode essentially accord with those for the scenario in which *neither* surface waves nor vertical stratification is present [case (iii)]. Heuristically, the fastest growing disturbance in case (v) is a hybrid LC/symmetric mode that is able to exploit multiple instability mechanisms for enhanced growth, with, for example, the destabilizing CL vortex torque neutralizing the stabilizing buoyancy torque associated with vertical stratification.

The predictions of the idealized analysis are broadly confirmed (albeit refined) by a full linear stability analysis in which the effects of vertical boundaries and diffusion are retained. We solve the full problem using a Chebyshev spectral collocation method (Trefethen 2000) with 40–50 modes. Figures 3, 4, and 5 show the real part of the growth rate of the fastest growing instability mode for Ri_{υ} = 0, 0.5, and 1, respectively, for a modest value of Ri_{h} = 0.15, both without (lower curves) and with (upper curves) surface wave effects. In these and all of the following results discussed below, La = 0.001 and Pe = 4000. For all parameter values considered, it is clear that surface wave effects enhance, in some cases by almost an order of magnitude, the growth rate of the dominant instability mode. As expected, growth rates are reduced as Ri_{υ} is increased, although much more so for the symmetric than for the hybrid mode. Notably, the full stability calculations confirm that the hybrid mode can exist in a parameter regime (Ri_{υ} ≥ 1) in which the symmetric mode is completely suppressed (see Fig. 5).

Also depicted in Figs. 3–5 are the eigenfunctions corresponding to the fastest-growing perturbation fields. Although the upper plot in Fig. 3 indicates that in the presence of surface waves growth rates are essentially unchanged by lateral stratification (cf. the red and black curves), the eigenfunction plots clearly reveal that the Langmuir cells nevertheless feel the influence of the lateral front by inclining to the vertical in the direction of increasing density to extract potential energy stored in the front, as anticipated. As described in section 4, this *cell tilting* has a crucial effect on the subsequent *nonlinear evolution* of the disturbance and, ultimately, on the stratification within the ML. Figure 4 confirms that, in the presence of stabilizing vertical stratification, the fastest-growing classical symmetric instability mode has streamlines that are roughly parallel to the basic-state isopycnals (middle plots), while the fastest-growing hybrid mode evidently can accomplish cross-isopycnal transport (bottom plots). Finally, for Ri_{υ} = 1, *two* hybrid modes appear with comparable maximum growth rates (see Fig. 5). Unlike the hybrid modes at smaller values of Ri_{υ}, the high wavenumber hybrid mode at Ri_{υ} = 1 shown in the bottom plots in Fig. 5 is a traveling instability. The lower wavenumber mode, which turns out to play a more important role in the nonlinear evolution of the ML stratification, is stationary.

## 4. Nonlinear simulation of hybrid LC/symmetric instability

Next, we investigate the finite-amplitude evolution of the hybrid LC/symmetric linear instability, as well as its impact on ML restratification, by numerically integrating the fully nonlinear system (3)–(6) subject to boundary conditions (7). For this purpose, we modified an existing, thoroughly validated pseudospectral code, developed in-house, for solving the 2D (nonhydrostatic) CL equations by incorporating Coriolis accelerations and horizontal and vertical density stratification through the Boussinesq approximation. A Fourier–Chebyshev-tau spatial discretization scheme was employed, and the discretized system was time advanced using a semi-implicit Crank–Nicolson/Adams–Bashforth algorithm. The simulations were initialized with the base profiles (8) and (9) plus a small-amplitude disturbance having the wavenumber of the fastest-growing mode predicted by the full linear stability calculations for the given parameters.

We report the results of three sets of simulations, each for La = 0.001, Pe = 4000, *S* = 1, and Ri_{h} = 0.15. Note that this La corresponds to an eddy viscosity *ν _{e}* = La

*H = O*(10

^{−3}) for a reasonable range of

*H*, in agreement with the

*sub-grid-scale*—that is, excluding the effects of LC—viscosity numerically computed in the LES of Langmuir turbulence by McWilliams et al. (1997). Our three simulations are distinguished by Ri

_{υ}, a measure of the

*initial*vertical stratification within the ML. In the first case, Ri

_{υ}= 0; in the second, Ri

_{υ}= 0.5; and in the third, Ri

_{υ}= 1. The panels in Figs. 6, 7, and 8 depict snapshots of the total fields (

*U*, Ω,

*ψ*, and

*T*, from top to bottom) at four different instants (with time increasing from left to right) during the evolution of the instability. In each case, the dynamics is broadly similar: a hybrid LC/symmetric instability mode, inclined to the vertical, is excited. As this instability is amplified, the cells become asymmetric and their inclination angle oscillates. After several convective time units, a cross-front shear flow is induced, driving light fluid over heavy fluid. The result of this sequence of events is a remarkably efficient, although time-dependent, vertical restratification of the ML—that is, there is an evident oscillation in the mean isopycnal slope following the time of maximum restratification, as can be more readily inferred from the time traces in Fig. 9 and, especially, in Fig. 10 and as discussed in more detail subsequently.

For Ri_{υ} = 0, the maximum mean vertical stratification, as measured by the nondimensional Brunt frequency squared, ^{−1}, this corresponds to just over 10 h; i.e., very fast restratification. For a range of scenarios, both with and without surface waves, Table 2 quantifies the value of the first maximum in the mean vertical stratification and the time at which that maximum is achieved; the value *after* the first maximum is also listed. Clearly, for the parameter regime investigated, the hybrid mode is far more effective (roughly a factor of four in terms of _{υ} = 0 and Ri_{υ} = 0.5) than the pure symmetric mode at restratifying the ML, and, again, is operative in a regime Ri_{υ} ≥ 1 in which classical symmetric instability is not.

_{h} = 0.15, La = 0.001, and Pe = 4000. *t*_{max} and *t*_{min} correspond to the nondimensional times at which *N*^{2}(*t* = 0) = Ri_{υ}.

_{υ}≥ 0.5, while the symmetric mode cannot. (In fact, the hybrid mode can effectively reduce the mean shear to zero during the initial development of the linear instability, which in turn yields a very large

_{h} = 0.15, La = 0.001, and Pe = 4000. *t*_{min} and *t*_{max} correspond to the nondimensional times at which Δ*U*_{min} and

*u*,

*ψ*, and −Ri

_{h}

*θ*, respectively, and integrate over the domain. In this way, we obtain, for example, the equation governing the evolution of the down-front perturbation kinetic energy (per unit mass),whereand

*L*= 2

*π*/

*k*is the width of the domain:Here,

*P*= −〈

_{U}*uw dU*/

_{B}*d*

*z*〉 accounts for the production of

*C*≡ −

*Ro*

^{−1}〈

*uυ*〉 represents energy transfers from down-front to cross-front flows by Coriolis forces, and

*y*average. Note that PE′ is one metric for restratification: specifically, the layer is restratifying when PE′ < 0. Considering the first term on the right-hand side of (19), we observe that when

*w*and

*θ*are positively correlated, as in Fig. 2, potential energy is released from the background lateral density gradient and converted to cross-front kinetic energy. The remaining terms vanish for

*linear*disturbances (

*υ*,

*θ*), which necessarily have zero horizontal average; that is, for small-amplitude disturbances, buoyancy production

*B*is the only factor that can alter the perturbation potential energy. However, during the nonlinear phase of the disturbance evolution, the potential energy may be altered by the last two terms in (19). Of these, the term involving advection of the background potential energy by horizontally averaged cross-front currents is the more important, since Pe

^{−1}≪ 1.

The time evolution of a subset of the various terms in the perturbation energy equations is shown in the series of plots in Fig. 9, which were taken from the simulations depicted in Figs. 6–8. Careful inspection of these energetics along with insights gleaned from the linear stability analysis suggests the following restratification mechanism.

- Linear instability: Hybrid LC/symmetric counterrotating cellular disturbances
*incline*to the vertical, in the direction of the density gradient, to exploit buoyancy production (i.e., to maximize positive or minimize negative*B*). - Nonlinear evolution: A “self–self” interaction of the dominant linear mode generates a nonzero
Reynolds stress (RS), since for tilted cells, *υ*′ and*w*′ (where the primes refer to fluctuations about the horizontal mean) are correlated. - Mean flow generation: The vertical divergence of this RS drives a horizontally averaged, vertically sheared cross-front mean flow:
- Mean flow advection: The sign of the induced shear is such that light fluid is carried over heavy fluid, restratifying the ML and, owing to the combined influence of cross-cellular shear and increased vertical stratification, shutting down the convection.
- Inertial oscillation: Coriolis forces imply that a nonzero
will induce a nonzero , sinceFor small La and weak or nonexistent convection, an inertial oscillation ensues.

*asymmetry*in each pair of counterrotating cells that, itself, is caused by the induced cross-front mean flow. Asymmetry in the structure of a given cell pair implies an asymmetry in the buoyancy and vortex torques experienced by the cells, which evidently tends to right the tilted cells (see, e.g., the second set of panels in Fig. 6). In contrast, the low-frequency part of the signal appears to be the manifestation of a simple inertial oscillation, driven by the action of the Coriolis force on the induced shear flow and having an

*O*(Ro

^{−1}) frequency. The phase relationships inherent in this oscillation are such that

*N*

^{2}is maximum approximately when

*maximum*; this phase relationship thus limits the maximum achievable Ri

_{υ}. For this reason,

_{υ}of the extent of restratification achieved by this instability mechanism.

More compelling evidence for the inertial oscillation is presented in Fig. 10; see, in particular, the time evolution of the perturbation potential energy PE′, advection of perturbation potential energy ADV, and down- and cross-front perturbation kinetic energies _{υ} = 0.5 simulation to a physical time that is unrealistically long in view of the time-independent forcing conditions and suppression of down-front variability. The effects of eddy diffusion also seem exaggerated over this longer time scale, as the inertial oscillation is more strongly damped than might be expected in reality. Note that from *t* ≈ 200 to *t* ≈ 500 the (nonmonotonic) decrease in the *magnitude* of PE′ is also attributable to exaggerated eddy diffusion in combination with the specification of fixed-flux thermal boundary conditions; that is, with these boundary conditions and in the *absence* of convection and cross-front flow (for 200 ≤ *t* ≤ 500), it is readily shown that *t* → ∞ on a time scale *O*(*π*^{2}/Pe). For weaker eddy diffusion of heat and momentum, however, the diffusive relaxation of *given* parameter regime apparently consists of intermittent bursts of convective activity, with amplification of residual hybrid LC/symmetric disturbances following nearly convection-free periods.

## 5. Conclusions

In this investigation, we have used a combination of linear stability theory and fully nonlinear numerical simulations to assess the influence of down-front-propagating surface waves on the evolution of submesoscale lateral fronts in the upper ocean. We briefly summarize our primary findings.

Through the action of the CL vortex force, surface waves fundamentally alter the linear stability of submesoscale fronts in (modified) thermal wind balance over inertial time scales. Unlike classical symmetric instability modes, the hybrid Langmuir circulation/symmetric mode accomplishes significant *cross-isopycnal* transport. Over a wide parameter regime, surface waves are strongly destabilizing, leading to growth rates much larger than those exhibited by the classical symmetric mode. In fact, our linear analysis shows and nonlinear simulations confirm that the hybrid mode is operative for vertical Richardson numbers Ri_{υ} ≥ 1, while symmetric instability is completely suppressed in this regime.

Our numerical simulations also suggest that the nonlinear evolution of the hybrid mode can drive vertical restratification within the mixed layer. This restratification process is efficient in that it has been shown to occur on a time scale of less than one day—faster than the restratification accomplished by the 3D “mixed layer instabilities” (MLI) discussed by Boccaletti et al. (2007). The restratification is also potentially significant in that maximum vertical Brunt frequencies up to an order of magnitude larger than those associated with the base state, and approximately 4 times larger than are attainable by classical symmetric instability, can be achieved. The key physical mechanism at work in the restratification process is a cross-front shear flow, driven by a nonlinear interaction of the inclined and counterrotating cells. Paradoxically, this result suggests that Langmuir circulation, which is generally viewed as a key vertical mixing mechanism within the upper ocean, may actually play a role in *supporting* vertical stratification.

Of course, these conclusions must be tempered by a number of factors. First, owing to Coriolis forces, the restratification is time-dependent. However, the *time-averaged* vertical stratification is indeed increased during the nonlinear stages of the instability. (For example, for Ri_{υ} = 0 and *S* = 1, the appropriately time-averaged *N*^{2} is roughly 2.7.) The crucial result of this investigation is that, in the presence of down-front-propagating surface waves, some fraction of the potential energy stored in the front can be accessed by the hybrid instability mode and converted into mean kinetic energy. In the absence of dissipative and three-dimensional (3D) effects, the oscillation in the vertical stratification would presumably continue unabated. Of course, this assertion also presumes that there is strictly one-way coupling between the submesoscale front and the convective instabilities, and that various submesoscale external forcings remain fixed indefinitely, including those responsible for the occurrence or lack of dynamic buoyancy forcing via Ekman drift. Thus, two key questions to be addressed in future studies include the role of three-dimensionality—that is, both the 3D evolution of the hybrid instability and its possible interaction with other 3D frontal instabilities, particularly the baroclinic MLIs—and the fully two-way coupling between submesoscale flows and convective instabilities, such as the hybrid mode studied here.

GPC gratefully acknowledges funding for this work from NSF CAREER Award 0348981, NSF CMG Award 0934827, and NASA ROSES08 NNX09AF38G subcontract. A portion of these funds was used to support KL and ZZ.

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