1. Introduction
The upper ocean, particularly at midlatitudes, is subject to intense, highly variable winds associated with synoptic atmospheric storms. These intermittent events energize nearly isotropic turbulence at length scales smaller than the mixed layer depth, which drives entrainment and mixing of pycnocline water into the mixed layer and thereby deepens the mixed layer and increases its density (e.g., Davis et al. 1981; Large and Crawford 1995; Dohan and Davis 2011; Forryan et al. 2015). In aggregate, storm-driven, small-scale turbulence contributes significantly to the seasonal increase in the mixed layer depth and mixed layer density during the autumn in midlatitudes (e.g., Large et al. 1986). Many previous studies have examined the upper-ocean response to storms using a one-dimensional framework (e.g., Pollard et al. 1972; Niiler and Kraus 1977; Price et al. 1978; Large et al. 1994). However, the upper ocean contains lateral variability associated with large-scale fronts, filaments, and eddies, which modify the evolution of upper-ocean stratification and small-scale turbulence during a storm.
Among the motions inducing lateral variability are submesoscales, anisotropic features with vertical scales similar to the mixed layer, horizontal scales between 0.1 and 10 km, and O(1) vorticity Rossby numbers (e.g., Thomas et al. 2008; Capet et al. 2008; McWilliams 2016), which are prevalent in the upper ocean (e.g., Munk et al. 2000; Shcherbina et al. 2013; Buckingham et al. 2016; Thompson et al. 2016). Submesoscales play an important role in restratifying the mixed layer (e.g., Haine and Marshall 1998; Lapeyre et al. 2006; Boccaletti et al. 2007; Mahadevan et al. 2010, 2012) and enhancing the exchange of water between the mixed layer and pycnocline (e.g., Lévy et al. 2001; Klein and Lapeyre 2009; Thomsen et al. 2016). In addition, submesoscales modify the energetics and fluxes associated with small-scale turbulence in the mixed layer (e.g., D’Asaro et al. 2011; Smith et al. 2016; Taylor 2016). For example, submesoscales transfer energy from large-scale geostrophic gradients to small-scale turbulence, while submesoscale stratification in the mixed layer locally inhibits turbulence.
Many submesoscale features are spawned from instabilities associated with horizontal density gradients or fronts (e.g., Haine and Marshall 1998; Boccaletti et al. 2007; Callies et al. 2016). These instabilities can be interpreted via stability analysis of an “Eady-like” baroclinic zone with parameters characteristic of the mixed layer (e.g., Stone 1966; Stamper and Taylor 2016). Depending on the gradient Richardson number Rig associated with the vertically sheared balanced flow, the fastest growing mode is one of two types: mixed layer baroclinic instability (MLI; when Rig > 0.95) or symmetric instability (SI; when Rig < 0.95). The most unstable normal mode of MLI is invariant in the cross-front direction and converts available potential energy associated with tilting isopycnals into kinetic energy and ultimately submesoscale eddies, while SI is invariant in the alongfront direction and draws its energy from the vertical shear. The net effect of both instabilities is to lower the center of mass of the fluid and increase the stable stratification in the mixed layer. However, submesoscales in the real ocean are a chaotic, nonlinearly interacting continuum rather than a discrete set of linear modes (e.g., Shcherbina et al. 2013).
Many of the numerical simulations upon which our understanding of nonlinear/turbulent submesoscale dynamics is based have either been unforced initial-value problems (e.g., Özgökmen et al. 2011; Skyllingstad and Samelson 2012; Stamper and Taylor 2016) or forced with steady surface cooling or winds (e.g., Taylor and Ferrari 2010; Thomas et al. 2013; Hamlington et al. 2014; Taylor 2016). One exception is a study of a storm event at the Gulf Stream front using observations and large-eddy simulations (LES) reported in Thomas et al. (2016). They found turbulent dissipation rates in excess of anticipated values and rapid restratification of the boundary layer and attributed these features to SI. Although they captured SI, the simulations in Thomas et al. (2016) had a limited domain size that excluded the possibility of MLI and hence submesoscale eddies.1
Despite the attention paid to submesoscales in recent years, the response of submesoscale eddies to storms is not well understood. Basic open questions remain, including the following: Can MLI maintain a stable stratification during intense storms? Are submesoscale eddies damped by small-scale turbulent mixing associated with strong winds? How is the small-scale turbulence in the mixed layer influenced by submesoscales during storms?
We address these questions using high-resolution LES, motivated by observations collected near 48.7°N, 16.3°W above the Porcupine Abyssal Plain during the Ocean Surface Mixing, Ocean Submesoscale Interaction Study (OSMOSIS), which reveal significant submesoscale activity throughout the year (Thompson et al. 2016; Buckingham et al. 2016). On 24–26 September 2012, during the deployment cruise, a storm passed over the field site and deepened the mixed layer (Rumyantseva et al. 2015). Glider profiles collected during the storm show that the mixed layer remained well stratified throughout the storm (their Fig. 4). An idealized representation of this event will be the basis for our analysis.
2. Model description
To elucidate the interaction between submesoscales and small-scale turbulence during the life cycle of a storm, we present results from a simulation in a large domain that captures the fastest growing MLI length scale, hence the associated energy source for submesoscale eddies, while simultaneously resolving small-scale turbulence. The domain is 1970 m by 1970 m by 80 m covered by a grid with 1024 by 1024 by 160 points that achieves a uniform resolution of 1.9 m by 1.9 m by 0.5 m in x and y and z, respectively. As in Taylor and Ferrari (2010) and Taylor (2016), the flow is expressed as a periodic (in x and y) perturbation from a fixed/constant mean horizontal density gradient 
The turbulent state at the onset of the storm (Fig. 1a) is obtained from a 3-day spinup simulation (Whitt 2017) that is forced by a constant air–sea (i.e., surface) buoyancy flux BA = 3 × 10−9 m2 s−3 (buoyancy b = −gρ/ρ0 is simulated, but this is roughly equivalent to a heat loss of 10 W m−2 to the atmosphere) and initialized with low-amplitude red noise on a vertical density profile based on Fig. 3b of Rumyantseva et al. (2015). The mixed layer depth HML, which is defined by an increase in the mean density 〈ρ〉x,y by 0.03 kg m−3 relative to the surface, is initially 35 m. The mixed layer is stratified: 
(a) Snapshots of density and (b) time series of wind stress magnitude (black) and vector components (dashed red and green) as well as the mixed layer buoyancy flux ratio RML (blue) [see (1)]. Black vectors in the snapshot at day 2.33 indicate the direction of the wind during the storm.
Citation: Journal of Physical Oceanography 47, 10; 10.1175/JPO-D-17-0130.1
(a) Snapshots of density and (b) time series of wind stress magnitude (black) and vector components (dashed red and green) as well as the mixed layer buoyancy flux ratio RML (blue) [see (1)]. Black vectors in the snapshot at day 2.33 indicate the direction of the wind during the storm.
Citation: Journal of Physical Oceanography 47, 10; 10.1175/JPO-D-17-0130.1
(a) Snapshots of density and (b) time series of wind stress magnitude (black) and vector components (dashed red and green) as well as the mixed layer buoyancy flux ratio RML (blue) [see (1)]. Black vectors in the snapshot at day 2.33 indicate the direction of the wind during the storm.
Citation: Journal of Physical Oceanography 47, 10; 10.1175/JPO-D-17-0130.1
The storm forcing during 24–26 September 2012 at the OSMOSIS site is represented by the idealized spatially uniform but time-dependent surface stress in Fig. 1b, which points 45° to the right of the mean geostrophic flow at the surface. Following the storm, the simulations continue for about 4 days without wind stress to elucidate the subsequent adjustment and restratification. To separate the effects of storm winds from storm buoyancy fluxes, the air–sea buoyancy flux is held constant at BA = 3 × 10−9 m2 s−3 during and after the simulated storm; this BA is about 10 times weaker than the buoyancy flux associated with the observed cooling during the storm (Rumyantseva et al. 2015).
To separate the influence of the front and submesoscales from the classic “one-dimensional” effects of the wind stress on the small-scale turbulence and stratification, the wind-forced simulation in the large domain is compared to a simulation in a small domain without a front or submesoscales. The small domain is 492.5 m by 492.5 m by 80 m and has the same grid resolution, the same surface boundary conditions, and the same mean density profile 〈ρ〉x,y(z) at day 0 as the large domain, but 〈M2〉x,y = 0.
All simulations are carried out with DIABLO (Taylor 2008), which solves the discrete, incompressible Boussinesq equations using a pseudospectral method for horizontal derivatives and second-order finite differences for vertical derivatives. Time stepping is accomplished using a third-order Runge–Kutta scheme for advection and the implicit Crank–Nicholson scheme for viscosity/diffusion. The LES solves a filtered version of the governing equations, which are closed using a modified Smagorinsky model to represent subgrid-scale stresses (Kaltenbach et al. 1994). The subgrid-scale diffusivity 
3. Results
At the onset of the storm, the density variance in the mixed layer of the large domain is dominated by submesoscales, although the domain contains variability at all resolved scales (Fig. 1a). In addition, submesoscale density variability remains a dominant feature of the mixed layer both during and after the storm. The following sections describe the simulated evolution of the mean stratification and small-scale turbulence as well as submesoscale variability within the mixed layer during and after the storm.
a. Mean stratification, shear, and dissipation
Both during and after the storm, the mixed layer is characterized by a stronger mean stratification 
Time series of horizontally averaged stratification 〈N2〉x,y and (equivalently) the balanced Richardson number 
Citation: Journal of Physical Oceanography 47, 10; 10.1175/JPO-D-17-0130.1
Time series of horizontally averaged stratification 〈N2〉x,y and (equivalently) the balanced Richardson number
Citation: Journal of Physical Oceanography 47, 10; 10.1175/JPO-D-17-0130.1
Time series of horizontally averaged stratification 〈N2〉x,y and (equivalently) the balanced Richardson number
Citation: Journal of Physical Oceanography 47, 10; 10.1175/JPO-D-17-0130.1
Despite the strong mean stratification and higher RiB throughout much of the mixed layer, the mixing layer depth HXL, where the dissipation rate 〈ε〉x,y > 10−8 W kg−1, is deeper during the storm in the wind-forced front compared to the wind-forced domain without a front or the front forced by a strong air–sea buoyancy flux. In addition, HXL remains deeper than HML for 0.5 days after the storm is over in the wind-forced front, unlike the other two strongly forced simulations (Fig. 2).
b. Spatial variability
The combination of a strongly stratified and turbulent mixing layer is paradoxical, but it can be explained by spatial variability associated with submesoscales. Both during and after the storm, the stratification N2 in the wind-forced front exhibits submesoscale variations of one to two orders of magnitude within the mixed layer at all depths (Figs. 3a, 4a). Regions of high stratification N2 
Snapshots of (a) stratification N2 and (b) small-scale turbulent kinetic energy at t = 2.72 days in the wind-forced front (just before the end of the storm, see Fig. 1b). Solid black contours of the large-scale density are overlaid. (top) The x–y slices are calculated as an average from z = −35.5 m to the surface in (a) and from z = −35.5 to −30.5 m in (b). (bottom) The x–z slices are calculated at the y location indicated by the dashed black lines in the x–y slices. Here, large scales are defined by applying a 150 m by 150 m square filter to the full fields at each vertical level, while small scales are defined as the difference between the full fields and the large-scale fields.
Citation: Journal of Physical Oceanography 47, 10; 10.1175/JPO-D-17-0130.1
Snapshots of (a) stratification N2 and (b) small-scale turbulent kinetic energy at t = 2.72 days in the wind-forced front (just before the end of the storm, see Fig. 1b). Solid black contours of the large-scale density are overlaid. (top) The x–y slices are calculated as an average from z = −35.5 m to the surface in (a) and from z = −35.5 to −30.5 m in (b). (bottom) The x–z slices are calculated at the y location indicated by the dashed black lines in the x–y slices. Here, large scales are defined by applying a 150 m by 150 m square filter to the full fields at each vertical level, while small scales are defined as the difference between the full fields and the large-scale fields.
Citation: Journal of Physical Oceanography 47, 10; 10.1175/JPO-D-17-0130.1
Snapshots of (a) stratification N2 and (b) small-scale turbulent kinetic energy at t = 2.72 days in the wind-forced front (just before the end of the storm, see Fig. 1b). Solid black contours of the large-scale density are overlaid. (top) The x–y slices are calculated as an average from z = −35.5 m to the surface in (a) and from z = −35.5 to −30.5 m in (b). (bottom) The x–z slices are calculated at the y location indicated by the dashed black lines in the x–y slices. Here, large scales are defined by applying a 150 m by 150 m square filter to the full fields at each vertical level, while small scales are defined as the difference between the full fields and the large-scale fields.
Citation: Journal of Physical Oceanography 47, 10; 10.1175/JPO-D-17-0130.1
As in Fig. 3, but at t = 3.33 days, just after the storm is over (see Fig. 1b).
Citation: Journal of Physical Oceanography 47, 10; 10.1175/JPO-D-17-0130.1
As in Fig. 3, but at t = 3.33 days, just after the storm is over (see Fig. 1b).
Citation: Journal of Physical Oceanography 47, 10; 10.1175/JPO-D-17-0130.1
As in Fig. 3, but at t = 3.33 days, just after the storm is over (see Fig. 1b).
Citation: Journal of Physical Oceanography 47, 10; 10.1175/JPO-D-17-0130.1
During the storm, the submesoscale variability lacks clear, coherent, vortical structures, but as the storm subsides, a coherent submesoscale cyclonic vortex quickly develops and can be seen by day 3.0 (snapshots at day 3.3 are shown in Fig. 4a). This vortex, which has a strongly stratified core and weakly stratified edges, qualitatively dominates the submesoscale variability after the storm (Fig. 4a). The vortex diameter is quantitatively consistent with the fastest growing MLI length scale (about 1 km), and it emerges on a time scale that is quantitatively consistent with the fastest growing MLI time scale (about half a day). However, the vortex forms during the storm and its growth may be significantly modified by the wind and the associated ageostrophic shear.
The small-scale (<150 m) turbulent kinetic energy exhibits spatial variations of one to two orders of magnitude within the mixed layer during and after the storm, and the pattern of variability of small-scale turbulence is qualitatively similar to the variability in stratification. As a result, strong turbulence penetrates to the mixed layer base in only a small fraction of the domain. Yet, this variability is sufficient to explain why the mixing layer depth HXL, defined using 〈ε〉x,y in Fig. 2a, penetrates deeply into the region of strong mean stratification. The cause of these deep penetrating events is not known but could be due to local interactions between the wind and the submesoscale fronts and filaments.
c. Energetics
The contributions of submesoscales and small-scale turbulence to the kinetic energy can be isolated using energy spectra. Here, we focus on the lower part of the mixed layer by presenting spectra at 30-m depth (about 3/4 of HML after the storm; see Fig. 2a). At this depth, the horizontal and vertical kinetic energy spectra have different slopes at large and small scales (Figs. 5a,b). In addition, the vertical kinetic energy spectra exhibit two local maxima: one at a wavenumber of about 1/1000 cycles per meter (near the fastest growing MLI mode) and one at a wavenumber between 1/50 and 1/100 cycles per meter. This motivates using a cutoff wavenumber kc = 1/150 cycles per meter, near the local minimum in the vertical kinetic energy spectra (see Fig. 5b), to separate large from small scales.
Time-averaged power spectra of (a) horizontal velocity Eh and (b) vertical velocity Eυ at z = −30 m as a function of radial horizontal wavenumber |kh|. Time series of (c) horizontal kinetic energy 
Citation: Journal of Physical Oceanography 47, 10; 10.1175/JPO-D-17-0130.1
Time-averaged power spectra of (a) horizontal velocity Eh and (b) vertical velocity Eυ at z = −30 m as a function of radial horizontal wavenumber |kh|. Time series of (c) horizontal kinetic energy
Citation: Journal of Physical Oceanography 47, 10; 10.1175/JPO-D-17-0130.1
Time-averaged power spectra of (a) horizontal velocity Eh and (b) vertical velocity Eυ at z = −30 m as a function of radial horizontal wavenumber |kh|. Time series of (c) horizontal kinetic energy
Citation: Journal of Physical Oceanography 47, 10; 10.1175/JPO-D-17-0130.1
Large-scale horizontal kinetic energy dominates the total kinetic energy in the wind-forced front. It grows during the storm and decays to about 25% of its late-storm maximum after the end of the storm (Fig. 5c). In contrast, large-scale horizontal kinetic energy rises only slightly in the front forced by a weak air–sea buoyancy flux and decays during forcing in the front forced by a strong air–sea buoyancy flux. Hence, the total kinetic energy is more than 10 times larger during the storm in the wind-forced front than in any of the other three simulations (Figs. 5c,d).
Large-scale vertical kinetic energy is about 10 times larger during the storm than before or after the storm in the simulation with a wind-forced front (Fig. 5d), which is qualitatively consistent with earlier studies that show wind enhances submesoscale vertical motions at fronts (e.g., Mahadevan and Tandon 2006; Thomas et al. 2008). However, the large-scale vertical kinetic energy is also enhanced during the storm in the simulation forced by a strong buoyancy flux (Fig. 5d, dashed red line). Comparing Figs. 5b and 5d, it is evident that the large scales are highly anisotropic at a wind-forced front (blue lines), while strong convective forcing (red lines) causes the flow to become more isotropic (although the large-scale horizontal kinetic energy is still more than 10 times larger than the vertical kinetic energy in this case.)
During the storm, the small-scale turbulent kinetic energy is similar in all three simulations with strong surface forcing (Fig. 5). However, after the storm, small-scale turbulence is less energetic, and small-scale spectral slopes are steeper for the wind-forced front compared to the simulation without the front (Fig. 5), presumably because the submesoscale restratification suppresses small-scale turbulence at 30-m depth in the simulation with the front (see Fig. 4). Yet, small-scale turbulence is more energetic in the large domain during a transition period just after the storm, for example, between days 2.75 and 4 (Figs. 5c,d), which explains why HXL remains deeper than HML after the storm (Fig. 2a) and suggests that mixing can decouple (in time) from wind forcing at fronts (as in Whitt et al. 2017).
4. Conclusions
It has been known for some time that submesoscales can have a significant impact on stratification and small-scale turbulence in the ocean mixed layer. This work expands our understanding of submesoscale dynamics by presenting high-resolution large-eddy simulations that elucidate the interaction between submesoscales and small-scale turbulence during the life cycle of a midlatitude storm. We find that submesoscales persist and even grow during strong winds. Contrary to existing theory and simulation results (Mahadevan et al. 2010), which suggest that submesoscale restratification should be overwhelmed by the destratifying effects of the Ekman buoyancy flux, our simulations show that submesoscales maintain strong mean stratification in the mixed layer even in the midst of strong downfront winds. Despite the strong mean stratification, small-scale turbulence intermittently penetrates to the mixed layer base because of strong modulation of mixed layer stratification on submesoscales. The small-scale turbulent kinetic energy is enhanced in regions of relatively weak stratification, both during and after the storm.
The persistence of strong, stable stratification during the storm, first reported by Rumyantseva et al. (2015) and confirmed here by the LES, challenges the prevailing description of submesoscales. Recent work has framed the description of the mixed layer depth and stratification as a competition between restratification by submesoscales associated with horizontal density gradients and mixing by small-scale turbulence associated with surface forcing (e.g., Mahadevan et al. 2010, 2012; Bachman and Taylor 2016; Taylor 2016). The results here suggest a more nuanced description where winds simultaneously energize small-scale turbulence and submesoscales. Notably, the submesoscale horizontal kinetic energy is significantly enhanced during the storm (see Fig. 5b). Despite the enhanced small-scale turbulence and the large destabilizing Ekman buoyancy flux and large values of the mixed layer buoyancy flux ratio RML, strong stratification persists in localized patches (Fig. 3a). The same level of stratification is not seen in a simulation with the same RML without wind forcing, suggesting that the enhancement of submesoscale activity by wind forcing is important for the evolution of mixed layer stratification.
These results raise several important questions for future work, including the following: Is MLI enhanced by small-scale (<1 km) buoyancy gradients and/or strong Ekman shear? Does the domain size constrain the dynamics of the submesoscales? Do surface waves, which are excluded here, modify the results? Finally, how do the results depend on the chosen parameters, including the horizontal and vertical density gradients, the wind stress, and the air–sea buoyancy flux?
Although only one set of parameters is considered here, this set of parameters is typical of the OSMOSIS site (Thompson et al. 2016) and presumably is relevant to other regions of the ocean. Moreover, the simulated strong stratification during the storm is qualitatively consistent with the observed mixed layer stratification at the OSMOSIS site during the September storm (Rumyantseva et al. 2015). Hence, the results, which challenge our current understanding of submesoscale dynamics, could provide insight into typical ocean conditions during the passage of a storm.
Acknowledgments
The authors thank Baylor Fox-Kemper, Liam Brannigan, and an anonymous reviewer for constructive suggestions. DBW was supported by the National Science Foundation via an NSF Postdoctoral Fellowship (1421125) and an NSF Polar Programs Grant (1501993). JRT was supported by the Natural Environment Research Council via Award NE/J010472/1. We would also like to acknowledge high-performance computing support from Yellowstone (ark:/85065/d7wd3xhc) provided by NCAR’s Computational and Information Systems Laboratory, sponsored by the National Science Foundation. The source code for DIABLO is publicly available (https://github.com/johnryantaylor/DIABLO), and the initial condition data necessary to reproduce the simulation is archived online (Whitt 2017).
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Skyllingstad et al. (2017), which was accepted for publication after the submission of this paper, present several large-eddy simulations of wind-forced fronts, expanding on Thomas et al. (2016). However, the analysis also focuses on domains that are too small to permit MLI.
