## 1. Introduction

The ocean’s main frontal systems, the Gulf Stream, Kuroshio, and Antarctic Circumpolar Current, underlie the midlatitude westerlies. As a consequence, the strongest wind work on the ocean circulation is found in these regions (Wunsch 1998). At the same time, the westerlies tend to lower the potential vorticity of the currents and make the fronts susceptible to symmetric instability (SI), an overturning instability that removes kinetic energy (KE) from the circulation (Thomas 2005; Thomas et al. 2013). Under steady, unidirectional winds, theory and large-eddy simulations (LES) predict that this sink of KE for the circulation scales with the so-called Ekman buoyancy flux, defined as the dot product of the Ekman transport and the surface buoyancy gradient (Thomas and Taylor 2010). Observations of upper-ocean turbulence made in the wind-forced Kuroshio when it was symmetrically unstable revealed enhanced turbulent dissipation at levels consistent with this theoretical prediction (D’Asaro et al. 2011).

While the findings are promising, extrapolating these results to estimate the global net sink of KE attributable to wind-forced SI might be ill-advised for several reasons. Principally, the conditions under which the theoretical prediction of Thomas and Taylor (2010) is formally applicable, that is, steady, unidirectional winds, are rarely met in the ocean’s main frontal systems. Here, the midlatitude westerlies coincide with the storm tracks and variable winds generate strong inertial motions (Alford 2003). How shifts in wind speed and direction and the resultant inertial motions affect the dynamics of symmetrically unstable fronts has not been investigated. A field campaign to the Gulf Stream during the late winter of 2012, described below, provided the ideal conditions to explore this physics. In this article, we will focus on one particular storm event that generated a symmetrically unstable flow with pronounced time variability. After highlighting the key elements of the experiment and methods (section 2), we describe the evolution of the upper ocean during the passage of the storm (section 3) and then present a dynamical explanation by comparing the data with a LES (section 4) and simple stability analysis of a time-dependent, symmetrically unstable flow (section 5).

## 2. Experiment and measurements

The Scalable Lateral Mixing and Coherent Turbulence (LatMix) 2012 field campaign (19 February–17 March 2012) studied submesoscale processes and their effect on mixing in the Gulf Stream and northern Sargasso Sea. For the work described here, two global class research vessels, the R/Vs *Knorr* and *Atlantis* surveyed around a subsurface, neutrally buoyant, acoustically tracked Lagrangian float (e.g., D’Asaro 2003) that was deployed in the middle of the strong front (North Wall) on the northern side of the Gulf Stream (see Fig. 1). The float was tracked using a Trackpoint II short-baseline acoustic tracking system mounted on the R/V *Knorr*. Because of the deep mixed layers, acoustic ray paths remained near the surface for longer distances than in our previous summertime experiments allowing acoustic tracking of the floats at ranges of 5–6 km.

As in D’Asaro et al. (2011), the Lagrangian float provided a reference frame for the measurements. The float moved along the front at an average speed of about 1.4 m s^{−1}. However, there were considerable spatial variations in the flow moving away from the float. Specifically, the velocity was strongly sheared in the horizontal, varying by ~±0.5 m s^{−1} within ±5 km of the track. Temperature and salinity measurements on the float show that the float remained in the front throughout the deployment. During this time, satellite IR images (not shown) illustrate that the front itself moves laterally about ±15 km, several times its own width. Thus, by measuring relative to the float, the effects of both downstream and cross-stream advection were minimized, and changes in frontal properties could be interpreted as temporal changes in a Lagrangian reference frame moving along the axis of the front.^{1} The vertical motion of the float within the boundary layer provided estimates of the turbulence intensity and dissipation rate (e.g., section 3c). The measurements were thus designed to study the properties of boundary layer turbulence within a strong front evolving in time.

Both vessels profiled velocity, salinity, and temperature. Both used 300- and 75-kHz underway ADCPs. Vertical sampling of the two vessels’ ADCPs was identical, spanning the range between 15 and 87 m with 4-m bin size for 300-kHz instruments and between 21.5 and 570 m with 8-m bin size for 75-kHz ADCPs. The 1-min ensemble averages were used, producing along-track resolution of about 0.2 km. Careful alignment of ADCP measurements was performed to minimize aliasing of ship speed into the measured velocities (Firing and Hummon 2010). A Triaxus-towed, undulating profiler collected measurements from the R/V *Knorr*. Triaxus profiled from the sea surface to 250-m depth at vertical speeds of 0.8–1.0 m s^{−1} and typical tow speeds of 6–7 knots (kt; 1 kt = 0.51 m s^{−1}). The profiler carried an extensive payload of physical and bio-optical sensors, including a Seabird SBE 9 plus CTD equipped with dual, pumped temperature (SBE 3plus) and conductivity (SBE 4C) sensors sampled at 24 Hz. Comparisons of pre- and postdeployment laboratory calibrations showed no evidence of sensor drift. Differences in temperature and conductivity sensor response times introduce noise in the derived salinities. Corrections were thus applied for lags introduced by plumbing and by the thermal mass of the conductivity cell (Lueck and Picklo 1990; Morison et al. 1994). The corrected data were time averaged to form 1-Hz time series, and, for the purposes of these analyses, further averaged into 2-m bins for both ascending and descending profiles, which themselves were subsequently averaged to create individual profiles. A moving vessel profiler (MVP) was deployed from the *Atlantis*. The MVP (Rolls Royce MVP 200) is a weighted CTD that freefalls at approximately 3 m s^{−1} and is returned to the surface by a winch. Casts to 200 m were recorded approximately every 800 m as the ship steamed at 8 kt, and only downcasts were used. The CTD data from the MVP is matched for temperature and conductivity cell response times.

This article focuses on data collected during a single float drift (5–9 March, yearday 64–68). Figure 1 shows the tracks of the ships and float during this drift superimposed on an image of the sea surface temperature representative of the conditions at the time of these measurements. The *Knorr* made tight sections closely following the float, while the *Atlantis* crossed a wider swath of the front so as to provide a larger-scale context. Both ships sampled hydrography at nominally 1-km resolution in the horizontal and less than 2 m in the vertical. The observations were made in the upper 200 m of the water column that was deep enough to capture both the surface boundary layer and the top of the pycnocline. Sections were completed on average every 1.3 h on the *Knorr* and 3.3 h on the *Atlantis* so that variability on time scales of an inertial period was well resolved. Note that since the water speeds (2 m s^{−1} was common) were comparable to the ship speeds, and the sections are approximately perpendicular to the front in a frame advected with the flow, they are not perpendicular to the front in the geographical coordinate system presented in Fig. 1.

The sections were transformed into a streamwise coordinate system, where the downstream direction (with velocity component *u* and coordinate *x*) is defined as the speed-weighted average direction of the current on the section. The cross-stream coordinate *y* is defined to be perpendicular to the downstream direction, increasing from the warm to cold side of the front, and is centered on the float. Once the streamwise coordinate was obtained, velocity and density data were mapped to cross-stream sections with a uniform grid by performing a one-dimensional cross-stream objective map at each vertical level. The form of the correlation function used in the mapping was Gaussian, with a RMS width of 1 km.

Air–sea fluxes were estimated using *Knorr* shipboard meteorological measurements and the COARE 3.5 bulk formula (Edson et al. 2013), using the wind speed relative to the mean water velocity between 10 and 30 m. The correction due to using the ocean currents averages −3.8%. The “3.5” modification of the COARE bulk stress calculation algorithm was developed from extensive direct wind stress observations during the Climate Variability and Predictability Program (CLIVAR) Mode Water Dynamics Experiment (CLIMODE; Marshall et al. 2009) in the Gulf Stream system during wintertime—the region and the conditions nearly identical to those experienced during LatMix. The RMS accuracy of COARE 3.5 wind stress estimation is 28.9%, the best among all the COARE variants to date (Edson et al. 2013). Of the two available anemometers (port, starboard), the one least attenuated by the superstructure was chosen. The selection was based on the comparison of anemometer readings for various relative wind directions. The RMS difference in wind speed between the two instruments was 9% with a mean bias of 0.6%. The air–sea buoyancy flux was calculated from heat flux by multiplying the heat flux by the appropriate conversion factor (i.e., 6.0 × 10^{−10} m^{4} s^{−3} W^{−1}). Neglecting buoyancy changes due to evaporation resulted in about 10% change in estimated buoyancy flux over yearday 65–66. Precipitation was negligible.

Strong and variable wind stress due to a rapidly moving low pressure system created a singular upper-ocean response during the 5–9 March measurements (Figs. 1, 4a). The low pressure resulted in intense air–sea fluxes of heat and momentum. At the storm’s peak the wind stress exceeded 1 N m^{−2} and rotated clockwise in time, swinging from the northwest to the southeast. The clockwise rotary nature of the winds, their rapid time evolution, and their downfront component suggest that both inertial motions and SI could be present in the front. In the next section, we describe observational evidence for both types of flows during the drift.

## 3. Evolution of the upper ocean within the Gulf Stream front

Sections of density and the downstream component of the vertical shear ∂*u*/∂*z* from the *Atlantis* are presented in Fig. 2. The front is seen as a region of nearly uniform strong lateral gradient approximately centered on 0 km, the float location. The boundary layer, extending to approximately 40 m (as determined by the vertical extent of the float’s trajectory) with relatively weak stratification compared with the thermocline, nonetheless exhibits both vertical and horizontal density stratification (contours, lower panels), with vertical shear in the downstream velocity in the same sense as the thermal wind shear. The shear and stratification increase through yearday 65.35 (Fig. 2h) and then decrease rapidly so that by yearday 65.61 (Fig. 2j) both the vertical shear and vertical stratification have become weak. This event is the main focus of the analysis presented here.

Figure 3 shows the evolution of stratification *N*^{2} = ∂*b*/∂*z* (where *b* = −*σ*_{θ}*g*/*ρ*_{o} is the buoyancy, *σ*_{θ} and *ρ*_{o} are the potential and reference densities, and *g* is the acceleration due to gravity), shear squared *S*^{2} = (∂*u*/∂*z*)^{2} + (∂*υ*/∂*z*)^{2}, and gradient Richardson number Ri = *N*^{2}/*S*^{2} following the float. Through yearday 65.4, the boundary layer is stably stratified (*N*^{2} ≈ 3 × 10^{−5} s^{−2}) except in the upper 10 m. The float trajectory (Fig. 3, white/gray trajectories) repeatedly cycles across this, indicating that active mixing is occurring to about 40-m depth. The Richardson number (Fig. 3c) is less than 1 but larger than 0.25. From yearday 65.4 to 65.6, the stratification and shear rapidly decrease to establish an unstratified, unsheared boundary layer. The float trajectories repeatedly traverse this layer showing that active mixing extends to about 80 m. The Richardson number remains near 1, except in the upper 20 m where the density is unstable. This pattern persists to about yearday 66.2; over the next day the stratification and shear increase, with much weaker mixing and a shallowing mixed layer. A stratified, actively mixing boundary layer with a Richardson number near 1 is inconsistent with turbulence associated with Kelvin–Helmoltz instability but can be present at fronts that are symmetrically unstable. For such fronts, the boundary layers are not horizontally homogeneous, and steady geostrophic flows can be unstable for Richardson numbers greater than 0.25 since vertical particle motions can avoid KE loss to mixing by moving slantwise along sloping isopycnals (Thomas and Taylor 2010; D’Asaro et al. 2011). Here, the time-dependent forcing and rapid boundary layer deepening near yearday 65.4 allows us to extend these concepts of SI turbulence to the unsteady regime.

### a. Ageostrophic shear and inertial motions in the boundary layer

^{−3}isopycnal surfaces). The vertical shear in the downstream direction (Fig. 4b)

*f*is the Coriolis parameter. Similarly, Fig. 4c shows the section-averaged stratification

Before yearday 65.2 the winds were weak and the section-averaged shear

Early in the record, the variations in shear had a similar time scale to inertial motions modified by the front’s vertical vorticity *ζ* = −∂*u*/∂*y*. Such motions oscillate at the effective inertial frequency *ζ* ≈ 0.6*f* (a value estimated from the cross-front-averaged vorticity in the boundary layer; e.g., Figs. 2a–e), yields an effective inertial period *T*_{i} = 2*π*/*f*_{eff} ≈ 0.6 days. Before yearday 65.6, the maximum and minimum in

*υ*

_{i}/∂

*z*|. If both of these quantities are constant, then the stratification in the boundary layer would follow the simple relationwhere

*φ*is a phase that makes the maxima in

^{−5}s

^{−2},

^{−7}s

^{−2}, and |∂

*υ*

_{i}/∂

*z*| = 0.003 s

^{−1}, we find that the simple physics encapsulated in (2) potentially explains the observed increases and subsequent decreases in shear and stratification through yearday 65.5, a hypothesis that we will explore more fully with the LES. Beyond this time, the inertial model predicts additional increases in both shear and stratification; instead, the boundary layer remains well mixed in density and momentum (Figs. 2, 4b–c). Potential vorticity and turbulence signatures, described in the next two sections, indicate that intense turbulence due to SI likely mixes the vertical shear, thereby quelling the sheared inertial oscillations.

### b. Evidence of a symmetrically unstable flow

*f*+

*ζ*) of the same sign of the Coriolis parameter, and stable stratification (Thomas et al. 2013). All these conditions were met in the Gulf Stream during these measurements. Vertical vorticity and PV were computed assuming that the flow was hydrostatic and two-dimensional, that is, that it did not vary in the downstream direction. Scaling arguments that justify this assumption are described in appendix A. Under these approximations,

*ζ*= −∂

*u*/∂

*y*and the PV isExpressing the PV as a sum of two constituents emphasizes the contrasting roles of vertical vorticity/stratification and baroclinicity (encompassed in the terms labeled

*q*

_{vert}and

*q*

_{bc}, respectively). If the flow is geostrophic, (3) can be simplified towhere the subscript “

*g*” specifies that

*q*

_{g}is associated with the geostrophic flow and with Richardson number Ri

_{B}=

*N*

^{2}/(∂

*u*

_{g}/∂

*z*)

^{2}, where the subscript

*B*denotes balanced, as in geostrophically balanced. Written in this form, (4) implies that a geostrophic flow is symmetrically unstable when its Richardson number drops below the critical value Ri

_{c}= (1 +

*ζ*/

*f*)

^{−1}. This value is typically greater than the threshold for Kelvin–Helmholtz instability (0.25), even for strong currents with cyclonic vorticity like the Gulf Stream (Stone 1966).

Figure 6 shows cross-stream sections of *q*_{g}, *q*, and density. The geostrophic PV in the boundary layer grew progressively more negative through the period of strong mixing (Figs. 6a–d). The vertical vorticity averaged over the top 60 m was mostly cyclonic with the absolute vorticity (*f* + *ζ*) always positive (Figs. 2a–e). The combination of stable to marginal stratification, positive absolute vorticity, and negative *q*_{g} indicates that the geostrophic flow in the boundary layer was symmetrically unstable during the measurement period.

The total PV *q* is similar to *q*_{g} at the start of the drift but diverges over time (Figs. 6f–j), being more negative at yearday 65.35, just before the mixing event, but less negative at 65.61, just after the mixing event. The difference is due to the ageostrophic shear, which increases the total shear before the mixing event and decreases it afterward (e.g., Fig. 4b). This is further evident in time series of the PV’s constituents *q*_{vert} are perfectly compensated by modifications in *q*_{bc} associated with the downstream component of the inertial shear throughout an inertial cycle. This is simply a manifestation of PV conservation when purely advective processes are involved. However, the presence of negative PV in the boundary layer cannot be explained by conservative processes alone and is consistent with removal of PV from the ocean due to atmospheric forcing.

**F**and Lagrangian changes in buoyancy

*Db*/

*Dt*that result in a flux of PV through the sea surface:where

*H*is the depth of the boundary layer,

*B*

_{o}is the air–sea buoyancy flux, and EBF =

**M**

_{e}⋅ ∇

_{h}

*b*is the Ekman buoyancy flux that quantifies changes in buoyancy caused by advection of density by the Ekman transport

**M**

_{e}(Thomas 2005; Thomas and Taylor 2010). The EBF was estimated over the drift using the downstream component of the wind stress

*y*-averaged, cross-stream buoyancy gradient

Both the EBF and buoyancy flux were positive over most of the drift, indicating that the atmospheric forcing was in the sense to reduce the PV in the boundary layer (Fig. 7a). Between yearday 65–67, the ocean was cooled by the atmosphere, with an average heat and buoyancy loss of 580 W m^{−2} and 7.0 × 10^{−7} m^{2} s^{−3}, respectively. During this same period, the EBF was on average positive with a mean value of 3.5 × 10^{−7} m^{2} s^{−3}; however, it experienced considerable temporal variability. For example, the EBF peaked at a value of 3.3 × 10^{−6} m^{2} s^{−3} near yearday 65.3 after ramping up from zero over a period of hours (Fig. 7a). These fluctuations in the EBF were caused primarily by changes in the wind not the front. Given the mean values of the EBF and air–sea buoyancy flux, and the observed mixed layer depth, the scaling (6) suggests that decreases in PV of order 1.0 × 10^{−9} s^{−3} in 0.5 days are to be expected. Changes in

### c. Characteristics of the boundary layer turbulence

*H*is twice the average float depth (Fig. 8b). The depth-averaged dissipation

*ε*

_{B}= 0.6

*B*

_{o}(Shay and Gregg 1986). This makes a small contribution to the overall dissipation (Fig. 8a, green line). Estimating the wind and wave contributions is more difficult as the dynamics of this forcing is still not well understood (D’Asaro 2014). Traditionally, the dissipation in the interior of a wind- and wave-driven boundary layer scales with

*τ*and water density

*ρ*. Higher values of dissipation, not sampled well by the float, are found in a wave-forced surface layer (Lombardo and Gregg 1989; Drennan et al. 1996) with dissipation rates decaying rapidly with depth. Furthermore, surface wave forcing through Stokes drift also does not scale exactly with

*A*and

*n*using the dataset from D’Asaro et al. (2014). These data have similar winds as at the data here but have little influence from fronts. Dissipation is computed using the same float-based methods. Figure 9 shows the results using the parameterizationThis yields an estimate of the depth-integrated dissipation rate due to wind and wave forcing (Fig. 8a, blue line). The buoyancy and wind forcing are summed to get the overall effect of air–sea forcing (red line). Although the proper way to combine these two effects is not well known, the uncertainty introduced by this is small since the buoyancy contribution is small. The sum roughly matches the overall shape of the dissipation curve, but on average falls below the measured values by about two standard deviations of the accuracy of (9). This suggests that additional forcing of the boundary layer may be needed. Simulations of steady wind-driven SI predict a boundary layer–averaged dissipation associated with SI turbulence of half the Ekman buoyancy flux (Thomas and Taylor 2010). Data similar to that shown here (D’Asaro et al. 2011) suggest that this mechanism explains excess dissipation observed in the Kuroshio front. The pattern of EBF (Fig. 8a, cyan line) is similar to that of the wind forcing but with a smaller magnitude. The sum of wind, buoyancy, and EBF forcing (Fig. 8a, magenta dots) matches the measurements within the estimated errors for most of the data, suggesting that here, as in the Kuroshio data, SI could explain the excess dissipation. This is tempered by the uncertainty in how to combine the three contributions. During the dissipation peak near yearday 65.4, the measured dissipation is clearly larger than that predicted by air–sea fluxes alone and also larger than predicted by air–sea fluxes and EBF. This additionally suggests that the unsteady aspects of SI could be important near the peak.

Thus, a semiempirical comparison of the observed dissipation with that expected from air–sea forcing alone and that expected from SI suggests that steady SI makes a significant (30%–50%) contribution away from the mixing event. Additional dissipation at the mixing event could be due to unsteady SI. We investigate these hypotheses further using an LES of a symmetrically unstable flow in unsteady conditions, as described in the next section.

## 4. Large-eddy simulation

### a. Model description

To examine how inertial oscillations might modify symmetric instability in the Gulf Stream, we conducted a series of large-eddy simulations. The numerical method and setup of these simulations is very similar to simulations that have been previously used to study symmetric instability in the Gulf Stream (Thomas et al. 2013). In particular, the code is fully nonhydrostatic and uses a modified constant Smagorinsky scheme to model the subgrid-scale fluxes, second-order finite differences in the vertical direction, a pseudospectral method in both horizontal directions, and a third-order accurate mixed implicit/explicit Crank–Nicolson/Runge–Kutta time-stepping algorithm. For details of the numerical method, see Taylor (2008).

The LES model is run in a “frontal zone” configuration with a prescribed background horizontal buoyancy gradient used previously in similar studies (e.g., Thomas et al. 2013; Taylor and Ferrari 2010). The departure from this background density, and all other quantities are periodic in both horizontal directions. The simulation parameters are given in Table 1. The computational domain size is 1 km in the cross-front direction, 500 m in the alongfront direction, and 120 m in the vertical, and a sponge-damping region is placed in the bottom 10 m of the computational domain to prevent spurious reflections of downward-propagating internal gravity waves.

Parameters for the large-eddy simulation.

The model is initialized with a stable density profile chosen to approximate observed conditions. The buoyancy profile is set so that the Richardson number of the geostrophic flow Ri_{B} is a piecewise linear function increasing with depth. In particular, from 0 < *z* < −80 m, Ri_{B} increases linearly from 0 at the surface to 1.5 at −80-m depth. From −80 < *z* < −120 m, Ri_{B} increases linearly again from 1.5 to 5, and Ri_{B} = 5 for *z* < −120 m. Note that since the mean vertical component of the relative vorticity is zero in the simulations due to the periodic boundary conditions, a portion of the upper layer is unstable to symmetric instability with Ri_{B} < 1. However, the LES does not capture a number of other physical processes that are likely to be important at the observational site. The alongfront domain size is too small to permit baroclinic instability, there is no horizontal shear associated with the initial flow, and the influence of surface gravity waves is not included. Therefore, although the LES allows us to examine the influence of high-frequency forcing and inertial oscillations on developing symmetric instability, it excludes baroclinic and barotropic instabilities and Langmuir turbulence.

### b. Comparison to observations

To compare more directly with the observations, two simulations (with and without a front) have been run forced with the observed surface wind stress and buoyancy flux (see Figs. 4a, 7a). The first simulation includes a background buoyancy gradient with *M*^{2} ≡ −∂*b*/∂*y* = 5 × 10^{−7} s^{−2}, while the second does not (*M*^{2} = 0). By comparing the two simulations, we can directly diagnose the influence of the front on the dynamical response. Both simulations are initialized at yearday 64.5, which was during a period of relatively weak forcing. This gives the simulations time to spin up before the strong storm that arrived at yearday 65.

Figure 10 shows time series of the stratification and shear averaged across the horizontal extent of the domain and from −60 < *z* −5 m. The upper 5 m was excluded from the average to compare more directly with observations and to exclude a thin boundary layer that forms in the simulations in response to the subgrid-scale LES viscosity. For comparison, the observed mean stratification and shear are also shown.

The agreement between the simulation with *M*^{2} = 5 × 10^{−7} s^{−2} and the observations is remarkable, particularly considering that aside from prescribing the initial density profile and forcing, the model is not tuned in any way to match the observations. A number of key features are accurately reproduced in the simulation. The shear and stratification both increase dramatically at yearday 65.25 in response to the strong wind forcing. By yearday 65.5, the stratification and shear are almost entirely eliminated in the upper 50 m. Then, the stratification and shear gradually return over the course of about a day. Notably, the simulation without a background front does not exhibit this restratification and increase in shear, suggesting that frontal dynamics are responsible for the restratification. We will now analyze the model output in more detail to quantify the roles of inertial motions and symmetric instability in modulating the stratification and energetics of the turbulence in the boundary layer.

### c. Wind-driven inertial motions in the boundary layer

*π*/

*f*= 0.78 days.

^{2}To determine if this is the case, we solved the equations governing the dynamics of wind-forced inertial motions averaged over the boundary layer depth

*H*:Solutions to (10) and (11) forced by the observed winds are compared to the ageostrophic flow from the LES in Fig. 11. We used a value of

*H*= 90 m and an initial condition of

*U*

_{i}=

*V*

_{i}= 0 at yearday 64.5 for the calculation. The good agreement in amplitude and phasing between this simple model and the LES suggests that the oscillations are wind-forced inertial motions. However, the model cannot capture the vertical variations of the inertial motions, which are pronounced especially earlier in the record and can affect the stratification in the boundary layer as described in section 3a and schematized in Fig. 5.

### d. Stratification budget

*M*

^{2}≠ 0 and are illustrated in Fig. 12. As with a typical mixed layer, differential mixing of buoyancy (N2MIX) is important. However, the rate of change in stratification follows more closely the differential horizontal advection of buoyancy (DHADV), indicating that the lateral density gradient of the front and inertial shear play an essential role in the restratification and destratification of the boundary layer in contrast to a standard mixed layer model. Lateral advection generally contributes to an increase in stratification, with one important exception: Just prior to the minimum in stratification near yearday 65.5, DHADV reduces the stratification at a rate greater than N2MIX, indicating that mixing alone cannot explain the destratification of the boundary layer at that time. In terms of the PV and its constituents,

*q*

_{vert}and

*q*

_{bc}[e.g., (3)], the reduction of the stratification and

*q*

_{vert}by DHADV at this time must be compensated by an increase in

*q*

_{bc}associated with inertial shear. Indeed, as illustrated in Fig. 11, near yearday 65.5 the inertial shear in the downstream direction is negative, which opposes the thermal wind shear and increases

*q*

_{bc}.

### e. Energetics of boundary layer turbulence

As in the observations, the storm that occurred during yearday 65 generated intense turbulence in the LES. Figure 13 shows a time series of the kinetic energy dissipation rate *ε*, diagnosed from the LES (solid blue line). For comparison, the average dissipation rate estimated from the vertical acceleration of the Lagrangian float is shown in blue circles, along with the 95% confidence intervals. During the storm peak, the LES dissipation rate agrees very well with the observations. Following the peak storm, from yearday 65.6–67, the LES dissipation rate is consistently smaller than the observations. Note that the LES neglects a number of physical processes, notably surface wave breaking and Langmuir turbulence that might contribute additional dissipation. Nevertheless, we can use the LES results to diagnose the sources and sinks of turbulent KE.

The depth-averaged AGSP and BFLUX in the simulation with *M*^{2} = 0 are also shown in Fig. 13 for comparison. It is evident that in addition to providing a new source of TKE production through the GSP, the front also significantly modifies the AGSP and BFLUX. In the simulation with a front (*M*^{2} = 5 × 10^{−7} s^{−2}), the maximum AGSP near yearday 65.3 is significantly enhanced relative to the simulation without a front. The development of stratification at the front concentrates the wind-driven shear in a relatively thin layer in the early stages of the storm, which appears to enhance the mean AGSP. Without the development of near-surface stratification, the simulation without a front also does not exhibit strong mixing (negative BFLUX) near yearday 65.3. In the later stages of the storm, following yearday 65.5, the AGSP remains positive in the simulation without a front, while it becomes a net sink of TKE in the simulation with a front. This highlights the qualitative change in the dominant energy pathways caused by the presence of a front as diagnosed from the LES. It should be noted, however, that near the peak of the storm the values of dissipation from the LES with and without a front are both consistent with the observed dissipation within the error bars of the estimate. Comparing this result to the findings illustrated in Fig. 10 suggests that while frontal dynamics is of secondary importance to the overall energy budget of the turbulence, it is critical to the evolution of the mean stratification and shear.

*H*is the depth of the layer with zero or negative PV. It then assumes that the buoyancy flux is a linear function of depth inside the so-called convective layer of thickness

*h*, defined by Taylor and Ferrari (2010), and zero belowUsing (16), the GSP can thus be parameterized asWhile (18) does not hold instantaneously, it may be valid in a time-averaged sense. Equation (16), which forms the basis of the parameterization, was derived based on a steady, turbulent Ekman balance, where accelerations are neglected. If the dominant acceleration is due to inertial motions, and if the time averaging window is longer than the inertial period, then the mean acceleration could be small, even if it is large instantaneously. If so, (18) could be skillful at predicting the time-mean GSP. To test this, the terms in the TKE equation were diagnosed from the LES, averaged in time, and compared to the predictions (16)–(18). However, to do so requires an estimate for the convective layer depth

*h*.

Taylor and Ferrari (2010) derived a scaling for *h*. They found that for steady forcing, turbulence driven by convection and downfront winds maintained a well-mixed layer for *z* > −*h*. When *h* was shallower than the layer with zero or negative PV (of thickness *H*), SI formed in the region −*h* < *z* < −*H*. Although the scaling derived in Taylor and Ferrari (2010) was for steady forcing, it is insightful to apply the scaling using the instantaneous surface wind and buoyancy flux. Here, *H* was diagnosed as the deepest location where Ri_{B} < 2.5.

The upper panel in Fig. 14 shows the time evolution of the horizontally averaged squared buoyancy frequency *N*^{2} from the LES with a front. This panel can be compared with the observed *N*^{2} time series in Fig. 3b, which shows many similar features. Notably, a region with very low stratification develops after the storm, starting from about 65.5, extending to a depth of approximately 75 m. The stratification then redevelops, starting at depth near the start of yearday 66 with the stable region extending increasingly higher in the water column. The weakly stratified region coincides with the low geostrophic Richardson number (Fig. 14, middle panel). Starting from about yearday 66.5, most of the boundary layer has developed a stable stratification with Ri_{B} ≃ 1, indicating a neutral state with respect to symmetric instability.

Taylor and Ferrari (2010) defined the convective layer as the region with a positive buoyancy flux *N*^{2} < 0 shown in purple in the top panel of Fig. 14.

The time-averaged EBF and air–sea buoyancy flux were used to predict the time-averaged convective layer depth and construct the parameterizations (17)–(18) that were compared to the time-mean GSP and BFLUX diagnosed from the LES (Fig. 15). The average covered three inertial periods starting at yearday 65. The buoyancy flux is positive in the upper 20 m, indicating that potential energy is converted to kinetic energy (i.e., convection) on average over these depths. The GSP is the dominant source of TKE, indicating that the thermal wind shear associated with the Gulf Stream provides most of the turbulent kinetic energy and, in turn, dissipation. The parameterizations (16)–(18) (dashed lines in the figure) match the LES results remarkably well, suggesting that they provide a skillful prediction for the time-averaged energy exchange terms associated with SI even when the surface forcing is strongly time dependent.

## 5. Transient energetics of symmetric instability in the presence of inertial shear

While the time-averaged energetics of SI under variable winds can be described by the theoretical scalings for steady forcing, the transient energetics of SI deviate significantly from the predictions. This is particularly evident near yearday 65.5 when the GSP in the LES reached its maximum value while the EBF dropped to a minimum (e.g., Fig. 13). Consistent with this discrepancy, around this time the float-based dissipation estimates exceeded the prediction of the theoretical scalings (Fig. 8). In this section, we explore how inertial shear in a symmetrically unstable front can influence the energetics of SI and potentially explain this discrepancy. To this end we performed a linear stability analysis on a basic state that captures the key features of the Gulf Stream front, that is, a flow with negative PV, stable stratification, and inertial shear.

### a. Basic state

*ϕ*are constants.

*x*) direction is assumed to scale with the thermal wind shear, differing by a factor of

*γ*. The lateral shear and hence vertical vorticity

*ζ*

_{g}of the geostrophic flow modifies the frequency of the inertial oscillation, shifting it from

*f*to

*f*

_{eff}. Changes in stratification are caused by the cross-front shear of the inertial oscillation that differentially advects buoyancy, as illustrated in Fig. 5. While the stratification, shear, and Richardson number change with time, the Ertel PV remains constant and equal toas required by PV conservation. It is important to note that at a front, the presence of an inertial oscillation can affect the value of the PV. The reason for this is that the horizontal component of the vorticity associated with the oscillation can project into the horizontal buoyancy gradient of the front. How large of a contribution this is depends on both the strength of the inertial shear (i.e.,

*γ*) and the phase of the oscillation

*ϕ*. If at

*t*= 0 the inertial shear is entirely in the downfront direction (i.e.,

*ϕ*= 0), then the PV is reduced relative to the case with no inertial oscillation since the inertial and thermal wind shears add. When the inertial shear is entirely cross-front at

*t*= 0, that is,

*ϕ*=

*π*/2, then the PV is unaffected by the oscillation. The fact that the PV depends on the properties of the inertial oscillation at the initial time might seem like a theoretical construct. However, the contribution to the PV from the inertial oscillation can be interpreted physically as the PV anomaly generated by the impulsive, presumably wind-driven, frictional torque needed to accelerate the horizontal component of the vorticity of the inertial motion by

*t*= 0+. With this interpretation, the phase

*ϕ*is determined by the direction of the impulsive force relative to the front, for example, if the force is downfront

*ϕ*= 0, while if it is upfront

*ϕ*=

*π*.

### b. Stability analysis

The basic state is perturbed with a 2D (i.e., invariant in the *x* direction) disturbance, with velocity, buoyancy, and pressure fields **u**′(*y*, *z*, *t*), *b*′(*y*, *z*, *t*), and *p*′(*y*, *z*, *t*). The perturbation that we investigate is characterized by streamlines in the *y*–*z* plane that run parallel to isopycnals and corresponds to the fastest growing mode for SI in a basic state with no inertial shear (i.e., *γ* = 0). The method of solving the evolution of the perturbations is described in appendix B. A basic state with parameters representative of the observations from the Gulf Stream, that is, ^{−10} s^{−3}, *M*^{2} = 5 × 10^{−7} s^{−2}, *f* = 9.2 × 10^{−5} s^{−1}, *ζ*_{g} – 0.6*f*, and *γ* = 0.67, is used in the calculation.

A time series of the kinetic energy per unit mass *γ* = 0) yet with the same PV (^{−10} s^{−3}) and geostrophic shear (*M*^{2} = 5 × 10^{−7} s^{−2}) but lower stratification (^{−5} s^{−2}) is also shown in the figure for comparison and exhibits exponential growth. Comparing the evolution of the KE for the two basic states reveals that SI in an inertial oscillation experiences periods of explosive growth. These occur at times when the stratification approaches its minimum (e.g., near *t* = 0.3 and 0.9 days), resulting in a 40-fold increase in KE in a tenth of a day. If the perturbation were allowed to develop secondary instabilities and turbulence, then presumably the period of explosive growth would correspond to a peak in turbulent dissipation. With these considerations in mind, we can interpret the timing of the maximum excess dissipation near yearday 65.5 seen in the observations (Fig. 8a) as being caused by a rapid growth of SI during the weakening stratification at this time (Fig. 4c).

**u**′ with (B1). KE can be changed by convergences/divergences of the energy flux (PWORK) and the release of potential energy via the BFLUX. The disturbances can also exchange KE with the background flow through shear production. In fact, given that SI does not induce pressure and buoyancy anomalies, the only way to change its KE is through shear production. The shear production is further decomposed into its geostrophic and ageostrophic parts (GSP and AGSP, respectively) with the latter representing the rate of KE extraction from the inertial oscillation.

During the period of explosive growth near 0.3 and 0.9 days, GSP > 0, while AGSP < 0, indicating that SI gains KE from the geostrophic flow while losing KE to the inertial oscillation (Fig. 16c). From this we can conclude that the enhanced growth is not associated with an extra energy source from the inertial shear. Instead, this difference in growth can be attributed to the temporal modulation of the stratification and GSP. In particular, the GSP intensifies as the stratification weakens. During these times isopycnals and hence perturbation streamlines steepen, leading to stronger vertical velocities and momentum fluxes and an amplification of the GSP. With this physics in mind, we interpret the maximum in GSP near yearday 65.5 seen in the LES (Fig. 13) as resulting from the interplay of inertial shear and the front that tilts isopycnals, reducing the stratification through differential horizontal advection, DHAV (Fig. 12), and leading to a more efficient extraction of KE from the geostrophic flow by SI.

## 6. Summary and discussion

Observations from the North Wall of the Gulf Stream made during the passage of a storm revealed a symmetrically unstable flow superposed with strongly sheared inertial motions. The event could be described in three phases: an initial phase where the stratification oscillated in time, a middle period where density and momentum in the boundary layer were well mixed, and a latter phase where the stratification and frontal vertical shear were restored to prestorm values. Turbulent dissipation estimates from a Lagrangian float cycling in the boundary layer were elevated relative to the expected TKE production by wind and air–sea buoyancy fluxes, implying that the frontal currents were an additional source of energy that was being tapped by SI. During the oscillatory stratification phase, however, the observed excess dissipation was significantly larger than that predicted by theoretical scalings for the energetics of SI under steady conditions.

The observational findings were interpreted using an LES configured with forcing and frontal characteristics taken from the observations and a linear stability analysis of a symmetrically unstable flow interacting with inertial motions. The LES illustrates how differential horizontal advection of buoyancy by inertial shear generated the oscillations in stratification during the initial passage of the storm. A stability analysis shows that at the phase of the oscillation when the stratification approaches its minimum, SI experiences explosive growth, extracting KE from the frontal flow at an enhanced rate relative to SI in steady conditions. This result is played out in the LES and might explain the excess dissipation seen in the observations during the period of oscillatory stratification.

While the energetics of SI driven by variable winds and interacting with inertial motions is transient, averaged over several inertial periods it is well predicted by parameterizations based on the theory developed for steady forcing. This suggests that these parameterizations could be used to estimate the global net sink of the ocean circulation’s KE by SI using wind fields averaged over a few inertial periods. Starting on a smaller scale, we attempt to assess the importance of the process to the overall energetics of the Gulf Stream. The time mean of the GSP averaged over the upper 50 m from the LES at the peak of the storm, that is, between yearday 65.3–65.5, is 2.3 × 10^{−6} m^{2} s^{−3} (Fig. 13). We can compare this to the baroclinic kinetic energy associated with the thermal wind shear, that is, ^{−7} s^{−2}, the baroclinic kinetic energy over the upper 50 m is 0.037 m^{2} s^{−2}. Without a source of energy to maintain the mean flow, the level of GSP during the peak of the storm would be able to entirely eliminate the thermal wind shear in less than 4.5 h. This is close to the duration of the period of intensified GSP, suggesting that SI could explain the near homogenization of momentum in the boundary layer subsequent to the initial passage of the storm.

After the boundary layer was mixed, the thermal wind shear and stratification were restored to prestorm values. The fact that the stratification remained relatively weak and the Richardson number near one suggests that the submesoscale mixed layer baroclinic instability (MLI) was not dominant during the drift (Boccaletti et al. 2007; Fox-Kemper et al. 2008). Indeed, the close correspondence between the observations and LES, which is not capable of simulating baroclinic instability, further supports this inference. It is not obvious why the front did not show clear signs of restratification by finite-amplitude MLI. However, it could simply be that the duration of the drift was not long enough for the effects of finite-amplitude MLI to be noticeable. For example, for 0 < Ri_{B} < 1, the *e*-folding time corresponding to the growth rate of the fastest growing mode of ageostrophic baroclinic instability is between 12 and 17 h (Stone 1970). Simulations of MLI at fronts with initial Richardson numbers in this range show that it takes several days, that is, longer than the duration of the drift, for finite-amplitude MLI to increase the mixed layer stratification beyond what is attributable to SI (e.g., Fig. 3 of Fox-Kemper et al. 2008).

The temporal evolution of the stratification and shear toward the end of the drift is reminiscent of geostrophic adjustment at a front, a problem that has been studied theoretically, primarily in the inviscid, adiabatic limit (e.g., Ou 1984; Tandon and Garrett 1994; Shakespeare and Taylor 2013). In this limit, PV conservation and geostrophy constrain the value of the time-mean stratification and shear, and inertial motions drive oscillations about this mean. The observations indicate, however, that PV is not conserved and changes sign over time (e.g., Figs. 7b, 8c). Furthermore in the latter phase of the record, the downstream shear asymptotes toward, rather than oscillates about, the thermal wind balance, suggesting that any sheared inertial motions that were present were damped (Fig. 10). These differences from the inviscid, adiabatic theory are likely attributable to SI that drives turbulence and entrains high PV water from the pycnocline into the boundary layer. A detailed study of geostrophic adjustment in a symmetrically unstable flow is beyond the scope of this work but will be the subject of future research.

## Acknowledgments

We are grateful to Jason Gobat, Eric Boget, Dave Winkel, Ben Jokinen, Mike Johnson, and the captains and crews of the R/Vs *Knorr* and *Atlantis* who made the collection of these observations in challenging wintertime conditions possible. This work was supported by ONR Grants N00014-09-1-0202 under the Scalable Lateral Mixing and Coherent Turbulence Departmental Research Initiative. Additional support came from ONR Grants N00014-09-1-0266 (C.M.L. and A.S.) and N00014-09-1-0172 (E.A.D. and A.S.).

## APPENDIX A

### Estimates of Downstream Variability

*υ*associated with along-stream variations that would not be seen in the LES. In the top panel of Fig. A1, we compare

*υ*from the

*Knorr*and

*Atlantis*observations and the LES. The standard deviations of

*υ*in the

*Knorr*observations and the LES results are the same, that is, 0.12 m s

^{−1}, while the standard deviation from the

*Atlantis*observations is 0.17 m s

^{−1}. This suggests that most of the variance in

*υ*is explained by inertial motions not balanced motions since the cross-stream velocity in the LES is dominated by the former. If the excess variance in the

*Atlantis*observations, ~0.05 m s

^{−1}, were associated with along-stream variations, and if the flow were isotropic ∂/∂

*x*~ ∂/∂

*y*~ 1/(10 km), then this would result in a vertical vorticity of ∂

*υ*/∂

*x*~ 1 × 10

^{−5}s

^{−1}that is an order of magnitude smaller than the vertical vorticity associated with the downstream component of the velocity −∂

*u*/∂

*y*. This is likely an upper bound for ∂

*υ*/∂

*x*since the flow at a front is far from isotropic, that is the characteristic length scale of the flow in the cross-stream direction is much smaller than that in the along-stream direction.

The second term in *q*_{res} involves a buoyancy gradient in the along-stream direction ∂*b*/∂*x*. If there were such a gradient, then it would be associated with a thermal wind shear in the cross-stream direction ∂*υ*_{g}/∂*z* = (1/*f*)∂*b*/∂*x*, which should be detectable in the observations of ∂*υ*/∂*z*. The histogram of ∂*υ*/∂*z* from the *Knorr* observations is shown in the bottom panel of Fig. A1. The mean of the distribution is 0.0016 s^{−1}. If this mean value were attributed to a flow in thermal wind balance it would correspond to an along-stream buoyancy gradient of 1.5 × 10^{−7} s^{−2}, which is ~⅓ the strength of the cross-front buoyancy gradient. If so, the contribution to the PV from this thermal wind shear would be −(1/*f*)(∂*b*/∂*x*)^{2} ~ −1 × 10^{−10} s^{−3}, which is an order of magnitude weaker than the PV anomaly associated with the cross-stream buoyancy gradient (e.g.,

In summary, we estimate that the terms in the PV and the vertical vorticity associated with alongfront variability are an order of magnitude weaker than the terms that we retained in our 2D approximation, and thus it is justifiable to neglect them.

## APPENDIX B

### Governing Equations and Solution Technique for Stability Analysis

**u**′(

*y*,

*z*,

*t*),

*b*′(

*y*,

*z*,

*t*), and

*p*′(

*y*,

*z*,

*t*) are governed by the incompressible, Boussinesq equations:where

*y*–

*z*plane can be expressed in terms of a streamfunction, that is,

*υ*′ = ∂

*ψ*/∂

*z*and

*w*′ = −∂

*ψ*/∂

*y*. Because of the lack of boundaries, and because of the spatially uniform gradients of the basic state, the method of Craik (1989) can be employed, that is, solutions of the form of plane wavesare sought, where

*φ*=

*ly*+

*mz*is the phase, and

**k**= (

*l*,

*m*) is the wavevector that is spatially uniform, yet varies with time, and where c.c. denotes the complex conjugate. The evolution of only a single plane wave is considered, which makes the nonlinear terms in (B1)–(B2) identically equal to zero. With the ansatz (B4), it follows that the phase does not change following the background flow, that is,

*Dφ*/

*Dt*= 0, yielding the following solution for the wavevector:where (

*l*

_{o},

*m*

_{o}) denotes its initial value.

**a**= [

*U*Ψ

*B*]

^{T}:where (

^{.}) denotes a time derivative and the matrix

*γ*= 0), a geostrophic background flow of the form (19) is symmetrically unstable when

*l*/

*m*=

*M*

^{2}/

*N*

^{2}(Taylor and Ferrari 2009). The effects of inertial oscillations on the dynamics of this particular mode for

*γ*≠ 0 can be explored by choosing initial components of the wavevector that satisfy the following relation:which forces −

*l*/

*m*=

*M*

^{2}/

*N*

^{2}for all times. With this initial condition it follows that

*E*

_{32}= 0, and thus the buoyancy anomaly of the perturbation is zero, that is,

*B*(

*t*) = 0. The evolution of the amplitude of the perturbation is governed bywhere the matrix

*F*

_{11}=

*E*

_{11},

*F*

_{12}=

*E*

_{12},

*F*

_{21}=

*E*

_{21}, and

*F*

_{22}=

*E*

_{22}and was solved numerically.

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