## 1. Introduction

The upper ocean with widespread fronts and filaments provides a conducive environment for the generation of SMS (typically between 0.1- and 10-km horizontal scale) flows. Fronts are elongated regions with lateral density gradient between two distinct water masses. A filament, on the other hand, represents an elongated water mass confined in surrounding water of different density; the lateral edges of the filament that separate the two water masses can be considered as fronts when the filament is wide enough that the edges evolve independently. The lateral density gradient in these systems is associated with stored available potential energy, which is released when instabilities, specifically baroclinic mixed layer instabilities (MLI), develop in them. The released potential energy thereby energizes SMS instabilities and motions, and the front restratifies. The upper ocean is also a host for finescale 3D turbulence, driven by surface forcing such as wind, waves, and cooling, which is responsible for mixing and homogenizing the surface layer. The SMS is largely quasi-2D but as the length scale within the SMS decreases so does rotational control since the local Rossby number increases to *O*(1). The two regimes—the quasi-2D submesoscale and the 3D finescale—thus have opposing tendencies but, acting together, they influence the upper-ocean properties through which the interior ocean and the atmosphere interact. In the present work, we focus on obtaining a detailed understanding of the two-way coupling between SMS motions originating from mixed layer baroclinic instability and convective boundary layer turbulence. Baroclinic instability (BI) growing at upper-ocean fronts, also known as mixed layer instability (MLI), is an important mechanism for generating SMS flows at 0.1–10 km in the horizontal and with time scale of the order of 1 day (Boccaletti et al. 2007; Capet et al. 2008a; Thomas et al. 2008; McWilliams 2016). Observations and models have revealed some interesting features of the dynamics (e.g., Mahadevan and Tandon 2006; Fox-Kemper et al. 2008; Özgökmen et al. 2011; Shcherbina et al. 2013; Stamper and Taylor 2017; D’Asaro et al. 2018; Verma et al. 2019) during the nonlinear evolution of MLI. Coherent SMS eddies develop as do coherent filaments which have convergent flow, significant local imbalance, large vertical vorticity and significant vertical velocity. In turbulence-resolving large-eddy simulations (LES) with *O*(1)-m grid size (e.g., Verma et al. 2019), the filaments are as thin as *O*(100) m, develop cyclonic vertical vorticity as large as *O*(20*f*), have instantaneous vertical velocity as large as 20 mm s^{−1}, and have sustained downward transport of Lagrangian tracer from the heavy side (Verma and Sarkar 2021). These filaments resemble the filamentary structures with large cyclonic vorticity and large downwelling velocity found in the ocean (D’Asaro et al. 2018) and the vorticity is positively skewed (e.g., Shcherbina et al. 2013). Despite their significant role in transport, the manner in which these coherent structures respond to boundary layer turbulence is not fully understood.

Recent observations in the northeast Atlantic ocean indicate strong seasonal variability of SMS motions in the mixed layer associated with the variability in surface forcing (Thompson et al. 2016; Buckingham et al. 2016; Yu et al. 2021). During fall, convective turbulence deepens the mixed layer suggesting that surface cooling dominates SMS restratification processes. During winter, the mixed layer depth is more variable indicating the dominance of restratification by SMS flow over convection driven by either surface cooling or Ekman buoyancy flux. The potential vorticity (PV) in the mixed layer during the late fall and winter shows significant negative values and the authors suggest occurrences of MLI and symmetric instability (SI) (Thompson et al. 2016). Mooring data from Yu et al. (2021) suggest strong SMS dynamics during winter with the mixed layer having two distinct zones: a near-surface convective layer where gravitational instability dominates and an underlying forced-SI layer in which SI is more prevalent. These new observations highlight a possibly strong coupling between SMS currents and surface cooling flux and motivate the present study.

There have been several theoretical studies that investigate the interaction between the SMS and the FS components (Shakespeare and Taylor 2013; Haney et al. 2015; McWilliams et al. 2015; Bodner and Fox-Kemper 2020; Crowe and Taylor 2020). The interaction between these two components has been numerically captured by recent turbulence resolving simulations (Hamlington et al. 2014; Whitt and Taylor 2017; Sullivan and McWilliams 2018, 2019; Verma et al. 2019; Skyllingstad and Samelson 2020). One aim of the present study is to investigate how surface cooling influences the energy interaction between these two components. Surface forcing can extract PV from the front that may assist the growth of forced SI (Thomas et al. 2013; D’Asaro et al. 2011; Ramachandran et al. 2018; Wenegrat et al. 2018). Several numerical models with strong winds and/or surface cooling indicate persistence of MLI, which maintains a stable stratification even in the presence of strong boundary layer turbulence (Haine and Marshall 1998; Fox-Kemper et al. 2008; Hamlington et al. 2014; Whitt and Taylor 2017; Callies and Ferrari 2018; Skyllingstad and Samelson 2020). In the presence of boundary layer turbulence, the vertical turbulent mixing of momentum plays a role to establish the ageostrophic secondary circulation (ASC) as per turbulent thermal wind (TTW) balance (McWilliams et al. 2015; Wenegrat and McPhaden 2016). At the small scales of LES, cross-front gradients of turbulent stresses come into play, e.g., for a dense filament (Sullivan and McWilliams 2018) or for the formation of gravity currents at strong fronts (Pham and Sarkar 2018). In the present study, we impose a surface cooling flux on an upper-ocean front and generate convective turbulence of varying strength. Gravitational instability gives rise to turbulent plumes that drive a positive vertical buoyancy flux to create a mixed layer. The plumes typically have comparable length scales in the horizontal and vertical, and have a characteristic velocity of (*B _{s}H*)

^{1/3}, where

*B*is the surface buoyancy flux and

_{s}*H*is the depth of the mixed layer (Deardorff 1980). In upper-ocean fronts with evolving SMS motions, there are several reasons for a two-way coupling between convection and SMS flows which can influence dynamics of both regimes. First, restratification by SMS currents can suppress the deepening of the mixed layer, albeit possibly with variation between coherent structures and the rest of the front. Second, convective turbulence can mix away the vertical gradient of thermal wind or enhance horizontal mixing so as to disrupt the frontal balance and modify the SMS motions. Third, surface cooling can enable conditions for the growth of forced instabilities, such as SI, by extracting potential vorticity from the front. These instabilities may suppress MLI, or they may coexist with MLI. Fourth, convection within the front occurs in the presence of thermal wind shear so that the properties of turbulence inside the front can change considerably relative to outside. We will quantitatively examine each of the aforementioned aspects of the SMS/convection coupling in the present paper. To do so, a suitable framework that decomposes the flow into SMS and FS components is beneficial. We adopt here the approach of Verma et al. (2019) for the flow decomposition using a 2D (

*x*–

*y*) low-pass filter and explicitly investigate the interaction between SMS flow features and the FS. In Verma et al. (2019), the finescales were generated by the growing MLI and were confined to a spatially sparse region, namely, inside the

*O*(100)-m-wide vortex filaments and the peripheries of eddies. However, in the simulations discussed here, the finescales that originate from the uniformly applied surface cooling occupy a spatially extensive region.

The problem of MLI in the presence of convection has been previously considered numerically by Whitt and Taylor (2017) and Callies and Ferrari (2018) using an idealized Eady model. Whitt and Taylor (2017) found that, during the passage of a modeled storm, the domain remained stratified in patches, suggesting the persistence of SMS dynamics. Both submesoscales as well as finescales became energized. Contrary to this finding, they observed a decrease in the submesoscale kinetic energy (KE) when a time-dependent surface buoyancy flux was applied to replicate the Ekman buoyancy flux generated by the storm. Callies and Ferrari (2018) studied the growth of baroclinic instability with convection in the equilibrium solution of a model problem by changing the imposed cooling/heating fluxes at the surface/bottom over two orders of magnitude. They observed that MLI grows in all cases. In cases with weak surface cooling, SMS dynamics appear to shut off convection, and the front remains stratified with positive PV over much of the vertical domain. With moderate cooling, there is slantwise convection and as cooling increases the plumes become more upright. In this paper, we expand on the studies of Whitt and Taylor (2017) and Callies and Ferrari (2018) and investigate how both the SMS currents and convective turbulence change as a result of their interaction by quantifying various aspects of the coupling as noted in the preceding paragraphs. Also, we explicitly address the role of surface cooling in the energetics of the system through separate balance equations for the SMS and the FS.

This paper is organized as follows. In section 2, the setup of the model warm filament (equivalently a double front) is described, the methods used to study the interaction between the SMS and the FS are given, and the flow evolution prior to applying the cooling flux is summarized. Velocity spectra are discussed and a length scale for separating the SMS from the FS is identified in section 3. Convection inside the front is contrasted with the outside region in section 4. In section 5, we describe the adjustment to convection of key features of SMS fronts such as coherent structures, downwelling/upwelling, restratification, and frontogenesis. The energetics of the SMS, the FS, and their coupling are investigated by examining energy balance equations at both scales in section 6. In section 7, the development of secondary instabilities is elaborated by analyzing PV, Richardson number, and Rossby number within the front, and their role in the extraction of energy from the SMS is examined. Finally, we offer conclusions and discuss implications of our results in section 8.

## 2. Problem formulation and methods

### a. Model setup

*x*direction with temperature varying in the lateral (

*y*) and vertical (

*z*) directions. The density (

*ρ*) and temperature (

*T*) deviations from reference values are related by

*ρ*/

*ρ*

_{0}=

*αT*, where

*α*is the coefficient of thermal expansion of seawater. The initial temperature profile is constructed by joining two profiles given by

The initial temperature field *T*(*y*, *z*) is symmetric around *y* = 0, and there are two fronts: a right front centered at *y* = −*Y*_{0} and a left front centered at *y* = *Y*_{0} = 4608 m. The width of each front is *L* = 2 km, the thickness of the surface layer containing the front is *H* = 70 m, and *δ _{H}* = 5 m controls the thickness of the transition layer between the upper layer and the thermocline. The weakly stratified surface layer and the pycnocline have squared buoyancy frequency

*M*

^{2}= −(

*g*/

*ρ*

_{0})(∂

*ρ*/∂

*y*), is nonuniform with a peak magnitude of

*y*= ±

*Y*

_{0}.

The fronts are initialized to be in thermal wind balance and the alongfront (*x*) velocity satisfies ∂*U*/∂*z* = −*M*^{2}/*f*, where *f* = 8.62 × 10^{−5} s^{−1} is the Coriolis parameter. The frontal jet has a peak speed of ^{−4} m s^{−1} is added to trigger the instabilities of the system. Key nondimensional parameters at the center of each front are as follows: Richardson number *U*_{0}/*fL* = 0.3.

The magnitude of the surface cooling flux (*Q _{s}*) is varied in a parametric study with four cases:

*Q*= 5, 15, 25, and 40 W m

_{s}^{−2}. Cooling is initiated at

*t*= 231 h when multiple coherent structures representative of MLI have formed. The surface buoyancy flux (

*B*) introduced by cooling competes with the restratifying vertical buoyancy flux of the SMS currents which has been parameterized by Fox-Kemper et al. (2008) as

_{s}*C*, where the dimensional group is

_{e}B_{f}*C*= 0.06. The relative strength of cooling can be measured by the ratio

_{e}*r*=

*B*/

_{s}*B*(Callies and Ferrari 2018) or, equivalently,

_{f}*C*(Mahadevan et al. 2010; Whitt and Taylor 2017) in the denominator. As shown in Table 1, the value of

_{e}B_{f}*r*varies from 0.01 to 0.1 and the ratio

The list of parameters for different cases.

We employ a LES model (Pham and Sarkar 2018; Verma et al. 2019) to numerically solve the incompressible, nonhydrostatic, Boussinesq Navier–Stokes equations for the velocity *u _{i}*, the temperature

*T*, and the dynamic pressure

*p*. The subgrid viscosity is parameterized following Ducros et al. (1996) and the subgrid Prandtl number of unity sets the subgrid diffusivity. The molecular viscosity is 10

^{−6}m

^{2}s

^{−1}and the molecular Prandtl number is 7. The model uses a mixed third-order Runge–Kutta (for advective fluxes) and Crank–Nicolson (for diffusive fluxes) scheme. Second-order finite difference discretization is used to compute spatial derivatives. The dynamic pressure is obtained by solving the Poisson equation with a multigrid iterative method.

The computational domain of 6147 m × 18 435 m × 161 m is discretized using a grid with about 1.2 billion points, which has 2050 × 6146 grid points in the horizontal with Δ*x* = Δ*y* = 3 m and 98 grid points in the vertical where Δ*z* varies from 1 m at the surface to 1.69 m at the bottom of the warm filament. The horizontal directions are periodic. The bottom boundary is a free-slip wall and has a constant heat flux which maintains the thermocline temperature gradient. A sponge region is set near the bottom boundary to prevent reflection of internal waves. The free-slip upper surface has a specified heat flux.

### b. Submesoscale-finesale decomposition

We adopt the decomposition, *ϕ*″ is the FS component of a field variable *ϕ*. Following Verma et al. (2019), which can be consulted for the details, *x*–*y*) Lanczos filter to *ϕ* and the FS is the remainder *k _{c}*) chosen for the filter corresponds to a change in the slope of energy spectra (discussed later in section 3) as well as the flow behavior from quasi-2D to 3D. Note that, if desired, the SMS field

*x*) and the deviation of

In later sections, we will quantitatively examine energy transfers to and between convective turbulence and the SMS flow. To do so, equations governing the SMS and FS derived in the appendix of this paper are utilized. The SMS equations are obtained by applying the homogeneous filter function with the SMS cutoff scale to the LES equations. Filtering gives rise to new terms in the SMS equations involving the residual stress

*B*) feed energy into the flow instabilities, and there are energy transfers between the SMS and FS. We will discuss the energy pathways in section 6 by the quantification of the KE and PE evolution equations for the two scales. The evolution equation for the submesoscale KE

_{s}We will show that, overall, the FS velocity is energized by the SMS motions through the transfer [*B*″ = *b*″*w*″) will be shown to be an additional third source which becomes increasingly important as surface cooling increases. The dissipation [

*E*= −

_{p}*zb*, where

*b*=

*αgT*is the buoyancy. Employing the equation for SMS temperature, the balance of submesoscale PE (

The SMS vertical buoyancy flux (

The FS contribution to the PE is −*zb*″. The horizontal average of −*zb*″ is close to zero and, besides, it will be seen that diagnostics of the submesoscale PE is sufficient to understand the energization of the SMS and FS velocity. Therefore, we do not further consider −*zb*″.

### c. Evolution prior to cooling

The simulation is run without surface cooling flux in the beginning, allowing baroclinic MLI at the fronts to evolve into its nonlinear stage. The evolution is similar at the two fronts, one at each edge of the warm filament. Therefore, we discuss only the right front in the schematic of Fig. 1 for the results to follow. Prior to surface cooling, the front evolves as in other previous studies of upper-ocean fronts. As MLI grows, the front meanders and develops vortex filaments with concentrated vertical vorticity. Further, the vortex filaments merge and grow in length as the instability progresses. Subsequently, the long filaments, comparable in length to the wavelength of the most dominant baroclinic mode, begin to wrap into coherent SMS eddies.

Surface cooling is initiated at *t* = 231 h when three SMS eddies have formed as shown in Figs. 2a and 2b. Significant restratification in the interior of the front has occurred as can be seen from the increase of

The right-front flow field at *t* = 231 h just before initiation of surface cooling. Horizontal sections shown at depth of 1 m: (a) temperature and (b) vertical vorticity. Vertical–alongfront sections shown at *y* = −5.2 km: (c) buoyancy frequency and (d) Richardson number. Velocity vectors in (a) and isocontours of temperature deviation in (c) and (d) are superposed. All figures hereafter are from the right front of the warm filament unless noted otherwise.

Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

The right-front flow field at *t* = 231 h just before initiation of surface cooling. Horizontal sections shown at depth of 1 m: (a) temperature and (b) vertical vorticity. Vertical–alongfront sections shown at *y* = −5.2 km: (c) buoyancy frequency and (d) Richardson number. Velocity vectors in (a) and isocontours of temperature deviation in (c) and (d) are superposed. All figures hereafter are from the right front of the warm filament unless noted otherwise.

Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

The right-front flow field at *t* = 231 h just before initiation of surface cooling. Horizontal sections shown at depth of 1 m: (a) temperature and (b) vertical vorticity. Vertical–alongfront sections shown at *y* = −5.2 km: (c) buoyancy frequency and (d) Richardson number. Velocity vectors in (a) and isocontours of temperature deviation in (c) and (d) are superposed. All figures hereafter are from the right front of the warm filament unless noted otherwise.

Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

## 3. Spectra and separation of scales

*u*,

*υ*,

*w*) are computed as follows

*x*direction wavenumber (

*k*) and 〈⋅〉

*is the average over the initial width of the right front, −5.8 <*

_{y}*y*< −3.8 km.

In every case, the energy spectra (shown at 20 m depth in Fig. 3) exhibits a peak at *k* ≈ 3.0 × 10^{−3} rad m^{−1} or *λ* ≈ 2 km, the scale of the fastest-growing baroclinic mode. Two different dynamical regimes are clearly distinguishable: a quasi-2D SMS regime with the horizontal components energetically dominating the vertical, and a 3D FS regime at <*O*(100) m where *u*, *υ*, and *w* have comparable energy. The spectral slope of *u* and *υ* in the SMS regime is between −2 and −3 for the unforced simulation Qs0, which is consistent with previous studies (e.g., Skyllingstad and Samelson 2012). Moreover, the spectral slope is approximately −5/3 (the inertial-range power law of 3D turbulence) over a small range of wavenumbers in the FS regime, before it falls off steeply. The spectra change gradually as the surface cooling increases in magnitude, especially in the FS regime, which becomes more energized. The energy at lower wavenumbers also increases. Overall, the spectral slope of the horizontal velocity components becomes shallower than in the unforced case. The change in *S ^{w}*(

*k*) is particularly striking, as the largest scales of

*w*become significantly more energetic in Qs40 compared to Qs0. Nevertheless,

*u*and

*υ*remain much more energetic than

*w*in all cases, and the flow behavior changes from quasi-2D, a characteristic of the SMS, to 3D turbulent finescales when the length scale drops below

*O*(100) m. This is also reflected in the spectral slope, which changes across the

*O*(100) m scale. Exploiting the change in flow behavior, we separate the two ranges, namely, the SMS and the FS, to study how they evolve and affect each other. We choose the cutoff length scale at

*λ*= 314 m, corresponding to a cutoff wavenumber

_{c}*k*= 0.02 rad m

_{c}^{−1}, from case Qs40 and apply a low-pass Lanczos filter in the horizontal to separate the SMS from the FS. A visualization of vertical velocity reveals that the chosen cutoff length scale generates a submesoscale field that is reasonably smooth in all cases and has well-separated upwelling and downwelling regions. Furthermore, integrating the spectra for

*k*>

*k*in case Qs40, we find that the FS has comparable energy in the three components:

_{c}Velocity spectra at 20-m depth and *t* ≈ 250 h: (a) *S ^{u}*(

*k*), (b)

*S*(

^{υ}*k*), and (c)

*S*(

^{w}*k*). The vertical line in each panel is plotted at

*k*= 0.02 rad m

_{c}^{−1}, depicting the separation between the SMS and the FS.

Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

Velocity spectra at 20-m depth and *t* ≈ 250 h: (a) *S ^{u}*(

*k*), (b)

*S*(

^{υ}*k*), and (c)

*S*(

^{w}*k*). The vertical line in each panel is plotted at

*k*= 0.02 rad m

_{c}^{−1}, depicting the separation between the SMS and the FS.

Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

Velocity spectra at 20-m depth and *t* ≈ 250 h: (a) *S ^{u}*(

*k*), (b)

*S*(

^{υ}*k*), and (c)

*S*(

^{w}*k*). The vertical line in each panel is plotted at

*k*= 0.02 rad m

_{c}^{−1}, depicting the separation between the SMS and the FS.

Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

## 4. Convection inside and outside the front

The frontal region evolves differently relative to that outside as illustrated by Fig. 4 for cases Qs5 and Qs40 at *t* ≈ 250 h, about 19 h after the initiation of cooling. Outside the front, the horizontal section at 10-m depth in both cases (Figs. 4a,b) shows FS convective plumes with significant *w*, both positive and negative. Inside the front, the plumes are limited to the vortex filaments and eddies in Qs5. Note that it is difficult to separate out regions with coherent downwelling inside the front and we will show in the next section that the SMS–FS decomposition is able to do so. The vertical cross section at the *x* = 3 km cross-front transect (Fig. 4c) shows that plumes within the front remain confined primarily near the surface. An exception is the vortex filament near the light edge (*y* = −3.6 km) where the plumes penetrate down to *z* ≈ −40 m, even deeper than the ≈−30-m value outside the front. Case Qs40 with a moderate surface cooling flux is very different. Plumes with significant *w* (Fig. 4b) are present everywhere at 10 m depth within the front. At the same time, the imprint of coherent SMS structures on the organization of the plumes is also visible. In contrast, the plumes outside the front are horizontally homogeneous. In Qs40, the plumes reach a depth of ∼25 m (Fig. 4d) in the front, excluding regions where coherent structures are present. In the vortex filaments and outside the front, plumes penetrate up to ∼40-m depth.

Effect of SMS eddies on convective turbulence for cases (a),(c),(e) Qs5 and (b),(d),(f) Qs40 is shown at 250 h. Vertical velocity is plotted on a horizontal plane at 10-m depth in (a) and (b) and on a lateral–vertical plane (*x* = 3 km) in (c) and (d). The dash–dotted line in (a) and (b) marks the lateral transect shown in (c) and (d). Panels (e) and (f) show the mixed zone depth (*h _{m}*), also shown by dash–dotted white lines in (c) and (d). The solid vertical lines in (c) depict the edges of the initial front.

Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

Effect of SMS eddies on convective turbulence for cases (a),(c),(e) Qs5 and (b),(d),(f) Qs40 is shown at 250 h. Vertical velocity is plotted on a horizontal plane at 10-m depth in (a) and (b) and on a lateral–vertical plane (*x* = 3 km) in (c) and (d). The dash–dotted line in (a) and (b) marks the lateral transect shown in (c) and (d). Panels (e) and (f) show the mixed zone depth (*h _{m}*), also shown by dash–dotted white lines in (c) and (d). The solid vertical lines in (c) depict the edges of the initial front.

Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

Effect of SMS eddies on convective turbulence for cases (a),(c),(e) Qs5 and (b),(d),(f) Qs40 is shown at 250 h. Vertical velocity is plotted on a horizontal plane at 10-m depth in (a) and (b) and on a lateral–vertical plane (*x* = 3 km) in (c) and (d). The dash–dotted line in (a) and (b) marks the lateral transect shown in (c) and (d). Panels (e) and (f) show the mixed zone depth (*h _{m}*), also shown by dash–dotted white lines in (c) and (d). The solid vertical lines in (c) depict the edges of the initial front.

Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

SMS currents at the front influence the deepening of the turbulent mixed layer. Because of the multiscale variability of the density in this flow, it is difficult to define a mixed layer depth using density. Here, a mixed zone with depth *h _{m}* is identified as a region with a weak stratification of

*h*(Figs. 4e,f). Within the front, the mixed layer is shallower outside vortex filaments and eddies. There is a distinct change of

_{m}*h*at the boundaries of these coherent structures, and the change is especially large when the surface cooling is strong as in Qs40.

_{m}Convective mixing continually competes with restratification and lateral advection by MLI. Figures 5a–c show *h _{m}*(

*x*,

*y*) at 3 different times for case Qs40. By

*t*= 238 h, there is significant deepening within the front, especially in the vortex filamentary roll-ups from the dense side (

*y*< 5.5 km), which deepen further by

*t*= 245 h. However, there are also regions that remain shallow at both times. Interestingly, at the later time of

*t*= 255 h in Fig. 5c, there is a qualitative change in that

*h*decreases relative to Figs. 5a and 5b within several regions of the front. The horizontally averaged value of

_{m}*h*(blue curve in Fig. 5d) confirms the late-time decrease of mixed layer depth when the average is confined to the front. Furthermore, the frontal average of

_{m}*h*≈ 25 m is significantly smaller than the average value of

_{m}*h*≈ 35 m outside the front. Discussion of Figs. 5e and 5f is deferred to section 5b on restratification.

_{m}Evolution of the mixed layer depth [*h _{m}*(

*x*,

*y*)] during the period of cooling in Qs40 is shown with the contours of

*h*(

_{m}*x*,

*y*) at times (a)

*t*= 238, (b)

*t*= 245, and (c)

*t*= 255 h. (d) Comparison of the evolution inside the front with that outside the front. (e),(f) Vertical buoyancy gradient

*x*–

*z*plane, respectively; the

*x*–

*z*cross section passes through

*y*= 5.5 km, and its intersection with a horizontal plane at 10-m depth is shown in Fig. 4f as a dashed black line. The dash–dotted black line in (e) and (f) depict the mixed layer depth.

Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

Evolution of the mixed layer depth [*h _{m}*(

*x*,

*y*)] during the period of cooling in Qs40 is shown with the contours of

*h*(

_{m}*x*,

*y*) at times (a)

*t*= 238, (b)

*t*= 245, and (c)

*t*= 255 h. (d) Comparison of the evolution inside the front with that outside the front. (e),(f) Vertical buoyancy gradient

*x*–

*z*plane, respectively; the

*x*–

*z*cross section passes through

*y*= 5.5 km, and its intersection with a horizontal plane at 10-m depth is shown in Fig. 4f as a dashed black line. The dash–dotted black line in (e) and (f) depict the mixed layer depth.

Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

Evolution of the mixed layer depth [*h _{m}*(

*x*,

*y*)] during the period of cooling in Qs40 is shown with the contours of

*h*(

_{m}*x*,

*y*) at times (a)

*t*= 238, (b)

*t*= 245, and (c)

*t*= 255 h. (d) Comparison of the evolution inside the front with that outside the front. (e),(f) Vertical buoyancy gradient

*x*–

*z*plane, respectively; the

*x*–

*z*cross section passes through

*y*= 5.5 km, and its intersection with a horizontal plane at 10-m depth is shown in Fig. 4f as a dashed black line. The dash–dotted black line in (e) and (f) depict the mixed layer depth.

Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

Convective turbulence at fronts is modified by the SMS currents as illustrated by vertical profiles of FS velocity variances (Fig. 6) in the strong cooling case Qs40. The vortex filaments, which are identified as regions having significant SMS downwelling with *u*″ and *υ*″ have comparable energy, which exceeds *w*″ over the entire depth of the front. This behavior stands in contrast to that in the region without frontal gradients (Fig. 6c), where *w*″ is much more energetic than *u*″ and *υ*″, except near the surface where *w* goes to zero because of the nonnormal flow boundary condition at the free-slip wall. The FS velocity variances within the front, but excluding the vortex filaments, are plotted in Fig. 6b. The vertical component is more energetic than the horizontal, akin to convective turbulence, in the upper layer before becoming less energetic below 25-m depth, suggesting a role of shear-driven turbulence below middepth. Comparing Figs. 6a and 6b with Fig. 6c clearly shows that the FS inside the front is modified, more so in the vortex filaments.

Vertical profiles of the FS velocity variances *y* < 2.5 km, where the lateral buoyancy gradient is negligible and the FS is purely due to convection.

Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

Vertical profiles of the FS velocity variances *y* < 2.5 km, where the lateral buoyancy gradient is negligible and the FS is purely due to convection.

Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

Vertical profiles of the FS velocity variances *y* < 2.5 km, where the lateral buoyancy gradient is negligible and the FS is purely due to convection.

Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

## 5. Adjustment in submesoscale dynamics

In section 4, it was shown that the presence of a lateral buoyancy gradient induces significant changes in upper-layer convection. In turn, convection modifies the SMS currents set up by the lateral buoyancy gradient as elaborated in this section. Note that SMS currents and associated restratification persist in all cases although they are substantially modified at the higher values of *r* or

### a. Effect on coherent structures

To assess the effect of convection on the filament/eddy structures at the front, we contrast the SMS properties at 10-m depth between cases Qs5 and Qs40 in Figs. 7a–f. The adopted Lanczos filter with the cutoff at 314 m meets our objective of separating the coherent field from the incoherent FS. For example, the small-scale patchiness in the full *w* field (Fig. 4b) is much reduced in its SMS counterpart

Submesoscale flow quantities on a horizontal plane at 10-m depth and *t* ≈ 250 h for (left) Qs5 and (right) Qs40: (a),(b) vertical velocity, (c),(d) vertical vorticity normalized by *f*, and (e),(f) horizontal buoyancy gradient magnitude.

Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

Submesoscale flow quantities on a horizontal plane at 10-m depth and *t* ≈ 250 h for (left) Qs5 and (right) Qs40: (a),(b) vertical velocity, (c),(d) vertical vorticity normalized by *f*, and (e),(f) horizontal buoyancy gradient magnitude.

Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

Submesoscale flow quantities on a horizontal plane at 10-m depth and *t* ≈ 250 h for (left) Qs5 and (right) Qs40: (a),(b) vertical velocity, (c),(d) vertical vorticity normalized by *f*, and (e),(f) horizontal buoyancy gradient magnitude.

Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

The organization of downwelling (*w* in the large (SMS) scales was also reflected in its spectrum *S ^{w}* (Fig. 3c). The organization of upwelling has also changed qualitatively; it has now become concentrated adjacent to the vortex filaments (Fig. 7b). These adjustments in

*ω*) in the vortex filaments is also stronger in Qs40 than Qs5 (Figs. 7c,d) as is the horizontal shear of

_{z}*u*and

*υ*. The enhanced downwelling and cyclonic vorticity of the vortex filaments in Qs40 are a product of SMS frontogenesis and imply strengthening of the process by convection. Nevertheless, the horizontal buoyancy gradient magnitude in the vortex filaments is relatively weaker and more diffused in Qs40 than Qs5, which is a result of relatively strong convective turbulence in the former.

The effect of convection on 〈*w*〉* _{xy}*(

*z*) is depicted in Fig. 8. For convenience, we split the velocity component as

*Q*increases. As noted earlier, this suggests enhancement in frontogenesis. This enhancement can be interpreted in the context of TTW, which predicts that vertical mixing by FS boundary layer turbulence leads to a secondary ageostrophic circulation that is frontogenetic near the surface. In the present simulations, TTW strengthens the secondary circulation at the front, further enhancing the sharpening of vortex filaments resulting from the growing MLI. Figure 8b shows profiles of the frontal mean of the FS vertical velocity

_{s}*w*″ in the regions with downwelling and upwelling SMS velocity, i.e.,

*w*″ can be larger than

Effect of convection on upwelling/downwelling among different cases: (a) the frontal mean of *w*″ in the downwelling (*x* domain and −6.2 km < *y* < −3.0 km in the lateral direction.

Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

Effect of convection on upwelling/downwelling among different cases: (a) the frontal mean of *w*″ in the downwelling (*x* domain and −6.2 km < *y* < −3.0 km in the lateral direction.

Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

Effect of convection on upwelling/downwelling among different cases: (a) the frontal mean of *w*″ in the downwelling (*x* domain and −6.2 km < *y* < −3.0 km in the lateral direction.

Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

### b. Restratification at the front

Convective turbulence is effective at eroding any vertical gradient of density or velocity, creating a mixed zone that deepens. Yet, SMS dynamics inhibit the growth of convective plumes in certain regions of the front (see Figs. 4e,f) even in Qs40, whose value of *h _{m}*) shows significant variability throughout the front. The vertical section of

*h*, there are stratified patches at middepth where

_{m}*h*correlates well with the SMS vertical velocity, being generally shallow in the region of upwelling

_{m}### c. Frontogenesis and horizontal convergence

Employing an asymptotic model, Barkan et al. (2019) show that frontogenetic tendency in SMS flows is inherently linked to the horizontal convergence of velocity. An important result is that the frontogenetic forcing given by *c*_{1} ≈ 1, validating the applicability of the result of Barkan et al. (2019) to the present case of a convectively forced front. The maximum value of

Scatterplots of the normalized frontogenetic forcing (

Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

Scatterplots of the normalized frontogenetic forcing (

Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

Scatterplots of the normalized frontogenetic forcing (

Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

## 6. Submesoscale and finescale energetics

In previous sections, we showed that uniform cooling leads to mixing depths and turbulence profiles inside the front which are different from those outside and also characterized the SMS adjustment to convective forcing. In this section, we investigate the SMS, FS, and their interaction by quantifying the SMS and FS energy equations.

### a. Submesoscale energy conversions

As per the balance of the submesoscale KE in Eq. (3), the SMS is energized by the SMS buoyancy production (

Profiles of horizontally averaged (denoted by 〈⋅〉* _{xy}*) buoyancy production are shown in Fig. 10 for different cases. As cooling strengthens, the peak in buoyancy production increases, and it moves away from the surface. For case Qs40, the peak has shifted to middepth (≈25 m), and the profile has become nearly symmetric about this depth. The depth of peak

*h*over the front. The increase in

_{m}The vertical profile of SMS buoyancy production, *t* ≈ 250 h. The curves are normalized by the vertical buoyancy flux scaling *B _{f}*.

Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

The vertical profile of SMS buoyancy production, *t* ≈ 250 h. The curves are normalized by the vertical buoyancy flux scaling *B _{f}*.

Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

The vertical profile of SMS buoyancy production, *t* ≈ 250 h. The curves are normalized by the vertical buoyancy flux scaling *B _{f}*.

Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

Based on the proposed parameterization of SMS buoyancy flux (Fox-Kemper et al. 2008) and the maximum value of *C _{e}* ≈ 0.005–0.035. The LES with no surface forcing in Skyllingstad and Samelson (2012) suggests

*C*≈ 0.005 (see their Fig. 12) comparable to that for Qs0. When including surface cooling with

_{e}*r*= 0.0024 (

*C*≈ 0.027 (see their Fig. 14) in their LES. The LES with surface wind forcing and without Stokes drift in Hamlington et al. (2014) shows a

_{e}*C*≈ 0.05 (see their Fig. 10). Our results and those from similar LES, all indicate a value range of

_{e}*C*that is smaller than 0.06 in Fox-Kemper et al. (2008). This implies that the observed restratification in these simulations is weaker than that predicted by the parameterization of Fox-Kemper et al. (2008). Moreover, the restratifying buoyancy flux depends on the strength of the boundary layer turbulence interacting with submesoscale currents, with magnitude consistently increasing with stronger turbulence, an effect not explicitly incorporated by Fox-Kemper et al. (2008).

_{e}The increase of the source (* _{xy}* with a difference of 10% but opposite in sign. Thus, the residual production acts to energize the FS. The direct dissipation (

The time-dependent effect of convection is assessed in Fig. 11. Surface cooling energizes the overall submesoscale KE. After an initial transient (∼10 h), the alongfront energy (Fig. 11a) increases. Lateral and vertical KE components (Figs. 11b,c) also increase in time. The amount of increase in all three KE components is enhanced by strengthening *Q _{s}*.

Time evolution of volume-averaged SMS quantities. Kinetic energy contributions: (a) alongfront,

Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

Time evolution of volume-averaged SMS quantities. Kinetic energy contributions: (a) alongfront,

Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

Time evolution of volume-averaged SMS quantities. Kinetic energy contributions: (a) alongfront,

Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

The two dominant terms in the KE budget, namely, the source provided by buoyancy (

*b*″

*w*″ appears with a positive sign in the finescale KE budget, we can interpret

*b*″

*w*″ as a term through which PE energizes the FS motions. Figure 11f shows the change in

### b. Finescale energy conversions

The finescale KE differs in energy content as well as its distribution among *u*, *υ*, and *w* inside the front relative to the purely convective turbulence outside, as was illustrated for Qs40 in Fig. 6 of section 4. The finescale KE balance [Eq. (4)] is examined here to better understand how surface cooling affects the sources and sinks of FS energy inside the front.

*B*″ which we find to be the primary source of finescale KE inside the front. The SMS KE is an additional source of finescale KE. In particular, the transfer term (Tr) extracts finescale KE from the SMS velocity. To distinguish the contributions from horizontal and vertical gradients of the SMS velocity, Tr is separated into Tr

*and Tr*

_{h}*as follows,*

_{υ}*consists of all the terms with the horizontal (*

_{h}*x*,

*y*) gradients of the SMS velocity, and Tr

*consists of those with the vertical gradient.*

_{υ}Figure 12 compares the vertical profiles of the dominant budget terms for the cases with surface cooling. The terms are normalized by the imposed surface buoyancy flux *B _{s}*. The magnitude of

*B*varies substantially taking values in Qs15, Qs25, and Qs40, which are 3, 5, and 8 times that of Qs5. The FS buoyancy flux (

_{s}*B*″ shown in Fig. 12a) is the dominant source and scaling with

*B*provides a reasonable collapse of its profiles, especially for cases Qs15, Qs25, and Qs40. The maximum value of

_{s}*B*″ is about 0.8

*B*in Qs40 showing that most of the surface buoyancy flux supplied by surface cooling appears in the FS buoyancy flux.

_{s}Dominant terms in the finescale KE balance equation at *t* ≈ 250 h: (a) FS buoyancy production (B-PROD), (b) Tr* _{υ}*, and (c) Tr

*. All budget terms are normalized by the surface buoyancy flux*

_{h}*B*. We note that

_{s}*B*in Qs15, Qs25, and Qs40 are, respectively, 3, 5, and 8 times stronger than in Qs5.

_{s}Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

Dominant terms in the finescale KE balance equation at *t* ≈ 250 h: (a) FS buoyancy production (B-PROD), (b) Tr* _{υ}*, and (c) Tr

*. All budget terms are normalized by the surface buoyancy flux*

_{h}*B*. We note that

_{s}*B*in Qs15, Qs25, and Qs40 are, respectively, 3, 5, and 8 times stronger than in Qs5.

_{s}Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

Dominant terms in the finescale KE balance equation at *t* ≈ 250 h: (a) FS buoyancy production (B-PROD), (b) Tr* _{υ}*, and (c) Tr

*. All budget terms are normalized by the surface buoyancy flux*

_{h}*B*. We note that

_{s}*B*in Qs15, Qs25, and Qs40 are, respectively, 3, 5, and 8 times stronger than in Qs5.

_{s}Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

The transfer Tr* _{υ}* by vertical shear is found to be a significant additional source of finescale KE in all the forced cases (Fig. 12b). Scaling by

*B*again provides an approximate collapse of the maxima in the Tr

_{s}*profiles among the cases, showing the importance of surface cooling to the transfer term. The location of the peak Tr*

_{υ}*follows the peak of the SMS buoyancy flux*

_{υ}*(Fig. 12c) also contributes as a source in the near-surface region. Inside the front, the volume average of Tr*

_{h}*+ Tr*

_{υ}*is as large as 50% of the volume-averaged*

_{h}*B*″ for case Qs40 at late times. The role played by SMS shear as a source of finescale KE explains the change of turbulence anisotropy inside the front from purely convective turbulence toward shear-driven turbulence.

The temporal evolution of the volume-averaged FS is investigated in Fig. 13a. In each case, the finescale KE increases with time as convection develops and eventually becomes approximately constant. The FS subgrid dissipation (Fig. 13c) evolves similarly. The time evolution of the buoyancy production (Fig. 13b), on the other hand, is somewhat different; in most cases, *B*″ decreases after its peak instead of approximately plateauing. The decrease in *B*″ is related to the generation of negative *B*″ at the bottom, corresponding to the entrainment of bottom fluid by convection, which is strongest for Qs40. Interestingly, during the time when *B*″ decreases, the production by Tr* _{υ}* (Fig. 13d) rises to compensate and allows the finescale KE to remain approximately constant.

The time evolution of volume-averaged quantities: (a) finescale KE, (b) FS buoyancy production, (c) subgrid dissipation, and (d) Tr* _{υ}*. The finescale KE budget terms from each case are normalized by the surface buoyancy flux

*B*, whereas the finescale KE curves are normalized by

_{s}*U*

_{0}

^{2}/2, same as the submesoscale KE. The thick dash–dotted lines in (b) depict

Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

The time evolution of volume-averaged quantities: (a) finescale KE, (b) FS buoyancy production, (c) subgrid dissipation, and (d) Tr* _{υ}*. The finescale KE budget terms from each case are normalized by the surface buoyancy flux

*B*, whereas the finescale KE curves are normalized by

_{s}*U*

_{0}

^{2}/2, same as the submesoscale KE. The thick dash–dotted lines in (b) depict

Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

The time evolution of volume-averaged quantities: (a) finescale KE, (b) FS buoyancy production, (c) subgrid dissipation, and (d) Tr* _{υ}*. The finescale KE budget terms from each case are normalized by the surface buoyancy flux

*B*, whereas the finescale KE curves are normalized by

_{s}*U*

_{0}

^{2}/2, same as the submesoscale KE. The thick dash–dotted lines in (b) depict

Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

The above results lead to the following overall picture (Fig. 14) of the energy pathways involved in the problem. The PE reservoir, through the mixed layer BI and the attendant SMS flux *B*″ *= b*″*w*″) which transfers energy from the PE reservoir to the finescale KE reservoir. There is also substantial production of the FS by shear (primarily vertical) of the SMS motions through the transfer term (Tr). We find, as will be demonstrated in section 7, that forced secondary instabilities such as SI and KH instability play a role in mediating this transfer from the SMS to the FS. The dissipation of KE is primarily at the FS by the modeled subgrid processes. Note that the SMS currents persist through extraction of the PE stored in the front because the lateral buoyancy gradient persists in the face of convective turbulence.

A schematic of energy pathways among the PE, submesoscale KE, and the finescale KE reservoirs.

Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

A schematic of energy pathways among the PE, submesoscale KE, and the finescale KE reservoirs.

Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

A schematic of energy pathways among the PE, submesoscale KE, and the finescale KE reservoirs.

Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

## 7. Secondary instabilities

As the SMS currents adjust to the imposed cooling, regions of enhanced vertical shear develop adjacent to the deepening convective plumes, and also there is an overall energy transfer from the SMS to the FS mediated by this shear. Thus, apart from the primary cooling-induced gravitational instability, there are regions of the front which could be unstable to secondary instabilities such as symmetric and inertial instabilities. These secondary instabilities can be diagnosed by analyzing potential vorticity (PV), along with Richardson and Rossby numbers (e.g., Thomas et al. 2013). PV is defined using the SMS fields as

The PV distribution at 20-m depth for Qs5 (Fig. 15a) reveals that it is positive in the vortex filaments and the eddy peripheries, and nearly zero in other regions of the front. The black lines overlaid in Fig. 15a are the contours of

(a),(b) SMS PV, (c),(d) Richardson number, and (e),(f) vertical vorticity are shown at 20-m depth and *t* ≈ 250 h for cases (left) Qs5 and (right) Qs40. (g),(h) Maps of different instabilities that develop in PV < 0 regions. The black solid lines in all panels show

Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

(a),(b) SMS PV, (c),(d) Richardson number, and (e),(f) vertical vorticity are shown at 20-m depth and *t* ≈ 250 h for cases (left) Qs5 and (right) Qs40. (g),(h) Maps of different instabilities that develop in PV < 0 regions. The black solid lines in all panels show

Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

(a),(b) SMS PV, (c),(d) Richardson number, and (e),(f) vertical vorticity are shown at 20-m depth and *t* ≈ 250 h for cases (left) Qs5 and (right) Qs40. (g),(h) Maps of different instabilities that develop in PV < 0 regions. The black solid lines in all panels show

Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

The values of *ϕ*_{Ri} and *ϕ*_{Ro} can be readily used in the negative PV regions to discriminate among SI, inertial instability (I), gravitational instability (G), or a combination of any two of the preceding instabilities (Thomas et al. 2013). The regions of positive PV are taken to be stable. The instability maps at 20-m depth (Figs. 15g,h) show that, in both Qs5 and Qs40, SI is present. In Qs40 with strong cooling, propensity to SI is widespread in regions adjacent to the coherent vortex filaments and eddies. In addition, there are thin regions within these coherent structures, e.g., at the edges of vortex filaments and the peripheries of eddies, that are also unstable to SI. Mixed I/SI and SI/G unstable regions are sparse. In Qs40, gravitationally unstable regions within vortex filaments and outside the front are present, with interspersed stable regions as denoted by the S regions in Figs. 15g and 15h. In contrast, the fluid at 20-m depth is largely stable to gravitational instability in Qs5.

Further insights into the generation of secondary instabilities in a gravitationally stable negative-PV region can be obtained by separating the baroclinic (horizontal) PV component, *ω _{a}* =

*ω*+

_{z}*f*, in these regions is close to zero, although the vertical buoyancy gradient may be significant (Fig. 16d). The horizontal buoyancy gradients are smaller in magnitude than the vertical (Figs. 16d–f), but combined with the horizontal vorticity components, whose magnitudes can be up to 20

*f*(not shown), they result in significant negative PV. Thus, consistent with the inference of the instability map shown in Fig. 15h, the above analysis also reveals that the negative PV regions are dominated primarily by SI.

Case Qs40 at 20-m depth. SMS PV components: (a) horizontal *z* component *x* component *y* component

Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

Case Qs40 at 20-m depth. SMS PV components: (a) horizontal *z* component *x* component *y* component

Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

Case Qs40 at 20-m depth. SMS PV components: (a) horizontal *z* component *x* component *y* component

Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

Secondary instabilities help generate the FS. In SI, the FS velocity is generated principally by the vertical shear of the horizontal velocity. In inertial instability, it is the horizontal shear that produces FS motions. In Fig. 17, FS generation by Tr* _{h}* and Tr

*as well as the subgrid dissipation are assessed at 20 m depth for cases Qs5 and Qs40. In both cases, FS generation is dominated by Tr*

_{υ}*rather than Tr*

_{υ}*(Figs. 17c–f), suggesting the presence of forced SI. In Qs5, FS generation is concentrated primarily at the peripheries of the eddies where vortex filaments wrap around these eddies. In Qs40, regions with negative PV of magnitude greater than*

_{h}*. The regions with*

_{υ}*.*

_{υ}The FS (a),(b) subgrid dissipation, (c),(d) the transfer term components Tr* _{υ}* and (e),(f) Tr

*are plotted at 20-m depth and*

_{h}*t*≈ 250 h for cases (left) Qs5 and (right) Qs40. The white solid lines in all panels show

Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

The FS (a),(b) subgrid dissipation, (c),(d) the transfer term components Tr* _{υ}* and (e),(f) Tr

*are plotted at 20-m depth and*

_{h}*t*≈ 250 h for cases (left) Qs5 and (right) Qs40. The white solid lines in all panels show

Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

The FS (a),(b) subgrid dissipation, (c),(d) the transfer term components Tr* _{υ}* and (e),(f) Tr

*are plotted at 20-m depth and*

_{h}*t*≈ 250 h for cases (left) Qs5 and (right) Qs40. The white solid lines in all panels show

Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

In both Qs5 and Qs40, the generation of finescales by the forced SI enhance subgrid dissipation, compared to the outside region dominated by convection alone (Figs. 17a,b). The FS generation by SI and enhanced dissipation (* _{h}*, which collects contributions from horizontal gradients of velocity, contribute to the FS generation primarily in the vortex filaments and outer edges of the eddies (Figs. 17e,f). However, excluding the near surface region where the frontogenetic tendency is the strongest, the contribution of Tr

*to the FS generation is weaker than Tr*

_{h}*.*

_{υ}Corresponding to the horizontal variability in the growth of convection at the front, the vertical organization of negative PV regions exhibits horizontal variability. PV is shown on lateral–vertical (*y*–*z*) sections, passing through *x* = 2.5 km (Fig. 18a) and *x* = 4.0 km (Fig. 18b) for the moderate cooling case Qs40. The section through *x* = 2.5 km is between two eddies and intercepts a vortex filament near the heavy edge of the front (*x* ≈ −6.8 km), whereas the section through *x* = 4.0 km passes through a coherent eddy as well as vortex filaments. The black solid line in Figs. 18a and 18b corresponds to *h _{m}*, the depth of nearly mixed surface water. In the section at

*x*= 2.5 km, adjacent to the vortex filament, there is an extensive region of negative PV, which is below

*h*and spreads over the entire depth of the front, slanting toward the bottom as it reaches the light edge. In contrast, the section through

_{m}*x*= 4.0 km exhibits relatively thin regions of negative PV regions which are adjacent to vortex filaments at the heavy and the light edges of the front. The eddy region, approximately from

*y*= −5.5 to −4.5 km in Fig. 18c, lies between two such regions of negative PV and has an upwelled dome of positive PV at its base. As shown in Figs. 18c and 18d, the generation of finescales by Tr

*is correlated with the negative PV region. Moreover, FS generation extends into the weakly stratified surface water, especially in the vortex filaments. This suggests that convection in the coherent filaments of the front also draws energy through vertical shear. Underneath the convective layer there are regions with 0.25 < Ri < 1 (Figs. 18e,f) where SI is supported, which is a crucial source of finescale KE at depth. There is significant turbulent dissipation (not shown) in both the convective layer and below it in the SI-unstable region.*

_{υ}Lateral–vertical (*y*–*z*) sections through (left) *x* = 2.5 km, taken between two eddies and intercepting a vortex filament at the heavy edge, and (right) *x* = 4.0 km, passing through a coherent eddy and filamentary regions at its edge. Quantities shown are as follows: (a),(b) potential vorticity, (c),(d) transfer term component Tr* _{υ}*, and (e),(f) gradient Richardson number Ri associated with the SMS flow. The black solid line in each panel denotes the mixed layer depth

*h*.

_{m}Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

Lateral–vertical (*y*–*z*) sections through (left) *x* = 2.5 km, taken between two eddies and intercepting a vortex filament at the heavy edge, and (right) *x* = 4.0 km, passing through a coherent eddy and filamentary regions at its edge. Quantities shown are as follows: (a),(b) potential vorticity, (c),(d) transfer term component Tr* _{υ}*, and (e),(f) gradient Richardson number Ri associated with the SMS flow. The black solid line in each panel denotes the mixed layer depth

*h*.

_{m}Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

Lateral–vertical (*y*–*z*) sections through (left) *x* = 2.5 km, taken between two eddies and intercepting a vortex filament at the heavy edge, and (right) *x* = 4.0 km, passing through a coherent eddy and filamentary regions at its edge. Quantities shown are as follows: (a),(b) potential vorticity, (c),(d) transfer term component Tr* _{υ}*, and (e),(f) gradient Richardson number Ri associated with the SMS flow. The black solid line in each panel denotes the mixed layer depth

*h*.

_{m}Citation: Journal of Physical Oceanography 52, 3; 10.1175/JPO-D-21-0148.1

## 8. Conclusions and discussion

In this paper, we examine the evolution of baroclinic mixed layer instability (MLI) at upper-ocean fronts in a background of convective boundary layer turbulence to understand their interaction. We utilize the model setup of a wide warm filament with edge fronts of width 2 km confined in the weakly stratified surface layer of depth 70 m. The fronts, initially in thermal wind balance, are unstable to MLI and evolve nearly independent of each other, developing SMS coherent vortex filaments and eddies. After the MLI structures have been established, we apply surface cooling to study their time-evolving interaction with boundary layer convective turbulence. The cooling flux takes various values between *Q _{s}* = 5 and 40 W m

^{−2}and is applied for a period of 25 h, larger than a night of cooling and about an inertial period.

Surface cooling affects the coherent eddy/filament structures of the SMS, which, in turn, affect the three-dimensional, turbulent fluctuations of the FS in different ways. A decomposition using a sharp filter in physical space successfully separates the SMS and the FS (including convective turbulence), allowing characterization of how surface cooling separately affects the SMS and FS ranges. By constructing energy balance equations for fluctuations in both ranges, we are able to quantitatively examine their individual energetics as well as their interaction.

This study extends the approach of decomposing the flow into mesoscale and SMS component using a low-pass filter in physical space employed by Mensa et al. (2013) and Haza et al. (2016) to SMS–FS decomposition, where the FS component is forced by applying surface cooling. In a study of SMS fronts with wind stress and Langmuir turbulence, Hamlington et al. (2014) perform spectral decomposition of the flow into low-pass and high-pass components with a cutoff length scale at 400 m. However, their analysis was limited to computing the multiscale fluxes of momentum and temperature. In the present analysis, the calculated residual stresses and heat fluxes include the influence of filtered finescale on the large-scale (SMS) dynamics. Furthermore, the approach utilized in this study enable us to examine the energy balance of both flow components explicitly.

The turbulent finescale shows strong horizontal variability across the front. Vortex filaments inside the surface layer of the unforced front have coherent downwelling velocity (Verma et al. 2019) leading to sustained downward vertical transport (Verma and Sarkar 2021) of tracer particles. In the present forced case with surface cooling, convective plumes deepen through the downwelling vortex filaments and reach larger depths than the plumes outside the front. In contrast, the depth of the mixed layer is suppressed in other parts of the front where continued restratification is maintained by MLI. In the regions with shallow mixed layer, the vertical velocity is typically upwelling and patches of strong shear and stratification are encountered below the mixed layer base. The energy in the finescale also shows strong horizontal variability. Relative to the convective turbulence outside the front, the cases with stronger cooling have more energetic horizontal components within the front and especially so within the vortex filaments.

The submesoscales are also strongly affected and shows strengthening of several aspects. The coherent downwelling through the vortex filaments increases as does the upwelling. The upwelling regions are spatially more concentrated near the vortex filaments relative to the unforced case. Associated with the strengthening of downwelling/upwelling motions, PE extraction by the SMS buoyancy flux (

Callies and Ferrari (2018), in their study of a doubly periodic reentrant channel with constant *M*^{2}/*f*^{2} = 1 and upward buoyancy flux at the top and bottom walls, found that MLIs are remarkably resilient to convection in the final equilibrated state. In the present problem of a finite-width front with larger initial *M*^{2}/*f*^{2} = 8, we similarly find that MLI persists in the face of convection during the time-evolving deepening of the mixed layer. Whitt and Taylor (2017) simulated the response of SMS motions to the passage of a storm with downfront (destabilizing Ekman buoyancy flux) wind. They find that the submesoscales are resilient and grow stronger in response to the wind. Significant energization of the SMS component was observed during the passing of a storm in the simulations of Whitt and Taylor (2017) but not when only cooling was applied. In the presence of wind-driven boundary turbulence and including the effect of Langmuir turbulence, Hamlington et al. (2014) showed that both SMS and FS buoyancy fluxes are enhanced. Interestingly, the SMS vertical buoyancy flux is enhanced in the present case of pure convection too, even though turbulence is generated differently.

The important energy pathways among the submesoscale and the finescale KE reservoirs and the PE reservoir in this problem are summarized by the simplified diagram of Fig. 14. Surface cooling leads to finescale plumes which drive a positive buoyancy flux *b*″*w*″ to energize the finescale KE. Most of the imposed surface cooling flux appears as the FS buoyancy flux with its horizontal average as large as 80% of *B _{s}*. As the magnitude of surface cooling is increased, the transfer term Tr by SMS velocity gradients becomes an increasingly important source of finescale KE and its volume-integrated value reaches about 50% of the volume-integrated

*b*″

*w*″. Overall, the transfer Tr

*by vertical gradients of SMS velocity dominates Tr*

_{υ}*by horizontal gradients of SMS velocity. Ultimately, the subgrid processes dissipate the finescale KE, which dominates the dissipation of the total KE at the front.*

_{h}FS generation by the transfer term in the unforced MLI study of Verma et al. (2019) was found to be weak relative to SMS buoyancy production. This suggested that the transfer of KE from the quasi-2D SMS to three-dimensional finescale turbulence was not effective. In contrast, we find in this study that a background of convective turbulence promotes energy extraction from the SMS velocity. PV becomes negative in the front within regions that can extend to below the mixed layer depth. The magnitude and spatial extent of negative PV increases with surface cooling. A map of instability criteria (e.g., Thomas et al. 2013) shows that the frontal regions, which are below the mixed layer and conducive to SI, become more prevalent when surface cooling is strong (e.g., Qs40 compared to Qs5). Since SI draws energy from the KE reservoir, the transfer term from submesoscale to finescale KE is correspondingly enhanced. Negative PV regions have been observed at upper-ocean fronts under wind (e.g., Ramachandran et al. 2018; D’Asaro et al. 2011). Nighttime cooling can lead to destabilizing buoyancy flux, which can also be a significant source of negative PV, as we have discussed in this study.

## Acknowledgments.

We are pleased to acknowledge the support of ONR Grant N00014-18-1-2137 and ONR Grant N00014-21-1-2869.

## Data availability statement.

The data used in this study will be openly available in a repository with a DOI locator.

## APPENDIX

### Governing Equations

#### a. Governing equations for the submesoscale and the finescale fields

*q*, respectively. Here, the stress tensor is defined as

_{j}*τ*= −(

_{ij}*ν*+

*ν*

^{sgs})[(∂

*u*/∂

_{i}*x*) + (∂

_{j}*u*/∂

_{j}*x*)] and encompasses both molecular and subgrid contributions. Similarly, the molecular and subgrid contributions constitute the heat flux vector, given by

_{i}*q*= −(

_{j}*κ*+

*κ*

^{sgs})(∂

*T*/∂

*x*). At the FS, the governing equations are given by

_{j}In the FS momentum and temperature transport equations above, the left-hand sides have two additional terms (second and third) along with the typical time rate of change (first) and the FS advective (fourth) terms. Note that the second term corresponds to the advection by the SMS velocity and the third term to distortion by the SMS gradient. On the right-hand sides of the equations, *τ _{ij}* and

*q*, respectively.

_{j}#### b. Submesoscale frontogenesis

*|*

_{h}b^{2}can be analyzed (e.g., Capet et al. 2008b). In vector notation, the transport equation is given by

*x*and

*y*directions, and

*D*/

*Dt*is the material derivative. The terms dotted with |∇

*|*

_{h}b^{2}on the right hand side are as follows. The first two terms

**Q**

*and*

_{s}**Q**

*are related to advection and are expressed as*

_{w}**Q**

*corresponds to straining by the horizontal velocity and*

_{s}**Q**

*to the titling of isopycnals by the horizontal gradient of vertical velocity, and the last two terms*

_{w}**Q**

*and*

_{dh}**Q**

*are related to diabatic processes driven by molecular, subgrid, and residual fluxes in the horizontal and the vertical:*

_{dυ}The forcing terms **Q** terms. The frontogenesis is commonly driven by

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