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  • View in gallery

    Initial cross-front sections of various fields in the Ri = 0.25 case: (a) salinity, (b) temperature, (c) alongfront velocity, (d) gradient Richardson number, and (e) potential vorticity. The solid lines in (a) and (b) are isohalines and isotherms, respectively. The Richardson number has a minimum value at the center of the front and increases toward the sides. The light side of the front has negative potential vorticity. The positive alongfront direction x is directed outward.

  • View in gallery

    Structure of ASC near the end of the simulations: Ri = (a) 0, (b) 0.13, (c) 0.25, and (d) 0.5. Contours denote x-averaged cross-front velocity , and solid lines are isohalines. In all cases, the ASC is composed of a surface current that flows from the light side toward the dense side of the front and a countercurrent that flows in the opposite direction at depth.

  • View in gallery

    Surface manifestation of (a) salinity, (b) dissipation rate, (c) alongfront velocity, and (d) cross-front velocity illustrate the development of the surface gravity current in the Ri = 0.25 case. The front loses geostrophic balance at approximately 10 h, after which the gravity current forms and sharpens the lateral salinity gradient at the nose. The gravity current propagates at a constant speed toward the dense side of the front. Intense mixing with a large value of dissipation occurs at the nose of the gravity current. All fields are averaged over the alongfront x direction.

  • View in gallery

    Initial development of x-averaged cross-front velocity in the Ri = 0.25 case at different times: t = (a) 5.9, (b) 8.3, and (c) 10.6 h during the initiation stage of ASC. At the surface, positive cross-front velocity develops on the side of the front, while negative cross-front velocity develops on the other side in (a). Positive velocity grows at depth on the side at the same time. The cross-front velocity on the side intensifies in (b) and (c) and spreads toward the side. Solid lines denote isohalines. Note that the color bar in this figure has a different range than that used in Fig. 2.

  • View in gallery

    Deviation from geostrophic balance in the Ri = 0.25 case. (a) Vertical profiles of the forces that balance the cross-front momentum equation at the initial time t = 0; (b) surface variability of at t = 0 shows a peak value at the center of the front; and (c) nonuniform increase of sea surface salinity between t = 0 and 3.86 h is due to the variability of across the front.

  • View in gallery

    Surface flow convergence at t = 5.8 h during the initiation stage of the ASC in the Ri = 0.25 case. (a) The surface cross-front velocity changes sign near the center of the front; (b) the imbalance between the Coriolis force and the lateral pressure gradient drives the cross-front velocity; and (c) the absolute vertical vorticity also has opposite signs across the front, as indicated by the filtered vorticity (red). The unfiltered absolute vertical vorticity (blue) exhibits strong spatial fluctuations due to broadband turbulence.

  • View in gallery

    Development of shear instability after the loss of geostrophic balance is shown by profiles at three different times during the initiation stage of the ASC in the Ri = 0.25 case: (a) cross-front velocity, (b) alongfront velocity, (c) cross-front shear component , (d) alongfront shear component , (e) stratification, and (f) gradient Richardson number. The profiles are taken on the light side of the front at y = 150 m. Because of the loss of balance, a shear layer forms near the surface and causes the magnitude of the vertical shear to increase and the stratification to decrease. As a result, the gradient Richardson decreases to <0.25, and the current induces shear instability.

  • View in gallery

    Development of a gravity current at t = 10.64 h after the onset of shear-driven turbulence in the surface shear layer in the Ri = 0.25 case. (a) A snapshot of salinity field shows turbulent mixing along slanted isopycnals on light side of the front; (b) the dominance of lateral pressure gradient in the momentum balance drives the gravity current to the left at the surface; and (c) the positive lateral gradient of turbulent stress increases the speed of the gravity current and sharpens the lateral density gradient at its nose. The dashed line in (a) indicates the location where the profiles are plotted in Fig. 7. The cross-front variation shown in (b) and (c) is taken at the surface.

  • View in gallery

    Development of countercurrent at t = 11.4 h after the onset of turbulence in the Ri = 0.25 case. (a) A snapshot of salinity field shows turbulence generation in the bottom half of the front; (b) the turbulence breaks the momentum balance between the pressure gradient and the Coriolis force ; (c) the negative lateral gradient of the turbulent stress on the right side of the turbulent patches contributes to the propagation of the countercurrent toward the light side of the front at depth.

  • View in gallery

    Development of the front into a gravity current in the Ri = 0 case is illustrated with various fields at the surface: (a) salinity, (b) dissipation rate, (c) alongfront velocity, and (d) cross-front velocity. After shear instability mixes up the geostrophic jet, the gravity current forms and sharpens the lateral salinity gradient. The gravity current propagates at a constant speed toward the dense side of the front. Intense mixing indicated by a large value of dissipation rate occurs at the nose of the gravity current.

  • View in gallery

    Shear instability disrupts geostrophic balance in the Ri = 0 case. (a) Profiles of the cross-front pressure gradient (red) and the Coriolis force (blue) at the center of the front at t = 0 (dashed) and 0.37 h (solid); (b) an xz plane of the alongfront velocity u at the center of the front at t = 0.37 h. The shear instability extracts energy from the geostrophic current. The resulting turbulence causes the front to slump.

  • View in gallery

    Cross-front sections of (a) salinity and (b) cross-front velocity as the front begins to slump in the Ri = 0 case at t = 2.1 h. Shear instability triggered by the vertical gradient of the cross-front velocity develops along the slanted isopycnal. The negative surface current and the positive countercurrent in the region below are the early manifestation of the ASC.

  • View in gallery

    Propagation of the surface gravity current on the dense side of the front at t = 7.4 h in the Ri = 0 case: (a) salinity, (b) dissipation rate ε, and (c) cross-front velocity. The gravity current in (a) and (b) has a turbulent nose with elevated dissipation rate. Subduction of surface water in the region ahead of the nose to depths below the gravity current can be seen in (c).

  • View in gallery

    Propagation of the countercurrent on the light side of the front at t = 7.4 h in the Ri = 0 case: (a) salinity, (b) dissipation rate, and (c) cross-front velocity. The nose of the subduction layer is quiescent; however, turbulence is seen in the region that is behind it.

  • View in gallery

    The gravity current propagates at different speeds among cases: Ri = (a) 0 and (b) 0.25. (top) Symbols mark the position of the maximum lateral salinity gradient taken to be the location of the nose. Dashed line in the top panel of (a) plots the stated expression, which relates the position of the gravity current to the intrinsic wave speed and a constant Froude number Fr = 0.3. The expression is valid in the Ri = 0 case, but not in the other cases in which the buoyancy difference across the nose changes during the propagation. (bottom) The curves show the density difference across the nose at times denoted by the filled symbols having the same color in the top panels. The density difference across the nose shows little change with increasing time in the Ri = 0 case, but it decreases in the Ri = 0.25 case.

  • View in gallery

    Veering of the flow with depth at the nose (black) of the gravity current, 100 m in front of the nose (red), and 100 m behind the nose (blue) in the cases: Ri = (a) 0 and (b) 0.25. The profiles correspond to the time marked by the green dot on the top panels of Fig. 15. Large symbols mark the surface current; smaller symbols denote the current at every 1.5-m depth. The hodographs use different nonunity aspect ratios due to the significantly larger alongfront velocity component. The nose of the gravity current is defined as the location with peak negative cross-front velocity.

  • View in gallery

    Temperature fields at depth show tracks of the countercurrent in the simulated cases: Ri = (a) 0, (b) 0.13, (c) 0.25, and (d) 0.5. The countercurrent carries warm water from the dense side across the front. Note that the temperature is taken at different depths among the cases since the vertical extent of the ASC is shallower in the cases with stronger stratification.

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Ageostrophic Secondary Circulation at a Submesoscale Front and the Formation of Gravity Currents

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  • 1 Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, California
  • | 2 Mechanical and Aerospace Engineering, and Scripps Institution of Oceanography, University of California, San Diego, La Jolla, California
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Abstract

Large-eddy simulations are performed to investigate the development of the ageostrophic secondary circulation (ASC) and associated transport in a submesoscale front. Based on the observations in the northern Bay of Bengal and in the Pacific cold tongue, the model front has a large cross-front density difference that is partially compensated with lateral temperature and salinity gradients. Vertical stratification is varied in different cases to explore its effect on the ASC. The evolution of the ASC differs with stratification. When the front is unstratified, shear instabilities, which develop from the geostrophic shear, cause the front to slump. Cold water from the light side propagates across the front on the surface, while warm water from the dense side spreads in the opposite direction at depth. In cases with stratifications, a shear layer driven by the cross-front pressure gradient forms at the surface to initiate the ASC. Shear-driven turbulence associated with the enhanced shear in the layer causes the front to slump, and the development of the ASC onward is similar to the unstratified case. Irrespective of the initial stratification of the strong fronts simulated here, the surface layer evolves into a gravity current. The ASC is composed of the surface gravity current and a countercurrent that are separated by a middle layer with enhanced stratification and a thermal inversion. Turbulent dissipation is enhanced at the nose of the gravity current and in a sheared region somewhat behind the leading edge of the countercurrent. The gravity current propagates at a speed proportional to the buoyancy difference across the front in the case with no stratification.

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Sutanu Sarkar, sarkar@ucsd.edu

Abstract

Large-eddy simulations are performed to investigate the development of the ageostrophic secondary circulation (ASC) and associated transport in a submesoscale front. Based on the observations in the northern Bay of Bengal and in the Pacific cold tongue, the model front has a large cross-front density difference that is partially compensated with lateral temperature and salinity gradients. Vertical stratification is varied in different cases to explore its effect on the ASC. The evolution of the ASC differs with stratification. When the front is unstratified, shear instabilities, which develop from the geostrophic shear, cause the front to slump. Cold water from the light side propagates across the front on the surface, while warm water from the dense side spreads in the opposite direction at depth. In cases with stratifications, a shear layer driven by the cross-front pressure gradient forms at the surface to initiate the ASC. Shear-driven turbulence associated with the enhanced shear in the layer causes the front to slump, and the development of the ASC onward is similar to the unstratified case. Irrespective of the initial stratification of the strong fronts simulated here, the surface layer evolves into a gravity current. The ASC is composed of the surface gravity current and a countercurrent that are separated by a middle layer with enhanced stratification and a thermal inversion. Turbulent dissipation is enhanced at the nose of the gravity current and in a sheared region somewhat behind the leading edge of the countercurrent. The gravity current propagates at a speed proportional to the buoyancy difference across the front in the case with no stratification.

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Sutanu Sarkar, sarkar@ucsd.edu

1. Introduction

Frontal regions, where the density of seawater varies laterally, are known to promote instabilities that affect the vertical and horizontal structure of the upper ocean and thereby influence the vertical heat flux across the ocean surface layer. The primary frontal instability is the development of eddies from baroclinic instability (Stone 1971; Haine and Marshall 1998). These eddies have a wide range of horizontal scales ranging from hundreds of kilometers (mesoscale) down to a few kilometers (submesoscale) (McWilliams 2016). During their nonlinear evolution, the eddies generate smaller-scale fronts and filaments having an increasingly larger lateral density gradient, a process known as frontogenesis (Hoskins and Bretherton 1972). As the fronts get thinner, the scale of the fronts gets closer to the scale of turbulent eddies, which are responsible for vertical transport in the upper-ocean layer. Recent numerical and theoretical studies suggest that at this range of scales, the flow physics is dominated by turbulent thermal wind (TTW) balance rather than geostrophic balance (Gula et al. 2014; McWilliams et al. 2015; McWilliams 2016; Wenegrat and McPhaden 2016; McWilliams 2017; Sullivan and McWilliams 2018).

Recent field observations suggest that fronts can be sharpened down to a lateral scale of a few meters. In the Bay of Bengal, Jinadasa et al. (2016) report a thin front where the salinity increases sharply over a few meters. The thin front occurs within a larger-scale front where the lateral density is significantly weaker and spans tens of kilometers. Similarly, Warner et al. (2018) describe features where the temperature in the surface layer of the Pacific cold tongue increases rapidly over a few meters. Since these sharp fonts occur in a region with strong tropical instability wave (TIW) activity, the authors hypothesize that the fronts are generated from the sharpening of the wider TIW-induced fronts. In both studies, the thin fronts are observed to propagate approximately at the intrinsic wave speed, which suggests that they have evolved into gravity currents. The observations show elevated dissipation rates in the leading region of the gravity current, similar to what is seen in laboratory experiments of lock exchange (Griffiths 1986; Thomas and Linden 2007) or in the observations of gravity currents in river plumes (Kilcher and Nash 2010). The new observations raise the following question: Can a submesoscale front sharpen to a lateral scale of a few meters from kilometer scale and evolve into a gravity current?

Early studies on frontal instabilities, which rely on quasigeostrophic (QG) and semigeostrophic (SG) theories, show that a barotropic external deformation field applied to a front with a uniform lateral density gradient can induce an ageostrophic secondary circulation (ASC; Hoskins and Bretherton 1972). The ASC in Hoskins and Bretherton (1972) arises to adjust the perturbed geostrophic flow back into a state that is in geostrophic balance. The circulation develops into a downwelling limb on the dense side and an upwelling limb on the light side of the front. Water from the light side is transported across the front on the ocean surface, while subduction carries water from the dense side in the opposite direction (Hoskins and Bretherton 1972; Hoskins 1982). The evolution of the ASC is known to be an important frontogenetic mechanism. In the present study, we hypothesize that as a geostrophic front is perturbed by turbulence, the surface component of an ASC can develop into a gravity current, similar to those observed by Jinadasa et al. (2016) and Warner et al. (2018), and demonstrate this hypothesis through a high-resolution LES process study.

The early two-dimensional SG model simulations of a mesoscale front by Thompson (2000) show that even in the absence of wind forcing and barotropic deformation, an Ekman flow driven by the action of eddy viscosity on geostrophic shear at the surface induces the ASC. In the model, which uses vertically varying eddy viscosity to parameterize turbulence, the surface density gradient increases, and the location of the maximum gradient moves toward the dense side of the front. The importance of turbulence on the evolution of ASC is further strengthened by the TTW analysis of Gula et al. (2014), McWilliams (2017), and Sullivan and McWilliams (2018).

Numerous modeling studies have illustrated how the evolution of ASC is sensitive to the parameterization of turbulence. Garrett and Loder (1981) use an eddy viscosity mixing parameterization in which the diffusion coefficient depends on the local buoyancy frequency to predict a sharpening of the lateral density gradient (i.e., frontogenesis) on the dense side of a front. The frontogenetic features of the flow are shown to depend on the mixing parameterization. Horizontal mixing of density and vertical momentum mixing are shown to be analogous with the ratio , where is the squared buoyancy frequency, and f is the Coriolis parameter. In the study of Thompson (2000), the development of the ASC varies considerably when vertical mixing is included. When an elevated constant value of vertical viscosity is included near the surface, the model shows the development of an Ekman flow that drives the ASC. The Ekman flow is absent when vertical mixing is not included in the model. The numerical experiments of ASC by Nagai et al. (2006) with three different mixing formulations show that the intensity of the ASC varies with the mixing schemes. Although these numerical simulations reveal important features of the ASC, owing to their coarse grid resolution, they are unable to answer the question as to whether the surface component of the ASC can evolve into a gravity current.

Recent high-resolution turbulence-resolving numerical simulations (Skyllingstad and Samelson 2012; Hamlington et al. 2014; Thomas and Lee 2005; Sullivan and McWilliams 2018) of submesoscale fronts and filaments suggest that frontogenesis is arrested by turbulent mixing. Sullivan and McWilliams (2018) show that the frontogenetic process is saturated at a scale of approximately 100 m by turbulent mixing. Skyllingstad and Samelson (2012) reveal the development of horizontal shear instabilities that prevent the front from sharpening further. In both of the studies, a gravity current (e.g., sharp front on a scale of a few meters) does not form. In the present study, we use high-resolution LES to demonstrate that an ASC can indeed develop into a gravity current in a submesoscale front when the lateral buoyancy gradient across the front is sufficiently large and Coriolis parameter f is sufficiently small. Here, is the buoyancy. In a geostrophically balanced front, the vertical shear , associated with the geostrophic current u, is directly proportional to and inversely proportional to f. As gets larger and f gets smaller, the vertical shear becomes large, making the front more susceptible to turbulence in the upper ocean. Numerical simulations of the ASC thus far have only considered fronts with a weak lateral gradient at mid-to-high latitude with relatively large Coriolis parameter. The gravity currents reported by Jinadasa et al. (2016) and Warner et al. (2018) are observed at low latitudes where f is small. Furthermore, recent observations in the Bay of Bengal indicate submesoscale fronts in the Bay of Bengal have a uniquely large lateral buoyancy gradient (Gordon et al. 2016; MacKinnon et al. 2016). The ratio of measured across multiple fronts with a width of a few kilometers can be as large as 500 (D. Sengupta and S. Lekha 2017, personal communication). Among the aforementioned numerical simulations, the cold water filament in the LES of Sullivan and McWilliams (2018) has the highest value of , which is still smaller by approximately an order of magnitude than the values observed in the Bay of Bengal. In the present study, we construct a model of a submesoscale front with parameters motivated by the observations in the Bay of Bengal to explore the possible dynamical link of the ASC with the formation of a gravity current. Specifically, we show that when the front has a strong lateral buoyancy gradient, the surface component of the ASC will evolve nonlinearly into a gravity current, which induces a sharp jump in the lateral density over a short distance of a just few meters, as observed by Jinadasa et al. (2016) and Warner et al. (2018).

The paper is structured as follows. Setup of the model front, the parametric study wherein upper-ocean stratification is changed among cases, and the LES model are presented in section 2. Section 3 contrasts the processes through which the ASC develops and intensifies in the different cases. In section 4, we focus on how the ASC transports water across the front through the formation of gravity currents at the surface and a countercurrent at depth. We conclude in section 5 with a discussion on how the gravity current and subduction observed in the Bay of Bengal can be outcomes of the ASC.

2. Model setup

The model front is shown in Fig. 1, where Cartesian coordinates x, y, and z correspond to the alongfront, cross-front, and vertical direction, respectively. The front has a width of L = 1 km and a depth of H = 20 m. The shallow depth H is chosen based on the shallow gravity currents and bores observed in the Bay of Bengal (Jinadasa et al. 2016). The relatively small width L is motivated by recent observations of sharp submesoscale fronts in the Bay (D. Sengupta and S. Lekha 2017, personal communication). The front is assumed to be homogeneous in the alongfront x direction (e.g., variation in the alongfront direction is insignificant relative to changes in the cross-front direction). The front is partially compensated with lateral gradients of salinity S and temperature T along the cross-front direction. The lateral buoyancy gradient is maintained in thermal wind balance by a geostrophic jet, which is directed in the negative x direction. In our model, since the ocean surface is assumed to be flat, thermal wind balance and geostrophic balance are equivalent. At initial time, the contribution of the salinity to the lateral buoyancy gradient and that of the temperature are constructed using as the thermal expansion coefficient and as the haline contraction coefficient. The subscripts S and T denote the contribution of salinity and temperature gradients, respectively. The total lateral buoyancy gradient is the sum of and . Note that the density across the front decreases in the positive y direction. The lateral buoyancy gradient is largest at the center of the front . As the front evolves, the ASC drives the maximum lateral buoyancy gradient away from the center.

Fig. 1.
Fig. 1.

Initial cross-front sections of various fields in the Ri = 0.25 case: (a) salinity, (b) temperature, (c) alongfront velocity, (d) gradient Richardson number, and (e) potential vorticity. The solid lines in (a) and (b) are isohalines and isotherms, respectively. The Richardson number has a minimum value at the center of the front and increases toward the sides. The light side of the front has negative potential vorticity. The positive alongfront direction x is directed outward.

Citation: Journal of Physical Oceanography 48, 10; 10.1175/JPO-D-17-0271.1

The front has a vertical stratification owing to salinity and temperature gradients. The stratification extends below the front into the pycnocline. Within the front, the stratification is uniform with a constant squared buoyancy frequency: , where the subscript u denotes the (upper) front layer. At depths , the stratification is given by a constant , where the subscript l denotes the (lower) pycnocline layer. The mathematical expressions used to construct the model front are given below.

  • Geostrophic velocity:
    eq1
  • Salinity:
    eq2
  • Temperature:
    eq3
The values of lateral and vertical gradients of temperature and salinity used in the above expressions are listed in Table 1.
Table 1.

Parameters used to construct the model font with depth H = 20 m and width L = 1 km. The lateral buoyancy gradient and the vertical buoyancy gradient are given in the units of . Subscripts T and S denote the contribution from temperature and salinity gradients, respectively. Subscripts u and l denote parameters in the upper frontal region and in the lower pycnocline , respectively. The Coriolis parameter is taken to be , corresponding to a latitude of 18°N. The Richardson number and the Burger number are computed at the center of the front . All cases have the same value of and the same Rossby number .

Table 1.

The model front is simulated by solving the nonhydrostatic Navier–Stokes equations under Boussinesq approximation. Alongfront velocity u, cross-front velocity υ, vertical velocity w, salinity S, temperature T, and dynamic pressure P are advanced in time t as follows:
e1
Here, is a reference density, is the gravitational acceleration, and is the Coriolis parameter at 18°N. The salinity and temperature are deviations from a reference value. The model assumes that the molecular viscosity ν and the molecular diffusivities of salinity and temperature are all equal to . Note that the LES produces subgrid viscosities and subgrid diffusivities that are at least an order of magnitude larger than the molecular counterparts in the frontal zone.

A periodic boundary condition is used in the alongfront x direction. Free-slip no-flux conditions are used at the ocean surface () and side boundaries. Free-slip and constant salt and heat flux conditions are used at the bottom boundary to maintain the uniform vertical gradients in the pycnocline and . Sponge regions are constructed at sidewalls () and at the bottom boundary () to prevent the reflection of internal waves from sidewalls and bottom boundary into the frontal region. The governing equations in Eqs. (1) are advanced in time using a mixed third-order Runge–Kutta and Crank–Nicolson scheme. A second-order central finite-difference method is used to compute spatial derivatives. The dynamic pressure is obtained by solving the Poisson equation via a multigrid iterative method.

LES is used to parameterize the subgrid fluxes denoted by superscript “sgs” in Eq. (1). Following Ducros et al. (1996), the subgrid viscosity is computed dynamically at every grid point (i, j, k) using a local velocity structure function F:
e2
where is the Kolmogorov constant, is the magnitude of the filter grid spacing, and
e3
Here, is the three-component velocity field that is obtained by applying a discrete Laplacian filter to the resolved velocity fields as follows:
e4

The model leads to subgrid viscosity that is significant only where there is a significant velocity difference across adjacent grid points. Readers are referred to Ducros et al. (1996) for further details in the implementation of the subgrid viscosity. The LES model also assumes that the subgrid diffusivities for salinity and temperature are equal to the subgrid viscosity.

When we scale Eq. (1) by the maximum geostrophic jet velocity , the depth H, and the buoyancy scale , the nondimensional parameters are the Ekman number , the nondimensional lateral buoyancy gradient , and the Richardson number . The nondimensional parameters used in the present study are listed in Table 1, noting that they are computed at the center of the front () in the front layer at the initial time. The large ratio is motivated by the sharp lateral density gradient observed in the Bay of Bengal. The stratification increases in four cases, such that the minimum Richardson numbers at the center of the front have values of 0, 0.13, 0.25, and 0.5. The different Ri cases illustrate how vertical mixing by shear-driven turbulence influences the evolution of the ASC. The Richardson number can also be written as , where is the Burger number, and is the Rossby number. The Burger number in the present study varies from 0 to 7. The chosen Rossby number Ro = 9.8 is large for mesoscale dynamics; however, it falls within the range of submesoscale fronts in which both rotation and shear turbulence contribute to the flow physics (McWilliams 2016).

Figure 1d provides a snapshot of the gradient Richardson number in the cross-front direction at initial time. The Richardson number has the minimum value at the center of the front, and it increases with distance away from the center (). In the cases Ri = 0 and 0.13, the vertical shear due to the geostrophic jet is susceptible to shear instability, while the strong stratification in the cases Ri = 0.25 and 0.5 stabilizes the jet shear. Unlike the Richardson number, which is symmetrical across the front, the potential vorticity (PV) is asymmetric. PV (Fig. 1f) has a negative value on the lighter side of the front. Here, the PV is computed using the resolved vorticity field and the resolved buoyancy field .

In addition to the jet velocity, salinity, and temperature fields that define the model front as shown in Figs. 1a–c, broadband velocity fluctuations are added to the initial conditions. The fluctuations have zero mean when averaged in the alongfront direction, and they have the following spectral energy content:
e5
where k denotes a wavenumber, and is set to correspond to a wavelength of 20 m. The fluctuations have a peak amplitude of at the surface and decrease to zero at the bottom of the front ().

The computational domain is a rectangular box bounded by , , and . A grid of 64 points × 4096 points × 192 points is used in the x, y, and z directions, respectively. A high-resolution grid is used to capture frontogenetic processes that sharpen the lateral density gradient. The grid spacing is uniform in the alongfront direction with . In the cross-front direction, the grid is uniform with in the region , and it is stretched at a rate of 0.5% in the region outside with a maximum grid size of 8 m at the far ends. The vertical grid spacing is uniform with at depths , and it is stretched at a rate of 2% in the region below with a maximum grid size of 0.8 m at the bottom boundary.

The use of a thin domain in the alongfront x direction implies that the large lateral wavelength of baroclinic instability will not be captured on the computational domain. A similar assumption has been used in SG model simulations (Thompson 2000; Thomas and Lee 2005; Nagai et al. 2006). The length scale and time scale of baroclinic instability are given by Stone (1966) as follows:
e6
With the parameters listed in Table 1, varies from 39.1 to 47.9 km, and varies from 20.3 to 24.8 h. The present study focuses on the physics of the ASC that develops at a much faster time scale than the growth rate of baroclinic instability. The large value of and low value of Ri of the present problem provide preference to the development of shear instabilities over the growth of baroclinic stability. The ASC develops in our model as the initial geostrophic flow is perturbed by turbulence at the ocean surface. In all cases, the ASC develops into a rotating gravity current and a countercurrent within a time period of 30 h, which is shorter than an inertial period. The development of baroclinic instability and the meandering of fronts are not considered since the model assumes homogeneity in the alongfront direction. As such, unlike many other studies, we do not decompose the flow fields into geostrophic and ageostrophic components in the analysis. Instead, we follow the method of Sullivan and McWilliams (2018) and average the flow fields in the alongfront direction. In the following discussion, the full three-dimensional flow field F is decomposed into an x-averaged mean component, denoted by angle brackets , and a fluctuating component, denoted by primes (′). For example:
e7
where F can be the velocity components, dynamics pressure, salinity, or temperature. The unsteady, three-dimensional fluctuation field denotes the contribution of turbulence.

3. Development of ageostrophic secondary circulation

Previous studies using QG or SG equations indicate that the ASC associated with mesoscale fronts develops into a single cell that consists of upwelling on the lighter side and downwelling on the denser side of the front (Thompson 2000; Nagai et al. 2006). Figure 2 illustrates that the structure of the ASC seen in the present study is also consistent with a cell that has upwelling/downwelling limbs. At the surface, negative cross-front velocity carries light water across the front. At depth, positive cross-front velocity drives a countercurrent, which carries dense water in the opposite direction. Because of the circulation, the front slumps and restratifies with isopycnals in the front being flatter than they are at the initial time. A new result of the present simulations is that the surface flow in the strong submesoscale front evolves into a turbulent gravity current with a sharp lateral density gradient at its nose, and the countercurrent at depth develops into a turbulent intrusion layer, both of which contribute to the complex stratification in the front. We defer discussion of the gravity current to section 4. In the remainder of the present section, we focus on the process that initially triggers development of the ASC and illustrate how vertical stratification alters the process.

Fig. 2.
Fig. 2.

Structure of ASC near the end of the simulations: Ri = (a) 0, (b) 0.13, (c) 0.25, and (d) 0.5. Contours denote x-averaged cross-front velocity , and solid lines are isohalines. In all cases, the ASC is composed of a surface current that flows from the light side toward the dense side of the front and a countercurrent that flows in the opposite direction at depth.

Citation: Journal of Physical Oceanography 48, 10; 10.1175/JPO-D-17-0271.1

a. Stratified front

In the cases with stratification (Ri ≠ 0), the evolution of the ASC does not vary significantly with stratification. The main difference is in the vertical extent of the circulation when it is fully developed. As the stratification increases, the circulation is constrained to shallower depths. We choose the simulation with Ri = 0.25 to illustrate the evolution of the ASC. Figure 3 depicts the surface evolution of the front as it evolves from a quiescent geostrophically balanced state into a turbulent gravity current. The evolution consists of two periods: an initiation stage of approximately 10 h, during which the cross-front velocity is weak (Fig. 3d), and the fully developed stage, during which a surface gravity current propagates at a finite speed toward the dense side of the front. The salinity field in Fig. 3a and the alongfront velocity field in Fig. 3c indicate that the frontal dynamics changes qualitatively after the gravity current emerges. The nose of the gravity current is marked by a sharp jump in salinity over a few meters in Fig. 3a, and the dissipation rate is elevated by orders of magnitude in Fig. 3b. Here, the dissipation rate is computed as

Fig. 3.
Fig. 3.

Surface manifestation of (a) salinity, (b) dissipation rate, (c) alongfront velocity, and (d) cross-front velocity illustrate the development of the surface gravity current in the Ri = 0.25 case. The front loses geostrophic balance at approximately 10 h, after which the gravity current forms and sharpens the lateral salinity gradient at the nose. The gravity current propagates at a constant speed toward the dense side of the front. Intense mixing with a large value of dissipation occurs at the nose of the gravity current. All fields are averaged over the alongfront x direction.

Citation: Journal of Physical Oceanography 48, 10; 10.1175/JPO-D-17-0271.1

To understand how the front can evolve into the gravity current, we examine the development of cross-front velocity during the initiation stage. Figure 4 shows snapshots of cross-front velocity in the cross-front plane at three different times during this stage. At an early time (Fig. 4a), the surface cross-front velocity is slightly negative on the light side of the front where the potential vorticity is negative. A positive velocity develops underneath the surface negative velocity to form a single cell, which is the early structure of the ASC. On the dense side of the front, the velocity is slightly positive so that the surface flow converges at the center of the front (). This cross-front flow structure persists throughout the initiation stage, and its amplitude increases in time (Figs. 4b,c) until turbulence develops in the front.

Fig. 4.
Fig. 4.

Initial development of x-averaged cross-front velocity in the Ri = 0.25 case at different times: t = (a) 5.9, (b) 8.3, and (c) 10.6 h during the initiation stage of ASC. At the surface, positive cross-front velocity develops on the side of the front, while negative cross-front velocity develops on the other side in (a). Positive velocity grows at depth on the side at the same time. The cross-front velocity on the side intensifies in (b) and (c) and spreads toward the side. Solid lines denote isohalines. Note that the color bar in this figure has a different range than that used in Fig. 2.

Citation: Journal of Physical Oceanography 48, 10; 10.1175/JPO-D-17-0271.1

The development of the cross-front velocity seen in Fig. 4 is driven by the loss of geostrophic balance at the surface. At the initial time, the mean component of the flow field is in geostrophic balance. The cross-front pressure gradient is balanced by the Coriolis force as in the following equation:
e8
Figure 5a illustrates the momentum balance at the center of the front (y = 0) at the initial time with vertical profiles of Coriolis force and pressure gradient. Taking the derivative of Eq. (8) with respect to the depth z leads to an equation for geostrophic shear:
e9
Because of the stress-free boundary condition at the surface, the front deviates from the geostrophic balance: , while at the surface. The frictional (turbulent and viscous) surface response acts to retard the geostrophic velocity and impose an effective geostrophic stress (in the positive x direction) that, in a rotating frame, gives rise to a surface current (Thomas and Rhines 2002). The velocity vector of the surface current veers to the right of the effective stress toward the dense side of the front, as indicated by the negative cross-front velocity on the light side of the front shown in Fig. 4. The development of the cross-front velocity and associated shear layer seen in the present study is similar to the Ekman layer described in the study of Wenegrat and McPhaden (2016), who use an analytical model with depth-varying eddy viscosity to show that even in the absence of surface wind, turbulent stress can initiate a surface current, which directs toward the dense side of the front. However, different from Wenegrat and McPhaden (2016), the uniquely large cross-front pressure gradient plays an important role in the evolution, not just Coriolis force and friction.
Fig. 5.
Fig. 5.

Deviation from geostrophic balance in the Ri = 0.25 case. (a) Vertical profiles of the forces that balance the cross-front momentum equation at the initial time t = 0; (b) surface variability of at t = 0 shows a peak value at the center of the front; and (c) nonuniform increase of sea surface salinity between t = 0 and 3.86 h is due to the variability of across the front.

Citation: Journal of Physical Oceanography 48, 10; 10.1175/JPO-D-17-0271.1

The surface imbalance triggers development of a surface turbulent stress, which, in the present simulations, initially arises from the subgrid LES model. Figure 5b shows the cross-front profile of subgrid viscosity at the surface with nonzero values in the front at the initial time. Because of the finite values of the subgrid viscosity, and thus finite-amplitude turbulent stress, the evolution of the cross-front circulation follows the TTW balance as discussed by Gula et al. (2014), McWilliams (2017), and Sullivan and McWilliams (2018). The equation for TTW balance takes the following form:
e10
where the stress components and include both turbulent (resolved and subgrid) and viscous contributions. They are defined as follows:
eq4
Since the subgrid viscosity is enhanced in Fig. 5b, the stresses have a finite value at the surface. In the studies of Thompson (2000) and Nagai et al. (2006), a large value of eddy viscosity is explicitly prescribed at the surface to induce a surface stress. Figure 5b shows the subgrid viscosity with a value of approximately at the center of the front. The viscosity produced by LES subgrid parameterization in the present study is smaller than the values used by Thompson (2000) and Nagai et al. (2006). In cases with realistic upper-ocean turbulence, the effective eddy viscosity would be significantly higher, and the imbalance would be more pronounced. We note that the velocity fluctuations added to the flow field at the initial time also generate a turbulent (resolved and subgrid) stress; however, the amplitude of the stress is significantly smaller than the stress that arises due to the enhanced subgrid viscosity at the surface seen in Fig. 5b.

As the subgrid stress reduces the speed of the geostrophic jet at the surface, the subgrid diffusivity (taken to be equal to the subgrid viscosity in the model) affects the surface salinity and temperature across the front. The subgrid diffusivity reduces the vertical stratification and thus increases the surface salinity throughout the front, as seen in Fig. 3a during the early hours of the initiation stage. The surface salinity does not increase uniformly across the front due to the cross-front variability of the subgrid viscosity/diffusivity. Figure 5b indicates that the subgrid viscosity has a cross-front variability with a peak at the center of the front (). As a result, the surface salinity increases fastest at the center and slower at the sides of the front (Fig. 5c). The nonuniform increase in salinity alters the cross-front pressure gradient . The magnitude of the pressure gradient is reduced on the side of the front while it becomes larger on the side. The change in the cross-front pressure gradient further contributes to the loss of geostrophic balance.

Because of the effects of the enhanced subgrid viscosity and diffusivity at the surface, both the alongfront velocity and the lateral pressure gradient are modified across the front, as shown in Fig. 6. On the dense side of the front (), the Coriolis force overtakes the reduced pressure gradient, and, according to Eq. (10), the cross-front velocity increases in time (i.e., ). This leads to a positive cross-front velocity on this side of the front. In contrast, the enhanced pressure gradient overtakes the Coriolis force and drives a negative cross-front velocity on the light side of the front () since . The development of the cross-front velocity further alters the flow field in the alongfront direction, as can been seen from the Reynolds-averaged alongfront momentum equation:
e11
Here, the stress components and are defined as follows:
eq5
As the amplitude of the cross-front velocity increases, the alongfront momentum equation is dominated by the first term on the right-hand side of Eq. (11). The term is the product of absolute vertical vorticity and the cross-front velocity . Figure 6c shows that the absolute vertical vorticity changes sign across the front: positive on the dense side and negative on the light side. Since the sign of the cross-front velocity is similar to that of the vorticity (Figs. 6a,c), the product of the two quantities is positive on both sides of the front. As a result, and the alongfront velocity at the surface increases (i.e., it becomes less negative) in time across the front. It is noted that the amplitude of the acceleration is larger on the side of the front because the cross-front velocity is considerably larger there.
Fig. 6.
Fig. 6.

Surface flow convergence at t = 5.8 h during the initiation stage of the ASC in the Ri = 0.25 case. (a) The surface cross-front velocity changes sign near the center of the front; (b) the imbalance between the Coriolis force and the lateral pressure gradient drives the cross-front velocity; and (c) the absolute vertical vorticity also has opposite signs across the front, as indicated by the filtered vorticity (red). The unfiltered absolute vertical vorticity (blue) exhibits strong spatial fluctuations due to broadband turbulence.

Citation: Journal of Physical Oceanography 48, 10; 10.1175/JPO-D-17-0271.1

The deviation from the geostrophic balance at the initial stage begins with the development of a shear layer, which is the precursor to the gravity current. As the surface subgrid viscosity increases in time, the shear layer thickens, as seen in the profiles of the cross-front and alongfront velocity components in Figs. 7a and 7b, respectively. The vertical profiles taken on the light side of the front at y = 150 m illustrate how the shear layer transforms into the gravity current. As the shear layer forms, the surface current veers to the right toward the dense side of the front. The alongfront velocity at the surface layer becomes less negative in Fig. 7b, while the cross-front velocity becomes more negative at the surface in Fig. 7a. As a result, the vertical shear of the cross-front velocity shown in Fig. 7c increases in magnitude at all depths in the front. Meanwhile, the alongfront shear in Fig. 7d shows a sign change: positive shear near the surface and negative shear at depth. At the initial time, the vertical shear associated with the geostrophic jet is negative. As the shear layer forms, as discussed above. The alongfront velocity at the surface becomes less negative when compared to the velocity at depth, which gives rise to a positive gradient of the alongfront velocity near the surface. Although the two components of the vertical shear have opposite signs near the surface, the magnitude of the combined vertical shear increases in time. At the same time, the stratification decreases inside the layer (Fig. 7e) owing to transport and mixing. With the increasing shear and decreasing stratification, the Richardson number reduces to <0.25 in the shear layer (Fig. 7f). The reduction in Ri does not occur throughout the vertical extent of the front. There are two layers of Ri < 0.25 that are separated by a layer of Ri > 0.25. The top layer of Ri < 0.25 subsequently evolves into the gravity current while the bottom layer develops into the countercurrent.

Fig. 7.
Fig. 7.

Development of shear instability after the loss of geostrophic balance is shown by profiles at three different times during the initiation stage of the ASC in the Ri = 0.25 case: (a) cross-front velocity, (b) alongfront velocity, (c) cross-front shear component , (d) alongfront shear component , (e) stratification, and (f) gradient Richardson number. The profiles are taken on the light side of the front at y = 150 m. Because of the loss of balance, a shear layer forms near the surface and causes the magnitude of the vertical shear to increase and the stratification to decrease. As a result, the gradient Richardson decreases to <0.25, and the current induces shear instability.

Citation: Journal of Physical Oceanography 48, 10; 10.1175/JPO-D-17-0271.1

As Ri decreases to <0.25, the shear in the top layer becomes unstable. Figure 8a indicates that turbulence arises in the shear layer on the light side of the front. The shear-driven turbulence develops along the slanted isohalines, and it spreads laterally and downward in time. The cross-front momentum balance in Fig. 8b indicates that as the shear layer becomes turbulent, the pressure gradient overtakes the Coriolis force to give rise to the negative cross-front velocity, which supports the development of the gravity current. Furthermore, according to Eq. (10), a positive cross-front gradient of the turbulent stress () acts to strengthen the surface gravity current. Figure 8c shows a positive at the surface in the immediate vicinity of the turbulent patch toward the light side of the front (0 < y < 0.15 km).

Fig. 8.
Fig. 8.

Development of a gravity current at t = 10.64 h after the onset of shear-driven turbulence in the surface shear layer in the Ri = 0.25 case. (a) A snapshot of salinity field shows turbulent mixing along slanted isopycnals on light side of the front; (b) the dominance of lateral pressure gradient in the momentum balance drives the gravity current to the left at the surface; and (c) the positive lateral gradient of turbulent stress increases the speed of the gravity current and sharpens the lateral density gradient at its nose. The dashed line in (a) indicates the location where the profiles are plotted in Fig. 7. The cross-front variation shown in (b) and (c) is taken at the surface.

Citation: Journal of Physical Oceanography 48, 10; 10.1175/JPO-D-17-0271.1

Similarly, the sheared lower layer with Ri < 0.25 in Fig. 7f also becomes turbulent. Figure 9a provides a snapshot of the turbulence at depth on the light side of the front. The turbulence in the figure extends from the surface down to the bottom of the front. We note that the cross-front velocity changes sign over the vertical extent of the turbulent patch in Fig. 7a. The patch includes the turbulence that gives rise to the gravity current with and also the turbulence that subsequently aids the development of the countercurrent. The turbulence disturbs the geostrophic balance between the Coriolis force and the pressure gradient at depth, as shown in Fig. 9b. Furthermore, the turbulent stress contributes to the development of the countercurrent. Figure 9c shows that the stress on the right side of the turbulent patch decreases by approximately over a cross-front distance of 50 m. The lateral gradient of the stress () is as large as the Coriolis force in Fig. 9b. According to Eq. (10), the two forces act together to produce a positive and thus drive the countercurrent.

Fig. 9.
Fig. 9.

Development of countercurrent at t = 11.4 h after the onset of turbulence in the Ri = 0.25 case. (a) A snapshot of salinity field shows turbulence generation in the bottom half of the front; (b) the turbulence breaks the momentum balance between the pressure gradient and the Coriolis force ; (c) the negative lateral gradient of the turbulent stress on the right side of the turbulent patches contributes to the propagation of the countercurrent toward the light side of the front at depth.

Citation: Journal of Physical Oceanography 48, 10; 10.1175/JPO-D-17-0271.1

As the countercurrent propagates at depth toward the light side of the front, there is an associated surface signal. In Fig. 3b, there is a thin strip of elevated dissipation rate, which forms at approximately 10 h near the center of the front and extends toward the light side in time. The mixing spreads faster than that at the nose of the gravity current on the dense side of the front. The strip is accompanied with elevated negative cross-front velocity in Fig. 3c. As the leading nose of the countercurrent propagates toward the light side at depth, the water immediately above the nose is pushed toward the dense side, creating the strip of negative velocity at the surface. Furthermore, the region behind the leading edge of the countercurrent is turbulent, as can be seen from Fig. 9a, and the surface manifestation of this turbulent region leads to finescale salinity disturbances that were seen in Fig. 3a after approximately 10 h in the region km.

b. Unstratified front

Unlike the stratified fronts discussed above, the ASC in the Ri = 0 case evolves in a different manner, despite having the same ASC structure at the end of the simulation. Figure 10 displays the evolution of the gravity current at the surface. In this case, the geostrophic balance disintegrates at an earlier time, and the gravity current develops after approximately 2 h. The loss of geostrophic balance is driven by intense turbulence that occurs symmetrically over a wide extent of the front, as shown in Fig. 10b. The turbulence generated during this initiation period is stronger than the turbulence seen at the nose of the gravity current that is subsequently formed. The strong turbulence and associated mixing reduce the alongfront velocity. Owing to the stronger mixing relative to the Ri = 0.25 case, the peak value of the alongfront velocity at h in Fig. 10c is considerably smaller than that in the Ri = 0.25 case (Fig. 3c).

Fig. 10.
Fig. 10.

Development of the front into a gravity current in the Ri = 0 case is illustrated with various fields at the surface: (a) salinity, (b) dissipation rate, (c) alongfront velocity, and (d) cross-front velocity. After shear instability mixes up the geostrophic jet, the gravity current forms and sharpens the lateral salinity gradient. The gravity current propagates at a constant speed toward the dense side of the front. Intense mixing indicated by a large value of dissipation rate occurs at the nose of the gravity current.

Citation: Journal of Physical Oceanography 48, 10; 10.1175/JPO-D-17-0271.1

Similar to the stratified cases, the front in the Ri = 0 case loses geostrophic balance initially at the surface due to the effect of enhanced subgrid viscosity. However, the absence of stratification modifies the development of the ASC in the following ways. First, the subgrid diffusivity does not increase the surface salinity, as was seen in the Ri = 0.25 case (Fig. 3a). As a result, the positive cross-front velocity on the heavy side and the negative cross-front velocity on the light side of the front that had developed during the initiation stage in Fig. 6a are not seen in this case. Second, since the gradient Richardson number is initially zero, the vertical shear is immediately susceptible to shear instabilities; the front quickly becomes turbulent. The turbulence patch that develops during the initiation stage in Fig. 10b is symmetric across the front due to the cross-front symmetry of the vertical shear of the geostrophic jet.

Figure 11 illustrates how the shear instability disrupts the geostrophic balance. The alongfront velocity on an xz plane at an early time, t = 0.37 h in Fig. 11b, shows coherent corrugations along the lines of constant alongfront velocity. The fluctuations indicate the development of shear instability into turbulence. At this time, the cross-front velocity is small, and the cross-front shear component does not contribute to the growth of shear instability. The instability extracts energy from the geostrophic jet and thus destroys geostrophic balance, as shown in Fig. 11a. At the initial time, the lateral pressure gradient is nearly in balance with the Coriolis force [e.g., Eq. (8)]. As the shear instability develops, the difference between the two terms becomes significant, and the front loses the geostrophic balance. The instability develops over the entire depth of the front and extends symmetrically away from the center. As the turbulence develops, the alongfront velocity is reduced (Fig. 10c). According to Eq. (10), when the lateral pressure gradient overtakes the Coriolis force at the surface, the flow induces a negative cross-front velocity. At the surface, water from the light side crosses over the center of the front to form the gravity current, while water from the dense side flows in the opposite direction to form the countercurrent at the bottom of the front.

Fig. 11.
Fig. 11.

Shear instability disrupts geostrophic balance in the Ri = 0 case. (a) Profiles of the cross-front pressure gradient (red) and the Coriolis force (blue) at the center of the front at t = 0 (dashed) and 0.37 h (solid); (b) an xz plane of the alongfront velocity u at the center of the front at t = 0.37 h. The shear instability extracts energy from the geostrophic current. The resulting turbulence causes the front to slump.

Citation: Journal of Physical Oceanography 48, 10; 10.1175/JPO-D-17-0271.1

As the turbulence that develops during the initiation stage dissipates, the front slumps, as illustrated in Fig. 12. The opposing cross-front currents in Fig. 12b induce a vertical shear . Shear instability associated with this cross-front shear component develops into overturns along the slanted isopycnals in the upper 5 m of the front. Shear instability also develops in the countercurrent region, . The development of shear instabilities divides the front vertically into three layers: a quiescent middle layer sandwiched by two turbulent layers at the top and bottom. The middle layer has a strong vertical shear and a strong stratification; however, turbulence is weak there. The turbulence in the top and bottom layers spreads vertically in time so that the middle layer gets thinner. Since the front is partially compensated with warmer water on the dense side, the countercurrent brings warmer water across the front. The temperature in the bottom layer is warmer than that in the top layer (i.e., the middle layer has a thermal inversion). The layered structure of the cross-front velocity and salinity in Fig. 12 gives rise to the ASC in this case.

Fig. 12.
Fig. 12.

Cross-front sections of (a) salinity and (b) cross-front velocity as the front begins to slump in the Ri = 0 case at t = 2.1 h. Shear instability triggered by the vertical gradient of the cross-front velocity develops along the slanted isopycnal. The negative surface current and the positive countercurrent in the region below are the early manifestation of the ASC.

Citation: Journal of Physical Oceanography 48, 10; 10.1175/JPO-D-17-0271.1

In this section, we have illustrated two different processes, both of which lead to the ASC. During the first process that arises in the stratified cases, the LES subgrid viscosity induces an ageostrophic near-surface flow in response to the zero-shear boundary condition at the surface. The shear layer formed by the ageostrophic flow thickens laterally and vertically, the local shear rate increases, and drops below 0.25 to drive resolved-scale turbulence that further strengthens the ASC. During the second process, which is illustrated in the Ri = 0 case, shear instability develops directly from the vertical shear of the geostrophic jet, and the shear-driven turbulence induces the ASC. Despite the different routes through which shear-driven turbulence forms to disintegrate the geostrophic balance, both processes ultimately result in a gravity current at the surface and a countercurrent at depth as components of the ASC. The development of the ASC after the onset of turbulence is qualitatively similar in all cases.

4. Transport by ageostrophic secondary circulation

In the previous section, we have illustrated the processes through which an ASC can form inside a strong submesoscale front. During the development, the ASC is shown to be composed of a gravity current at the surface and a countercurrent at depth. In this section, we examine the structure as well as the propagation of the gravity current and the countercurrent to highlight how the ASC transports water laterally and vertically across the front. Since the development of the ASC after the onset of turbulence and the emergence of the gravity current is similar in all cases, we focus mainly on the case Ri = 0.

Figure 13a depicts the nose of the gravity current where the surface salinity changes rapidly over just a few meters. The sharp salinity jump corresponds to an enhanced with values as high as 5000, which is two orders of magnitude larger than the initial value of at the submesoscale front. The shear-driven turbulence inside the nose remains strong during the cross-front propagation. The elevated dissipation rate in Fig. 13b is at least three orders of magnitude larger than the value in the surrounding area. Subduction occurs in the region ahead of the nose, and positive velocity is seen in the region below the gravity current. The structure of the ASC takes a final form when the gravity current reaches the dense end of the front.

Fig. 13.
Fig. 13.

Propagation of the surface gravity current on the dense side of the front at t = 7.4 h in the Ri = 0 case: (a) salinity, (b) dissipation rate ε, and (c) cross-front velocity. The gravity current in (a) and (b) has a turbulent nose with elevated dissipation rate. Subduction of surface water in the region ahead of the nose to depths below the gravity current can be seen in (c).

Citation: Journal of Physical Oceanography 48, 10; 10.1175/JPO-D-17-0271.1

On the light side of the front, the bottom turbulent layer (early time structure shown in Fig. 12) develops into a countercurrent at late time, as depicted in Fig. 14a. Isohalines near the bottom of the front converge to form the nose of the countercurrent. Unlike the nose of the gravity current, the nose of the countercurrent is quiescent. The strong vertical shear in the region immediately trailing the nose in Fig. 14c is stabilized by the strong stratification there, as shown in Fig. 14a. Turbulence develops in the region that is somewhat behind the nose. In this region, the stratification is weaker, and the shear is stronger. The dissipation rate in Fig. 14c indicates that the turbulence inside the countercurrent layer is considerably stronger than that in the surface layer. As the ASC develops to its full lateral extent, turbulence in the region behind the nose of the countercurrent becomes equally strong as that in the nose of surface gravity current.

Fig. 14.
Fig. 14.

Propagation of the countercurrent on the light side of the front at t = 7.4 h in the Ri = 0 case: (a) salinity, (b) dissipation rate, and (c) cross-front velocity. The nose of the subduction layer is quiescent; however, turbulence is seen in the region that is behind it.

Citation: Journal of Physical Oceanography 48, 10; 10.1175/JPO-D-17-0271.1

Theoretical studies and laboratory experiments suggest that a gravity current with a constant value of lateral buoyancy difference propagates at a speed of , where is the buoyancy difference across the gravity current, and Fr is the Froude number (Benjamin 1968). From this relation, the position of the nose of the gravity current can be described by the following expression:
e12
where and are the reference position and time when the nose of the gravity current is first identified by the sharp lateral salinity gradient previously shown in Fig. 3. The top panel of Fig. 15a marks the position of the gravity current across the front at different times in the Ri = 0 case. The position, which is identified by the maximum lateral salinity gradient, indicates that the gravity current propagates at a constant speed with Fr = 0.3. To estimate Fr, we compute the speed by fitting a straight line through the position of the noses at the various times in the top panel of Fig. 15a to find Ug = −0.037 m s−1. The dashed line in the panel shows the position of the gravity current as described by Eq. (12), in which H = 20 m, and is the buoyancy difference across the front at initial time. In laboratory experiments of lock exchange, Marino et al. (2005) report that the values of Fr range from 0.4 to 0.5 when the gravity current does not rotate as it propagates. In experiments of a rotating gravity current, Thomas and Linden (2007) find that Fr varies between 0.1 and 1.51. The value of Fr = 0.3 in the present simulation is slightly smaller than the value reported by Marino et al. (2005) but well within the range of values reported by Thomas and Linden (2007).
Fig. 15.
Fig. 15.

The gravity current propagates at different speeds among cases: Ri = (a) 0 and (b) 0.25. (top) Symbols mark the position of the maximum lateral salinity gradient taken to be the location of the nose. Dashed line in the top panel of (a) plots the stated expression, which relates the position of the gravity current to the intrinsic wave speed and a constant Froude number Fr = 0.3. The expression is valid in the Ri = 0 case, but not in the other cases in which the buoyancy difference across the nose changes during the propagation. (bottom) The curves show the density difference across the nose at times denoted by the filled symbols having the same color in the top panels. The density difference across the nose shows little change with increasing time in the Ri = 0 case, but it decreases in the Ri = 0.25 case.

Citation: Journal of Physical Oceanography 48, 10; 10.1175/JPO-D-17-0271.1

When the stratification increases, the propagation speed of the gravity current is considerably smaller. Equation (12) can no longer be applied due to the complex background that emerges from the involved mixing processes. The top panel of Fig. 15b illustrates the propagation in the Ri = 0.25 case. The propagation speed is approximately a factor of 3 smaller than that in the Ri = 0 case. The slower speed is due to the decreasing density difference that drives the gravity current. In the Ri = 0 case, the lateral density difference across the nose of the gravity current remains the same at different times, as shown in the bottom panel of Fig. 15a. The density difference decreases in time in the case Ri = 0.25 (the bottom panel of Fig. 15b). Vertical mixing increases the surface density behind the nose and, therefore, reduces the lateral density difference.

Because of the effect of rotation, the gravity current exhibits significant veering, as illustrated by the hodographs in Fig. 16 for the cases with Ri = 0 and 0.25. The hodograph for each value of Ri is plotted at the time marked by the green symbol in the corresponding upper panel of Fig. 15 and at three locations: at the nose of the gravity currents and at 100 m in front and behind the nose. In the case Ri = 0, the background flow in the front of the nose consists of a weak, slightly negative cross-front υ velocity (red profile in Fig. 16a), while the alongfront u velocity is relatively larger, with its amplitude decreasing with depth. The positive υ with weak u at depth marks the trailing end of the countercurrent. At the nose (black profile), the surface current is dominated by negative u (i.e., along the direction of the geostrophic jet) and has negative υ. The subsurface flow at the nose exhibits a relatively thick region with positive υ, counter to the surface υ. The veering of the flow becomes stronger behind the nose (blue profile). Relative to the surface current, the current at depth veers to the right and creates a spiral structure with similarities to an Ekman spiral driven by a surface wind stress in a stratified fluid (Pham and Sarkar 2017). In the present study with no wind stress, the rightward veering reflects the effect of the Coriolis force on the propagation of the gravity currents. The gravity current in the case Ri = 0.25 also exhibits a similar spiral structure, as shown in Fig. 16b, despite having a smaller magnitude of cross-front velocity.

Fig. 16.
Fig. 16.

Veering of the flow with depth at the nose (black) of the gravity current, 100 m in front of the nose (red), and 100 m behind the nose (blue) in the cases: Ri = (a) 0 and (b) 0.25. The profiles correspond to the time marked by the green dot on the top panels of Fig. 15. Large symbols mark the surface current; smaller symbols denote the current at every 1.5-m depth. The hodographs use different nonunity aspect ratios due to the significantly larger alongfront velocity component. The nose of the gravity current is defined as the location with peak negative cross-front velocity.

Citation: Journal of Physical Oceanography 48, 10; 10.1175/JPO-D-17-0271.1

As the surface gravity current carries cool freshwater toward the dense side of the front, the subsurface flow transports warm saline water in the opposite direction at depth. Figure 17 illustrates the lateral spreading of warm water at the bottom of the ASC in the four simulated cases. Unlike the gravity current at the surface, the temperature at depth does not show a sharp jump over a lateral scale of a few meters. The lateral temperature change in the countercurrent is relatively more gradual. At the nose of the surface gravity current, shear instability develops only along the bottom interface between light and dense fluid so that the light water behind the gravity current can reach the nose of the gravity current so as to create a sharp density jump at the surface. In contrast, the countercurrent propagates across the front at depth with a vertical profile of cross-front velocity that resembles a jet. The jet has shear instabilities that develop on both its upper and lower flanks so that the resulting turbulence prevents the warm water in the trail of the countercurrent from reaching the leading nose. We note that the thin strip of cold water on the light side of the front in the Ri = 0 case (Fig. 17a) is created by the intense turbulent mixing in the lower flank of the countercurrent jet (previously shown in Fig. 14), which brings cold water upward from the pycnocline into the surface mixed layer. The turbulent mixing is not as strong in the other cases where there is vertical stratification of the front in the initial conditions. The spreading of warm water at depth creates a thermal inversion layer where temperature in the countercurrent is higher than in the surface gravity current. This thermal inversion layer is capped by the middle layer where the stratification is considerably stronger. Heat from the surface on the dense side of the front can be transported downward and trapped at depth by the ASC.

Fig. 17.
Fig. 17.

Temperature fields at depth show tracks of the countercurrent in the simulated cases: Ri = (a) 0, (b) 0.13, (c) 0.25, and (d) 0.5. The countercurrent carries warm water from the dense side across the front. Note that the temperature is taken at different depths among the cases since the vertical extent of the ASC is shallower in the cases with stronger stratification.

Citation: Journal of Physical Oceanography 48, 10; 10.1175/JPO-D-17-0271.1

5. Summary and discussion

In the present study, we have performed high-resolution LES of a submesoscale front to show that an ageostrophic secondary circulation (ASC) in a submesoscale front can develop into a gravity current. The model front, which is initially in geostrophic balance, has a strong lateral density gradient across the front and a small Coriolis parameter. The front is partially compensated with dense, warm water on one side and light, cold water on the other. The lateral buoyancy gradient is two orders of magnitude larger than the squared inertial frequency . The time scale associated with the initial vertical shear is also considerably smaller than the inertial time scale. As a result, the front is susceptible to shear instabilities and turbulence, which are nonhydrostatic dynamics that cannot be described by quasigeostrophic (QG) or semigeostrophic (SG) theories. Our choice of the model front is motivated by observations in the northern Bay of Bengal where the inertial frequency f is relatively small and where the lateral density gradient between fresh river water and the ocean water is large. Vertical stratification is varied in four cases to illustrate its effect on the flow and the resultant distribution of salt and heat across the front.

An ASC with a downwelling limb on the dense side and upwelling limb on the light side of the front develops in time. The LES shows how the instabilities associated with the establishment of the ASC are influenced by the stratification. When the front is unstratified, shear instability associated with the vertical shear of the geostrophic jet grows and breaks the momentum balance. Subsequently, a gravity current forms near the surface, and a countercurrent develops in the bottom half of the front. The gravity current and the countercurrent propagate in opposite directions toward the opposite ends of the front to form the ASC. In cases with stratification, a shear layer driven by the cross-front pressure gradient forms at the surface. The layer is initiated by the degradation of geostrophic balance at the surface by the LES subgrid stress. The shear layer associated with the ageostrophic current becomes unstable with turbulence that is resolved by these high-resolution simulations, and, subsequently, a gravity current and an underlying countercurrent develop to form the ASC.

The simulations illustrate that the surface gravity current and the countercurrent below are highly turbulent and lead to multiscale variation of salinity and temperature (Sarkar et al. 2016). The nose of the gravity current exhibits large density overturns and strong dissipation rate. The speed of the gravity current is shown to be proportional to the lateral buoyancy difference across the nose in the case with no stratification. While the gravity current and the countercurrent propagate in opposite directions, a middle layer with a strong vertical salinity gradient forms between them. The ASC is shown to bring cold water from the light side toward the dense side of the front at the surface. At depth, the ASC carries warm water from the dense side toward the light side. The cross-front transport of heat creates a thermal inversion in the middle layer where the water is warmer than at the surface.

The development of ASC in our simulations reveals features that are similar to recent observations at frontal regions in the Bay of Bengal as well as in the Pacific cold tongue. They include the following:

  • Jinadasa et al. (2016) observe a nonlinear bore with a sharp jump of lateral salinity gradient and enhanced turbulent dissipation. The surface gravity current in the model also sharpens the lateral density gradient to a scale of a few meters and enhances the turbulent dissipation. The observed bore shows evidence of subduction in the region ahead of the bore. In our model, similar subduction occurs as the gravity current reaches the dense side of the front.

  • Warner et al. (2018) illustrate the observation of a gravity current propagating in the surface layer of the Pacific equatorial Ocean. The gravity current is shown to develop from a sharp front associated with tropical instability waves. The gravity current brings freshwater from the north toward the equator and creates a barrier layer in its path. In our model, as the surface gravity current and the countercurrent propagate in opposite directions, a barrier layer is also formed between the two currents. This barrier in the computational model has a large vertical stratification due to the salinity gradient that separates fresh, cool water in the gravity current from the warm, saline water in the countercurrent.

The study of Nagai et al. (2006), which is based on QG and SG theories, indicates vertical mixing influences the development of ASC at mesoscale fronts. Using a constant horizontal eddy diffusivity, three different vertical mixing formations including KPP parameterization were investigated by Nagai et al. (2006) to show that vertical mixing can intensify the ASC. In submesoscale fronts, turbulent mixing becomes an important term in the cross-front momentum balance. At the submesoscale, the front is no longer in geostrophic balance but rather in TTW balance, a state in which turbulence can sharpen the lateral density gradient and arrest the front at the same time (Gula et al. 2014; McWilliams 2017). Results from the present simulations further demonstrate that both vertical and horizontal gradients of the turbulent mixing are important to the development of the ASC. As the surface gravity current and the subducting countercurrent propagate across the front, the associated turbulence varies highly in the cross-front direction. The development of the gravity current and the subduction of warm water to depths are driven by the lateral gradient of turbulent fluxes, which must be adequately represented by the computational model.

An important feature of ASC is its capability to sharpen the lateral density gradient to generate smaller-scale fronts (Thompson 2000). The LES of Sullivan and McWilliams (2018) show that the ASC in a submesoscale front can generate fronts as thin as 100 m, a scale at which turbulence arrests the frontogenetic process. In the present study, the frontogenesis associated with the development of the ASC is not arrested at this scale. The lateral density gradient continues to sharpen down to the scale of a few meters. At this fine scale, the surface component of the ASC evolves into a gravity current. We attribute the difference to the uniquely large density gradient across the fronts and the small value of Coriolis parameter prescribed in the model. The ratio of used in the present study is an order of magnitude larger than the value used by Sullivan and McWilliams (2018). Another possible reason is differences in the initial turbulence prescribed at the front. In the present study, the initial turbulence added to the flow field is weak. Sullivan and McWilliams (2018) initialize their models with significantly stronger turbulence, such that the front is in TTW balance at the onset of the simulation. In their model, the turbulence is maintained throughout the simulations by surface wind and cooling. While convective cooling can arrest fronts by enhancing surface turbulence, the effect of wind on frontogenesis depends on the wind direction (Thomas and Lee 2005). A future study of the competition between frontal turbulence and turbulence generated by wind is necessary to determine the role of wind forcing on the development of ASC of a strong front.

Acknowledgments

We are grateful for the support provided by Office of Naval Research Grant N00014-15-1-2613 and N00014-17-1-2735.

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