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  • View in gallery

    Simulation setup for the DNS runs showing the size of the nondimensionalized domain, direction of the rotation, buoyancy forcing at the top surface in color, and surface wind stress magnitude and direction in the panel above the domain.

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    (a) Density distribution at the surface over the Southern Ocean from ECCO2 output and from buoyancy boundary condition imposed in the simulations in this paper as a function of latitude. (b) Surface wind stress profiles for each of the simulations as a function of latitude. WF1 and WF2 are sinusoidal profiles with symmetric easterlies and westerlies. WF3 is a polynomial fit of the ECCO2 wind stress profile over the Southern Ocean, resulting in stronger westerlies. WF4 is a polynomial fit of doubled wind stress from ECCO2. The black axes (bottom and left) represent the parameters in the dimensional form to compare with the ocean values, and the blue axes (top and right) represent the nondimensionalized values used in the DNS.

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    Evolution of the APE and KE reservoirs over the nondimensional simulation time. All energy budget terms are averaged over the final 80 time units when KE reaches steady state.

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    Instantaneous 3D structures of (a) reduced gravity and (b) Ro^=ζ/f for WF3 simulation at a time within the statistic steady state period at selected slices of the domain. Submesoscale eddies, for which Ro~O(1), are resolved (in contrast, mesoscales have Ro ≪ 1).

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    (left) Zonally and temporally averaged fields of g^0.5 (reduced gravity with Δg^/2 subtracted to emphasize lighter and denser fluids) and (right) density–latitude overturning streamfunction Ψyb remapped onto y and pseudo-z coordinates for (a),(f) BF; (b),(g) WF1; (c),(h) WF2; (d),(i) WF3; (e),(j) WF4. Black contours in (a)–(e) are isopycnals. In (f)–(j), positive cells have clockwise circulation, and negative cells counterclockwise circulation.

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    Ekman pumping vertical velocities for each simulation at the surface computed from the wind stress profiles in Fig. 2b as functions of simulation latitude y. Black dashed line represents the zero axis.

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    Density–latitude overturning circulation function separated int (left)o mean and (right) turbulent components for (a),(b) WF2; (c),(d) WF3; and (e),(f) WF4. Positive cells have clockwise circulation, and negative cells have counterclockwise circulation.

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    Sensitivity of the transport along outcropping isopycnals (Ψupper in red circles), the deep northward transport of dense water (ΨAABW in blue squares), and the southward transport of surface water (ΨAASW in black triangles), and to (a) maximum westerlies wind stress (order of simulations: BF, WF1, WF2, WF3, WF4) and (b) maximum easterlies wind stress (order of simulations: BF, WF1, WF3, WF2, WF4); (c) the ratio of ΨAABW to Ψupper, which indicates the strength of buoyancy driven to wind driven regimes, as functions of both westerlies (teal squares; order of simulations: BF, WF1, WF2, WF3, WF4) and easterlies (purple circles; order of simulations: BF, WF1, WF3, WF2, WF4) maximum wind stresses.

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    Energy diagram showing sources, sinks of and exchanges between KE and APE reservoirs based on (9)(12) in text.

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    Energy budget terms from Fig. 9 plotted as functions of KE generation for each simulation. (a) KE, (b) APE, (c) mean and turbulent vertical buoyancy fluxes, (d) APE generation rates, (e) conversion rates between mean and turbulent KE, (f) conversion rates between mean and turbulent KE, (g) KE dissipation rates, and (h) APE dissipation rates. For (g) and (h), the dissipation values calculated explicitly from the expressions in (9)(12) are marked with solid lines, and the values calculated as residuals from steady state with dashed lines. The order of simulations is labeled in (d).

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    Time-averaged distribution of (left) logD(EKf) and (right) logD(EAt) for WF3 for (a),(b) zonal averages and (c),(d) surface distribution. Both KE and APE dissipation rates are high near the surface and in the convective region near the southern end. KE dissipation is also amplified by the surface wind stress.

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    Temporally and zonally averaged distribution of logD(EAf) as a function of latitude y for BF (dashed), WF2 (dash–dotted), WF3 (dotted), and WF4 (solid) at (a) surface, (b) average over top third of domain, (c) average over middle third of domain, and (d) average over bottom third of domain.

  • View in gallery

    Depth–density overturning circulation functions: (left) total, (center) mean, (right) turbulent for each simulation, (a)–(c) BF, (d)–(f) WF2, (g)–(i) WF3, and (j)–(l) WF4. WF1 is omitted here because it is substantially similar to BF. Positive values indicate conversion from KE to APE, and negative values indicate conversion from APE to KE.

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    The accumulated diapycnal transport in (left) latitude–density space and (right) depth–density space. (a),(b) BF; (c),(d) WF3; (e),(f) WF4; (g),(h) difference between WF3 and BF; and (i),(j) difference between WF3 and WF4.

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    Mixing efficiency η as a function of KE generation (simulations are marked for each value).

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Energetics of a Rotating Wind-Forced Horizontal Convection Model of a Reentrant Channel

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  • 1 University of North Carolina at Chapel Hill, Chapel Hill, North Carolina
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Abstract

We present numerical results for an idealized rotating, buoyancy- and wind-forced channel as a simple model for the Southern Ocean branch of the meridional overturning circulation (MOC). Differential buoyancy forcing is applied along the top horizontal surface, with surface cooling at one end (to represent the pole) and surface warming at the other (to represent the equatorial region) and a zonally re-entrant channel to represent the Antarctic Circumpolar Current (ACC). Zonally uniform surface wind forcing is applied with a similar pattern to the westerlies and easterlies with varying magnitude relative to the buoyancy forcing. The problem is solved numerically using a 3D direct numerical simulations (DNS) model based on a finite-volume solver for the Boussinesq Navier–Stokes equations with rotation. The overall dynamics, including large-scale overturning, baroclinic eddying, turbulent mixing, and resulting energy cascades, are studied by calculating terms in the energy budget using the local available potential energy framework. The basic physics of the overturning in the Southern Ocean are investigated at multiple scales and the output from the fully resolved DNS simulations is compared with the results from previous studies of the global (ECCO2) and Southern Ocean eddy-permitting state estimates. We find that both the magnitude and shape of the zonal wind stress profile are important to the spatial pattern of the overturning circulation. However, the available potential energy budget and the diapycnal mixing are not significantly affected by the surface wind stress and are primarily set by the buoyancy forcing at the surface.

© 2021 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: Varvara E. Zemskova, zemskova@live.unc.edu

Abstract

We present numerical results for an idealized rotating, buoyancy- and wind-forced channel as a simple model for the Southern Ocean branch of the meridional overturning circulation (MOC). Differential buoyancy forcing is applied along the top horizontal surface, with surface cooling at one end (to represent the pole) and surface warming at the other (to represent the equatorial region) and a zonally re-entrant channel to represent the Antarctic Circumpolar Current (ACC). Zonally uniform surface wind forcing is applied with a similar pattern to the westerlies and easterlies with varying magnitude relative to the buoyancy forcing. The problem is solved numerically using a 3D direct numerical simulations (DNS) model based on a finite-volume solver for the Boussinesq Navier–Stokes equations with rotation. The overall dynamics, including large-scale overturning, baroclinic eddying, turbulent mixing, and resulting energy cascades, are studied by calculating terms in the energy budget using the local available potential energy framework. The basic physics of the overturning in the Southern Ocean are investigated at multiple scales and the output from the fully resolved DNS simulations is compared with the results from previous studies of the global (ECCO2) and Southern Ocean eddy-permitting state estimates. We find that both the magnitude and shape of the zonal wind stress profile are important to the spatial pattern of the overturning circulation. However, the available potential energy budget and the diapycnal mixing are not significantly affected by the surface wind stress and are primarily set by the buoyancy forcing at the surface.

© 2021 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: Varvara E. Zemskova, zemskova@live.unc.edu

1. Introduction

The Southern Ocean has been shown to play important roles in global meridional transport (Marshall and Speer 2012), oxygenation of the deep waters via the formation of Antarctic Bottom Water (AABW) and atmospheric carbon sequestration (Sabine et al. 2004; Toggweiler et al. 2006). The energy in the Southern Ocean is drawn both from the strong and consistent zonal winds and differential surface buoyancy forcing. Many studies using eddy-permitting ocean models have shown that the overturning circulation is sensitive to changes both in the wind stress magnitude and location of the wind stress profile (Meredith and Hogg 2006; Farneti et al. 2010; Abernathey et al. 2011; Hogg et al. 2017) and in the local buoyancy forcing in the upwelling midlatitude region (Morrison et al. 2011) and the dense water formation region near Antarctica (Snow et al. 2016).

The circulation of the Southern Ocean is typically divided into two cells: the wind-driven upper cell where dense waters flowing in from the Northern Hemisphere upwell along the outcropping isopycnals, and the abyssal lower cell where dense water originates through convective processes (Toggweiler and Samuels 1993). The increased magnitude of the westerlies and its southward shift, observed both from recent climatological data (Thompson and Solomon 2002) and from paleoclimatology (Toggweiler et al. 2006), have been subjects of recent studies of the Southern Ocean (e.g., Farneti et al. 2010; Abernathey et al. 2011; Barkan et al. 2015; Hogg et al. 2017) with primary focus on the upper cell [see Gent (2016) for a comprehensive review of such studies]. However, Stewart and Thompson (2012, 2015) have shown that changes in the magnitude of the polar easterlies affect the transport in the deep cell of the MOC, a sensitivity that is not commonly included in the Southern Ocean circulation models and was not considered in the direct numerical simulations (DNS) studies of the Southern Ocean transport by Sohail et al. (2018, 2019). The region encompassed by the polar easterlies is of importance, in particular as the increase of atmospheric temperatures and glacial melting continues. For instance, Crampton et al. (2016) found that the eastward wind stress over the open ocean during the interglacial period contributed to the diatom community composition.

The previous modeling efforts of the Southern Ocean and the MOC, which are discussed in detail by Gent (2016), found eddy compensation of the increased kinetic energy input (Henning and Vallis 2005; Hallberg and Gnanadesikan 2006; Meredith and Hogg 2006; Wolfe and Cessi 2010; Abernathey et al. 2011). These studies showed that as the eastward wind stress was increased in the models, mesoscale eddy production via baroclinic instabilities increased to balance the increased wind-driven Ekman mean flow. Their results were contrary to the previous findings of the Drake Passage effect in models with coarser resolution (e.g., Toggweiler and Samuels 1993, 1995; Farneti et al. 2010), where an increase in wind input resulted in an increase in the overturning circulation. However, although the fine-resolution eddy-resolving ocean models consistently observe the eddy compensation, these studies find that the degree of compensation is dependent on many factors, including the choice of surface boundary conditions (Gent 2016).

In this paper, we study the relative contributions from buoyancy and wind forcing through separating the energetics into kinetic and available potential energy fields. The separation of the total energy budget into kinetic energy (KE) and available potential energy (APE) allows for the separation of reversible adiabatic mixing (exchange between KE and APE reservoirs) and irreversible diabatic mixing (loss of APE to background potential energy, BPE) (Winters et al. 1995; Hughes et al. 2009). We use the energetic framework developed for stratified fluids by Scotti and White (2014), which has been previously applied to the eddy-permitting MITgcm ocean model ECCO2 by Zemskova et al. (2015). We focus on the response of the overturning circulation and the energy budget dynamics in the Southern Ocean to changes in wind stress magnitude and understanding the relative role of easterlies and westerlies through DNS.

While certain oceanic parameters (especially aspect ratio and Rayleigh number) cannot be matched in DNS, it can achieve significant scale separation between the turbulent scale and the basin scale and explicitly resolve the rate of energy transfer at dissipative scales (Arneborg 2002). In GCMs (such as MITgcm), parameterizations schemes (e.g., KPP) make it difficult to interpret the effects of the small-scale processes. As such, the results from GCMs limits the application of energetic arguments to the overturning circulation (Hogg et al. 2017). Furthermore, the results from GCMs are highly sensitive to the resolution and parameterization schemes (Jayne 2009). For instance, Thoppil et al. (2011) showed that the ocean model’s simulated KE increases with greater resolution, and the dependence of eddy compensation of the Southern Ocean MOC on the model resolution is discussed in the review by Gent (2016).

DNS, which have been previously used to model idealized ocean basins by studies such as Barkan et al. (2013, 2015), Gayen et al. (2014), Vreugdenhil et al. (2016), and Sohail et al. (2018, 2019), resolve processes at all scales, so that all energetics, including the viscous dissipation and diapycnal mixing rates, can be calculated directly, unlike eddy-resolving ocean models that do not resolve submesoscale processes. Because of the large computational resources required for DNS to reach the steady state, we are unable to incorporate some of the real-ocean features that have been demonstrated to affect the circulation in the Southern Ocean, such as the bathymetry of the Antarctic shelf (Foldvik et al. 2004; Stewart and Thompson 2015), rough bottom topography linked to internal waves (Garabato et al. 2004; Nikurashin et al. 2014), and temporal variability of the surface fluxes (Chen et al. 2016; Roberts et al. 2017). Nevertheless, the results from this paper can be illuminating to the physical mechanisms that currently remain unresolved in the global ocean models. We strive to replicate realistic surface boundary conditions by approximating surface density and surface wind stress over the Southern Ocean from ECCO2, and we verify that our results qualitatively agree with more realistic ocean models and observations. We are able to reproduce the major features of the ocean residual overturning circulation using this highly idealized DNS model. This base case is used to study the interplay between surface buoyancy and wind forcing and the effects of surface forcing on the overturning circulation and the ocean energy budget.

We discuss the DNS setup, including the boundary conditions and pertinent nondimensional parameters, in section 2a. The DNS results are presented in section 3 for energy budget terms computed according to equations in section 2b and for overturning circulation dynamics, according to equations in section 2c. We discuss the implications of our findings in section 4.

2. Methods

a. Problem setup

We solve incompressible, nonhydrostatic Navier–Stokes equations with added rotation in the Boussinesq approximation:
ut=(uu)p+νΔug^kfk×u,g^t=(ug^)+κΔg^,u=0,
where u = (u, υ, w) is velocity; g^=g(ρρ0)/ρ0 is reduced gravity, or negative buoyancy (we refer to it through text as “buoyancy” for simplicity); p is pressure; f = 2|Ω| is the Coriolis parameter; k is a vertical unit vector positive upward; ν is kinematic viscosity; and κ is diffusivity. These equations are solved with DNS using Stratified Ocean Model with Adaptive Mesh Refinement (SOMAR), which uses finite difference schemes for spatial discretization of viscous terms and finite volume method for advection terms (Santilli and Scotti 2015; Chalamalla et al. 2017). This code is derived from the Chombo adaptive mesh refinement (Adams et al. 2015) modified to allow for anisotropic grid spacing and mesh refinement, which are useful for geophysical fluid dynamics problems with very small aspect ratios (Santilli and Scotti 2011).
For this problem, we consider a rectangular channel, shown in Fig. 1, as an idealized representation of a portion of the Antarctic Circumpolar Current between 80° and 40°S. The channel has width lx = 5H and length ly = 10H, where H is the depth of the channel. While the aspect ratio H/ly = 0.1 is several orders of magnitude larger than that of the real ocean, it is smaller than previous studies of rotating horizontal convection (Gayen et al. 2014; Vreugdenhil et al. 2016; Sohail et al. 2018). Equations are made nondimensional using H for length scales, H/U for time scales where U=gH is the characteristic velocity for gravity currents, and reduced gravity g′ for mass. In all of the simulations we use the f-plane approximation, with the Coriolis parameter f < 0 consistent in sign with the Southern Hemisphere. The flow is periodic in zonal (x) direction, and no-slip and no-flux boundary conditions are imposed on the bottom and meridional (y) sides. At the top surface, we consider a combination of buoyancy and wind stress boundary conditions. All simulations have a prescribed zonally uniform buoyancy distribution B at the top, which varies with latitude:
B(y)=1.5elog(3)y/100.5.
This buoyancy distribution is taken as an approximation of the time-averaged surface density profile over the Southern Ocean based on the ocean-state simulations from MITgcm model ECCO2, as shown in Fig. 2a, where the buoyancy distribution is shown redimensionalized in density units for easier comparison with the real ocean values. The approximation, based on the observations (Karsten and Marshall 2002), matches in its form the exponential profile employed for a sponge layer at the northern boundary of the reentrant channel used in previous MITgcm studies (Abernathey et al. 2011; Stewart and Thompson 2012). Connecting the surface buoyancy distribution with the vertical buoyancy stratification in the Equatorial ocean interior via thermal wind balance was previously used in a theoretical model of the MOC in an idealized basin with circumpolar channel by Nikurashin and Vallis (2011). In our study, we employ no-flux boundary conditions for buoyancy at the northern boundary rather than explicitly specifying the distribution. Dirichlet boundary conditions at the surface for buoyancy were chosen instead of Neumann boundary condition because the density distribution was found to be less time dependent, in particular season dependent, than heat flux over the Southern Ocean based on the ECCO2 output.
Fig. 1.
Fig. 1.

Simulation setup for the DNS runs showing the size of the nondimensionalized domain, direction of the rotation, buoyancy forcing at the top surface in color, and surface wind stress magnitude and direction in the panel above the domain.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-19-0169.1

Fig. 2.
Fig. 2.

(a) Density distribution at the surface over the Southern Ocean from ECCO2 output and from buoyancy boundary condition imposed in the simulations in this paper as a function of latitude. (b) Surface wind stress profiles for each of the simulations as a function of latitude. WF1 and WF2 are sinusoidal profiles with symmetric easterlies and westerlies. WF3 is a polynomial fit of the ECCO2 wind stress profile over the Southern Ocean, resulting in stronger westerlies. WF4 is a polynomial fit of doubled wind stress from ECCO2. The black axes (bottom and left) represent the parameters in the dimensional form to compare with the ocean values, and the blue axes (top and right) represent the nondimensionalized values used in the DNS.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-19-0169.1

We ran a total of five simulations: One with only surface buoyancy forcing (referred throughout the text as BF) and four with buoyancy forcing and various surface wind stress distributions (referred throughout the text as WF1, WF2, WF3, and WF4). Surface wind stress τ is applied through the Neumann boundary conditions on u velocity following Barkan et al. (2015), such that
τi=νuz|z=0=νFτ(y),
where i is the unit vector in the zonal direction and ν is kinematic viscosity. The wind stress distributions Fτ(y), recast in dimensional units and projected to real ocean latitudes, are shown for each of the simulations in Fig. 2b. The wind stress profiles for both WF1 and WF2 are sinusoidal, such that the easterlies and westerlies are equal over the ACC:
Fτ(y)=τmaxsin(2πy/ly),
and in nondimensional terms τmax = 0.2, 2 for WF1 and WF2, respectively. Such sinusoidal profile has been used to approximate westerlies in several previous studies of the Southern Ocean (Abernathey et al. 2011; Barkan et al. 2015). WF3 profile is a Fourier series fit to the time-averaged wind stress profile over the Southern Ocean from ECCO2 simulations with asymmetric strength of easterlies and westerlies, such that the westerlies are stronger than WF2, but the easterlies are weaker. WF4 is a Fourier series fit to the doubled wind stress obtained from ECCO2 simulations. The maximum magnitudes for the easterlies and the westerlies for each simulation are summarized in Table 2. We also define a nondimensional parameter S following Sohail et al. (2018, 2019) in order to represent the strength of mechanical forcing:
S=ντmaxgH.
The values for S are also shown in Table 2 and are calculated based on the maximum strength of the westerlies for the simulations with asymmetric wind stress profiles.
We define several nondimensional parameters (Rossby, Rayleigh, Ekman, and Prandtl numbers, aspect ratio) to characterize these simulations:
Ro=ΔgHfLy,Ra=BmaxLy3νκ,EkH=νfH2,EkL=νfLy2,Pr=νκ,α=HL,
where κ is diffusivity. For all simulations, the aspect ratio is α = 0.1. Here, EkH is the vertical Ekman number and EkL is the horizontal Ekman number. For all of the simulations, f = −4 based on our nondimensionalization, and we obtain other nondimensional parameters consistent with ocean parameters as discussed in Barkan et al. (2015): Ro = 0.025, Ra = 1011, EKH = 6.6 × 10−5, EKL = 6.6 × 10−7, and Pr = 7. These parameters are within the range of a strong rotation and a geostrophic regime similar to the previous DNS studies, such as Vreugdenhil et al. (2016) and Sohail et al. (2018), and laboratory studies (Park and Whitehead 1999), and, with the exception of Ra, they match oceanic values.

Table 1 shows the hierarchy of length scales in terms of values that are representative for the Southern Ocean and general circulation models (GCMs) and for our DNS setup, starting from the smallest Kolmogorov scale up to the basin scale. The parameters for the Southern Ocean are taken from the previously reported values (Garabato et al. 2004; Thompson et al. 2007; Lenn and Chereskin 2009; Sohail et al. 2018). For DNS simulations, we show the length scales in the dimensional form using depth H = 4 km, Coriolis parameter f = 10−4 s−1 as length and time scales for an easier comparison with the values used in the GCMs.

Table 1.

Length scales and nondimensional parameters for representative Southern Ocean values and for the DNS runs (length scales are dimensionalized using depth H = 4000 m and the Coriolis parameter f = 10−4 s−1 for length and time scales for easier comparison). Here, ν is viscosity, κ is diffusivity, ε is KE dissipation, N is Brunt–Väisälä frequency, f is the Coriolis parameter, Δb = gΔρ/ρ the reduced gravity, H is depth, and L is horizontal length. For parameters with *, the value is shown for a representative WF3 run.

Table 1.

In the ocean, there are two crucial ranges of scales: 1) [LO, Rd], which contains the upper bound of 3D turbulence to submesoscales, and 2) [Rd, L], which contains the mesoscale eddies to the basin-scale overturning circulation. There is O(1000) scale separation within each of these ranges, which would require the allocation of O(106) grid points in one direction (in particular, zonal or meridional) for sufficient resolution. However, given the modern computational resources, it is possible to have only O(1000) grid points. Global ocean models like SOSE (Mazloff et al. 2010), as shown by their resolution in Table 1, choose to resolve the [Rd, L] range of scales, and processes that occur at subgrid length scales have to be parameterized. As a result, submesoscale processes, and small-scale processes including lee waves generated over bottom topography and turbulent mixing, are not resolved (Storch et al. 2012; Chen et al. 2014).

In DNS, we can resolve both ranges of scales, resolving down to the Kolmogorov scale [see from Table 1 that (Δx, Δy, Δz)max < η] by compressing them into smaller intervals. In our DNS configuration, we have O(50) scale separation between each of the dynamic ranges, [η, Rd] and [Rd, L], which allows us to capture the energy transfer from basin-scale circulation to baroclinic mesoscale eddies and from submesoscales to dissipative scales. As a result, the DNS approach allows us to directly analyze the full energy cascade by directly computing kinetic energy dissipation and mixing, which cannot be achieve using GCMs.

To place our results in the scope of other studies of rotating horizontal convection, we define a nondimensional parameter Q which was first defined by Hignett et al. (1981). This parameter is a ratio of thermal boundary layer thickness to Ekman layer thickness, and following (Vreugdenhil et al. 2016) for a large Ra regime:
Q=(RaPr)2/5EkL1.
Hignett et al. (1981) identified different regimes based on Q: 1) Q = 0 nonrotating, 2) Q ≪ Pr−1 ≪ 1 very weak rotation, 3) Pr−1Q ≪ 1 weak rotation, 4) Q ~ 1 moderate rotation, 5) 1 ≪ Q ≪ Ra4/15 strong rotation, and 6) Q ≫ Ra4/15 extreme rotation. The strong regime (5) is associated with the geostrophic thermal wind balance, and Vreugdenhil et al. (2016) approximates the range of Q relevant to the ocean dynamics to be Q ≈ 4–22. For the simulations in this paper, Q = 27.6, which is approximately in such range.

Simulations are run on a grid with resolution (Nx, Ny, Nz) = (512, 1024, 128) to a statistical steady state, as shown in Fig. 3. All analysis calculations are done for time average over the last 80 nondimensional time units. The adequacy of the resolution and the statistical steady state were confirmed by the accurate closure of the energy budget by comparing the kinetic energy dissipation calculated directly from the simulation results and implicitly as a residual of the energy budget (Gayen et al. 2014; Vreugdenhil et al. 2016). We can also confirm that all scales are resolved by comparing (Δx, Δy, Δz)max to the Batchelor number ηb=(ν3/ε*)1/4Pr1/2=ηkPr1/2, where ε* is the local dissipation rate and ηk is the Kolmogorov length scale. The Batchelor number is the relevant small-scale parameter for this problem because Pr > 1 (Stevens et al. 2010; Vreugdenhil et al. 2016). In our simulations, (Δx, Δy, Δz)maxπηb as shown in Table 2, meaning that all scales are properly resolved. The resolved small-scale dynamics are illustrated in Fig. 4, which shows an instantaneous 3D structure of reduced gravity and Ro^=ζ/f (an alternative definition for Rossby number) for WF3, where ζ is vertical vorticity. Geostrophic flow and mesoscales eddies have Ri = (∂g′/∂z)/(∂u/∂z)2 ≫ 1 and subsequently Ro^1, whereas we observe many structures with Ro^~O(1) indicative of ageostrophic submesoscale motions, which have Ri < 1. Similarly large-eddy structures with values of Ro^1 have been observed in previous studies (McWilliams 2016).

Fig. 3.
Fig. 3.

Evolution of the APE and KE reservoirs over the nondimensional simulation time. All energy budget terms are averaged over the final 80 time units when KE reaches steady state.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-19-0169.1

Table 2.

Parameters (Batchelor scale ηb and maximum nondimensional magnitude of westerlies and easterlies) specific to each simulation, the model resolution in nondimensional units, parameter S, which quantifies the strength of the wind to buoyancy forcing, and N/f (near the surface and bottom boundaries), which quantifies the strength of the stratification. The estimates for the Southern Ocean for S (Sohail et al. 2019) and N/f (Garabato et al. 2004) are also shown.

Table 2.
Fig. 4.
Fig. 4.

Instantaneous 3D structures of (a) reduced gravity and (b) Ro^=ζ/f for WF3 simulation at a time within the statistic steady state period at selected slices of the domain. Submesoscale eddies, for which Ro~O(1), are resolved (in contrast, mesoscales have Ro ≪ 1).

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-19-0169.1

b. Energy budget terms

The energetic framework for the flow separated into mean (large-scale) and turbulent (small-scale, fluctuating) components has been presented in many previous studies (Winters et al. 1995; Huang 2005; Hughes et al. 2009; Storch et al. 2012; Scotti and White 2014). In this paper, we use the mean and turbulent energy budgets for KE and APE derived by Scotti and White (2014) and previously used by Zemskova et al. (2015) to calculate the ocean energy budgets using ECCO2 output data. Following the convention, we partition buoyancy and each velocity component: x=x¯+x, where ¯ is the mean flow averaged zonally and temporally, and primed quantity is the turbulent component calculated as deviation from the mean. The cross product of fluctuating terms is calculated as xy¯=xy¯x¯y¯.

APE is the difference between total potential energy Ep and the background potential energy EBPE, which is the minimum potential energy of the system found if all fluid parcels are reordered adiabatically based on their densities (Winters et al. 1995). For each instantaneous three-dimensional field, we can find the reference reduced gravity profile g^*(z) by resorting the fluid parcels, such that density increases monotonically with depth. The height of each fluid parcel in the reference background state is z*(g^), and it varies in both space and time because of the dependence on the instantaneous local reduced gravity g^(x,t).

We define KE and APE, respectively, as volume integrals:
EK=12ρ0Vu2+υ2+w2dV,EA=EpEBPE=ρ0Vg^(zz*)dV.
The mean and turbulent kinetic energy budgets, respectively, are given by
dEKmdt=V(g^¯g^*)w¯dV+Vui¯xjuiuj¯dV+Su¯dSνV(ui¯xj)2dV=C(EKm,EAm)C(EKm,EKf)+G(EKm)D(EKm),
and
dEKfdt=Vg^w¯dVVui¯xjuiuj¯dV+Sτiuj¯dSνV(uixj)2¯dV=C(EKf,EAm)+C(EKm,EKf)+G(EKf)D(EKf).
Here, C(EKm,EAm) is the transfer of energy between mean KE and mean APE, and C(EKf,EAm) between turbulent KE and mean APE reservoirs, which represent the reversible conversion of energy via vertical buoyancy fluxes. The term C(EKm,EKf) is the rate of production of turbulent KE from mean shear. The terms G(EKm) and G(EKf) represent the generation rates of mean and turbulent KE. Here, G(EKf)=0 because we only consider the mean component of the wind forcing. The terms D(EKm) and D(EKf) are dissipation rates of mean and turbulent KE; these terms will be calculated both directly and as residuals at steady state.
The mean and turbulent APE energy budgets are, respectively,
dEAmdt=V(g^¯g^*)w¯dV+Vg^w¯dVVg^ui¯g^¯dz*dg^|g^¯dV+κSzsg^¯zdS+κSz*(g^¯)g^¯zdS+κA(g^¯|z=0g^¯|z=H)+κV(g^¯z)2dz*db|g^¯dV=C(EKm,EAm)+C(EKf,EAm)C(EAm,EAf)+G(Ep)+G(EAm)+CI(Ep)D(EAm),
and
dEAfdt=Vg^ui¯g^¯dz*dg^|g^¯dV+κSz*g^z¯z*(g^¯)g^¯zSκV(g^z)2dz*dg^¯+(g^¯z)2dz*dg^|g^¯dV=C(EAm,EAf)+G(EAf)D(EAf).
The mean and turbulent vertical buoyancy fluxes are C(EKm,EAm) and C(EKf,EAm), defined as above in the KE budget. The term C(EAm,EAf) is the conversion rate between mean and turbulent APE. The generation rates of mean and turbulent APE are represented by G(EAm) and G(EAf) calculated as the rate of supply of buoyancy into the background potential field through the surface. The term G(Ep) represents the net buoyancy flux along the surface, and is identically zero both through the choice of zs = 0 and the fact that no net buoyancy input occurs at steady state (Hughes et al. 2009). The term CI(Ep) is the conversion rate of internal to potential energy (Winters et al. 1995; Hughes et al. 2009), and in the limit of Paparella and Young (2002), it must balance the sum of mean and turbulent vertical buoyancy fluxes C(EKm,EAm)C(EKf,EAm). Finally, D(EAm) and D(EAf) are the dissipation rates of mean and turbulent APE, respectively. Both of these terms are positive definite because dz*/dg^<0 (heavier fluid parcels are at lower depth when the density field is adiabatically resorted). All of the sources and sinks of KE and APE and the exchanges between the four reservoirs are shown in an energy diagram in Fig. 9.

c. Overturning streamfunction calculation

The overturning streamfunction calculated using Eulerian zonal mean flow is known to have a strong Deacon cell in the ACC (Döös and Webb 1994), which is unphysical as it can be shown that the fluid parcels follow isopycnals rather than these streamlines. To accurately represent the physical circulation and eliminate the Deacon cell, we compute the overturning streamfunction in density–latitude space (Nurser and Lee 2004; Zika et al. 2013; Hogg et al. 2017):
Ψyb=1t2t1t1t2g˜g^υ(x,y,z,t)dzdxdt,
where the temporal averaging is performed over the interval [t1, t2]. We further project the density–latitude streamfunction on to geographic coordinates by computing pseudodepth coordinates for the isopycnals at each latitude based on g¯ following Zika et al. (2013).
We also calculate the depth–density streamfunction (Zika et al. 2013; Hogg et al. 2017):
Ψbztotal=1t2t1t1t2g˜g^w(x,y,z,t)dxdydt.
This streamfunction identifies the conversion between the KE and APE reservoirs, and it can be decomposed into mean and turbulent components:
Ψbzmean=g˜g^w¯(x,y,z)dxdyandΨbzturb=ΨbztotalΨbzmean.
The mean and turbulent components correspond to the vertical buoyancy fluxes, C(EKm,EAm) and C(EAm,EKf), as noted by Nycander et al. (2007), such that they can be used to analyze the contributions from the mean winds and baroclinic adjustment.

3. Results

a. Overturning circulation

Figure 5 shows the zonally and temporally averaged reduced gravity g′ (left column) and density–latitude overturning circulation function Ψyb defined in (13) remapped onto y and pseudo-z coordinates for easier comparison with geographic coordinates (right column) for all simulations in order from BF (top) to WF4 (bottom). The reduced gravity fields have Δg′/2 = 0.5 subtracted to emphasize the lighter and denser fluids in the domain. Consistent with the findings by Sohail et al. (2019), there is little change in stratification with increased wind stress. The nondimensional parameter N/f near the surface and bottom (N is zonally, meridionally, and temporally average buoyancy frequency) for all simulations is shown in Table 2. While the values among all simulations are close, indicating that there is little difference in stratification, the stratification in the simulations is much smaller than in the real ocean, where N/f ~ 3–50, as approximated from measurements (Garabato et al. 2004; Nikurashin and Ferrari 2013). The stratification is set by many aspects in the Southern Ocean, which were not incorporated into the DNS setup, including the shelf dynamics, smaller-scale topography and the exchange with the northern basin (Barkan et al. 2015). In addition, N/f is the inverse of ratio of the depth to radius of deformation. As shown in Table 1, there is a smaller scale separation for the DNS, such that this aspect ratio is greater compared with that of the ocean, and subsequently the stratification is weaker in the simulations.

Fig. 5.
Fig. 5.

(left) Zonally and temporally averaged fields of g^0.5 (reduced gravity with Δg^/2 subtracted to emphasize lighter and denser fluids) and (right) density–latitude overturning streamfunction Ψyb remapped onto y and pseudo-z coordinates for (a),(f) BF; (b),(g) WF1; (c),(h) WF2; (d),(i) WF3; (e),(j) WF4. Black contours in (a)–(e) are isopycnals. In (f)–(j), positive cells have clockwise circulation, and negative cells counterclockwise circulation.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-19-0169.1

The positive cells in Figs. 5f–j are thermally indirect with clockwise circulation, where upwelling waters are denser than downwelling waters. The negative cells are thermally direct with counterclockwise circulation indicating the downwelling of dense waters (Hogg et al. 2017). In the simulation with both realistic surface buoyancy and wind stress distributions (WF3), there are three distinct overturning cells (Fig. 5i), consistent with the Southern Ocean described in many previous studies, such as Lumpkin and Speer (2007): 1) negative cell with downwelling of dense water in the south, analogous to AABW formation, 2) positive cell with waters flowing along isopycnals at middepth outcropping in the middle of the domain like the NADW upwelling in the ACC, and 3) subtropical cell with southward flow near the surface. As the wind stress increases, we observe the progression from buoyancy-driven regime for BF and WF1 (Figs. 5f and 5g, respectively) to a wind-dominated regime for WF4 (Fig. 5j) where the dense water formation cell at the southern end effectively disappears. This progression suggests that the current overturning structure of the Southern Ocean is sensitive to the particular balance between wind stress and buoyancy forcing, as we are able to reproduce its main features in a simple, highly idealized box model using DNS without any complex bathymetry, also validating the application of our simulations to ocean dynamics.

While the values of the parameter S, which represents the strength of mechanical forcing relative to the buoyancy differential, are comparable to those in the simulations within the turbulent convective regime by Sohail et al. (2019), our simulations impose both the easterly and westerly wind stresses, whereas Sohail et al. (2019) only considers the westerlies overlying the dense water formation region. This difference in the surface buoyancy forcing distribution may in part explain that Sohail et al. (2019) obtain a thermally indirect cell, which is primarily wind driven, that is much weaker and more spatially confined compared with the thermally direct dense-water formation cell, whereas our simulations yield a substantially strong thermally indirect cell.

The overturning streamfunctions also differ between WF2 and WF3 even though the KE generation is roughly equal, suggesting the importance of the antisymmetry in strength of the easterlies and westerlies in the ocean. For the simulation with the symmetric wind stress profile (WF2), the deep dense water formation cell is shut down similarly to WF4, and the isopycnals upwell further south than in WF3, with a more prominent subtropical cell. The Ekman pumping velocities wE, calculated from wind stress curl, are shown in Fig. 6 for all the simulations. In comparison to WF3, vertical velocities for WF2 are weaker in the westerlies upwelling region (6 ≤ y ≤ 8), but are stronger or equal in the easterlies upwelling region (3 ≤ y ≤ 5.5), leading to the isopycnals upwelling further south. The downwelling velocities at the northern and southern ends are overall weaker for WF2 wind stress profile than for WF3, but for WF2 there is no zero wind curl at the transition between the westerlies and the trade winds and at the southern boundary unlike in WF3 and the real ocean (Farneti et al. 2010).

Fig. 6.
Fig. 6.

Ekman pumping vertical velocities for each simulation at the surface computed from the wind stress profiles in Fig. 2b as functions of simulation latitude y. Black dashed line represents the zero axis.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-19-0169.1

The difference in wE profiles with latitude and in the resulting overturning circulation suggests a problem with representing the westerlies as sine function in the models of the Southern Ocean (cf. Abernathey et al. 2011; Barkan et al. 2015). The zero curl at the transitions between the westerlies and easterlies and between the westerlies and the trade winds plays an important role in the overturning dynamics, as also noted in Farneti et al. (2010): it affects the latitude of isopycnal upwelling and can be incorporated into models by approximating the observed surface wind stress using sums (as in this paper) or products Stewart and Thompson (2012) of trigonometric functions.

The Ekman pumping velocities in the convective region are stronger for WF2 and WF4 than WF3, so one would expect the buoyancy driven flow to be enhanced, which is contrary to the overturning streamfunctions in Fig. 5. These overturning streamfunctions are the residual balance between the large mean and eddy components of the flow, which are calculated analogously to the mean and turbulent components of Ψbz in Eq. (15). These components are shown for simulations WF2, WF3, and WF4 in Fig. 7. In the mean field (left panel), we observe that the flow is primarily wind driven. The negative counterclockwise buoyancy-driven cells are additionally enhanced where Ekman pumping resulting from the easterlies wind stress curl is strong (where wE minimum occurs): near the southern end for WF2 and around y ≈ 2 for WF4. However, these negative cells are balanced by the positive circulation in the eddy streamfunctions (right panels) for both WF2 and WF4, whereas no such compensation occurs in the case of WF3.

Fig. 7.
Fig. 7.

Density–latitude overturning circulation function separated int (left)o mean and (right) turbulent components for (a),(b) WF2; (c),(d) WF3; and (e),(f) WF4. Positive cells have clockwise circulation, and negative cells have counterclockwise circulation.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-19-0169.1

To quantify the effects of wind stress magnitude and distribution on the overturning circulation, we calculate the transport terms following the analysis by Stewart and Thompson (2012, 2015) of the Southern Ocean transport dynamics using MITgcm. The term Ψupper is the maximum transport along the outcropping isopycnals in the positive cell, which is similar to the upper MOC cell discussed in previous studies (Abernathey et al. 2011; Zika et al. 2013; Hogg et al. 2017). The term ΦAABW is the maximum of the northward transport of dense water formed at the southern end of the domain, similar to the transport of AABW in the ocean. The term ΦAASW is the maximum southward transport at near the surface of the waters that upwell along the isopycnals [this is analogous to the Antarctic Surface Waters transport in Stewart and Thompson (2015)]. This term is calculated around y ≈ 2.5 near the maximum of the eastward wind stress and subsequently the maximum southward Ekman transport in the southern half of the domain. The term ΦAASW is not necessarily equal to the lower cell transport ΦAABW because it represents the transport in the near-surface cell primarily driven by Ekman transport, in contrast to the deep cell that is primarily buoyancy driven.

Figure 8 shows the sensitivity of each of the transport terms to the maximum magnitude of the westerlies (τmaxwest) (Fig. 8a) and easterlies (τmaxeast) (Fig. 8b) for each of the simulations. In Fig. 8a, the simulations are in the order of BF, WF1, WF2, WF3, and WF4 from smallest to largest τmaxwest, and in Fig. 8b, the order is BF, WF1, WF3, WF2, and WF4 from smallest to largest τmaxeast. The only term that correlates with increased westward wind stress is Ψupper (shown in red circles), increasing consistently with the results from Stewart and Thompson (2012) and Hogg et al. (2017). This transport also generally increases with the increasing eastward wind stress magnitude, but it is suppressed for WF2 and it saturates at large wind stress magnitudes (in Fig. 8b). The saturation of Ψupper, even as the magnitude of the wind stress doubles between WF3 and WF4, is reminiscent of the insensitivity of the ACC to changes in wind stress observed by Böning et al. (2008) and shown via the eddy-saturated regime in ocean models (Hallberg and Gnanadesikan 2006; Meredith and Hogg 2006).

Fig. 8.
Fig. 8.

Sensitivity of the transport along outcropping isopycnals (Ψupper in red circles), the deep northward transport of dense water (ΨAABW in blue squares), and the southward transport of surface water (ΨAASW in black triangles), and to (a) maximum westerlies wind stress (order of simulations: BF, WF1, WF2, WF3, WF4) and (b) maximum easterlies wind stress (order of simulations: BF, WF1, WF3, WF2, WF4); (c) the ratio of ΨAABW to Ψupper, which indicates the strength of buoyancy driven to wind driven regimes, as functions of both westerlies (teal squares; order of simulations: BF, WF1, WF2, WF3, WF4) and easterlies (purple circles; order of simulations: BF, WF1, WF3, WF2, WF4) maximum wind stresses.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-19-0169.1

The term ΨAASW (shown in black triangles) generally increases with τmaxwest (in Fig. 8a), but this southward transport in stronger in the case of the symmetric westerlies and easterlies distribution (WF2) than in the antisymmetric case (WF3). This terms increases linearly with the increased strength of the easterlies (in Fig. 8b), consistent with findings of Stewart and Thompson (2015), who demonstrated that this transport is correlated mainly with the wind-driven Ekman transport. Because WF2 and WF4 have stronger easterlies than WF3, the southward transport is greater in the region 3 ≤ y ≤ 5 and the positive overturning circulation cell extends further south, leaving less space for the negative deep overturning cell. The response of the transport in the upper MOC cell to the westward wind stress (in Fig. 8a) and the increase in near-surface southward transport with greater eastward wind stress (in Fig. 8b) are consistent with the findings in previous studies of the Southern Ocean models, validating our DNS models as means to study the physical mechanisms of the real ocean.

The strength of the deep MOC cell transport, ΨAABW shown in Figs. 8a and 8b in blue squares, does not have a clear pattern with changing westward and eastward wind stresses. Previous studies have found variable patterns in the sensitivity of the transport of this cell when considering changes in the westerlies and the easterlies separately. Abernathey et al. (2011) found that the lower cell transport weakly increases with increasing westerlies wind stress, whereas Stewart and Thompson (2015) showed that this transport of AABW does not change with an increase in the easterlies. In this model, we consider changes in both the easterlies and the westerlies concurrently, and the lower cell transport does not have a clear response to the changes to both wind stresses simultaneously. However, we can measure the shift from buoyancy-driven to wind-driven regimes using the ratio of ΨAABW to Ψupper. This ratio linearly decreases, as shown in Fig. 8c, with both maximum westerlies (in teal squares) and easterlies (in purple circles) wind stress, quantifying the progression we observed in the streamfunctions in Figs. 5f–j. However, WF2 is an outlier for both of these linear relationships with ΨAABWupper → 0, whereas ΨAABWupper ≈ 1 for WF3. The symmetry of the westward and eastward wind stresses coupled with the absence of zero wind stress curl at the transition between the easterlies and the westerlies lead to the wind driven circulation overwhelming the buoyancy-driven circulation.

b. Energy budget

We calculate the energy fluxes terms and the total amount of energy in each of the mean and turbulent KE and APE reservoirs as shown in the energy diagram in Fig. 9. Because of our choice of time-invariant boundary conditions, we find that at steady state G(EAf)=0, G(EKm)=Sτx¯u¯dS, as there is only zonal wind stress component, and G(EKf)=0. All values were calculated as volume integrals over the entire domain and averaged temporally over the steady state time interval, and are shown as functions of G(EKm) in Fig. 10 to illustrate the sensitivity of the generation and exchange terms to changes in wind stress magnitude. WF2 has slightly higher KE generation rate [G(EKm)=0.021] than WF3 [G(EKm)=0.020], so the order of simulations shown in each plot in Fig. 10 is BF, WF1, WF3, WF2, and WF4. We verify that the simulations have converged by computing dissipation of KE and APE, shown respectively in Figs. 10g and 10h, both explicitly from the expressions in (9)(12), marked with solid lines, and as residuals from steady state by setting dE/dt = 0, marked with dashed lines. The approximate agreement between the explicit and implicit dissipation terms indicates the closure of the energy budget and appropriate choice of resolution for the computational model.

Fig. 9.
Fig. 9.

Energy diagram showing sources, sinks of and exchanges between KE and APE reservoirs based on (9)(12) in text.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-19-0169.1

Fig. 10.
Fig. 10.

Energy budget terms from Fig. 9 plotted as functions of KE generation for each simulation. (a) KE, (b) APE, (c) mean and turbulent vertical buoyancy fluxes, (d) APE generation rates, (e) conversion rates between mean and turbulent KE, (f) conversion rates between mean and turbulent KE, (g) KE dissipation rates, and (h) APE dissipation rates. For (g) and (h), the dissipation values calculated explicitly from the expressions in (9)(12) are marked with solid lines, and the values calculated as residuals from steady state with dashed lines. The order of simulations is labeled in (d).

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-19-0169.1

The total KE in the system increases with greater wind stress (Fig. 10a) mainly due to the increase in the turbulent KE, as expected from previous works (Marshall and Radko 2003; Meredith and Hogg 2006), while the mean KE stays relatively constant. As wind stress increases, most of the excess KE generation is converted from mean KE into turbulent KE (Fig. 10e), and then dissipated through turbulent KE (Fig. 10g) such that C(EKm,EKf) and D(EKf) grow linearly with wind stress, while D(EKm) only moderately increases. As shown by Cessi et al. (2006), work from zonal wind stress is mostly dissipated via bottom drag with the small remainder, “useful wind work,” which converted to APE and is available to drive the baroclinic eddies. In our energy budget analysis, we see that as the mean mechanical energy input increases, the conversion from the mean to turbulent KE also increases, while the “useful wind work” component C(EKm,EAm) does not significantly change. Most of the KE is dissipated through the turbulent field rather than the mean field, such that a significant portion of the KE dissipation occurs within the interior of the baroclinic eddies (Fig. 11a) near the locations of the extrema of the wind stress curl. Such large eddies with high KE dissipation rate have been observed in previous DNS studies in a domain with similar rotation rate (Vreugdenhil et al. 2016, 2019). Because the scale separation between the Ekman layer depth, the Rossby radius of deformation, and the domain depth is significantly more compressed in the DNS compared to the real ocean, it could give rise to these eddies. KE dissipation also is high at the surface, within the eddies at the surface and also at the southern end within the convective region, as shown in Fig. 11c.

Fig. 11.
Fig. 11.

Time-averaged distribution of (left) logD(EKf) and (right) logD(EAt) for WF3 for (a),(b) zonal averages and (c),(d) surface distribution. Both KE and APE dissipation rates are high near the surface and in the convective region near the southern end. KE dissipation is also amplified by the surface wind stress.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-19-0169.1

The conversion between the mean and turbulent KE reservoirs is in the opposite direction than previously reported in ocean models, such as Storch et al. (2012) and Zemskova et al. (2015), mainly because we do not have time-variable wind stress in this model, and in the ocean the turbulent KE generation rate is the same order of magnitude as that of the mean KE. However, when Storch et al. (2012) decomposed the global ocean energy budgets to calculate localized dynamics, they found that in the upper layer of the Southern Ocean (above the depth of 3000 m) the mean KE is converted to turbulent KE, opposite of the global budgets and consistent with our DNS model. These results suggest that the bulk of the ACC is dominated by the time-mean winds that then generate shear instabilities, and our model is a good representation of the upper 3000 m of the Southern Ocean.

The total APE reservoir increases only marginally with greater KE generation rate (Fig. 10b), which can be observed from the APE generation (Fig. 10d) and APE dissipation (Fig. 10h) rates remaining mostly constant with increasing wind stress. When the APE reservoir is decomposed into mean and turbulent components, the mean APE is greater than turbulent APE in the buoyancy-driven regime when the wind stress is very small (BF, WF1), but when the winds get stronger, the turbulent APE reservoir becomes bigger. Most of the mean APE generated at the surface is converted to turbulent APE (Fig. 10f), and the turbulent APE dissipation is significantly larger than the mean APE dissipation.

The diapycnal mixing, represented by the APE dissipation D(EAf), is largest for WF3, and then decreases as G(EKm) increases from WF2 to WF4. The APE dissipation depends on the magnitude of the density gradients, both negative (as in the dense water formation region) and positive (as in the upwelling isopycnal region), and is particularly strongest in the convective region, as shown in Fig. 11. The distribution of diapycnal mixing is qualitatively similar to Sohail et al. (2018), who showed it being strongest in the convection region and near surface. In their model, the cooling region coincides with the westerlies region, such that both the isopycnal upwelling and the densest surface water occur near the southern boundary of the domain.

Figure 12 shows APE dissipation [in terms of logD(EAf)] as a function of latitude y averaged temporally and zonally at different depths for BF, WF2, WF3, and WF4 runs. At the surface (Fig. 12a) and in the top third of the domain (Fig. 12b), the diapycnal mixing is approximately equal among all simulations, as it is primarily controlled by the buoyancy distribution. The APE dissipation is highest within the convective plume at all depths. In the middle third of the domain (Fig. 12c), there are elevated values in the isopycnal upwelling region (2 < y < 4 for WF2 and WF4 and 4 < y < 5.5 for WF3 as shown in the overturning circulation functions in Fig. 5) and in the westerlies region.

Fig. 12.
Fig. 12.

Temporally and zonally averaged distribution of logD(EAf) as a function of latitude y for BF (dashed), WF2 (dash–dotted), WF3 (dotted), and WF4 (solid) at (a) surface, (b) average over top third of domain, (c) average over middle third of domain, and (d) average over bottom third of domain.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-19-0169.1

In the deepest third of the domain, the APE dissipation is much larger for WF3, in particular in two regions: 1 < y < 3 and 7 < y < 10. In the first region, the easterlies play an important role for WF3 and support the formation of deep MOC cell, whereas isopycnals upwell in this region for WF2 and WF4. For WF2 and WF4, there is no deep MOC cell that propagates to the latitude of the second region, but the deep MOC cell extends along the bottom to the northern end of the domain for WF3 (refer to overturning circulation functions in Fig. 5) meaning that the density stratification is greater in this region for WF3 than for other runs. It suggests that the particular surface forcing balance in WF3 maximizes the APE dissipation rate when the buoyancy-driven convective transport (ΨAABW) is approximately equal to the intermediate water upwelling in the upper MOC cell (Ψupper) (see Fig. 8c for ΨAABWupper versus τmax). However, the diapycnal mixing rate diminishes when ΨAABWupper > 1 as in buoyancy-driven regime (BF, WF1), where there is no significant upper MOC upwelling, or ΨAABWupper < 1 as in the wind-driven regime (WF2, WF4), where the deep convective cell is constrained.

For all simulations, we observe a balance between the mean and turbulent vertical buoyancy fluxes, as shown in Fig. 10c. This balance maintains the Paparella and Young limit, CI(EP)C(EKm,EAm)+C(EAm,EKf)0, and there is no net transfer of APE to the KE budget. Both C(EKm,EAm) and C(EAm,EKf) generally increase with greater G(EKm), and the balance between the two terms is in line with the eddy compensation view of the Southern Ocean that suggests that increased input of mechanical energy increases the tilt of the isopycnals converting KE to APE, and the eddies then release this APE to the turbulent KE reservoir by flattening isopycnals (Marshall and Radko 2003; Farneti et al. 2010; Wolfe and Cessi 2010). However, both mean and turbulent vertical buoyancy fluxes are largest for WF3, similar to the amplification of APE generation and dissipation for this simulation. In the next subsection, we discuss in detail the distribution of this adiabatic conversion term using the depth–density overturning circulation function for each of the simulations.

c. Conversion between KE and APE

The breakdown of total (left column), mean (center column), and turbulent (right column) depth–density overturning circulation functions for each of the simulations is shown in Fig. 13. Each of the components of the overturning circulation are calculated from Eq. (14), with positive values indicating thermally indirect cells where KE is converted to APE, and negative values indicating thermally direct cells where APE is converted to KE (Hogg et al. 2017). These overturning circulation functions show the total, mean and turbulent vertical buoyancy fluxes in depth–density space, because, as pointed out by Nycander et al. (2007), the depth–density overturning streamfunction describes a purely adiabatic flow under steady state conditions. In this framework, we do not see the counterclockwise near-surface cell that appears in the density–latitude overturning streamfunction shown in Figs. 5f–j (Zika et al. 2013).

Fig. 13.
Fig. 13.

Depth–density overturning circulation functions: (left) total, (center) mean, (right) turbulent for each simulation, (a)–(c) BF, (d)–(f) WF2, (g)–(i) WF3, and (j)–(l) WF4. WF1 is omitted here because it is substantially similar to BF. Positive values indicate conversion from KE to APE, and negative values indicate conversion from APE to KE.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-19-0169.1

We observe largely compensation of the mean field by the turbulent field with the values of the total overturning circulation functions at least an order of magnitude smaller than either of the mean or the turbulent components. The primarily wind-driven component, which is the positive cell in Ψbzmean that corresponds to the upper MOC and the conversion term C(EKm,EAm), increases with greater wind stress. Compared with the results for BF, Ψbzmean is smaller, or more negative, for denser fluid (g^0.5), and larger for lighter fluid (g^0.5) for the symmetric wind stress cases (WF1 and WF2), primarily because the convergent wind stress at the southern end increases downwelling of denser water decreasing the APE of the region, and convergent wind stress at the northern end increases downwelling of lighter water creating more APE. In the asymmetric wind stress cases (WF3 and WF4), Ψbzmean is greater, or more positive, than that of BF almost everywhere because downward Ekman vertical velocity is even larger at the northern end so there is more downwelling of lighter water and there is no downward Ekman vertical velocity at the very southern end of the domain, where the densest surface waters are, due to zero wind stress curl.

For WF3 (Fig. 13h), C(EKm,EAm) is amplified throughout all depths, whereas for WF4 this conversion is strongest in the upper half of the domain (Fig. 13k), consistent with greater contribution of near-surface Ekman pumping due to stronger wind stress. In particular, the greater near-surface conversion of KE to APE occurs due to the downwelling of lighter waters at the northern end and upwelling of denser waters further south in the domain for WF4 than for WF3. As a result, the total conversion between KE and APE is primarily positive for WF4 (Fig. 13j) with a small signature of negative lower MOC cell at middepth, while Ψbztotal is more balanced between the negative and positive cells for WF3 (Fig. 13g) and the dense-water formation cell extends all the way to the bottom of the domain.

The downwelling due to Ekman pumping in the southern half of the domain is amplified for WF4 in comparison to WF3, and for WF2, it occurs closer to the southern end where the densest water lies at the surface. As a result, it would be expected to have a stronger dense water formation cell for WF2 and WF4 than for WF3, but we observe the opposite in the total density–latitude overturning streamfunctions (Figs. 5h–j) where this cell is suppressed in WF2 and WF4. We observe that this negative cell in the mean component of Ψbz is stronger for WF2 and WF4 in Figs. 13e and 13k, respectively, than for WF3 (Fig. 13h). However, this cell is compensated by the conversion of APE to turbulent KE via eddy production, such that in WF2 and WF4, the negative counterclockwise cell does not appear in Ψbztotal (cf. Figs. 13d and 13j) and similarly in the total Ψyb. In contrast, eddies do not fully compensate the wind-driven conversion of KE to APE for WF3, and there is a strong negative cell in the total overturning streamfunction (Fig. 13g).

For WF2 Ψbztotal is overall positive. In the mean field, the conversion of KE to APE is stronger for WF2 in the lighter density classes near the surface, corresponding to the stronger downwelling near the northern end and stronger upwelling in the middle of the domain, highlighting the problem with modeling the westerlies as a sine function with no zero curl at the transition zones to the trade winds to the north and to the easterlies to the south.

From these depth–density overturning circulation functions, we can conclude that the mean field of the adiabatic fluxes for all simulations is primarily wind driven and compensated by the eddy fluxes, and the total rate of energy conversion between KE and APE is small. However, the basin-integrated values for the mean and turbulent components of these vertical buoyancy fluxes are not simply functions of KE generation rate, or the strength of the westerlies or the easterlies. The distribution of the residual circulation (sum of mean and eddy components) and the basin-integrated values depend on the latitudinal wind stress profiles, and in particular are sensitive to whether zero wind stress curl is imposed in the transition regions between the trade winds and the westerlies, and the westerlies and the easterlies.

d. Diapycnal transport

To quantify the effect of wind stress on the water mass modification, we compute the diffusive flux in a form analogous to the overturning streamfunctions for the adiabatic flow. We use the definition for the diapycnal flux across an isoscalar (in this case, reduced gravity) surface derived by Winters and D’Asaro (1996) for density–latitude and depth–density streamfunctions:
Ψybdia=1t2t1t1t2g˜g^κdz*dg^(g^)2dxdz,Ψbzdia=1t2t1t1t2g˜g^g˜g^κdz*dg^(g^)2dxdz.
The first expression represents the accumulated diapycnal transport at a given latitude, and the second the accumulated diapycnal transport at a given depth in buoyancy coordinates. Figure 14 shows this transport for BF (Figs. 14a,b), WF3 (Figs. 14c,d), and WF4 (Figs. 14e,f). Because dz*/dg^<0 by definition, the diapycnal transport is always positive. To highlight the effect of the wind stress, we show the differences between WF3 and BF in Figs. 14g and 14h and between WF3 and WF4 in Figs. 14i and 14j.
Fig. 14.
Fig. 14.

The accumulated diapycnal transport in (left) latitude–density space and (right) depth–density space. (a),(b) BF; (c),(d) WF3; (e),(f) WF4; (g),(h) difference between WF3 and BF; and (i),(j) difference between WF3 and WF4.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-19-0169.1

The diapycnal transport is the strongest in the dense plume region (y < 4 in all simulations), especially amplified near the surface (z > −0.2), which is consistent with the regions of the greatest APE dissipation. While only the vertical diapycnal fluxes κ(g^/z)2 contribute to the APE dissipation rate, the diapycnal transport includes the contribution from the horizontal gradients. However, as the regions of the highest diapycnal transport and the APE dissipation rate are approximately the same, the vertical fluxes are the main contributor to the diapycnal transport and water mass formation.

The peak diapycnal transport for the BF run is centered around y = 2, whereas it is strong further near the southern end for the WF3 run, due to the dense plume extending further south pushed by the surface Ekman transport from the easterlies. The wind stress also increases the diapycnal transport in the region of upwelling isopycnals (around y = 5) and near the northern edge of the domain where the downward Ekman pumping of light surface water creates unstable density gradients. While the diapycnal transport occurs primarily within the convective plume created by the surface buoyancy flux, it is overall stronger when the wind is present compared with the buoyancy-forcing only simulation.

Comparing WF3 with WF4, the diapycnal transport is greater within the convective plume, which is stronger for WF3 as shown by the residual overturning streamfunctions Ψyb and Ψbz in Figs. 5 and 13. It is also intensified in the isopycnal upwelling regions (positive values in Fig. 14e around y = 5 for WF3 upwelling and negative values for 2 < y < 4 for WF4 upwelling). As the dense plume propagates along the bottom all the way to the northern end for WF3, the diapycnal transport is also greater (8 < y < 10). The diapycnal transport plotted in the depth–density space shows that it is overall greater for WF3 than WF4, consistent with the greater APE dissipation rate and the strong dense water formation region in the residual overturning circulation. However, near the surface for the intermediate densities, WF4 has stronger diapycnal transport. This amplification is consistent with the stronger transport in the region 5.5 < y < 7 where the Ekman upwelling is greater for WF4 than WF3, which creates local buoyancy gradients.

The distribution of diapycnal transport indicates that it is primarily controlled by the dense water mass formation due to surface buoyancy flux. However, the presence of the wind stress enhances the diapycnal fluxes, which agrees with the findings of increased diapycnal mixing with wind stress magnitude by Sohail et al. (2018). However, the surface wind stress profiles play an important role locally creating density gradients.

The domain-averaged mixing efficiency η, which is defined as the ratio of diapycnal mixing to the total mechanical energy sink (dissipation of both APE and KE), can be calculated as (Peltier and Caulfield 2003; Sohail et al. 2018)
η=D(EA)CI(EP)D(EA)CI(EP)+D(EK).
The mixing efficiency values for each of the simulations are shown in Fig. 15. As the wind stress and input of KE into the domain increases, the mixing efficiency decreases because KE dissipation increases with G(KE) while APE dissipation increases only slightly (see Figs. 10g,h). These results are qualitatively similar to Sohail et al. (2018), whose scaling theory predicted that the dissipation rate increases with wind stress as D(EK) ~ S3/2, whereas the diapycnal mixing rate increases at a significantly lower rate with D(EA) ~ S1/4. As a result, the mixing efficiency η decreases with increasing wind stress.
Fig. 15.
Fig. 15.

Mixing efficiency η as a function of KE generation (simulations are marked for each value).

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-19-0169.1

The buoyancy-driven simulations (BF, WF1) have very high mixing efficiency (η > 0.8), as previously found for horizontal convection simulations (Scotti and White 2011; Gayen et al. 2014) and shown by the theoretical scaling (Vreugdenhil et al. 2016; Sohail et al. 2018). The mixing efficiency values obtained in this DNS study for the simulations with significant surface wind stress (WF2, WF3, WF4) are smaller than 0.75–0.80 range found by Sohail et al. (2018). However, their aspect ratio (H/L = 0.4) is larger than that in this study (H/L = 0.1) and we imposed a more gradual change with latitude in surface buoyancy. As a result, in Sohail et al. (2018, 2019), the sinking plume, where most of the diapycnal mixing occurs, occupies a greater portion of the domain and the transport of the abyssal cell is much stronger than the weak and small upper cell, compared with the overturning circulation in our simulations. We also find that within the sinking plume and near the surface the local mixing efficiency, ηl values are quite high (ηl > 0.8). The mixing efficiency values for WF3 and WF2 that we find are within the range (0.3–0.5) found by the previous observations and DNS to provide enough energy for the lower MOC, as discussed in Mashayek et al. (2017).

4. Discussion and conclusions

In this study, we use DNS of an idealized ocean basin to assess the sensitivity of the overturning circulation in the Southern Ocean to the magnitude and distribution of the surface wind stress profile. The DNS allows us to fully resolve the energy cascade and compute processes that occur at turbulent scales directly. The circulation in the Southern Ocean is complicated by many factors, including topography, seasonality of surface fluxes, near-coastal processes around Antarctica, and the interhemispheric interaction with the overturning in the Northern Hemisphere. Nevertheless, we were able to reproduce the major features of the residual overturning circulation using a rotating horizontal convection model in a highly idealized domain by imposing surface wind stress and density distribution approximated from the Southern Ocean data. This qualitative agreement validates the application of the results from our model to the real ocean phenomena to help understand the physical mechanisms of the interplay between surface buoyancy and wind forcing.

As the wind stress is increased from the simulation with only surface buoyancy forcing (BF) to the simulation with a wind stress magnitude that is double the present-day levels (WF4), the residual overturning circulation shifts from the dominant buoyancy-driven counterclockwise cell to the dominant wind-driven clockwise upwelling cell. When the wind stress is much stronger than the present-day level, the upwelling upper MOC cell shifts further south and the lower MOC cell of dense water formation becomes insignificant in the residual circulation. While the setup is idealized in this work, this result has climatological implications for atmospheric carbon sequestration, which relies on the formation of dense water at the surface that sinks into the abyss below the thermocline.

An important distinction between this study and previous analyses of the Southern Ocean sensitivity via GCMs (Farneti et al. 2010; Abernathey et al. 2011; Hogg et al. 2017) and DNS (Barkan et al. 2015; Sohail et al. 2018) is the inclusion of polar easterlies into the wind forcing. This setup allows us to study both the lower and the upper MOC cells, whereas previously studies primarily focused on the strength of the upper cell. Intuitively, it would be expected for stronger polar easterlies to amplify the lower MOC cell via larger downwelling Ekman velocities in the convective plume region. However, it is important to note that the residual overturning circulation is the difference between the mean and turbulent components. We find that while the lower MOC cell is amplified by the downwelling Ekman velocities of stronger polar easterlies (WF4 and WF2 compared with WF3), this increase is compensated by the turbulent eddies, leading to a weaker residual dense water formation cell.

A remarkable result from our simulations is that the residual overturning circulation function for the WF3 run forced with a surface density and wind stress distributions fitted to the real ocean data includes both strong lower MOC and upper MOC cells. All simulations show primarily eddy compensation of the mean circulation indicated by the balance between domain-integrated vertical buoyancy fluxes (Fig. 10c) and by the residual component of the depth–density overturning being much smaller than the mean and turbulent components (Fig. 14). However, the deeper analysis of both density–latitude and depth–density streamfunctions shows that the overturning dynamics, in particular the dense water formation cell, are dependent on the wind stress magnitude and distribution.

From the basin-integrated energetics, we find that while the KE reservoir is sensitive to the surface wind stress, the APE terms are not significantly affected. The net KE dissipation rate increases linearly with the KE generation rate as a result of the balance of the mean and turbulent conversion rates between KE and APE reservoirs. In contrast, the generation and dissipation rates of APE do not appreciably change with an increase in energy input from the surface wind. Calculation of APE dissipation locally shows that the values are largest at the surface and within the convective plume, consistent with the DNS results of Sohail et al. (2018). Furthermore, the APE dissipation rates at the surface and within the plume are primarily independent of the surface wind forcing, as shown in Fig. 12, and the APE generation rate at the surface and basin-integrated dissipation rate are equal, in line with the theoretical arguments of Hughes et al. (2009). These results suggest that diapycnal mixing is strongly driven by surface buoyancy fluxes that control the water mass formation at the surface, as the highest rates of mixing occur in the regions of strong density gradients and large APE generation rates.

However, the APE dissipation rate is maximized for the WF3 run, in particular because of the larger values in the northern half of the domain near the bottom boundary. Unlike the other wind-driven runs, WF3 has a deep MOC cell that propagates from the dense water formation region at the southern end to the northern end along the bottom, creating stronger density stratification in that part of the domain. Thus, while the basin-integrated values may be largely independent of the surface wind forcing, locally diapycnal mixing can be significantly affected by the overturning circulation dynamics that emerge from the differences in wind stress and the degree of eddy compensation of the wind-driven mean circulation. If we extrapolate this result, with caution, to the circulation in the real ocean, it implies that the particular balance between the strong buoyancy-driven AABW cell and the wind-driven upwelling NADW cell maximizes diapycnal mixing in the ocean.

The important effect of the strength and extent of the AABW cell on diapycnal mixing is further relevant in the ocean. In the Southern Ocean, the abyssal mixing is enhanced by the radiation and breaking of the internal waves (in particular, lee waves) resulting from the geostrophic bottom flow over rough topography (Polzin 2004, 2009; Nikurashin and Ferrari 2010; St. Laurent et al. 2012; MacKinnon 2013). This study focused on the contribution of the wind and buoyancy forcing to the energy budget and the circulation, but the effect of bottom topography on the turbulent dissipation and diapycnal mixing rates needs to be investigated in the future DNS work.

Acknowledgments

This research was supported by NSF Physical Oceanography Grants OCE-1155558 and OCE-1736989. This research is also part of the Blue Waters sustained-petascale computing project, which is supported by the National Science Foundation (Awards OCI-0725070 and ACI-1238993) and the state of Illinois. Blue Waters is a joint effort of the University of Illinois at Urbana-Champaign and its National Center for Supercomputing Applications.

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