1. Introduction and motivation
The general circulation of the ocean is forced by surface fluxes of momentum, heat, and freshwater at basin scales. A large fraction of the kinetic energy Ek associated with the large-scale forcing must be dissipated at molecular scales in order for the circulation to remain approximately steady. The Ek pathways across this wide range of scales remain poorly understood (Ferrari and Wunsch 2009). Possible routes to dissipation include nonlinear internal wave interactions in the interior, instability of geostrophic motions, and direct interactions with side and bottom boundaries (Müller et al. 2005; Ferrari and Wunsch 2010). About 90% of the Ek in the oceans is stored in the geostrophic eddy field (Ferrari and Wunsch 2009) and, given this large fraction, we focus here on the “instability” pathway to dissipation.
Geostrophic eddies are formed, primarily, via baroclinic instability of large-scale ocean currents that are in approximate thermal wind balance (Gill et al. 1974). These geostrophic currents and eddies are considered “balanced” in the sense that they are in geostrophic and hydrostatic balance, and the associated velocity and buoyancy fields are entirely determined by the potential vorticity (Ford 1993). The quasi-two-dimensional geostrophic eddy field is expected, based on geostrophic turbulence theory, to transfer energy to larger scales (Salmon 1980). Kinetic energy dissipation, however, is observed to take place at small scales (Wunsch and Ferrari 2004 and references therein). Instability mechanisms leading to loss of balance (LOB) are therefore hypothesized to take place in order for some of the kinetic energy of the geostrophic eddies (EKE) to be transferred to small scales where it can be dissipated.
Recent numerical studies (Capet et al. 2008c; Molemaker et al. 2010) have suggested that loss of balance due to submesoscale instabilities can provide an efficient route to small-scale dissipation. To explore this hypothesis, we pose an idealized problem of flow in a flat-bottomed reentrant channel, externally forced by surface buoyancy fluxes and wind stress, in a configuration that resembles the Antarctic Circumpolar Channel (ACC). The flow is allowed to reach a statistical steady state at which point it exhibits both forward and inverse energy cascades. Flow interactions with irregular bathymetry are excluded so that bottom drag is the sole mechanism available to dissipate the upscale energy cascade. This simple setting allows us to compare the Ek energy pathways along two main routes: the “inverse” route—in which EKE is transferred to larger scales where it is dissipated via bottom drag—and the “forward” route—in which EKE cascades to small dissipative scales, presumably via loss of balance. We analyze these competing routes to dissipation for two different forcing scenarios and evaluate the relative efficiency of each pathway.
Capturing both the inverse and forward pathways while computing kinetic energy dissipation accurately requires a wide range of scales to be resolved and is a challenging numerical task. We are therefore forced to make compromises in the problem setup. We evaluate the validity of our numerical simulations in representing realistic oceanic flows based on five nondimensional parameters (see section 2 for details) and find them to be in quantitative agreement with the oceanic dynamical regime. In addition, we diagnose the mean buoyancy structure and the mean and eddy-driven circulations and find them to be in qualitative agreement with zonal-mean theories of the ACC (see section 3 for details). This suggests that understanding the physical mechanisms illuminated by this process study is helpful in improving our dynamical understanding of real ocean phenomena.
The paper is organized as follows: in section 2 we discuss the problem setup; in section 3 we describe the overall flow features; in section 4 we show the Ek balance; in section 5 we provide spectral analyses of velocity variance, Ek equation, and Ek fluxes; in section 6 we evaluate loss of balance in relation to EKE dissipation; and finally, in section 7 we provide a summary and discussion.
2. Problem setup
(left) Time-averaged buoyancy field for BF. Straight white arrows indicate the sense of the buoyancy-driven circulation; wavy arrows illustrate the importance of baroclinic eddies in transporting buoyancy meridionally. (right) Time-averaged zonal velocity u for WBF. The wind-induced barotropic jet is in the positive x direction. The equation 
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0068.1
(left) Time-averaged buoyancy field for BF. Straight white arrows indicate the sense of the buoyancy-driven circulation; wavy arrows illustrate the importance of baroclinic eddies in transporting buoyancy meridionally. (right) Time-averaged zonal velocity u for WBF. The wind-induced barotropic jet is in the positive x direction. The equation
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0068.1
(left) Time-averaged buoyancy field for BF. Straight white arrows indicate the sense of the buoyancy-driven circulation; wavy arrows illustrate the importance of baroclinic eddies in transporting buoyancy meridionally. (right) Time-averaged zonal velocity u for WBF. The wind-induced barotropic jet is in the positive x direction. The equation
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0068.1
Both simulations were run for ~0.25 of a diffusive time (H2/κ) by which [(8a, below)] had reached a statistical steady state (as shown in Fig. 5, described later). Time averages and all analyses presented in this paper are based on the last 0.05 diffusive time, during which all physical variables were stored every 6.5 inertial periods (f−1) for a total of 58 snapshots. Throughout the manuscript we present probability distribution functions (PDFs) and cumulative distribution functions (CDFs) of various diagnostic parameters computed for mean and total quantities at selected regions of the domain. The PDFs and CDFs are computed using all grid points in the specified domain for all snapshots. For example, if a PDF is computed based on the entire volume V then the total number of samples is 65 × 1025 × 1024 × 58 ~ O(109). The PDFs in Figs. 2, 4, and 17 (as seen below) are normalized to lie between zero and one.
PDF of (7) for BF (dashed) and WBF (solid) computed over the entire volume. The PDF is normalized by the maximal number of observed samples so that values range from 0 to 1.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0068.1
PDF of (7) for BF (dashed) and WBF (solid) computed over the entire volume. The PDF is normalized by the maximal number of observed samples so that values range from 0 to 1.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0068.1
PDF of (7) for BF (dashed) and WBF (solid) computed over the entire volume. The PDF is normalized by the maximal number of observed samples so that values range from 0 to 1.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0068.1
The model Eqs. (1) for both simulations are solved using the nonhydrostatic three-dimensional pseudospectral model flow_solve. See the appendix for more details on the model and problem setup.
Throughout the paper 〈( )〉 denotes a volume average, 
3. Overall flow features
Figure 1 (left) shows the time-averaged buoyancy field for BF. The specified flux boundary condition [(2)] corresponds to buoyancy gain near y = 0 and buoyancy loss near y = Ly. Figure 3 shows the time- and zonal-mean isopycnal height (top), Eulerian-mean streamfunction 
Time- and zonal-mean isopycnal height normalized by (top) BmaxH/κ, (middle) Eulerian-mean streamfunction 
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Time- and zonal-mean isopycnal height normalized by (top) BmaxH/κ, (middle) Eulerian-mean streamfunction
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0068.1
Time- and zonal-mean isopycnal height normalized by (top) BmaxH/κ, (middle) Eulerian-mean streamfunction
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PDF of (11) for WBF computed in the range (0.8 < y/H < 14.2 and 0.06 < z/H < 0.94). The PDF is normalized by the maximal number of observed samples so that values range from 0 to 1.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0068.1
PDF of (11) for WBF computed in the range (0.8 < y/H < 14.2 and 0.06 < z/H < 0.94). The PDF is normalized by the maximal number of observed samples so that values range from 0 to 1.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0068.1
PDF of (11) for WBF computed in the range (0.8 < y/H < 14.2 and 0.06 < z/H < 0.94). The PDF is normalized by the maximal number of observed samples so that values range from 0 to 1.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0068.1
4. Kinetic energy balance
Figure 5 shows the terms in (8a) demonstrating that, in both cases, the Ek reservoir has reached a statistical steady state. The ratio 
The terms in (8a) normalized by Bmax.
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The terms in (8a) normalized by Bmax.
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The terms in (8a) normalized by Bmax.
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The decomposition of 〈wb〉, ϵ, and ϵd in (8a) into the mean and eddy components. The Ekman velocity is wE = −(2f)−1dτs/dy. 
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The decomposition of 〈wb〉, ϵ, and ϵd in (8a) into the mean and eddy components. The Ekman velocity is wE = −(2f)−1dτs/dy.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0068.1
The decomposition of 〈wb〉, ϵ, and ϵd in (8a) into the mean and eddy components. The Ekman velocity is wE = −(2f)−1dτs/dy.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0068.1
The ratio 
5. Spectral analysis
a. Horizontal velocity wavenumber spectra
Figure 7 shows the time-averaged horizontal velocity (u and υ) wavenumber spectra in the kx (black) and ky (gray) directions, computed both near the surface (solid lines) and at depth (dashed line). For both BF and WBF, the surface spectral shapes are nearly flat at low wavenumbers, exhibit a rolloff, and become linear over more than a decade of wavenumbers and then steepen further at higher wavenumber where horizontal isotropy is lost and the flow becomes more three-dimensional (the horizontally isotropic regions are gray shaded). The spectral shapes at depth closely resemble those at the surface, although the spectral level is reduced by about a decade over the entire wavenumber range (in WBF the amount of energy at the lowest meridional wavenumber is independent of depth, a signature of the barotropic zonal jet).
The y-averaged horizontal velocity wavenumber spectra (black) and x-averaged horizontal velocity wavenumber spectra (gray) depth averaged in the ranges 0.85 < z/H < 1 (solid lines) and 0.3 < z/H < 0.45 (dashed lines). Straight lines indicate −2 (dotted–dashed) and −3 (dashed) spectral slopes. The shaded gray rectangles indicate the wavenumber range where the flow is approximately isotropic. Spectra are normalized by BmaxLy/f.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0068.1
The y-averaged horizontal velocity wavenumber spectra (black) and x-averaged horizontal velocity wavenumber spectra (gray) depth averaged in the ranges 0.85 < z/H < 1 (solid lines) and 0.3 < z/H < 0.45 (dashed lines). Straight lines indicate −2 (dotted–dashed) and −3 (dashed) spectral slopes. The shaded gray rectangles indicate the wavenumber range where the flow is approximately isotropic. Spectra are normalized by BmaxLy/f.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0068.1
The y-averaged horizontal velocity wavenumber spectra (black) and x-averaged horizontal velocity wavenumber spectra (gray) depth averaged in the ranges 0.85 < z/H < 1 (solid lines) and 0.3 < z/H < 0.45 (dashed lines). Straight lines indicate −2 (dotted–dashed) and −3 (dashed) spectral slopes. The shaded gray rectangles indicate the wavenumber range where the flow is approximately isotropic. Spectra are normalized by BmaxLy/f.
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For BF, the spectral slope after the rolloff is close to k−3, whereas in WBF it is closer to k−2. A k−3 spectral slope is typically associated with an enstrophy inertial subrange in quasigeostrophic turbulence theory (Salmon 1980). A k−2 spectral slope has previously been observed in submesoscale-resolving simulations (Capet et al. 2008a, their Fig. 6) and was interpreted as an indication of a significant amount of kinetic energy in the submesoscale range. This suggests that the energy reservoir in the wavenumber band associated with submesoscale features is larger for WBF.
b. Kinetic energy spectral balance
Figure 8 shows the individual terms in the spectral energy balance (9). The cospectra are vertically integrated, time averaged, and are computed as a function of the horizontal wavenumber 
The Ek balance in spectral space [(9)]. Terms are conversion to/from potential energy (blue), advective flux divergence (green), bottom drag dissipation (red), Laplacian dissipation (black), and wind work (magenta). Solid lines denote total fields and dashed lines denote eddy fields. Shaded gray boxes correspond to the wavenumber range in Fig. 7, where the flow is isotropic. Vertical dashed lines denote the most unstable baroclinic mode. All terms are normalized by 
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0068.1
The Ek balance in spectral space [(9)]. Terms are conversion to/from potential energy (blue), advective flux divergence (green), bottom drag dissipation (red), Laplacian dissipation (black), and wind work (magenta). Solid lines denote total fields and dashed lines denote eddy fields. Shaded gray boxes correspond to the wavenumber range in Fig. 7, where the flow is isotropic. Vertical dashed lines denote the most unstable baroclinic mode. All terms are normalized by
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0068.1
The Ek balance in spectral space [(9)]. Terms are conversion to/from potential energy (blue), advective flux divergence (green), bottom drag dissipation (red), Laplacian dissipation (black), and wind work (magenta). Solid lines denote total fields and dashed lines denote eddy fields. Shaded gray boxes correspond to the wavenumber range in Fig. 7, where the flow is isotropic. Vertical dashed lines denote the most unstable baroclinic mode. All terms are normalized by
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0068.1
For BF, there is very little difference between total and eddy fields. Positive 
In contrast, for WBF there is a clear difference between total and eddy fields at low wavenumbers (khLy < 2.5), where most of the energy is in the mean flow. Wind work (magenta line) supplies energy at large scales, primarily to the barotropic zonal jet (Fig. 1) whose energy is dissipated by bottom drag (red line). Negative values of 
c. Kinetic energy spectral fluxes: Forward and inverse cascades
(top right) Spectral Ek flux Π(kh) [(13)] normalized by Bmax. Shaded gray boxes correspond to the wavenumber range in Fig. 7, where the flow is isotropic. The vertical dashed line denotes the most unstable baroclinic mode. (top left) A snapshot of the depth-averaged enstrophy (ζ/f)2, where ζ ≡ υx − uy is the vertical vorticity. (a)–(c) Bandpass-filtered enstrophy in the horizontal wavenumber range indicated on the spectral flux plot. BF simulation.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0068.1
(top right) Spectral Ek flux Π(kh) [(13)] normalized by Bmax. Shaded gray boxes correspond to the wavenumber range in Fig. 7, where the flow is isotropic. The vertical dashed line denotes the most unstable baroclinic mode. (top left) A snapshot of the depth-averaged enstrophy (ζ/f)2, where ζ ≡ υx − uy is the vertical vorticity. (a)–(c) Bandpass-filtered enstrophy in the horizontal wavenumber range indicated on the spectral flux plot. BF simulation.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0068.1
(top right) Spectral Ek flux Π(kh) [(13)] normalized by Bmax. Shaded gray boxes correspond to the wavenumber range in Fig. 7, where the flow is isotropic. The vertical dashed line denotes the most unstable baroclinic mode. (top left) A snapshot of the depth-averaged enstrophy (ζ/f)2, where ζ ≡ υx − uy is the vertical vorticity. (a)–(c) Bandpass-filtered enstrophy in the horizontal wavenumber range indicated on the spectral flux plot. BF simulation.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0068.1
As in Fig. 9, but for WBF simulation.
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As in Fig. 9, but for WBF simulation.
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As in Fig. 9, but for WBF simulation.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0068.1
For BF, band a is associated with large-scale motions in the deep convection region. Band b corresponds to “mesoscale” geostrophic eddies that, in agreement with geostrophic turbulence theory, are seen to transfer energy to larger scales. Note that EKE dissipation due to both bottom drag and Laplacian dissipation have maxima (khLy = 4.2) near the transition wavenumber (khLy ~ 3.5) between regions a and b (Fig. 8, top). Our interpretation is that the maximum in Laplacian EKE dissipation is an energy sink associated with the forward transfer in the deep convection region, and the maximum in bottom drag EKE dissipation corresponds to the arrest of the inverse energy cascade associated with the geostrophic eddies. Band c corresponds to the sharp fronts that develop at the edges of the eddies and deep convection filaments, and we argue in section 6 that the flow features in this wavenumber range are associated with the loss of balance.
For WBF (Fig. 10), the dynamics associated with wavenumber bands b and c and the corresponding bandpass-filtered enstrophies are qualitatively similar to BF. Quantitatively, the ratio of forward to inverse energy fluxes is larger than in BF, suggesting that loss of balance at these scales is more vigorous in WBF. This is in agreement with the shallower spectral slope of WBF compared with that of BF (Fig. 7). The dynamics in band a are now dominated by the large-scale wind-driven jet. As indicated by Fig. 8 (bottom left), the total kinetic energy at these scales (associated with the zonal jet) is removed by bottom drag (solid red line). As in BF, we interpret the maximum bottom drag EKE dissipation (dashed red line in bottom-right panel) observed in the transition wavenumber between bands a and b (khLy = 4.2) to be associated with the halting of the inverse energy cascade. Laplacian dissipation (dashed black line) is more evenly distributed across the wavenumber with a much broader maximum than in BF. We note, however, that inferring dynamics solely from spectral fluxes can be misleading, particularly in situations where the energy transfers are nonlocal in wavenumber space.
6. Loss of balance
Figures 11a,b and 12a,b show snapshots of the Richardson number 
(a) Richardson number 
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(a) Richardson number
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(a) Richardson number
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As in Fig. 11, but for WBF. Note the different colorbar range in (c).
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As in Fig. 11, but for WBF. Note the different colorbar range in (c).
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0068.1
As in Fig. 11, but for WBF. Note the different colorbar range in (c).
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0068.1
N2 changes sign (becomes negative);
the absolute vorticity in isopycnal coordinates A = f + ζ changes sign (becomes negative for f > 0);
change of sign of A − |S| (becomes negative for f > 0), where S2 = (uX − υY)2 + (υX + uY)2 is the variance of the horizontal strain rate in isopycnal coordinates; or
the Richardson number Ri < ¼.
Other forms of submesoscale instabilities are clearly present in our numerical simulations. Mixed layer instability (MLI) (Fox-Kemper et al. 2008) is an obvious one, as indicated by the O(1) Richardson number values in Figs. 11a and 12a. MLI is important for the restratification of the mixed layer (Boccaletti et al. 2007) and may develop into finite-amplitude mixed layer eddies that can lead to intense frontogenesis, a precondition for LOB. Nevertheless, it is a balanced instability that does not lead directly to LOB (Molemaker et al. 2005; Thomas et al. 2008); that is, additional instability mechanisms, such as the ones described above, must take place for LOB to occur. The indirect pathways to LOB due to MLI and other balanced phenomena are therefore assumed to be accounted for by the instability mechanisms i–iv above.
Figures 13 and 14 show surface slices of the conditions for LOB (i–iii) in BF and WBF, respectively. The contour plot in Figs. 11a and 12a shows condition iv for LOB. Regions where N2 < 0 are not plotted in conditions ii–iv. For SI, only the regions where PV < 0 and PVV > 0 are plotted. Similarly, for INI, only regions where the PV < 0 and PVH > 0 are plotted.
Horizontal slices of (a) (N/f)2, (b) PV/f3 associated with symmetric instability (see text for detail), (c) (A − |S|)/f, (d) PV/f3 associated with inertial instability (see text for detail), (e) 
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Horizontal slices of (a) (N/f)2, (b) PV/f3 associated with symmetric instability (see text for detail), (c) (A − |S|)/f, (d) PV/f3 associated with inertial instability (see text for detail), (e)
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0068.1
Horizontal slices of (a) (N/f)2, (b) PV/f3 associated with symmetric instability (see text for detail), (c) (A − |S|)/f, (d) PV/f3 associated with inertial instability (see text for detail), (e)
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0068.1
As in Fig. 13, but for WBF.
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As in Fig. 13, but for WBF.
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As in Fig. 13, but for WBF.
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Qualitatively, in BF, on large scales, LOB occurs because of CI in the deep convection region near y/H > 12 (Fig. 13a). At smaller scales, within the deep convection region, LOB occurs in some locations due to SI (Fig. 13b) and more extensively due to AAI, as indicated by the fine, blue filaments in Fig. 13c. The condition for INI is never satisfied (Fig. 13d). The condition KHI is rarely satisfied, as indicated by the hardly visible black contours in Fig. 11a.
In WBF, LOB due to SI and more so due to AAI occurs over a much greater portion of the domain, extending throughout the thermocline region (y/H < 5). The condition for INI is, again, never satisfied. LOB due to CI is still observed in the deep convection region (now y/H > 10) and, in addition, across the jet (6 < y/H < 9) at much smaller scales. In both flows, there are regions, especially near fronts, where PV ~ A > 0 but A − |S| < 0, emphasizing the importance of elevated strain variance in triggering LOB in these locations. The condition for KHI is satisfied more extensively as well, primarily in the jet region (contour lines in Fig. 12a). Note, however, that the level of EKE in the jet region is low compared with levels at the thermocline and deep convection regions (Fig. 12c). Visually, regions of unbalanced, nonhydrostatic flow (Figs. 14e and 14f) correspond spatially to the regions of elevated Laplacian dissipation shown in Fig. 12d. This is consistent with the notion that the forward energy cascade at frontal scales (wavenumber band c in Figs. 9 and 10) and the corresponding Laplacian dissipation are dynamically linked with LOB.
Figure 15 shows the CDF of (16), denoted by 
CDF of 
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CDF of
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CDF of
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Figure 16 shows the time- and zonal-mean Laplacian EKE dissipation 
Time and x-averaged EKE dissipation. The dashed horizontal line in the bottom panel denotes the base of the surface layer. Note the different colorbar ranges.
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Time and x-averaged EKE dissipation. The dashed horizontal line in the bottom panel denotes the base of the surface layer. Note the different colorbar ranges.
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Time and x-averaged EKE dissipation. The dashed horizontal line in the bottom panel denotes the base of the surface layer. Note the different colorbar ranges.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0068.1
For WBF, 
To quantify which of the LOB instabilities is most probable we have computed the CDFs of each of the LOB conditions (i–iv). In both simulations, the CDFs were computed near the top (0.92 ≤ z/H ≤ 1), extending over the full zonal and meridional extents. For WBF, we have also computed the CDFs in the same vertical and zonal ranges, however, limiting the meridional extent to the thermocline region. Table 1 summarizes the results of the CDF analysis. Figure 17 shows the CDFs for the WBF simulations.
The probability P of conditions i–iv in the BF and WBF simulations. CI is P(N2 < 0). SI is P(PV < 0 ∩ PVV > 0 ∩ N2 > 0). INI is P(PV < 0 ∩ PVH > 0 ∩ N2 > 0). AAI is P(A − |S| < 0 ∩ N2 > 0). KHI is P(Ri < 0.25 ∩ N2 > 0). PVH and PVV are the horizontal and vertical components of the potential vorticity [(14)]. CDFs are computed based on the entire zonal and meridional extents in the range (0.92 ≤ z/H ≤ 1). WBF–thermocline is (0.92 ≤ z/H ≤ 1) and (0 < y/H < 5).
CI is the CDF of (N/f)2; SI is the CDF of PV/f3 associated with SI; AAI is the PDF of (A − |S|)/f; and KHI is the CDF of Ri (see text and Table 1 for details on how the probabilities were computed). Circles are the CDF computed near the top of the domain (0.92 ≤ z/H ≤ 1). Squares are the CDF computed near the top of the domain (0.92 ≤ z/H ≤ 1) in the thermocline region (0 < y/H < 5). Note that the bin resolution is much higher than shown by the symbols. For SI, the corresponding PDFs are also shown (inset). For KHI, Ri values greater than a thousand were set to 1000 to decrease the bin resolution needed to resolve the range 0 ≤ Ri ≤ 1 (inset). The vertical dashed lines denote the transition values to instability (refer to Table 1 for actual values). WBF simulation.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0068.1
CI is the CDF of (N/f)2; SI is the CDF of PV/f3 associated with SI; AAI is the PDF of (A − |S|)/f; and KHI is the CDF of Ri (see text and Table 1 for details on how the probabilities were computed). Circles are the CDF computed near the top of the domain (0.92 ≤ z/H ≤ 1). Squares are the CDF computed near the top of the domain (0.92 ≤ z/H ≤ 1) in the thermocline region (0 < y/H < 5). Note that the bin resolution is much higher than shown by the symbols. For SI, the corresponding PDFs are also shown (inset). For KHI, Ri values greater than a thousand were set to 1000 to decrease the bin resolution needed to resolve the range 0 ≤ Ri ≤ 1 (inset). The vertical dashed lines denote the transition values to instability (refer to Table 1 for actual values). WBF simulation.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0068.1
CI is the CDF of (N/f)2; SI is the CDF of PV/f3 associated with SI; AAI is the PDF of (A − |S|)/f; and KHI is the CDF of Ri (see text and Table 1 for details on how the probabilities were computed). Circles are the CDF computed near the top of the domain (0.92 ≤ z/H ≤ 1). Squares are the CDF computed near the top of the domain (0.92 ≤ z/H ≤ 1) in the thermocline region (0 < y/H < 5). Note that the bin resolution is much higher than shown by the symbols. For SI, the corresponding PDFs are also shown (inset). For KHI, Ri values greater than a thousand were set to 1000 to decrease the bin resolution needed to resolve the range 0 ≤ Ri ≤ 1 (inset). The vertical dashed lines denote the transition values to instability (refer to Table 1 for actual values). WBF simulation.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0068.1
For BF, in agreement with Fig. 13, CI is the most probable instability condition followed by AAI and SI. The conditions for INI and KHI are essentially never satisfied (for INI the probability is, in fact, identically zero).
For WBF, the probability for all conditions increases by an order of magnitude, however, the condition for INI is rarely satisfied. CI is the most probable instability condition followed by AAI, SI, and KHI. In the thermocline region, however, the probability for AAI exceeds all other conditions by an order of magnitude. The reduction in CI is clear from Fig. 14a that shows that N2 > 0 in the thermocline region. CI is thus most probable in the jet and convection regions. SI is a “forced” instability and can only be sustained if it is maintained by winds and/or buoyancy fluxes (Thomas and Lee 2005). Otherwise, SI will tend to restratify the surface layer and adjust the background flow to a state of marginal stability with PV ≈ 0, thus damping itself out (Taylor and Ferrari 2009). This is clearly seen in the PDFs of SI (inset in Fig. 17), where the most probable PV values are positive and very close to zero. The reduction in the probability of SI in the thermocline region compared with the entire domain suggests that it is sustained, primarily, by buoyancy loss in the deep convection region. Finally, as depicted in Fig. 12a (contours), the Richardson number is less than a quarter, primarily in the jet region. This explains the reduction in the probability for KHI within the thermocline region.
7. Summary and discussion
We have investigated two idealized DNS of channel flow with different external forcing: one forced solely by surface buoyancy fluxes (BF) and the other by both surface buoyancy fluxes and wind stress (WBF). In both experiments there are two possible routes to Ek dissipation: upscale energy cascade leading to dissipation due to bottom drag (the inverse route) and downscale energy cascade, catalyzed by loss of balance (LOB), leading to Laplacian dissipation at small scales (the forward route).
In WBF, a strong barotropic zonal jet is directly forced by the wind stress. As a result, the total kinetic energy reservoir is 7 times larger than in BF. Only a fraction of that wind work, however, that is, useful wind work (Cessi et al. 2006), is converted to available potential energy and then drained to excite geostrophic eddies. Consequently, the eddy kinetic energy (EKE) reservoir in WBF is only twice that of BF. Furthermore, in both simulations, the amount of EKE dissipation due to Laplacian dissipation at small scales (ϵ′) exceeds that due to bottom drag 
Analyses of the Rossby number, Richardson number, violation of the different balance constraints, and of Ek spectral fluxes indicate that ϵ′ is directly linked to ageostrophic motions (Ro ≥ 1, Ri ≤ 1) that undergo LOB instabilities and exhibit a forward energy cascade. Spatial maps of time- and zonal-mean EKE dissipation are correlated with regions of LOB at frontal (submeso) scales. As a result, for BF, EKE dissipation is strongest in the upper domain within the deep convection region. For WBF, where the forward energy cascade at submesoscales and the degree of LOB are much larger, EKE dissipation is still strongest near the surface (e.g., in the “surface layer”) and particularly within the thermocline region. In both cases a CDF of the parameter quantifying the degree of LOB [(16)] indicates that LOB occurs much more frequently in the surface layer than in the interior.
Analyses of the CDFs and spatial maps of the different LOB conditions (section 6) that lead to the forward energy transfers indicate that, in BF, convective instability (CI) is most probable on the large scales followed by ageostrophic anticyclonic instability (AAI) and to a lesser extent symmetric instability (SI), both of which act at submesoscales. Inertial instability (INI) and Kelvin Helmholts instability (KHI) are essentially never found. All instability mechanisms in this case take place within the deep convection region.
For WBF, the probability for all conditions is increased by an order of magnitude compared with BF. In addition, different instabilities are more/less probable at different meridional regions. In the deep convection region (10 < y/H), CI is most probable at large scales, followed by AAI and SI that act at submesoscales. In the jet region (6 < y/H < 9), CI, AAI, and KHI are all diagnosable, although the amount of EKE there is very low. In the thermocline region (y/H < 5), where most of the EKE is found and where most of the EKE dissipation takes place, AAI is the most probable instability, exceeding all others by an order of magnitude. This suggests that AAI is a more prevalent mechanism for LOB at these submesoscales. It is important to note that because of our model output frequency (≈6.5 × f−1), we may underestimate the importance of SI that equilibrates on much faster time scales than AAI, thereby wiping itself out as it resets PV to zero. Nevertheless, we argue that it is the lack of external forcing needed to sustain symmetric instability that causes the probability reduction in the thermocline region. The fact that in the deep convection region where buoyancy loss can sustain SI the probabilities are high supports this argument.
It is noteworthy that a variety of balanced processes, such as mixed layer instability, are important to catalyze frontogenesis and thus precondition the flow to be more susceptible to LOB. Nevertheless, additional instability mechanisms, such as the ones described above, must take place for LOB to occur. Therefore, we have not explicitly analyzed all possible preconditioning processes and assumed LOB to be accounted for by the specific instability mechanisms listed in section 6.
Oceanographic implications
The numerical challenge of resolving the wide range of scales necessary to capture both forward and inverse energy cascades while computing dissipation accurately is considerable, particularly for an externally forced problem that requires a long time to equilibrate. We therefore had to make compromises in the problem setup. The results of such an exercise are intended to provide insight into the physics of nearly balanced flows that are dynamically similar to oceanic flows and that drive both up- and down-scale energy cascades. The similarity to oceanic flows has been demonstrated by diagnosing the quasi-steady simulations to be nearly hydrostatically balanced and with small characteristic values of the Rossby, Burger, and Ekman numbers. Though the nearly adiabatic nature of the WBF simulation and the overall thermal structure in the thermocline region are in reasonable qualitative agreement with flow in the ACC (Marshall and Speer 2012, and references therein), the strength of the stratification produced in the simulation is weaker than that observed. This is partially because of processes associated with bathymetric detail and lateral exchange with other basins that are not included in our model. Whether or not higher stratification would alter the relative transfer rates of the forward and inverse energy cascades by suppressing certain frontal instability mechanisms is not clear. In the deep convection zone the dynamics are perhaps more relevant to weakly stratified regions subject to deep convection such as the Labrador Sea (Jones and Marshall 1997).
The useful wind work concept lead us to propose the nondimensional parameter SEKE [(12)] to account for the relative contribution of wind to buoyancy forcing to EKE generation. The parameter SEKE ~ 100 for representative ACC parameters, suggesting that wind forcing is responsible for most of the EKE generation, with buoyancy forcing providing only 1%. In our WBF simulation, SEKE is much smaller (~6). Figures 12a and 12b show that wind forcing greatly increases/decreases the Ro/Ri values at the fronts. This suggests that higher SEKE values may, in fact, lead to higher/lower values of Ro/Ri, that is, stronger ageostrophic motions, which will have even greater susceptibility to LOB and forward energy cascade.
One can better relate our simulations to ocean scales based on our approximation kdLy ≈ 10 (see section 5b for detail), meaning that the domain in our simulations contains roughly 10 Rossby radii of deformation. This estimate is in agreement with the enstrophy structures associated with bandpass filter (b) in Figs. 9 and 10. Our horizontal grid spacing is thus ≈Rd/100 and that for high latitudes (Rd = 15–25 km) would be about 150–250 m. The vertical resolution needs to be sufficient to resolve the elevated EKE dissipation in the surface “mixed” layer (dashed line in Fig. 16). In our case, seven grid points were required to properly resolve the surface layer. Note that a low-order finite difference model would require many more grid points than a spectral model to accurately resolve the same flow features. Finally, Fig. 14e shows that hydrostatic balance is partially lost in the sharp frontal features that develop at the edge of the baroclinic eddies. This suggests that nonhydrostatic dynamics may be important at these small scales and can affect the energy pathways and LOB instabilities, as has previously been noted by Mahadevan (2006). Magaldi and Haine (2015), for example, have found the amount of strain variance to be much larger in nonhydrostatic versus hydrostatic simulations, a feature that would undoubtedly increase the likelihood for AAI.
Figure 18 shows horizontal and vertical slices of a local region around a representative front in WBF (rectangular box in Fig. 12). Figure 18a shows the buoyancy field at the front with the predominantly geostrophic flow (solid arrows) in the vicinity of the high pressure, positively buoyant eddy. Figures 18c and 18e show the vertical velocity and EKE dissipation (log scale) in the same region. EKE dissipation is strongest at the front and correspond well to downwelling regions. Vertical slices of the same fields at x/H = 5.2 are shown in Figs. 18b, 18d, and 18f. As expected from classical theory (Hoskins and Bretherton 1972), ageostrophic circulation with downwelling on the cold side of the front and upwelling on the warm side is observed. The correlation between elevated dissipation and downwelling is found all through the thermocline region. The largest values of dissipation, however, are confined to the surface layer (dashed line), as expected from Fig. 16.
A local region (rectangular box in Fig. 12) showing a typical submesoscale frontogenesis in WBF: (a),(b) buoyancy normalized by BmaxH/κ; (c),(d) vertical velocity normalized by BmaxH/(κf); and (e),(f) log of Laplacian EKE dissipation normalized by Bmax. Left panels are in the (x, y) plane. Right panels are in the (y, z) plane corresponding to the solid lines in (a)–(c). Solid black arrows in (a) and (b) denote velocity vectors. Horizontal dashed line in (f) indicates the base of the surface layer (see Fig. 16).
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0068.1
A local region (rectangular box in Fig. 12) showing a typical submesoscale frontogenesis in WBF: (a),(b) buoyancy normalized by BmaxH/κ; (c),(d) vertical velocity normalized by BmaxH/(κf); and (e),(f) log of Laplacian EKE dissipation normalized by Bmax. Left panels are in the (x, y) plane. Right panels are in the (y, z) plane corresponding to the solid lines in (a)–(c). Solid black arrows in (a) and (b) denote velocity vectors. Horizontal dashed line in (f) indicates the base of the surface layer (see Fig. 16).
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0068.1
A local region (rectangular box in Fig. 12) showing a typical submesoscale frontogenesis in WBF: (a),(b) buoyancy normalized by BmaxH/κ; (c),(d) vertical velocity normalized by BmaxH/(κf); and (e),(f) log of Laplacian EKE dissipation normalized by Bmax. Left panels are in the (x, y) plane. Right panels are in the (y, z) plane corresponding to the solid lines in (a)–(c). Solid black arrows in (a) and (b) denote velocity vectors. Horizontal dashed line in (f) indicates the base of the surface layer (see Fig. 16).
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0068.1
Although we do not attempt to perform a thorough analysis of all possible mechanisms that may lead to downwelling and small-scale dissipation, our results suggest that AAI may play an important role in this process. Understanding the dynamics associated with AAI is an active area of research (Ménesguen et al. 2012). It has been previously identified in high-resolution numerical studies of variable configurations and complexities (Mahadevan and Tandon 2006; Capet et al. 2008b; Molemaker et al. 2010), although its significance as a mechanism leading to LOB was not quantified.
The fact that EKE is dissipated preferentially at small scales near the surface, rather than at large scales via bottom drag suggests that the forward route associated with LOB may provide an important route to dissipation for the kinetic energy stored in the geostrophic eddy field. Although we do not incorporate LOB mechanisms associated with topographic interactions, except as a parameterized bottom drag, the spatial structure of small-scale EKE dissipation suggests that LOB caused by frontal instabilities near the ocean surface can be very efficient, independent of boundary effects.
It has been shown in models (Chen and Kamenkovich 2013) and in observations (Zajaczkovski and Gille 2014, manuscript submitted to J. Geophys. Res.) that baroclinic eddies in the ACC are typically generated in localized regions downstream of topography. Whether this localization would affect the degree of LOB or the instability mechanisms leading to it is unclear. Furthermore, Nikurashin et al. (2013) have demonstrated in numerical simulations that include realistic Southern Ocean topography that topographically generated internal waves play an important role in the total Ek dissipation. However, because they did not distinguish between mean and eddy fields it is hard to evaluate what fraction of the internal waves was excited by the mean flow and what fraction was excited by the eddies. In the ocean, LOB due to both frontal instability near the surface and internal wave generation near topography may provide a route leading to EKE dissipation. To our knowledge, no study has yet quantified which of the routes dominates.
Other phenomena such as downfront winds (Thomas and Lee 2005) and convection due to the diurnal cycle (Taylor and Ferrari 2010) may trigger or suppress certain frontal instability mechanisms. In addition, inertial gravity waves (Thomas 2012) and Stokes drift associated with the surface gravity wave field (McWilliams and Fox-Kemper 2013) may affect the frontal processes and complicate the picture further. These complexities, however, only emphasize the importance of understanding the physics of fronts, which lie at the cusp between balanced and unbalanced motions near the ocean surface.
Acknowledgments
RB thanks George Carnevale for useful discussion on spectral fluxes. Comments from two anonymous referees greatly impoved this manuscript. This work was supported by the National Science Foundation, Grant Numbers OCE-0926481 and OCE-1259580. XSEDE computing resources at the National Institute of Computational Sciences, University of Tennessee were made available under Grant TG-OCE120004.
APPENDIX
Details of the Numerical Simulations
The model Eqs. (1) are solved using the nonhydrostatic, three-dimensional, pseudospectral model flow_solve (Winters and de la Fuente 2012) on an equally spaced grid with (nz × ny × nx) = (65 × 1025 × 1024) grid points. The grid spacing is dx = dy = dz, ensuring that nonhydrostatic motions are completely resolved.
To verify that all simulated fields were resolved down to the buoyancy variance dissipation scale, the spectra of the second derivative (highest derivative in the equations of motion) of buoyancy in each direction were analyzed. The value of Pr was 7 for both simulations, so the buoyancy field exhibits the smallest scales in these simulations and the second derivative of buoyancy is the hardest computed field to resolve. The decay in the spectra of the second derivative down to the highest wavenumber was the criterion used to ensure sufficient resolution. Table 2 provides the dimensional scales of the variables used to set up the simulations. Finally, the Kolmogorov microscale lko = (ν3/EKEflux)¼ based on the EKE flux measured in our WBF simulation (Fig. 6) is lko ≈ 3dy. This is an indication that dissipative scales are adequately resolved. Furthermore, it illustrates that the forward cascade is reasonably inertial with peak values being ≈13lko (Fig. 10).
Parameters used in the numerical simulations. For BF τmax = 0.
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