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## Abstract

An equivalent-barotropic fluid on the *β* plane, forced at small scales by random stirring and dissipated by linear heat and vorticity drag, is considered as a local model for flow in the weather layer of internally forced planetary atmospheres. The combined presence of *β,* a finite deformation scale, and large-scale dissipation produce novel dynamics with possible relevance to the giant gas planets, which are apparently driven by small-scale convective stirring. It is shown that in order for anisotropy to form, one must have *β*(*ϵλ*
^{5})^{−1/3} ≳ 3.9, where *ϵ* is the (convectively driven) energy generation rate, *λ* is the deformation wavenumber, and *β* is the Coriolis gradient. The critical value above is not equivalent to the barotropic stability criterion, and numerical simulations demonstrate that anisotropic flow with average zonal velocities that are supercritical with respect to the latter can form. The formation of jets (a different matter) is not implied by the excess of zonal kinetic energy, and is instead sensitive to the relevant stability criterion for the flow geometry at hand. When *β* is sufficiently large that anisotropy does form, the flow scale and rms zonal velocity are set by a combination of Rossby wave cascade inhibition, the total energy constraint imposed by the large-scale dissipation, and the partitioning between available potential and kinetic energies. The resulting theory demonstrates that a relatively narrow range of parameters will allow for the formation of anisotropic flow with scale larger than the deformation scale. This is consistent with observations that indicate little separation between the jet scales and deformation scales on Jupiter and Saturn.

## Abstract

An equivalent-barotropic fluid on the *β* plane, forced at small scales by random stirring and dissipated by linear heat and vorticity drag, is considered as a local model for flow in the weather layer of internally forced planetary atmospheres. The combined presence of *β,* a finite deformation scale, and large-scale dissipation produce novel dynamics with possible relevance to the giant gas planets, which are apparently driven by small-scale convective stirring. It is shown that in order for anisotropy to form, one must have *β*(*ϵλ*
^{5})^{−1/3} ≳ 3.9, where *ϵ* is the (convectively driven) energy generation rate, *λ* is the deformation wavenumber, and *β* is the Coriolis gradient. The critical value above is not equivalent to the barotropic stability criterion, and numerical simulations demonstrate that anisotropic flow with average zonal velocities that are supercritical with respect to the latter can form. The formation of jets (a different matter) is not implied by the excess of zonal kinetic energy, and is instead sensitive to the relevant stability criterion for the flow geometry at hand. When *β* is sufficiently large that anisotropy does form, the flow scale and rms zonal velocity are set by a combination of Rossby wave cascade inhibition, the total energy constraint imposed by the large-scale dissipation, and the partitioning between available potential and kinetic energies. The resulting theory demonstrates that a relatively narrow range of parameters will allow for the formation of anisotropic flow with scale larger than the deformation scale. This is consistent with observations that indicate little separation between the jet scales and deformation scales on Jupiter and Saturn.

## Abstract

As in the midlatitude atmosphere, midocean eddies are primarily generated by baroclinically unstable mean currents. In contrast to the atmosphere, however, oceanic currents are significantly nonzonal. Even weak nonzonal currents are linearly unstable since *β* does not suppress growing meridional waves. Theories for the nonlinear equilibration of baroclinic instability, and hence theories for the amplitudes of midocean eddies, must therefore take into account the different dynamics of nonzonal flow. It is shown here that the amplitude of fully developed baroclinic turbulence due to nonzonal shears differs from that due to zonal shears primarily in the nature of the eddy generation. Since *β* will act to create large-scale zonal jet structures regardless of the generation source, the nature of eddy fluxes of potential vorticity (the source of eddy energy) in the zonal and meridional directions are fundamentally different. The cross-jet mixing has been shown previously to obey a mixing-length scaling, and this corresponds to the generation due to unstable zonal flow. The along-jet mixing, which corresponds to the generation due to the meridional shear, is shown here to be best described by a shear dispersion model. The resulting flux is orders of magnitude higher than in the cross-jet direction, and thus eddy energies driven by baroclinically unstable mean flows with a nonzero meridional component are much larger. This provides an explanation for recently reported results. Moreover, given recent observational and modeling studies showing the ubiquitous presence of zonal jets in the oceans, the results presented here indicate a powerful source of eddy energy.

## Abstract

As in the midlatitude atmosphere, midocean eddies are primarily generated by baroclinically unstable mean currents. In contrast to the atmosphere, however, oceanic currents are significantly nonzonal. Even weak nonzonal currents are linearly unstable since *β* does not suppress growing meridional waves. Theories for the nonlinear equilibration of baroclinic instability, and hence theories for the amplitudes of midocean eddies, must therefore take into account the different dynamics of nonzonal flow. It is shown here that the amplitude of fully developed baroclinic turbulence due to nonzonal shears differs from that due to zonal shears primarily in the nature of the eddy generation. Since *β* will act to create large-scale zonal jet structures regardless of the generation source, the nature of eddy fluxes of potential vorticity (the source of eddy energy) in the zonal and meridional directions are fundamentally different. The cross-jet mixing has been shown previously to obey a mixing-length scaling, and this corresponds to the generation due to unstable zonal flow. The along-jet mixing, which corresponds to the generation due to the meridional shear, is shown here to be best described by a shear dispersion model. The resulting flux is orders of magnitude higher than in the cross-jet direction, and thus eddy energies driven by baroclinically unstable mean flows with a nonzero meridional component are much larger. This provides an explanation for recently reported results. Moreover, given recent observational and modeling studies showing the ubiquitous presence of zonal jets in the oceans, the results presented here indicate a powerful source of eddy energy.

## Abstract

Temperature–salinity profiles from the region studied in the North Atlantic Tracer Release Experiment (NATRE) show large isopycnal excursions at depths just below the thermocline. It is proposed here that these thermohaline filaments result from the mesoscale stirring of large-scale temperature and salinity gradients by geostrophic turbulence, resulting in a direct cascade of thermohaline variance to small scales. This hypothesis is investigated as follows: Measurements from NATRE are used to generate mean temperature, salinity, and shear profiles. The mean stratification and shear are used as the background state in a high-resolution horizontally homogeneous quasigeostrophic model. The mean state is baroclinically unstable, and the model produces a vigorous eddy field. Temperature and salinity are stirred laterally in each density layer by the geostrophic velocity and vertical advection is by the ageostrophic velocity. The simulated temperature–salinity diagram exhibits fluctuations at depths just below the thermocline of similar magnitude to those found in the NATRE data. It is shown that vertical diffusion is sufficient to absorb the laterally driven cascade of tracer variance through an amplification of filamentary slopes by small-scale shear. These results suggest that there is a strong coupling between vertical mixing and horizontal stirring in the ocean at scales below the deformation radius.

## Abstract

Temperature–salinity profiles from the region studied in the North Atlantic Tracer Release Experiment (NATRE) show large isopycnal excursions at depths just below the thermocline. It is proposed here that these thermohaline filaments result from the mesoscale stirring of large-scale temperature and salinity gradients by geostrophic turbulence, resulting in a direct cascade of thermohaline variance to small scales. This hypothesis is investigated as follows: Measurements from NATRE are used to generate mean temperature, salinity, and shear profiles. The mean stratification and shear are used as the background state in a high-resolution horizontally homogeneous quasigeostrophic model. The mean state is baroclinically unstable, and the model produces a vigorous eddy field. Temperature and salinity are stirred laterally in each density layer by the geostrophic velocity and vertical advection is by the ageostrophic velocity. The simulated temperature–salinity diagram exhibits fluctuations at depths just below the thermocline of similar magnitude to those found in the NATRE data. It is shown that vertical diffusion is sufficient to absorb the laterally driven cascade of tracer variance through an amplification of filamentary slopes by small-scale shear. These results suggest that there is a strong coupling between vertical mixing and horizontal stirring in the ocean at scales below the deformation radius.

## Abstract

Recent studies indicate that altimetric observations of the ocean’s mesoscale eddy field reflect the combined influence of surface buoyancy and interior potential vorticity anomalies. The former have a surface-trapped structure, while the latter are often well represented by the barotropic and first baroclinic modes. To assess the relative importance of each contribution to the signal, it is useful to project the observed field onto a set of modes that separates their influence in a natural way. However, the surface-trapped dynamics are not well represented by standard baroclinic modes; moreover, they are dependent on horizontal scale.

Here the authors derive a modal decomposition that results from the simultaneous diagonalization of the energy and a generalization of potential enstrophy that includes contributions from the surface buoyancy fields. This approach yields a family of orthonormal bases that depend on two parameters; the standard baroclinic modes are recovered in a limiting case, while other choices provide modes that represent surface and interior dynamics in an efficient way.

For constant stratification, these modes consist of symmetric and antisymmetric exponential modes that capture the surface dynamics and a series of oscillating modes that represent the interior dynamics. Motivated by the ocean, where shears are concentrated near the upper surface, the authors consider the special case of a quiescent lower surface. In this case, the interior modes are independent of wavenumber, and there is a single exponential surface mode that replaces the barotropic mode. The use and effectiveness of these modes is demonstrated by projecting the energy in a set of simulations of baroclinic turbulence.

## Abstract

Recent studies indicate that altimetric observations of the ocean’s mesoscale eddy field reflect the combined influence of surface buoyancy and interior potential vorticity anomalies. The former have a surface-trapped structure, while the latter are often well represented by the barotropic and first baroclinic modes. To assess the relative importance of each contribution to the signal, it is useful to project the observed field onto a set of modes that separates their influence in a natural way. However, the surface-trapped dynamics are not well represented by standard baroclinic modes; moreover, they are dependent on horizontal scale.

Here the authors derive a modal decomposition that results from the simultaneous diagonalization of the energy and a generalization of potential enstrophy that includes contributions from the surface buoyancy fields. This approach yields a family of orthonormal bases that depend on two parameters; the standard baroclinic modes are recovered in a limiting case, while other choices provide modes that represent surface and interior dynamics in an efficient way.

For constant stratification, these modes consist of symmetric and antisymmetric exponential modes that capture the surface dynamics and a series of oscillating modes that represent the interior dynamics. Motivated by the ocean, where shears are concentrated near the upper surface, the authors consider the special case of a quiescent lower surface. In this case, the interior modes are independent of wavenumber, and there is a single exponential surface mode that replaces the barotropic mode. The use and effectiveness of these modes is demonstrated by projecting the energy in a set of simulations of baroclinic turbulence.

## Abstract

The quasigeostrophic equations consist of the advection of linearized potential vorticity coupled with advection of temperature at the bounding upper and lower surfaces. Numerical models of quasigeostrophic flow often employ greater (scaled) resolution in the horizontal than in the vertical (the two-layer model is an extreme example). In the interior, this has the effect of suppressing interactions between layers at horizontal scales that are small compared to *Nδz*/*f* (where *δz* is the vertical resolution, *N* the buoyancy frequency, and *f* the Coriolis parameter). The nature of the turbulent cascade in the interior is, however, not fundamentally altered because the downscale cascade of potential enstrophy in quasigeostrophic turbulence and the downscale cascade of enstrophy in two-dimensional turbulence (occurring layerwise) both yield energy spectra with slopes of −3. It is shown here that a similar restriction on the vertical resolution applies to the representation of horizontal motions at the surfaces, but the penalty for underresolving in the vertical is complete suppression of the surface temperature cascade at small scales and a corresponding artificial steepening of the surface energy spectrum. This effect is demonstrated in the nonlinear Eady model, using a finite-difference representation in comparison with a model that explicitly advects temperature at the upper and lower surfaces. Theoretical predictions for the spectrum of turbulence in the nonlinear Eady model are reviewed and compared to the simulated flows, showing that the latter model yields an accurate representation of the cascade dynamics. To accurately represent dynamics at horizontal wavenumber *K* in the vertically finite-differenced model, it is found that the vertical grid spacing must satisfy *δz* ≲ 0.3*f*/(*NK*); at wavenumbers *K* > 0.3*f*/(*Nδz*), the spectrum of temperature variance rolls off rapidly.

## Abstract

The quasigeostrophic equations consist of the advection of linearized potential vorticity coupled with advection of temperature at the bounding upper and lower surfaces. Numerical models of quasigeostrophic flow often employ greater (scaled) resolution in the horizontal than in the vertical (the two-layer model is an extreme example). In the interior, this has the effect of suppressing interactions between layers at horizontal scales that are small compared to *Nδz*/*f* (where *δz* is the vertical resolution, *N* the buoyancy frequency, and *f* the Coriolis parameter). The nature of the turbulent cascade in the interior is, however, not fundamentally altered because the downscale cascade of potential enstrophy in quasigeostrophic turbulence and the downscale cascade of enstrophy in two-dimensional turbulence (occurring layerwise) both yield energy spectra with slopes of −3. It is shown here that a similar restriction on the vertical resolution applies to the representation of horizontal motions at the surfaces, but the penalty for underresolving in the vertical is complete suppression of the surface temperature cascade at small scales and a corresponding artificial steepening of the surface energy spectrum. This effect is demonstrated in the nonlinear Eady model, using a finite-difference representation in comparison with a model that explicitly advects temperature at the upper and lower surfaces. Theoretical predictions for the spectrum of turbulence in the nonlinear Eady model are reviewed and compared to the simulated flows, showing that the latter model yields an accurate representation of the cascade dynamics. To accurately represent dynamics at horizontal wavenumber *K* in the vertically finite-differenced model, it is found that the vertical grid spacing must satisfy *δz* ≲ 0.3*f*/(*NK*); at wavenumbers *K* > 0.3*f*/(*Nδz*), the spectrum of temperature variance rolls off rapidly.

## Abstract

The horizontal wavenumber spectra of wind and temperature near the tropopause have a steep −3 slope at synoptic scales and a shallower −5/3 slope at mesoscales, with a transition between the two regimes at a wavelength of about 450 km. Here it is demonstrated that a quasigeostrophic model driven by baroclinic instability exhibits such a transition near its upper boundary (analogous to the tropopause) when surface temperature advection at that boundary is properly resolved and forced. To accurately represent surface advection at the upper and lower boundaries, the vertical structure of the model streamfunction is decomposed into four parts, representing the interior flow with the first two neutral modes, and each surface with its Green’s function solution, resulting in a system with four prognostic equations. Mean temperature gradients are applied at each surface, and a mean potential vorticity gradient consisting both of *β* and vertical shear is applied in the interior. The system exhibits three fundamental types of baroclinic instability: interactions between the upper and lower surfaces (Eady type), interactions between one surface and the interior (Charney type), and interactions between the barotropic and baroclinic interior modes (Phillips type). The turbulent steady states that result from each of these instabilities are distinct, and those of the former two types yield shallow kinetic energy spectra at small scales along those boundaries where mean temperature gradients are present. When both mean interior and surface gradients are present, the surface spectrum reflects a superposition of the interior-dominated −3 slope cascade at large scales, and the surface-dominated −5/3 slope cascade at small scales. The transition wavenumber depends linearly on the ratio of the interior potential vorticity gradient to the surface temperature gradient, and scales with the inverse of the deformation scale when *β* = 0.

## Abstract

The horizontal wavenumber spectra of wind and temperature near the tropopause have a steep −3 slope at synoptic scales and a shallower −5/3 slope at mesoscales, with a transition between the two regimes at a wavelength of about 450 km. Here it is demonstrated that a quasigeostrophic model driven by baroclinic instability exhibits such a transition near its upper boundary (analogous to the tropopause) when surface temperature advection at that boundary is properly resolved and forced. To accurately represent surface advection at the upper and lower boundaries, the vertical structure of the model streamfunction is decomposed into four parts, representing the interior flow with the first two neutral modes, and each surface with its Green’s function solution, resulting in a system with four prognostic equations. Mean temperature gradients are applied at each surface, and a mean potential vorticity gradient consisting both of *β* and vertical shear is applied in the interior. The system exhibits three fundamental types of baroclinic instability: interactions between the upper and lower surfaces (Eady type), interactions between one surface and the interior (Charney type), and interactions between the barotropic and baroclinic interior modes (Phillips type). The turbulent steady states that result from each of these instabilities are distinct, and those of the former two types yield shallow kinetic energy spectra at small scales along those boundaries where mean temperature gradients are present. When both mean interior and surface gradients are present, the surface spectrum reflects a superposition of the interior-dominated −3 slope cascade at large scales, and the surface-dominated −5/3 slope cascade at small scales. The transition wavenumber depends linearly on the ratio of the interior potential vorticity gradient to the surface temperature gradient, and scales with the inverse of the deformation scale when *β* = 0.

## Abstract

Satellite altimetric observations of the ocean reveal surface pressure patterns in the core of the Antarctic Circumpolar Current (ACC) that propagate downstream (eastward) but slower than the mean surface current by about 25%. The authors argue that these observations are suggestive of baroclinically unstable waves that have a steering level at a depth of about 1 km. Detailed linear stability calculations using a hydrographic atlas indeed reveal a steering level in the ACC near the depth implied by the altimetric observations. Calculations using a nonlinear model forced by the mean shear and stratification observed close to the core of the ACC, coinciding with a position where mooring data and direct eddy flux measurements are available, reveal a similar picture, albeit with added details. When eddy fluxes are allowed to adjust the mean state, computed eddy kinetic energy and eddy stress are close to observed magnitudes with steering levels between 1 and 1.5 km, broadly consistent with observations.

An important result of this study is that the vertical structure of the potential vorticity (PV) eddy diffusivity is strongly depth dependent, implying that the diffusivity for PV and buoyancy are very different from one another. It is shown that the flow can simultaneously support a PV diffusivity peaking at 5000 m^{2} s^{−1} or so near the middepth steering level and a buoyancy diffusivity that is much smaller, of order 1000 m^{2} s^{−1}, exhibiting less vertical structure. An effective diffusivity calculation, using an advected and diffused tracer transformed into area coordinates, confirms that the PV diffusivity more closely reflects the mixing properties of the flow than does the buoyancy diffusivity, and points explicitly to the need for separating tracer and buoyancy flux parameterizations in coarse-resolution general circulation models. Finally, implications for the eddy-driven circulation of the ACC are discussed.

## Abstract

Satellite altimetric observations of the ocean reveal surface pressure patterns in the core of the Antarctic Circumpolar Current (ACC) that propagate downstream (eastward) but slower than the mean surface current by about 25%. The authors argue that these observations are suggestive of baroclinically unstable waves that have a steering level at a depth of about 1 km. Detailed linear stability calculations using a hydrographic atlas indeed reveal a steering level in the ACC near the depth implied by the altimetric observations. Calculations using a nonlinear model forced by the mean shear and stratification observed close to the core of the ACC, coinciding with a position where mooring data and direct eddy flux measurements are available, reveal a similar picture, albeit with added details. When eddy fluxes are allowed to adjust the mean state, computed eddy kinetic energy and eddy stress are close to observed magnitudes with steering levels between 1 and 1.5 km, broadly consistent with observations.

An important result of this study is that the vertical structure of the potential vorticity (PV) eddy diffusivity is strongly depth dependent, implying that the diffusivity for PV and buoyancy are very different from one another. It is shown that the flow can simultaneously support a PV diffusivity peaking at 5000 m^{2} s^{−1} or so near the middepth steering level and a buoyancy diffusivity that is much smaller, of order 1000 m^{2} s^{−1}, exhibiting less vertical structure. An effective diffusivity calculation, using an advected and diffused tracer transformed into area coordinates, confirms that the PV diffusivity more closely reflects the mixing properties of the flow than does the buoyancy diffusivity, and points explicitly to the need for separating tracer and buoyancy flux parameterizations in coarse-resolution general circulation models. Finally, implications for the eddy-driven circulation of the ACC are discussed.

## Abstract

The linear wave and baroclinic instability properties of various geostrophic models valid when the Rossby number is small are investigated. The models are the “*L*
_{1}” dynamics, the “geostrophic potential vorticity” equations, and the more familiar quasigeostrophic and planetary geostrophic equations. Multilayer shallow water equations are used as a control. The goal is to determine whether these models accurately portray linear baroclinic instability properties in various geophysically relevant parameter regimes, in a highly idealized and limited set of cases. The *L*
_{1} and geostrophic potential vorticity models are properly balanced (devoid of inertio-gravity waves, except possibly at solid boundaries), valid on the *β* plane, and contain both quasigeostrophy and planetary geostrophy as limits in different parameter regimes; hence, they are appropriate models for phenomena that span the deformation and planetary scales of motion. The *L*
_{1} model also includes the “frontal geostrophic” equations as a third limit. In fact, the choice to investigate such relatively unfamiliar models is motivated precisely by their applicability to multiple scales of motion.

The models are cast in multilayer form, and the dispersion properties and eigenfunctions of wave modes and baroclinic instabilities produced are found numerically. It is found that both the *L*
_{1} and geostrophic potential vorticity models have sensible linear stability properties with no artifactual instabilities or divergences. Their growth rates are very close to those of the shallow water equations in both quasigeostrophic *and* planetary geostrophic parameter regimes. The growth rate of baroclinic instability in the planetary geostrophic equations is shown to be generally less than the growth rate of the other models near the deformation radius. The growth rate of the planetary geostrophic equations diverges at high wavenumbers, but it is shown how this is ameliorated by the presence of the relative vorticity term in the geostrophic potential vorticity equations.

## Abstract

The linear wave and baroclinic instability properties of various geostrophic models valid when the Rossby number is small are investigated. The models are the “*L*
_{1}” dynamics, the “geostrophic potential vorticity” equations, and the more familiar quasigeostrophic and planetary geostrophic equations. Multilayer shallow water equations are used as a control. The goal is to determine whether these models accurately portray linear baroclinic instability properties in various geophysically relevant parameter regimes, in a highly idealized and limited set of cases. The *L*
_{1} and geostrophic potential vorticity models are properly balanced (devoid of inertio-gravity waves, except possibly at solid boundaries), valid on the *β* plane, and contain both quasigeostrophy and planetary geostrophy as limits in different parameter regimes; hence, they are appropriate models for phenomena that span the deformation and planetary scales of motion. The *L*
_{1} model also includes the “frontal geostrophic” equations as a third limit. In fact, the choice to investigate such relatively unfamiliar models is motivated precisely by their applicability to multiple scales of motion.

The models are cast in multilayer form, and the dispersion properties and eigenfunctions of wave modes and baroclinic instabilities produced are found numerically. It is found that both the *L*
_{1} and geostrophic potential vorticity models have sensible linear stability properties with no artifactual instabilities or divergences. Their growth rates are very close to those of the shallow water equations in both quasigeostrophic *and* planetary geostrophic parameter regimes. The growth rate of baroclinic instability in the planetary geostrophic equations is shown to be generally less than the growth rate of the other models near the deformation radius. The growth rate of the planetary geostrophic equations diverges at high wavenumbers, but it is shown how this is ameliorated by the presence of the relative vorticity term in the geostrophic potential vorticity equations.