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## Abstract

The equilibrium statistics and predictability properties of one- and two-layer quasi-geostrophic flow are examined with the aid of a numerical model. The effect of beta in one-layer flow is to slow the transfer of energy into larger scales and to increase the predictability. In two-layer flow, when beta is zero, energy caters the system via baroclinic instability of the mean Row at very large scales and most energy transfer is confined to low wavenumbers. When beta is non-zero, energy enters at higher wavenumber (in baroclinic modes mainly) before cascading preferentially to lower wavenumber zonal barotropic modes. The predictability of two-layer flow is not significantly altered by beta, because beta increases the range of wavenumber over which significant nonlinear energy transfer occurs. The predictability times of the long waves are found to be always larger than those of the short waves, even when the initial error is spread evenly acres wavenumbers. Reducing the mean baroclinicity increases the predictability time. Two-layer flow is lest predictable than one-layer flow of the same barotropic energy, because of the effects of barolinic instability and the transfer of energy from baroclinic modes.

## Abstract

The equilibrium statistics and predictability properties of one- and two-layer quasi-geostrophic flow are examined with the aid of a numerical model. The effect of beta in one-layer flow is to slow the transfer of energy into larger scales and to increase the predictability. In two-layer flow, when beta is zero, energy caters the system via baroclinic instability of the mean Row at very large scales and most energy transfer is confined to low wavenumbers. When beta is non-zero, energy enters at higher wavenumber (in baroclinic modes mainly) before cascading preferentially to lower wavenumber zonal barotropic modes. The predictability of two-layer flow is not significantly altered by beta, because beta increases the range of wavenumber over which significant nonlinear energy transfer occurs. The predictability times of the long waves are found to be always larger than those of the short waves, even when the initial error is spread evenly acres wavenumbers. Reducing the mean baroclinicity increases the predictability time. Two-layer flow is lest predictable than one-layer flow of the same barotropic energy, because of the effects of barolinic instability and the transfer of energy from baroclinic modes.

## Abstract

The spectral integration of the quasi-geostrophic equations is reexamined for simple boundary conditions in Cartesian geometry. For doubly-periodic flow, it is shown that the mean shear must be constant in time or its evolution specified; grid transform methods are then appropriate. In a channel model, a suitable choice of spectral expansion allows the mean shear to be determined internally, i.e., from the model equations, and also allows the meridional gradient of potential vorticity at the boundaries to be specified. However, the use of conventional transform techniques will lead to aliasing and energy nonconservation, and use must be made either of interaction coefficients or a combination of appended zero grid transforms and analytic Fourier expansions.

## Abstract

The spectral integration of the quasi-geostrophic equations is reexamined for simple boundary conditions in Cartesian geometry. For doubly-periodic flow, it is shown that the mean shear must be constant in time or its evolution specified; grid transform methods are then appropriate. In a channel model, a suitable choice of spectral expansion allows the mean shear to be determined internally, i.e., from the model equations, and also allows the meridional gradient of potential vorticity at the boundaries to be specified. However, the use of conventional transform techniques will lead to aliasing and energy nonconservation, and use must be made either of interaction coefficients or a combination of appended zero grid transforms and analytic Fourier expansions.

## Abstract

Geostrophic balance is shown to be the minimum energy state, for a given linear potential vorticity field, for small deviations of the height field around a resting state, in the shallow-water equations. This includes (but is not limited to) the linearized shallow-water equations. Quasigeostrophic motion is evolution on the slow manifold defined by advection of linear potential vorticity by the velocity field that minimizes that energy. Other linear and nonlinear arguments suggest that geostrophic adjustment is a process whereby the energy of a flow is minimized consistent with the maintenance of the potential vorticity field. A variational calculation that minimizes energy for a given potential vorticity field leads to a balance relationship that for the unapproximated shallow-water equations is similar but not identical to geostrophic balance. Preliminary numerical evidence, involving the inversion of potential vorticity for a simple model, indicates that this balance is a somewhat better approximation to the primitive equations than geostrophy.

It is also shown how the process of geostrophic adjustment may be significantly accelerated, or parameterized, in the primitive equations by the addition of certain terms to the equations of motion. Application of the parameterization to an unbalanced state in a primitive equation model is very effective in achieving a balanced state and in continuously filtering gravity waves. It is more accurate and less sensitive to tunable parameters than pure divergence damping, and may also be a useful and much simpler alternative to nonlinear normal-mode schemes whenever those may be inappropriate.

## Abstract

Geostrophic balance is shown to be the minimum energy state, for a given linear potential vorticity field, for small deviations of the height field around a resting state, in the shallow-water equations. This includes (but is not limited to) the linearized shallow-water equations. Quasigeostrophic motion is evolution on the slow manifold defined by advection of linear potential vorticity by the velocity field that minimizes that energy. Other linear and nonlinear arguments suggest that geostrophic adjustment is a process whereby the energy of a flow is minimized consistent with the maintenance of the potential vorticity field. A variational calculation that minimizes energy for a given potential vorticity field leads to a balance relationship that for the unapproximated shallow-water equations is similar but not identical to geostrophic balance. Preliminary numerical evidence, involving the inversion of potential vorticity for a simple model, indicates that this balance is a somewhat better approximation to the primitive equations than geostrophy.

It is also shown how the process of geostrophic adjustment may be significantly accelerated, or parameterized, in the primitive equations by the addition of certain terms to the equations of motion. Application of the parameterization to an unbalanced state in a primitive equation model is very effective in achieving a balanced state and in continuously filtering gravity waves. It is more accurate and less sensitive to tunable parameters than pure divergence damping, and may also be a useful and much simpler alternative to nonlinear normal-mode schemes whenever those may be inappropriate.

## Abstract

The combined effects of wind, geometry, and diffusion on the stratification and circulation of the ocean are explored by numerical and analytical methods. In particular, the production of deep stratification in a simply configured numerical model with small diffusivity is explored.

In the ventilated thermocline of the subtropical gyre, the meridional temperature gradient is mapped continously to a corresponding vertical profile, essentially independently of (sufficiently small) diffusivity. Below this, as the vertical diffusivity tends to zero, the mapping becomes discontinuous and is concentrated in thin diffusive layers or internal thermoclines. It is shown that the way in which the thickness of the main internal thermocline (i.e., the diffusive lower part of the main thermocline), and the meridional overturning circulation, scales with diffusivity differs according to the presence or absence of a wind stress. For realistic parameter values, the ocean is in a scaling regime in which wind effects are important factors in the scaling of the thermohaline circulation, even for the single hemisphere, flat-bottomed case.

It is shown that deep stratification may readily be produced by the combined effects of surface thermodynamic forcing and geometry. The form of the stratification, but not its existence, depends on the diffusivity. Such deep stratification is efficiently produced, even in single-basin, single-hemisphere simulations, in the presence of a partially topographically blocked channel at high latitudes, provided there is also a surface meridional temperature gradient across the channel. For sufficiently simple geometry and topography, the abyssal stratification is a maximum at the height of the topography. In the limit of small diffusivity, the stratification becomes concentrated in a thin diffusive layer, or front, whose thickness appears to scale as the one-third power of the diffusivity. Above and below this diffusive abyssal thermocline are thick, largely adiabatic and homogeneous water masses. In two hemisphere integrations, the water above the abyssal thermocline may be either “intermediate” water from the same hemisphere as the channel, or “deep” water from the opposing hemisphere, depending on whether the densest water from the opposing hemisphere is denser than the surface water at the equatorward edge of the channel. The zonal velocity in the channel is in thermal wind balance, thus determined more by the meridional temperature gradient across the channel than by the wind forcing. If the periodic channel extends equatorward past the latitude of zero wind-stress curl, the poleward extent of the ventilated thermocline, and the surface source of the mode water, both then lie at the equatorial boundary of the periodic circumpolar channel, rather than where the wind stress curl changes sign.

## Abstract

The combined effects of wind, geometry, and diffusion on the stratification and circulation of the ocean are explored by numerical and analytical methods. In particular, the production of deep stratification in a simply configured numerical model with small diffusivity is explored.

In the ventilated thermocline of the subtropical gyre, the meridional temperature gradient is mapped continously to a corresponding vertical profile, essentially independently of (sufficiently small) diffusivity. Below this, as the vertical diffusivity tends to zero, the mapping becomes discontinuous and is concentrated in thin diffusive layers or internal thermoclines. It is shown that the way in which the thickness of the main internal thermocline (i.e., the diffusive lower part of the main thermocline), and the meridional overturning circulation, scales with diffusivity differs according to the presence or absence of a wind stress. For realistic parameter values, the ocean is in a scaling regime in which wind effects are important factors in the scaling of the thermohaline circulation, even for the single hemisphere, flat-bottomed case.

It is shown that deep stratification may readily be produced by the combined effects of surface thermodynamic forcing and geometry. The form of the stratification, but not its existence, depends on the diffusivity. Such deep stratification is efficiently produced, even in single-basin, single-hemisphere simulations, in the presence of a partially topographically blocked channel at high latitudes, provided there is also a surface meridional temperature gradient across the channel. For sufficiently simple geometry and topography, the abyssal stratification is a maximum at the height of the topography. In the limit of small diffusivity, the stratification becomes concentrated in a thin diffusive layer, or front, whose thickness appears to scale as the one-third power of the diffusivity. Above and below this diffusive abyssal thermocline are thick, largely adiabatic and homogeneous water masses. In two hemisphere integrations, the water above the abyssal thermocline may be either “intermediate” water from the same hemisphere as the channel, or “deep” water from the opposing hemisphere, depending on whether the densest water from the opposing hemisphere is denser than the surface water at the equatorward edge of the channel. The zonal velocity in the channel is in thermal wind balance, thus determined more by the meridional temperature gradient across the channel than by the wind forcing. If the periodic channel extends equatorward past the latitude of zero wind-stress curl, the poleward extent of the ventilated thermocline, and the surface source of the mode water, both then lie at the equatorial boundary of the periodic circumpolar channel, rather than where the wind stress curl changes sign.

## 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.

## Abstract

Quasigeostrophic turbulence theory and numerical simulation are used to study the mechanisms determining the scale, structure, and equilibration of mesoscale ocean eddies. The present work concentrates on using freely decaying geostrophic turbulence to understand and explain the vertical and horizontal flow of energy through a stratified, horizontally homogeneous geostrophic fluid. It is found that the stratification profile, in particular the presence of a pycnocline, has significant, qualitative effects on the efficiency and spectral pathways of energy flow. Specifically, with uniform stratification, energy in high baroclinic modes transfers directly, quickly (within a few eddy turnaround times), and almost completely to the barotropic mode. By contrast, in the presence of oceanlike stratification, kinetic energy in high baroclinic modes transfers intermediately to the first baroclinic mode, whence it transfers inefficiently (and incompletely) to the barotropic mode. The efficiency of transfer to the barotropic mode is reduced as the pycnocline is made increasingly thin. The *β* effect, on the other hand, improves the efficiency of barotropization, but for oceanically realistic parameters this effect is relatively unimportant compared to the effects of nonuniform stratification. Finally, the nature of turbulent cascade dynamics is such as to lead to a concentration of first baroclinic mode kinetic energy near the first radius of deformation, which, in the case of a nonuniform and oceanically realistic stratification, has a significant projection at the surface. This may in part explain recent observations of surface eddy scales by TOPEX/Poseidon satellite altimetry, which indicate a correlation of surface-height variance with the scale of the first deformation radius.

## Abstract

Quasigeostrophic turbulence theory and numerical simulation are used to study the mechanisms determining the scale, structure, and equilibration of mesoscale ocean eddies. The present work concentrates on using freely decaying geostrophic turbulence to understand and explain the vertical and horizontal flow of energy through a stratified, horizontally homogeneous geostrophic fluid. It is found that the stratification profile, in particular the presence of a pycnocline, has significant, qualitative effects on the efficiency and spectral pathways of energy flow. Specifically, with uniform stratification, energy in high baroclinic modes transfers directly, quickly (within a few eddy turnaround times), and almost completely to the barotropic mode. By contrast, in the presence of oceanlike stratification, kinetic energy in high baroclinic modes transfers intermediately to the first baroclinic mode, whence it transfers inefficiently (and incompletely) to the barotropic mode. The efficiency of transfer to the barotropic mode is reduced as the pycnocline is made increasingly thin. The *β* effect, on the other hand, improves the efficiency of barotropization, but for oceanically realistic parameters this effect is relatively unimportant compared to the effects of nonuniform stratification. Finally, the nature of turbulent cascade dynamics is such as to lead to a concentration of first baroclinic mode kinetic energy near the first radius of deformation, which, in the case of a nonuniform and oceanically realistic stratification, has a significant projection at the surface. This may in part explain recent observations of surface eddy scales by TOPEX/Poseidon satellite altimetry, which indicate a correlation of surface-height variance with the scale of the first deformation radius.

## Abstract

The statistical dynamics of midocean eddies, generated by baroclinic instability of a zonal mean flow, are studied in the context of homogeneous stratified quasigeostrophic turbulence. Existing theory for eddy scales and energies in fully developed turbulence is generalized and applied to a system with surface-intensified stratification and arbitrary zonal shear. The theory gives a scaling for the magnitude of the eddy potential vorticity flux, and its (momentum conserving) vertical structure. The theory is tested numerically by varying the magnitude and mode of the mean shear, the Coriolis gradient, and scale thickness of the stratification and found to be partially successful. It is found that the dynamics of energy in high (*m* > 1) baroclinic modes typically resembles the turbulent diffusion of a passive scalar, regardless of the stratification profile, although energy in the first mode does not. It is also found that surface-intensified stratification affects the baroclinicity of flow: as thermocline thickness is decreased, the (statistically equilibrated) baroclinic energy levels remain nearly constant but the statistically equilibrated level of barotropic eddy energy falls. Eddy statistics are found to be relatively insensitive to the magnitude of linear bottom drag in the small drag limit. The theory for the magnitude and structure of the eddy potential vorticity flux is tested against a 15-layer simulation using profiles of density and shear representative of those found in the mid North Atlantic; the theory shows good skill in representing the vertical structure of the flux, and so might serve as the basis for a parameterization of eddy fluxes in the midocean. Finally, baroclinic kinetic energy is found to concentrate near the deformation scale. To the degree that surface motions represent baroclinic eddy kinetic energy, the present results are consistent with the observed correlation between surface eddy scales and the first radius of deformation.

## Abstract

The statistical dynamics of midocean eddies, generated by baroclinic instability of a zonal mean flow, are studied in the context of homogeneous stratified quasigeostrophic turbulence. Existing theory for eddy scales and energies in fully developed turbulence is generalized and applied to a system with surface-intensified stratification and arbitrary zonal shear. The theory gives a scaling for the magnitude of the eddy potential vorticity flux, and its (momentum conserving) vertical structure. The theory is tested numerically by varying the magnitude and mode of the mean shear, the Coriolis gradient, and scale thickness of the stratification and found to be partially successful. It is found that the dynamics of energy in high (*m* > 1) baroclinic modes typically resembles the turbulent diffusion of a passive scalar, regardless of the stratification profile, although energy in the first mode does not. It is also found that surface-intensified stratification affects the baroclinicity of flow: as thermocline thickness is decreased, the (statistically equilibrated) baroclinic energy levels remain nearly constant but the statistically equilibrated level of barotropic eddy energy falls. Eddy statistics are found to be relatively insensitive to the magnitude of linear bottom drag in the small drag limit. The theory for the magnitude and structure of the eddy potential vorticity flux is tested against a 15-layer simulation using profiles of density and shear representative of those found in the mid North Atlantic; the theory shows good skill in representing the vertical structure of the flux, and so might serve as the basis for a parameterization of eddy fluxes in the midocean. Finally, baroclinic kinetic energy is found to concentrate near the deformation scale. To the degree that surface motions represent baroclinic eddy kinetic energy, the present results are consistent with the observed correlation between surface eddy scales and the first radius of deformation.

## Abstract

Time averaged fields produced by a two-level, quasi-geostrophic, nonlinear time-dependent model of large-scale flow over topography are compared to results from stationary linear theory in order to assess the influence of transient eddies. It is shown that stationary linear theory predicts excessive amplitudes and has a low phase correlation with these time-averaged fields. Addition of the stationary nonlinear terms gives only a slight improvement. The transient eddy fluxes are responsible for reducing the amplitude of the linear solutions. Resonant effects evident in the linear models are highly damped, but still noticeable, in the turbulent solutions. The energetics of the stationary flow show that the transfer of stationary to transient energy is significant. Instability analyses of the time-averaged flow suggest that unstable perturbations are likely to arise which have structures qualitatively similar to time-averaged variance fields. We conclude that the time averages in such turbulent models depend both upon the stationary forcings and the instabilities that arise and that neglect of transient fluxes will lead to unrealistic, results.

## Abstract

Time averaged fields produced by a two-level, quasi-geostrophic, nonlinear time-dependent model of large-scale flow over topography are compared to results from stationary linear theory in order to assess the influence of transient eddies. It is shown that stationary linear theory predicts excessive amplitudes and has a low phase correlation with these time-averaged fields. Addition of the stationary nonlinear terms gives only a slight improvement. The transient eddy fluxes are responsible for reducing the amplitude of the linear solutions. Resonant effects evident in the linear models are highly damped, but still noticeable, in the turbulent solutions. The energetics of the stationary flow show that the transfer of stationary to transient energy is significant. Instability analyses of the time-averaged flow suggest that unstable perturbations are likely to arise which have structures qualitatively similar to time-averaged variance fields. We conclude that the time averages in such turbulent models depend both upon the stationary forcings and the instabilities that arise and that neglect of transient fluxes will lead to unrealistic, results.

## Abstract

Low-order primitive equation and balanced models are compared by evaluating the correlation dimension of each over a range of Rossby numbers. The models are the nine-component primitive equation model of Lorenz and the corresponding three-component balance model. Both models display behavior ranging from stable fixed points and limit cycles to chaotic dynamics. At low Rossby number, the correlation dimensions of the models are (to the accuracy of the calculation) very similar, even in the presence of strange attractors. At higher Rossby number, the behavior differs: in some regions where the balance model goes into a limit cycle the primitive equation model displays chaotic behavior, with a correlation dimension greater than three. This appears to be caused by the (somewhat intermittent) appearance of gravity waves. Since here the calculated correlation dimension is higher than the number of slow modes, the gravity waves cannot be slaved to the slower geostrophic activity.

## Abstract

Low-order primitive equation and balanced models are compared by evaluating the correlation dimension of each over a range of Rossby numbers. The models are the nine-component primitive equation model of Lorenz and the corresponding three-component balance model. Both models display behavior ranging from stable fixed points and limit cycles to chaotic dynamics. At low Rossby number, the correlation dimensions of the models are (to the accuracy of the calculation) very similar, even in the presence of strange attractors. At higher Rossby number, the behavior differs: in some regions where the balance model goes into a limit cycle the primitive equation model displays chaotic behavior, with a correlation dimension greater than three. This appears to be caused by the (somewhat intermittent) appearance of gravity waves. Since here the calculated correlation dimension is higher than the number of slow modes, the gravity waves cannot be slaved to the slower geostrophic activity.

## Abstract

The anticipated potential Vorticity method (APV) is a parameterization of the effects of subgrid or unresolved scales on those explicitly resolved for barotropic, quasi-geostrophic and certain types of primitive equation models. One novelty of the method lies in the fact that it exactly conserves energy, while still dissipating the enstrophy. (Diffusion of potential Vorticity, on the other hand, conserves neither.) We have numerically evaluated the effective eddy diffusivity associated with such parameterizations. Additionally, we have evaluated the effective eddy diffusivity explicitly, i.e., by direct numerical simulation. We find, in accord with closure calculations, that explicit simulations give cusplike behavior near the cut-off wavenumber *k*
_{max}. This is induced by the continuous interaction of scales on both sides of *k*
_{max}, transferring enstrophy to higher wavenumbers. The APV method reproduces this, provided that the lag or anticipation times of the vorticity are suitably (but perhaps arbitrarily) chosen. In particular, it is necessary for the anticipation time to increase rapidly with wavenumber. This in turn necessitates extra boundary conditions at walls. At low wavenumbers, the eddy viscosity produced by the APV method, predicted by closure theory and directly calculated are all negative. The test-field model prescribes a saturation toward a constant negative eddy viscosity for *k* ≪ *k*
_{max}. This is qualitatively verified by explicit simulations. The APV method is consistent for wavenumbers in the inertial range. For the very lowest wavenumber, when the energy at the lowest wavenumbers is small, the method produces large negative values, producing another cusp at the largest scales resolved by the model. Explicit simulations show similar behavior. (Diffusion of potential vorticity, on the other hand, is similar to a constant, positive, eddy viscosity.) Some numerical simulations of baroclinic forced dissipative flows are presented. At medium and high resolutions the APV method is successfully able to produce a fairly “flat” inertial range. At low resolutions, when the maximum wavenumber is in the energy containing range, the APV method either produces unrealistically high energy levels at low wavenumbers, or else the simulation is too energetic at all scales, depending on the strength of the parameterization. An energy conserving parameterization is not necessarily appropriate here. Overall, in terms of eddy viscosities, the APV method performs as well as or better than more conventional schemes using prescribed eddy diffusivities. If the resolution of the model extends into the inertial range, the APV method apparently performs very well, although the effects of a lack of Galilean invariance remain unresolved.

## Abstract

The anticipated potential Vorticity method (APV) is a parameterization of the effects of subgrid or unresolved scales on those explicitly resolved for barotropic, quasi-geostrophic and certain types of primitive equation models. One novelty of the method lies in the fact that it exactly conserves energy, while still dissipating the enstrophy. (Diffusion of potential Vorticity, on the other hand, conserves neither.) We have numerically evaluated the effective eddy diffusivity associated with such parameterizations. Additionally, we have evaluated the effective eddy diffusivity explicitly, i.e., by direct numerical simulation. We find, in accord with closure calculations, that explicit simulations give cusplike behavior near the cut-off wavenumber *k*
_{max}. This is induced by the continuous interaction of scales on both sides of *k*
_{max}, transferring enstrophy to higher wavenumbers. The APV method reproduces this, provided that the lag or anticipation times of the vorticity are suitably (but perhaps arbitrarily) chosen. In particular, it is necessary for the anticipation time to increase rapidly with wavenumber. This in turn necessitates extra boundary conditions at walls. At low wavenumbers, the eddy viscosity produced by the APV method, predicted by closure theory and directly calculated are all negative. The test-field model prescribes a saturation toward a constant negative eddy viscosity for *k* ≪ *k*
_{max}. This is qualitatively verified by explicit simulations. The APV method is consistent for wavenumbers in the inertial range. For the very lowest wavenumber, when the energy at the lowest wavenumbers is small, the method produces large negative values, producing another cusp at the largest scales resolved by the model. Explicit simulations show similar behavior. (Diffusion of potential vorticity, on the other hand, is similar to a constant, positive, eddy viscosity.) Some numerical simulations of baroclinic forced dissipative flows are presented. At medium and high resolutions the APV method is successfully able to produce a fairly “flat” inertial range. At low resolutions, when the maximum wavenumber is in the energy containing range, the APV method either produces unrealistically high energy levels at low wavenumbers, or else the simulation is too energetic at all scales, depending on the strength of the parameterization. An energy conserving parameterization is not necessarily appropriate here. Overall, in terms of eddy viscosities, the APV method performs as well as or better than more conventional schemes using prescribed eddy diffusivities. If the resolution of the model extends into the inertial range, the APV method apparently performs very well, although the effects of a lack of Galilean invariance remain unresolved.