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- Author or Editor: Timothy E. Dowling x

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## Abstract

We present evidence from analysis of *Voyager* data and numerical experiments that in Jupiter's troposphere at midlatitudes the potential vorticity is given simply by the reciprocal of the streamfunction. This relationship agrees with the results of a vortex-tube stretching analysis of the *Voyager* wind-field data of the Great Red Spot and White Oval BC, whereas other published models do not. The derivative of streamfunction with respect to potential vorticity is negative definite, and in the quasigeostrophic limit the relationship is neutrally stable with respect to Arnol'd's second stability theorem. Numerical experiments indicate that the relationship is also neutrally stable in the primitive shallow-water system. This resolves a long-standing question as to how Jupiter's cloud-top winds are able to violate the Rayleigh-Kuo stability criterion, and constrains the two-layer model to a single free parameter.

## Abstract

We present evidence from analysis of *Voyager* data and numerical experiments that in Jupiter's troposphere at midlatitudes the potential vorticity is given simply by the reciprocal of the streamfunction. This relationship agrees with the results of a vortex-tube stretching analysis of the *Voyager* wind-field data of the Great Red Spot and White Oval BC, whereas other published models do not. The derivative of streamfunction with respect to potential vorticity is negative definite, and in the quasigeostrophic limit the relationship is neutrally stable with respect to Arnol'd's second stability theorem. Numerical experiments indicate that the relationship is also neutrally stable in the primitive shallow-water system. This resolves a long-standing question as to how Jupiter's cloud-top winds are able to violate the Rayleigh-Kuo stability criterion, and constrains the two-layer model to a single free parameter.

## Abstract

Layer thickness variations in Jupiter's atmosphere are investigated by treating potential vorticity as a conserved tracer. Starting with the horizontal velocity field measured from *Voyager* images, fluid trajectories around the Great Red Spot (GRS) and White Oval BC are calculated. The flow is assumed to be frictionless, adiabatic, hydrostatic, and steady in the reference frame of the vortex. Absolute vorticity is followed along each trajectory; its magnitude is assumed to vary directly as the thickness, which is defined as the mass per unit area between potential temperature surfaces. To the accuracy of the observations. the inferred thickness is a separable function of trajectory and latitude. The latitude dependence has positive curvature near the GRS and BC. The relative variations of thickness with respect to latitude are generally larger than the relative variations of Coriolis parameter with respect to latitude—the beta effect. The data are a useful diagnostic which will help differentiate between models, of Jovian vortices. The present analysis employs a quasi-geostrophic model in which a thin upper weather layer, which contains the vortex, is supported hydrostatically by a much deeper lower layer. In this model, the upper free surface does not contribute to the observed variation of thickness along trajectories. Such variations are due exclusively to bottom topography—flow of the deep lower layer relative to the vortex. The observation are used to infer the form of the deep zonal velocity profile vs. latitude. The magnitude of the profile depends on the unknown static stability. The principal result is the existence of horizontal shear in the deep layer zonal velocity profile, i.e., the lower layer is not in solid body rotation and does not act like a flat solid surface. In this respect the data support the hypothesis of Ingersoll and Cuong concerning motions in the deep layer. However at some latitudes the data violate Ingersoll and Cuong's criterion governing the compactness of the vortices. At these latitudes the topography allows standing Rossby waves (wakes) extending far downstream to the west. Observed wavelike features, the filamentary regions, are possibly formed by this mechanism.

## Abstract

Layer thickness variations in Jupiter's atmosphere are investigated by treating potential vorticity as a conserved tracer. Starting with the horizontal velocity field measured from *Voyager* images, fluid trajectories around the Great Red Spot (GRS) and White Oval BC are calculated. The flow is assumed to be frictionless, adiabatic, hydrostatic, and steady in the reference frame of the vortex. Absolute vorticity is followed along each trajectory; its magnitude is assumed to vary directly as the thickness, which is defined as the mass per unit area between potential temperature surfaces. To the accuracy of the observations. the inferred thickness is a separable function of trajectory and latitude. The latitude dependence has positive curvature near the GRS and BC. The relative variations of thickness with respect to latitude are generally larger than the relative variations of Coriolis parameter with respect to latitude—the beta effect. The data are a useful diagnostic which will help differentiate between models, of Jovian vortices. The present analysis employs a quasi-geostrophic model in which a thin upper weather layer, which contains the vortex, is supported hydrostatically by a much deeper lower layer. In this model, the upper free surface does not contribute to the observed variation of thickness along trajectories. Such variations are due exclusively to bottom topography—flow of the deep lower layer relative to the vortex. The observation are used to infer the form of the deep zonal velocity profile vs. latitude. The magnitude of the profile depends on the unknown static stability. The principal result is the existence of horizontal shear in the deep layer zonal velocity profile, i.e., the lower layer is not in solid body rotation and does not act like a flat solid surface. In this respect the data support the hypothesis of Ingersoll and Cuong concerning motions in the deep layer. However at some latitudes the data violate Ingersoll and Cuong's criterion governing the compactness of the vortices. At these latitudes the topography allows standing Rossby waves (wakes) extending far downstream to the west. Observed wavelike features, the filamentary regions, are possibly formed by this mechanism.

## Abstract

Most current models of Jupiter's Great Red Spot (GRS) are cast in terms of a two-layer model, where a thin upper weather layer, which contains the vortex, overlies a much deeper layer, which is meant to represent the neutrally stratified deep atmosphere. Any motions in the deep layer are assumed to be zonal and steady. This two-layer model is dynamically equivalent to a one-layer model with meridionally varying solid bottom topography, called the reduced-gravity model. Specifying the motions, or lack thereof, in the lower layer of the two-layer model is equivalent to specifying the bottom topography, and hence the far-field potential vorticity, in the reduced-gravity model. Current models of the GRS start by guessing the deep motions and then proceed to study vortices using the implied bottom topography. Here, using the GRS cloud-top velocity data, we derive the bottom topography, up to a constant that depends on the unknown radius of deformation (or equivalently, the product of the reduced gravity and the mean thickness of the upper layer). The bottom topography is inferred from three quantities derived from the velocity data—Bernoulli streamfunction, kinetic energy per unit mass, and absolute vorticity—all of which are functions only of horizontal position in the reference frame of the vortex. Far from the vortex, potential vorticity versus latitude is calculated from the observed cloud-top zonal velocity and the derived bottom topography. The results show that the deep atmosphere is in differential motion and that the far-field potential vorticity gradient changes sign at several latitudes. Numerical shallow water experiments are performed, using both the derived bottom topography and the bottom topographies prescribed by current models. The results of three published studies are reproduced in our numerical experiments. Each of these models is successful in maintaining a long-lived, isolated vortex, but only the present model yields absolute vorticity profiles along streamlines that agree with those observed for the GRS by Dowling and Ingersoll. In all the models, large vortices form by merging with smaller vortices. In the present, observationally based model, and in one other published model, the smaller vortices arise spontaneously because the observed cloud-top zonal velocity profile is unstable. These two models require an additional momentum source to maintain the upper-layer zonal velocity profile. In the other two models, the bottom topography stabilizes the zonal velocity profile. If dissipation is present, the latter two models require an additional source of smaller vortices to maintain the larger one. A crucial unanswered question for the present model, and for Jupiter itself, is how the cloud-top zonal velocity profile is maintained in its present unstable state.

## Abstract

Most current models of Jupiter's Great Red Spot (GRS) are cast in terms of a two-layer model, where a thin upper weather layer, which contains the vortex, overlies a much deeper layer, which is meant to represent the neutrally stratified deep atmosphere. Any motions in the deep layer are assumed to be zonal and steady. This two-layer model is dynamically equivalent to a one-layer model with meridionally varying solid bottom topography, called the reduced-gravity model. Specifying the motions, or lack thereof, in the lower layer of the two-layer model is equivalent to specifying the bottom topography, and hence the far-field potential vorticity, in the reduced-gravity model. Current models of the GRS start by guessing the deep motions and then proceed to study vortices using the implied bottom topography. Here, using the GRS cloud-top velocity data, we derive the bottom topography, up to a constant that depends on the unknown radius of deformation (or equivalently, the product of the reduced gravity and the mean thickness of the upper layer). The bottom topography is inferred from three quantities derived from the velocity data—Bernoulli streamfunction, kinetic energy per unit mass, and absolute vorticity—all of which are functions only of horizontal position in the reference frame of the vortex. Far from the vortex, potential vorticity versus latitude is calculated from the observed cloud-top zonal velocity and the derived bottom topography. The results show that the deep atmosphere is in differential motion and that the far-field potential vorticity gradient changes sign at several latitudes. Numerical shallow water experiments are performed, using both the derived bottom topography and the bottom topographies prescribed by current models. The results of three published studies are reproduced in our numerical experiments. Each of these models is successful in maintaining a long-lived, isolated vortex, but only the present model yields absolute vorticity profiles along streamlines that agree with those observed for the GRS by Dowling and Ingersoll. In all the models, large vortices form by merging with smaller vortices. In the present, observationally based model, and in one other published model, the smaller vortices arise spontaneously because the observed cloud-top zonal velocity profile is unstable. These two models require an additional momentum source to maintain the upper-layer zonal velocity profile. In the other two models, the bottom topography stabilizes the zonal velocity profile. If dissipation is present, the latter two models require an additional source of smaller vortices to maintain the larger one. A crucial unanswered question for the present model, and for Jupiter itself, is how the cloud-top zonal velocity profile is maintained in its present unstable state.

## Abstract

Three-dimensional numerical simulations of freely evolving stratified geostrophic turbulence on the *β* plane are presented as a simplified model of zonal jet formation on Jupiter. This study samples the parameter space that covers the low, middle, and high latitudes of Jupiter by varying the central latitude of the *β* plane. The results show that robust zonal jets can emerge from initial small-scale random turbulence through the upscale redistribution of the kinetic energy in the spectral space. The resulting flow’s sensitivities to the flow’s deformation radius *L _{D}
* and the two-dimensional Rhines length

*L*=

_{β}*U*/

*β*

*U*is the characteristic turbulence velocity and

*β*is the meridional gradient of the planetary vorticity) are tested, revealing that whether the outcome of the upscale energy transfer becomes dominated by jets or vortices depends on the relative values of

*L*and

_{D}*L*. The values of

_{β}*L*and

_{β}*L*are varied by tuning the

_{D}*β*-plane parameters, and it is found that the flow transitions from a jet-dominated regime in

*L*≲

_{β}*L*to a vortical flow in

_{D}*L*≳

_{β}*L*. A height-to-width ratio equal to

_{D}*f*/

*N*, the Coriolis parameter divided by the Brunt–Väisälä frequency, has previously been established for stable vortices, and this paper shows that this aspect ratio also applies to the zonal jets that emerge in these simulations.

## Abstract

Three-dimensional numerical simulations of freely evolving stratified geostrophic turbulence on the *β* plane are presented as a simplified model of zonal jet formation on Jupiter. This study samples the parameter space that covers the low, middle, and high latitudes of Jupiter by varying the central latitude of the *β* plane. The results show that robust zonal jets can emerge from initial small-scale random turbulence through the upscale redistribution of the kinetic energy in the spectral space. The resulting flow’s sensitivities to the flow’s deformation radius *L _{D}
* and the two-dimensional Rhines length

*L*=

_{β}*U*/

*β*

*U*is the characteristic turbulence velocity and

*β*is the meridional gradient of the planetary vorticity) are tested, revealing that whether the outcome of the upscale energy transfer becomes dominated by jets or vortices depends on the relative values of

*L*and

_{D}*L*. The values of

_{β}*L*and

_{β}*L*are varied by tuning the

_{D}*β*-plane parameters, and it is found that the flow transitions from a jet-dominated regime in

*L*≲

_{β}*L*to a vortical flow in

_{D}*L*≳

_{β}*L*. A height-to-width ratio equal to

_{D}*f*/

*N*, the Coriolis parameter divided by the Brunt–Väisälä frequency, has previously been established for stable vortices, and this paper shows that this aspect ratio also applies to the zonal jets that emerge in these simulations.

## Abstract

We investigate the relationship between Ertel potential vorticity *Q* and Bernoulli potential *B* on orthobaric density surfaces in the Antarctic Circumpolar Current (ACC), using the Southern Ocean State Estimate. Similar to the extratropical atmospheres of Earth and Mars, *Q* and *B* correlate in the ACC in a function-like manner with modest scatter. Below the near-surface, the underlying function relating *Q* and *B* appears to be nearly linear. Nondimensionalizing its slope yields “Ma,” a “Mach” number for long Rossby waves, the ratio of the local flow speed to the intrinsic long Rossby wave speed. We empirically estimate the latter using established and novel techniques that yield qualitatively consistent results. Previous work related “Ma” to the degree of homogeneity of *Q* and to Arnol’d’s shear stability criteria. Estimates of “Ma” for the whole ACC are notably positive, implying inhomogeneous *Q*, on all circumpolar buoyancy surfaces studied. Upper layers generally exhibit “Ma” slightly less than unity, suggesting that shear instability may operate within these layers. Deep layers exhibit “Ma” greater than unity, implying stability. On surfaces shallower than 1000 m just north of the ACC, the *Q* versus *B* slope varies strongly on subannual and interannual time scales, but “Ma” hovers near unity. We also study spatial variability: the ACC is speckled with hundreds of small-scale features with “Ma” near unity, whereas away from the ACC “Ma” is more commonly negative or above unity, both corresponding to stability. Maps of the time-mean “Ma” show stable regions occupy most of the Southern Ocean, except for several topographically controlled hotspots where “Ma” is always near unity.

## Abstract

We investigate the relationship between Ertel potential vorticity *Q* and Bernoulli potential *B* on orthobaric density surfaces in the Antarctic Circumpolar Current (ACC), using the Southern Ocean State Estimate. Similar to the extratropical atmospheres of Earth and Mars, *Q* and *B* correlate in the ACC in a function-like manner with modest scatter. Below the near-surface, the underlying function relating *Q* and *B* appears to be nearly linear. Nondimensionalizing its slope yields “Ma,” a “Mach” number for long Rossby waves, the ratio of the local flow speed to the intrinsic long Rossby wave speed. We empirically estimate the latter using established and novel techniques that yield qualitatively consistent results. Previous work related “Ma” to the degree of homogeneity of *Q* and to Arnol’d’s shear stability criteria. Estimates of “Ma” for the whole ACC are notably positive, implying inhomogeneous *Q*, on all circumpolar buoyancy surfaces studied. Upper layers generally exhibit “Ma” slightly less than unity, suggesting that shear instability may operate within these layers. Deep layers exhibit “Ma” greater than unity, implying stability. On surfaces shallower than 1000 m just north of the ACC, the *Q* versus *B* slope varies strongly on subannual and interannual time scales, but “Ma” hovers near unity. We also study spatial variability: the ACC is speckled with hundreds of small-scale features with “Ma” near unity, whereas away from the ACC “Ma” is more commonly negative or above unity, both corresponding to stability. Maps of the time-mean “Ma” show stable regions occupy most of the Southern Ocean, except for several topographically controlled hotspots where “Ma” is always near unity.