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- Author or Editor: Andrew P. Ingersoll x

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## Abstract

A homogeneous fluid is bounded above and below by horizontal plane surfaces in rapid rotation about a vertical axis. An obstacle is attached to one of the surfaces, and at large distances from the obstacle the relative velocity is steady and horizontal. Solutions are obtained as power series expansions in the Rossby number, uniformly valid as the Taylor number approaches infinity.

If the height of the obstacle is greater than the Rossby number times the depth, a stagnant region (Taylor column) forms over the obstacle. Outside this region there is a net circulation in a direction opposite the rotation. The shape of the stagnant region and the circulation are uniquely determined as part of the solution.

Possible geophysical applications are discussed, and it is shown that stratification renders Taylor columns unlikely on earth, but that the Great Red Spot of Jupiter may be an example of this phenomenon, as Hide has suggested.

## Abstract

A homogeneous fluid is bounded above and below by horizontal plane surfaces in rapid rotation about a vertical axis. An obstacle is attached to one of the surfaces, and at large distances from the obstacle the relative velocity is steady and horizontal. Solutions are obtained as power series expansions in the Rossby number, uniformly valid as the Taylor number approaches infinity.

If the height of the obstacle is greater than the Rossby number times the depth, a stagnant region (Taylor column) forms over the obstacle. Outside this region there is a net circulation in a direction opposite the rotation. The shape of the stagnant region and the circulation are uniquely determined as part of the solution.

Possible geophysical applications are discussed, and it is shown that stratification renders Taylor columns unlikely on earth, but that the Great Red Spot of Jupiter may be an example of this phenomenon, as Hide has suggested.

## Abstract

Expressions are derived for the potential energy of a fluid whose density depends on three variables: temperature, pressure, and salinity. The thermal expansion coefficient is a function of depth, and the application is to thermobaric convection in the oceans. Energy conservation, with conversion between kinetic and potential energies during adiabatic, inviscid motion, exists for the Boussinesq and anelastic approximations but not for all approximate systems of equations. In the Boussinesq/anelastic system, which is a linearization of the thermodynamic variables, the expressions for potential energy involve thermodynamic potentials for salinity and potential temperature. Thermobaric instability can occur with warm salty water either above or below cold freshwater. In both cases the fluid may be unstable to large perturbations even though it is stable to small perturbations. The energy per mass of this finite-amplitude instability varies as the square of the layer thickness. With a 4-K temperature difference and a 0.6-psu salinity difference across a layer that is 4000 m thick, the stored potential energy is ∼0.3 m^{2} s^{−2}, which is comparable to the kinetic energy of the major ocean currents. This potential could be released as kinetic energy in a single large event. Thermobaric effects cause parcels moving adiabatically to follow different neutral trajectories. A cold fresh parcel that is less dense than a warm salty parcel near the surface may be more dense at depth. Examples are given in which two isopycnal trajectories cross at one place and differ in depth by 1000 m or more at another.

## Abstract

Expressions are derived for the potential energy of a fluid whose density depends on three variables: temperature, pressure, and salinity. The thermal expansion coefficient is a function of depth, and the application is to thermobaric convection in the oceans. Energy conservation, with conversion between kinetic and potential energies during adiabatic, inviscid motion, exists for the Boussinesq and anelastic approximations but not for all approximate systems of equations. In the Boussinesq/anelastic system, which is a linearization of the thermodynamic variables, the expressions for potential energy involve thermodynamic potentials for salinity and potential temperature. Thermobaric instability can occur with warm salty water either above or below cold freshwater. In both cases the fluid may be unstable to large perturbations even though it is stable to small perturbations. The energy per mass of this finite-amplitude instability varies as the square of the layer thickness. With a 4-K temperature difference and a 0.6-psu salinity difference across a layer that is 4000 m thick, the stored potential energy is ∼0.3 m^{2} s^{−2}, which is comparable to the kinetic energy of the major ocean currents. This potential could be released as kinetic energy in a single large event. Thermobaric effects cause parcels moving adiabatically to follow different neutral trajectories. A cold fresh parcel that is less dense than a warm salty parcel near the surface may be more dense at depth. Examples are given in which two isopycnal trajectories cross at one place and differ in depth by 1000 m or more at another.

## Abstract

Radiative-convective equilibrium models of planetary atmospheres are discussed for the case when the infrared opacity is due to a vapor in equilibrium with its liquid or solid phase. For a grey gas, or for a gas which absorbs at all infrared wavelengths, equilibrium is impossible when the solar constant exceeds a critical value. Equilibrium therefore requires that the condensed phase evaporates into the atmosphere.

Moist adiabatic and pseudoadiabatic atmospheres in which the condensing vapor is a major atmospheric constituent are considered. This situation would apply if the solar constant were supercritical with respect to an abundant substance such as water. It is shown that the condensing gas would be a major constituent at all levels in such an atmosphere. Photodissociation of water in the primordial Venus atmosphere is discussed in this context.

## Abstract

Radiative-convective equilibrium models of planetary atmospheres are discussed for the case when the infrared opacity is due to a vapor in equilibrium with its liquid or solid phase. For a grey gas, or for a gas which absorbs at all infrared wavelengths, equilibrium is impossible when the solar constant exceeds a critical value. Equilibrium therefore requires that the condensed phase evaporates into the atmosphere.

Moist adiabatic and pseudoadiabatic atmospheres in which the condensing vapor is a major atmospheric constituent are considered. This situation would apply if the solar constant were supercritical with respect to an abundant substance such as water. It is shown that the condensing gas would be a major constituent at all levels in such an atmosphere. Photodissociation of water in the primordial Venus atmosphere is discussed in this context.

## Abstract

Exactly solving the absolute minimum potential energy state (Lorenz reference state) is a difficult problem because of the nonlinear nature of the equation of state of seawater. This problem has been solved recently but the algorithm comes at a high computational cost. As the first part of this study, the authors develop an algorithm that is ~10^{3}–10^{5} times faster, making it useful for energy diagnosis in ocean models. The second part of this study shows that the global patterns of Lorenz available potential energy (APE) density are distinct from those of eddy kinetic energy (EKE). This is because the Lorenz APE density is based on the entire domainwide parcel rearrangement, while mesoscale eddies, if related to baroclinic instability, are typically generated through local parcel rearrangement approximately around the eddy size. Inspired by this contrast, this study develops a locally defined APE framework: the eddy size–constrained APE density based on the strong constraint that the parcel rearrangement/displacement to achieve the minimum potential energy state should not exceed the local eddy size horizontally. This concept typically identifies baroclinically unstable regions. It is shown to be helpful to detect individual eddies/vortices and local EKE patterns, for example, around the Southern Ocean fronts and subtropical western boundary currents. This is consistent with the physical picture that mesoscale eddies are associated with a strong signature in both the velocity field (i.e., EKE) and the stratification (i.e., local APE). The new APE concept may be useful in parameterizing mesoscale eddies in ocean models.

## Abstract

Exactly solving the absolute minimum potential energy state (Lorenz reference state) is a difficult problem because of the nonlinear nature of the equation of state of seawater. This problem has been solved recently but the algorithm comes at a high computational cost. As the first part of this study, the authors develop an algorithm that is ~10^{3}–10^{5} times faster, making it useful for energy diagnosis in ocean models. The second part of this study shows that the global patterns of Lorenz available potential energy (APE) density are distinct from those of eddy kinetic energy (EKE). This is because the Lorenz APE density is based on the entire domainwide parcel rearrangement, while mesoscale eddies, if related to baroclinic instability, are typically generated through local parcel rearrangement approximately around the eddy size. Inspired by this contrast, this study develops a locally defined APE framework: the eddy size–constrained APE density based on the strong constraint that the parcel rearrangement/displacement to achieve the minimum potential energy state should not exceed the local eddy size horizontally. This concept typically identifies baroclinically unstable regions. It is shown to be helpful to detect individual eddies/vortices and local EKE patterns, for example, around the Southern Ocean fronts and subtropical western boundary currents. This is consistent with the physical picture that mesoscale eddies are associated with a strong signature in both the velocity field (i.e., EKE) and the stratification (i.e., local APE). The new APE concept may be useful in parameterizing mesoscale eddies in ocean models.

## Abstract

The Madden–Julian oscillation (MJO) is the dominant mode of intraseasonal variability in the tropics. Despite its primary importance, a generally accepted theory that accounts for fundamental features of the MJO, including its propagation speed, planetary horizontal scale, multiscale features, and quadrupole structures, remains elusive. In this study, the authors use a shallow-water model to simulate the MJO. In this model, convection is parameterized as a short-duration localized mass source and is triggered when the layer thickness falls below a critical value. Radiation is parameterized as a steady uniform mass sink. The following MJO-like signals are observed in the simulations: 1) slow eastward-propagating large-scale disturbances, which show up as low-frequency, low-wavenumber features with eastward propagation in the spectral domain, 2) multiscale structures in the time–longitude (Hovmöller) domain, and 3) quadrupole vortex structures in the longitude–latitude (map view) domain. The authors propose that the simulated MJO signal is an interference pattern of westward and eastward inertia–gravity (WIG and EIG) waves. Its propagation speed is half of the speed difference between the WIG and EIG waves. The horizontal scale of its large-scale envelope is determined by the bandwidth of the excited waves, and the bandwidth is controlled by the number density of convection events. In this model, convection events trigger other convection events, thereby aggregating into large-scale structures, but there is no feedback of the large-scale structures onto the convection events. The results suggest that the MJO is not so much a low-frequency wave, in which convection acts as a quasi-equilibrium adjustment, but is more a pattern of high-frequency waves that interact directly with the convection.

## Abstract

The Madden–Julian oscillation (MJO) is the dominant mode of intraseasonal variability in the tropics. Despite its primary importance, a generally accepted theory that accounts for fundamental features of the MJO, including its propagation speed, planetary horizontal scale, multiscale features, and quadrupole structures, remains elusive. In this study, the authors use a shallow-water model to simulate the MJO. In this model, convection is parameterized as a short-duration localized mass source and is triggered when the layer thickness falls below a critical value. Radiation is parameterized as a steady uniform mass sink. The following MJO-like signals are observed in the simulations: 1) slow eastward-propagating large-scale disturbances, which show up as low-frequency, low-wavenumber features with eastward propagation in the spectral domain, 2) multiscale structures in the time–longitude (Hovmöller) domain, and 3) quadrupole vortex structures in the longitude–latitude (map view) domain. The authors propose that the simulated MJO signal is an interference pattern of westward and eastward inertia–gravity (WIG and EIG) waves. Its propagation speed is half of the speed difference between the WIG and EIG waves. The horizontal scale of its large-scale envelope is determined by the bandwidth of the excited waves, and the bandwidth is controlled by the number density of convection events. In this model, convection events trigger other convection events, thereby aggregating into large-scale structures, but there is no feedback of the large-scale structures onto the convection events. The results suggest that the MJO is not so much a low-frequency wave, in which convection acts as a quasi-equilibrium adjustment, but is more a pattern of high-frequency waves that interact directly with the convection.

## Abstract

The Madden–Julian oscillation (MJO), also known as the intraseasonal oscillation (ISO), is a planetary-scale mode of variation in the tropical Indian and western Pacific Oceans. Basic questions about the MJO are why it propagates eastward at ∼5 m s^{−1}, why it lasts for intraseasonal time scales, and how it interacts with the fine structure that is embedded in it. This study will test the hypothesis that the MJO is not a wave but a wave packet—the interference pattern produced by a narrow frequency band of mixed Rossby–gravity (MRG) waves. As such, the MJO would propagate with the MRG group velocity, which is eastward at ∼5 m s^{−1}. Simulation with a 3D model shows that MRG waves can be forced independently by relatively short-lived, eastward- and westward-moving disturbances, and the MRG wave packet can last long enough to form the intraseasonal variability. This hypothesis is consistent with the view that the MJO is episodic, with an irregular time interval between events rather than a periodic oscillation. The packet is defined as the horizontally smoothed variance of the MRG wave—the rectified MRG wave, which has features in common with the MJO. The two-dimensional Fourier analysis of the NOAA outgoing longwave radiation (OLR) dataset herein indicates that there is a statistically significant correlation between the MJO amplitude and wave packets of MRG waves but not equatorial Rossby waves or Kelvin waves, which are derived from the Matsuno shallow water theory. However, the biggest absolute value of the correlation coefficient is only 0.21, indicating that the wave packet hypothesis explains only a small fraction of the variance of the MJO in the OLR data.

## Abstract

The Madden–Julian oscillation (MJO), also known as the intraseasonal oscillation (ISO), is a planetary-scale mode of variation in the tropical Indian and western Pacific Oceans. Basic questions about the MJO are why it propagates eastward at ∼5 m s^{−1}, why it lasts for intraseasonal time scales, and how it interacts with the fine structure that is embedded in it. This study will test the hypothesis that the MJO is not a wave but a wave packet—the interference pattern produced by a narrow frequency band of mixed Rossby–gravity (MRG) waves. As such, the MJO would propagate with the MRG group velocity, which is eastward at ∼5 m s^{−1}. Simulation with a 3D model shows that MRG waves can be forced independently by relatively short-lived, eastward- and westward-moving disturbances, and the MRG wave packet can last long enough to form the intraseasonal variability. This hypothesis is consistent with the view that the MJO is episodic, with an irregular time interval between events rather than a periodic oscillation. The packet is defined as the horizontally smoothed variance of the MRG wave—the rectified MRG wave, which has features in common with the MJO. The two-dimensional Fourier analysis of the NOAA outgoing longwave radiation (OLR) dataset herein indicates that there is a statistically significant correlation between the MJO amplitude and wave packets of MRG waves but not equatorial Rossby waves or Kelvin waves, which are derived from the Matsuno shallow water theory. However, the biggest absolute value of the correlation coefficient is only 0.21, indicating that the wave packet hypothesis explains only a small fraction of the variance of the MJO in the OLR data.

## Abstract

A nonlinear numerical model of long-lived Jovian vortices has been constructed. We assume that the measured zonal velocity profile ū(*y*) extends into the adiabatic interior, but that the eddies and large oval structures are confined to a shallow stably stratified upper layer. Each vortex is stationary with respect to the shear flow ū(*y*) at a critical latitude *y*
_{c}, that is close to the latitude of the vortex center, in agreement with observed flows on Jupiter. Our model differs from the solitary wave model of Maxworthy and Redekopp in that the stratification is not large in our model (the radius of deformation is less than the latitudinal scale of the shear flow), and therefore stationary linear wave solutions, neutral or amplified, do not exist. The solutions obtained are strongly nonlinear in contrast to the solitary wave solutions which are the weakly nonlinear extensions of ultralong linear waves. Both stable and unstable vortices are found in the numerical experiments. When two stable vortices collide, they merge after a short transient phase to form a larger stable vortex. This merging, rather than the non-interaction behavior predicted by the solitary wave theory, is more in agreement with observations of Jovian vortices. We suggest that the long-lived Jovian vortices maintain themselves against dissipation by adsorbing smaller vortices which are produced by convection.

## Abstract

A nonlinear numerical model of long-lived Jovian vortices has been constructed. We assume that the measured zonal velocity profile ū(*y*) extends into the adiabatic interior, but that the eddies and large oval structures are confined to a shallow stably stratified upper layer. Each vortex is stationary with respect to the shear flow ū(*y*) at a critical latitude *y*
_{c}, that is close to the latitude of the vortex center, in agreement with observed flows on Jupiter. Our model differs from the solitary wave model of Maxworthy and Redekopp in that the stratification is not large in our model (the radius of deformation is less than the latitudinal scale of the shear flow), and therefore stationary linear wave solutions, neutral or amplified, do not exist. The solutions obtained are strongly nonlinear in contrast to the solitary wave solutions which are the weakly nonlinear extensions of ultralong linear waves. Both stable and unstable vortices are found in the numerical experiments. When two stable vortices collide, they merge after a short transient phase to form a larger stable vortex. This merging, rather than the non-interaction behavior predicted by the solitary wave theory, is more in agreement with observations of Jovian vortices. We suggest that the long-lived Jovian vortices maintain themselves against dissipation by adsorbing smaller vortices which are produced by convection.

## Abstract

We examine the evolution of baroclinic vortices in a time-dependent, nonlinear numerical model of a Jovian atmosphere. The model uses a normal-mode expansion in the vertical, using the barotropic and first two baroclinic modes. Results for the stability of baroclinic vortices on an *f* plane in the absence of a mean zonal flow are similar to results of Earth vortex models, although the presence of a fluid interior on the Jovian planets shifts the stability boundaries to smaller length scales. The presence of a barotropic mean zonal flow in the interior stabilizes vortices against instability and significantly modifies the finite amplitude form of baroclinic instabilities. The effect of a zonal flow on a form of barotropic instability produces periodic oscillations in the latitude and longitude of the vortex as observed at the level of the cloud tops. This instability may explain some, but not all, observations of longitudinal oscillations of vortices on the outer planets. Oscillations in aspect ratio and orientation of stable vortices in a zonal shear flow are observed in this baroclinic model, as in simpler twodimensional models. Such oscillations are also observed in the atmospheres of Jupiter and Neptune. The meridional propagation and decay of vortices on a *β* plane is inhibited by the presence of a mean zonal flow. The direction of propagation of a vortex relative to the mean zonal flow depends upon the sign of the meridional potential vorticity gradient; combined with observations of vortex drift rates, this may provide a constraint on model assumption for the flow in the deep interior of the Jovian planets.

## Abstract

We examine the evolution of baroclinic vortices in a time-dependent, nonlinear numerical model of a Jovian atmosphere. The model uses a normal-mode expansion in the vertical, using the barotropic and first two baroclinic modes. Results for the stability of baroclinic vortices on an *f* plane in the absence of a mean zonal flow are similar to results of Earth vortex models, although the presence of a fluid interior on the Jovian planets shifts the stability boundaries to smaller length scales. The presence of a barotropic mean zonal flow in the interior stabilizes vortices against instability and significantly modifies the finite amplitude form of baroclinic instabilities. The effect of a zonal flow on a form of barotropic instability produces periodic oscillations in the latitude and longitude of the vortex as observed at the level of the cloud tops. This instability may explain some, but not all, observations of longitudinal oscillations of vortices on the outer planets. Oscillations in aspect ratio and orientation of stable vortices in a zonal shear flow are observed in this baroclinic model, as in simpler twodimensional models. Such oscillations are also observed in the atmospheres of Jupiter and Neptune. The meridional propagation and decay of vortices on a *β* plane is inhibited by the presence of a mean zonal flow. The direction of propagation of a vortex relative to the mean zonal flow depends upon the sign of the meridional potential vorticity gradient; combined with observations of vortex drift rates, this may provide a constraint on model assumption for the flow in the deep interior of the Jovian planets.

## Abstract

We propose a nonlinear, quasi-geostrophic, baroclinic model of Jovian atmospheric dynamics, in which vertical variations of velocity are represented by a truncated sum over a complete set of orthogonal functions obtained by a separation of variables of the linearized quasi-geostrophic potential vorticity equation. A set of equations for the time variation of the mode amplitudes in the nonlinear case is then derived. We show that for a planet with a neutrally stable, fluid interior instead of a solid lower boundary, the baroclinic mode represents motions in the interior, and is not affected by the baroclinic modes. One consequence of this is that a normal-mode model with one baroclinic mode is dynamically equivalent to a one layer model with solid lower topography. We also show that for motions in Jupiter's cloudy lower troposphere, the stratosphere behaves nearly as a rigid lid, so that the normal-mode model is applicable to Jupiter. We test the accuracy of the normal-mode model for Jupiter using two simple problem forced, vertically propagating Rossby waves, using two and three baroclinic modes and baroclinic instability, using two baroclinic modes. We find that the normal-road model provide qualitatively correct results, even with only a very limited number of vertical degrees of freedom.

## Abstract

We propose a nonlinear, quasi-geostrophic, baroclinic model of Jovian atmospheric dynamics, in which vertical variations of velocity are represented by a truncated sum over a complete set of orthogonal functions obtained by a separation of variables of the linearized quasi-geostrophic potential vorticity equation. A set of equations for the time variation of the mode amplitudes in the nonlinear case is then derived. We show that for a planet with a neutrally stable, fluid interior instead of a solid lower boundary, the baroclinic mode represents motions in the interior, and is not affected by the baroclinic modes. One consequence of this is that a normal-mode model with one baroclinic mode is dynamically equivalent to a one layer model with solid lower topography. We also show that for motions in Jupiter's cloudy lower troposphere, the stratosphere behaves nearly as a rigid lid, so that the normal-mode model is applicable to Jupiter. We test the accuracy of the normal-mode model for Jupiter using two simple problem forced, vertically propagating Rossby waves, using two and three baroclinic modes and baroclinic instability, using two baroclinic modes. We find that the normal-road model provide qualitatively correct results, even with only a very limited number of vertical degrees of freedom.

## 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.