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## Abstract

A homogenization technique is used to study the change in the frequency of planetary Rossby waves that results from their interaction with a small-scale two-dimensional topography. The frequency change is computed explicitly for a topography consisting of a random distribution of well-separated cylindrical seamounts; it corresponds to a phase-speed increase (decrease) when the flat-bottom Rossby wave frequency is larger (smaller) than a typical topographic frequency. The topography is also shown to lead to a finite damping of the Rossby waves, even in the limit of infinitesimally small Ekman friction.

## Abstract

A homogenization technique is used to study the change in the frequency of planetary Rossby waves that results from their interaction with a small-scale two-dimensional topography. The frequency change is computed explicitly for a topography consisting of a random distribution of well-separated cylindrical seamounts; it corresponds to a phase-speed increase (decrease) when the flat-bottom Rossby wave frequency is larger (smaller) than a typical topographic frequency. The topography is also shown to lead to a finite damping of the Rossby waves, even in the limit of infinitesimally small Ekman friction.

## Abstract

Application of the stability theorems for multilayer quasigeostrophic flows reveals that the three-layer model may be nonlinearly unstable while in linearly subcritical conditions, the instability being then due to explosive resonant interaction of Rossby waves. This contrasts with the Phillips two-layer model for which linear theory suffices to explain any instability and motivates this study of the nonlinear saturation of instability in the three-layer model.

A rigorous bound on the disturbance eddy energy is calculated using Shepherd's method for a wide range of basic shear and channel width. The method is applied using stable basic flows whose stability is established by either Arnol'd's first or second theorem. For flows unstable through explosive interaction only, the bound indicates that the disturbance energy can attain as much as 40% of the basic flow energy, the maximum disturbance energy being obtained for flows close to linear instability.

With regard to linear instability, an important difference between two- and three-layer flows is the disappearing of the short-wave cutoff for certain basic shears in the three-layer model. The significance of this phenomenon in the context of saturation is discussed.

## Abstract

Application of the stability theorems for multilayer quasigeostrophic flows reveals that the three-layer model may be nonlinearly unstable while in linearly subcritical conditions, the instability being then due to explosive resonant interaction of Rossby waves. This contrasts with the Phillips two-layer model for which linear theory suffices to explain any instability and motivates this study of the nonlinear saturation of instability in the three-layer model.

A rigorous bound on the disturbance eddy energy is calculated using Shepherd's method for a wide range of basic shear and channel width. The method is applied using stable basic flows whose stability is established by either Arnol'd's first or second theorem. For flows unstable through explosive interaction only, the bound indicates that the disturbance energy can attain as much as 40% of the basic flow energy, the maximum disturbance energy being obtained for flows close to linear instability.

With regard to linear instability, an important difference between two- and three-layer flows is the disappearing of the short-wave cutoff for certain basic shears in the three-layer model. The significance of this phenomenon in the context of saturation is discussed.

## Abstract

The gravity waves (GWs) produced by three-dimensional potential vorticity (PV) anomalies are examined under the assumption of constant vertical shear, constant stratification, and unbounded domain. As in the two-dimensional case analyzed in an earlier paper, the disturbance near the PV anomaly is well modeled by quasigeostrophic theory. At larger distances the nature of the disturbance changes across the two inertial layers that are located above and below the anomaly, and it takes the form of a vertically propagating GW beyond these.

For a horizontally monochromatic PV anomaly of infinitesimal depth, the disturbance is described analytically using both an exact solution and a WKB approximation; the latter includes an exponentially small term that captures the change of the solution near the PV anomaly induced by the radiation boundary condition in the far field. The analytical results reveal a strong sensitivity of the emission to the Richardson number and to the orientation of the horizontal wavenumber: the absorptive properties of the inertial layers are such that the emission is maximized in the Northern Hemisphere for wavenumbers at negative angles to the shear.

For localized PV anomalies, numerical computations give the temporal evolution of the GW field. Analytical and numerical results are also used to establish an explicit form for the Eliassen–Palm flux that could be used to parameterize GW sources in GCMs. The properties of the Eliassen–Palm flux vector imply that in a westerly shear, the GWs exert a drag in a southwest direction in the upper inertial layer, and in a northwest direction at the altitudes where the GWs dissipate aloft.

## Abstract

The gravity waves (GWs) produced by three-dimensional potential vorticity (PV) anomalies are examined under the assumption of constant vertical shear, constant stratification, and unbounded domain. As in the two-dimensional case analyzed in an earlier paper, the disturbance near the PV anomaly is well modeled by quasigeostrophic theory. At larger distances the nature of the disturbance changes across the two inertial layers that are located above and below the anomaly, and it takes the form of a vertically propagating GW beyond these.

For a horizontally monochromatic PV anomaly of infinitesimal depth, the disturbance is described analytically using both an exact solution and a WKB approximation; the latter includes an exponentially small term that captures the change of the solution near the PV anomaly induced by the radiation boundary condition in the far field. The analytical results reveal a strong sensitivity of the emission to the Richardson number and to the orientation of the horizontal wavenumber: the absorptive properties of the inertial layers are such that the emission is maximized in the Northern Hemisphere for wavenumbers at negative angles to the shear.

For localized PV anomalies, numerical computations give the temporal evolution of the GW field. Analytical and numerical results are also used to establish an explicit form for the Eliassen–Palm flux that could be used to parameterize GW sources in GCMs. The properties of the Eliassen–Palm flux vector imply that in a westerly shear, the GWs exert a drag in a southwest direction in the upper inertial layer, and in a northwest direction at the altitudes where the GWs dissipate aloft.

## Abstract

The gravity waves (GWs) generated by potential vorticity (PV) anomalies in a rotating stratified shear flow are examined under the assumptions of constant vertical shear, two-dimensionality, and unbounded domain. Near a PV anomaly, the associated perturbation is well modeled by quasigeostrophic theory. This is not the case at large vertical distances, however, and in particular beyond the two inertial layers that appear above and below the anomaly; there, the perturbation consists of vertically propagating gravity waves. This structure is described analytically, using an expansion in the continuous spectrum of the singular modes that results from the presence of critical levels.

Several explicit results are obtained. These include the form of the Eliassen–Palm (EP) flux as a function of the Richardson number *N* ^{2}/Λ^{2}, where *N* is the Brunt–Väisälä frequency and Λ the vertical shear. Its nondimensional value is shown to be approximately exp(−*πN*/Λ)/8 in the far-field GW region, approximately twice that between the two inertial layers. These results, which imply substantial wave–flow interactions in the inertial layers, are valid for Richardson numbers larger than 1 and for a large range of PV distributions. In dimensional form they provide simple relationships between the EP fluxes and the large-scale flow characteristics.

As an illustration, the authors consider a PV disturbance with an amplitude of 1 PVU and a depth of 1 km, and estimate that the associated EP flux ranges between 0.1 and 100 mPa for a Richardson number between 1 and 10. These values of the flux are comparable with those observed in the lower stratosphere, which suggests that the mechanism identified in this paper provides a substantial gravity wave source, one that could be parameterized in GCMs.

## Abstract

The gravity waves (GWs) generated by potential vorticity (PV) anomalies in a rotating stratified shear flow are examined under the assumptions of constant vertical shear, two-dimensionality, and unbounded domain. Near a PV anomaly, the associated perturbation is well modeled by quasigeostrophic theory. This is not the case at large vertical distances, however, and in particular beyond the two inertial layers that appear above and below the anomaly; there, the perturbation consists of vertically propagating gravity waves. This structure is described analytically, using an expansion in the continuous spectrum of the singular modes that results from the presence of critical levels.

Several explicit results are obtained. These include the form of the Eliassen–Palm (EP) flux as a function of the Richardson number *N* ^{2}/Λ^{2}, where *N* is the Brunt–Väisälä frequency and Λ the vertical shear. Its nondimensional value is shown to be approximately exp(−*πN*/Λ)/8 in the far-field GW region, approximately twice that between the two inertial layers. These results, which imply substantial wave–flow interactions in the inertial layers, are valid for Richardson numbers larger than 1 and for a large range of PV distributions. In dimensional form they provide simple relationships between the EP fluxes and the large-scale flow characteristics.

As an illustration, the authors consider a PV disturbance with an amplitude of 1 PVU and a depth of 1 km, and estimate that the associated EP flux ranges between 0.1 and 100 mPa for a Richardson number between 1 and 10. These values of the flux are comparable with those observed in the lower stratosphere, which suggests that the mechanism identified in this paper provides a substantial gravity wave source, one that could be parameterized in GCMs.

## 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

Anticyclonic vortices focus and trap near-inertial waves so that near-inertial energy levels are elevated within the vortex core. Some aspects of this process, including the nonlinear modification of the vortex by the wave, are explained by the existence of trapped near-inertial eigenmodes. These vortex eigenmodes are easily excited by an initial wave with horizontal scale much larger than that of the vortex radius. We study this process using a wave-averaged model of near-inertial dynamics and compare its theoretical predictions with numerical solutions of the three-dimensional Boussinesq equations. In the linear approximation, the model predicts the eigenmode frequencies and spatial structures, and a near-inertial wave energy signature that is characterized by an approximately time-periodic, azimuthally invariant pattern. The wave-averaged model represents the nonlinear feedback of the waves on the vortex via a wave-induced contribution to the potential vorticity that is proportional to the Laplacian of the kinetic energy density of the waves. When this is taken into account, the modal frequency is predicted to increase linearly with the energy of the initial excitation. Both linear and nonlinear predictions agree convincingly with the Boussinesq results.

## Abstract

Anticyclonic vortices focus and trap near-inertial waves so that near-inertial energy levels are elevated within the vortex core. Some aspects of this process, including the nonlinear modification of the vortex by the wave, are explained by the existence of trapped near-inertial eigenmodes. These vortex eigenmodes are easily excited by an initial wave with horizontal scale much larger than that of the vortex radius. We study this process using a wave-averaged model of near-inertial dynamics and compare its theoretical predictions with numerical solutions of the three-dimensional Boussinesq equations. In the linear approximation, the model predicts the eigenmode frequencies and spatial structures, and a near-inertial wave energy signature that is characterized by an approximately time-periodic, azimuthally invariant pattern. The wave-averaged model represents the nonlinear feedback of the waves on the vortex via a wave-induced contribution to the potential vorticity that is proportional to the Laplacian of the kinetic energy density of the waves. When this is taken into account, the modal frequency is predicted to increase linearly with the energy of the initial excitation. Both linear and nonlinear predictions agree convincingly with the Boussinesq results.