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## 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) 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

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.