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## Abstract

The effects of spherical geometry on the nonlinear evolution of baroclinic waves are investigated by comparing integrations of a two-layer primitive equation (PE) model in spherical and Cartesian geometry. To isolate geometrical effects, the integrations use basic states with nearly identical potential vorticity (PV) structure.

Although the linear normal modes are very similar, significant differences develop at finite amplitude. Anticyclones (cyclones) in spherical geometry are relatively stronger (weaker) than those in Cartesian geometry. For this basic state, the strong anticyclones on the sphere are associated with anticyclonic wrapping of high PV in the upper layer (i.e., high PV air is advected southward and westward relative to the wave). In Cartesian geometry, large quasi-barotropic cyclonic vortices develop, and no anticyclonic wrapping of PV occurs. Because of their influence on the synoptic-scale flow, spherical geometric effects also lead to significant differences in the structure of mesoscale frontal features.

A standard midlatitude scale analysis indicates that the effects of sphericity enter in the next-order correction to β-plane quasigeostrophic (QG) dynamics. At leading order these spherical terms only affect the PV inversion operator (through the horizontal Laplacian) and the advection of PV by the nondivergent wind. Scaling arguments suggest, and numerical integrations of the barotropic vorticity equation confirm, that the dominant geometric effects are in the PV inversion operator. The dominant metric in the PV inversion operator is associated with the equatorward spreading of meridians on the sphere, and causes the anticyclonic (cyclonic) circulations in the spherical integration to become relatively stronger (weaker) than those in the Cartesian integration.

This study demonstrates that the effects of spherical geometry can be as important as the leading-order ageostrophic effects in determining the structure of evolution of dry baroclinic waves and their embedded mesoscale structures.

## Abstract

The effects of spherical geometry on the nonlinear evolution of baroclinic waves are investigated by comparing integrations of a two-layer primitive equation (PE) model in spherical and Cartesian geometry. To isolate geometrical effects, the integrations use basic states with nearly identical potential vorticity (PV) structure.

Although the linear normal modes are very similar, significant differences develop at finite amplitude. Anticyclones (cyclones) in spherical geometry are relatively stronger (weaker) than those in Cartesian geometry. For this basic state, the strong anticyclones on the sphere are associated with anticyclonic wrapping of high PV in the upper layer (i.e., high PV air is advected southward and westward relative to the wave). In Cartesian geometry, large quasi-barotropic cyclonic vortices develop, and no anticyclonic wrapping of PV occurs. Because of their influence on the synoptic-scale flow, spherical geometric effects also lead to significant differences in the structure of mesoscale frontal features.

A standard midlatitude scale analysis indicates that the effects of sphericity enter in the next-order correction to β-plane quasigeostrophic (QG) dynamics. At leading order these spherical terms only affect the PV inversion operator (through the horizontal Laplacian) and the advection of PV by the nondivergent wind. Scaling arguments suggest, and numerical integrations of the barotropic vorticity equation confirm, that the dominant geometric effects are in the PV inversion operator. The dominant metric in the PV inversion operator is associated with the equatorward spreading of meridians on the sphere, and causes the anticyclonic (cyclonic) circulations in the spherical integration to become relatively stronger (weaker) than those in the Cartesian integration.

This study demonstrates that the effects of spherical geometry can be as important as the leading-order ageostrophic effects in determining the structure of evolution of dry baroclinic waves and their embedded mesoscale structures.

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## Abstract

Atmospheric predictability is measured by the average difference (or “error”) within an ensemble of forecasts starting from slightly different initial conditions. The spatial scale of the error field is a fundamental quantity; for meteorological applications, the error field typically varies with latitude and longitude and so requires a two-dimensional (2D) spectral analysis. Statistical predictability theory is based on the theory of homogeneous, isotropic turbulence, in which spectra are circularly symmetric in 2D wavenumber space. One takes advantage of this circular symmetry to reduce 2D spectra to one-dimensional (1D) spectra by integrating around a circle in wavenumber polar coordinates. In recent studies it has become common to reduce 2D error spectra to 1D by computing spectra in the zonal direction and then averaging the results over latitude. It is shown here that such 1D error spectra are generically fairly constant across the low wavenumbers as the amplitude of an error spectrum grows with time and therefore the error spectrum is said grow “up-amplitude.” In contrast computing 1D error spectra in a manner consistent with statistical predictability theory gives spectra that are peaked at intermediate wavenumbers. In certain cases, this peak wavenumber is decreasing with time as the error at that wavenumber increases and therefore the error spectrum is said to grow “upscale.” We show through theory, simple examples, and global predictability experiments that comparisons of model error spectra with the predictions of statistical predictability theory are only justified when using a theory-consistent method to transform a 2D error field to a 1D spectrum.

## Abstract

Atmospheric predictability is measured by the average difference (or “error”) within an ensemble of forecasts starting from slightly different initial conditions. The spatial scale of the error field is a fundamental quantity; for meteorological applications, the error field typically varies with latitude and longitude and so requires a two-dimensional (2D) spectral analysis. Statistical predictability theory is based on the theory of homogeneous, isotropic turbulence, in which spectra are circularly symmetric in 2D wavenumber space. One takes advantage of this circular symmetry to reduce 2D spectra to one-dimensional (1D) spectra by integrating around a circle in wavenumber polar coordinates. In recent studies it has become common to reduce 2D error spectra to 1D by computing spectra in the zonal direction and then averaging the results over latitude. It is shown here that such 1D error spectra are generically fairly constant across the low wavenumbers as the amplitude of an error spectrum grows with time and therefore the error spectrum is said grow “up-amplitude.” In contrast computing 1D error spectra in a manner consistent with statistical predictability theory gives spectra that are peaked at intermediate wavenumbers. In certain cases, this peak wavenumber is decreasing with time as the error at that wavenumber increases and therefore the error spectrum is said to grow “upscale.” We show through theory, simple examples, and global predictability experiments that comparisons of model error spectra with the predictions of statistical predictability theory are only justified when using a theory-consistent method to transform a 2D error field to a 1D spectrum.

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## Abstract

This study investigates gravity wave generation and propagation from jets within idealized vortex dipoles using a nonhydrostatic mesoscale model. Two types of initially balanced and localized jets induced by vortex dipoles are examined here. These jets have their maximum strength either at the surface or in the middle levels of a uniformly stratified atmosphere. Within these dipoles, inertia–gravity waves with intrinsic frequencies 1–2 times the Coriolis parameter are simulated in the jet exit region. These gravity waves are nearly phase locked with the jets as shown in previous studies, suggesting spontaneous emission of the waves by the localized jets. A ray tracing technique is further employed to investigate the propagation effects of gravity waves. The ray tracing analysis reveals strong variation of wave characteristics along ray paths due to variations (particularly horizontal variations) in the propagating environment.

The dependence of wave amplitude on the jet strength (and thus on the Rossby number of the flow) is examined through experiments in which the two vortices are initially separated by a large distance but subsequently approach each other and form a vortex dipole with an associated amplifying localized jet. The amplitude of the stationary gravity waves in the simulations with 90-km grid spacing increases as the square of the Rossby number (Ro), when Ro falls in a small range of 0.05–0.15, but does so significantly more rapidly when a smaller grid spacing is used.

## Abstract

This study investigates gravity wave generation and propagation from jets within idealized vortex dipoles using a nonhydrostatic mesoscale model. Two types of initially balanced and localized jets induced by vortex dipoles are examined here. These jets have their maximum strength either at the surface or in the middle levels of a uniformly stratified atmosphere. Within these dipoles, inertia–gravity waves with intrinsic frequencies 1–2 times the Coriolis parameter are simulated in the jet exit region. These gravity waves are nearly phase locked with the jets as shown in previous studies, suggesting spontaneous emission of the waves by the localized jets. A ray tracing technique is further employed to investigate the propagation effects of gravity waves. The ray tracing analysis reveals strong variation of wave characteristics along ray paths due to variations (particularly horizontal variations) in the propagating environment.

The dependence of wave amplitude on the jet strength (and thus on the Rossby number of the flow) is examined through experiments in which the two vortices are initially separated by a large distance but subsequently approach each other and form a vortex dipole with an associated amplifying localized jet. The amplitude of the stationary gravity waves in the simulations with 90-km grid spacing increases as the square of the Rossby number (Ro), when Ro falls in a small range of 0.05–0.15, but does so significantly more rapidly when a smaller grid spacing is used.

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## Abstract

Quasigeostrophic theory is an approximation of the primitive equations in which the dynamics of geostrophically balanced motions are described by the advection of potential vorticity. Quasigeostrophic theory also represents a leading-order theory in the sense that it is derivable from the primitive equations in the asymptotic limit of zero Rossby number. Building upon quasigeostrophic theory, and the centrality of potential vorticity, the authors have recently developed a systematic asymptotic framework from which balanced, next-order corrections in Rossby number can be obtained. The approach is illustrated here through numerical solutions pertaining to unstable waves on baroclinic jets. The numerical solutions using the full primitive equations compare well with numerical solutions to our equations with accuracy one order beyond quasigeostrophic theory; in particular, the inherent asymmetry between cyclones and anticyclones is captured. Explanations of the latter and the associated asymmetry of the warm and cold fronts are given using simple extensions of quasigeostrophic– potential-vorticity thinking to next order.

## Abstract

Quasigeostrophic theory is an approximation of the primitive equations in which the dynamics of geostrophically balanced motions are described by the advection of potential vorticity. Quasigeostrophic theory also represents a leading-order theory in the sense that it is derivable from the primitive equations in the asymptotic limit of zero Rossby number. Building upon quasigeostrophic theory, and the centrality of potential vorticity, the authors have recently developed a systematic asymptotic framework from which balanced, next-order corrections in Rossby number can be obtained. The approach is illustrated here through numerical solutions pertaining to unstable waves on baroclinic jets. The numerical solutions using the full primitive equations compare well with numerical solutions to our equations with accuracy one order beyond quasigeostrophic theory; in particular, the inherent asymmetry between cyclones and anticyclones is captured. Explanations of the latter and the associated asymmetry of the warm and cold fronts are given using simple extensions of quasigeostrophic– potential-vorticity thinking to next order.

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## Abstract

Quasigeostrophic theory is an approximation of the primitive equations in which the dynamics of geostrophically balanced motions are described by the advection of potential vorticity. Quasigeostrophy also represents a leading-order theory in the sense that it is derivable from the full primitive equations in the asymptotic limit of zero Rossby number. Building upon quasigeostrophy, and the centrality of potential vorticity, a systematic asymptotic framework is developed from which balanced, next-order corrections in Rossby number are obtained. The simplicity of the approach is illustrated by explicit construction of the next-order corrections to a finite-amplitude Eady edge wave.

## Abstract

Quasigeostrophic theory is an approximation of the primitive equations in which the dynamics of geostrophically balanced motions are described by the advection of potential vorticity. Quasigeostrophy also represents a leading-order theory in the sense that it is derivable from the full primitive equations in the asymptotic limit of zero Rossby number. Building upon quasigeostrophy, and the centrality of potential vorticity, a systematic asymptotic framework is developed from which balanced, next-order corrections in Rossby number are obtained. The simplicity of the approach is illustrated by explicit construction of the next-order corrections to a finite-amplitude Eady edge wave.

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## Abstract

Previous simulations of dipole vortices propagating through rotating, stratified fluid have revealed small-scale inertia–gravity waves that are embedded within the dipole near its leading edge and are approximately stationary relative to the dipole. The mechanism by which these waves are generated is investigated, beginning from the observation that the dipole can be reasonably approximated by a balanced quasigeostrophic (QG) solution. The deviations from the QG solution (including the waves) then satisfy linear equations that come from linearization of the governing equations about the QG dipole and are forced by the residual tendency of the QG dipole (i.e., the difference between the time tendency of the QG solution and that of the full primitive equations initialized with the QG fields). The waves do not appear to be generated by an instability of the balanced dipole, as homogeneous solutions of the linear equations amplify little over the time scale for which the linear equations are valid. Linear solutions forced by the residual tendency capture the scale, location, and pattern of the inertia–gravity waves, although they overpredict the wave amplitude by a factor of 2. There is thus strong evidence that the waves are generated as a forced linear response to the balanced flow. The relation to and differences from other theories for wave generation by balanced flows, including those of Lighthill and Ford et al., are discussed.

## Abstract

Previous simulations of dipole vortices propagating through rotating, stratified fluid have revealed small-scale inertia–gravity waves that are embedded within the dipole near its leading edge and are approximately stationary relative to the dipole. The mechanism by which these waves are generated is investigated, beginning from the observation that the dipole can be reasonably approximated by a balanced quasigeostrophic (QG) solution. The deviations from the QG solution (including the waves) then satisfy linear equations that come from linearization of the governing equations about the QG dipole and are forced by the residual tendency of the QG dipole (i.e., the difference between the time tendency of the QG solution and that of the full primitive equations initialized with the QG fields). The waves do not appear to be generated by an instability of the balanced dipole, as homogeneous solutions of the linear equations amplify little over the time scale for which the linear equations are valid. Linear solutions forced by the residual tendency capture the scale, location, and pattern of the inertia–gravity waves, although they overpredict the wave amplitude by a factor of 2. There is thus strong evidence that the waves are generated as a forced linear response to the balanced flow. The relation to and differences from other theories for wave generation by balanced flows, including those of Lighthill and Ford et al., are discussed.

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## Abstract

Results from homogeneous, isotropic turbulence suggest that predictability behavior is linked to the slope of a flow’s kinetic energy spectrum. Such a link has potential implications for the predictability behavior of atmospheric models. This article investigates these topics in an intermediate context: a multilevel quasigeostrophic model with a jet and temperature perturbations at the upper surface (a surrogate tropopause). Spectra and perturbation growth behavior are examined at three model resolutions. The results augment previous studies of spectra and predictability in quasigeostrophic models, and they provide insight that can help interpret results from more complex models. At the highest resolution tested, the slope of the kinetic energy spectrum is approximately

## Abstract

Results from homogeneous, isotropic turbulence suggest that predictability behavior is linked to the slope of a flow’s kinetic energy spectrum. Such a link has potential implications for the predictability behavior of atmospheric models. This article investigates these topics in an intermediate context: a multilevel quasigeostrophic model with a jet and temperature perturbations at the upper surface (a surrogate tropopause). Spectra and perturbation growth behavior are examined at three model resolutions. The results augment previous studies of spectra and predictability in quasigeostrophic models, and they provide insight that can help interpret results from more complex models. At the highest resolution tested, the slope of the kinetic energy spectrum is approximately

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## Abstract

Normal modes of a linear vertical shear (Eady shear) are studied within the linearized primitive equations for a rotating stratified fluid above a rigid lower boundary. The authors' interest is in modes having an inertial critical layer present at some height within the flow. Below this layer, the solutions can be closely approximated by balanced edge waves obtained through an asymptotic expansion in Rossby number. Above, the solutions behave as gravity waves. Hence these modes are an example of a spatial coupling of balanced motions to gravity waves.

The amplitude of the gravity waves relative to the balanced part of the solutions is obtained analytically and numerically as a function of parameters. It is shown that the waves are exponentially small in Rossby number. Moreover, their amplitude depends in a nontrivial way on the meridional wavenumber. For modes having a radiating upper boundary condition, the meridional wavenumber for which the gravity wave amplitude is maximal occurs when the tilts of the balanced edge wave and gravity waves agree.

## Abstract

Normal modes of a linear vertical shear (Eady shear) are studied within the linearized primitive equations for a rotating stratified fluid above a rigid lower boundary. The authors' interest is in modes having an inertial critical layer present at some height within the flow. Below this layer, the solutions can be closely approximated by balanced edge waves obtained through an asymptotic expansion in Rossby number. Above, the solutions behave as gravity waves. Hence these modes are an example of a spatial coupling of balanced motions to gravity waves.

The amplitude of the gravity waves relative to the balanced part of the solutions is obtained analytically and numerically as a function of parameters. It is shown that the waves are exponentially small in Rossby number. Moreover, their amplitude depends in a nontrivial way on the meridional wavenumber. For modes having a radiating upper boundary condition, the meridional wavenumber for which the gravity wave amplitude is maximal occurs when the tilts of the balanced edge wave and gravity waves agree.

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## Abstract

In the course of adapting a nonhydrostatic cloud model [or primitive-equation model (PE)] for simulations of large-scale baroclinic waves, we have encountered systematic discrepancies between the PE solutions and those of the semigeostrophic (SG) equations. Direct comparisons using identical, uniform potential vorticity jets show that 1) the linear modes of the PE have distinctively different structure than the SG modes; 2) at finite amplitude, the PE pressure field develops lows that are deeper, and highs that are weaker, than in the SG solution; and 3) the nonlinear PE wave produces a characteristic “cyclonic wrapping” of the temperature contours on both horizontal boundaries and has an associated “bent-back” frontal structure at the surface, while in the SG solutions (for this particular basic state jet) there is an equal tendency to pull temperature contours anticyclonically around highs and cyclonically around lows. An analysis of the vorticity and potential vorticity equations for small Rossby number reveals that the SG model errs in its treatment of terms involving the ageostrophic vorticity. Simulations based on an equation set that includes the leading-order dynamical contributions of the ageostrophic vorticity agree more closely with the PE simulations.

## Abstract

In the course of adapting a nonhydrostatic cloud model [or primitive-equation model (PE)] for simulations of large-scale baroclinic waves, we have encountered systematic discrepancies between the PE solutions and those of the semigeostrophic (SG) equations. Direct comparisons using identical, uniform potential vorticity jets show that 1) the linear modes of the PE have distinctively different structure than the SG modes; 2) at finite amplitude, the PE pressure field develops lows that are deeper, and highs that are weaker, than in the SG solution; and 3) the nonlinear PE wave produces a characteristic “cyclonic wrapping” of the temperature contours on both horizontal boundaries and has an associated “bent-back” frontal structure at the surface, while in the SG solutions (for this particular basic state jet) there is an equal tendency to pull temperature contours anticyclonically around highs and cyclonically around lows. An analysis of the vorticity and potential vorticity equations for small Rossby number reveals that the SG model errs in its treatment of terms involving the ageostrophic vorticity. Simulations based on an equation set that includes the leading-order dynamical contributions of the ageostrophic vorticity agree more closely with the PE simulations.