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- Author or Editor: Timothy J. Dunkerton x

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## Abstract

When the time-averaging operator is applied to the Generalized Lagrangian Mean equations of motion there results a conservation law involving a total static energy invariant which contains the so-called “pseudoenergy”. This invariant is analogous to the Kelvin or Bjerknes circulation which is conserved as an invariant in the zonal averaging case. An approximate pseudoenergy is also derived which is applicable in cases where quadratic “available” potential energy is of interest.

In the small-amplitude limit, the pseudoenergy may be evaluated as an Eulerian diagnostic in terms of the perturbation potential vorticity and entropy fields.

As in the zonal averaging case, the Lagrangian time mean leads to conservation laws not containing any kind of artificial conversion of energy which appears in the conventional Eulerian mean formulation. Hence the Lagrangian mean provides a static energy invariant analogous to the Kelvin or Bjerknes circulation which may be of use in the study of nonlinear waves on time-mean flows in the lower atmosphere.

## Abstract

When the time-averaging operator is applied to the Generalized Lagrangian Mean equations of motion there results a conservation law involving a total static energy invariant which contains the so-called “pseudoenergy”. This invariant is analogous to the Kelvin or Bjerknes circulation which is conserved as an invariant in the zonal averaging case. An approximate pseudoenergy is also derived which is applicable in cases where quadratic “available” potential energy is of interest.

In the small-amplitude limit, the pseudoenergy may be evaluated as an Eulerian diagnostic in terms of the perturbation potential vorticity and entropy fields.

As in the zonal averaging case, the Lagrangian time mean leads to conservation laws not containing any kind of artificial conversion of energy which appears in the conventional Eulerian mean formulation. Hence the Lagrangian mean provides a static energy invariant analogous to the Kelvin or Bjerknes circulation which may be of use in the study of nonlinear waves on time-mean flows in the lower atmosphere.

## Abstract

The variability of zonally averaged stratospheric circulation is examined using daily gridded analyses from the U.K. Met. Office for 1991–99, corresponding to the period observed by the *Upper Atmosphere Research Satellite.* Application of rotated principal component analysis to the dataset reveals dominant modes of variability consisting of annual, semiannual, and quasi-biennial oscillations, together with intraseasonal and interannual variability in the winter hemisphere. In the upper stratosphere during northern winter, poleward propagating zonal wind anomalies at the stratopause and a sudden deceleration of the subtropical mesospheric jet in each midwinter are observed. The high-latitude flow is more variable, and the data suggest two contrasting types of wintertime evolution in the polar stratosphere. One is characterized in high latitudes by relatively strong flow in early winter and a significantly weakened flow after solstice, the other by relatively weak flow in early winter and a strong positive flow anomaly after solstice. In both, the subtropical deceleration is accompanied by high-latitude acceleration. In the second type, polar westerlies remain long after solstice, decaying gradually, while in the first type, polar easterlies appear after 10–30 days. In two winters of the first type, the subtropical deceleration is unusually abrupt, followed by brief reacceleration of the polar vortex and a spectacular breakdown after 30 days. Multivariate EOF analysis incorporating temperature data separates deceleration events in northern winter affecting the subtropical jet, with midlatitude warming, from those affecting the polar night jet, with polar warming.

## Abstract

The variability of zonally averaged stratospheric circulation is examined using daily gridded analyses from the U.K. Met. Office for 1991–99, corresponding to the period observed by the *Upper Atmosphere Research Satellite.* Application of rotated principal component analysis to the dataset reveals dominant modes of variability consisting of annual, semiannual, and quasi-biennial oscillations, together with intraseasonal and interannual variability in the winter hemisphere. In the upper stratosphere during northern winter, poleward propagating zonal wind anomalies at the stratopause and a sudden deceleration of the subtropical mesospheric jet in each midwinter are observed. The high-latitude flow is more variable, and the data suggest two contrasting types of wintertime evolution in the polar stratosphere. One is characterized in high latitudes by relatively strong flow in early winter and a significantly weakened flow after solstice, the other by relatively weak flow in early winter and a strong positive flow anomaly after solstice. In both, the subtropical deceleration is accompanied by high-latitude acceleration. In the second type, polar westerlies remain long after solstice, decaying gradually, while in the first type, polar easterlies appear after 10–30 days. In two winters of the first type, the subtropical deceleration is unusually abrupt, followed by brief reacceleration of the polar vortex and a spectacular breakdown after 30 days. Multivariate EOF analysis incorporating temperature data separates deceleration events in northern winter affecting the subtropical jet, with midlatitude warming, from those affecting the polar night jet, with polar warming.

## Abstract

The symmetric instability due to horizontal shear on an equatorial beta-plane exhibits two distinct modes of instability. The classical monotonic non-oscillatory instability exists for all Prandtl numbers but is favored when the Prandtl number is approximately less than 3/2. For values of Prandtl number approximately larger than this we find that an oscillating “overstability” is the preferred mode of instability. This result contrasts with the baroclinic centrifugally stable case in which overstabilities exist but are never preferred. Similar results can be demonstrated analytically on an artificially bounded *f*-plane which mimics the finite latitudinal scale imposed by the equatorial beta-plane geometry. Radiative relaxation would favor the monotonic mode, but the effect might be insignificant if breaking internal gravity waves are present.

## Abstract

The symmetric instability due to horizontal shear on an equatorial beta-plane exhibits two distinct modes of instability. The classical monotonic non-oscillatory instability exists for all Prandtl numbers but is favored when the Prandtl number is approximately less than 3/2. For values of Prandtl number approximately larger than this we find that an oscillating “overstability” is the preferred mode of instability. This result contrasts with the baroclinic centrifugally stable case in which overstabilities exist but are never preferred. Similar results can be demonstrated analytically on an artificially bounded *f*-plane which mimics the finite latitudinal scale imposed by the equatorial beta-plane geometry. Radiative relaxation would favor the monotonic mode, but the effect might be insignificant if breaking internal gravity waves are present.

## Abstract

It is shown that slowly varying linear equatorial Rossby-gravity waves in a barotropically neutral mean-wind profile near the equator accelerate the mean flow in a stabilizing sense there. This indicates that the Rossby-gravity wave, believed to be the driving force in the easterly acceleration phase of the quasi-biennial oscillation, cannot force a barotropically unstable mean flow near the equator. Mean flows generated near the equator in the easterly phase of the oscillation in then context of these approximations will therefore resemble or be approximately bounded by a parabola of curvature β, where β is the planetary vorticity gradient. This result does not depend upon a “barotropic adjustment” process, although the latter has been suggested in the past and would yield the same result, but over a broader latitudinal area.

## Abstract

It is shown that slowly varying linear equatorial Rossby-gravity waves in a barotropically neutral mean-wind profile near the equator accelerate the mean flow in a stabilizing sense there. This indicates that the Rossby-gravity wave, believed to be the driving force in the easterly acceleration phase of the quasi-biennial oscillation, cannot force a barotropically unstable mean flow near the equator. Mean flows generated near the equator in the easterly phase of the oscillation in then context of these approximations will therefore resemble or be approximately bounded by a parabola of curvature β, where β is the planetary vorticity gradient. This result does not depend upon a “barotropic adjustment” process, although the latter has been suggested in the past and would yield the same result, but over a broader latitudinal area.

## Abstract

Instabilities arising on a latitudinally sheared mean flow fall into one of at least two classes: inertial instabilities associated with a reversed potential vorticity and barotropic instabilities associated with a reversed meridional gradient of potential vorticity. Both types of instability are described by the generalized Laplace tidal equation, a horizontal structure equation that explicitly includes the effect of horizontal divergence on the disturbances. The effect of horizontal divergence on barotropic instability has not been extensively studied. A systematic investigation of the eigenfunctions of the generalized Laplace tidal equation for monotonic mean zonal wind profiles having a single, narrow region of reversed vorticity gradient in tropical latitudes reveals that, in the limit of low planetary zonal wavenumber, the modes of barotropic instability bifurcate into weakly divergent modes of hemispheric scale, and strongly divergent, “internal” modes trapped about the source region, i.e., equatorially trapped. Disturbances in the second category penetrate into the deep tropics—the side of the critical latitude with *positive* intrinsic frequency—as a Kelvin wave type of behavior not previously seen in this context.

These results suggest, first, that hemispheric barotropic instability need not be purely nondivergent. In fact, the growth of weakly divergent modes is preferred. Their equivalent depth is similar to that of free neutral modes of the homogeneous vertical structure equation. Second, the existence of equatorially trapped divergent barotropic instability may be of interest in the tropical troposphere and mesosphere. The equatorial amplitude of these disturbances can be significant, and their frequency, which is generally less than that of a dry Kelvin wave, is determined by a critical latitude in the region of reversed vorticity gradient.

## Abstract

Instabilities arising on a latitudinally sheared mean flow fall into one of at least two classes: inertial instabilities associated with a reversed potential vorticity and barotropic instabilities associated with a reversed meridional gradient of potential vorticity. Both types of instability are described by the generalized Laplace tidal equation, a horizontal structure equation that explicitly includes the effect of horizontal divergence on the disturbances. The effect of horizontal divergence on barotropic instability has not been extensively studied. A systematic investigation of the eigenfunctions of the generalized Laplace tidal equation for monotonic mean zonal wind profiles having a single, narrow region of reversed vorticity gradient in tropical latitudes reveals that, in the limit of low planetary zonal wavenumber, the modes of barotropic instability bifurcate into weakly divergent modes of hemispheric scale, and strongly divergent, “internal” modes trapped about the source region, i.e., equatorially trapped. Disturbances in the second category penetrate into the deep tropics—the side of the critical latitude with *positive* intrinsic frequency—as a Kelvin wave type of behavior not previously seen in this context.

These results suggest, first, that hemispheric barotropic instability need not be purely nondivergent. In fact, the growth of weakly divergent modes is preferred. Their equivalent depth is similar to that of free neutral modes of the homogeneous vertical structure equation. Second, the existence of equatorially trapped divergent barotropic instability may be of interest in the tropical troposphere and mesosphere. The equatorial amplitude of these disturbances can be significant, and their frequency, which is generally less than that of a dry Kelvin wave, is determined by a critical latitude in the region of reversed vorticity gradient.

## Abstract

Selective transmission of gravity waves into the upper mesosphere and lower thermosphere leads to the generation of mean flows opposite to those below. This interaction is addressed in the context of a simplified transient, stochastic, conservative wave model. The dominant phase speed in the spectrum appears to determine the magnitude of the upper level flow, while the height of interaction is determined by the forcing amplitude. Observed features of the upper atmosphere are efficiently explained by this model, and the results compare very well to recent steady wave models, despite the differing formulations.

The model also provides a tentative explanation of the semiannual oscillation in the tropical upper mesosphere.

## Abstract

Selective transmission of gravity waves into the upper mesosphere and lower thermosphere leads to the generation of mean flows opposite to those below. This interaction is addressed in the context of a simplified transient, stochastic, conservative wave model. The dominant phase speed in the spectrum appears to determine the magnitude of the upper level flow, while the height of interaction is determined by the forcing amplitude. Observed features of the upper atmosphere are efficiently explained by this model, and the results compare very well to recent steady wave models, despite the differing formulations.

The model also provides a tentative explanation of the semiannual oscillation in the tropical upper mesosphere.

## Abstract

Reasons underlying the asymmetry in shear-zone intensity in the observed and simulated quasi-biennial oscillations are investigated. It is shown that much of the incorrect model asymmetry originates in the differing equipartition laws of the Kelvin and Rossby gravity waves. The observed asymmetry cannot entirely be explained by vertical advection due to the residual mean meridional circulation. It is suggested that latitudinal shear plays a role in the observed shear zone asymmetry by reducing the degree of inflection in the dependence of Rossby-gravity wave vertical group velocity on intrinsic frequency via a curvature-induced change in the effective planetary vorticity gradient. The experiments are suggestive of a possible mechanical dissipation of the Rossby-gravity wave.

## Abstract

Reasons underlying the asymmetry in shear-zone intensity in the observed and simulated quasi-biennial oscillations are investigated. It is shown that much of the incorrect model asymmetry originates in the differing equipartition laws of the Kelvin and Rossby gravity waves. The observed asymmetry cannot entirely be explained by vertical advection due to the residual mean meridional circulation. It is suggested that latitudinal shear plays a role in the observed shear zone asymmetry by reducing the degree of inflection in the dependence of Rossby-gravity wave vertical group velocity on intrinsic frequency via a curvature-induced change in the effective planetary vorticity gradient. The experiments are suggestive of a possible mechanical dissipation of the Rossby-gravity wave.

## Abstract

The quasi-biennial oscillation is simulated in two dimensions using a WKB approach for the equatorial waves in which analytic approximations to the wave-induced body forces are inserted as forcing terms in a primitive equation model of the zonally-averaged flow. Realistically large amplitude oscillations are obtained with this method.

Examination of one such oscillation in this paper clarifies the linear and nonlinear role of the residual mean meridional circulation: at low latitudes due to its vertical advection of momentum, and at higher latitudes on account of the Coriolis torque. Because this circulation also advects ozone, the primary source of radiative heating in this region, coupling between the ozone and dynamical oscillations will be significant in determining the strength of this meridional circulation.

Also of importance for the waves themselves are the scale-dependent radiative damping rate and, in the Rossby-gravity wave, the effect of latitudinal shear.

## Abstract

The quasi-biennial oscillation is simulated in two dimensions using a WKB approach for the equatorial waves in which analytic approximations to the wave-induced body forces are inserted as forcing terms in a primitive equation model of the zonally-averaged flow. Realistically large amplitude oscillations are obtained with this method.

Examination of one such oscillation in this paper clarifies the linear and nonlinear role of the residual mean meridional circulation: at low latitudes due to its vertical advection of momentum, and at higher latitudes on account of the Coriolis torque. Because this circulation also advects ozone, the primary source of radiative heating in this region, coupling between the ozone and dynamical oscillations will be significant in determining the strength of this meridional circulation.

Also of importance for the waves themselves are the scale-dependent radiative damping rate and, in the Rossby-gravity wave, the effect of latitudinal shear.

## Abstract

Local shear and convective instabilities of internal inertia-gravity waves (IGW) are examined assuming a steady, plane-parallel flow with vertical profiles of horizontal velocity and static stability resembling an IGW packet in a basic state at rest, without mean vertical shear. The eigenproblem can be described in terms of a nondimensional rotation rate *R* = *f*/*ω̂*_{0} < 1*f* is the Coriolis parameter, *ω̂*_{0}*a,* such that *a* = 1 for convectively neutral waves. In the nonrotating case, shear instability is possible only for convectively supercritical waves, with horizontal wavevector aligned parallel or nearly parallel to the plane of IGW propagation. Transverse convection, with wavevector aligned perpendicular to the plane of IGW propagation, displays faster growth than parallel shear or convective instability at any horizontal wavenumber. For intermediate *R,* eigenmodes in supercritical IGW are characterized at small horizontal wavenumber *k* by a transverse mode of convective instability and a parallel mode of shear instability. The transverse mode again has larger growth rate at small *k* but is suppressed at high wavenumbers where parallel convection prevails. Shear production of perturbation kinetic energy in transverse instability is positive (negative) at intermediate or large (small) *R.* For *R* approaching unity, shear instability takes precedence over convective instability at all azimuths regardless of *a.* In this limit, growth of the most unstable mode is almost independent of azimuth. It is shown that the parallel shear instabilities of an IGW are analogous to the unstable modes of a stratified jet.

## Abstract

Local shear and convective instabilities of internal inertia-gravity waves (IGW) are examined assuming a steady, plane-parallel flow with vertical profiles of horizontal velocity and static stability resembling an IGW packet in a basic state at rest, without mean vertical shear. The eigenproblem can be described in terms of a nondimensional rotation rate *R* = *f*/*ω̂*_{0} < 1*f* is the Coriolis parameter, *ω̂*_{0}*a,* such that *a* = 1 for convectively neutral waves. In the nonrotating case, shear instability is possible only for convectively supercritical waves, with horizontal wavevector aligned parallel or nearly parallel to the plane of IGW propagation. Transverse convection, with wavevector aligned perpendicular to the plane of IGW propagation, displays faster growth than parallel shear or convective instability at any horizontal wavenumber. For intermediate *R,* eigenmodes in supercritical IGW are characterized at small horizontal wavenumber *k* by a transverse mode of convective instability and a parallel mode of shear instability. The transverse mode again has larger growth rate at small *k* but is suppressed at high wavenumbers where parallel convection prevails. Shear production of perturbation kinetic energy in transverse instability is positive (negative) at intermediate or large (small) *R.* For *R* approaching unity, shear instability takes precedence over convective instability at all azimuths regardless of *a.* In this limit, growth of the most unstable mode is almost independent of azimuth. It is shown that the parallel shear instabilities of an IGW are analogous to the unstable modes of a stratified jet.

## Abstract

Rawinsonde data from tropical Pacific stations were examined for westward-propagating 3–6-day meridional wind oscillations in the troposphere and lower stratosphere, 1973–1992. Four types were identified from cross- spectrum and principal component analysis. 1) The dominant oscillation, near 250 mb, had a period slightly greater than 5 days, zonal wavenumber 4–6, and modified Rossby-gravity structure near the date line. 2) In the western Pacific lower troposphere there was broadband activity with short zonal scale, coupled to upper- tropospheric waves in NH summer. 3) In the central Pacific, during NH autumn, there was a well-defined ∼4½-day oscillation with maximum amplitude in the lower troposphere and baroclinic phase tilt above. The vertical structure suggested coupling to deep tropical convection; this interpretation was supported by correlation of meridional wind with antisymmetric outgoing longwave radiation. 4) In the stratosphere, Rossby-gravity waves had periods ≤4 days and zonal wavenumber 3-4. Unlike tropospheric waves, these disturbances were coherent in a *shallow* layer, largest in west phase of QBO and annual cycle (NH winter-spring).

## Abstract

Rawinsonde data from tropical Pacific stations were examined for westward-propagating 3–6-day meridional wind oscillations in the troposphere and lower stratosphere, 1973–1992. Four types were identified from cross- spectrum and principal component analysis. 1) The dominant oscillation, near 250 mb, had a period slightly greater than 5 days, zonal wavenumber 4–6, and modified Rossby-gravity structure near the date line. 2) In the western Pacific lower troposphere there was broadband activity with short zonal scale, coupled to upper- tropospheric waves in NH summer. 3) In the central Pacific, during NH autumn, there was a well-defined ∼4½-day oscillation with maximum amplitude in the lower troposphere and baroclinic phase tilt above. The vertical structure suggested coupling to deep tropical convection; this interpretation was supported by correlation of meridional wind with antisymmetric outgoing longwave radiation. 4) In the stratosphere, Rossby-gravity waves had periods ≤4 days and zonal wavenumber 3-4. Unlike tropospheric waves, these disturbances were coherent in a *shallow* layer, largest in west phase of QBO and annual cycle (NH winter-spring).