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Timothy J. Dunkerton

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.

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Timothy J. Dunkerton

Abstract

Internal gravity waves, and the stress divergence and turbulence induced by them, are essential components of the atmospheric and oceanic general circulations. Theoretical studies have not yet reached a consensus as to how gravity waves transport and deposit momentum. The two best-known theories, resonant interaction and Eikonal saturation, yield contradictory answers to this question. In resonant interaction theory, an energetic, high-frequency, low-wavenumber wave is unstable to two waves of approximately half the frequency and is backscattered by a low-frequency wave or mean finestructure of twice the vertical wavenumber. By contrast, the Eikonal saturation model, as it is commonly used, ignores reflection by assuming a slowly varying basic state and does not question the longevity of the primary wave in the presence of local Kelvin–Helmboltz or convective instabilities. The resonant interaction formalism demands that the interactions be weakly nonlinear. The Eikonal saturation model allows strong, “saturated” waves but ignores reflection and eliminates nonlinear instability with respect to other horizontal wavenumbers by invoking the linear or quasi-linear assumption.

To help bridge the gap between the two theories, results from prototype, nonlinear numerical simulations are presented. Attention is directed at the nonlinear instability of gravity waves in a slowly varying basic state. Parametric instability theory yields a group trajectory length scale for the primary wave expressed in terms of the dominant vertical wavelength and degree of convective saturation. This result delimits the range of validity for the Eikonal saturation model: a low-amplitude wave introduced into an undisturbed slowly varying basic state easily traverses many vertical wavelengths; conversely, a convectively neutral wave soon undergoes decay through nonlinear instability provided that some noise is present initially or created in situ by off-resonant interactions.

The numerical results establish the existence of a cascade in wavenumber space, which for hydrostatic waves proceeds toward both higher and lower horizontal wavenumbers, in accord with theory. Substantial reductions in momentum flux are found relative to the linear values.

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Timothy J. Dunkerton

Abstract

Advection of angular momentum by the mean meridional circulation is important in the quasi-biennial and semiannual oscillations of the tropical middle atmosphere. The advection is nonlinear, implying a finite horizontal or vertical displacement of angular momentum surfaces. Horizontal advection contributes to the easterly phase of the semiannual oscillation, and is sensitive to extratropical body forces. The mean meridional circulation may be thought of as a hybrid Hadley/body-force circulation driven by radiative heating and Eliassen-Palm flux convergence. Realistic steady states are obtained when a mesospheric friction layer, representing gravity wave drag, is included in the problem. This device resolves an ambiguity in the inviscid theory of the middle atmosphere Hadley circulation. Nonlinear advection is also important in the quasi-biennial oscillation; it is responsible, in part, for the strong asymmetry between east and west phases. Diabatic advection of westerly shear displaces angular momentum surfaces downward at the equator in agreement with observations. From this initial condition, it is shown that a self-propagating westerly jet is excited that differs substantially from the linear-diffusive propagation discussed by Dickinson.

These results are derived from high-resolution, two-dimensional models of the atmosphere. Realistic simulations of the quasi-biennial and stratopause semiannual oscillations are obtained without ad hoc forcing of semiannual easterlies. It is argued, however, that a spectrum of Kelvin or gravity waves may be necessary for the westerly acceleration phase. A novel result is that the period of the quasi-biennial oscillation is increased by extratropical body forces, due to the time mean Brewer-Dobson circulation.

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Timothy J. Dunkerton

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.

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Timothy J. Dunkerton

Abstract

The propagation and refraction of stationary inertia–gravity waves in the winter stratosphere is examined with ray tracing. Due to their smaller vertical group velocity these waves experience more lateral ray movement and horizontal refraction that the simple gravity waves recently discussed by Dunkerton and Butchart. Stationary waves are rotated by the transverse horizontal shear and propagate into the polar night jet. Circumstances are found in which the mean flow shear has enhanced unstable wavebreaking by compressing, the wave packet and decreasing the absolute value of wave action density required for breaking. In some other places, reflection from the caustic is more likely.

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Timothy J. Dunkerton

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.

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Timothy J. Dunkerton

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.

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Timothy J. Dunkerton

Abstract

For barotropic flow in spherical geometry, the ideal potential vorticity staircase with flat steps and vertical risers exhibits a relationship between prograde jet strength and spacing such that, for regular spacing, the distance between adjacent jets is given by a suitably defined “Rhines scale” multiplied by a positive constant equal to . This result was obtained previously by the author in the equatorial limit of spherical geometry and by others in periodic beta-plane geometry. An improved asymptotic method has been devised to explain the strength–spacing relationship in sphere-filling solutions. This analysis explains the approximate validity of the equatorial asymptotics and yields new insight on minimum energy states and staircase mode transitions simulated in the presence of random, persistent energy inputs at high horizontal wavenumber.

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Timothy J. Dunkerton

Abstract

Interannual variability of trace constituents in the stratosphere is examined using methane, water vapor, and ozone data from the Halogen Occultation Experiment aboard the Upper Atmosphere Research Satellite in 1992–99. Application of rotated principal component analysis to the dataset reveals dominant modes of variability consisting of annual, semiannual, and quasi-biennial oscillations (QBOs), together with “subbiennial” variations evidently due to nonlinear interaction between the annual cycle and QBO. The structure of quasi-biennial variability is approximately symmetric about the equator, while subbiennial variability, with certain exceptions, is approximately antisymmetric and confined mostly to the subtropics. The vertical structure and downward propagation of the ozone QBO at the equator is described by a pair of symmetric EOFs having separate amplitude maxima in the lower and upper stratosphere. A second pair of EOFs explains the seasonal dependence of subtropical ozone anomalies. For each tracer, the subtropical anomaly is larger in the Northern Hemisphere.

A novel “phase diagram” illustrates the joint seasonal and QBO dependence of tracer anomalies. A pair of principal components are used to define the phase of the dynamical QBO. When plotted against the phase of the annual cycle, the QBO follows a diagonal trajectory with regular phase progression except for an occasional slowing of easterly shear-zone descent near 50 hPa. Tracer principal components of symmetric and antisymmetric EOFs, plotted along this trajectory, display the distinct signatures of quasi-biennial and subbiennial variation. Tracer anomalies reconstructed using an idealized representation of QBO and subbiennial harmonics display the seasonal synchronization and decadal modulation characteristic of QBO–annual cycle interaction.

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Timothy J. Dunkerton

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).

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