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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

Observations of ascending water vapor anomalies in the tropical lower stratosphere—the so-called tape recorder effect—have been used to infer profiles of vertical mean motion, vertical constituent diffusivity, and lateral in-mixing rate in this region. The magnitude of vertical wave flux required to drive the quasi-biennial oscillation (QBO) in the presence of mean upwelling, along with other related aspects of QBO dynamics, is examined in light of the tape recorder results using a simple one-dimensional model. As found in a previous study, it is necessary that wave fluxes significantly exceed the “classic” values associated with long-period Kelvin and Rossby-gravity waves; the extra fluxes are presumably associated with a continuous spectrum of short-period gravity and inertia–gravity waves. Larger wave fluxes, when used in connection with traditional wave-transport parameterizations developed for the QBO, require a larger vertical diffusivity of momentum in order to prevent the formation of unrealistically strong vertical wind shear. The profile of constituent diffusivity derived from the tape recorder effect, however, is much smaller everywhere than the momentum diffusivity assumed in previous QBO modeling studies.

Realistic shear is obtained in the model using a momentum-conserving “shear adjustment” scheme representing the effect of unresolved shear instabilities and other processes not included in the wave-transport parameterization. This device, together with a QBO amplitude profile based on the equatorial-wave phase speeds, motivates an (otherwise inviscid) analytic QBO solution in the underdamped, quasi-compressible case. The simple analytic solutions replicate most aspects of the numerical solution, display a similar dependence on wave flux at the lower boundary, and provide reasonably accurate estimates of QBO period in the inviscid limit.

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Timothy J. Dunkerton

Abstract

A simple theoretical model was developed to investigate the inertial instability of zonally nonuniform, non-parallel flow near the equator. The basic state was independent of height and time but included cross-equatorial shear with longitudinal variation, as observed in the tropical mesosphere and elsewhere. Numerical solutions were obtained for the most unstable modes.

It is shown that, in addition to previously known “global” (symmetric and nonsymmetric) modes of inertial instability, there exist “local” modes within regions of anomalous potential vorticity. Local modes may be exactly stationary or display zonal phase propagation, but are distinguished from global modes by their zero group velocity and concentration of amplitude within, or downstream from, the region of most unstable flow. Local stationary instability has the largest growth rate and occurs in strong inhomogeneous shear when the in situ mean flow is near zero, that is, quasi-stationary with respect to the (stationary) basic-state pattern. This situation is expected in an equatorial Rossby wave critical layer.

The local mode has properties similar to those of “absolute” instability of nonparallel flow as discussed elsewhere in fluid dynamics.

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Timothy J. Dunkerton

Abstract

A theory of inertial instability on the equatorial beta-plane is developed with application to the inertial stability of the equatorial middle atmosphere at the solstices. It is shown that the stability of this region depends primarily on two unknowns. First, there is the question of whether eddy diffusion can be regarded as stabilizing, or whether this diffusion actually arises from the instability itself. Second, because the diabatic circulation would appear to induce a cross-equatorial shear much greater than that observed, or than that modeled in Holton and Wehrbein (1980), it appears that the gravity wave-induced decelerations would he crucial to the stability of this flow. Unfortunately, the parameterization scheme of Leovy (1964) designed to mimic this effect obscures the issue, since this “frictional drag” concept is invalid on a local basis (Lindzen, 1981).

The expected structure and vertical wavelength of the equatorial inertial instability is discussed in the context of this simple model. Predicted vertical wavelengths also depend on the unknown factors listed above. The greatest likelihood of an observable inertial instability would be in the winter tropical mesosphere, within 10° of the equator.

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Timothy J. Dunkerton

Abstract

The Holton-Lindzen (1972) theory of the quasi-biennial oscillation is reevaluated in the light of some recent developments in the theory of wave, mean-flow interaction due to Andrews and Mclntyre (1976a, b). These developments suggest that wave transience should be regarded as the primary cause of the oscillation, in a chronological sense, with wave absorption providing an essential, but chronologically secondary cause. The spontaneous formation and descent of shear zones anticipated in the analytic theory of Part I is here applied to the theory of the oscillation, by extending the numerical calculations of that paper to include equatorial waves of the type observed in the equatorial stratosphere.

Some remarks are also made concerning probable causes of the observed QBO variability.

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Timothy J. Dunkerton

Abstract

Vertically propagating internal waves give rise to mean flow accelerations in an atmosphere due to the effects of wave transience resulting from compressibility and vertical group velocity feedback. Such accelerations appear to culminate in the spontaneous formation and descent of regions of strong mean wind shear. Both analytical and numerical solutions are obtained in an approximate quasi-linear model which describes this effect.

The numerical solutions display mean flow accelerations due to Kelvin waves in the equatorial stratosphere. Wave absorption alters the transience mechanism in some significant respects, particularly in causing the upper atmospheric mean flow acceleration to be very sensitive to the precise magnitude and distribution of the damping mechanisms.

Part II of this series discusses numerical simulations of transient equatorial waves in the quasi-biennial oscillation. These results are of sufficient qualitative interest to merit attention in this paper, and this is done with the help of a simpler, prototype standing-wave model (Plumb, 1977).

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Timothy J. Dunkerton

Abstract

The subtropical mesospheric jet observed by the Nimbus 7 Limb Infrared Monitor of the Stratosphere in late 1978 was flanked to the north and south by regions of reversed potential vorticity gradient. In mid-December, enhanced planetary wave activity propagating upward into the mesosphere led to visible overreflection from the low-latitude reversed gradient region and rapid deceleration of the jet. It is argued, first, that the overreflection visible in the geopotential height field was probably genuine, and not likely to have been due to Rossby waves incident on an inertially unstable region. Nor was it due to the opposing mean meridional circulation. Second, the observed dominance of wave 1 in the overreflected flux may have been attributable to hemispheric barotropic instability: a low-wavenumber type of instability on the sphere related to the midlatitude modes discovered by Hartmann. In comparison to the barotropically unstable eigenmodes for higher zonal wavenumbers, the wave 1 mode has a slower growth rate but larger spatial extent. For practical purposes, it is a radiating mode excitable by sources in the far field. Equally important, the phase speed of the eigenmodes can be made exactly zero when the mean flow vanishes just within this region, as observed in mid-December 1978. Resonant excitation is therefore possible.

Realistic opposing mean meridional advection has only a slight effect on the barotropic eigenmode, provided that high-wavenumber oscillations are filtered out of the calculation, acting to reduce the growth rate and shift the subtropical secondary amplitude maximum a few degrees towards the pole.

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Timothy J. Dunkerton

Abstract

The decelerating effect of enhanced upper tropospheric wavedriving (DF) in winter and early spring induces a “reverse” component of the residual mean meridional circulation in the polar lower stratosphere opposite to that induced by radiative cooling. The cooling, in turn, is maintained by the decelerating effect of stratospheric DF. If the upper-tropospheric wavedriving is increased, and the stratosphere wavedriving is sufficiently reduced, the change in the mean circulation will include upwelling in the polar lower stratosphere. Analytic and numerically derived properties of this generalized residual mean “body form” circulation are discussed.

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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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