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R. Alan Plumb

Abstract

A locally applicable (nonzonally-averaged) conservation relation is derived for quasi-geostrophic stationary waves on a zonal flow, a generalization of the Eliassen-Palm relation. The flux which appears in this relation constitutes, it is argued, a useful diagnostic of the three-dimensional propagation of stationary wave activity. This is illustrated by application to a simple theoretical model of a forced Rossby wave train and to a Northern Hemisphere winter climatology. Results of the latter procedure suggest that the major forcing of the stationary wave field derives from the orographic effects of the Tibetan plateau and from nonorographic effects (diabatic heating and/or interaction with transient eddies) in the western North Atlantic and North Pacific Oceans and Siberia. No evidence is found in the data for wave trains of tropical origin; forcing by the orographic effects of the Rocky mountains seems to be of secondary importance.

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R. Alan Plumb

Abstract

This note reports the derivation of an alternative form of Andrews' conservation law for small-amplitude, quasi-geostrophic, transient eddies on a steady but spatially nonuniform flow.

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R. Alan Plumb

Abstract

Observations of the “quasi-two-day” wave are reviewed and it is concluded that the weight of evidence indicates that the wave is a solstical phenomenon, with maximum amplitudes in low latitudes of the summer mesosphere. It is suggested that the properties of this wave and of a similar wave found in a numerical model of the middle atmosphere may be consistent with an origin via baroclinic instability of the easterly jet in the summer mesosphere. This suggestion is supported by the results of a stability analysis of a one-dimensional model of the summer mesospheric flow.

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R. Alan Plumb

Abstract

The structure of the. meridional circulation driven by steady, zonally symmetric thermal or mechanical forcing is analyzed via the techniques of tidal theory. The relevant Hough functions are derived and, together with consideration of their associated vertical structure, these are used as the basis for discussion of forced meridional circulations in the middle atmosphere. Thus, the global extent of the complementary cooling associated with stratospheric warmings in the high-latitude winter hemisphere is explained as a manifestation of the global nature of the associated (residual) mean circulation. However, the response to equatorial forcing is more localized and it appears that the high-latitude manifestations of the quasi-biennial oscillation cannot be accounted for as a result of coupling, via a mean circulation, with the tropical stratosphere.

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R. Alan Plumb

Abstract

This paper is a continuation of a study of the response of a two-layer baroclinic fluid to weak, near resonant planetary wave forcing. Unlike Part I (Plumb, 1979), the fluid is baroclinically stable and viscous damping is incorporated into the model. With no damping, periodic solutions are found which are similar to but simpler than those of Part 1. With the addition of damping, multiple steady solutions are possible, in agreement with results of earlier studies, notably Charney and DeVore (1979). The importance of nonlinear resonance, the relationships between inviscid and viscous dynamics, and the time scales on which these processes act are investigated by numerical integration of a series of initial value problems.

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R. Alan Plumb

Abstract

A nonacceleration theorem is derived for small-amplitude, transient quasi-geostrophic eddies on a three-dimensional time-mean flow. This theorem states that the divergence of the eddy potential vorticity flux—and hence the forcing by the eddies of the mean geostrophic flow—vanishes to leading order under conditions that (i) the eddies and mean flow are conservative, (ii) the eddy enstrophy density and the quantity Fu/|ū| (where Fu is the component of the eddy potential vorticity flux in the direction of the time-mean flow ū) are constant along time-mean streamlines, and (iii) the boundary conditions on the mean geostrophic flow are independent of the eddies.

The requirement of downstream–constant eddy amplitudes parallels that of steadiness of eddy amplitudes in the equivalent theorem for eddies on a zonal-mean flow. In general, when this condition is not met, the divergence of the transient eddy flux of potential vorticity is nonzero. Thus, unlike in the zonal-mean problem, small-amplitude, conservative, transient eddies propagating on a steady, three-dimensional mean flow will, in some if not in most cases, influence the mean flow in a nontrivial way, even though their amplitudes are steady in time. There are, however, some constraints on the nature of this interaction; conservative eddies do not impact on the global time-mean enstrophy budget, while small-amplitude conservative eddies on a conservative mean flow make no explicit contribution to the global budget of time-mean energy.

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R. Alan Plumb

Abstract

An approximate theory is developed of small-amplitude transient eddies on a slowly varying time-mean flow. Central to this theory is a flux MT, which in most respects constitutes a generalization of the Eliassen–Palm flux to three dimensions; it is a conservable measure of the flux of eddy activity (for small amplitude transients) and is parallel to group velocity for an almost-plane wave train. The use of this flux as a diagnostic of transient eddy propagation is demonstrated by application of the theory to a ten-year climatology of the Northern Hemisphere winter circulation. Results show the anticipated concentration of eddy flux along the major storm tracks.

While, in a suitably transformed system, MT may be regarded as a flux of upstream momentum, it is not a complete description of the eddy forcing of the mean flow; additional effects arise due to downstream transience (i.e., spatial inhomogeneity in the direction of the time-mean flow) of the eddy amplitudes.

The relation between MT and the “E-vector” of Hoskins et al. is discussed.

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R. Alan Plumb

Abstract

The properties and stability of forced planetary waves in a continuous baroclinc shear are investigated gated using a weakly nonlinear analysis. A steady solution exists which under certain conditions may be unstable via interaction with a free traveling wave of the same wavenumber. This wave then grows at the expense of the zonal mean available potential energy. The parameter range in which instability occurs is extended by finite-amplitude perturbations.

Theoretical predictions are compared with results from a highly truncated model of the winter atmosphere. Agreement is good, even after assumptions of the theory have broken down. The theory does become inapplicable, however, following a mean wind reversal (stratosphere warming); in an cases studied, such warming were found to occur subsequent to the instability. The theory appears to explain many features of observed and model warmings.

Results of the study suggest that warmings arise from the intrinsic instability of the winter atmosphere. They can occur in the presence of steady forcing—no precursor tropospheric pulse of planetary wave energy is necessary. In these experiments, interaction with critical lines (of zero wind speed) plays no part in the generation of warmings.

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R. Alan Plumb

Abstract

The traditional derivation of the energy cycle is reviewed and some paradoxical properties of the energy conversion and flux terms under non-acceleration conditions (steady, conservative motion) are noted. An alternative scheme is derived, based on the transformed Eulerian-mean system of Andrews and McIntyre (1976). The structure of this scheme is somewhat different from that of the traditional form (e.g., the only mean-to-eddy conversion term is through the kinetic energies) and the individual definitions of energy conversion and flux terms, and of eddy potential energy, differ from their counterparts in the traditional scheme. Thew differences are manifested most dramatically under non-acceleration conditions, but the qualitative differences are substantial even when these conditions are not met.

Comparison between the two schemes provides insight into the limitations of energy diagnostics. It is argued that the physical interpretations sometimes associated with individual components of the energy cycle and with the overall structure of the cycle are not generally valid.

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Aditi Sheshadri and R. Alan Plumb

Abstract

The two leading empirical orthogonal functions (EOFs) of zonal-mean zonal wind describe north–south fluctuations, and intensification and narrowing, respectively, of the midlatitude jet. Under certain circumstances, these two leading EOFs cannot be regarded as independent but are in fact manifestations of a single, coupled, underlying mode of the dynamical system describing the evolution in time of zonal wind anomalies. The true modes are revealed by the principal oscillation patterns (POPs). The leading mode and its associated eigenvalue are complex, its structure involves at least two EOFs, and it describes poleward (or equatorward) propagation of zonal-mean zonal wind anomalies. In this propagating regime, the principal component (PC) time series associated with the two leading EOFs decay nonexponentially, and the response of the system to external forcing in a given EOF does not depend solely on the PC decorrelation time nor on the projection of the forcing onto that EOF. These considerations are illustrated using results from an idealized dynamical core model. Results from Southern Hemisphere ERA-Interim data are partly consistent with the behavior of the model’s propagating regime. Among other things, these results imply that the time scale that determines the sensitivity of a model to external forcing might be different from the decorrelation time of the leading PC and involves both the rate of decay of the dynamical mode and the period associated with propagation.

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