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Brian F. Farrell

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Brian F. Farrell

Abstract

Inclusion of Ekman damping in baroclinic models severely limits the range of unstable wavenumbers as well as the growth rate of the instabilities that remain. In contrast, there is much less reduction by the same dissipation of the transient growth of perturbations chosen to resemble those associated with observations of the initial stages of cyclogenesis. It is shown here that the Charney problem with Ekman dissipation included provides a realistic model of damped instability, that the growth rates of the unstable waves are small compared both with observed deepening rates and with deepening rates for initial value problems, and that vertical discretization is likely to produce spurious instabilities in damped models.

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Brian F. Farrell

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The asymptotic development at large time of waves arising from localized disturbances in a baroclinic flow is examined.

Vertical structures unlike those associated with the more commonly examined temporal normal modes are found both for the pulse confined to a channel as previously examined and for the unconfined pulse on an infinite β-plane. These structures and their implied transports are compared to observations in the regions of storm tracks.

It is also found that the meridional extent of the asymptotic solution becomes large compared to observed cyclone wavetrains, emphasizing the importance of flow inhomogeneity and sphericity effects in determining the latitudinal structure of eddies.

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Brian F. Farrell

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The asymptotic response of the Charney baroclinic instability problem to a localized perturbation is determined using the formalism of Briggs (1964) and exploiting a recently obtained highly accurate WKB approximate dispersion relation (Lindzen and Rosenthal, 1981). Comparison is made with previous results for two-level and Eady models.

Small scales and rapid growth characteristic of the initial stages of cyclogenesis are found and the linear dispersion relation, which can be obtained from observation of zonal wind and stability, emerges as a forecast tool for prediction of geographically local cyclogenesis.

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Brian F. Farrell

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Explaining the growth of disturbances superimposed on mean flows is a central problem in meteorology. Most widely studied models of the development process involve perturbations to shear flows with shear restricted to the meridional direction. Recently the importance of zonal variation of the mean flow was recognized and the study of shear flows extended to include zonal variation in shear. These studies found that the eigenfunctions associated with unstable modes in the simple shear problem are highly sensitive to zonal variation of the mean flow. However, there also exists another mechanism for development in a zonally inhomogeneous flow field: transient growth not associated with exponential instability. Properly configured perturbations exhibit transient growth in deformation fields associated with regions of confluence and diffluence at rates comparable to development in shear flow.

In this work analytic solution of the linear initial value problem for the barotropic vorticity equation in deformation flow is used to construct local perturbations that undergo rapid transient development. Implications for cyclogenesis and block formation are discussed.

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Brian F. Farrell

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Forecast reliability is known to be highly variable and this variability can be traced in part to differences in the innate predictability of atmospheric flow regimes. These differences in turn have traditionally been ascribed to variation in the growth rate of exponential instabilities supported by the flow. More recently, drawing on modern dynamical systems theory, the asymptotic divergence of trajectories in phase space of the nonlinear equations of motion has been cited to explain the observed loss of predictability. In this report it is shown that increase in error on synoptic forecast time scales is controlled by rapidly growing perturbations that are not of normal mode form. It is further noted that unpredictable regimes are not necessarily associated with larger exponential growth rates than are relatively more predictable regimes. Moreover, model problems illustrating baroclinic and barotropic dynamics suggest that asymptotic measures of divergence in phase space, while applicable in the limit of infinite time, may not be appropriate over time intervals addressed by present synoptic forecast.

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Brian F. Farrell

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Development of perturbations in a baroclinic flow can arise both from exponential instability and from the transient growth of favorably configured disturbances that are not of normal mode form. The transient growth mechanism is able to account for development of neutral and damped waves as well as for an initial growth of perturbations asymptotically dominated by unstable modes at significantly greater than their asymptotic exponential rates. Unstable modes, which are the eigenfunctions of a structure equation, are discrete and typically few in number. In contrast, disturbances favorable for transient growth form a large subset of all perturbations. To assess the potential of transient growth to account for a particular phenomena it is useful to obtain from this subset the initial condition that gives the maximum development in a well-defined sense. These optimal perturbations have a role in the theory of transient development analogous to that of the normal modes in exponential instability theory; for instance they are the structures that the theory predicts should be found to precede rapid development.

In this work optimal perturbations for the excitation of baroclinic stable and unstable waves are found. The optima are obtained for the formation of synoptic scale cyclones as well as for the development of planetary scale stationary and transient baroclinic Rossby waves. It is argued from these examples that optimal perturbations are likely to limit predictability on time scales relevant to the short and medium range forecast problem and that unstable modes, if present, dominate the long range forecast.

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Brian F. Farrell

Abstract

The growth of perturbations in a baroclinic flow is examined as an initial value problem. Although the long time asymptotic behavior is dominated by discrete exponentially growing normal modes when they exist, these do not form a complete set and initial intensification is shown to be dependent on the continuous spectrum. The vertical structure of perturbations emerges as an important influence on initial growth, and physically realistic disturbances are shown to grow to amplitudes where nonlinear effects are important before obtaining normal mode form.

Connection is made with the work of Arnol'd (1965) and Blumen (1968) and the numerical experiments of Simmons and Hoskins (1979). Application of these results to cyclogenesis in geographically fixed areas is suggested and implied constraints on numerical models discussed.

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Brian F. Farrell

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As the record of past climate becomes clearer, the existence of regimes has emerged as a primary characteristic of the climate system. Present climate is now known to represent one regime among others including glacial climates such as characterized recent intervals of the Pleistocene and the much warmer equable climates of the Eocene and Cretaceous. An important test of climate theory is the ability to explain the record of climate change in terms of atmosphere-ocean dynamics, and the response of the dynamics to internal and external changes.

The hypothesis is advanced that variation in the strength and extent of the symmetric circulation is an important mechanism mediating climate change. Using a simplified model we explore dynamical variables that influence the symmetric circulation and the physical processes that may have modified these dynamical variables to maintain the warm equable climates.

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Eyal Heifetz and Brian F. Farrell

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This work continues the generalized stability theory (GST) analysis of baroclinic shear flow in the primitive equations (PE), focusing on the regime in which the mean baroclinic shear and the stratification are of the same order. The Eady model basic state is used and solutions obtained using the PE are compared to quasigeostrophic (QG) solutions.

Similar optimal growth is obtained in the PE and QG frameworks for eddies with horizontal scale equal to or larger than the Rossby radius, although PE growth rates always exceed QG growth rates. The primary energy growth mechanism is the conventional baroclinic conversion of mean available potential energy to perturbation energy mediated by the eddy meridional heat flux. However, for eddies substantially smaller than the Rossby radius, optimal growth rates in the PE greatly exceed those found in the QG. This enhanced growth rate in the PE is dominated by conversion of mean kinetic energy to perturbation kinetic energy mediated by the vertical component of zonal eddy momentum flux. This growth mechanism is filtered in QG. In the intermediate Richardson number regime mixed Rossby–gravity modes are nonorthogonal in energy, and these participate in the process of energy transfer from the barotropic source in the mean shear to predominantly baroclinic waves during the transient growth process.

The response of shear flow in the intermediate Richardson number regime to spatially and temporally uncorrelated stochastic forcing is also investigated. It is found that a comparable amount of shear turbulent variance is maintained in the rotational and mixed Rossby–gravity modes by such unbiased forcing suggesting that any observed dominance of rotational mode energy arises from restrictions on the effective forcing and damping.

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