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Brian Farrell

Abstract

The origin and growth to moderate amplitude of disturbances in shear flow has been traditionally ascribed to the linear modal instability of the flow. Recent work on initial value problems has suggested that nonmodal growth of perturbations may be of equal and perhaps greater importance, at least in cases of rapid development. Examples of robust growth in model problems which support no instabilities and in baroclinic flows with realistic Ekman damping for which the exponential modes have zero or negative growth rates have been shown. Such examples have focused attention on the perturbations which are configured to tap the energy of mean flows. Here a critical examination of these favorably configured perturbations is given, making use of the simple constant free shear barotropic model which allows construction of exact two-dimensional isolated wave packet solutions.

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Brian Farrell

Abstract

Properly configured disturbances are known to be effective in transferring the kinetic energy of a mean shear flow to neutrally stable modal and nonmodal waves. Consideration of perturbation energetics requires that such disturbances produce down-gradient momentum fluxes which are associated with perturbation phase lines oriented against the mean shear. Initial conditions chosen arbitrarily, except that they satisfy this requirement, have been shown to result in robust excitation of neutral waves. A question naturally arising from such studies is whether there exists, in some well-defined sense, a best or most effective choice of initial conditions which optimally excites the waves. This question is addressed as a variational problem, and examples of optimal initial conditions are identified for the barotropic β-plane channel. These examples include the most effective excitation of a given neutral Rossby mode and the most rapidly growing perturbation for a given time period without restriction on spectra composition.

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Brian Farrell

Abstract

A solution of the linear initial value problem for the model of Eady with the inclusion of Ekman damping is presented. This model exhibits large transient growth of perturbations for synoptic cyclone Spatial scales and a realistic value of the vertical turbulent viscosity coefficient despite the fact that all normal modes are exponentially decaying. Similar results are found for the model of Charney, implying that exponential instability cannot, in general, serve to explain the occurrence of cyclone scale disturbances in midlatitudes. Rather these are seen to arise additionally and perhaps predominantly from the release of mean flow potential energy by favorably configured initial perturbations. The Petterssen criterion for midlatitude cyclogenesis results naturally from this development as does its extension to the formation of subtropical monsoon depressions. Implications for the maintenance of midlatitude temperature gradients are discussed.

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Brian Farrell

Abstract

Solution of the initial-value problem for the Eady model is presented. In the presence of boundaries, normal mode waves as well as non-modal waves exist. Energy extracted from the mean flow during the initial development of a perturbation is found to excite the persistent normal modes. It is suggested that this process may be important to cyclogenesis and in providing energy to neutral or near-neutral normal modes. In particular, the Petterssen criterion for cyclogenesis is clarified.

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Brian Farrell and Ian Watterson

Abstract

A barotropic Rossby wave incident to a region of increasing mean flow velocity opposing the wave group velocity undergoes a reversal of direction at a stopping point where the mean flow velocity and local wave group velocities are equal and opposite. Incident wave amplitude increases approaching this stopping point, which may be referred to as a group velocity critical layer, but eventually suffers a decrease along its trajectory so that the reflected wave amplitude and energy tend to zero on approach to a phase velocity critical layer located where the opposing flow vanishes.

Some applications of this process to observations of synoptic scale Rossby waves are suggested and an example of the interaction of an incident wave with a barotropic model of the Hadley circulation is presented to illustrate these ideas.

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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

Abstract

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

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

Abstract

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

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