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Michael Ghil, Boris Shkoller, and Victor Yangarber

Abstract

We derive a system of diagnostic equations for the velocity field, or “wind laws,” for a barotropic primitive-equation model of large-scale atmospheric flow. The derivation is mathematically exact and does not involve any physical assumptions, such as nondivergence or vanishing of derivatives of the divergence, which are not already present in the prognostic equations. Therefore, initial states computed by solving these diagnostic equations should be compatible with the type of motion described by the prognostic equations of the model, and should not generate initialization shocks when inserted into the prognostic model.

Based on the diagnostic system obtained, we are able to give precise meaning to the question whether the wind field is determined by the mass field and by its time history. The answer to this important question is affirmative, in the precise formulation we provide.

The diagnostic system corresponding to the chosen barotropic model is a generalization of the classical balance equation. The ellipticity condition for this system is derived and given a physical interpretation. Numerical solutions of the diagnostic system are exhibited, including cases in-which the system is of mixed elliptic-hyperbolic type.

Such diagnostic systems can be obtained for other primitive equation models. They are valid for all atmospheric scales and regions for which the prognostic models from which they are derived hold. Some problems concerning the possibility of implementing such a system in operational numerical weather prediction are discussed.

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Guennadi I. Soloviev, Vitali D. Shapiro, Richard C. J. Somerville, and Boris Shkoller

Abstract

The tilting instability is an instability of a two-dimensional fluid that transforms convective motion into shear flow. As a generalization of previous analytical work on the tilting instability in an ideal fluid, the authors investigate the instability with thermal buoyancy included as a source supporting convection against viscous dissipation. The results show two distinct instabilities: for large Rayleigh numbers, the instability is similar to the tilting instability in an inviscid fluid; for small Rayleigh numbers, it resembles a dissipative (i.e., viscous) instability driven by thermal buoyancy. This paper presents a linear stability analysis together with numerical solutions describing the nonlinear evolution of the flow for both types of instabilities. It is shown that the tilting instability develops for values of the aspect ratio (the ratio of the horizontal spatial scale to the vertical scale) that are less than unity. In the case of an ideal fluid, the instability completely transforms the convection into a shear flow, while the final stage of the dissipative instability is one of coexisting states of convection and horizontal shear flow. This study is confined to two dimensions, and the role of the tilting instability in three dimensions remains a subject for future research. In two dimensions, however, the tilting instability can readily generate shear flows from convective motions, and this mechanism may well be important in the interpretation of the results of two-dimensional numerical simulations.

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