The process of internal wave generation by the interaction of an oscillatory background flow (U0 cos (ω0t), V0 sin (ω0t), W0 sin (ω0t)) over a uniform slope is investigated. The stratification is assumed to be uniform and the fluid of infinite depth. Analytical solutions are obtained which give the energy flux in the radiating internal wave field. Since waves are generated not only at the fundamental frequency ω0, but also at all the harmonic frequencies less than the buoyancy frequency, the energy flux for both low and high frequency waves is considered. The acoustic limit approximation (the limiting case in which the tidal excursion, U0/ω0, is much less than the scale of the topography) gives a reasonable approximation to the energy flux.

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Linearized stability of incompressible viscous fluid flows in a thin spherical shell is studied by using the two-dimensional Navier-Stokes equations on a sphere. The stationary flow on the sphere has two singularities (a sink and a source) at the North and South poles of the sphere. We prove analytically for the linearized Navier-Stokes equations that the stationary flow is asymptotically stable. When the spherical layer is truncated between two symmetrical rings, we study eigenvalues of the linearized equations numerically by using power series solutions and show that the stationary flow remains asymptotically stable for all Reynolds numbers. (2000). 76D05, 76E20, 34B24, 34L16. Mathematics Subject ClassificationKeywords. Navier-Stokes equations on a sphere, associated Legendre equation, asymptotic stability of stationary flow, numerical approximation of eigenvalues.

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We study the nonlinear incompressible fluid flows within a thin rotating spherical shell. The model uses the two-dimensional Navier-Stokes equations on a rotating three-dimensional spherical surface and serves as a simple mathematical descriptor of a general atmospheric circulation caused by the difference in temperature between the equator and the poles. Coriolis effects are generated by pseudoforces, which support the stable west-to-east flows providing the achievable meteorological flows rotating around the poles. This work addresses exact stationary and non-stationary solutions associated with the nonlinear Navier-Stokes. The exact solutions in terms of elementary functions for the associated Euler equations (zero viscosity) found in our earlier work are extended to the exact solutions of the Navier-Stokes equations (non-zero viscosity). The obtained solutions are expressed in terms of elementary functions, analyzed, and visualized.