ALBERT

Description: A method of numerically integrating the Navier-Stokes equations for certain three-dimensional incompressible flows is described. The technique is presented through application to the particular problem of describing thermal convection in a rotating annulus. The equations, in cylindrical polar co-ordinate form, are integrated with respect to time by a marching process, together with the solving of a Poisson equation for the pressure. A suitable form of the finite difference equations gives a computationally-stable long-term integration with reasonably faithful representation of the spatial and temporal characteristics of the flow.Trigonometric interpolation techniques provide accurate (discretely exact) solutions to the Poisson equation. By using an auxiliary algorithm for rapid evaluation of trigonometric transforms, the proportion of computation needed to solve the Poisson equation can be reduced to less than 25% of the total time needed to’ advance one time step. Computing on a UNIVAC 1108 machine, the flow can be advanced one time-step in 2 sec for a 14 × 14 × 14 grid upward to 96 sec for a 60 × 34 × 34 grid.As an example of the method, some features of a solution for steady wave flow in annulus convection are presented. The resemblance of this flow to the classical Eady wave is noted.

Description: The thermally driven motion of water contained in a rotating annulus of square cross-section and having a free surface is investigated by numerical integration of the three-dimensional non-linear Navier–Stokes equations. The nature of steady wave flow is examined in detail and a comparison made with the corresponding axisymmetric solution in parameter space. The steady wave solution proves to be consistent kinematically, dynamically and energetically with Lorenz's hypothesis that the wave can be attributed to the baroclinic instability mechanism. The deviaboric The deviation from the zonal mean. The wave motion is almost completely independent of the side boundary layers which make little contribution to the characteristics and energetics of the deviakoric flow. These side layers are approximately axisymmetric and appear qualitatively indistinguishable from their counterparts in the axisymmetric solution. However, significant Ekman layer features appear in the deviatoric wave structure. Away from the boundaries the dynamical balance of terms is hydrostatic and quasi-geostrophic with changes of vertical vorticity influenced by stretching and viscous diffusion. Heat conduction is completely unimportant except in the side boundary layers. The angular momentum transport by the deviatoric motion is largest at the free surface and is mainly against the angular momentum gradient. A strong outward deviatoric flux of momentum is found in the Ekman layer. The dissipation of deviatoric kinetic energy occurs in the Ekman layer and jet whilst most of the dissipation of the mean kinetic energy occurs in the boundary layer of the inner wall. The large differences between the axisymmetric and zonal mean states indicate that linear baroclinic instability analysis of the axisymmetric state is not strictly relevant to an understanding of the wave formation. The character of the wave suggests that the mean environment with which the deviatoric wave interacts is the wave-present zonal mean state. Only a non-linear finite amplitude baroclinic instability analysis (as yet undeveloped) could possibly explain the wave formation. © 1971, Cambridge University Press. All rights reserved.