Modal Analysis of Convection in a Rotating Fluid

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Abstract
The Boussinesq equations for convection in a plane layer of fluid rotating about a vertical axis are expanded in the planform functions of linear theory. The case where only one horizontal mode is retained is studied in detail for a wide range of parameters and for free and rigid boundaries. For large Rayleigh numbers numerical techniques are used and the results compared with an asymptotic solution. For large Rayleigh numbers, moderate Prandtl numbers and rigid boundaries, steady solutions are found which display non-monotonic dependence of beat flux on rotation rate even when the horizontal wavenumber is fixed. This behavior is suggestive of Rossby's experimental results and the appearance of Ekrman-like boundary layers in the solution seems to confirm his proposed explanation. At lower Rayleigh numbers, the effect diminishes markedly and an appeal to a variable horizontal wavenumber still seems needed. However, the heat flux varies monotonically with rotation rate when the boundaries are free, which result disagrees with that of Morgan. We discuss the cause of this difference. For the low Prandtl numbers typical for mercury, the numerical solutions are time-dependent and be- become periodic at large time. The periods are in reasonable agreement with those reported by Rossby when the horizontal wavenumber is chosen so that the heat flux is the same as his.

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