On The Spectrum Of Linearized Operator Of Rotating Bènard Problem When (?) And (?) Act In Different Directions

Consider a horizontal layer of fluid in which an adverse temperature gradient is maintained by heating the underside. Because of thermal expansion, the fluid at the bottom expands as it becomes hotter, but, for low temperature gradients, viscosity prevents the onset of convective motions, and heat is transported through the fluid only by conduction. When the temperature gradient reaches a critical value, the buoyancy overcomes viscosity, and the fluid gives rise to a regular pattern of convective motion. This phenomenon is now called Benard convection which is a standard model used to study the thermal convection.For more general situation, the importance of Benard convection with rotation in atmospheric and oceanic flow has led to a significant theoretical and experimental interest in this problem. The stability of Benard convection with rotation about a vertical axis has been studied by many specialists in past years. But it doesn't get enough attention whenΩand g act in different directions.In this paper we apply numerical method to the linearized spectral problem of rotating Benard problem whenΩand g act in different directions with stress-free and rigid boundary conditions. Letξ0 be the minimum value of the real parts of the eigenvaluesσin the spectrum problem (ξ0=min{Reσ}). In this paperσis the decay rate rather than the growth rate in physics, soξ0 indicates the smallest lower bound of the decay rate of the perturbations.The dependence ofξ0 and the critical Rayleigh number RC on the rotating angleβbetweenΩand g is given. It's shown that bothξ0 and RC decrease with the growth of the angleβ. That means the convection becomes more and more unstable with the growth ofβ. Moreover,ξ0 is dependent on Prandtl number.