On The Spectrum Of Linearized Operator Of Rotating Bénard Problem

A rotating fluid layer heated from below is a standard model used to study the thermal convection.This is so called the rotating Benard problem.In this paper,we consider an infinite horizontal layer R²Ã—(0,d)in a Cartesian reference frame Oxyz with unit vector k=(0,0,1)^T for the z-axis being against gravity.The layer,rotating about z-axis,is filled with an incompressible fluid and heated from below.Because of the inhibiting effect of viscosity and gravity of fluid,the fluid remains at rest(i.e.basic state)when the temperature difference between the bottom and top of the layer is small. However,on account of buoyancy,the basic state will become unstable and the thermal convection sets in when the temperature difference exceeds a critical value.We start this paper with Boussinesq-equations satisfied by the aforesaid problem,then the linearized spectral problem of rotating Benard problem is considered for stress-free and rigid boundary conditions.Letξ₀ be the minimum value of the real parts of the eigenvaluesσin the spectrum problem(ξ₀=min{Reσ}).Here,σis the decay rate rather than the growth rate in Physics,soξ₀ indicates the smallest lower bound of the decay rate of the perturbations.In this paper the dependence ofξ₀ on the rotation rateΩand the Rayleigh number R is given for some parameters.It follows thatξ₀ increases with the growth of the rotation rate and the limit ofξ₀ exists as the rotation rate approaches +∞.The values of the limit depend on Prandtl number and the boundary conditions.Otherwise,ξ₀ decreases with the growth of the Rayleigh number.