Unstable spectrum of a Rayleigh–Bénard system with variable viscosity

This Note studies a Rayleigh–Bénard system in an infinite layer, in the case of temperature-dependent viscosity, with rigid boundary conditions for the velocity at the bottom and free-slip at a top of the layer. It states the linearized problem in the relevant functional operator set-up and identifies, for each nonzero transverse frequency $k$ and Rayleigh number $R$ the (finite) number of modes which are unstable in time. This number is equal to the number of eigenvalues of a particular operator which are smaller than $R$.