The Gravitational Instability of a Diffusive Boundary Layer; Towards a Theoretical Minimum for Time of Onset of Convecti

In this paper we extend previous work in on the linearized analysis of gravitational instability of a diffusive boundary layer in a semi-infinite anisotropic homogenous porous medium. We express the time derivative of the square of the standard L^2-norm of a given perturbation as a time dependent quadratic form on an appropriate Hilbert space . Numerical analysis of the spectra of these quadratic forms give rise to results qualitatively similar to previous results in the litterature. We demonstrate that after the time of instability only perturbations having a non-zero projection onto a one-dimensional subspace of are unstable. We also find that the space of neutrally stable perturbations before onset of instability form a large subspace of the space of possible perturbations, where numerical analysis strongly indicate that this subspace is infinite dimensional. Error estimates for a certain part of the numerical analysis are not yet rigorous. In particular, estimating the spectrum of unbounded linear operators using finite matrix approximations still lacks a theoretical basis. However, the largest eigenvalues of larger and larger matrices approximating the operator converge quickly to well defined values, and it is conjectured that the given critical values are the correct ones for the problem at hand.