  The aim of this paper is to investigate the onset of penetrative convection
in a Darcy-Brinkmann porous medium under the hypothesis of local therma
non-equilibrium. For the problem at stake, the strong form of the principle of
exchange of stabilities has been proved, i.e. convective motions can occur only
through a secondary stationary motion. We perform linear and nonlinear
stability analyses of the basic state motion, with particular regard to the
behaviour of the stability thresholds with respect to the relevant physical
parameters characterizing the model. The Chebyshev-$\tau$ method and the
shooting method are employed and implemented to solve the differential
eigenvalue problems arising from linear and nonlinear analyses to determine
critical Rayleigh numbers. Numerical simulations prove the stabilising effect
of upper bounding plane temperature, Darcy's number and the interaction
coefficient characterising the local thermal non-equilibrium regime.
