A Lorenz-like model for the horizontal convection flow

In this work we study a finite dynamical system for the description of the bifurcation pattern of the convection flow of a fluid between two parallel horizontal planes which undergoes a {\em horizontal} gradient of temperature ({\em horizontal} convection flow). Although in the two-dimensional case developed here,literature reports as well a long list of analytical and numerical solutions to this problem, the peculiar aim of this work makes it worthwhile. Actually we develop the route that Saltzman (1962) \cite{Sal62} and Lorenz (1963) \cite{Lor63} proposed for the {\em vertical} convection flow that started successfully the approach to finite dynamical systems. We obtain steady-to-steady and steady-to-periodic bifurcations in qualitative agreement with already published results. At first we adopt the non-dimensional scheme used by Saltzman and Lorenz; as it admits also physically meaningless solutions, we introduce a different set of reference quantities so overcoming this drawback.