Coincidence of the linear and non-linear stability bounds in a horizontal thermal convection problem

We deal with the emergence of the horizontal three-dimensional convection flow from an asymptotic mechanical equilibrium in a parallelepipedic box with rigid walls and a very small horizontal temperature gradient. The non-linear stability bound is associated with a variational problem. It is proved that this problem is equivalent to the eigenvalue problem governing the linear stability pf the asymptotic basic conduction state and so the two bounds, the linear one and the non-linear one, coincide. Finally, the eigenvalue problem is reduced to a system consisting of a polynomial equation and a trascendental equation. The numerical solution of this system yields the common stability bound. Its physical interpretation and comparison with the three-dimensional case is provided for various aspect ratios in the two-dimensional horizontal directions. (C) 1999 Elsevier Science Ltd. All rights reserved.