ON FINITE-AMPLITUDE PATTERNS OF CONVECTION NEAR A LATERAL BOUNDARY

A model equation is studied as a means of simulating patterns of convection in a fluid uniformly heated from below. A roll pattern parallel to a lateral boundary is unstable to cross-rolls in the vicinity of the boundary. Normal modes of instability are analysed and a stable finite-amplitude structure which consists of a combination of rolls parallel and perpendicular to the boundary is proposed. The theory incorporates the presence of a small boundary imperfection. In the limit of vanishing imperfection the extent of the perpendicular cross-rolls increases and the solution of a coupled pair of amplitude equations corresponds to a continual logarithmically slow drift of the roll pattern with time.