Effect of Lateral Boundaries on Large-scale Mode: Linear Stability of Square Cell Flows in Rectangular Regions

Abstract

Linear stability of the square cell flow represented by the stream function: Ψ = sin x sin y is investigated numerically in various bounded region D = [0, M π] ×[0, N π]. The disturbances are limited to two-dimensional ones and a perfect slip condition is assumed to be applied. Special attention is paid to clarify how the critical long-wave mode (or large-scale mode) of the flow in unbounded region is modified by lateral boundaries. It is shown that the critical modes are classified into three cases according to the configuration ( M , N ): (i) M =1, (ii) ( M , N )=(2, odd numbers), (3, 4) and (3, 5), and (iii) the others. The last one is the most typical cases and is related to the long-wave mode in the unbounded region. The structure of the mode is one stationary vortex with the system size, which we call global rotation, for M ∼ N while it is a series of stationary counter-rotating vortices for M ≪ N . In case (ii) the critical modes are oscillatory though they are related to case (iii). In case (i) (l...