Pattern selection in rotating convection with experimental boundary conditions.

present the results of weakly nonlinear calculations that enable us to compute accurately both the critical rotation rate beyond which rolls are Kuppers-Lortz unstable as well as the angle with which the instability sets in. These quantities are a sensitive function of the Prandtl number, and here the results differ considerably from those with more idealized boundary conditions. The remainder of the paper is concerned with weakly nonlinear oscillatory convection. The calculations are restricted to two spatial dimensions so that the nonlinear problem concerns only traveling and standing waves. We determine the relative stability between these two patterns and show that either may be stable, depending on the system parameters. In this discussion we are careful to consider the impact of distant sidewalls in preventing the appearance of the mean flows that are associated with traveling patterns in unbounded systems. As in the Kiippers-Lortz calculation we consider stability only with respect to perturbations of the same wave number as the basic pattern. Although this procedure excludes a large class of potentially important perturbations, prior experience suggests that the selected perturbations are the most dangerous. In addition this restriction enables us to rely on results obtained from equivariant bifurcation theory. This technique [6] allows us to classify the possible weakly nonlinear and spatially periodic states and establish their relative stability. In addition it identifies precisely the computations that have to be carried out on the partial diiferential equations in order to discriminate among the possible scenarios. The paper is organized as follows. In the following section we summarize the relevant bifurcation theory results. In Sec. III we formulate the hydrodynamical equations and describe the technique we use to solve them. In Sec. IV we solve the linear stability problem for the conduction solution and identify regions in parameter space where oscillatory and steady-state convection takes place. In Sec. V we restrict the region of stable smallamplitude rolls by computing the location of the onset of