Hidden symmetries and pattern formation in Lapwood convection

摘要

We study Lapwood convection (convection of a fluid in a porous medium) on a two-dimensional rectangular domain. The linearized eigenmodes are symmetric pxq cellular patterns, which we call (p, q) modes. Numerical calculations of the branching structure near mode interaction points have derived bifurcation diagrams for the (3, 1)/(1, 1) and (3, l)/(2, 2) mode interactions which are non-generic, even when the rectangular symmetry of the domain is taken into account. This has raised questions about the accuracy of the numerical method used, a finite-element Galerkin approximation implemented using Harwell's ENTWIFE code. We show that this apparent lack of genericity is partly a consequence of 'hidden' translational symmetries, which arise when the problem is extended to one with periodic boundary conditions. This extension procedure has become standard for partial differential equations (PDEs) with Neumann or Dirichlet boundary conditions, and it reveals restrictions on the Liapunov-Schmidt reduced bifurcation equations and the resulting singularity-theoretic normal forms. Its application to Lapwood convection is unusual in that the PDE involves a mixture of both Neumann and Dirichlet boundary conditions. Specifically, on the vertical sidewalls the stream function satisfies Dirichlet boundary conditions (is zero), but the temperature satisfies Neumann (no-flux) boundary conditions. Nevertheless, we show that for abstract group-theoretical reasons the same symmetry constraints that occur for purely Neumann boundary conditions are imposed on the Liapunov-Schmidt reduced bifurcation equations, and therefore the same list of normal forms is valid. The hidden symmetries force certain terms in the reduced bifurcation equations to be zero and change the generic branching geometry. With the aid of MACSYMA, we determine a small number of low-order coefficients of the reduced bifurcation equations which are needed to find the correct normal form. We show that in some cases the normal form is more degenerate than might be anticipated, but that when these degeneracies are taken into account the resulting branching geometry reproduces that found in the earlier numerical approach. In particular, we obtain an analytic vindication of the numerical method.