ABSTRACT


Consider a shallow layer of high Prandtl number fluid in a large aspect ratio elliptical container with heated sidewalls. The Rayleigh number is in the range for which rolls/stripes are the stable local planform. The heated sidewalls mean that the rolls next to the boundary will be parallel to it. The challenge is to fill in the rest of the pattern. This problem is one of the simpler examples of natural patterns, namely patterns in translationally and rotationally invariant twodimensional systems with preferred wavelength but orientational degeneracy. The resulting textures are complicated, consisting of a mosaic of patches of rolls with almost constant orientation mediated by line and point defects. The goal of theory is to describe such patterns and their defects from a macroscopic viewpoint. Using the example of convection in an elliptical container, we will see how the theory gives a fairly accurate prediction of the stationary patterns which are realized and how it makes contact with and extends the class of minimization problems associated with harmonic maps. Return to Applied Math Colloquium home page 