GLOBAL MACROSCOPIC DESCRIPTION OF PATTERNS FAR FROM ONSET

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 two-dimensional
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.