Global description of patterns far from onset: a case study

The Cross–Newell phase diffusion equation τ(k)ΘT=−∇·k→B(k),k→=∇Θ,|k→|=k, and its regularization describe patterns and defects far from onset in large aspect ratio systems with translational and rotational symmetry. In this paper we show how director field solutions of this equation can be used to describe features of global patterns. The ideas are illustrated in the context of a non-trivial case study of high Prandtl number convection in a large aspect ratio, shallow, elliptical container with heated sidewalls, for which we also have the results of simulation and experiment.