Improving Spectral Models By Unfolding Their Singularities

Abstract

Maximally truncated spectral models have been used recently by fluid and atmospheric dynamicists to study nonlinear behavior of the governing partial differential system. However, too few external control parameters may be available in the truncated model to describe adequately the steady states near singular parameter values at which two or more stationary solutions meet. These missing parameters correspond in many cases to small but significant physical effects whose inclusion may be critically important for the model results to be realistic. We apply to truncated spectral models a recently developed contact catastrophe method that allows determination of the crucial physical effects that govern the steady states of a fluid system. Spectral systems of three different fluid flow models of interest in atmospheric science are considered. Two parameters are necessary for modeling Rayleigh-Bénard convection. One represents the magnitude of the horizontal component, the other the magnitude of the vertical component of the externally imposed heating. Four parameters are required for modeling axisymmetric flow in either a rotating annulus or the atmosphere if the Prandtl number σ and the aspect ratio a are related by σa < 1. These are the horizontal and vertical components of the external heating, the Coriolis parameter, and either the inclination angle of the vessel (annulus) or the Newtonian heating rate (atmosphere). Four parameters are essential for modeling quasi-geostrophic flow in a channel. They are the three Fourier coefficients of the Newtonian heating rate and the amplitude of a superimposed time-independent zonal current.