Low-order spectral models of the atmospheric circulation: A survey

A quasi-geostrophic potential vorticity equation is derived from the Navier-Stokes equations for atmospheric motions. It describes the evolution of a quasi-horizontally flow on time scales of a few days and more. The associated boundary-value problem is analyzed by projection of the equation onto orthonormal eigenfunctions (modes) of a Sturm-Liouville operator. The result is a spectral model, consisting of an infinite number of nonlinear ordinary differential equations for the evolution of the mode amplitudes. Low-order spectral models, in which only a few modes are resolved, appear to have properties which agree with observations of the atmospheric circulation. However, little justification is available for truncating the spectral expansion at low resolution numbers. It is argued that stochastic forcing terms should be added to the equations, but it is not a priori clear how they should be specified. A derivation is presented of a specific low-order spectral model of the quasi-geostrophic potential vorticity equation. Some of its subsystems are analyzed for their physical and mathematical properties. It appears that topography can act as a triggering mechanism to generate multiple equilibria. The corresponding flow patterns resemble preference states of the atmospheric circulation. The systems can vacillate between three characteristic regimes with transitions provided either by external or internal mechanisms. A discussion is presented on the validity of stochastically forced spectral models and deterministic chaotic models for the atmospheric circulation.