Steady states and stability properties of a low-order barotropic system with forcing and dissipation

Abstract

A low-order barotropic system with Newtonian forcing and dissipation is investigated. The system has one component in the zonal flow characterized by a Legendre function and two components in the eddy flow having the same longitudinal wave number, but different meridional indices and described by two associated Legendre functions. Five real amplitudes characterizing the system and a system of five ordinary, non-linear differential equations describe the behaviour of the system.

The steady states of the system can be determined by a numerical solution of a 9th degree equation for a given intensity of the forcing. Section 3 contains a description of the determination of the steady states and shows that up to five steady states may exist for a given forcing. The most complicated of the steady states have energy in all components of the system and the nonlinear exchange of energy among the components is essential for maintaining the state. Other steady states are either zonal or contain energy in one component only.

A general procedure for the determination of the stability of a given steady state is outlined in Section 4 together with a determination of the stability of all steady states within a given parameter range in the forcing. It turns out that among the steady states which have energy in the wave components we find both stable and unstable configurations. It would thus appear that this result is in general agreement with the synoptic experience that some wave configurations can exist in a relatively unchanged form for a long time while others, presumably corresponding to the unstable configurations, will change rapidly. A close correspondence with real atmospheric conditions is impossible in a low-order system.

The final part of Section 4 gives the result of some numerical experiments with the simple system to illustrate that the type of initial disturbance given to a steady, unstable state will determine the asymptotic steady state in a case where two or more such states exist.