Non-linear, low order interactions

A study of the advection equation with Newtonian forcing and dissipation is presented. Only a few components in a spectral expansion of the simple advection equation are retained, and no rotation is included. Without forcing and dissipation the equation is thus only energy conserving.

A two component system is analysed in detail with respect to the number of steady states, their stability and the non-linear behaviour around the steady states. Depending on the forcing we obtain one or three steady states and their stability properties are forcing dependent. For some forcing values an unstable limit cycle develops around one of the steady states.

For a three component system the properties mentioned above are investigated for the conservative case and for the forced and dissipative case. Forcing on the largest scale, only, gives rise to one stable steady state. A more complicated behaviour is obtained when forcing is introduced on the smaller scales. Stable limit cycles may develop, as well as “catastrophes” i.e. a sudden change of the number of steady states for only a small parameter variation.

Generalizations from the behaviour of these simple systems to that of the atmosphere are difficult if not impossible. It is nevertheless believed that an analysis such as the one presented here may increase our understanding of the behaviour of multi-component systems.