Previous: The wave forcing Next: Conclusion Up: Ext. Abst.

A minimal model

The minimal model captures the basic physics of the vacillations and llustrates the basic principles behind the downward propagation: the competition between wave-driving and the radiative damping. It is in agreement with the conclusion from Fig. 5 that the vertical wave flux dominates. Stripping the transformed Eulerian-mean equation to its back-bone results in:

The zonal acceleration is given by a relaxation towards u_r and  by the wave forcing . In addition we have put in a diffusion term partly to account for the non-localness of the original equation.

The Eliassen-Palm flux represents the propagation of eddy activity. In the optical limit the vertical propagation of  F is given by

where F₀ is the wave forcing at the lower boundary, z=0, taken as the tropopause. The resistance g depends on the mean flow  u and diverge when u gets close to the group velocity c of the waves. The exact functional form depends on the wave type. Here we take

The system resembles  the model of the quasi-biennial oscillation studied by Plumb (1977), which is driven by two waves at the lower boundary and which does not include the relaxation. In the present work only one wave is needed. As the Holton-Mass model  (Holton and Mass, 1976) this minimal model undergoes a Hopf-bifurcation from a quiescent state to an oscillating  state when the forcing at the lower boundary is increased above a threshold value. The Hopf-bifurcation has also been identified in the GCM (Christiansen, 1999).