On forced and unforced triadic models of atmospheric flow

The lowest-order spectral model for barotropic, non-divergent flow on the sphere is studied both in the forced and unforced case. It is found that the unforced, inviscid oscillations are typically non-periodic, the periodicity of the spectra notwithstanding. Such non-periodicity is characteristic of the spectral representation on the sphere, in the sense that typical solutions of the equivalent plane model are exactly periodic. By an analytic treatment, it is also found that for some triad configurations, the limit set of the corresponding non-conservative model with axisymmetric forcing may contain a limit cycle together with the stable, zonal-flow state, while only one attractor (limit cycle or zonal flow) is implied by the equivalent plane model. Numerical solutions are reported which confirm the above analysis.