A theory is developed to describe the weakly nonlinear dynamics which applies in the simultaneous presence of several, long, baroclinic waves. The geometry is flat (i.e. β = 0) and dissipation is modelled by Ekman friction in the context of the quasi-geostrophic two-layer model. Three main problems are discussed.
For free, unstable waves it is shown that the wave which is realized in finite amplitude is not the linearly most unstable wave. Rather a longer wave, capable of achieving the single largest steady amplitude, is favoured in the competition for the potential energy of the basic state. This result is shown necessary if the end state is steady and numerous numerical calculations indicate the pre-eminence of the same wave if the final state is vacillatory. The notion of conjugate waves, capable of identical final amplitude, is also discussed.If the free waves are subject to time-varying supercriticality so that intervals of stability ensue, the response is asymmetric over the period of the forcing. Sufficiently rapid ‘seasonal’ forcing leads to long-term aperiodic response.If each wave in the spectrum is directly forced a wave hysteresis phenomenon occurs. Sudden jumps in the wave amplitude at critical values of the forcing are intrinsic to the wave response. Again, sufficiently rapid wave forcing produces an aperiodic response.The forced wave problem exhibits multiple equilibria. Each solution branch corresponds to a different dominant wave. The determination of the realized branch depends on the relative stability criteria developed for the free waves.
Weakly nonlinear dynamics in reaction–diffusion systems with Lévy flights
