DOUBLE HOPF BIFURCATION AND QUASI-PERIODIC FLOW IN A MODEL FOR BAROCLINIC INSTABILITY.

Abstract

The interaction between pairs of dispersive waves is studied with different zonal wavenumbers in the beta -plane model for baroclinic instability. We find that degenerate (codimension 2) double Hopf bifurcations occur at isolated points in the (F**2/r**2,F** minus **1) parameter space. Here F is Froude number and r is an effective viscosity. Using perturbation methods, we drive the phase and amplitude equations for the interacting modes, which have distinct frequencies. These equations take the standard normal form for a nonresonant double Hopf bifurcation and can readily be analyzed using phase plane methods. The authors compute numerical values of the coefficients that determine stability in several cases and find that both pure and mixed (quasi-periodic) modes exist and that these modes can be stable or unstable, depending upon the zonal wavenumbers and other parameters.