Double Hopf bifurcations in the differentially heated rotating annulus

We study a mathematical model of the differentially heated rotating fluid annulus experiment. In particular, we analyze the double Hopf bifurcations that occur along the transition between axisymmetric steady solutions and nonaxisymmetric rotating waves. The model uses the Navier-Stokes equations in the Boussinesq approximation. At the bifurcation points, center manifold reduction and normal form theory are used to deduce the local behavior of the full system of partial differential equations from a low-dimensional system of ordinary differential equations. It is not possible to compute the relevant eigenvalues and eigenfunctions analytically. Therefore, the linear part of the equations is discretized, and the eigenvalues and eigenfunctions are approximated from the resulting matrix eigenvalue problem. However, the projection onto the center manifold and reduction to normal form can be done analytically. Thus, a combination of analytical and numerical methods is used to obtain numerical approximations of the normal form coefficients, from which the dynamics are deduced. The results indicate that, close to the transition, there are regions in parameter space where there are multiple stable waves. Hysteresis of these waves is predicted. The validity of the results is shown by their consistency with experimental observations.