An explicit example of Hopf bifurcation in fluid mechanics

Abstract

It is observed that a complete and explicit example of Hopf bifurcation appears not to be known in fluid mechanics. Such an example is presented for the rotating Bénard problem with free boundary conditions on the upper and lower faces, and horizontally periodic solutions. Normal modes are found for the linearization, and the Veronis computation of the wave numbers is modified to take into account the imposed horizontal periodicity. An invariant subspace of the phase space is found in which the hypotheses of the Joseph-Sattinger theorem are verified, thus demonstrating the Hopf bifurcation. The criticality calculations are carried through to demonstrate rigorously, that the bifurcation is subcritical for certain cases, and to demonstrate numerically that it is subcritical for all the cases in the paper.