Computation of azimuthal waves and their stability in thermal convection in rotating spherical shells with application to the study of a double-Hopf bifurcation

摘要

A methodology to compute azimuthal waves, appearing in thermal convection of a pure fluid contained inna rotating spherical shell, and to study their stability is presented. It is based on continuation, Newton-Krylov,nand Arnoldi methods. An application to the study of a double-Hopf bifurcation of the basic state is shownnfor Ekman and Prandtl numbers E = 10−4 and σ = 0.1, respectively, radius ratios η ∈ [0.32,0.35], Rayleighnnumbers R ∈ [1.8 × 105,6 × 105], and nonslip and perfectly conducting boundary conditions. The knowledgenof the bifurcation diagrams, including the unstable solutions, allows one to understand the coexistence of stablenthermal Rossby waves of different azimuthal wave numbers at some parameter regions, and the origin of somennew intermittent solutions found, as trajectories close to heteroclinic chains. Moreover, the structure of theneigenfunctions at the secondary bifurcations explains the existence of the amplitude and shape modulated waves.