A numerical study of several time integration methods for solving the threedimensional Boussinesq thermal convection equations in rotating spherical shells is presented. Implicit and semi-implicit time integration techniques based on backward differentiation and extrapolation formulae are considered. The use of Krylov techniques allows the implicit treatment of the Coriolis term with low storage requirements. The codes are validated with a known benchmark, and their efficiency is studied. The results show that the use of high order methods, especially those with time step and order control, increase the efficiency of the time integration, and allows to obtain more accurate solutions.

Rotating waves (RW) bifurcating from the axisymmetric basic magnetized spherical Couette (MSC) flow are computed by means of Newton-Krylov continuation techniques for periodic orbits. In addition, their stability is analysed in the framework of Floquet theory. The inner sphere rotates whilst the outer is kept at rest and the fluid is subjected to an axial magnetic field. For a moderate Reynolds number Re = 10 3 (measuring inner rotation) the effect of increasing the magnetic field strength (measured by the Hartmann number Ha) is addressed in the range Ha ∈ (0, 80) corresponding to the working conditions of the HEDGEHOG experiment at Helmholtz-Zentrum Dresden-Rossendorf. The study reveals several regions of multistability of waves with azimuthal wave number m = 2, 3, 4, and several transitions to quasiperiodic flows, i.e modulated rotating waves (MRW). These nonlinear flows can be classified as the three different instabilities of the radial jet, the return flow and the shear-layer, as found in previous studies. These two flows are continuously linked, and part of the same branch, as the magnetic forcing is increased. Midway between the two instabilities, at a certain critical Ha, the nonaxisymmetric component of the flow is maximum. arXiv:1805.06750v2 [physics.flu-dyn]

A methodology to compute azimuthal waves, appearing in thermal convection of a pure fluid contained in a rotating spherical shell, and to study their stability is presented. It is based on continuation, Newton-Krylov, and Arnoldi methods. An application to the study of a double-Hopf bifurcation of the basic state is shown for Ekman and Prandtl numbers E = 10 −4 and σ = 0.1, respectively, radius ratios η ∈ [0.32,0.35], Rayleigh numbers R ∈ [1.8 × 10 5 ,6 × 10 5 ], and nonslip and perfectly conducting boundary conditions. The knowledge of the bifurcation diagrams, including the unstable solutions, allows one to understand the coexistence of stable thermal Rossby waves of different azimuthal wave numbers at some parameter regions, and the origin of some new intermittent solutions found, as trajectories close to heteroclinic chains. Moreover, the structure of the eigenfunctions at the secondary bifurcations explains the existence of the amplitude and shape modulated waves.