Ferran Garcia

On the onset of low-Prandtl-number convection in rotating spherical shells : non-slip boundary conditions

Accurate numerical computations of the onset of thermal convection in wide rotating spherical shells are presented. Low-Prandtl-number (σ ) fluids, and non-slip boundary conditions are considered. It is shown that at small Ekman numbers (E), and very low σ values, the well-known equatorially trapped patterns of convection are superseded by multicellular outer-equatorially-attached modes. As a result, the convection spreads to higher latitudes affecting the body of the fluid, and increasing the internal viscous dissipation. Then, from E< 10 −5 , the critical Rayleigh number (Rc) fulfils a power-law dependence Rc ∼ E −4/3 , as happens for moderate and high Prandtl numbers. However, the critical precession frequency (|ωc|) and the critical azimuthal wavenumber (mc) increase discontinuously, jumping when there is a change of the radial and latitudinal structure of the preferred eigenfunction. In addition, the transition between spiralling columnar (SC), and outer-equatorially-attached (OEA) modes in the (σ , E)-space is studied. The evolution of the instability mechanisms with the parameters prevents multicellular modes being selected from σ 0.023. As a result, and in agreement with other authors, the spiralling columnar patterns of convection are already preferred at the Prandtl number of the liquid metals. It is also found that, out of the rapidly rotating limit, the prograde antisymmetric (with respect to the equator) modes of small mc can be preferred at the onset of the primary instability.

A comparison of high-order time integrators for thermal convection in rotating spherical shells

A numerical study of several time integration methods for solving the three-dimensional 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.

Exponential versus IMEX high-order time integrators for thermal convection in rotating spherical shells

We assess the accuracy and efficiency of several exponential time integration methods coupled to a spectral discretization of the three-dimensional Boussinesq thermal convection equations in rotating spherical shells. Exponential methods are compared to implicit-explicit (IMEX) multi-step methods already studied previously in [1]. The results of a wide range of numerical simulations highlight the superior accuracy of exponential methods for a given time step, especially when employed with large time steps and at low Ekman number. However, presently available implementations of exponential methods appear to be in general computationally more expensive than those of IMEX methods and further research is needed to reduce their computational cost per time step. A physically justified extrapolation argument suggests that some exponential methods could be the most efficient option for integrating flows near Earths outer core conditions.

Continuation and stability of convective modulated rotating waves in spherical shells

Modulated rotating waves (MRW), bifurcated from the thermal-Rossby waves that arise at the onset of convection of a fluid contained in a rotating spherical shell, and their stability, are studied. For this purpose, Newton-Krylov continuation techniques are applied. Nonslip boundary conditions, an Ekman number E=10^{-4}, and a low Prandtl number fluid Pr=0.1 in a moderately thick shell of radius ratio η=0.35, differentially heated, are considered. The MRW are obtained as periodic orbits by rewriting the equations of motion in the rotating frame of reference where the rotating waves become steady states. Newton-Krylov continuation allows us to obtain unstable MRW that cannot be found by using only time integrations, and identify regions of multistability. For instance, unstable MRW without any azimuthal symmetry have been computed. It is shown how they become stable in a small Rayleigh-number interval, in which two branches of traveling waves are also stable. The study of the stability of the MRW helps to locate and classify the large sequence of bifurcations, which takes place in the range analyzed. In particular, tertiary Hopf bifurcations giving rise to three-frequency stable solutions are accurately determined.

A Comparison of High-Order Time Integrators for Highly Supercritical Thermal Convection in Rotating Spherical Shells

The efficiency of implicit and semi-implicit time integration codes based on backward differentiation and extrapolation formulas for the solution of the three-dimensional Boussinesq thermal convection equations in rotating spherical shells was studied in Garcia et al. (J Comput Phys 229:7997–8010, 2010) at weakly supercritical Rayleigh numbers R, moderate (10−3) and low (10−4) Ekman numbers, E, and Prandtl number σ = 1. The results presented here extend the previous study and focus on the effect of σ and R by analyzing the efficiency of the methods for obtaining solutions at \(E = 1{0}^{-4}\), σ = 0. 1 and low and high supercritical R. In the first case (quasiperiodic solutions) the decrease of one order of magnitude does not change the results significantly. In the second case (spatio-temporal chaotic solutions) the differences in the behavior of the semi-implicit codes due to the different treatment of the Coriolis term disappear because the integration is dominated by the nonlinear terms. As in Garcia et al. (J Comput Phys 229:7997–8010, 2010), high order methods, either with or without time step and order control, increase the efficiency of the time integrators and allow to obtain more accurate solutions.

Numerical study of the onset of thermosolutal convection in rotating spherical shells

The influence of an externally enforced compositional gradient on the onset of convection of a mixture of two components in a rotating fluid spherical shell is studied for Ekman numbers E = 10−3 and E = 10−6, Prandtl numbers σ = 0.1, 0.001, Lewis numbers τ = 0.01, 0.1, 0.8, and radius ratio η = 0.35. The Boussinesq approximation of the governing equations is derived by taking the denser component of the mixture for the equation of the concentration. Differential and internal heating, an external compositional gradient, and the Soret and Dufour effects are included in the model. By neglecting these two last effects, and by considering only differential heating, it is found that the critical thermal Rayleigh number Rec depends strongly on the direction of the compositional gradient. The results are compared with those obtained previously for pure fluids of the same σ. The influence of the mixture becomes significant when the compositional Rayleigh number Rc is at least of the same order of magnitude as the ...

Equatorial symmetry breaking and the loss of dipolarity in rapidly rotating dynamos

Abstract Numerical studies of convection driven dynamos in rotating spherical shells exhibit a transition from steady dipolar to reversing multipolar dynamos as the forcing is increased. The dipolar-multipolar transition has so far been characterized using purely hydrodynamic parameters (Christensen and Aubert, Geophys. J. Int. 2006, 166, 97–114, Soderlund et al., Earth Planet. Sci. Lett. 2012, 333–334, 9–20, Oruba and Dormy, Geophys. Res. Lett. 2014, 41, 7115–7120). Motivated by these earlier descriptions, we investigate the hydrodynamic transitions occurring at the critical parameters. We show that the loss of dipolarity in dynamos is associated with a purely hydrodynamic transition, characterized by a breaking of the flow equatorial symmetry. Contrary to earlier expectations, we show by varying the Prandtl number that the transition is not necessarily associated with a degradation of the flow helicity.

Oscillatory convection in rotating spherical shells: low Prandtl number and non-slip boundary conditions

A five-degree model, which reproduces faithfully the sequence of bifurcations and the type of solutions found through numerical simulations of the three-dimensional Boussinesq thermal convection equations in rotating spherical shells with fixed azimuthal symmetry, is derived. A low Prandtl number fluid of

Antisymmetric polar modes of thermal convection in rotating spherical fluid shells at high Taylor numbers

\sigma=0.1

Critical torsional modes of convection in rotating fluid spheres at high Taylor numbers

subject to radial gravity, filling a shell of radius ratio