The onset of thermo-compositional convection in rotating spherical shells

Abstract

Double-diffusive convection driven by both thermal and compositional buoyancy in a rotating spherical shell can exhibit a rather large number of behaviours often distinct from that of the single diffusive system. In order to understand how the differences in thermal and compositional molecular diffusivities determine the dynamics of thermo-compositional convection we investigate numerically the linear onset of convective instability in a double-diffusive setup. We construct an alternative equivalent formulation of the non-dimensional equations where the linearised double-diffusive problem is described by an effective Rayleigh number, Ra, measuring the amplitude of the combined buoyancy driving, and a second parameter, α, measuring the mixing of the thermal and compositional contributions. This formulation is useful in that it allows for the analysis of several limiting cases and reveals dynamical similarities in the parameters space which are not obvious otherwise. We analyse the structure of the critical curves in this Ra−α space, explaining asymptotic behaviours in α, transitions between inertial and diffusive regimes, and transitions between large scale (fast drift) and small scale (slow drift) convection. We perform this analysis for a variety of diffusivities, rotation rates and shell aspect ratios showing where and when new modes of convection take place.

Regimes of thermo-compositional convection and related dynamos in rotating spherical shells

“…Systematic parameter studies of the linear onset of double-diffusive convection in rotating spherical shells were undertaken by Net et al (2012) and recently by Silva et al (2019). Following Simitev (2011), the study of Silva et al (2019) confirmed that due to distinct "double-diffusive" eigenmodes, critical curves for the onset of thermo-compositional instability are generally multi-valued and form what may be described as "pockets" of instability protruding regions of quiescence. Situations thus arise whereby increasing the thermal or the compositional driving leads to onset of instability at the entry of a pocket and a subsequent return to quiescence as driving is further increased to exit the pocket.…”

Effects of Shell Thickness on Cross-Helicity Generation in Convection-Driven Spherical Dynamos

“…The onset of thermal convection in rapidly rotating spherical shells has been extensively studied, for example, most recently as a special case of the onset of thermo-compositional convection [48]. In general, two major regimes are found at onset-columnar convection and equatorially-attached convection.…”

“…To determine accurate values for the critical parameters at onset we use our open source numerical code [50]. The code implements a Galerkin spectral projection method due to Zhang and Busse [51] to solve the linearised versions of Equations (6a) and (6c).…”

“…The method leads to a generalised eigenvalue problem for the critical Rayleigh number R c and frequency ω c of the most unstable mode of thermal convection at specified other parameter values and at specified azimuthal wave number m of the convective perturbation. Numerical extremisation and continuation problems then are tackled in order to follow the marginal stability curve in the parameter space as detailed in [48]. The critical values thus obtained are shown in Figure 3.…”

“…It is interesting to compare and validate these results against theoretical results for the onset convection in rapidly rotating systems. The asymptotic analysis of this problem has a long and distinguished history of local and global linear stability analysis [52][53][54][55][56], see also [48] for a brief overview. Converting results of Yano [57] to our dimensionless parameters, length and time scales, we obtain…”