# The onset of thermo-compositional convection in rotating spherical shells

## TL;DR

This paper investigates the linear onset of thermo-compositional convection in rotating spherical shells, revealing how differences in thermal and compositional diffusivities influence the transition between various convective regimes.

## Contribution

It introduces an effective Rayleigh number and a mixing parameter to analyze the linear stability of double-diffusive convection, providing new insights into the dynamics across different regimes.

## Key findings

- Identification of critical curves in the Ra-α space
- Analysis of asymptotic behaviors and regime transitions
- Mapping of convection modes across diffusivities and rotation rates

## 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, $\text{Ra}$, measuring the amplitude of the combined buoyancy driving, and a second parameter, $\alpha$, 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 $\text{Ra}-\alpha$ space, explaining asymptotic behaviours in $\alpha$, 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.