  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.
