Appendix A: The governing equations Convection in a rotating cylindrical annulus with fixed heat flux boundaries
$\displaystyle\Dinv{P}\DP{}{t} \Dlapla_2 \psi + \Dinv{P}J(\psi, \Dlapla_2 \psi) - \eta \DP{\psi}{x} = R \DP{\theta}{x} + \Dlapla_2 \Dlapla_2 \psi,$ (1)
$\displaystyle\DP{\theta}{t} + J(\psi, \theta) - \DP{\psi}{x} = \Dlapla_2 \theta.$ (2)

The governing equations are non-dimensionalized using the thickness of layer d as length scale, thermal diffusion time $d^2/\kappa$ as time scale, $\kappa/d$ as velocity scale, and $\Gamma d$ as temperature scale. $\kappa$ is the thermal diffusivity and $\Gamma$ is the absolute value of temperature gradient of the basic state determined by the heat flux at the boundaries. $\psi, \theta$ are stream function and temperature disturbance, respectively. ${\displaystyle \Dlapla_2 \equiv \DP[2]{}{x}+\DP[2]{}{z}}$ is the two-dimensional Laplacian operator, and ${\displaystyle J(f,g) = \DP{f}{x}\DP{g}{z} - \DP{f}{z}\DP{g}{x}}$ is the Jacobian. Non-dimensional numbers appearing in the system are the Rayleigh number ${\displaystyle R = \frac{\alpha g d^4\Gamma}{\kappa\nu}}$, the Prandtl number ${\displaystyle P = \frac{\nu}{\kappa} }$, and the parameter of topographic b effect ${\displaystyle \eta = \frac{4 \eta_0 d}{l E}}$, where $\alpha$ is the thermal expansion coefficient, g is the gravitational acceleration, $\nu$ is the kinematic viscosity, $\eta_0$ is inclination of the boundaries, l is the size in the direction of the rotation axis, and E is the Ekman number ${\displaystyle E = \frac{\nu}{d^2\Omega} }$.

Appendix A: The governing equations Convection in a rotating cylindrical annulus with fixed heat flux boundaries