Boussinesq thermal dynamo convection in a spherical shell

The model equations are

$$ (\partial_t - Pm \nabla^2){\bf u} = {\bf u}\times(\nabla\times{\bf u}) -\nabla \Pi +\frac{Pm^2}{E}\widetilde{Ra} \Theta \frac{\bf r}{r_o} - \frac{Pm}{E}\hat{\bf z}\times{\bf u} + \frac{Pm}{E}(\nabla\times{\bf B})\times {\bf B}, $$

$$ (\partial_t - \frac{Pm}{Pr}\nabla^2)\Theta =- ({\bf u}\cdot \nabla) \Theta -u_r \frac{d T_c}{dr}, $$

$$ (\partial_t - \nabla^2) {\bf B} = \nabla\times({\bf u} \times {\bf B }.) $$

with the parameters defined as

$$ Pr = \frac{\nu}{\kappa};\quad Pr = \frac{\nu}{\eta};\quad E=\frac{\nu}{2\Omega d^2};\quad Ra=\frac{ \alpha g_o \mathcal{T} d}{2\nu\Omega} $$

where $\mathcal{T}$ is the temperature nondimensionalisation value and depends on the temperature boundary conditions and heating mode.

The system is solved using a Toroidal/Poloidal decomposition of the velocity

$$ {\bf u} = \nabla\times T_u {\bf r} + \nabla\times\nabla\times P_u {\bf r} . $$

and the magnetic field:

$$ {\bf B} = \nabla\times T_B {\bf r} + \nabla\times\nabla\times P_B {\bf r} . $$