Boussinesq rotating thermal convection(RTC) in a spherical shell

The model equations are

$$ (\partial_t -\Delta){\bf u}= {\bf u}\times(\nabla\times{\bf u})- \frac{1}{E}\hat{\bf z}\times{\bf u} -\nabla \Pi + \frac{Ra}{E} \left(\frac{d}{r_o}\right)\Theta{\bf r}$$

$$ (\partial_t -\frac{1}{Pr}\Delta)\Theta = -{\bf u}\cdot\nabla\Theta + \mathcal{H}{\bf u}\cdot{\bf r} $$

with the parameters defined as

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

where $\nu$, $\kappa$, $\alpha$ are the kinematic viscosity, thermal conductivity, and thermal expansion coefficents; $\Omega$ is the rotation rate; $r_o$ is the outer radius of the spherical shell and $d$ is its thickness; $g_o$ is defined via the gravitational acceleration, ${\bf g} = -g_o {\bf r}/r_o$. $\mathcal{T}$ is a characteristic temperature and $\mathcal{H}$ is the effective temperature background. The spherical shell has an inner radius of $r_i$, outeer radius $r_o$ and thickness $d=r_o-r_i$.

The model equations are non-dimensionalised using $d$ as the characteristic length-scale and the viscous time-scale $\nu/d^2$. Note that in the model equation, the factor $d/r_o$ as written in the buoyancy term, involves dimensional quantities. If the outer radius $r_o$ is already non-dimensional, that factor is simply $1/r_o$.

The definition of both $\mathcal{T}$ and $\mathcal{H}$ depend on the mode of thermal driving (differential or internal heating, controlled by the parameter heating) and on the boundary conditions (fixed-temperature or fixed-flux):

• For fixed temperature boundary conditions, the background temperature is as given in Eq. 1.6 of Dormy et al., 2004 (DOI: 10.1017/S0022112003007316). Then $\mathcal{T} = \Delta T = T_i - T_o$ (where $T$ is the temperature of the resulting reference state) and:

  • $\mathcal{H}=$bg_eff for uniform internal heating (heating=0)
  • $\mathcal{H}=$ bg_eff $/r^3$ for differential heating (heating=1)
    where bg_eff is a pre-factor, function of $r_o$ and $d$.

• The fixed-flux case is not currently implemented cleanly. For fixed-flux at either one of the boundaries, the relevant thermal scale is defined by a thermal gradient $\beta$, so that the characteristic temperature is $\mathcal{T}=\beta d$. The exact definition of $\beta$ (and therefore of the Rayleigh number) depends on the mode of heating, but still:

  • $\mathcal{H}=$bg_eff for uniform internal heating (heating=0)
  • $\mathcal{H}=$ bg_eff $/r^3$ for differential heating (heating=1)
    Currently the correct bg_eff is not implemented for fixed-flux conditions, but the code can still be used to calculate the onset of convection. The resulting Rayleigh number needs, however, to be corrected to account for this discrepancy.

At present, only uniform heating is implemented in the internally heated case.

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

$$ {\bf u} = \nabla\times T {\bf r} + \nabla\times\nabla\times P {\bf r} . $$