Boundary Conditions

Velocity

\(\vec{u}\) is calculated in the dynamo code using a poloidal toroidal decomposition:

\[ \vec{u} = \mathbf{\nabla} \times(\mathbb{T}\vec{r}) + \mathbf{\nabla} \times\mathbf{\nabla} \times(\mathbb{P}\vec{r}). \]

No slip BCs:

\[ \mathbb{T}=0, \quad \mathbb{P}=0, \quad \partial_r\mathbb{P}=0. \]

Stress-free

\[ \left(\partial_{r} - \,\frac1{r}\right)\mathbb{T} = 0, \quad \mathbb{P}=0, \quad \partial_{rr} \mathbb{P}=0. \]

The condition \(P=0\) appears for both and corresponds to the no-penetration condition \(u_r=0\).

Codensity/Temperature

Called codensity in the code, can act as a codensity term for single component convection, or temperature if separate temperature and compositional fields are required.

Fixed Temperature

Solving the conduction problem

\[ \frac{\partial T}{\partial r} = \frac{A}{r^2}, \]

gives

\[ T = \frac{-A}{r} + B. \]

we set \(T|_{r_i}=1\) and \(T|_{r_o}=0\), giving a background conduction state of

\[ \frac{\partial{T}}{{\partial r}} = \frac{1}{r^2}\frac{r_ir_o}{r_i-r_o.} \]

Fixed Flux

\[ {\frac{\partial T}{\partial r}\bigg|}_{r=r_i} = \text{d_qi} \]

\[ {\frac{\partial T}{\partial r}\bigg|}_{r=r_o} = \text{d_qo} \]

Composition

Solving the conduction problem

\[ \frac{\partial \xi}{\partial r} = \frac{A}{r^2}, \]

gives

\[ \xi = \frac{-A}{r} + B. \]

we set \(\xi|_{r_i}=1\) and \(\xi|_{r_o}=0\), giving a background conduction state of

\[ \frac{\partial{\xi}}{{\partial r}} = \frac{1}{r^2}\frac{r_ir_o}{r_i-r_o.} \]

Fixed Flux

\[ {\frac{\partial \xi}{\partial r}\bigg|}_{r=r_i} = \text{d_qi_comp} \]

\[ {\frac{\partial \xi}{\partial r}\bigg\vert}_{r=r_o} = \text{d_qo_comp} \]

Magnetic Field

For the decomposition

\[ \vec{B} = \mathbf{\nabla} \times(\mathcal{T}\vec{r}) + \mathbf{\nabla} \times\mathbf{\nabla} \times(\mathcal{P}\vec{r}). \]

Insulating BCs

\[ {\mathcal T}=0, \quad \left(\partial_{r}-\,\frac{l}{r}\right){\mathcal P}=0 \quad \text{ on }~~ r_i, \]

\[ {\mathcal T}=0, \quad \left(\partial_{r}+\,\frac{l+1}{r}\right){\mathcal P}=0 \quad \text{ on } r_o. \]

Conducting inner Core

\[ \left[{\mathcal T}\right]=0, \quad \left[{\mathcal P}\right]=0, \quad \left[\partial_{r}{\mathcal P}\right]=0, \quad \nabla^2_1 \left[\partial_{r}{\mathcal T}\right] = \frac1{r}\,\hat{\vec{r}}\cdot\mathbf{\nabla} \times \left(B_r\left[\vec{u}\right]\right) \quad \text{ on } r_i \]

where \([\,\cdot\,]\) is the jump across the interface. The last condition is valid for an inner core of the same magnetic diffusivity, and for no-slip boundaries simplifies to \([\partial_{r}{\mathcal T}]=0\).