Control Parameters

The dynamo code uses the following non-dimensional parameter definitions:

Ekman Number

$$ \text{Ek} = \frac{\nu}{\Omega D^2}, $$ where $\nu$ is kinematic viscosity, $\Omega$ is the rate of rotation, and $D$ is the shell thickness.

Prandtl Number

$$ \text{Pr} = \frac{\nu}{\kappa}, $$ where $\nu$ is kinematic viscosity, and $\kappa$ is thermal diffusivity.

Set using d_Pr.

Schmidt Number (Compositional Prandtl)

$$ Sc = \frac{\nu}{\kappa_\xi} $$ where $\nu$ is kinematic viscosity, and $\kappa_\xi$ is the compositional diffusivity.

Set using d_Sc.

Magnetic Prandtl Number

$$ \text{Pm} = \frac{\nu}{\eta}, $$ where $\nu$ is kinematic viscosity, and $\eta$ is magnetic diffusivity.

Set using d_Pm.

Thermal Rayleigh Number

Determines the amount of thermal driving in the momentum equation. Set using d_Ra.

Fixed Temperature

\[ \text{Ra}_T = \frac{\alpha g_0 \Delta T D^4}{r_o\kappa \nu }, \]

where $\alpha$ is the coefficient of thermal expansion, $g_0$ is acceleration due to gravity, $\Delta T$ is the temperature difference between inner and outer boundaries, $D$ is the shell thickness, $r_o$ is the dimensional outer radius, $\kappa$ is the thermal diffusivity, and $\nu$ is the kinematic viscosity.

Fixed Flux

\[ \text{Ra}_T = \frac{\alpha g_0 \beta D^3}{r_o\kappa \nu }, \]

where $\alpha$ is the coefficient of thermal expansion, $g_0$ is acceleration due to gravity, $\beta$ is a temperature scale divided by the shell thickness, $D$ is the shell thickness, $r_o$ is the dimensional outer radius, $\kappa$ is the thermal diffusivity, and $\nu$ is the kinematic viscosity.

$\beta$ is used in the fixed flux case as $\Delta T$ can not be known A priori, it is derived from solving the conduction problem in the sphere.

Compositional Rayleigh Number

Set using d_Ra_comp.

Fixed Composition

\[ \text{Ra}_\xi = \frac{\alpha_\xi g_0 \Delta \xi D^4}{r_o\kappa_\xi \nu }, \]

where $\alpha_\xi$ is the coefficient of chemical expansion, $g_0$ is acceleration due to gravity, $\Delta \xi$ is the compositional difference between inner and outer boundaries, $D$ is the shell thickness, $r_o$ is the dimensional outer radius, $\kappa_\xi$ is the compositional diffusivity, and $\nu$ is the kinematic viscosity.

Fixed flux

\[ \text{Ra}_\xi = \frac{\alpha_\xi g_0 \beta_\xi D^3}{r_o\kappa_\xi \nu }, \]

where $\alpha_\xi$ is the coefficient of chemical expansion, $g_0$ is acceleration due to gravity, $\beta_\xi$ is a compositional scale divided by the shell thickness, $D$ is the shell thickness, $r_o$ is the dimensional outer radius, $\kappa_\xi$ is the compositional diffusivity, and $\nu$ is the kinematic viscosity.

$\beta_\xi$ is used in the fixed flux case as $\Delta \xi$ can not be known A priori, it is derived from solving the diffusion problem in the sphere.