Definitions

Geometric

Outer radius

$r_o$ , distance from sphere origin to outer boundary. For the geodynamo, this is the core-mantle-boundary (CMB).

SI units $\text{m}$.

Inner radius

$r_i$ , distance from sphere origin to inner boundary. For the geodynamo, this is the inner-core-boundary (ICB).

Shell Thickness

\[ D = r_o - r_i \]

distance between inner and outer boundaries, where $r_o$ is the outer radius and $r_i$ is the inner radius.

Shell aspect ratio

$r_i/r_o$, ratio of inner radius and outer radius.

Controlled with d_rratio.

For Earth like geometry, approximate that $r_i/r_o \approx 0.35$. This then sets the non-dimensional value of $r_i^* \approx 0.53$ and $r_o^* \approx 1.53$.

Shell volume

The integral over the volumn of the spherical shell is defined as $$ \int{\dots }dV = \int_0^{2\pi}\int_0^\pi \int _{r_i}^{r_o}\dots r^2 \sin\theta dr d\theta d\phi $$ where $r_o$ and $r_i$ are the outer and inner radii, $\theta$ is the co-latitude and $\phi$ is the azimuth.

For Earth like geometry, take $r_i/r_o \approx 0.35$. The non-dimensional volume of the shell is then

\[ V^* = \frac{4}{3}\pi\left((r_o^*)^3 - (r_i^*)^3\right) \approx 14.5988. \]

Gravity

We assume a linear gravity profile, $$ \boldsymbol{g} = - g_0 \frac{r}{r_o} \hat{\boldsymbol{r}} $$ where $g_0$ is reference gravity at the outer radius, $r$ is the radial distance, and $\hat{\boldsymbol{r}}$ is the unit vector in the radial direction.

Rotation Rate

Amount of rotation per unit time. Symbol $\boldsymbol{\Omega}$.

SI units of $\text{rad s}^{-1}$.

Moment of Inertia

Moment of inertia of solid sphere is $I=\frac{2}{5}mr^2$.

Fluid Properties

Density

Mass per unit volume. Symbol $\rho$.

SI units of $\text{kg m}^{-3}$.

Dynamic Viscosity

Resistance of material to deformation, symbol $\mu$.

SI units $\text{kg }\text{m}^{-1}\text{s}^{-1}$.

Kinematic Viscosity

\[ \nu = \frac{\mu}{\rho_0}, \]

$\nu$ is the kinematic viscosity, ratio of the dynamic viscosity $\mu$ to the density $\rho_0$.

SI units $\text{m}^2 \text{s}^{-1}$.

Characteristic Velocity

$U$, estimated in the dynamo code as $\sqrt{2E^*_\text{kin}/V^*}$.

Heat Transport Properties

Heat Capacity

Measure of how much heat is required to increase temperature of a material. Symbol $C_p$.

SI units $\text{J}\text{K}^{-1}\text{kg}^{-1}$.

Temperature Difference

$\Delta T = T|_{r_i} - T|_{r_o}$

Thermal conductivity

Material property that approximates how quickly a material will allow heat energy will flow through it. Symbol $k$.

SI units of $\text{W}\text{m}^{-1}\text{K}^{-1}$.

Thermal Diffusivity

Ratio of the thermal conductivity $k$ to the volumetric heat capacity ($\rho~C_p$).

$$ \kappa = \frac{k}{\rho C_p} $$ where $k$ is thermal conductivity, $\rho$ is density, and $C_P$ is the heat capacity.

SI units $\text{m}^2\text{s}^{-1}$.

Coefficient thermal expansion

How much a material changes in volume per change in temperature (at constant pressure).

Symbol $\alpha$.

SI units $\text{K}^{-1}$.

Beta

$\beta$, used in the definition of the Rayleigh number for thermal convection with fixed-flux boundary conditions. It is derived as an integration constant from the conduction problem for the spherical shell $$ \kappa\nabla^2 T = 0 \rightarrow \kappa \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial T}{\partial r}\right) = 0, $$

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

Estimating that

\[ \frac{\partial T}{\partial r} = \frac{\beta}{r^2} \rightarrow \frac{\Delta T}{D} \sim \frac{\beta}{D^2}, \]

we can replace $\Delta T$ with $\beta/D$.

Internal Heat source

A custom heat source can be put into the code through the term $S$ in the temperature equation

\[ \frac{Pr}{Pm}\left(\frac{\partial T}{\partial t} + \boldsymbol{u}\cdot\nabla T \right) = \nabla^2 T + S, \]

to simulate e.g. radiogenic heating. This can be either a constant term, or a radially varying source term $S(r)$, and is set using i_source_load.

Compositional Properties

Compositional Diffusivity

$\kappa_\xi$.

SI units $\text{m}^2\text{s}^{-1}$.

Composition Difference

\[ \Delta \xi = \xi|_{r_i} - \xi|_{r_o} \]

Coefficient Chemical Expansion

How much a material changes in volume per change in light element proportion.

Symbol $\alpha_\xi$.

Internal Composition source

A custom composition source can be put into the code through the term $S_\xi$ in the composition equation

\[ \frac{Sc}{Pm}\left(\frac{\partial \xi}{\partial t} + \boldsymbol{u}\cdot\nabla \xi \right) = \nabla^2 \xi + S_\xi. \]

This can be either a constant term, or a radially varying source term $S_\xi(r)$, and is set using i_source_comp_load.

Composition Beta

$\beta_\xi$, used in the definition of the compositional Rayleigh number for compositional convection with fixed-flux boundary conditions. It is derived as an integration constant from the diffusion problem for the spherical shell $$ \kappa_\xi\nabla^2 \xi = 0 \rightarrow \kappa_\xi \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial \xi}{\partial r}\right) = 0, $$

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

Estimating that

\[ \frac{\partial \xi}{\partial r} = \frac{\beta_\xi}{r^2} \rightarrow \frac{\Delta \xi}{D} \sim \frac{\beta_\xi}{D^2}, \]

we can replace $\Delta \xi$ with $\beta_\xi/D$.

Magnetic Properties

Magnetic Permeability of free space

$\mu_0 = \boldsymbol{B}/\boldsymbol{H}$. Magnetization produced in a material in response to a magnetic field - the ratio of magnetic induction $\boldsymbol{B}$ to field intensity $\boldsymbol{H}$.

Magnetic Diffusivity

$$ \eta = 1/\mu_0\sigma, $$ where $\mu_0$ is magnetic permeability, and $\sigma$ is the electrical conductivity. Has dimensions of $\text{L}^2\text{T}^{-1}$.

Electrical Conductivity

$\sigma=1/\rho$, where $\rho$ is the resitivity of the material, measured in ohm metre.

Geodynamo Properties

Dipole Decay time

$\tau_\text{d}^\text{dip} = r_o^2/\pi^2\eta = \tau_\text{d}/\pi^2$.

Time for the main dipole field of the Earth to decay.

Magnetic Diffusion time

$\tau_\text{d} = r_o^2/\eta$.

Characteristic timescale for diffusion of the magnetic field.

Overturn time

Gauss Coefficients

The Gauss coefficients $g_l^m$ and $h_l^m$ are used in the representation of a planetary magnetic field. Outside the dynamo region, the magnetic field satisfies $\boldsymbol{B}=-\nabla V$, where $V$ is the magnetic potential and $\boldsymbol{B}$ is the magnetic field. Potential of the Earth's magnetic field can be written as

\[ V = r_\text{S} \sum_{l=1}^\infty \sum_{m=0}^l { \left(\frac{r_\text{S}}{r} \right)^{l+1}P_l^m(\cos\theta)\left(g_l^m\cos m\phi + h_l^m \sin m\phi \right), } \]

$r_\text{S}$ is the radius of the surface of the Earth, $P_l^m$ is Schmidt semi-normalised associated Legendre function of degree $l$ and order $m$, e.g. Olson (2008).

Dipole moment

(Axial Dipole Moment?)

See eg Olson (2008)

Defined as

\[ \boldsymbol{M} = (M_x \hat{\boldsymbol{x}}+ M_y \hat{\boldsymbol{y}}+ M_z \hat{\boldsymbol{z}}) = \frac{4\pi r_S^3}{\mu_0}\left(g_1^1 \hat{\boldsymbol{x}} + h_1^1 \hat{\boldsymbol{y}} + g_1^0 \hat{\boldsymbol{z}}\right) \]

where $M_x$, $M_y$ and $M_z$ are the components of the Dipole moment vector in the cartesian unit vectors $\hat{\boldsymbol{x}}$, $\hat{\boldsymbol{y}}$ and $\hat{\boldsymbol{z}}$, and $g_1^1$, $h_1^1$ and $g_1^0$ are gauss coefficients.

Magnitude of Dipole Moment

Defined as

\[ M = \frac{4 \pi r_S^3}{\mu_0} g_1 \]

where $r_S$ is the distance from the centre to the surface, $\mu_0$ is the magnetic permeability, and

\[ g_1 = \sqrt{\left(g_1^1\right)^2 + \left(h_1^1\right)^2 + \left(g_1^0\right)^2} \]

is the dipole part of the magnetic field calculated from the dipolar gauss coefficients $g_1^1$, $g_1^0$ and $h_1^1$.

Roberts Number

Used in alternative casting of input parameters, Roberts number $q=\kappa/\eta$, where $\kappa$ is thermal diffusivity, and $\eta$ is magnetic diffusivity.