Non-Dimensional Scales

Length

Non-dimensionalised based on the shell thickness $D=r_o-r_i$, where $r_o$ and $r_i$ are the inner and outer radii measured from the origin.

Timescale

Time $t$ is non-dimensionalised on the magnetic diffusion timescale $\tau_\eta=D^2/\eta$, where $\eta$ is the magnetic diffusivity.

Non-Magnetic

For non-magnetic runs, the timescale can be set to either the viscous or thermal diffusion times, by setting one of them to be the same a the magnetic diffusion time. For the visous diffusion time, set Pm=1. To use the use the thermal diffusion time, set Pm=Pr.

Temperature

Fixed Temperature

\[ T^* = \frac{T}{\Delta T} \]

Non-dimensionalised on $\Delta T = T|_{r_i} - T|_{r_o}$, the temperature difference between the bottom and top boundaries. Satisfies the equation $$ \frac{\partial T}{\partial t} = \kappa \nabla^2 T+ S, $$ where $T$ is the temperature relative to the base conductive temperature, $\kappa=k/\rho_0C_p$ is the thermal diffusivity, $k$ is thermal conductivity, $\rho_0$ is mean density, and $C_p$ is the heat capacity. $S$ is the internal heat source.

Fixed flux

\[ T^* = \frac{T}{\beta/D} \]

For fixed flux temperature boundary conditions with no source, we take $\Delta T \sim \beta/D$, where $\beta$ is a constant of integration from the solution to the conduction problem $$ \frac{\partial}{\partial r}\left(r^2\frac{\partial T}{\partial r}\right)=0, $$

and $D$ is the shell thickness.

Chemical Composition

We define $\xi$ as the light element concentration in the outer core.

Fixed Compositional Gradient

\[ \xi^* = \frac{\xi}{\Delta \xi} \]

Non-dimensionalised on $\Delta \xi = \xi|_{r_i} - \xi|_{r_o}$, the composition difference between the bottom and top boundaries. Satisfies the equation $$ \frac{\partial \xi}{\partial t} = \kappa_\xi \nabla^2 \xi+ S_\xi, $$ where $\xi$ is the composition relative to the base diffusive profile, $\kappa_\xi$ is the compositional diffusivity, $S_\xi$ is the internal composition source.

Fixed Flux

\[ \xi^* = \frac{\xi}{\beta_\xi/D} \]

For fixed flux composition boundary conditions with no source, we take $\Delta \xi \sim \beta_\xi/D$, where $\beta_\xi$ is a constant of integration from the solution to the static equation $$ \frac{\partial}{\partial r}\left(r^2\frac{\partial \xi}{\partial r}\right)=0, $$

and $D$ is the shell thickness.

Magnetic Field

$$ \boldsymbol{B^*} = \frac{\boldsymbol{B}}{(2\Omega\rho_0\mu_0\eta)^{1/2}}, $$ where $\boldsymbol{B^*}$ is the non-dimensional magnetic field that occurs in the dynamo code, $\Omega$ is the rotation rate, $\rho_0$ is the density at $r_o$, $\mu_0$ is the magnetic permeability, $\eta$ is the magnetic diffusivity, and $\boldsymbol{B}$ is the dimensional magnetic field.

Velocity

$$ \boldsymbol{u^*} = \frac{\boldsymbol{u}}{\eta/ D}, $$ where $\boldsymbol{u^*}$ is the non-dimensional velocity appearing in the dynamo code, $\boldsymbol{u}$ is dimensional velocity, $\eta$ is magnetic diffusivity, and $D=r_o-r_i$ is the shell thickness.

Pressure

\[ P^* = \frac{P}{\Omega\rho_0\eta} \]

where $\Omega$ is the rotation rate, $\rho_0$ is the density, and $\eta$ is magnetic diffusivity.

Energy

$$ E^* = \frac{E}{\rho_0 D \eta^2}, $$ where $E^*$ is non-dimensional energy appearing in dynamo code, $E$ is dimensional energy, $\rho_0$ is density, $D=r_o-r_i$ is shell thickness, and $\eta$ is magnetic diffusivity (which is set equal to kinematic viscosity or thermal diffusivity in non-magnetic cases).

Dissipation

Viscous ($D_\nu$) and ohmic ($D_\Omega$) dissipation are both scaled with $\rho_0 \eta^3/D$, where $\rho_0$ is density, $\eta$ is magnetic diffusivity, and $D$ is shell thickness.

Viscous Torque

Dimensional defined as $$ \Gamma_\nu = \int{\rho_0 \nu \partial_r\left(\frac{u_\phi}{r}\right)r^2\sin\theta dS}, $$ where $\rho_0$ is the density, $\nu$ is kinematic viscosity $u_\phi$ is the $\phi$ component of dimensional velocity, and $S$ is the spherical surface of the inner boundary.

Non-dimensionalised by $\rho_0 D\eta^2$ in the code, becomes

\[ \Gamma_\nu^*=\text{Pm}\int{\partial_r^*\left(\frac{u^*_\phi}{r^*}\right)(r^*)^2\sin\theta} dS^*, \]

where all the variables are now their non-dimensionalised versions, and Pm is the magnetic Prandtl number.

Magnetic Torque

Column 4 of rot_torque.dat

Dimensional defined as $$ \Gamma_\eta = \frac{1}{\mu_0}\int{B_r B_\phi r \sin\theta}dS $$ where $\mu_0$ is magnetic permeability, $B_r$ and $B_\phi$ are the dimensional radial and $\phi$ components of the magnetic field, and $S$ is the spherica surface of the inner boundary.

Non-dimensionalised by $\rho_0 D\eta^2$ in the code, becomes

\[ \Gamma_\eta^* = \frac{2\text{Pm}}{\text{Ek}}\int{B_r^*B_\phi^* r^* \sin\theta}dS^*, \]

where Pm is the magnetic Prandtl, and Ek is the Ekman Number.