Equations- Boussinesq

The non-dimensionalised governing equations for the dynamo code. Unless explicitly stated, all variables here are non-dimensionalised, and the $^*$ has been dropped. Non-dimensionalisation information can be found here.

Momentum Equation

Controlled using b_vel_tstep

\[ \begin{aligned} \frac{\partial \boldsymbol{u}}{\partial t} &= \text{Pm}\nabla^2\boldsymbol{u} -\nabla\left(\frac{\text{Pm}}{\text{Ek}}P + \frac{1}{2}|\boldsymbol{u}|^2\right) + \boldsymbol{u}\times \left(\nabla \times \boldsymbol{u}\right)\\ &- \frac{2\text{Pm}}{\text{Ek}}\hat{\boldsymbol{z}}\times \boldsymbol{u} + \text{Pm}^2\left( \frac{\text{Ra}_T ~T}{\text{Pr}} + \frac{\text{Ra}_\xi ~\xi}{\text{Sc}}\right)\boldsymbol{r} + \frac{2\text{Pm}}{\text{Ek}}\left(\nabla\times \boldsymbol{B}\right)\times\boldsymbol{B} \end{aligned} \]

where $\boldsymbol{u}$ is velocity, $P$ is pressure, $\hat{\boldsymbol{z}}$ is the unit vector in the direction of the rotation axis, $T$ is the temperature, $\xi$ is the composition, $\boldsymbol{B}$ is the magnetic field, Pm is the magnetic Prandtl number, Ek is the Ekman number, $\text{Ra}_T$ is the thermal Rayleigh number, Pr is the thermal Prandtl number, $\text{Ra}_\xi$ is the compositional Rayleigh number, and Sc is the Scmidt (compositional Prandtl) number.

Temperature Equation

Controlled using b_cod_tstep

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

where Pr is the Prandtl number, Pm is the magnetic Prandtl number, $T$ is the temperature, $t$ is time, $\boldsymbol{u}$ is velocity, and $S$ is the internal heat source.

Composition Equation

Controlled using b_comp_tstep

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

where Sc is the Schmidt number, Pm is the magnetic Prandtl number, $\xi$ is the composition, $t$ is time, $\boldsymbol{u}$ is velocity, and $S_\xi$ is the internal composition source.

Induction Equation

Controlled using b_mag_tstep $$ \frac{\partial \boldsymbol{B}}{\partial t} = \nabla \times \left(\boldsymbol{u}\times\boldsymbol{B}\right) + \nabla^2\boldsymbol{B}, $$ where $\boldsymbol{B}$ is the magnetic field, $t$ is time, and $\boldsymbol{u}$ is velocity. No non-dimensional parameters appear in the induction equation, due to the choice of magnetic diffusive timescale.

Solenoidal Magnetic Field

\[ \nabla \cdot \boldsymbol{B} = 0. \]

Incompressibility Constraint

\[ \nabla \cdot \boldsymbol{u} = 0. \]

Inner Core Rotation

Controlled using b_rot_tstep. Written out to column 2 of rot_torque.dat

The moment of inertia of a solid sphere is $I=\frac{2}{5}mr^2$. Assuming the same density as the outer core, the non-dimensional moment of inertia of the inner core is $I_i=\frac{2}{5}(\frac{4}{3} \pi r_i^3)r_i^2 = \frac{8}{15} \pi r_i^5$.

The rotation of the inner core ($\Omega$) is due to the viscous torque ($\Gamma_\nu$) and Lorentz torque ($\Gamma_\eta$), and is calculated as $$ I_i \frac{\partial \Omega}{\partial t} = \Gamma_\nu + \Gamma_\eta. $$

Inner core velocity

$$ \boldsymbol{u} = \Omega r \sin\theta \hat{\boldsymbol{\phi}} $$ where $\Omega$ is inner core rotation rate, and $\hat{\boldsymbol{\phi}}$ is unit vector in $\phi$ direction.

Inner core magnetic field

Assuming the same diffusivity as the outer core,

\[ \frac{\partial \boldsymbol{B}}{\partial t} - \nabla^2\boldsymbol{B} = -\left[\frac{\partial B_r}{\partial \phi},\frac{\partial B_\theta}{\partial \phi}\frac{\partial B_\phi}{\partial \phi}\right]. \]

Viscous Torque

$$ \Gamma_\nu=\text{Pm}\int{\partial_r\left(\frac{u_\phi}{r}\right)r^2\sin\theta} dS, $$ where $\Gamma_\nu$ is viscous torque, $u_\phi$ is the $\phi$ component of the velocity, $r$ is the radial distance, and Pm is the magnetic Prandtl number. Column 3 of rot_torque.dat

Magnetic Torque

\[ \Gamma_\eta = \frac{2\text{Pm}}{\text{Ek}}\int{B_rB_\phi r \sin\theta}dS \]

where $\Gamma_\eta$ is magnetic torque, Pm is the magnetic Prandtl, Ek is the Ekman Number, $B_r$ and $B_\phi$ are the $r$ and $\phi$ components of the magnetic field, $r$ is the radial distance, and $S$ is the spherical surface of the inner boundary.

Found in column 4 of rot_torque.dat