Non Dimensionalisation
Momentum Equation
In spherical polar coordinates \((r,\theta,\phi)\), the dimensional momentum equation is
\[
\rho \left(\frac{\partial \boldsymbol{u}}{\partial t} + \boldsymbol{u}\cdot \nabla \boldsymbol{u} \right)
+
2\rho\boldsymbol{\Omega}\times\boldsymbol{u}
=
- \nabla P
+
\rho \boldsymbol{g}
+
\frac{\left(\nabla \times \boldsymbol{B}\right)\times \boldsymbol{B}}{\mu_0}
+
\rho\nu\nabla^2\boldsymbol{u},
\]
\(\rho\) is the density,
where \(\boldsymbol{u}\) is velocity,
\(t\) is time,
\(\boldsymbol{\Omega} = \Omega\hat{\boldsymbol{z}}\) is the rotation,
\(\Omega\) is the rotation rate,
\(\hat{\boldsymbol{z}}\) is the unit vector in the direction of the rotation axis,
\(P\) is pressure,
\(\boldsymbol{g}\) is gravity,
\(\mu_0\) is the magnetic permeability,
\(\boldsymbol{B}\) is the magnetic field,
and \(\nu\) is the kinematic viscosity.
Boussinesq Approximation
In the Boussinesq approximation, density perturbations are only important in the
buoyancy term, and are assumed to be linear, of the form
\[
\rho = \rho_0\left(1-\alpha(T-T_0) - \alpha_\xi(\xi-\xi_0) \right)
\]
where
\(\rho_0\) is the reference density at \(r_o\),
\(\alpha\) is the thermal expansivity,
\(T\) is the temperature,
\(T_0\) is the reference temperature at \(r_o\),
\(\alpha_\xi\) is the compositional expansivity,
\(\xi\) is the composition,
and \(\xi_0\) is the reference composition at \(r_0\).
Defining gravity as \(\boldsymbol{g} = -(g_0 r/r_0)\hat{\boldsymbol{r}}\),
the momentum equation becomes
\[\begin{align*}
\rho_0 \left(\frac{\partial \boldsymbol{u}}{\partial t} + \boldsymbol{u}\cdot \nabla \boldsymbol{u} \right)
+
2\rho_0\boldsymbol{\Omega}\times\boldsymbol{u}
=
- \nabla P& \\
+
\rho_0\left(1-\alpha(T-T_0) - \alpha_\xi(\xi-\xi_0) \right)& (-(g_0 r/r_0)\hat{\boldsymbol{r}}) \\
&+
\frac{\left(\nabla \times \boldsymbol{B}\right)\times \boldsymbol{B}}{\mu_0}
+
\rho_0\nu\nabla^2\boldsymbol{u},
\end{align*}\]
\[\begin{align*}
\rho_0 \left(\frac{\partial \boldsymbol{u}}{\partial t} + \boldsymbol{u}\cdot \nabla \boldsymbol{u} \right)
+
2\rho_0\boldsymbol{\Omega}&\times\boldsymbol{u}
=
-
\nabla P
-
\rho_0(1+\alpha T_0 + \alpha\xi_0)\frac{g_0 r}{r_0}\hat{\boldsymbol{r}}
\\
&+\rho_0\left(\alpha T + \alpha_\xi\xi \right) \frac{g_0 r}{r_0}\hat{\boldsymbol{r}}
+
\frac{\left(\nabla \times \boldsymbol{B}\right)\times \boldsymbol{B}}{\mu_0}
+
\rho_0\nu\nabla^2\boldsymbol{u}.
\end{align*}\]
We can absorb the hydrostatic balance into a modified pressure and divide through by \(\rho_0\) to obtain
\[
\frac{\partial \boldsymbol{u}}{\partial t} + \boldsymbol{u}\cdot \nabla \boldsymbol{u}
+
2\boldsymbol{\Omega}\times\boldsymbol{u}
=
- \frac{\nabla P}{\rho_0}
+
\left(\alpha T + \alpha_\xi\xi \right) \frac{g_0 r}{r_0}\hat{\boldsymbol{r}}
+
\frac{\left(\nabla \times \boldsymbol{B}\right)\times \boldsymbol{B}}{\rho_0\mu_0}
+
\nu\nabla^2\boldsymbol{u}.
\]
Non-dimensionalisation
Using the non-dimensional scales \(D\) for length,
\(D^2/\eta\) for time,
\(\Delta T\) or \(\beta/D\) for temperature,
\(\Delta \xi\) or \(\beta_\xi/D\) for composition,
\((2\Omega\rho\mu_0\eta)^{1/2}\) for magnetic field,
\(\Omega\rho_0\eta\) for pressure,
and \(\eta/D\) for velocity,
and denoting non-dimensional variables with \(x^*\),
the equation becomes
\[\begin{align*}
\frac{\eta/D}{D^2/\eta}\frac{\partial \boldsymbol{u^*}}{\partial t^*} + \frac{\eta^2/D^2}{D}\boldsymbol{u^*}\cdot \nabla^* \boldsymbol{u^*}
\\ +
\frac{\eta}{D} 2\Omega\hat{\boldsymbol{z}}\times\boldsymbol{u^*}
= \\
-\frac{\Omega\rho_0\eta}{D} \frac{\nabla^* P^*}{\rho_0}
\\+
\left(\Delta T D \alpha T^* + \Delta \xi D \alpha_\xi\xi^* \right) \frac{g_0 r^*}{r_0}\hat{\boldsymbol{r}}
\\+
\frac{2\Omega\rho_0\mu_0\eta}{D}\frac{\left(\nabla^* \times \boldsymbol{B^*}\right)\times \boldsymbol{B^*}}{\rho_0\mu_0}
\\+
\frac{\eta/D}{D^2}\nu\nabla^{*2}\boldsymbol{u^*},
\end{align*}\]
tidying:
\[\begin{align*}
\frac{\eta^2}{D^3}\frac{\partial \boldsymbol{u^*}}{\partial t^*} + \frac{\eta^2}{D^3}\boldsymbol{u^*}\cdot \nabla^* \boldsymbol{u^*}
\\ +
\frac{\eta\Omega}{D} 2\hat{\boldsymbol{z}}\times\boldsymbol{u^*}
= \\
-\frac{\Omega\eta}{D} \nabla^* P^*
\\+
\left(\frac{\Delta T D \alpha g_0}{r_0} T^* + \frac{\Delta \xi D \alpha_\xi g_0}{r_0} \xi^* \right) r^*\hat{\boldsymbol{r}}
\\+
\frac{2\Omega\eta}{D} \left(\nabla^* \times \boldsymbol{B^*}\right)\times \boldsymbol{B^*}
\\+
\frac{\eta\nu}{D^3}\nabla^{*2}\boldsymbol{u^*},
\end{align*}\]
rearranging:
\[\begin{align*}
\frac{\partial \boldsymbol{u^*}}{\partial t^*} + \boldsymbol{u^*}\cdot \nabla^* \boldsymbol{u^*}
\\ +
\frac{\Omega D^2}{\eta} 2\hat{\boldsymbol{z}}\times\boldsymbol{u^*}
= \\
-\frac{\Omega D^2}{\eta} \nabla^* P^*
\\+
\left(\frac{\Delta T D^4 \alpha g_0}{\eta^2 r_0} T^* + \frac{\Delta \xi D^4 \alpha_\xi g_0}{r_0 \eta^2} \xi^* \right) r^*\hat{\boldsymbol{r}}
\\+
\frac{2\Omega D^2}{\eta} \left(\nabla^* \times \boldsymbol{B^*}\right)\times \boldsymbol{B^*}
\\+
\frac{\nu}{\eta}\nabla^{*2}\boldsymbol{u^*},
\end{align*}\]
expanding for non-dimensional parameters:
\[\begin{align*}
\frac{\partial \boldsymbol{u^*}}{\partial t^*} + \boldsymbol{u^*}\cdot \nabla^* \boldsymbol{u^*}
\\ +
\frac{\Omega D^2}{\nu} \frac{\nu}{\eta} 2\hat{\boldsymbol{z}}\times\boldsymbol{u^*}
= \\
-\frac{\Omega D^2}{\nu} \frac{\nu}{\eta} \nabla^* P^*
\\+
\left(\frac{\Delta T D^4 \alpha g_0}{\kappa \nu r_0} \frac{\kappa}{\nu} \frac{\nu^2}{\eta^2} T^* + \frac{\Delta \xi D^4 \alpha_\xi g_0}{\kappa_\xi\nu r_0} \frac{\kappa_\xi}{\nu} \frac{\nu^2}{\eta^2} \xi^* \right) r^*\hat{\boldsymbol{r}}
\\+
\frac{2\Omega D^2}{\nu} \frac{\nu}{\eta} \left(\nabla^* \times \boldsymbol{B^*}\right)\times \boldsymbol{B^*}
\\+
\frac{\nu}{\eta}\nabla^{*2}\boldsymbol{u^*},
\end{align*}\]
finally substituting for \(Ra\),
\(Ek\),
\(Ra_\xi\),
\(Pr\),
\(Pm\),
\(Sc\):
\[\begin{align*}
\frac{\partial \boldsymbol{u^*}}{\partial t^*} + \boldsymbol{u^*}\cdot \nabla^* \boldsymbol{u^*}
\\ +
\frac{Pm}{Ek} 2\hat{\boldsymbol{z}}\times\boldsymbol{u^*}
= \\
-\frac{Pm}{Ek}\nabla^* P^*
\\+
\left(\frac{Ra_T Pm^2}{Pr} T^* + \frac{Ra_\xi Pm^2}{Sc} \xi^* \right) r^*\hat{\boldsymbol{r}}
\\+
\frac{2Pm}{Ek} \left(\nabla^* \times \boldsymbol{B^*}\right)\times \boldsymbol{B^*}
\\+
Pm\nabla^{*2}\boldsymbol{u^*},
\end{align*}\]
Fixed Flux boundary conditions
In the case that fixed flux boundary conditions are used for either temperature or chemical composition,
the temperature and composition are scaled with \(\beta/D\) and \(\beta_\xi/D\), respectively.
This leads to the buoyancy term in the momentum equation becoming
\[
\left(\frac{(\beta/D) D^4 \alpha g_0}{\kappa \nu r_0} \frac{\kappa}{\nu} \frac{\nu^2}{\eta^2} T^* + \frac{(\beta_\xi/D) D^4 \alpha_\xi g_0}{\kappa_\xi\nu r_0} \frac{\kappa_\xi}{\nu} \frac{\nu^2}{\eta^2} \xi^* \right) r^*\hat{\boldsymbol{r}},
\]
and the flux Rayleigh numbers are used instead of the standard Rayleigh numbers:
\[
\left(\frac{Ra_T^F Pm^2}{Pr} T^* + \frac{Ra_\xi^F Pm^2}{Sc} \xi^* \right) r^*\hat{\boldsymbol{r}}.
\]
Either, neither, or both of the buoyancy terms can be flux or standard, depending on the boundary conditions.
Both flux and standard Rayleigh numbers are input to the code using d_Ra or d_Ra_comp,
but they are defined differently.
Induction Equation
We start with the dimensional equation
\[
\frac{\partial \boldsymbol{B}}{\partial t} = \nabla \times \left(\boldsymbol{u}\times\boldsymbol{B}\right) + \eta\nabla^2\boldsymbol{B},
\]
where \(\boldsymbol{B}\) is the magnetic field,
\(t\) is time,
\(\boldsymbol{u}\) is velocity,
and \(\eta\) is the magnetic diffusivity.
We non-dimensionalise with non-dimensional scales \(D\) for length,
\(D^2/\eta\) for time,
\((2\Omega\rho_0\mu_0\eta)^{1/2}\) for magnetic field,
and \(\eta/D\) for velocity,
and denoting non-dimensional variables with \(x^*\),
\[
\frac{(2\Omega\rho_0\mu_0\eta)^{1/2}}{D^2/\eta}\frac{\partial \boldsymbol{B^*}}{\partial t^*}
=
\frac{ (\eta/D) (2\Omega\rho_0\mu_0\eta)^{1/2} }{D}\nabla^* \times \left(\boldsymbol{u^*}\times\boldsymbol{B^*}\right)
+
\frac{(2\Omega \rho_0\mu_0\eta)^{1/2}}{D^2}\eta\nabla^{*2}\boldsymbol{B^*},
\]
tidying
\[
\frac{\eta(2\Omega\rho_0\mu_0\eta)^{1/2}}{D^2}\frac{\partial \boldsymbol{B^*}}{\partial t^*}
=
\frac{ \eta (2\Omega\rho_0\mu_0\eta)^{1/2} }{D^2}\nabla^* \times \left(\boldsymbol{u^*}\times\boldsymbol{B^*}\right)
+
\frac{\eta(2\Omega \rho_0\mu_0\eta)^{1/2}}{D^2}\eta\nabla^{*2}\boldsymbol{B^*},
\]
simplifies to
\[
\frac{\partial \boldsymbol{B^*}}{\partial t^*}
=
\nabla^* \times \left(\boldsymbol{u^*}\times\boldsymbol{B^*}\right)
+
\nabla^{*2}\boldsymbol{B^*},
\]
Temperature Equation
\[
\frac{\partial T}{\partial t} + \boldsymbol{u}\cdot\nabla T = \kappa \nabla^2 T + S
\]
We non-dimensionalise with non-dimensional scales \(D\) for length,
\(D^2/\eta\) for time,
\(\Delta T\) and \(\beta/D\) for fixed temperature and fixed flux Temperature boundary
conditions respectively, and \(\eta/D\) for velocity. The source term is non-dimensionalised in terms of temperature diffusion.
Denoting non-dimensional variables with \(x^*\) we have
Fixed Temperature
\[
\frac{\Delta T}{D^2/\eta}\frac{\partial T^*}{\partial t^*}
+
\frac{(\eta/D) (\Delta T)}{D}\boldsymbol{u^*}\cdot\nabla^* T^*
=
\frac{\Delta T}{D^2}\kappa \nabla^{*2} T^*
+
S^*,
\]
tidying,
\[
\frac{\Delta T \eta}{D^2}\frac{\partial T^*}{\partial t^*}
+
\frac{\eta\Delta T}{D^2}\boldsymbol{u^*}\cdot\nabla^* T^*
=
\frac{\Delta T \kappa}{D^2} \nabla^{*2} T^*
+
S^*,
\]
cancelling
\[
\frac{\eta}{\kappa}\left(\frac{\partial T^*}{\partial t^*}
+
\boldsymbol{u^*}\cdot\nabla^* T^* \right)
=
\nabla^{*2} T^*
+
S^*,
\]
substituting for \(Pm\), \(Pr\)
\[
\frac{Pr}{Pm}\left(\frac{\partial T^*}{\partial t^*}
+
\boldsymbol{u^*}\cdot\nabla^* T^* \right)
=
\nabla^{*2} T^*
+
S^*.
\]
Fixed Flux
\[
\frac{\beta/D}{D^2/\eta}\frac{\partial T^*}{\partial t^*}
+
\frac{(\eta/D) (\beta/D)}{D}\boldsymbol{u^*}\cdot\nabla^* T^*
=
\frac{\beta/D}{D^2}\kappa \nabla^{*2} T^*
+
S^*,
\]
tidying,
\[
\frac{(\beta/D) \eta}{D^2}\frac{\partial T^*}{\partial t^*}
+
\frac{\eta (\beta/D)}{D^2}\boldsymbol{u^*}\cdot\nabla^* T^*
=
\frac{(\beta/D) \kappa}{D^2} \nabla^{*2} T^*
+
S^*,
\]
cancelling
\[
\frac{\eta}{\kappa}\left(\frac{\partial T^*}{\partial t^*}
+
\boldsymbol{u^*}\cdot\nabla^* T^* \right)
=
\nabla^{*2} T^*
+
S^*,
\]
substituting for \(Pm\), \(Pr\)
\[
\frac{Pr}{Pm}\left(\frac{\partial T^*}{\partial t^*}
+
\boldsymbol{u^*}\cdot\nabla^* T^* \right)
=
\nabla^{*2} T^*
+
S^*.
\]
Chemical Composition Equation
\[
\frac{\partial \xi}{\partial t} + \boldsymbol{u}\cdot\nabla \xi = \kappa_\xi \nabla^2 \xi + S_\xi
\]
We non-dimensionalise with non-dimensional scales \(D\) for length,
\(D^2/\eta\) for time,
\(\Delta \xi\) and \(\beta_\xi/D\) for fixed composition and fixed flux composition boundary
conditions respectively, and \(\eta/D\) for velocity. The source term is non-dimensionalised in terms of composition diffusion.
Denoting non-dimensional variables with \(x^*\) we have
Fixed Composition
\[
\frac{\Delta \xi}{D^2/\eta}\frac{\partial \xi^*}{\partial t^*}
+
\frac{(\eta/D) (\Delta \xi)}{D}\boldsymbol{u^*}\cdot\nabla^* \xi^*
=
\frac{\Delta \xi}{D^2}\kappa_\xi \nabla^{*2} \xi^*
+
S_\xi^*,
\]
tidying,
\[
\frac{\Delta \xi \eta}{D^2}\frac{\partial \xi^*}{\partial t^*}
+
\frac{\eta\Delta \xi}{D^2}\boldsymbol{u^*}\cdot\nabla^* \xi^*
=
\frac{\Delta \xi \kappa_\xi}{D^2} \nabla^{*2} \xi^*
+
S_\xi^*,
\]
cancelling
\[
\frac{\eta}{\kappa_\xi}\left(\frac{\partial \xi^*}{\partial t^*}
+
\boldsymbol{u^*}\cdot\nabla^* \xi^* \right)
=
\nabla^{*2} \xi^*
+
S_\xi^*,
\]
substituting for \(Pm\), \(Sc\)
\[
\frac{Sc}{Pm}\left(\frac{\partial \xi^*}{\partial t^*}
+
\boldsymbol{u^*}\cdot\nabla^* \xi^* \right)
=
\nabla^{*2} \xi^*
+
S_\xi^*.
\]
Fixed Flux
\[
\frac{\beta_\xi/D}{D^2/\eta}\frac{\partial \xi^*}{\partial t^*}
+
\frac{(\eta/D) (\beta_\xi/D)}{D}\boldsymbol{u^*}\cdot\nabla^* \xi^*
=
\frac{\beta_\xi/D}{D^2}\kappa_\xi \nabla^{*2} \xi^*
+
S_\xi^*,
\]
tidying,
\[
\frac{(\beta_\xi/D) \eta}{D^2}\frac{\partial \xi^*}{\partial t^*}
+
\frac{\eta (\beta_\xi/D)}{D^2}\boldsymbol{u^*}\cdot\nabla^* \xi^*
=
\frac{(\beta_\xi/D) \kappa_\xi}{D^2} \nabla^{*2} \xi^*
+
S_\xi^*,
\]
cancelling
\[
\frac{\eta}{\kappa_\xi}\left(\frac{\partial \xi^*}{\partial t^*}
+
\boldsymbol{u^*}\cdot\nabla^* \xi^* \right)
=
\nabla^{*2} T^*
+
S_\xi^*,
\]
substituting for \(Pm\), \(Sc\)
\[
\frac{Sc}{Pm}\left(\frac{\partial \xi^*}{\partial t^*}
+
\boldsymbol{u^*}\cdot\nabla^* \xi^* \right)
=
\nabla^{*2} \xi^*
+
S_\xi^*.
\]