Non-dimensionalization

For numerical convenience we use dimensionless variables when solving the Navier Stokes equation, the induction equation, and the heat equation. A non-dimensionalization procedure is always possible thanks to the Buckingham-\(\pi\) theorem.

The momentum equation

In a reference frame rotating with angular speed \(\Omega\), the dimensional form of the linear momentum equation (i.e. the Navier-Stokes equation) describing the acceleration \(\partial_t \mathbf{u}\) of a small fluid parcel with density \(\rho\), including buoyancy and the Lorentz force, is:

\[ \rho\partial_t \mathbf{u} +2\rho\,\mathbf{\Omega}\times\mathbf{u}=-\nabla p +\frac{1}{\mu_0}(\nabla \times \mathbf{B})\times \mathbf{B} + \rho\mathbf{g}+\rho\nu\nabla^2{\mathbf{u}}, \]

where \(p\) is the reduced pressure, \(\mathbf{B}\) the magnetic field, and \(\nu\) is the kinematic viscosity of the fluid. Assume small density perturbations \(\rho'\) following

\[ \rho=\rho_0+\rho'=\rho_0-\rho_0\alpha\theta, \]

where \(\alpha\) is the fluid's thermal expansion coefficient and \(\theta\) is the temperature variation from the isentropic temperature profile \(T(r)\). Assume also a gravitational acceleration following

\[ \mathbf{g}=-g_0\frac{r}{R}\mathbf{\hat r}. \]

Within the Boussinesq approximation the density variations enter only through the buoyancy force, so

\[ \rho\mathbf{g} \longrightarrow \rho' \mathbf{g} = \rho_0 \alpha g_0 \frac{r}{R} \theta \mathbf{\hat r}. \]

Now we make the dimensional units explicit. With \(L\) being the unit of length, \(\tau\) the unit of time, \(\theta^*\) the unit of temperature, \(P^*\) the unit of pressure, and \(\mathbf{\hat z}\) the unit vector along \(\mathbf{\Omega}\), then the momentum equation, after dividing by \(\rho_0\), is

\[ \frac{L}{\tau^2}\,\partial_t \mathbf{u} +L\frac{\Omega}{\tau}\, 2\mathbf{\hat z}\times\mathbf{u}=-\frac{P^*}{\rho_0 L} \nabla p + \frac{B_0^2}{L\rho_0 \mu_0}(\nabla \times \mathbf{B})\times \mathbf{B} + \frac{\alpha g_0 \theta^*}{R} L r \theta\mathbf{\hat r}+\frac{\nu }{\tau L}\nabla^2 \mathbf{u}, \]

where it is understood that the variables \(r, t, \mathbf{u}, p, \theta, \mathbf{B}\) are now dimensionless. Multiply the equation by \(\tau^2/L\) and get

\[ \partial_t \mathbf{u} + 2\,\Omega\tau\,\mathbf{\hat z}\times\mathbf{u}=- \nabla p + \frac{\tau^2 B_0^2}{L^2\rho_0 \mu_0}(\nabla \times \mathbf{B})\times \mathbf{B} + \tau^2 \frac{\alpha g_0 \theta^*}{R} r \theta \mathbf{\hat r}+\tau\frac{\nu}{L^2}\nabla^2 \mathbf{u}. \]

Above we have chosen the pressure scale as \(P^*=\rho_0 L^2/\tau^2\). For the time being we leave the temperature scale \(\theta^*\) unspecified, and define the Ekman number \(E\) as

\[ E \equiv \frac{\nu}{\Omega L^2}, \]

The Lehnert number \(Le\) as

\[ Le \equiv \frac{B_0}{\Omega L \sqrt{\rho_0\mu_0}}, \]

the Rayleigh number \(Ra\) as

\[ Ra \equiv \frac{\alpha g_0 \theta^* L^4}{\nu\kappa R}, \]

and the Prandtl number \(Pr\) as

\[ Pr \equiv \frac{\nu}{\kappa}, \]

where \(\kappa\) is the thermal diffusivity of the fluid So, the momentum equation becomes

\[ \partial_t \mathbf{u} + 2\,(\Omega\tau)\,\mathbf{\hat z}\times\mathbf{u}=-\nabla p + (\Omega\tau)^2 Le^2(\nabla \times \mathbf{B})\times \mathbf{B} + (\Omega\tau)^2 E^2\frac{Ra}{Pr}\,\theta\, r\, \mathbf{\hat r}+(\Omega\tau)\,E\,\nabla^2 \mathbf{u}. \]

Alternatively, if we deal with problems without viscosity or thermal diffusion where the Rayleigh number diverges, it is better to define a reference Brunt-Väisälä frequency \(N_0\) such that

\[ N_0^2 \equiv -\frac{\alpha g_0 \theta^*}{R}. \]

If the Rayleigh number is finite then we can write

\[ E^2\frac{Ra}{Pr} = -\frac{N_0^2}{\Omega^2}. \]

The momentum equation, using the reference Brunt-Väisälä frequency reads

\[ \partial_t \mathbf{u} + 2\,(\Omega\tau)\,\mathbf{\hat z}\times\mathbf{u}=-\nabla p + (\Omega\tau)^2 Le^2(\nabla \times \mathbf{B})\times \mathbf{B} - (\Omega\tau)^2 \frac{N_0^2}{\Omega^2}\,\theta\, r\, \mathbf{\hat r}+(\Omega\tau)\,E\,\nabla^2 \mathbf{u}. \]

A common choice for the time scale is the rotation time scale, so \(\tau=1/\Omega\) and the \((\Omega\tau)\) factors go away. Another choice is the viscous diffusion time scale, with \(\tau=L^2/\nu\), in which case \(\Omega\tau=1/E\). Yet another choice is the Alfvén wave time scale, with \(\tau=L \sqrt{\mu_0\rho_0}/B_0\) so that \(\Omega\tau=1/Le\).

The induction equation

The induction equation in dimensional form is

\[ \partial_t \mathbf{B} = \nabla \times (\mathbf{u} \times \mathbf{B}) + \eta \nabla^2 \mathbf{B}, \]

where \(\eta\) is the magnetic diffusivity. Making the dimensional scale factors explicit we get

\[ \frac{B_0}{\tau} \partial_t \mathbf{B} = \frac{B_0}{\tau} \nabla \times (\mathbf{u} \times \mathbf{B}) + \eta \frac{B_0}{L^2} \nabla^2\mathbf{B}, \]

where \(\mathbf{u}, \mathbf{B}, t\) are now dimensionless. Multiply now by \(\tau/B_0\) and obtain

\[ \partial_t \mathbf{B} = \nabla \times (\mathbf{u} \times \mathbf{B}) + (\Omega\tau)E_\eta \nabla^2 \mathbf{B}, \]

where \(E_\eta\) is the magnetic Ekman number defined as

\[ E_\eta \equiv \frac{\eta}{\Omega L^2}. \]

The heat equation

The heat equation in its dimensional form is

\[ \partial_t \theta=-\mathbf{u}\cdot\nabla T+\kappa \nabla^2 \theta. \]

We assume that an isentropic temperature background \(T(r)\) exists, which is only a function of radius. Its gradient is then \(\nabla T=\mathbf{\hat r}\,\mathrm{d}T/\mathrm{d}r\). Now we write \(\mathrm{d}T/\mathrm{d}r=C\,f(r)\), where \(C\) is a scale factor for the gradient (can be negative) and \(f(r)\) is a dimensionless function of \(r\) (with \(r\) also dimensionless). Then we can write the heat equation, using this time dimensionless variables exclusively as

\[ \partial_t \theta=-\frac{LC}{\theta^*}\,u_r\,f(r)+(\Omega\tau)\frac{E}{Pr} \nabla^2 \theta. \]

In kore we choose always the temperature scale as \(\theta^*=-LC\) so that the heat equation reads simply

\[ \partial_t \theta=u_r\,f(r)+(\Omega\tau)\frac{E}{Pr} \nabla^2 \theta. \]

A temperature profile with a linear gradient is common in the literature. In that case \(\partial_r T=-\beta r\) (dimensional). In dimensionless variables this is \(-\beta L r\) (\(r\) now dimensionless), so that the temperature scale is \(\theta^*=\beta L^2\). And if the length scale is the CMB radius, i.e. \(L=R\), then the Rayleigh number becomes

\[ Ra = \frac{\alpha g_0 \beta R^5}{\nu\kappa}. \]