We can define a galactic magnetic field as a mean $(\bar{B})$ plus a random $(b)$ component:

$$ \begin{aligned} B = \bar{B} + b \end{aligned} $$

If we assume the random field component to be small compared to the mean field, and plug this into the induction equation, given by

$$ \begin{equation} \frac{\partial B}{\partial t} = \nabla \times (V \times B - \eta_T \nabla \times B) \end{equation} $$

where $V$ is the galaxy velocity and $\eta_T$ is the magnetic diffusivity, we get the mean field galactic dynamo equation, given by

$$ \begin{equation} \frac{\partial \bar{B}}{\partial t} = \nabla \times (\bar{V} \times \bar{B}) + \nabla \times (\alpha \bar{B}) - \nabla \times (\eta_T \nabla \times \bar{B}) \end{equation} $$

where $\alpha$ is the mean kinetic helicity density, which describes how helical flow can stretch and twist the toroidal fields to amplify the poloidal fields in a galactic disk.

In this project, my task is to solve the dynamo equations along $z$, so all radial derivatives will be ignored. Moreover, globally, the $\bar{B}_z$ component is typically negligible for thin disks, so we will only solve for $\bar{B}r$ and $\bar{B}\phi$. For an axially symmetric disk, with axially symmetric magnetic fields, we can write the $\bar{B}r$ and $\bar{B\phi}$ components of the dynamo equation, but first we scale all the parameters as given in the following table:

Table 1

$$ \begin{array}{| c |c | c |}\hline \text{Parameter} & \text{ Before } & \text{ After scaling }\\ \hline \hline \text{Radius} & r & r/R_0 \\ \hline \text{Height} & z & z/h_0 \\ \hline \text{Field} & B & B/B_0 \\ \hline \text{Alpha} & \alpha & \alpha/\alpha_0 \\ \hline \text{Angular velocity} & \Omega & \Omega/\Omega_0 \\ \hline \text{Angular Sheer rate} & S = r\frac{\partial \Omega}{\partial r} & S/S_0 \\ \hline \text{Time} & t & t/t_0 \\ \hline \text{Diffusivity} & \eta_T & 1 \\ \hline \end{array} $$

where the scaling constants are specified in the methods. The dynamo equations then become:

$$ \begin{equation} \frac{\partial B_r}{\partial t} = -R_\alpha \frac{\partial}{\partial z} (\alpha B_\phi) + \frac{\partial^2 B_r}{\partial z^2} \end{equation} $$

$$ \begin{equation} \frac{\partial B_\phi}{\partial t} = R_\omega S B_r + R_\alpha \frac{\partial}{\partial z} (\alpha B_r)+ \frac{\partial^2 B_\phi}{\partial z^2} \end{equation} $$

where

$$ \begin{equation} R_\omega = S_0 h_0^2/\eta_T \hspace{1cm} \text{and} \hspace{1cm} R_\alpha=\alpha_0 h_0/\eta_T \end{equation} $$