We are solving for the $\alpha^2-\omega$ dynamo equations:

$$ \begin{equation}

\boxed{\frac{\partial B_r}{\partial t} = -\frac{\partial}{\partial z} (\alpha B_\phi) + \frac{\partial^2 B_r}{\partial z^2} }

\end{equation} $$

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

Generally, $\alpha \equiv \alpha(z, r)$, but we will consider the case where $\alpha \equiv \alpha(z) = sin(\pi z/h)$. Note that it is important that $\alpha$ changes sign as $z$ goes from $-h$ to $+h$ because the helicity of the galaxy changes in the norther and southern hemispheres. Recall that $B_r$ is scaled by $R_\alpha$. For a typical disk galaxy, the scaling parameters, as listed in table (1), are chosen as

Table 2

$$ \begin{array}{|c | c | c |}\hline \text{Scaling Constant} & \text{ Value } & \text{Unit} \\ \hline R_0 & 10 & kpc \\ \hline h_0 & 0.5 & kpc \\ \hline B_0 & 1e-6 & G \\ \hline \alpha_0 & 2000 & cm s^{-1}\\ \hline \Omega_0 & 100 & km s^{-1} Mpc^{-1} \\ \hline S_0 & -50 & km s^{-1} Mpc^{-1} \\ \hline \eta_T & 1e26 & cm^2 s^{-1} \\ \hline t_0 = h_0^2/\eta_T & 0.745 & Gyr \\ \hline \end{array} $$

Boundary Conditions

To solve these differential equations, we need boundary conditions. If we assume electromagentic vaccum outside the disk, we see that the electric fields vanish outside the galaxy. Hence in the limit $|z| = \pm h_0$, we see that $\nabla \times B = 0$. Therefore, for an axially symmetric system, if we neglect $\partial \phi/\partial r$ terms, then we can choose Dirichlet boundary conditions:

$$ \begin{equation} B_r(\pm h) = B_\phi (\pm h) = 0 \end{equation} $$

Also, the dynamo equations have a symmetry associated with them. Note that $\alpha(z) = -\alpha(-z)$ and $\Omega(z) = \Omega(-z)$. These conditions, it turns out, lead to Neumann boundary conditions:

$$ \begin{equation} \frac{\partial B_r}{\partial z}(\pm h) = \frac{\partial B_\phi}{\partial z} (\pm h) = 0 \end{equation} $$

I develop my code to allow both kinds of boundary conditions. In my work on Task 1, I showed the differences between the eigenfunctions of the dynamo equations for these two types of BCs.

Evolution Timescales

To numerically evolve the magnetic fields that satisfy the dynamo equations, we start with a seed field. The scaling parameter $t_0$ is the diffusion timescale of the galactic dynamo, and this is the typical timescale required for the fields to converge to the eigenvalue solutions of the dynamo equations.

Rotation Curve

For galaxies, typically, the rotation curve is given by

$$ \begin{equation} \Omega = \frac{\Omega_0}{\sqrt{1+\frac{r^2}{r_\Omega^2}}} \end{equation} $$