Magnetohydrodynamics Equations

The Euler equations describe compressible flow in the absence of dissipative effects (like viscosity). If we add magnetic fields, we get the equations of (ideal) magnetohydrodynamics. In one-dimension, they are:

\[\begin{split} \begin{align*} \frac{\partial \rho}{\partial t} + \frac{\partial (\rho u)}{\partial x} &= 0 \\ \frac{\partial (\rho u)}{\partial t} + \frac{\partial}{\partial x} (\rho u^2 + p + \frac{B^2}{2} - \vec{B} \cdot \vec{B}) &= 0 \\ \frac{\partial (\rho E)}{\partial t} + \frac{\partial}{\partial x} \left [ (\rho E + p) u - \vec{B} (\vec{u}\cdot \vec{B})\right ] &= 0 \\ \frac{\partial \vec{B}}{\partial t} + \frac{\partial}{\partial x} (\vec{B}\vec{u} -\vec{u}\vec{B}) &= 0 \\ \end{align*} \end{split}\]

Here, \(E\) is the specific total energy which is related to the specific internal energy as:

\[ \rho E = \rho e + \frac{1}{2} \rho u^2, \]

\(\vec{B}\) is the magnetic field vector, and the system is closed by an equation of state:

\[p = p(\rho, e)\]

A common equation of state is a gamma-law EOS:

\[ p = \rho e (\gamma - 1)\]

where \(\gamma\) is a constant. For an ideal gas, \(\gamma\) is the ratio of specific heats, \(c_p / c_v\).

Conservative form

As expressed above, the ideal MHD equations are in conservative form. We can define the conservative state, \({\bf U}\) as:

\[\begin{split}{\bf U} = \left ( \begin{array}{c} \rho \\ \rho u \\ \rho E \\ \vec{B}\end{array} \right )\end{split}\]

and the flux, \({\bf F}({\bf U})\) as:

\[\begin{split}{\bf F} = \left ( \begin{array}{c} \rho u \\ \rho u^2 + p + \frac{B^2}{2} - \vec{B} \cdot \vec{B} \\ (\rho E + p) u - \vec{B} (\vec{u}\cdot \vec{B}) \\ \vec{B}\vec{u} - \vec{u}\vec{B} \end{array} \right )\end{split}\]

and then our system in conservative form is:

\[{\bf U}_t + [{\bf F}({\bf U})]_x = 0\]

and we can do the same technique of discretizing the domain into cells and integrating over the volume of a cell to get the finite-volume conservative update for the system:

\[\frac{\partial {\bf U}_i}{\partial t} = - \frac{1}{\Delta x} \left [ {\bf F}({\bf U}_{i+1/2}) - {\bf F}({\bf U}_{i+1/2}) \right ]\]

Primitive variable form

We again can express the MHD equations in terms of the primitive variables, \({\bf q}\):

\[\begin{split}{\bf q} = \left ( \begin{array}{c} \rho \\ u \\ p \end{array} \right )\end{split}\]

and the evolution equations are:

\[{\bf q}_t + {\bf A}({\bf q}) {\bf q}_x = 0\]

For Euler, we defined the speed of sound as:

\[c_s = \sqrt{\frac{\Gamma_1 p}{\rho}}\]

In MHD, we can also define the Alfven velocity:

\[c_A = \sqrt{\frac{B}{\sqrt{\mu_0 \rho}}}\]

and the fast magnetosonic speed

\[c_f = \sqrt{v_A^2+c_s^2}\]

Characteristic form

In MHD we get a more complex set of Eigenvalues. In total, we have 7: $$

(1)\[\begin{align} \lambda^{(-)} &= u - c_f \\ \lambda^{(-)} &= u - v_A \\ \lambda^{(-)} &= u - c_s \\ \lambda^{(0)} &= u \\ \lambda^{(+)} &= u + c_s \\ \lambda^{(+)} &= u + v_A \\ \lambda^{(+)} &= u + c_f \\ \end{align}\]

$$

These are the speeds at which information propagates in our system.