# idealMhdEosEqn¶

Ideal MHD system. Eigensystem, eigenvalues and fluxes for the ideal MHD system.

\notag \begin{align} \frac{\partial}{\partial t}\left( \begin{array}{c} \rho \\ \rho\,u_{x} \\ \rho\,u_{y} \\ \rho\,u_{z} \\ e \\ B_{x} \\ B_{y} \\ B_{z} \\ \end{array} \right) + \nabla\cdot\left( \begin{array}{ccc} \rho\,u_{x} & \rho\,u_{y} & \rho\,u_{z} \\ \rho\,u_{x}^{2}+P - \frac{1}{\mu_{0}}B_{x}^{2} + \frac{1}{2\,\mu_{0}}B^{2}& \rho\,u_{x}\,u_{y}-\frac{1}{\mu_{0}}B_{x}\,B_{y} & \rho\,u_{x}\,u_{z}-\frac{1}{\mu_{0}}B_{x}\,B_{z} \\ \rho\,u_{y}\,u_{x}-\frac{1}{\mu_{0}}B_{x}\,B_{y} & \rho\,u_{y}\,u_{y} + P - \frac{1}{\mu_{0}}B_{y}^{2} + \frac{1}{2\,\mu_{0}}B^{2} & \rho\,u_{y}\,u_{z}-\frac{1}{\mu_{0}}B_{y}\,B_{z} \\ \rho\,u_{z}\,u_{x}-\frac{1}{\mu_{0}}B_{x}\,B_{z} & \rho\,u_{z}\,u_{y} -\frac{1}{\mu_{0}}B_{z}\,B_{y} & \rho\,u_{z}\,u_{z} + P - \frac{1}{\mu_{0}}B_{z}^{2} + \frac{1}{2\,\mu_{0}}B^{2} \\ u_{x}\left(e+P+\frac{1}{2\mu_{0}}B^{2}\right)-\frac{1}{\mu_{0}}B_{x}\,B\cdot u & u_{y}\left(e+P+\frac{1}{2\mu_{0}}B^{2}\right)-\frac{1}{\mu_{0}}B_{y}\,B\cdot u & u_{z}\left(e+P+\frac{1}{2\mu_{0}}B^{2}\right)-\frac{1}{\mu_{0}}\,B_{z}B\cdot u \\ 0 & -\left(u_{x}\,B_{y}-u_{y}\,B_{x}\right) & \left(u_{x}\,B_{z}-u_{z}\,B_{x}\right) \\ \left(u_{x}\,B_{y}-u_{y}\,B_{x}\right) & 0 & -\left(u_{y}\,B_{x}-u_{x}\,B_{y}\right) \\ -\left(u_{x}\,B_{z}-u_{z}\,B_{x}\right) & \left(u_{y}\,B_{x}-u_{x}\,B_{y}\right) & 0 \\ \end{array} \right) = 0 \end{align}

## Primary Variables (8)¶

8 primary variables

1. $$\rho$$ mass density
2. $$\rho\,u_{x}$$ x momentum density
3. $$\rho\,u_{y}$$ y momentum density
4. $$\rho\,u_{z}$$ z momentum density
5. $$e$$ energy density
6. $$B_{x}$$ x magnetic field
7. $$B_{y}$$ y magnetic field
8. $$B_{z}$$ z magnetic field

## Auxiliary variables (2)¶

1st auxiliary variable is the fluid pressure (1 component)

1. $$P$$ total fluid pressure (not magnetic pressure included)

2nd auxiliary variable is an estimate of the gas dynamic sound speed (1 component)

1. $$a$$ estimate of the fluid sound speed

## Parameters¶

mu0 (float)

Is the permeability of free space

basementPressure (float)

The minimum pressure allowed. Defaults to 0.

basementDensity (float)

The minimum density allowed. Defaults to 0.

Note

basementPressure and basementDensity are only used if correct=true

correctNans (boolean)

Tells whether or not nans occuring as a result of numerical eigensystem de-normalization should be corrected

correct (boolean)

Tells whether or not densities or pressures should be corrected when they fall below basement pressures or basement densities. When set to true pressure=max(basementPressure, pressure) and density = max(basementDensity, density)

## Example¶

An example idealMhdEos equation block is given below:

<Equation mhd>
kind = idealMhdEosEqn
basementDensity = $0.1*BASEMENT_DENSITY$
basementPressure = $0.1*BASEMENT_PRESSURE$
mu0 = MU0
</Equation>