twoTemperatureMhdEosEqn

Two temperature MHD model with with general equation of state. The equations are solved in conservative form.

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

Primary Variables (9)

  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. \(e\) total energy density
  6. \(B_{x}\) x magnetic field
  7. \(B_{y}\) y magnetic field
  8. \(B_{z}\) z magnetic field
  9. \(e_{e}\) electron energy density

Auxiliary variables (6)

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

  1. \(P\) total pressure

2nd auxiliary variable is the electron pressure (1 component)

  1. \(P_{e}\) electron pressure

3rd auxiliary variable is the sound speed (1 component)

  1. \(a\) estimate of the speed of sound for determining the time step

4th auxiliary variable is current density (3 components)

  1. \(J_{x}\) x current density
  2. \(J_{y}\) y current density
  3. \(J_{z}\) z current density

5th auxiliary variable is electric field (3 components)

  1. \(E_{x}\) x electric field
  2. \(E_{y}\) y electric field
  3. \(E_{z}\) z electric field

6th auxiliary variable is current density (3 components)

  1. \(Z\) charge state

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.

fundamentalCharge (float)

elementary charge

ionMass (float)

mass of the ion

Example

An example twoTemperatureMhdEos equation block is given below::

<Equation twoTemperatureMhdEos>
  kind = twoTemperatureMhdEosEqn
  mu0 = MU0
  fundamentalCharge = CHARGE
  ionMass = MI
</Equation>