mhdSrc

Many of the MHD system require source terms as the hyperbolic part is not sufficient to describe the simple system. This source contains the source terms that should be added to these models.

Sources include:

The idealMHD source terms. The ideal MHD case only applies when one does not convert the source terms to a conservative flux. Note that we do not need separate source for general equation of state since the source terms are independent of the relation between internal energy and pressure.

\[\notag \begin{align} s=\left( \begin{array}{c} 0\\ J_{y}B_{z}-J_{z}B_{y}\\ J_{z}B_{x}-J_{x}B_{z}\\ J_{x}B_{y}-J_{y}B_{x}\\ J_{x}E_{x}+J_{y}E_{y}+J_{z}E_{z}\\ 0\\ 0\\ 0\\ \end{array} \right) \end{align}\]

The gasDynamicMhdEqn source terms (which include an electron energy equation)

\[\notag \begin{align} s=\left( \begin{array}{c} 0\\ J_{y}B_{z}-J_{z}B_{y}\\ J_{z}B_{x}-J_{x}B_{z}\\ J_{x}B_{y}-J_{y}B_{x}\\ J_{x}E_{x}+J_{y}E_{y}+J_{z}E_{z}\\ 0\\ 0\\ 0\\ 0\\ J_{x\,e}E_{x}+J_{y\,e}E_{y}+J_{z\,e}E_{z}\\ \end{array} \right) \end{align}\]

The twoTemperatureMhdEqn source terms

\[\notag \begin{align} s=\left( \begin{array}{c} 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ J_{x\,e}E_{x}+J_{y\,e}E_{y}+J_{z\,e}E_{z}\\ \end{array} \right) \end{align}\]

Parameters (twoTemperatureMhdEqn and twoTemperatureMhdEosEqn)

ionMass (float)
The mass of an ion
fundamentalCharge
proton charge

Parent Updater Data (idealMhd)

in (string vector, required)

1st Variable

  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

2nd Variable (current density)

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

3rd Variable (electric field)

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

Parent Updater Data (twoTemperatureMhdEqn and gasDynamicMhdEqn)

in (string vector, required)

1st Variable

  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
  9. \(Phi\) correction potential
  10. \(e_{e}\) electron energy

2nd Variable (current density)

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

3rd Variable (electric field)

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

4th Variable (charge state)

  1. \(Z\) charge state

Parent Updater Data (twoTemperatureMhdEosEqn)

in (string vector, required)

1st Variable

  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
  9. \(e_{e}\) electron energy equation

2nd Variable (current density)

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

3rd Variable (electric field)

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

4th Variable (charge state)

  1. \(Z\) charge state

Example

<Source axisymmetricSource>
  kind = mhdSrc
  model = gasDynamicMhdSrc
</Source>