lorentzForce

Computes the lorentz force given from fluid variables, particle mass, charge and permittivity. This lorentz force would be used as a source term for fluid equations.

\[\notag \begin{align} s=\rho\frac{q}{m}\left( \begin{array}{c} 0\\ E_{x}+u_{y}B_{z}-u_{z}B_{y}\\ E_{y}+u_{z}B_{x}-u_{x}B_{z}\\ E_{z}+u_{x}B_{y}-u_{y}B_{x}\\ u_{x}\,E_{x}+u_{y}\,E_{y}+u_{z}\,E_{z}\\ \end{array} \right) \end{align}\]

where \(q\) is the species charge, \(m\) is the species mass \(\epsilon_{0}\) is the permittivity, \(\rho\) is the fluid mass density, \(u_{x}\) is the fluid x velocity, \(u_{y}\) is the fluid y velocity, \(u_{z}\) is the fluid z velocity, \(E_{x}\) is the x electric field, \(E_{y}\) is the y electric field, \(E_{z}\) is the z electric field, \(B_{x}\) is the x magnetic field, \(B_{y}\) is the y magnetic field and \(B_{z}\) is the z magnetic field.

In the case where the user wants the Lorentz term for the two-fluid form twoFluidEqn the source is written as

\[\notag \begin{align} s=\left( \begin{array}{c} 0\\ \rho_{c}E_{x}+j_{y}B_{z}-j_{z}B_{y}\\ \rho_{c}E_{y}+j_{z}B_{x}-j_{x}B_{z}\\ \rho_{c}E_{z}+j_{x}B_{y}-j_{y}B_{x}\\ 0\\ (r_{i}^{2}\rho_{i}+r_{e}^{2}\rho_{e})E_{x}+(r_{i}^{2}\rho_{i}u_{y\,i}+r_{e}^{2}\rho_{e}u_{y\,e})B_{z}-(r_{i}^{2}\rho_{i}u_{z\,i}+r_{e}^{2}\rho_{e}u_{z\,e})B_{y}\\ (r_{i}^{2}\rho_{i}+r_{e}^{2}\rho_{e})E_{y}+(r_{i}^{2}\rho_{i}u_{z\,i}+r_{e}^{2}\rho_{e}u_{z\,e})B_{x}-(r_{i}^{2}\rho_{i}u_{x\,i}+r_{e}^{2}\rho_{e}u_{x\,e})B_{z}\\ (r_{i}^{2}\rho_{i}+r_{e}^{2}\rho_{e})E_{z}+(r_{i}^{2}\rho_{i}u_{x\,i}+r_{e}^{2}\rho_{e}u_{x\,e})B_{y}-(r_{i}^{2}\rho_{i}u_{y\,i}+r_{e}^{2}\rho_{e}u_{y\,e})B_{x}\\ j_{x\,i}\,E_{x}+j_{y\,i}\,E_{y}+j_{z\,i}\,E_{z}\\ j_{x\,e}\,E_{x}+j_{y\,e}\,E_{y}+j_{z\,e}\,E_{z}\\ \end{array} \right) \end{align}\]

and this source can be chosen by choosing type=twoFluidEqn. The variables are defined as follows, \(r_{i}=q_{i}/m_{i}\) and \(r_{e}=q_{e}/m_{e}\) where \(q_{e}\) is the electron charge, \(q_{i}\) is the ion charge, \(m_{e}\) is the electron mass and \(m_{i}\) is the ion mass. In addition the variables \((\rho_{\alpha},u_{x\,\alpha},u_{y\,\alpha},u_{x\,\alpha})\) are the species mass density, species x velocity, species y velocity, and species z velocity. In this case \(\alpha\) represents the species, either \(e\) for electron or \(i\) for ion. In addition \((j_{x},j_{y},j_{z})\) are the total current densities in the x, y and z directions.

Parameters common to all systems

type (string)
The type of source is split5 (the default), or twoFluidEqn

Parameters (type=split5)

mass (float)
The mass of the fluid species
charge (float)
The charge of the fluid species

Parameters (type=twoFluidEqn)

electronMass (float)
The electron mass
ionMass (float)
The ion mass
electronCharge (float)
The electron charge
ionCharge (float)
The ion charge

Parent Updater Data (type=split5) Default

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

2nd Variable

  1. \(e_{x}\) x electric field
  2. \(e_{y}\) y electric field
  3. \(e_{z}\) z electric field
  4. \(b_{x}\) x magnetic field
  5. \(b_{y}\) y magnetic field
  6. \(b_{z}\) z magnetic field
out (string vector, required)

The output variable is a length 5 vector, but the first component is 0 so that it works simply as a fluid source for the euler equations.

1st Variable

  1. \(0.0\) mass density. No contribution from Lorentz force
  2. \(L_{x}\) x momentum density contribution of Lorentz force
  3. \(L_{y}\) y momentum density contribution of Lorentz force
  4. \(L_{z}\) z momentum density contribution of Lorentz force
  5. \(E\cdot J\) energy density contribution of Lorentz force

Parent Updater Data (type=twoFluidEqn)

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. \(\rho_{c}\) total charge density
  6. \(j_{x}\) x current density
  7. \(j_{y}\) y current density
  8. \(j_{z}\) z current density
  9. \(e_{i}\) ion energy density
  10. \(e_{e}\) electron energy density

2nd Variable

  1. \(e_{x}\) x electric field
  2. \(e_{y}\) y electric field
  3. \(e_{z}\) z electric field
  4. \(b_{x}\) x magnetic field
  5. \(b_{y}\) y magnetic field
  6. \(b_{z}\) z magnetic field

Example

<Source lorentzIon>
  kind = lorentzForce
  mass = ION_MASS
  charge = ION_CHARGE
</Source>

<Source lorentz>
  kind = lorentzForce
  type = twoFluidEqn
  ionMass = ION_MASS
  electronMass = ELECTRON_MASS
  ionCharge = ION_CHARGE
  electronCharge = ELECTRON_CHARGE
</Source>