# 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>