generalizedOhmsLaw (1d, 2d, 3d)

The generalizedOhmsLaw updater computes the electric field determined by parameters in the generalized Ohm’s law.

\[\notag \begin{equation} \mathbf{E}=\eta\mathbf{J}+\left[-\left(\mathbf{J}-\mathbf{J}_{\mathrm{ion}}\right)\times \mathbf{B}+\nabla P_{e}\right] \left(n_{e} q_{e} \right)^{-1} \end{equation}\]

Data

in (string vector of 3, required)
Vector of Conserved Quantities (nodalArray, 9-components, required)

The vector of conserved quantities, \(\mathbf{q}\) has 9 entries:

  1. \(\rho\): mass density
  2. \(\rho\,u_{\hat{\mathbf{i}}} = \rho \mathbf{u} \cdot \hat{\mathbf{i}}\): momentum density in the \(\hat{\mathbf{i}}\) direction
  3. \(\rho\,u_{\hat{\mathbf{j}}} = \rho \mathbf{u} \cdot \hat{\mathbf{j}}\): momentum density in the \(\hat{\mathbf{j}}\) direction
  4. \(\rho\,u_{\hat{\mathbf{k}}} = \rho \mathbf{u} \cdot \hat{\mathbf{k}}\): momentum density in the \(\hat{\mathbf{k}}\) direction
  5. \(E = \frac{P}{\gamma -1} + \tfrac{1}{2}\rho|\mathbf{u}|^2 + \tfrac{1}{2}|\mathbf{b}|^2\): total energy density
  6. \(b_{\hat{\mathbf{i}}} = \mathbf{b} \cdot \hat{\mathbf{i}} = \mu^{-1/2}_{0} \mathbf{B} \cdot \hat{\mathbf{i}}\): magnetic field normalized by permeability of free-space in the \(\hat{\mathbf{i}}\) direction
  7. \(b_{\hat{\mathbf{j}}} = \mathbf{b} \cdot \hat{\mathbf{j}} = \mu^{-1/2}_{0} \mathbf{B} \cdot \hat{\mathbf{j}}\): magnetic field normalized by permeability of free-space in the \(\hat{\mathbf{j}}\) direction
  8. \(b_{\hat{\mathbf{k}}} = \mathbf{b} \cdot \hat{\mathbf{k}} = \mu^{-1/2}_{0} \mathbf{B} \cdot \hat{\mathbf{k}}\): magnetic field normalized by permeability of free-space in the \(\hat{\mathbf{k}}\) direction
Current Density (nodalArray, 3-components, required)

Vector of plasma currents:

  1. \(J_{\hat{\mathbf{i}}} = \mathbf{J} \cdot \hat{\mathbf{i}}\): current in the \(\hat{\mathbf{i}}\) direction
  2. \(J_{\hat{\mathbf{j}}} = \mathbf{J} \cdot \hat{\mathbf{j}}\): current in the \(\hat{\mathbf{j}}\) direction
  3. \(J_{\hat{\mathbf{k}}} = \mathbf{J} \cdot \hat{\mathbf{k}}\): current in the \(\hat{\mathbf{k}}\) direction
Charge State (nodalArray, 1-component, required)
\(Z\), the ionization state.
out (string vector, required)
Electric field (nodalArray, 3-components, required)

Vector of electric fields:

  1. \(E_{\hat{\mathbf{i}}} = \mathbf{E} \cdot \hat{\mathbf{i}}\): electric field in the \(\hat{\mathbf{i}}\) direction
  2. \(E_{\hat{\mathbf{j}}} = \mathbf{E} \cdot \hat{\mathbf{j}}\): electric field in the \(\hat{\mathbf{j}}\) direction
  3. \(E_{\hat{\mathbf{k}}} = \mathbf{E} \cdot \hat{\mathbf{k}}\): electric field in the \(\hat{\mathbf{k}}\) direction
resistivity (nodalArray, 1-component, optional)
Scalar resistivity: if this term exists, then the resistive term is included in the evaluation of E.
electronPressureDerivative (nodalArray, 3-components, optional)
\(\nabla P_e\) If this term is set then the diamagnetic drift term is used in determining the electric field.

Parameters

The generalizedOhmsLaw updater accepts the parameters below, in addition to those required by Updater:

idealTerm (boolean)
Set to false if the ideal term \(\mathbf{u}\times\mathbf{B}\) should be ignored, otherwise set to true. Defaults to true.
hallTerm (boolean)
Set to false if the Hall term should be ignored, otherwise set to true. Defaults to false.
fundamentalCharge (float)
The charge of a proton
ionMass (float)
mass of the ion. Currently assumes only one ion species
electronMass (float)
mass of the electron.
boltzmannConstant (float)
boltzmann’s constant

Example

<Updater computeE>
  kind = generalizedOhmsLaw1d
  onGrid = domain

  in = [q, J, Zbar]
  out = [E]
  electronPressureDerivative = gradPe
  resistivity = etaJ

  hallTerm = true

  fundamentalCharge = CHARGE
  ionMass = MI
  electronMass = ME
  boltzmannConstant = KB
</Updater>