Ohmic current density

The ohmic current density is tracked with the unknown j_ohm, and is always calculated self-consistently from other unknowns.

Unless the ohmic current is resolved on the kinetic grid, it is given by

\[\frac{j_\mathrm{ohm}}{B} = \sigma \frac{ \langle \boldsymbol{E}\cdot\boldsymbol{B} \rangle }{ \langle B^2 \rangle }\]

where \(\sigma\) denotes the conductivity, and depends on the plasma parameters (most sensitively to the cold electron temperature T_cold). When the ohmic current is resolved by the distribution function, it is defined as

\[\frac{j_\mathrm{ohm} + j_\mathrm{hot}}{B} = \frac{2\pi e}{B_\mathrm{min}} \int \mathrm{d}p \mathrm{d}\xi_0 \,\frac{\mathcal{V}'}{V'} \Theta v f_\mathrm{hot},\]

where \(\Theta\) is zero for trapped particles and equals unity for passing particles, and the contribution from the hot-electron current is denoted j_hot (typically defined as a similar integral moment, but integrated for momenta above some cutoff).

Conductivity formulas

The conductivity can be calculated using different models for the conductivity. DREAM implements the model described in O Sauter, C Angioni and YR Lin-Liu (1999), which is a numerical fit of the neoclassical conductivity to Fokker-Planck simulations across all collisionality regimes (and not just the banana regime, which is available with kinetic simulations in DREAM). In the Sauter paper, the classical conductivity (in the absence of neoclassical effects, i.e. in cylindrical geometry) appears as a free parameter, which we have chosen as the relativistic conductivity given in BJ Braams and CFF Karney (1989), where we interpolate in the values they provide in Table 1.

The conductivity model can be controlled with the following settings:

Name	Description
CONDUCTIVITY_MODE_BRAAMS	Uses the cylindrical conductivity, ignoring all neoclassical effects
CONDUCTIVITY_MODE_SAUTER_COLLISIONLESS	Uses the Sauter formula in the collisionless limit, representing the banana-regime conductivity
CONDUCTIVITY_MODE_SAUTER_COLLISIONAL	Uses the full expression given in Eqs (13a-b) in the Sauter paper

Note

The default conductivity mode in DREAM is CONDUCTIVITY_MODE_SAUTER_COLLISIONLESS as this corresponds to the same approximations as used in the kinetic equation.

Corrected conductivity

When resolving ohmic current kinetically in DREAM, an order-unity error is incurred from the fact that we do not include the electron-electron field-particle collision operator. At an effective charge of \(Z_\mathrm{eff} = 1\), the error is approximately a factor of 2, but approaches 1 as the effective charge goes to infinity.

In order to correct for this error, DREAM allows the option to add an ohmic current correction on top of the kinetic contribution to j_ohm. An extensive numerical scan of kinetic simulations has revealed that the ratio of numerical conductivity (obtained in DREAM with a test-particle collision operator) to real (as described by the Braams-Karney model) is well described by

\[\frac{\sigma_\mathrm{numerical}}{\sigma} \approx 1 + \frac{1}{b+Z_\mathrm{eff}},\]

where we numerically determined \(a=-1.406\) and \(b=1.888\), yielding a maximum relative error less than \(1\%\) across a large range of plasma temperatures and charge.

The conductivity corretion used in DREAM is controlled with the following settings:

Name	Description
CORRECTED_CONDUCTIVITY_DISABLED	No correction is added.
CORRECTED_CONDUCTIVITY_ENABLED	A correction is added of the form \(\Delta j_\mathrm{hot} = (\sigma - \sigma_\mathrm{numerical})\frac{ \langle \boldsymbol{E}\cdot\boldsymbol{B} \rangle }{ \langle B^2 \rangle}\)

Note

The conductivity correction is only active when the hot-tail grid is enabled, when using COLLISION_FREQENCY_MODE_FULL and when resolving the pitch-angle distribution.

Example

The conductivity mode and correction can be set according to the following example:

import DREAM.Settings.Equations.OhmicCurrent as JOhm

ds = DREAMSettings()

ds.eqsys.j_ohm.setConductivityMode(JOhm.CONDUCTIVITY_MODE_SAUTER_COLLISIONAL)
ds.eqsys.j_ohm.setCorrectedConductivity(JOhm.CORRECTED_CONDUCTIVITY_ENABLED)