23.4 Thermoelectric Coefficients

The thermoelectric unit physics/materialProperies/Thermoelectric provides routines for computing thermoelectric coefficients which govern the thermoelectric transport of both magnetic fields and heat. The coefficients included are: $\beta_\parallel$, $\beta_\perp$, and $\beta_\wedge$. These appear in Ohm's Law as follows:

$$e{\bf E}/k = -\beta_\parallel {\bf\nabla} T - (\beta_\perp - \bet... ...es {\bf\nabla} T) \times {\bf b} - \beta_\wedge ({\bf b} \times {\bf\nabla} T),$$ (23.7)

where $e$ is electron charge, $k$ is Boltzmann's constant, ${\bf b}$ is a unit vector in the direction of the B-field and $T$ is the electron temperature. The nomenclature in the literature is confusing, but in FLASH, we refer to these terms as Seebeck, CrossField, and Nernst, respectively. Each of these coefficients also appears in the electron heat flux equation, with transport driven by current rather than temperature gradients. There is also a fourth coefficient $\beta_{new}$ that replaces the combination of $(\beta_\parallel - \beta_\perp) / \chi$ in the CrossField term when the effective atomic number is greater than or equal to 1. All four coefficients are returned as dimensionless. See Sec:non_ideal_MHD for more details on how these coefficients are implemented.

23.4.1 Constant thermoelectric coefficients

This implementation returns constant thermoelectric coefficients as set by the following runtime parameters:

• thel_constantParallel: $\beta_{\parallel}$
• thel_constantPerpendicular: $\beta_{\perp}$
• thel_constantCross: $\beta_{\wedge}$
• thel_constantNew: $\beta_{new}$

The default value for all of these is zero. To activate the Constant thermoelectric implementation, simply add the appropriate path in the Config file of your simulation directory.

23.4.2 DaviesWen thermoelectric coefficients

This implementation returns thermoelectric coefficients based on the work by Davies, et al. (Phys. Plasmas, 2021). These coefficients are dependent on temperature, ionization, atomic number, and electron Hall parameter. They represent the only physically realistic implementation in FLASH, thus should be used in MHD HEDP applications where thermoelectric transport is important.

To activate the DaviesWen thermoelectric implementation, simply add the appropriate path in the Config file of your simulation directory.