Induction equation

The magnetic field is updated via the induction equation:

\[\frac{\partial {\bf B}}{\partial t} = \nabla \times \left[{ {\bf v} \times {\bf B}} \, - \underbrace{ \eta_{\rm O} {\bf J}}_{\text{Ohmic diffusion}} \, - \underbrace{\eta_{\rm H} \left({\bf J} \times {\bf \hat{B}}\right)}_{\text{Hall effect}} + \underbrace{\eta_{\rm A} \left({\bf J}\times {\bf \hat{B}} \right)\times {\bf \hat{B}} }_{\text{Ambipolar diffusion}} \right]\]

where \({\bf J} = \nabla \times {\bf B}/\mu_0\) is the current density, and \(\hat{\bf B}\) is the unit vector in the direction of \({\bf B}\). The labeled terms correspond to the non-ideal MHD terms.

Both the Ohmic and Ambipolar diffusion terms are added to the EMFs before updating the magnetic field (see Benitez-Llambay & Masset (2016) for details). The Hall term is treated with the method described in appendix A Krapp et al. (2018) (see also Bai (2014)).

Warning

To activate the non-ideal MHD terms, -DOHMICDIFFUSION, -DAMBIPOLARDIFFUSION or -DHALLEFFECT must be defined in the .opt file.

Diffusivities

The user is responsible for defining the diffusivities \(\eta_{\rm O}, \eta_{\rm H}, \eta_{\rm A}\). These are defined as the Fields EtaOhm, EtaHall, EtaAD, and filled in the source files:

• src/nimhd_ohmic_diffusion_coeff.c

• src/nimhd_hall_effect_coeff.c

• src/nimhd_ambipolar_diffusion_coeff.c

The resistivities are then filled every time step in src/main.c, before calling the CFL condition. In the standard implementation, the diffusitivies are filled with a “constant” value given by the parameters:

• OhmicDiffusionCoeff (global variable OHMICDIFFUSIONCOEFF)

• HallEffectCoeff (global variable HALLEFFECTCOEFF)

• AmbipolarDiffusionCoeff (global variable AMBIPOLARDIFFUSIONCOEFF)

The value of these parameters should be defined in the .par file.

Note

If the diffusivities do not evolve with time, the user can add a return call after the first initialization (see e.g., setups/mri/nimhd_ohmic_diffusion_coeff.c).