Beyond ideal MHD:

Resistive, reactive and relativistic plasmas

• Ian Hawke
• Kyriaki Dionysopoulou
• Nils Andersson
• Greg Comer

• @IanHawke
• github.com/IanHawke
• orcid.org/0000-0003-4805-0309
• STAG, University of Southampton

1. Extend single fluid MHD with source;
2. Treat each species as separate fluid.

State of the art:

	Single fluid	Multi fluid
Phenomenological resistivity	GR: Palenzuela et al; Dionysopoulou et al	SR: Amano et al; Barkov et al
Physical closure resistivity	GR: Khanna et al	GR: This work.

Measure particle number $n^a_{\text{x}}$ using reference space volume. Master function $\Lambda (n^2_\text{xy}, A_b)$.

Conjugate momenta $\bar{\mu}_a^{\text{x}}$ need not be parallel to flux $n^a_{\text{x}}$ (entrainment).

Allowing reference space interactions gives resistive, dissipative terms.

Equations of motion

Variational calculation: $$ \begin{align} \nabla_a n^a_{\text{x}} &= \Gamma_{\text{x}}, \\ f^{\text{x}}_a + \Gamma_{\text{x}} \bar{\mu}^{\text{x}}_a &= R^{\text{x}}_a. \end{align} $$

Force $f^{\text{x}}_a$ balances reactions $\Gamma_{\text{x}}$ and resistivity $R^{\text{x}}_a.$

Variational constraints:

1. Resistivity written as projection of velocity differences;
2. Reactions fixed by resistivity;
3. EM gauge invariance, second law, fix symmetry of coefficients.
4. No consistent 2-fluid resistive form without reactions.

• 3d, GR, nonlinear.
• Based on Einstein Toolkit.
• No entrainment.
• Phenomenological resistivity.
• Explicit or IMEX in time.
• ES-WENO.
• Attains expected convergence rates.