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

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

MHD approximation good for NS interior and exterior

- on average;
- for different reasons.

Going beyond the ideal case needed to

- deal with highly dynamical situations;
- consistently link interior and exterior.

- Extend single fluid MHD with source;
- 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.

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:

- Resistivity written as projection of velocity differences;
- Reactions fixed by resistivity;
- EM gauge invariance, second law, fix symmetry of coefficients.
- 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.

- Multifluid formalism with coupled reference spaces gives
- equations of motions with dissipation and resistivity;
- constraints on form of resistive terms;
- in principle values of all terms.

- Code implementation in 3d GR
- is completed in simplified model;
- is practical for moderate sized problems;
- shows the (in)consistencies of phenomenological approaches.