Beyond ideal MHD:

Resistive, reactive and relativistic plasmas

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


MHD is not enough

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.


  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.

Multifluid comparison

Neutron star collapse

Neutron star collapse


  • 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.