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3.3 Going further

Like any approximation, ideal MHD has its own range of validity. This is mainly set by the relevant length and time scales of the system under study. The former needs to be typically much larger than the Larmor radius of the plasma particles, while the latter needs to be much shorter than the time scale of the collisions among the plasma particles. Furthermore, it is assumed that such collisions yield no resistivity, which permits one to neglect any diffusion of the magnetic field lines. There are nevertheless examples in astrophysics in which resistivity effects may become important and ideal MHD may not be valid to describe the dynamics of the plasma, typically in situations involving dilute accretion flows. For example, the Hall effect in protostellar disks is one of those effects, with significant implications regarding the magneto-rotational instability, the mechanism believed to regulate angular momentum transport in disks through sustained accretion. In addition, reconnection of magnetic field lines (in, e.g., solar flares) is another important nonideal MHD effect, which may appear in turbulent plasmas near the scale of energy dissipation. When the fluid is no longer a perfect conductor, kinetic effects become important and the particle-in-cell (or Monte Carlo) methods are usually employed instead of finite-volume methods.

Magnetic reconnection and dissipation processes are in particular recognized to be very important in high-energy astrophysics. Such processes not only affect the global dynamics of the plasma in scenarios involving compact objects, but are also believed to account for the production of the high-energy emission. Magnetic-field–line reconnection has actually been already observed in the existing numerical simulations based upon ideal GRMHD. However, this can only be explained with purely numerical reasons due to the nonvanishing amount of numerical artificial resistivity inherent to every computer code. Only very recently have a few approaches been put forward to develop numerical schemes to handle relativistic plasma with physical resistivity, involving the formulation and solution of the resistive relativistic MHD equations [409204].

Another interesting area in which new results are likely to be found may come from the coupling of GRMHD codes with particle codes. While the former type of codes give a good description of the large-scale dynamic processes, the latter are particularly suited to capturing the radiative processes operative at a microphysical scale in highly-relativistic objects, whose correct modelling may help extract and interpret observational diagnostics. Monte Carlo methods are extremely powerful in such aspects, as recently shown in the context of GRBs by [141], who confirmed the generation of strong electromagnetic fields by a Weibel-like instability in collisionless shocks and demonstrated their macroscopic propagation. When suitably coupled to GRMHD codes, particle codes could use the motivated initial and boundary conditions supplied by the magneto-fluid model to perform local particle-in-cell simulations.

Finally, it is worth noting that in some situations it may suffice to solve the equations of force-free electrodynamics as an alternative to solving the full set of GRMHD equations. In an astrophysical context this approach, valid in the low inertia limit of MHD, has been considered by, e.g., [203Jump To The Next Citation Point250] to model neutron star and black-hole magnetospheres. On the one hand, the force-free equations of motion can also be cast as a set of conservation laws (amenable to be solved using the same techniques as for the full GRMHD system) and may be easier to integrate in regions where the rest-mass energy is much smaller than the magnetic energy density. On the other hand, if the inertia of the plasma particles is not negligible, the force-free approximation is inaccurate as it cannot capture the thermal energy evolution of the particles as well as their motion along field lines. Finally, the question of the likelihood of this approach to hold within current sheets, due to the tearing mode instability of magnetic field lines reconnection, should also be investigated.

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