**7. References**


54 Topics in Magnetohydrodynamics

Each of the three models tries to mimic one aspect of the complete theory of SCSs. The metriplectic MHD presents the non-Hamiltonian algebrization; the SFT for the resistive MHD is characterized by the presence of noise yielding a path integral approach; the fractal model of reconnection admits the irregular nature of MHD fields, involving the fractional

A large amount of work must still be done to imagine how those three approaches could be combined in a unique framework, the invoked "SCS Theory", reducing to the three models in different limits: this further research is for sure out of the subject of the work here, in which a flavour had to be given about some characteristics that this "SCS Theory" should

As a final remark, we underline that the self-consistent "SCS Theory" should present a sort of scale-covariance, because all the phenomena concerning plasma ISCs do involve multiscale dynamics. The technique of Renormalization Group will then be naturally applied to such a thory (see e.g. Chang et al., 1992 and references therein). A first direct application of such technique, using the exact full dynamic differential renormalization group for critical dynamics can be found in Chang et al. (1978). The use of Renormalization Group techniques to predict physical quantities to be compared with real spacecraft data is already well established (see e.g. Chang, 1999; Chang et al., 2004), and the results are very encouraging,

The authors are grateful to Philip J. Morrison of the Institute for Fusion Studies of the University of Texas in Austin, for useful discussions and criticism. The work of Massimo Materassi has been partially supported by EURATOM through the "Contratto di Associazione Euratom-Enea-CNR". The work of Emanuele Tassi was partially supported by the Agence Nationale de la Recherche (ANR GYPSI n. 2010 BLAN 941 03). This work was also supported by the European Community under the contract of Association between EURATOM, CEA, and the French Research Federation for fusion study. The views and opinions expressed herein do not necessarily reflect those of the European Commission. Emanuele Tassi acknowledges also fruitful discussions with the "Equipe de Dynamique

Giuseppe Consolini and Massimo Materassi did this work as a part of ISSI Team n. 185 "Dispersive cascade and dissipation in collisionless space plasma turbulence – observations

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**0**

**3**

Ravi Samtaney

*Kingdom of Saudi Arabia*

**Implicit Numerical Methods for**

*King Abdullah University of Science and Technology, Thuwal*

A fluid description of the plasma is obtained by taking velocity moments of the kinetic equations (Vlasov or Fokker-Planck equations) for electrons and ions and employing certain closure assumptions. A hierarchy of MHD models can be derived. Generally, if the time scales of interest are larger than the electron-ion collision time scales, then one may model the plasma as a single fluid. Furthermore, the fluid description of a plasma is valid when the length scales under investigation are larger than the Debye length; and the frequencies are smaller than the cyclotron frequency. The Debye length argument can also be cast in terms of a frequency: namely the plasma frequency. In addition, it is a standard assumption that the speeds involved are much smaller than the speed of light. The oft-used term "resistive MHD" is a single-fluid model of a plasma in which a single velocity and pressure describe both the electrons and ions. The resistive MHD model of a magnetized plasma does not include finite Larmor radius (FLR) effects, and is based on the simplifying limit in which

the particle collision length is small compared with the macroscopic length scales.

The scientific literature has numerous instances of methods and techniques to solve the MHD system of equations. To limit the scope of this chapter, we focus our discussion to single fluid resistive and ideal MHD. Although single fluid resistive (or ideal) MHD is in a sense the simplest fluid model for a plasma, these equations constitute a system of nonlinear partial differential equations, and hence pose many interesting challenges for numerical methods and simulations. In particular, there is a vast amount of literature devoted to numerical methods and simulations of resistive MHD wherein the time stepping method is explicit or semi-implicit. For example, in simulating MHD flows with shocks, shock-capturing methods from hydrodynamics have been tailored to MHD and have been very successfully used (see for example Reference Samtaney et al. (2005)). Such aforementioned shock-capturing methods almost exclusively employ explicit time stepping. This is entirely sensible given that the flow speeds are of the same order as, or exceed the fast wave speeds. In several physical situations, the diffusive time scales are much larger than the advective time scale. In these cases, the Lundquist number is large (*S* >> 1) and the diffusion terms are usually much smaller than the hyperbolic or wave-dominated terms in the equations. Usually the diffusion terms become important in thin boundary layers or thin current sheets within the physical domain. We are interested in computing such flows but with the additional constraint that

**1. Introduction**

**1.1 Scope of this chapter**

**Magnetohydrodynamics**

