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2 Equations and Phenomenology

In this section, we present the basic equations that are used to describe charged fluid flows, and the basic phenomenology of low-frequency turbulence. Readers interested in examining closely this subject can refer to the very wide literature on the subject of turbulence in fluid flows, as for example the recent books by, e.g., Pope (2000Jump To The Next Citation Point), McComb (1990), Frisch (1995Jump To The Next Citation Point) or many others, and the less known literature on MHD flows (Biskamp, 1993Jump To The Next Citation PointBoyd and Sanderson, 2003Biskamp, 2003). Plasma is seen as a continuous collisional medium so that all quantities are functions of space r and time t. Apart for the required quasi-neutrality, the basic assumption of MHD is that fields fluctuate on the same time and length scale as the plasma variables, say wtH -~ 1 and kLH -~ 1 (k and w are, respectively, the wave number and the frequency of the fields, while tH and LH are the hydrodynamic time and length scale, respectively). Since the plasma is treated as a single fluid, we have to take the slow rates of ions. A simple analysis shows also that the electrostatic force and the displacement current can be neglected in the non-relativistic approximation. Then, MHD equations can be derived as shown in the following sections.


 2.1 The Navier-Stokes equation and the Reynolds number
 2.2 The coupling between a charged fluid and the magnetic field
 2.3 Scaling features of the equations
 2.4 The non-linear energy cascade
 2.5 The inhomogeneous case
 2.6 Dynamical system approach to turbulence
 2.7 Shell models for turbulence cascade
 2.8 The phenomenology of fully developed turbulence: Fluid-like case
 2.9 The phenomenology of fully developed turbulence: Magnetically-dominated case
 2.10 Some exact relationships

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