Magnetohydrodynamic Turbulence
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Introduction Turbulence remains one of the last unresolved problems of classical physics. Turbulent flows in electrically conducting media represent an important aspect of this problem, because of their general importance for the evolution of astroand geophysical plasmas [10, 87]. Turbulence in plasmas, i.e. ionized gases, also offers valuable insights into the not yet fully understood nonlinear dynamics of spectral cascades and structure formation due to the presence or generation of magnetic fields (see, e.g. [73, 95, 96]). These allow additional diagnostic access to the underlying nonlinear interaction of turbulent fluctuations. In experimental devices for thermonuclear fusion the magnetically confined hot plasma is basically collisionless and requires kinetic treatment. Exceptions are the thin and comparably cool edge layer near the vessel boundaries [94] and plasmas in reversed-field pinch configurations [71]. In contrast turbulent plasmas in or beyond the earth often allow a fluid description due to the immense size of the dynamical regions and associated time-scales of interest compared to the effective mean-free-path and the frequencies related to the plasma particles [3]. Since plasma turbulence is a fully nonlinear problem comprising the dynamics of many interacting degrees of freedom, the relatively simple single fluid description of magnetohydrodynamics (MHD) represents a sensible starting point for theoretical and numerical investigations. While it is often desirable to include additional and more complex physical components in the model of the turbulent medium, we will refrain from doing so in this chapter not to obfuscate the inherent properties of MHD turbulence. The interest in these properties lies mainly in their potential universality, that is to say the inherent properties of turbulence might well be important for the dynamics of systems involving additional physics, e.g. gravity, radiation, rotation, or convection. For additional simplicity of the MHD description, the mass density of the magnetofluid is assumed to be constant in time and spatially uniform, ρ = ρ0 = 1. In addition relativistic effects are neglected and fluid velocities are M¨ uller, W.-C.: Magnetohydrodynamic Turbulence. Lect. Notes Phys. 756, 223–254 (2009) c Springer-Verlag Berlin Heidelberg 2009 DOI 10.1007/978-3-540-78961-1 6
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assumed to be much smaller than the magnetosonic speeds in the plasma. The flow can therefore be regarded as incompressible, dρ/dt = 0 (cf. for example, [92]). This condition though rarely fulfilled in realistic plasma flows yields another simplification of the problem by reducing the continuity equation, d ρ + ρ∇ · v = 0 dt to a simple solenoidality constraint on the velocity field, ∇ · v = 0. The dimensionless incompressible MHD equations governing the motions of an electrically conducting fluid on large space- and timescales on which fluctuations of the electrical charge density are levelled out quasi-instantaneously and the kinetic nature of the medium becomes invisible then are ˆΔv , ∂t v = −v
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