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13.1 Statistical description of MHD turbulence

When an MHD fluid is turbulent, it is impossible to know the detailed behavior of velocity field v(x,t) and magnetic field b(x,t), and the only description available is the statistical one. Very useful is the knowledge of the invariants of the ideal equations of motion for which the dissipative terms 2 m \~/ b and 2 n \~/ v are equal to zero because the magnetic resistivity m and the viscosity n are both equal to zero. Following Frisch et al. (1975) there are three quadratic invariants of the ideal system which can be used to describe MHD turbulence: total energy E, cross-helicity Hc, and magnetic helicity Hm. The above quantities are defined as follows:
1 2 2 E = -< v + b >, (55) 2
Hc =< v .b >, (56)
Hm =< A .B >, (57)
where v and b are the fluctuations of velocity and magnetic field, this last one expressed in Alfvén units (b- --> V~ -b-) 4pr, and A is the vector potential so that B = \~/ × A. The integrals of these quantities over the entire plasma containing regions are the invariants of the ideal MHD equations:
integral 1- 2 2 3 E = 2 (v + b )d x, (58)
1 integral 3 Hc = 2 (v .b)d x, (59)
integral 3 Hm = (A .B)d x, (60)

In particular, in order to describe the degree of correlation between v and b, it is convenient to use the normalized cross-helicity sc:

s = 2Hc-, (61) c E
since this quantity simply varies between +1 and - 1.
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