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:

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:

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

since this quantity simply varies between $+1$ and $- 1$ .