### 13.1 Statistical description of MHD turbulence

When an MHD fluid is turbulent, it is impossible to know the detailed behavior of velocity field
and magnetic field , 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 and are equal to zero because the magnetic resistivity and the
viscosity 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 ,
cross-helicity , and magnetic helicity . The above quantities are defined as follows:
where and are the fluctuations of velocity and magnetic field, this last one expressed in Alfvén
units , and is the vector potential so that . 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 and , it is convenient to use
the normalized cross-helicity :

since this quantity simply varies between and .