### 2.9 The phenomenology of fully developed turbulence: Magnetically-dominated case

The phenomenology of the magnetically-dominated case has been investigated by Iroshnikov (1963) and
Kraichnan (1965), then developed by Dobrowolny et al. (1980b) to tentatively explain the occurrence of
the observed Alfvénic turbulence, and finally by Carbone (1993) and Biskamp (1993) to get scaling laws
for structure functions. It is based on the Alfvén effect, that is, the decorrelation of interacting
eddies, which can be explained phenomenologically as follows. Since non-linear interactions
happen only between opposite propagating fluctuations, they are slowed down (with respect
to the fluid-like case) by the sweeping of the fluctuations across each other. This means that
but the characteristic time required to efficiently transfer energy from an
eddy to another eddy at smaller scales cannot be the eddy-turnover time, rather it is increased
by a factor ( is the Alfvén time), so that . Then,
immediately
This means that both modes are transferred at the same rate to small scales , namely ,
and this is the conclusion drawn by Dobrowolny et al. (1980b). In reality, this is not fully correct, namely
the Alfvén effect yields to the fact that energy transfer rates have the same scaling laws for modes
but, we cannot say anything about the amplitudes of and (Carbone, 1993). Using the usual
scaling law for fluctuations, it can be shown that the scaling behavior holds . Then, when the
energy transfer rate is constant, we found a scaling law different from that of Kolmogorov and, in particular,

Using this phenomenology the high-order moments of fluctuations are given by . Even in this
case, results to be a linear function of the order . The pseudo-energy spectrum can be easily
found to be
This is the Iroshnikov-Kraichnan spectrum.