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

The phenomenology of the magnetically-dominated case has been investigated by Iroshnikov (1963Jump To The Next Citation Point) and Kraichnan (1965Jump To The Next Citation Point), then developed by Dobrowolny et al. (1980bJump To The Next Citation Point) to tentatively explain the occurrence of the observed Alfvénic turbulence, and finally by Carbone (1993Jump To The Next Citation Point) and Biskamp (1993Jump To The Next Citation Point) 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 ± ( ±)2 ± e ~ dzl /Tl but the characteristic time ± Tl 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 t±l /tA (tA ~ l/cA < t±l is the Alfvén time), so that Tl± ~ (t±l )2/tA. Then, immediately
± ± ± [dzl-]2[dzl ]2 e ~ lcA .

This means that both ± modes are transferred at the same rate to small scales , namely e+ ~ e- ~ e, and this is the conclusion drawn by Dobrowolny et al. (1980bJump To The Next Citation Point). 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 e+ and e- (Carbone, 1993Jump To The Next Citation Point). Using the usual scaling law for fluctuations, it can be shown that the scaling behavior holds e-- > c1- 4he'. Then, when the energy transfer rate is constant, we found a scaling law different from that of Kolmogorov and, in particular,

dz±l ~ (ecA)1/4l1/4. (24)
Using this phenomenology the high-order moments of fluctuations are given by (p) p/4 Sl ~ l. Even in this case, qp = p/4 results to be a linear function of the order p. The pseudo-energy spectrum can be easily found to be
1/2 -3/2 E(k) ~ (ecA) k . (25)
This is the Iroshnikov-Kraichnan spectrum.
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