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Incompressible MHD is described by two solenoidal vector fields, the velocity, v , and the magnetic intensity, B . These must simultaneously satisfy the Navier Stokes and induction equations.
It might seem that the addition of the induction equation would make MHD turbulence a much harder problem than HD turbulence.
It is certainly richer, offering a greater variety of solutions than HD turbulence. However, in some ways it is also simpler to analyze. Unlike HD turbulence, MHD turbulence has a weak limit which is amenable to perturbation theory.
This is a consequence of the existence of linear MHD waves.
In the incompressible limit, these are of two types, referred to as shear and pseudo Alfvén waves. The latter is the incompressible vestige of the slow mode of compressible MHD.
The displacement vector of a shear Alfvén wave is perpendicular to the plane defined by its wave vector, k , and a uniform background magnetic field, B0 , whereas that of a pseudo Alfvén wave lies in this plane.
The two wave modes share the dispersion relation, ?2=(k?B0)24???(k?vA)2,(1)
and propagate with group velocity, vA , either parallel or antiparallel to B0 depending upon the sign of k? .
Three properties of incompressible MHD form the bulwark of current models of MHD turbulence. In the limit of vanishing viscosity and resistivity:
Wavepackets which propagate in one direction along the magnetic field are exact solutions of the nonlinear equations of MHD.
Collisions between oppositely directed wave packets cause distortions that give rise to the turbulent cascade.
Wavepackets do not exchange energy when they collide.
The first and second items are immediate consequences of the equations of ideal MHD written in terms of Elsässer variables, z±=v±b , where b is the magnetic fluctuation about a uniform background field expressed in velocity units according to b??B/(4??)1/2 .
