8 - Two-dimensional turbulence  pp. 161-182

Two-dimensional turbulence

By Dieter Biskamp

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In the previous chapters we assumed, explicitly or tacitly, that turbulence is isotropic, in particular that there is no mean magnetic field, so that spectra depend only on the modulus of the wave vector k and structure functions only on the distance l between two points. We also assumed that the kinetic and magnetic energies have similar magnitudes, which implies that the magnetic field is distributed in a space-filling way. However, even if the turbulence is globally isotropic, the local dynamics is not, differing strongly between the directions parallel and perpendicular to the local magnetic field. As discussed in Section 5.3.3, the small-scale fluctuations are dominated by perpendicular modes, and Alfvén waves propagating parallelly are only weakly excited, which gives rise to the observed Kolmogorov energy spectrum k−5/3 instead of the IK spectrum k−3/2.

In nature magnetic turbulence often occurs about a mean magnetic field, just as hydrodynamic turbulence occurs about a mean flow. However, whereas the latter can be eliminated by transforming to a moving coordinate system, the presence of a mean magnetic field has a strong effect on the turbulent dynamics. If this field is much larger than the fluctuation amplitude, turbulence becomes essentially 2D in the plane perpendicular to the field, since the stiffness of field lines suppresses magnetic fluctuations, Alfvén waves, with short wavelengths along the field. Hence turbulent motions tend to simply displace field lines without bending them. To deal with this situation quantitatively we derive a set of equations for a plasma embedded in a strong magnetic field B0 = B0ez.