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Incompressible turbulent flows are investigated in the framework of ideal magnetohydrodynamics. All the field quantities vary with only two spatial dimensions. Equilibrium canonical distributions are determined in a phase space whose co-ordinates are the real and imaginary parts of the Fourier coefficients for the field variables. In the geometry considered, the magnetic field and fluid velocity have variable x and y components, and all field quantities are independent of z. Three constants of the motion are found (one of them new) which survive the truncation in Fourier space and permit the construction of canonical distributions with three independent temperatures. Spectral densities are calculated. One of the more novel physical effects is the appearance of macroscopic structures involving long wavelength, self-generated, magnetic fields (‘magnetic islands’) for a wide range of initial parameters. Current filaments show a tendency toward consolidation in much the same way that vorticity filaments do in the guiding-centre plasma case. In the presence of finite dissipation, energy cascades to higher wavenumbers can be accompanied by vector potential cascades to lower wavenumbers, in much the same way as, in the fluid dynamic (Navier-Stokes) case, energy cascades to lower wavenumbers accompany enstrophy cascades to higher wavenumbers. It is suggested that the techniques may be relevant to theories of the magnetic dynamo problem and to the generation of megagauss magnetic fields when pellets are irradiated by lasers.
Fyfe et al. (Fri,) studied this question.