Turbulent transfer rates from small-scale MHD excitations to large-scale Fourier modes are calculated algebraically using the method of Biskamp and Welter (1983). Three cases are considered: two-dimensional Navier-Stokes flows, two-dimensional incompressible MHD, and the weakly three-dimensional Strauss equations. In all cases, an initially large spectral gap, between the small-scale and large-scale excitations is assumed, and attention focuses on the initial values of the back-transfer rates. The sign of the transfer is determined by the sign of an analytically calculable eddy viscosity and/or anomalous resistivity. The authors confirm the results of Biskamp and Welter for the case of two-dimensional MHD, but find some differences for the case of the Strauss equations. It is argued that the Strauss equations may not exhibit an inverse cascade phenomenon for the spatially periodic case unless their initial spectra are such that the behaviour is essentially that of two-dimensional MHD.
Validation and verification of a quasi two-dimensional turbulence model and an eddy detection method for liquid metal magnetohydrodynamics flows