Parametric investigation of self-similar decay laws in MHD turbulent flows

An investigation of the deat laws of energy and of higher moments of ht Els asser fields z(+/-) = v +/- b in the self-similar regime of magnetohydrodyn amic (MHD) turbulence is presented, using phenomenological models as well a s two-dimensional numerical simulations with periodic boundary conditions a nd up to 2048(2) grid points. The results are compared with the generalizat ion of the parameter-free model derived by Galtier et al. [Phys. Rev. Lett. 79, 2807 (1997)], which takes into account the slowing down of the dynamic s due to the propagation of Alfven waves. The new model developed here allo ws for a study in terms of one parameter governing the wavenumber dependenc e of the energy spectrum at scales of the order of (and larger than) the in tegral scale of the flow. The one-dimensional compressible case is also dea lt with in two of its simplest configurations. Computations are performed f or a standard Laplacian diffusion as well as with a hyperdiffusive algorith m. The results are sensitive to the amount of correlation between the veloc ity and the magnetic field, but rather insensitive to all other parameters such as the initial ratio of kinetic to magnetic energy or the presence or absence of a uniform component of the magnetic field. In all cases, the dec ay is significantly slower than for neutral fluids in a way that favours fo r MHD flows the phenomenology of Iroshnikov [Soviet Astron. 7, 566 (1963)] and Kraichnan [Phys. Fluids 8, 1385 (1965)] as opposed to that of Kolmogoro v [Dokl. Akad. Nauk. SSSR 31, 538 (1941)]. The temporal evolution of q-mome nts of the generalized vorticities <\omega(+/-)\(q)> = <\omega +/- j\(q)> u p to order q = 10 is also given, and is compared with the prediction of the model. Less agreement obtains as q grows - a fact probably due to intermit tency and the development of coherent structures in the form of eddies and of vorticity and current sheets.