An investigation of the decay laws of energy and of higher moments of the
Elsässer fields z±=v±b
in the self-similar regime of magnetohydrodynamic
(MHD) turbulence is presented, using phenomenological models as well as two-dimensional numerical simulations with periodic boundary conditions and up
to 20482 grid points. The results are compared with the generalization 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 dynamics due to the
propagation of Alfvén waves. The new model developed here allows for a study
in terms of one parameter governing the wavenumber dependence of the energy
spectrum at scales of the order of (and larger than) the integral scale of the flow.
The one-dimensional compressible case is also dealt with in two of its simplest
configurations. Computations are performed for a standard Laplacian diffusion
as well as with a hyperdiffusive algorithm. The results are sensitive to the
amount of correlation between the velocity 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 decay is significantly slower than for neutral fluids in a way that
favours for MHD flows the phenomenology of Iroshnikov
[Soviet Astron.7, 566
(1963)] and Kraichnan [Phys. Fluids8, 1385 (1965)]
as opposed to that of
Kolmogorov [Dokl. Akad. Nauk. SSSR31, 538 (1941)].
The temporal evolution of q-moments of the generalized vorticities
〈[mid ]ω±[mid ]q〉
=〈[mid ]ω±j[mid ]q〉
up 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 intermittency and
the development of coherent structures in the form of eddies, and of vorticity
and current sheets.