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.