  We re-analyze the Olesen arguments on the self-similarity properties of
freely evolving, nonhelical magnetohydrodynamic turbulence. We find that a
necessary and sufficient condition for the kinetic and magnetic energy spectra
to evolve self-similarly is that the initial velocity and magnetic field are
not homogeneous functions of space of different degree, to wit, the initial
energy spectra are not simple powers of the wavenumber with different slopes.
If, instead, they are homogeneous functions of the same degree, the evolution
is self-similar, it proceeds through selective decay, and the order of
homogeneity fixes the exponents of the power laws according to which the
kinetic and magnetic energies and correlation lengths evolve in time. If just
one of them is homogeneous, the evolution is self-similar and such exponents
are completely determined by the slope of that initial spectrum which is a
power law. The latter evolves through selective decay, while the other spectrum
may eventually experience an inverse transfer of energy. Finally, if the
initial velocity and magnetic field are not homogeneous functions, the
evolution of the energy spectra is still self-similar but, this time, the
power-law exponents of energies and correlation lengths depend on a single free
parameter which cannot be determined by scaling arguments. Also in this case,
an inverse transfer of energy may in principle take place during the evolution
of the system.
