## 5 Numerical Simulations

Numerical simulations currently represent one of the main source of information about non-linear
evolution of fluid flows. The actual super-computers are now powerful enough to simulate equations
(NS or MHD) that describe turbulent flows with Reynolds numbers of the order of in
two-dimensional configurations, or in three-dimensional one. Of course, we are far from achieving
realistic values, but now we are able to investigate turbulence with an inertial range extended for
more than one decade. Rather the main source of difficulties to get results from numerical
simulations is the fact that they are made under some obvious constraints (say boundary conditions,
equations to be simulated, etc.), mainly dictated by the limited physical description that we are
able to use when numerical simulations are made, compared with the extreme richness of the
phenomena involved: numerical simulations, even in standard conditions, are used tout court
as models for the solar wind behavior. Perhaps the only exception, to our knowledge, is the
attempt to describe the effects of the solar wind expansion on turbulence evolution by Velli
et al. (1989, 1990). Even with this far too pessimistic point of view, used here solely as a few words of
caution, simulations in some cases were able to reproduce some phenomena observed in the solar
wind.
Nevertheless, numerical simulations have been playing a key role, and will continue to do so in our
seeking an understanding of turbulent flows. Numerical simulations allows us to get information that cannot
be obtained in laboratory. For example, high resolution numerical simulations provide information at every
point on a grid and, for some times, about basic vector quantities and their derivatives. The number of
degree of freedom required to resolve the smaller scales is proportional to a power of the Reynolds number,
say to , although the dynamically relevant number of modes may be much less. Then one of the
main challenge remaining is how to handle and analyze the huge data files produced by large simulations (of
the order of Terabytes). Actually a lot of papers appeared in literature on computer simulations related to
MHD turbulence. The interested reader can look at the book by Biskamp (1993) and the reviews by
Pouquet (1993, 1996).