Mean fields and fluctuations in compressible magnetohydrodynamic flows
  We apply Gaussian smoothing to obtain mean magnetic field, density, velocity,
and magnetic and kinetic energy densities from our numerical model of the
interstellar medium, based on three-dimensional magnetohydrodynamic equations
in a shearing box $1 \times 1 \times 2 \, {\rm kpc}$ in size. The interstellar
medium is highly compressible, as the turbulence is transonic or supersonic; it
is thus an excellent context in which to explore the use of smoothing to
represent physical variables in a compressible medium in terms of their mean
and fluctuating parts. Unlike alternative averaging procedures, such as
horizontal averaging, Gaussian smoothing retains the three-dimensional
structure of the mean fields. Although Gaussian smoothing does not obey the
Reynolds rules of averaging, physically meaningful and mathematically sound
central statistical moments are defined as suggested by Germano (1992). We
discuss methods to identify an optimal smoothing scale $\ell$ and the effects
of this choice on the results. From spectral analysis of the magnetic, density
and velocity fields, we find a suitable smoothing length for all three fields,
of $\ell \approx 75 \, {\rm pc}$. Such a smoothing scale is likely to be
sensitive to the choice of simulation parameters; this may be considered in
future work, but here we just explore the methodology. We discuss the
properties of third-order statistical moments in fluctuations of kinetic energy
density in compressible flows, and suggest their physical interpretation. The
mean magnetic field, amplified by a mean-field dynamo, significantly alters the
distribution of kinetic energy in space and between scales, reducing the
magnitude of kinetic energy at intermediate scales. This intermediate-scale
kinetic energy is a useful diagnostic of the importance of SN-driven outflows.
