Mean-field diffusivities in passive scalar and magnetic transport in
  irrotational flows
  Certain aspects of the mean-field theory of turbulent passive scalar
transport and of mean-field electrodynamics are considered with particular
emphasis on aspects of compressible fluids. It is demonstrated that the total
mean-field diffusivity for passive scalar transport in a compressible flow may
well be smaller than the molecular diffusivity. This is in full analogy to an
old finding regarding the magnetic mean-field diffusivity in an electrically
conducting turbulently moving compressible fluid. These phenomena occur if the
irrotational part of the motion dominates the vortical part, the P\`eclet or
magnetic Reynolds number is not too large, and, in addition, the variation of
the flow pattern is slow. For both the passive scalar and the magnetic cases
several further analytical results on mean-field diffusivities and related
quantities found within the second-order correlation approximation are
presented, as well as numerical results obtained by the test-field method,
which applies independently of this approximation. Particular attention is paid
to non-local and non-instantaneous connections between the turbulence-caused
terms and the mean fields. Two examples of irrotational flows, in which
interesting phenomena in the above sense occur, are investigated in detail. In
particular, it is demonstrated that the decay of a mean scalar in a
compressible fluid under the influence of these flows can be much slower than
without any flow, and can be strongly influenced by the so-called memory
effect, that is, the fact that the relevant mean-field coefficients depend on
the decay rates themselves.
