Derivation and precision of mean field electrodynamics with mesoscale
  fluctuations
  Mean field electrodynamics (MFE) facilitates practical modeling of secular,
large scale properties of astrophysical or laboratory systems with
fluctuations.Practitioners commonly assume wide scale separation between mean
and fluctuating quantities, to justify equality of ensemble and spatial or
temporal averages.Often however, real systems do not exhibit such scale
separation. This raises two questions: (I) what are the appropriate generalized
equations of MFE in the presence of mesoscale fluctuations? (II) how precise
are theoretical predictions from MFE? We address both by first deriving the
equations of MFE for different types of averaging, along with mesoscale
correction terms that depend on the ratio of averaging scale to variation scale
of the mean. We then show that even if these terms are small, predictions of
MFE can still have a significant precision error. This error has an intrinsic
contribution from the dynamo input parameters and a filtering contribution from
differences in the way observations and theory are projected through the
measurement kernel.Minimizing the sum of these contributions can produce an
optimal scale of averaging that makes the theory maximally precise.The
precision error is important to quantify when comparing to observations because
it quantifies the resolution of predictive power. We exemplify these principles
for galactic dynamos, comment on broader implications, and identify
possibilities for further work.
