Derivation and precision of mean field electrodynamics with mesoscale fluctuations

TL;DR

This paper derives generalized mean field electrodynamics equations accounting for mesoscale fluctuations and quantifies the resulting prediction precision errors, improving the understanding of large-scale astrophysical magnetic field modeling.

Contribution

It introduces mesoscale correction terms into MFE equations and analyzes the intrinsic and filtering sources of prediction errors, proposing an optimal averaging scale for maximal precision.

Findings

Mesoscale correction terms depend on the ratio of averaging to variation scales.

Prediction errors can be minimized by choosing an optimal averaging scale.

Intrinsic and filtering errors significantly impact the predictive power of MFE.

Abstract

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.