Eddy-damped quasinormal Markovian closure: a closure for magnetohydrodynamic turbulence?

Abstract

A set of nonlinear differential equations are developed that are analogous to the spectral evolution equations of incompressible magnetohydrodynamics (MHD). Because these equations possess little detail of MHD, apart from salient symmetry properties, they provide a toy model in which aspects of turbulent MHD can be understood readily. In the context of this model, the eddy-damped quasinormal Markovian (EDQNM) closure often used in Navier–Stokes turbulence is demonstrated to provide physically realizable spectra for magnetohydrodynamic turbulence, if the eddy-damping functions are chosen to satisfy certain symmetry properties. The requirements of physical realizability are more demanding in MHD than in fluid turbulence. In the absence of mean fields, this model demonstrates that the components of not only the turbulent kinetic energy spectrum, but also the magnetic energy spectra, never become negative. Another condition for realizability possessed by this model is that the components of the turbulent cross-helicity spectrum always satisfy a Schwarz inequality with respect to the corresponding components of the kinetic and magnetic energy spectra.