Inverse energy cascade in a nearly two-dimensional turbulence

Abstract

Spectral structures of homogeneous axisymmetric turbulence subjected to a uniform magnetic field B0 and a randomly stirred two-dimensional force fed in a narrow band of wavenumbers is studied, using the energy spectrum equation that is closed by the eddy-damped quasinormal Markovian (EDQNM) approximation. When the interaction parameter N=σB20L/(ρ0u) is large (where σ is the electric conductivity, ρ0 the uniform fluid density, L the integral scale, and u the root mean square velocity), the existence of the inverse energy cascade toward large scales with a k−5/3 energy spectrum and the enstrophy cascade toward small scales with k−3 energy spectrum are observed in the evolution of the energy of velocity components transverse to B0. The energy of the velocity components parallel to B0 shows a k−1 energy spectrum, which agrees with the k−1 variance spectrum predicted by Lesieur and Herring [J. Fluid Mech. 161, 77 (1985)] for a passive scalar convected by two-dimensional isotropic turbulence. This is of great interest because it is found that in the limit of two-dimensional turbulence the equation for the parallel velocity components is reduced to the same equation as that for a passive scalar convected by two-dimensional isotropic turbulence.