Waves and non-propagating mode in stratified and rotating magnetohydrodynamic turbulence

In this study, we consider a freely decaying, stably stratified, and rotating homogeneous magneto-hydrodynamic (MHD) turbulent plasma with a vertical background magnetic field ( B-0=B(0)z), aligned with the density gradient (with a constant Brunt-Vaisala frequency N) viewed in a frame rotating uniformly around the vertical axis ( Omega(0)=Omega(0)z). Quasi-linear theory is used to analyze the flow dynamics for an inviscid and non-diffusive Boussinesq fluid. We perform a normal mode decomposition emphasizing three types of motions: a non-propagating (NP) mode, which is no longer a vortex mode, and slow and fast magneto-inertia-gravity waves. The total energy as well as the L-2 norm, say Gamma, of the magnetic induction potential scalar (MIPS), which remains similar to the potential enstrophy for non-magnetized rotating and stratified flows, are inviscid invariants. In contrast with the potential vorticity for non-magnetized rotating and stratified flows, the MIPS is not affected by system rotation in the quasi-linear limit, and this is the effect of rotation which presumes an inverse cascade of energy in the equilibrium statistical mechanics. We characterized the system setting up our investigation from the point of view of equilibrium statistical mechanics in the limit of small Froude number and small Alfv & eacute;n-Mach number. In this limit, the non-propagating quantity Gamma can be approximated by its quadratic part that explicitly depends only on the vertical component of the fluctuating magnetic field and the density fluctuations. We demonstrate that the partition function restricted to the non-propagating manifold does not indicate an inverse cascade of energy.