Transfer equations for spectral densities of inhomogeneous MHD turbulence

On the basis of the dynamic equations governing the evolution of magneto-hydrodynamic fluctuations expressed in terms of Elsässer variables and of their correlation functions derived by Marsch and Tu, a new set of equations is presented here describing the evolutions of the energy spectrum e± and of the residual energy spectra e^R and e^s of MHD turbulence in an inhomogeneous magnetofluid. The nonlinearities associated with triple correlations in these equations are analysed in detail and evaluated approximately. The resulting energy-transfer functions across wavenumber space are discussed. For e± they are shown to be approximately energy-conserving if the gradients of the flow speed and density are weak. New cascading functions are heuristically determined by an appropriate dimensional analysis and plausible physical arguments, following the standard phenomenology of fluid turbulence. However, for e^R the triple correlations do not correspond to an ‘energy’ conserving process, but also represent a nonlinear source term for e_R. If this source term can be neglected, the spectrum equations are found to be closed. The problem of dealing with the nonlinear source terms remains to be solved in future investigations.