Quantification of the Cross-helicity Turbulent Cascade in Compressible MHD Simulations

In plasma turbulence, energy and cross helicity are transferred across scales at a constant rate as a consequence of nonlinear interactions. In incompressible magnetohydrodynamics (MHD), the energy cascade rate of both quantities can be computed by means of the temporal evolution of second-order structure functions, known as Karman-Howarth-Monin (KHM) equations. In the present work, we derive the KHM equation to compute the energy cascade rate of cross helicity in compressible MHD. Using three-dimensional direct numerical simulations, we validate the equation and use it to measure the cross-helicity turbulence properties. Our results show a slower development of the cross-helicity cascade with respect to the energy one and the presence of inverse cascades of energy and cross helicity at large scales when in the presence of a strong mean field. We propose the relation of these phenomena with the longer duration of geomagnetic storms after the arrival of solar winds with large cross helicity and the observation of patchy inertial ranges displaying positive and negative cascade rates for certain solar wind intervals.