Quantification of the cross-helicity cascade with Karman-Howarth-Monin and Spectral transfer equations

Spectral transfer equations allow to  quantify the value of the energy flux of a turbulent flow across concentric shells in Fourier space. Karman-Howarth-Monin equations serve as a complement to the Spectral Transfer analysis, since they  quantify  as well the energy transfer rate of turbulence across scales via third-order structure functions, but also provide information on the directionality of the flux. We have extended the use of these methods to study the cascade of cross-helicity and compare it to the energy cascade  in 3D compressible MHD simulations. Our results show that the cross-helicity cascade reaches stationarity after the energy cascade, thus indicating a slower turbulence development for this invariant. Once fully developed, the cross-helicity cascade matches the main features of the energy one.