Non-conservative cascades in MHD turbulence

In fully developed turbulence, according to Kolmogorovs hypothesis, an extended range of scales, called the inertial range, exists, in which neither external nor viscous forces play an appreciable role, and all statistical properties are determined by the dissipation rate of kinetic energy. Turbulent flows, in which the influence of external forces on scales corresponding to the inertial interval is of fundamental importance, are common in nature. Then, the usual integrals of motion, such as the energy of velocity pulsations, hydrodynamic helicity, etc. cease to be so and other quadratic quantities that include scale dependence become integrals of motion. At the same time, from the point of view of the usual integrals of motion, cascade processes become non-conservative. In this work, this idea is developed in frame of shell models of MHD-turbulence. The approach introduced allows us to describe some special regimes of cascade processes in MHD-turbulence.