A non-local shell model of hydrodynamic and magnetohydrodynamic turbulence

We derive a new shell model of magnetohydrodynamic ( MHD) turbulence in which the energy transfers are not necessarily local. Like the original MHD equations, the model conserves the total energy, magnetic helicity, cross-helicity and volume in phase space ( Liouville's theorem) apart from the effects of external forcing, viscous dissipation and magnetic diffusion. The model of hydrodynamic ( HD) turbulence is derived from the MHD model setting the magnetic field to zero. In that case the conserved quantities are the kinetic energy and the kinetic helicity. In addition to a statistically stationary state with a Kolmogorov spectrum, the HD model exhibits multiscaling. The anomalous scaling exponents are found to depend on a free parameter a that measures the non-locality degree of the model. In freely decaying turbulence, the infra-red spectrum also depends on a. Comparison with theory suggests using alpha = -5/2. In MHD turbulence, we investigate the fully developed turbulent dynamo for a wide range of magnetic Prandtl numbers in both kinematic and dynamic cases. Both local and non-local energy transfers are clearly identified.