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A non-local shell model of hydrodynamic and magnetohydrodynamic turbulence

Abstract : 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 α that measures the non-locality degree of the model. In freely decaying turbulence, the infra-red spectrum also depends on α. Comparison with theory suggests using α = −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.
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https://hal-insu.archives-ouvertes.fr/insu-00347599
Contributor : Pascale Talour <>
Submitted on : Tuesday, December 16, 2008 - 11:51:37 AM
Last modification on : Friday, September 25, 2020 - 3:14:01 AM

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Franck Plunian, Rodion Stepanov. A non-local shell model of hydrodynamic and magnetohydrodynamic turbulence. New Journal of Physics, Institute of Physics: Open Access Journals, 2007, 9 (art. no. 294), pp.NIL_52-NIL_77. ⟨10.1088/1367-2630/9/8/294⟩. ⟨insu-00347599⟩

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