Journal
JOURNAL OF FLUID MECHANICS
Volume 773, Issue -, Pages 154-177Publisher
CAMBRIDGE UNIV PRESS
DOI: 10.1017/jfm.2015.232
Keywords
high-Hartmann-number flows; MHD and electrohydrodynamics; MHD turbulence
Categories
Funding
- US National Science Foundation (NSF) [OCE-0824636, OCE-1332750]
- Office of Naval Research
- NSF [PHY-1205219]
- grant TURBA from Labex PALM [ANR-10-LABX-0039]
- Simons Foundation
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [1440415, 1515161] Funding Source: National Science Foundation
- Division Of Physics
- Direct For Mathematical & Physical Scien [1205219] Funding Source: National Science Foundation
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We investigate the behaviour of flows, including turbulent flows, driven by a horizontal body force and subject to a vertical magnetic field, with the following question in mind: for a very strong applied magnetic field, is the flow mostly two-dimensional, with remaining weak three-dimensional fluctuations, or does it become exactly 2-D, with no dependence along the vertical direction? We first focus on the quasi-static approximation, i. e. the asymptotic limit of vanishing magnetic Reynolds number, Rm << 1: we prove that the flow becomes exactly 2-D asymptotically in time, regardless of the initial condition and provided that the interaction parameter N is larger than a threshold value. We call this property absolute two-dimensionalization: the attractor of the system is necessarily a (possibly turbulent) 2-D flow. We then consider the full magnetohydrodynamic (MHD) equations and prove that, for low enough Rm and large enough N, the flow becomes exactly 2-D in the long-time limit provided the initial vertically dependent perturbations are infinitesimal. We call this phenomenon linear two-dimensionalization: the (possibly turbulent) 2-D flow is an attractor of the dynamics, but it is not necessarily the only attractor of the system. Some 3-D attractors may also exist and be attained for strong enough initial 3-D perturbations. These results shed some light on the existence of a dissipation anomaly for MHD flows subject to a strong external magnetic field.
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