Exact two-dimensionalization of low-magnetic-Reynolds-number flows subject to a strong magnetic field
Basile Gallet, Charles R. Doering
Session: Magnetohydrodynamics 2
Session starts: Tuesday 25 August, 15:00
Presentation starts: 15:00
Room: Room F
Basile Gallet (Laboratoire SPHYNX, Service de Physique de l'Etat Condensé, DSM, CEA Saclay, CNRS, 91191 Gif-sur-Yvette, France)
Charles R. Doering (Department of Physics, Department of Mathematics, and Center for the Study of Complex Systems, University of Michigan, Ann Arbor, MI 48109, USA)
We investigate the behavior 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 very strong applied magnetic field, is the flow mostly two-dimensional, with remaining weak three-dimensional fluctuations, or does it become exactly 2D, with no dependence along the vertical?
We restrict attention to low-magnetic-Reynolds number (Rm) flow. Because liquid metals have low magnetic Prandtl number, such low-$Rm$ flows can have a kinetic Reynolds number as large as one million and therefore be strongly turbulent.
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 2D asymptotically in time, regardless of the initial condition and provided 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) 2D flow.
We then consider the full-magnetohydrodynamic equations and we prove that, for low enough Rm and large enough N, the flow becomes exactly two-dimensional in the long-time limit provided the initial vertically-dependent perturbations are infinitesimal. We call this phenomenon linear two-dimensionalization: the (possibly turbulent) 2D flow is an attractor of the dynamics, but it is not necessarily the only attractor of the system. Some 3D attractors may also exist and be attained for strong enough initial 3D perturbations.
These results shed some light on the existence of a dissipative anomaly for magnetohydrodynamic flows subject to a strong external magnetic field.