Notes on Collapse in Magnetic Hydrodynamics
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Notes on Collapse in Magnetic Hydrodynamics E. A. Kuznetsova,b,c,d,* and E. A. Mikhailove a Lebedev
b
Physical Institute, Russian Academy of Sciences, Moscow, 119991 Russia Landau Institute of Theoretical Physics, Russian Academy of Sciences, Chernogolovka, Moscow oblast, 142432 Russia c Skolkovo Institute of Science and Technology, Skolkovo, Moscow oblast, 143026 Russia d Space Research Institute, Russian Academy of Sciences, Moscow, 117997 Russia e Moscow State University, Moscow, 119991 Russia *e-mail: [email protected] Received May 14, 2020; revised May 14, 2020; accepted May 14, 2020
Abstract—We consider the magnetic collapse as a possible process of the magnetic field singularity formation in a finite time in the framework of ideal magnetohydrodynamics for incompressible f luids, which is important for various astrophysical applications (in particular, as the mechanism of formation of magnetic filaments in the convective zone of the Sun). The possibility of collapse is associated with the compressibility of the continuously distributed magnetic field lines. The well-known example of magnetic filament formation in the kinematic dynamo approximation with a given velocity field, which was considered for the first time by Parker in 1963, rather indicates that the magnetic field increases with time exponentially. In the case of the kinematic approximation for the induction equation, the filaments are formed in regions with a hyperbolic velocity profile. DOI: 10.1134/S106377612009006X
CONTENTS Introduction.......................................................... Convection in Astrophysics.................................... Compression of Field Lines and Attractor.............. Convective Cell and Boundary Conditions............. Filamentation........................................................
496 497 498 499 500
Numerical Simulation................................................. Effect of Viscosity on the Magnetic Field Evolution..... Conclusions................................................................ References..................................................................
1. INTRODUCTION Collapse as the formation of singularities for smooth initial conditions is a key point in explaining the nature of hydrodynamic turbulence as well as magnetohydrodynamic (MHD) turbulence. The Kolmogorov—Obukhov theory [1, 2] of developed hydrodynamic turbulence for large Reynolds numbers (Re ≫ 1) in the inertial interval predicts the divergence of vorticity fluctuations δω with scale , for small , like ,−2/3 , indicating the connection between the Kolmogorov turbulence and collapse. Numerical experiments performed at the end of 1990s seemed to indicate the observation of the collapse, but detailed analysis revealed its absence (the discussion of this problem can be found in [3, 4]). This problem still remains unsolved despite the numerical experiments indicating the formation of singularities at the solid wall in the framework of the 3D Euler equations [5]. The collapse in the 2D Euler hydrodynamics
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