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The Okubo–Weiss-type topological criteria in two-dimensional magnetohydrodynamic flows

Published online by Cambridge University Press:  16 April 2024

B.K. Shivamoggi*
Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA
G.J.F. van Heijst
J. M. Burgers Centre and Fluid Dynamics Laboratory, Eindhoven University of Technology, NL-5600MB Eindhoven, The Netherlands
L.P.J. Kamp
J. M. Burgers Centre and Fluid Dynamics Laboratory, Eindhoven University of Technology, NL-5600MB Eindhoven, The Netherlands
Email address for correspondence:


The Okubo–Weiss (Okubo, Deep-Sea Res., vol. 17, issue 3, 1970, pp. 445–454; Weiss, Physica D, vol. 48, issue 2, 1991, pp. 273–294) criterion has been widely used as a diagnostic tool to divide a two-dimensional (2-D) hydrodynamical flow field into hyperbolic and elliptic regions. This paper considers extension of these ideas to 2-D magnetohydrodynamic (MHD) flows, and presents an Okubo–Weiss-type criterion to parameterize the magnetic field topology in 2-D MHD flows. This ensues via its topological connections with the intrinsic metric properties of the underlying magnetic flux manifold, and is illustrated by recasting the Okubo–Weiss-type criterion via the 2-D MHD stationary generalized Alfvénic state condition to approximate the slow-flow-variation ansatz imposed in its derivation. The Okubo–Weiss-type parameter then turns out to be related to the sign definiteness of the Gaussian curvature of the magnetic flux manifold. A similar formulation becomes possible for 2-D electron MHD flows, by using the generalized magnetic flux framework to incorporate the electron-inertia effects. Numerical simulations of quasi-stationary vortices in 2-D MHD flows in the decaying turbulence regime are then given to demonstrate that the Okubo–Weiss-type criterion is able to separate the MHD flow field into elliptic and hyperbolic field configurations very well.

Research Article
Copyright © The Author(s), 2024. Published by Cambridge University Press

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Babiano, A., Boffetta, G., Provenzale, A. & Vulpiani, A. 1994 Chaotic advection in point vortex models and two-dimensional turbulence. Phys. Fluids 6 (7), 24652474.CrossRefGoogle Scholar
Babiano, A. & Provenzale, A. 2007 Coherent vortices and tracer cascades in two-dimensional turbulence. J. Fluid Mech. 574, 429448.CrossRefGoogle Scholar
Banerjee, D. & Pandit, R. 2014 Statistics of the inverse-cascade regime in two-dimensional magnetohydrodynamic turbulence. Phys. Rev. E 90 (1), 013018.CrossRefGoogle ScholarPubMed
Banerjee, D. & Pandit, R. 2019 Two-dimensional magnetohydrodynamic turbulence with large and small energy-injection length scales. Phys. Fluids 31 (6), 065111.CrossRefGoogle Scholar
Brachet, M.E., Meneguzzi, M., Politano, H. & Sulem, P.L. 1988 The dynamics of freely decaying two-dimensional turbulence. J. Fluid Mech. 194, 333349.CrossRefGoogle Scholar
Elhmaïdi, D., Provenzale, A. & Babiano, A. 1993 Elementary topology of two-dimensional turbulence from a Lagrangian viewpoint and single-particle dispersion. J. Fluid Mech. 257, 533558.CrossRefGoogle Scholar
Giovanelli, R.G. 1949 XVII. Electron energies resulting from an electric field in a highly ionized gas. Lond. Edinb. Dub. Phil. Mag. J. Sci. 40 (301), 206214.CrossRefGoogle Scholar
Goedbloed, J.P.H. & Poedts, S. 2004 Principles of Magnetohydrodynamics: With Applications to Laboratory and Astrophysical Plasmas. Cambridge University Press.CrossRefGoogle Scholar
Gordeev, A.V., Kingsep, A.S. & Rudakov, L.I. 1994 Electron magnetohydrodynamics. Phys. Rep. 243 (5), 215315.CrossRefGoogle Scholar
Greene, J.M. 1993 Reconnection of vorticity lines and magnetic lines. Phys. Fluids B 5 (7), 23552362.CrossRefGoogle Scholar
Hasegawa, A. 1985 Self-organization processes in continuous media. Adv. Phys. 34 (1), 142.CrossRefGoogle Scholar
Kinney, R., McWilliams, J.C. & Tajima, T. 1995 Coherent structures and turbulent cascades in two-dimensional incompressible magnetohydrodynamic turbulence. Phys. Plasmas 2 (10), 36233639.CrossRefGoogle Scholar
Kuznetsov, E.A., Naulin, V., Nielsen, A.H. & Rasmussen, J.J. 2007 Effects of sharp vorticity gradients in two-dimensional hydrodynamic turbulence. Phys. Fluids 19 (10), 105110.CrossRefGoogle Scholar
Mcwilliams, J.C. 1984 The emergence of isolated coherent vortices in turbulent flow. J. Fluid Mech. 146, 2143.CrossRefGoogle Scholar
Núñez, M. 2007 Long time convergence of magnetohydrodynamic flows to alfvénic states. J. Plasma Phys. 73 (6), 947955.CrossRefGoogle Scholar
Ohkitani, K. 1991 Wave number space dynamics of enstrophy cascade in a forced two-dimensional turbulence. Phys. Fluids A 3 (6), 15981611.CrossRefGoogle Scholar
Okubo, A. 1970 Horizontal dispersion of floatable particles in the vicinity of velocity singularities such as convergences. Deep-Sea Res. 17 (3), 445454.Google Scholar
Ouellette, N.T. & Gollub, J.P. 2007 Curvature fields, topology, and the dynamics of spatiotemporal chaos. Phys. Rev. Lett. 99 (19), 194502.CrossRefGoogle ScholarPubMed
Rollins, D.K. & Shivamoggi, B.K. 2007 Current sheet formation near a hyperbolic magnetic neutral line in a variable-density plasma: an exact solution. Phys. Lett. A 366 (1), 97100.CrossRefGoogle Scholar
Shivamoggi, B.K. 1986 Evolution of current sheets near a hyperbolic magnetic neutral point. Phys. Fluids 29 (3), 769772.CrossRefGoogle Scholar
Shivamoggi, B.K. 1999 Current-sheet formation near a hyperbolic magnetic neutral line in the presence of a plasma flow with a uniform shear-strain rate: an exact solution. Phys. Lett. A 258 (2), 131134.CrossRefGoogle Scholar
Shivamoggi, B.K. 2011 Characteristics of plasma Beltrami states. Eur. Phys. J. D 64 (2), 393404.CrossRefGoogle Scholar
Shivamoggi, B.K. 2015 a Beltrami states in 2D electron magnetohydrodynamics. arXiv:1506.06094.Google Scholar
Shivamoggi, B.K. 2015 b Electron magnetohydrodynamic turbulence: universal features. Eur. Phys. J. D 69 (2), 55.CrossRefGoogle Scholar
Shivamoggi, B.K. 2016 Plasma relaxation and topological aspects in electron magnetohydrodynamics. Phys. Plasmas 23 (7), 072302.CrossRefGoogle Scholar
Shivamoggi, B.K. & Michalak, M. 2019 Topological implications of the total generalized electron-flow magnetic helicity invariant in electron magnetohydrodynamics. Phys. Plasmas 26 (4), 44501.CrossRefGoogle Scholar
Shivamoggi, B.K., van Heijst, G.J.F. & Kamp, L.P.J. 2016 The Okubo–Weiss criteria in two-dimensional hydrodynamic and magnetohydrodynamic flows. arXiv:1110.6190.Google Scholar
Shivamoggi, B.K., Van Heijst, G.J.F. & Kamp, L.P.J. 2022 The Okubo–Weiss criterion in hydrodynamic flows: geometric aspects and further extension. Fluid Dyn. Res. 54 (1), 015505.CrossRefGoogle Scholar
Toda, M. 1974 Instability of trajectories of the lattice with cubic nonlinearity. Phys. Lett. A 48 (5), 335336.CrossRefGoogle Scholar
Weiss, J. 1991 The dynamics of enstrophy transfer in two-dimensional hydrodynamics. Physica D 48 (2), 273294.CrossRefGoogle Scholar