Okubo-Weiss Type Criteria in Hydrodynamic and MHD Flows and Topological Interpretations

Due to its conceptual simplicity and easy implementation, The Okubo[1]-Weiss[2] criterion has been extensively used as a diagnostic tool for dividing a 2D hydrodynamic flow field into hyperbolic and elliptic regions. We give its topological connections with the intrinsic metric properties of the underlying vorticity manifold. The Okubo-Weiss criterion is recast (Shivamoggi et al. [3]) by using the 2D hydrodynamic Beltrami flow condition (Shivamoggi et al. [4]) that approximates the slow flow-variation ansatz imposed in its derivation. This connects the Okubo-Weiss criterion with the sign definiteness of the Gaussian curvature of the underlying vorticity manifold. An Okubo-Weiss type parameter is given (Shivamoggi et al. [5]) to provide a useful diagnostic tool to characterize the magnetic field topology in 2D MHD flows and is validated via several real plasma examples and numerical simulations of MHD flows showing separation into elliptic and hyperbolic flow regions very well, as predicted by the Okubo- Weiss type parameter.

References:

[1] A. Okubo: Deep Sea Res. 17, 445, (1970).

[2] J. Weiss: Physica D 48, 273, (1991).

[3] B.K. Shivamoggi, G.J.F. van Heijst, and J. Juul Rasmussen:. Phys. Lett A 374, 2309, (2010). [4] B.K. Shivamoggi, G.J.F. van Heijst, and LP.J. Kamp: Fluid Dyn. Res. 54, 015505, (2022).

[5] B.K. Shivamoggi, G.J.F. van Heijst, and L.P.J. Kamp: J. Plasma Phys. 90, 905900214, (2024).