Generalized Eulerian-Lagrangian description of Navier-Stokes and resistive MHD dynamics

TL;DR

This paper introduces a generalized formalism for describing Navier-Stokes and MHD dynamics using Weber-Clebsch potentials, validated through numerical simulations that reveal vortex and magnetic reconnection phenomena.

Contribution

The paper develops a new generalized equations of motion for Weber-Clebsch potentials applicable to both Navier-Stokes and MHD, incorporating a novel parameter and demonstrating their effectiveness through simulations.

Findings

Captures vortex reconnection in Navier-Stokes flows.

Detects magnetic reconnection in MHD flows.

Correlates activity in magnetic enstrophy with reconnection events.

Abstract

New generalized equations of motion for the Weber-Clebsch potentials that describe both the Navier-Stokes and MHD dynamics are derived. These depend on a new parameter, which has dimensions of time for Navier-Stokes and inverse velocity for MHD. Direct numerical simulations are performed. For Navier-Stokes, the generalized formalism captures the intense reconnection of vortices of the Boratav, Pelz and Zabusky flow, in agreement with the previous study by Ohkitani and Constantin. For MHD, the new formalism is used to detect magnetic reconnection in several flows: the 3D Arnold, Beltrami and Childress (ABC) flow and the (2D and 3D) Orszag-Tang vortex. It is concluded that periods of intense activity in the magnetic enstrophy are correlated with periods of increasingly frequent resettings. Finally, the positive correlation between the sharpness of the increase in resetting frequency and the spatial localization of the reconnection region is discussed.