Abstract
This paper aims to explore the complex interactions of topological vortex rings in fluid mechanics and establish a novel topological framework for them. Vortex rings, as coherent structures carrying circulation and possessing stable topological structures, undergo interactions (such as collision, threading, and merging) that are inherently processes of topological transformation. We introduce the concept of the Möbius loop to describe the twisted intrinsic property (i.e., "Twist") of a single vortex ring and employ the Hopf link—the simplest linked ring topological invariant—to characterize the state of two vortex rings threading through each other (i.e., "Writhe"). The focus is on analyzing the evolution of the total circulation, energy, and topological structure (Knots) during the superposition of co-rotating vortex rings and counter-rotating vortex rings. Research indicates that the superposition of co-rotating rings tends to form higher-order Hopf links (e.g., Solomon's link) or merge into a single ring, their topological structure tending towards stability; whereas the superposition of counter-rotating rings can generate complex knotted structures through reconnection processes, or annihilate each other, a process accompanied by drastic changes in the system's topological charge. This study combines profound topological mathematics (e.g., torus, linking number, self-linking number) with fluid dynamics, providing a new perspective for understanding the topological essence of vortex dynamics.
Keywords: topological vortex ring; Möbius loop; Hopf link; knot; vortex reconnection; writhe/twist
1. Introduction
Vortex rings are a common and important type of topological defect in fluids, superfluids, and even quantum field theory. Their stability stems from topological protection: a vortex ring consists of a toroidal singularity core surrounded by a vorticity field, and its circulation Γ is conserved by Kelvin's circulation theorem, endowing it with a property analogous to a "topological charge." The interaction of two or more vortex rings is a central issue in fields such as turbulence, biological propulsion, and quantum fluids.
Traditional dynamical analysis focuses on the Navier-Stokes equations but often struggles to capture the global topological properties of interactions. This paper proposes that vortex ring interactions can be elegantly described using concepts from low-dimensional topology:
a. Hopf Link: As the simplest (linking number ±1) non-trivial link of two rings, it is an ideal model for describing two vortex rings threading through each other without being knotted.[3]
b. Möbius Loop: A twisted loop with a single-sided property, used as a metaphor for the potential Twist inherent in the vortex ring core itself. This twist can originate from the spin of the vortex core or initial generation conditions.[5]
The superposition of co- and counter-rotating vortex rings is essentially the vector superposition of two topological charges (circulation), the outcome of which profoundly influences the overall topological structure (whether knotted or not) and dynamical fate of the system.
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