A Study on Topological Vortex Ring Interactions Based on...

A Study on Topological Vortex Ring Interactions Based on Möbius Loop and Hopf Link Concepts (2)

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2. Theoretical Foundation: Topological Invariants

2.1 Linking Number (Lk) For two simple, unknotted but mutually entwined closed curves (like an ideal Hopf link), the linking number Lk is a topological invariant representing the total number of times (with sign) one loop winds around the other. For two vortex rings, Lk quantitatively describes the topological strength of their mutual threading.

2.2 Writhe and Twist Number (Wr & Tw) According to the Călugăreanu–White formula, for a twisted ring, its total linking number (if linked with another ring) or self-linking number can be decomposed as [1,8]:

a. Writhe (Wr): Describes the degree of distortion and coiling of the curve's global geometric configuration, related to the crossing points in its projection onto a plane. For two vortex rings, their mutual spatial configuration contributes to the writhe.

b. Twist (Tw): Describes the degree to which the curve rotates around its own centerline. This is precisely where the concept of the "Möbius loop" comes into play. A vortex ring with high Tw has its vorticity field direction rotating along the circumference, analogous to the abstract concept of the direction flipping after traversing a Möbius strip once. Tw is an intrinsic property of the vortex core.

3. Superposition and Interaction of Co-rotating Vortex Rings

When two vortex rings with the same circulation direction (co-rotating) collide head-on or move side-by-side [6]:

(1) Topological Description: Their interaction can be seen as the superposition of two co-oriented Hopf links. The total topological charge (total circulation) of the system increases.

(2) Dynamical Process: Due to their mutually reinforcing induced velocity fields, the two rings tend to repel each other or orbit a common axis. Under specific conditions (e.g., matched size, speed), they can:

a. Stable Linking: Form a stable, higher-order linked state (linking number Lk > 1), analogous to the "Solomon's link" in topology.

b. Merging: Through viscous dissipation or vortex reconnection, the two rings merge into a single, larger vortex ring potentially carrying higher Tw (twist). This process may reduce the overall topological complexity (the link disappears) but increase the intrinsic twist.

(3) Energy and Topology: Co-rotating superposition generally leads to a decrease in the system's kinetic energy (more stable), and the topological structure (although possibly more complex in linking) remains overall ordered and stable [7].

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