Golden Ratio and Vortices

Golden Ratio appears in vortices?

---
layout: post
read_time: true
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title: "Golden Ratio in Vortices"
date: 2022-10-21
img: posts/20221021/point_vortex.jpg
tags: [Hydrodynamics, Dynamical System, Scientific Computation]
author: Hanchun Wang
description: "point vortex on the half plane"
---

Summary

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Prof. Boris Khesin

In this two point vortices system on the half plane, we show that \(W\) is a non-dimensional parameter, and \(W=\phi, 1, \frac{1}{\phi}\) are three bifurcation values in this system. Where \(\phi\) is the Golden Ratio.

There are two mechanisms applied on a point vortex.

1. Move along the boundary of the half plane
2. rotate on the other vortex

These two mechanisms compete and lead to different types of the trajectories.

Vortex Pair

Two point vortices same strength is a vortex pair. Leapfrogging motions appear when $$W<1$$, and cusp motion appear at $$W=1/\phi$$.

Vortex Dipole

A vortex dipole are two point vortices with same strength but negative sign. Cusp motion appear at $$W=\phi$$, and the vortex dipole will escape from the boundary when $$W<1$$.

Introduction

A single point vortex on the halfplane can be regarded as a vortex dipole that symmetric on the boundary \((y=0)\) in the full plane. Thus an \(N\) point vortex system in the halfplane is equivalent to a \(2N\) point vortex system in the full plane which are \(N\) point vortices and their \(N\) images.

The Green’s function on the half-plane \(\mathbb{R}_+^2\) is

\[G_{\mathbb{R}^2_+}\left( {z,z'} \right) = - \frac{1}{2\pi}\log ||z - z'||+\frac{1}{2\pi}\log ||z - z'^*||\]

and the Hamiltonian is

\[H=\frac{1}{4 \pi} \log \left(\left(2 y_1\right)^{\Gamma_1^2}\left(2 y_2\right)^{\Gamma_2^2}\left(\frac{\left(x_1-x_2\right)^2+\left(y_1+y_2\right)^2}{\left(x_1-x_2\right)^2+\left(y_1-y_2\right)^2}\right)^{\Gamma_1 \Gamma_2}\right)\]

The first term represents the interaction between different vortices with their images; and the second term represents the interaction between a vortex and its image.

For the two point vortices system on the half plane, the Hamiltonian is The motion of equations are

\[\dot{x}_1=\frac{\Gamma_1}{4 \pi} \frac{1}{y_1}+\frac{\Gamma_2}{4 \pi}\left(\frac{2\left(y_1+y_2\right)}{\left(x_1-x_2\right)^2+\left(y_1+y_2\right)^2}-\frac{2\left(y_1-y_2\right)}{\left(x_1-x_2\right)^2+\left(y_1-y_2\right)^2}\right)\]

and

\[\dot{y}_1=\frac{-\Gamma_2}{4 \pi}\left(\frac{2\left(x_1-x_2\right)}{\left(x_1-x_2\right)^2+\left(y_1+y_2\right)^2}-\frac{2\left(x_1-x_2\right)}{\left(x_1-x_2\right)^2+\left(y_1-y_2\right)^2}\right)\]

To study vortex bifurcations we normalize their strengths by setting \(\Gamma_1=\Gamma_2=1\) for a vortex pair and \(\Gamma_1=-\Gamma_2=1\) for a dipole. Furthermore, introduce the following dimensionless parameter

\[W:=(P / \Gamma)^2 \exp \left(-4 \pi \mathcal{H} / \Gamma^2\right)\]

measuring the vortex interaction where \(\Gamma:=\Gamma_1=\pm \Gamma_2\). As we will see below, the increase of \(W\) corresponds to the weakening of the interaction between the point vortices.

More GIF

Vortex Pair

When two point vortices have same strength, we have the following four scenarios.

Vortex Dipole

When two point vortices have opposite strength \(\Gamma_1=-\Gamma_2\), we have the following four scenarios.