"Devil's Double Vortex" Puzzle Solved: UNIST and Seoul National University End 50-Year Mystery

UNIST and Seoul National University Prove Mathematical Existence of Sadovskii Vortex Patch
Expected to Aid Aircraft and Ship Wake Analysis, Typhoon Research... Published in Annals of PDE

The double vortex phenomenon, which remained unsolved for 50 years and was dubbed the "devil's vortex" among scientists, has finally been resolved by a Korean research team.

The research team has mathematically proven that a pair of vortices, called the Sadovskii patch, can exist within an ideal fluid.

This comes more than 50 years after the model structure was first proposed.

On December 2, Professor Kyu-Dong Choi and student Youngjin Shim from the Department of Mathematical Sciences at UNIST announced, together with Professor Jeong In-Ji from Seoul National University, that they have proven the Sadovskii patch can exist as a solution to the Euler equations.

Research team, Professor Kyu-Dong Choi and Researcher Youngjin Shim (right). Provided by UNIST

The Sadovskii patch is a special pair of vortices, each spinning in opposite directions with equal rotational intensity, that move together while remaining in complete contact. Although they resemble the vortices that form at the tips of airplane wings or behind ships, they are theorized to exist in ideal fluids, which allows them to maintain their shape and move in a straight line indefinitely, unlike vortices in real water or air.

In 1971, Russian mathematician V. S. Sadovskii first proposed this model through numerical simulations, but noted in his paper that mathematically proving the existence of this patch would be extremely difficult.

Existing approaches to studying Sadovsky-type vortices.

The most definitive way to mathematically prove its existence is to find a function that simultaneously describes both the shape and motion of the Sadovskii patch-that is, to directly find a solution to the Euler equations, which govern fluid motion. However, finding such a function is generally considered nearly impossible. Instead, mathematicians have attempted to logically demonstrate the 'existence' of a solution to the equations, but even this has proven challenging due to the unique structure in which the two vortices must move together in perfect contact along a symmetry axis without any discontinuity.

The research team overcame this challenge using the variational method. The variational method is a technique that finds the function which maximizes or minimizes a given value among all possible functions that satisfy certain conditions.

The team first set the distance between the vortices to be small and imposed an upper limit on the rotational intensity of the vortices. Within these conditions, they identified the vortex pair with the maximum kinetic energy. Through a step-by-step analysis of the structure of this maximum-energy vortex pair, they proved that its shape matches the patch proposed by Sadovskii.

Variational method-based proof technique used by the research team.

Professor Kyu-Dong Choi explained, "We have been competing with Professor Huang-Tong's team at Peking University to prove the mathematical existence of the Sadovskii patch. Unlike their research, our study not only demonstrates the mathematical existence of the Sadovskii patch but also verifies its mechanical validity, meaning its physical stability, which sets our work apart." Physical stability means that the existence of the patch is not only logically consistent but can also persist at a level that is observable in reality.

The research team added that this achievement broadens the foundation of hydrodynamic understanding in fields such as turbulence research, analysis of aircraft and ship wakes, and studies of interactions between atmospheric and oceanic vortices, such as the Fujiwhara effect. The Fujiwhara effect refers to the interference phenomenon that occurs when two or more typhoons are in close proximity, first described by Japanese meteorologist Sakuhei Fujiwhara in 1921.

The research results were published in the December issue of the Annals of PDE, one of the top journals in the field of mathematics specializing in partial differential equations.

This research was supported by the National Research Foundation of Korea under the Ministry of Science and ICT.

This content was produced with the assistance of AI translation services.