A Korean research team has mathematically proved that a vortex pair called the "Sadovskii patch" can exist in an ideal fluid. It comes more than 50 years after the model structure was proposed.
A team led by Choi Gyudong, a professor of mathematics at Ulsan National Institute of Science and Technology, said on the 2nd that, together with Seoul National University Professor Jeong Inji, it proved that the Sadovskii patch can exist as a solution to the Euler equations. The findings were published in the December issue of the Annals of Partial Differential Equations (Annals of PDE), an international mathematics journal.
The Sadovskii patch is a special vortex pair in which two vortices rotating in opposite directions, with equal rotation strength, move while fully attached. It looks similar to vortices that form at airplane wingtips or behind ships, but unlike real water or air, because an ideal fluid is assumed, it can maintain its shape and move straight forever.
In 1971, Russian mathematician V. S. Sadovskii first proposed the model through numerical simulations, but he noted in his paper that mathematically proving the existence of this patch would not be easy.
The most definitive way to prove existence mathematically is to directly find a function that simultaneously describes the shape and motion of the Sadovskii patch, that is, a solution to the Euler equations, the laws of motion for fluids. But obtaining such a function explicitly is generally close to impossible. Instead, mathematicians logically establish the "existence" of a solution to the equations, yet even that was difficult, due to the special structure in which two vortices must move continuously while in complete contact along a symmetry axis.
The team solved this anew using the calculus of variations. The calculus of variations is a method for finding, among many possible functions that satisfy certain conditions, the function that maximizes or minimizes a given value.
The team first set a small gap between the vortices and imposed a condition that bounded the rotation strength of the vortices, then found the vortex pair with the largest kinetic energy within that set. By analyzing the structure of this maximal-energy vortex pair step by step, they proved that its shape matches the form of the patch proposed by Sadovskii.
Professor Choi Gyudong said, "We have been competing with Peking University Professor Huang-Tong's team in research to establish the mathematical existence of the Sadovskii patch, and unlike their work, our study verified not only the mathematical existence of the Sadovskii patch but also its dynamical validity, that is, physical stability, which sets it apart." Physical stability means not only that the patch's existence is logically consistent, but also that it can persist at a level observable in reality.
The team added that the achievement broadens the foundation of hydrodynamic understanding in areas such as turbulence research, wake analysis of aircraft and ships, and studies of interactions among atmospheric and oceanic vortices such as the Fujiwhara effect. The Fujiwhara effect is an interference phenomenon that occurs when two or more typhoons are adjacent, proposed by Japanese meteorologist Fujiwhara Sakuhei in 1921.
References
Annals of PDE (2025), DOI: https://doi.org/10.1007/s40818-025-00212-4