Yasuhide Fukumoto (Institute of Mathematics for Industry, Kyushu University & Osaka Central Advanced Mathematical Institute)
Title: Global time evolution of a counter-rotating vortex pair of finite thickness in a viscous fluid
Abstract: Vortex pairs act to produce lift and thrust of aircrafts and animals. They move by themselves, entraining surrounding fluid, and are capable of transporting neutrally buoyant materials over long distances. We develop a method for calculating self-induced motion of a counter-rotating vortex pair over the entire time range, including the decaying stage, with focus on the influence of viscosity. Initial-value problems are solved for a vortex pair embedded in a viscous fluid. For high-Reynolds-number motion, the method of matched asymptotic expansions in powers of the ratio of the core radius to the distance between vortices, a small parameter, is used to derive the traveling speed of the pair at early times [1]. This gives the upper bound on the traveling speed in an early stage.
As an alternative means, we adapt, to the two-dimensional vortical flow, Helmholtz-Lamb's formula for an axisymmetric vortex ring. This method saves, to a large extent, manipulations of asymptotic expansions, and is widely extensible to magnetic vortex rings say [2]. In the large-Reynolds-number limit, a topological method (Kelvin-Benjamin's kinematic variational principle) achieves a further simplification [3, 4].
For low-Reynolds-number motion, the vorticity obeying the Stokes equation is substituted into the 2D-version of Helmholtz-Lamb's formula. At early times, this gives the lower bound on the traveling speed. An analysis at large times reveals the decaying behavior of the traveling speed, being distinct from the axisymmetric case [5].
References
[1] U. Habibah, H. Nakagawa and Y. Fukumoto, "Finite-thickness effect on speed of a counter-rotating viscous vortex pair", Fluid Dyn. Res. Vol.50, 031401 (2018).
[2] Y. Fukumoto, S. Oshiro and T. Ruzi, "Motion of magnetic vortex rings subject to Hall effect", Phys. Rev. Fluids Vol.10, 124703 (2025).
[3] Y. Fukumoto and H. K. Moffatt, "Kinematic variational principle for motion of vortex rings", Physica D Vol.237, pp.2210-2217 (2008).
[4] Y. Fukumoto, "Global time evolution of viscous vortex rings", Theor. Comp. Fluid Dyn. Vol.24, pp.335-347 (2010).
[5] Y. Fukumoto and F. Kaplanski, "Global time evolution of an axisymmetric vortex ring at low Reynolds numbers", Phys. Fluids Vol.20, 053103 (2008).
Mitsuo Higaki (Kobe University)
Title: An Arnold-type stability analysis of vortex filaments
Abstract: I will present ongoing work with Masashi Aiki (Tokyo Univ. Sci.) on the shape stability of vortex filaments evolving under the localized induction equation. Our approach is based on Arnold’s variational method [Arnold 1965] and provides a unified framework for the stability of arc-shaped and circular filaments.
A central step in Arnold’s method is the construction of a suitable Lagrange functional from conserved quantities. A key ingredient in our setting is the conserved fluid impulse, studied in [Arms and Hama 1965] and [Fukumoto 1987].
Xiaoran Liu (Westlake University)
Title: Inverse Energy Cascade and Non-uniqueness for the Advection-Diffusion Equation
Abstract: We prove non-uniqueness for the advection-diffusion equation with a drift in a logarithmically near-critical class below $L_t^2L_x^\infty$. The first construction is a diffusion-assisted inverse cascade of scalar energy that diverges as $t \to 0$. The second uses compressed inverse mixing, with diffusion acting perturbatively, and produces a bounded parabolic solution with the energy jump at the initial time. In both constructions the drift and scalar are smooth for positive time. We also consider the drift in $L^{2,\infty}_tL^\infty_x$ and solutions satisfying the energy inequality starting from almost every time. In this setting, we prove uniqueness of solutions with bounded energy, and construct non-unique solution with energy blowing up as time goes to zero.
Young-Jin Sim (KIAS)
Title: Geometric Rigidity of Variational Sadovskii Vortices
Abstract: In this talk, I will explain how energy maximization constrains the geometry of Sadovskii vortices, pairs of oppositely rotating vortices that touch along a symmetry axis and travel without changing shape. For the two-dimensional incompressible Euler equations, we study a family of those vortices obtained by maximizing kinetic energy subject to constraints on the impulse and the $L^p$-norm of the vorticity. The exponent $1<p\leq\infty$ parameterizes this family, which includes the classical Sadovskii vortex patch at $p=\infty$ and the Lamb-Chaplygin dipole at $p=2$.
For every exponent in this range, we prove that each upper vortex core is simply connected. Its free boundary has a nondegenerate quadratic tangency at the top and meets the symmetry axis orthogonally, with quantitative estimates on the local boundary behavior. For $2<p\leq\infty$, the core becomes strictly narrower as the height increases. For $2<p<\infty$, every nonempty positive vorticity superlevel set is a symmetric, simply connected region.
Numerical computations will also illustrate the vortex shapes and vorticity distributions across the family, including evidence for a point-vortex limit as $p$ approaches $1$. I will also compare our variational computations with Huang and Tong’s fixed-point computations of the Sadovskii patch. This talk is based on joint work with Guolin Qin (CAS).