Nonlinear Scale-Local Geometric Deformations of Vortex Rings in Smooth Euler Flows via Bayesian Optimization and Adjoint Methods

TL;DR

This paper develops a geometric Lagrangian framework and a hybrid Bayesian-adjoint optimization method to analyze nonlinear deformations of vortex rings in Euler flows, revealing intrinsic mechanisms behind vortex structure changes.

Contribution

It introduces a novel geometric Lagrangian approach and a hybrid optimization framework combining Bayesian exploration with adjoint methods for vortex analysis.

Findings

Identifies nonlinear mechanisms driving vortex deformation.

Develops a wave equation for swirling particle axes.

Demonstrates the hybrid optimization's effectiveness in complex landscapes.

Abstract

We consider the incompressible three-dimensional Euler equations for a vortex ring with Kelvin waves undergoing radially expanding Lagrangian transport. To clarify the fundamental mechanisms underlying nonlinear scale-local deformations of the vortex structure, we develop a geometric Lagrangian framework that avoids singular integral representations of the pressure and yields a novel wave equation governing the axis of swirling particles. Within this framework, we identify intrinsic nonlinear mechanisms that drive scale-local deformations of the vortex structure, supported by a machine-learning-based analysis. Specifically, we propose a hybrid optimization framework that combines Bayesian global exploration with adjoint-based local refinement. The resulting optimization problem exhibits a highly non-convex loss landscape, in which the adjoint method alone fails to escape local minima.