Dimensional reduction of continuum models for vortex filament motion.

Scott Strong

sstrong@mines.edu

In a 2021 MURF, we constructed the scaling limits for describing potential flows of inviscid fluids with the Gross–Pitaevskii equation and began the construction of its corresponding Biot-Savart integral, which provides a representation for the velocity fields induced by vortex lines/filaments. In a 2022 SURF, we generalized a transformation of the induced velocity field to a class of nonlinear integrodifferential Schrodinger equations providing an evolution of the vortex geometry as described by its curvature and torsion, which shows that the fluid's vortex skeleton is an inherently nonlinear medium supporting wave motion. The current project aims to complete this narrative by finalizing the connection between the fluid continuum and the Biot-Savart representation of the induced velocity field generating autonomous vortex filament dynamics. Additionally, we intend to continue our existing studies of integral regularization techniques to draw connections between the induced flow parameters and those nonlinear waves predicted by our family of Schrodinger evolutions. Filament structures appear in various interesting and important contexts, e.g., quantum liquids, finite wing theories, hypothetical fluid computers, liquid crystals, supercoiling in DNA, and cell motility. Maybe the most provocative perspective comes from the open conjectures and questions raised in the 1960s by Feynmann and Onsager, which assert that turbulent states, characterized by interacting vortex filaments, are expected to relax to a non-turbulent state by allowing energy transfer between length scales mediated by nonlinear waves. Significant theoretical and experimental advancements have occurred in the last 10-15 years. A recent paper published in Nature this March 2023, titled Rotating quantum wave turbulence, reports on experimental advancements in observing this wave turbulence. Completing this research project will aid in future development by quantifying the appropriate scaling for the myriad of mathematical models in the field.

Grand Challenge: Not applicable.

Geometric quantum hydrodynamics: modeling and simulation of vortex rings in ideal fluids, Rudge et al., 2021 https://www.mines.edu/undergraduate-research/project/geometric-quantum-hydrodynamics-modeling-and-simulation-of-vortex-rings-in-ideal-fluids/ [Additional resources here.] Hasimoto transformation of general flows expressed in the Frenet frame, Hofer et al., 2023, Applied Numerical Mathematics,  https://scholar.google.com/citations?view_op=view_citation&hl=en&user=_LiI3FwAAAAJ&sortby=pubdate&citation_for_view=_LiI3FwAAAAJ:LI9QrySNdTsC Derivation of the Biot-Savart equation from the nonlinear Schrödinger equation, Bustamante et al., Phys. Rev. E, 2015, https://journals.aps.org/pre/abstract/10.1103/PhysRevE.92.053019 Rotating quantum wave turbulence, Mäkinen et al., Nature, 2023, https://www.nature.com/articles/s41567-023-01966-z

Scott Strong, sstrong@mines.edu