Project Info

Dimensional reduction of continuum models for vortex filament motion.

Scott Strong

Project Goals and Description:

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.

More Information:

Grand Challenge: Not applicable.
Geometric quantum hydrodynamics: modeling and simulation of vortex rings in ideal fluids, Rudge et al., 2021 [Additional resources here.] Hasimoto transformation of general flows expressed in the Frenet frame, Hofer et al., 2023, Applied Numerical Mathematics, Derivation of the Biot-Savart equation from the nonlinear Schrödinger equation, Bustamante et al., Phys. Rev. E, 2015, Rotating quantum wave turbulence, Mäkinen et al., Nature, 2023,

Primary Contacts:

Scott Strong,

Student Preparation


Minimum qualifications: Completion of MATH111, 112, 213, 225, PHGN 100, 200. Ideal qualifications: Additionally, completion of MATH332 and/or MATH455. Experience in scientific computing and/or quantum mechanics.




Practice with learning in the context of academic research, e.g., record keeping, iterative processes, regulation through open-ended problems, and diagnostics of ill-posed problems. Mathematical modeling with vector analysis, differential equations, linear algebra, differential geometry applied to space curves, continuum fluids, and their idealization in quantum fluids.


The successful applicant will be expected to learn how to work independently and keep records to efficiently report progress. To support this endeavor, the principle investigator will mentor the undergraduate researcher through weekly meetings where reporting, reflection, and planning will be emphasized. In addition, we will use weekly meetings to discuss technical/supplemental content, participate in collaborative problem-solving, and prepare scientific communications/presentations.


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