Non-Perturbative Interfacial Waves

University Of Missouri-Columbia, Columbia MO

Investigators

Abstract

Interfacial waves are waves that propagate along a boundary separating two substances or regions. They are found throughout nature, ranging from internal waves in the ocean to spreading contagions in biology and defect patterns in material science. Much is now known about waves that are perturbative in some sense. For instance, if the interface is not greatly disturbed, its motion can be well predicted using simpler linear or weakly nonlinear models. However, many important phenomena fall far outside this category. As one example, following the 2022 volcanic eruption in Tonga, tsunami-like waves were observed moving much faster than linear theory predicts in shallow water and even seemed to amplify in deep water. This project aims to deepen the mathematical understanding of these and similar non-perturbative waves for which the underlying systems exhibit intrinsically nonlinear dynamics, surface singularities, or resonance phenomena. Progress in this direction will be benefit the larger mathematics and science communities, and society more broadly as they will bring predictive capabilities improvements and ultimately help mitigate future disasters. The project will provide research training opportunities for undergraduate and graduate students. Internal bores are fronts that propagate along pycnoclines in stratified bodies of water. They play a geophysically significant role in mixing, energy transport, and oceanic circulation. Using global bifurcation theory and free boundary elliptic regularity theory methods, this project seeks to prove the existence of overhanging internal gravity waves and verify a conjecture of von Karman regarding the existence and form of exact gravity currents. Another objective concerns the time evolution of interfacial hydrodynamic waves. As part of this project, the investigator will use techniques from infinite-dimensional Hamiltonian systems to characterize the spectral stability of multimodal internal capillary-gravity solitary waves, and prove that solitary waves with strong surface tension are orbitally unstable with respect to transverse perturbations. Understanding meteotsunamis --- atmospherically generated tsunamis --- created by the Tonga eruption is another central objective. One prominent explanation for their formation is based on three-wave resonance in a two-phase lightly compressible Euler system. This project will give the first rigorous treatment to this theory by constructing steady axisymmetric three-dimensional solutions as models for the initial atmospheric shockwaves and exact three-dimensional doubly-periodic steady solutions for exhibiting the resonant triad. The final aim of the project concerns localized vortical structures carried by waves. A hollow vortex is a bounded region of constant pressure encircled by a vortex sheet and suspended inside a perfect fluid; their existence and stability in the plane have been studied since the 19th century. The investigator will construct exact water waves with a submerged hollow vortex and ascertain their orbital stability, giving insight into wave-vortex interaction. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.