Unstable stokes waves in constant vorticity flows

We delve into the modulational instability of small-amplitude, non-zero vorticity Stokes waves of the classical water-wave problem, which concerns the two-dimensional incompressible flow of a perfect fluid of unit depth under the force of gravity. The stability analysis method is based on a periodic Evans function approach developed recently by Hur and Yang for irrotational Stokes waves [1, 2] and center manifold mechanism to reduce the infinite-dimensional hydrodynamic problem to a four-dimensional space. We construct a modulational instability index, denoted indlow(ω, κ), depending on the vorticity ω and the wave number κ, and prove instability when indlow(ω, κ) > 0. Our result reproduces the Benjamin–Feir instability for κ > 1.3627827 . . . when vorticity is zero. For the non-zero vorticity case, we depict the stability/instability region. This modulational instability result rigorously justifies the NLS approximation of Thomas, Kharif, and Manna [3]

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2024

Author dc:creator

• Hsiao, Ting-Yang

Contributors dc:contributor

• Hur, Vera Mikyoung
• Bronski, Jared
• Tzirakis, Nikolaos
• Zharnitsky, Vadim

• Water Waves
• Rotational Stokes Waves
• Modulational Instability

Statement dc:rights

• Copyright 2024 Ting-Yang Hsiao

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/127490

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F127490