Stokes Waves in a Constant Vorticity Flow

Sergey A. Dyachenko, Vera Mikyoung Hur

Research output: Chapter in Book/Report/Conference proceedingChapter


The Stokes wave problem in a constant vorticity flow is formulated via conformal mapping as a modified Babenko equation. The associated linearized operator is self-adjoint, whereby efficiently solved by the Newton-conjugate gradient method. For strong positive vorticity, a fold develops in the wave speed versus amplitude plane, and a gap as the vorticity strength increases, bounded by two touching waves, whose profile contacts with itself, enclosing a bubble of air. More folds and gaps follow as the vorticity strength increases further. Touching waves at the beginnings of the lowest gaps tend to the limiting Crapper wave as the vorticity strength increases indefinitely, while a fluid disk in rigid body rotation at the ends of the gaps. Touching waves at the boundaries of higher gaps contain more fluid disks.

Original languageEnglish (US)
Title of host publicationNonlinear Water Waves
Subtitle of host publicationAn Interdisciplinary Interface
EditorsDavid Henry, Konstantinos Kalimeris, Emilian I Părău, Jean-Marc Vanden-Broeck, Erik Wahlén
Number of pages16
ISBN (Print)9783030335359, 9783030335366
StatePublished - Dec 2019

Publication series

NameTutorials, Schools, and Workshops in the Mathematical Sciences
ISSN (Print)2522-0969
ISSN (Electronic)2522-0977


  • Conformal
  • Constant vorticity
  • Numerical
  • Stokes wave

ASJC Scopus subject areas

  • General Mathematics
  • Physics and Astronomy (miscellaneous)


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