Shallow water models with constant vorticity

Research output: Contribution to journalArticle

Abstract

We modify the nonlinear shallow water equations, the Korteweg–de Vries equation, and the Whitham equation, to permit constant vorticity, and examine wave breaking, or the lack thereof. By wave breaking, we mean that the solution remains bounded but its slope becomes unbounded in finite time. We show that a solution of the vorticity-modified shallow water equations breaks if it carries an increase of elevation; the breaking time decreases to zero as the size of vorticity increases. We propose a full-dispersion shallow water model, which combines the dispersion relation of water waves and the nonlinear shallow water equations in the constant vorticity setting, and which extends the Whitham equation to permit bidirectional propagation. We show that its small amplitude and periodic traveling wave is unstable to long wavelength perturbations if the wave number is greater than a critical value, and stable otherwise, similarly to the Benjamin–Feir instability in the irrotational setting; the critical wave number grows unboundedly large with the size of vorticity. The result agrees with that from a multiple scale expansion of the physical problem. We show that vorticity considerably alters the modulational stability and instability in the presence of the effects of surface tension.

Original languageEnglish (US)
Pages (from-to)170-179
Number of pages10
JournalEuropean Journal of Mechanics, B/Fluids
Volume73
DOIs
StatePublished - Jan 1 2019

Keywords

  • Breaking
  • Constant vorticity
  • Modulational instability
  • Shallow water

ASJC Scopus subject areas

  • Mathematical Physics
  • Physics and Astronomy(all)

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