Stokes Waves in a Constant Vorticity Flow
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.
KeywordsStokes wave Constant vorticity Conformal Numerical
Mathematics Subject Classification (2000)Primary 76B15; Secondary 76B07 30C30 65T50
VMH is supported by the US National Science Foundation under the Faculty Early Career Development (CAREER) Award DMS-1352597, and SD is supported by the National Science Foundation under DMS-1716822. VMH is grateful to the Erwin Schrödinger International Institute for Mathematics and Physics for its hospitality during the workshop Nonlinear Water Waves. This work was completed with the support of our TE X-pert.
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