Dispersion Equation for Water Waves with Vorticity and Stokes Waves on Flows with Counter-Currents

Abstract

The two-dimensional free-boundary problem of steady periodic waves with vorticity is considered for water of finite depth. We investigate how flows with small-amplitude Stokes waves on the free surface bifurcate from a horizontal parallel shear flow in which counter-currents may be present. Two bifurcation mechanisms are described: for waves with fixed Bernoulli's constant and fixed wavelength. In both cases the corresponding dispersion equations serve for defining wavelengths from which Stokes waves bifurcate. Necessary and sufficient conditions for the existence of roots of these equations are obtained. Two particular vorticity distributions are considered in order to illustrate general results.Mathematics Subject Classification (2010) 76B15 · 35Q35

Two-dimensional stability of finite-amplitude gravity waves on water of finite depth with constant vorticity

“…Since the pioneering work of Constantin & Strauss (2004), there is a huge literature that concerns the formulation and the mathematical properties of steady periodic surface water waves with vorticity (existence, unicity, bifurcation). For a review on the recent rigorous results, the reader can refer to Constantin & Varvaruca (2011) and Kozlov & Kuznetsov (2014). Among the authors using asymptotic methods or purely numerical methods, on can cite Tsao (1959), Dalrymple (1974), Brevik (1979), Simmen & Saffman (1985), Teles da Silva & Peregrine (1988) , Kishida & Sobey (1988), Vanden-Broeck (1996), Swan & James (2001), Ko & Strauss (2008), Pak & Chow (2009), Cheng, Cang & Liao (2009 , Moreira & Chacaltana (2015), Hsu, Francius, Montalvo & Kharif (2016), Ribeiro-Jr, Milewski & Nachbin (2017).…”

Steady water waves with vorticity: an analysis of the dispersion equation

“…At these wavenumbers, branches of small-amplitude Stokes waves bifurcate from the shear flow, as is shown by Constantin & Strauss (2004 for unidirectional flows and by Kozlov & Kuznetsov (2014) for flows in which, generally speaking, countercurrents may be present; in both cases, general vorticity distributions are considered. We demonstrate that for unidirectional flows the approach to the dispersion equation developed in the latter paper is equivalent to that of Constantin & Strauss (2004; the latter is based on a certain Sturm-Liouville problem.…”

“…Background Kozlov & Kuznetsov (2014) briefly characterized the plethora of results obtained for the problem under consideration and a similar one dealing with waves on water of infinite depth; further details can be found in the survey article by Strauss (2010). Therefore, we focus only on the work concerning various forms of the dispersion equation.…”

Hamiltonian Formulation for Wave-Current Interactions in Stratified Rotational Flows

“…These are surfaces where the phase speed of wave propagation equals the mean flow speed-see [9,10,20,31] for a discussion in the simpler context of travelling waves in a homogeneous fluid. The extraction of momentum from the mean flow feeds the growth of critical layers, triggering instability mechanisms (see [22]).…”