Large-amplitude steady rotational water waves

Analyticity of periodic traveling free surface water waves with vorticity

“…For Stokes waves one can show that the maximal value of the horizontal fluid velocity in the flow is attained at the wave crest and for regular waves the wave speed exceeds this maximal value, while for the waves of greatest height these two values are equal (and consequently the wave crest is a stagnation point since the vertical fluid velocity there is zero) [1], [20], [21]. For rotational waves the existence of waves of this type is currently at the level of conjectures supported by formal considerations and numerical simulations (see the discussions in [7], [19], [15], [16]). Homogeneity (constant density) implies the equation of mass conservation…”

Dispersion Relations for Steady Periodic Water Waves of Fixed Mean-Depth with an Isolated Bottom Vorticity Layer

“…remarkably, we will see that the dispersion relations for the two systems coincide, which suggests that the main difference in the two frameworks is observed for larger amplitude waves [24] (see [3] for analogous dispersion relations for the fixed mass-flux system). The small-amplitude solutions which were obtained in [19], although they are in fact nonlinear, are compatible with the linear regime for water waves, and one of the most important, and most physically interesting, properties of linear water waves is their dispersion relation [25].…”

“…In [19], the mean-depth of the fluid above the flat bed is a fixed quantity for the continuum of solutions of the governing equations which are given by the bifurcation curve, in [11] the mass-flux is a fixed quantity. It was observed in [24] that, for fixed mass flux, as we move along the bifurcation curve the mean-depth of the flows actually varies, and so we must make a choice between the different frameworks. Since the mean-depth of the fluid may reasonably be regarded as the more inherent physical property of a fluid flow, the framework which we adopt here is that of [19] where the mean depth is fixed.…”

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

“…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). Although the recent important theoretical developments have confirmed that periodic waves can exist over flows with arbitrary vorticity, it appears that their stability to infinitesimal disturbances and their subsequent nonlinear evolution have not been studied extensively so far.…”