The fixed points and the manifolds in a second order Stokes wave

Abstract

Here, we present an analysis of the flow properties of second order Stokes waves in water. The description of the flow field is developed using the concept of fixed points and manifolds, which is commonly employed for the analysis of a nonlinear dynamic system. We find that the components of the velocity field are related to each other by an elliptic correlation, where the center of the ellipse represents the fixed points. Since an ellipse is not likely to pass through its center, the estimation of the fixed points in a second order Stokes wave seems challenging. However, we find that the fixed points can be found out in a degenerate case of the ellipse; such a case is observed at the bottom surface that is found to host all the fixed points. The vertical lines passing through the fixed points represent the manifolds. We find that, interestingly, the fixed points and the corresponding manifolds are not fixed but rather move with a speed that equals the wave celerity. Here, we show that the deformation of the free surface requires straining. The flow field evolves in a manner to sustain such straining. Despite the rigid nature, the flow straining is also observed at the bottom surface. Such straining is found to be generated by the fixed points at the bottom surface. The vertically oriented manifolds are found acting as the guides to mediate such flow and straining exchange between the free and bottom surface.