ABSTRACT Stagnation points beneath solitary gravity-capillary waves in the weakly nonlinear weakly dispersive regime in a sheared channel with finite depth and constant vorticity are investigated. A Korteweg-de Vries equation that incorporates the surface tension and the vorticity effects is obtained asymptotically from the full Euler equations. The velocity field in the bulk fluid is approximated which allow us to compute stagnation points in the solitary wave moving frame. We show that stagnation points bellow the crest of elevation solitary waves exist for large values of the vorticity and Bond numbers less than a critical value that depends on the vorticity. Remarkably, the positions of these stagnation points do not depend on the surface tension. Besides, we show that when there are two stagnation points located at the bottom of the channel, they are pulled towards the horizontal coordinate of the solitary wave crest as the Bond number increases until its critical value. gravitycapillary gravity capillary investigated Kortewegde Korteweg de equations frame Remarkably Besides

ABSTRACT In this work we present a numerical study of surface water waves over variable topographies for the linear Euler equations based on a conformal mapping and Fourier transform. We show that in the shallow-water limit the Jacobian of the conformal mapping brings all the topographic effects from the bottom to the free surface. Implementation of the numerical method is illustrated by a MATLAB program. The numerical results are validated by comparing them with exact solutions when the bottom topography is flat, and with theoretical results for an uneven topography.

ABSTRACT In this work, we investigate the propagation of rotational solitary waves over a submerged obstacle in a vertically sheared shallow water channel with constant vorticity. In the weakly nonlinear, weakly dispersive regime the problem is formulated in the forced Korteweg-de Vries equation framework. The initial value problem for this equation is solved numerically using a Fourier pseudospectral method with an integrating factor. Solitary waves are taken as initial data and their interactions with an obstacle are analysed. We identify three types of regimes according to the intensity of the vorticity. A rotational solitary wave can bounce back and forth over the obstacle remaining trapped for large times, it can pass over the obstacle without reversing its direction or the wave can be blocked, i.e., it bounces back and forth above the obstacle until reaching a steady state. Such behaviour resembles the classical damped spring-mass system.

ABSTRACT Nonlinear steady rotational waves in a low-pressure region are investigated. The problem is formulated in a simplified canonical domain through the use of a conformal mapping, which flattens the free surface. Steady waves are computed numerically using a Newton’s method and classified into three types. Besides, our results indicate that there is a region in which steady waves do not exist. The thickness of this region is compared with the one predicted by the weakly nonlinear, weakly dispersive regime.