Strongly nonlinear long gravity waves in uniform shear flows

Long surface gravity waves of finite amplitude in uniform shear flows are considered by using an asymptotic model derived under the assumption that the aspect ratio between wavelength and water depth is small. Since its derivation requires no assumption on wave amplitude, the model can be used to describe arbitrary amplitude waves. It is shown that the simple model captures the interesting features of strongly nonlinear solitary waves observed in previous numerical solutions. When compared with the case of zero vorticity, the solitary wave in uniform shear flows is wider when propagating upstream (opposite to the direction of surface drift), while it is narrower when propagating downstream. For the upstream-propagating solitary wave, a stationary recirculating eddy appears at the bottom when wave amplitude exceeds the critical value. For the case of downstream propagation, no eddy forms at the bottom but the solitary wave becomes more peaked, yielding a cusp at the critical wave amplitude, beyond which the solitary wave has a round wave profile. Although the appearance of the derivative singularity is inconsistent with the long-wave assumption in the model, round wave profiles away from the singularity are qualitatively similar to numerical solutions and observation.