Abstract
Abstract
Highlights
In the present paper, we consider the problem describing two-dimensional gravity waves travelling on a flow of finite depth
A new feature of laminar flows with constant vorticity is that there are flows with critical levels
We apply the spatial dynamics method to the scaled problem in the same way as in Kozlov & Lokharu (2019). This allows us to reduce the problem to a finite-dimensional Hamiltonian system; it has one degree of freedom and admits a homoclinic orbit describing a solitary wave of elevation in the original coordinates
Summary
We consider the problem describing two-dimensional gravity waves travelling on a flow of finite depth. Wheeler (2013) examined waves of large amplitude, but, like in the irrotational case, all solitary-wave solutions have the same structure of streamlines, that is, are symmetric and of positive elevation (see Hur. 2008; Wheeler 2015; Kozlov, Kuznetsov & Lokharu 2015, 2017). A new feature of laminar flows with constant vorticity (compared with irrotational ones) is that there are flows with critical levels It was shown by Wahlén (2009) that small perturbations of these parallel flows are periodic waves with arrays of cat’s-eye vortices – regions where closed streamlines surround stagnation points. A new family of solitary waves is constructed for large negative values of the constant vorticity All these waves have a remarkable property: the corresponding flow is unidirectional at both infinities, but there is a cat’s-eye vortex centred below the wave crest (see figure 1).
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