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Elliptic and hyperelliptic functions describing the particle motion beneath small-amplitude water waves with constant vorticity
We provide analytic solutions of the nonlinear differential equation system describing the particle paths below small-amplitude periodic gravity waves travelling on a constant vorticity current. We show that these paths are not closed curves. Some solutions can be expressed in terms of Jacobi elliptic functions, others in terms of hyperelliptic functions. We obtain new kinds of particle paths. We make some remarks on the stagnation points which could appear in the fluid due to the vorticity.
In this paper we make a detailed analysis of the short-wavelength instability method for barotropic incompressible fluids. We apply this method to edge waves in stratified water. These waves are unstable to short-wavelength perturbations if their steepness exceeds a specific threshold.
Small-amplitude equatorial water waves with constant vorticity: Dispersion relations and particle trajectories
We consider the two-dimensional equatorial water-wave problem with constant vorticity in the $f$-plane approximation. Within the framework of small-amplitude waves, we derive the dispersion relations and we find the analytic solutions of the nonlinear differential equation system describing the particle paths below such waves. We show that the solutions obtained are not closed curves. Some remarks on the stagnation points are also provided.
We describe the physical hypotheses underlying the derivation of an approximate model of water waves. For unidirectional surface shallow water waves moving over an irrotational flow as well as over a non-zero vorticity flow, we derive the Camassa-Holm equation by an interplay of variational methods and small-parameter expansions.
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