CPAA
Elliptic and hyperelliptic functions describing the particle motion beneath small-amplitude water waves with constant vorticity
Delia Ionescu-Kruse
Communications on Pure & Applied Analysis 2012, 11(4): 1475-1496 doi: 10.3934/cpaa.2012.11.1475
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.
keywords: Small-amplitude water waves elliptic functions hyperelliptic functions. vorticity particle paths
DCDS
Short-wavelength instabilities of edge waves in stratified water
Delia Ionescu-Kruse
Discrete & Continuous Dynamical Systems - A 2015, 35(5): 2053-2066 doi: 10.3934/dcds.2015.35.2053
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.
keywords: localized instability analysis stratification Short-wavelength method edge waves explicit solutions.
DCDS
Small-amplitude equatorial water waves with constant vorticity: Dispersion relations and particle trajectories
Delia Ionescu-Kruse Anca-Voichita Matioc
Discrete & Continuous Dynamical Systems - A 2014, 34(8): 3045-3060 doi: 10.3934/dcds.2014.34.3045
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.
keywords: constant vorticity equatorial f-plane approximation small-amplitude waves particle paths Geophysical water waves disperion relations.
DCDS
Variational derivation of the Camassa-Holm shallow water equation with non-zero vorticity
Delia Ionescu-Kruse
Discrete & Continuous Dynamical Systems - A 2007, 19(3): 531-543 doi: 10.3934/dcds.2007.19.531
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.
keywords: variational methods flows with non-zero vorticity. Camassa-Holm equation shallow water waves

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