Linear water waves with vorticity: Rotational features and particle paths

Department of Mathematics, Lund University, PO Box 118, 221 00 Lund, Sweden
Journal of Differential Equations (Impact Factor: 1.68). 01/2008; 244(8):1888-1909. DOI: 10.1016/j.jde.2008.01.012
Source: arXiv


We study steady linear gravity waves of small amplitude travelling on a current of constant vorticity. For positive vorticity the situation resembles that of Stokes waves, but if the vorticity is large enough the particle trajectories are affected. For negative vorticity we show that there may appear internal waves and vortices, wherein the particle trajectories are not ellipses.

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Available from: Mats Ehrnström, Oct 10, 2015
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    • "Spurred by the above results there has also been interest in studying the properties and dynamics of these waves below the surface [11] [37]. This had been done for linear waves in [14]. Several other avenues have also been considered: We mention heterogeneous waves both with [39] [40] and without [38] surface tension, a variational approach [2] and Hamiltonian formulation with center manifold reduction [19]. "
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    ABSTRACT: We construct small-amplitude solitary traveling gravity-capillary water waves with a finite number of point vortices along a vertical line, on finite depth. This is done using a local bifurcation argument. The properties of the resulting waves are also examined: We find that they depend significantly on the position of the point vortices in the water column.
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    • "The effect of vorticity on the modulational stability of periodic surface waves was investigated in [25]. In the presence of vorticity, even the linear problem still provides challenges, as testified to by the recent study [5] on particle paths in rotational linear surface waves, and the article [26] which focuses on the linear dispersion relation for surface waves in the context of non-constant vorticity. "
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    ABSTRACT: The effect of constant background vorticity on the pressure beneath steady long gravity waves at the surface of a fluid is investigated. Using an asymptotic expansion for the streamfunction, we derive a model equation and a formula for the pressure in a flow with constant vorticity. The model equation was previously found by Benjamin (1962), [3], and is given in terms of the vorticity ω0ω0, and three parameters Q,RQ,R and SS representing the volume flux, total head and momentum flux, respectively.The focus of this work is on the reconstruction of the pressure from solutions of the model equation and the behavior of the surface wave profiles and the pressure distribution as the strength of the vorticity changes. In particular, it is shown that for strong enough vorticity, the maximum pressure is no longer located under the wave crest, and the fluid pressure near the surface can be below atmospheric pressure.
    European Journal of Mechanics - B/Fluids 01/2013; 37:187–194. DOI:10.1016/j.euromechflu.2012.09.009 · 1.66 Impact Factor
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    • "Therefore, such a layer, generally speaking, separates two other layers with opposite directions of flow. First, the existence of steady waves on a flow of constant vorticity in which a critical layer (and so, a counter-current) is present was established by Ehrnström and Villari [30], who studied streamlines and particle paths in the framework of linear theory. It was Wahlén [68], who proved the existence of small-amplitude Stokes waves with a constant vorticity and discovered that in the reference frame moving with the wave there is a critical layer. "
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    ABSTRACT: The two-dimensional free-boundary problem of steady periodic waves with vorticity is considered for water of finite depth. We investigate how a flow with small-amplitude Stokes waves on the free surface bifurcates from a horizontal parallel shear flow in which counter-currents may be present. The bifurcation mechanism is described in terms of a dispersion equation; namely, wavelengths of Stokes waves bifurcate from the values defined by the roots of this equation. The latter generalizes that for irrotational waves and involves only quantities given on the horizontal free surface of the initial parallel shear flow. Sufficient conditions guaranteeing the existence of roots of the dispersion equation are obtained. Two particular vorticity distributions are considered in order to illustrate general results.
    Archive for Rational Mechanics and Analysis 07/2012; 214(3). DOI:10.1007/s00205-014-0787-0 · 2.22 Impact Factor
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