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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    • "The shear current of constant vorticity with a free surface has recently also attracted much interest in the mathematical community , e.g. [14] [16] [17] [18] and further references therein. Recently Constantin [19] proved that when the vorticity is constant for a shear with a free surface, wave propagation must be aligned either exactly upstream or exactly downstream, i.e., the flow must be effectively twodimensional . "
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    ABSTRACT: In the study of surface waves in the presence of a shear current, a useful and much studied model is that in which the shear flow has constant vorticity. Recently it was shown by Constantin [Eur. J. Mech. B/Fluids 30 (2011) 12-16] that a flow of constant vorticity can only permit waves travelling exactly upstream or downstream, but not at oblique angles to the current, and several proofs to the same effect have appeared thereafter. Physical waves cannot possibly adhere to such a restriction, however. We resolve the paradox by showing that an oblique plane wave propagating atop a current of constant vorticity according to the linearized Euler equation carries with it an undulating perturbation of the vorticity field, hence is not prohibited by the Constantin theorem since vorticity is not constant. The perturbation of the vorticity field is readily interpreted in a Lagrangian perspective as the wave motion gently shifting and twisting the vortex lines as the wave passes. In the special case of upstream or downstream propagation, the wave advection of vortex lines does not affect the Eulerian vorticity field, in accordance with the theorem. We conclude that the study of oblique waves on shear currents requires a formalism allowing undulating perturbations of the vorticity field, and the constant vorticity model is helpful only in certain 2D systems.
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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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