Oblique waves on a vertically sheared current are rotational

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.

6 pages, 1 figure, Accepted for publication in Eur. J. Mech. B/Fluids

References in corpus (5)

• Ship waves on uniform shear current at finite depth: wave resistance and critical velocity
• Initial surface disturbance on a shear current: the Cauchy--Poisson problem with a twist
• Waves from an oscillating point source with a free surface in the presence of a shear current
• Oscillating line source in a shear flow with a free surface: critical layer-like contributions
• Water Waves from General, Time-Dependent Surface Pressure Distribution in the Presence of a Shear Current

Cited by in corpus (11)

• Approximate dispersion relations for waves on arbitrary shear flows
• Ship waves on uniform shear current at finite depth: wave resistance and critical velocity
• Surface waves on arbitrary vertically-sheared currents
• Statistics of weakly nonlinear waves on currents with strong vertical shear
• Observation of surface wave patterns modified by sub-surface shear currents
• Weakly non-linear transient waves on a shear current: Ring waves and skewed Langmuir rolls
• Waves from an oscillating point source with a free surface in the presence of a shear current
• Oscillating line source in a shear flow with a free surface: critical layer-like contributions
• Sheared free-surface flow over three-dimensional obstructions of finite amplitude
• Turbulence-induced anti-Stokes flow: experiments and theory
• Oscillating sources in a shear flow with a free surface