Long ring waves in a stratified fluid over a shear flow

Abstract

Oceanic waves registered by satellite observations often have curvilinear fronts and propagate over various currents. In this paper, we study long linear and weakly-nonlinear ring waves in a stratified fluid in the presence of a depth-dependent horizontal shear flow. It is shown that despite the clashing geometries of the waves and the shear flow, there exists a linear modal decomposition (different from the known decomposition in Cartesian geometry), which can be used to describe distortion of the wavefronts of surface and internal waves, and systematically derive a 2+1 - dimensional cylindrical Korteweg-de Vries - type equation for the amplitudes of the waves. The general theory is applied to the case of the waves in a two-layer fluid with a piecewise-constant current, with an emphasis on the effect of the shear flow on the geometry of the wavefronts. The distortion of the wavefronts is described by the singular solution (envelope of the general solution) of the nonlinear first-order differential equation, constituting generalisation of the dispersion relation in this curvilinear geometry. There exists a striking difference in the shape of the wavefronts of surface and interfacial waves propagating over the same shear flow.

25 pages, 11 figures

References in corpus (4)

• Nonlinear shallow ocean wave soliton interactions on flat beaches
• Ship waves in the presence of uniform vorticity
• Coupled Ostrovsky equations for internal waves in a shear flow
• Initial surface disturbance on a shear current: the Cauchy--Poisson problem with a twist

Cited by in corpus (8)

• Soliton solutions to the fifth-order Korteweg - de Vries equation and their applications to surface and internal water waves
• Observation of surface wave patterns modified by sub-surface shear currents
• Nonlinear ring waves in a two-layer fluid
• Weakly non-linear transient waves on a shear current: Ring waves and skewed Langmuir rolls
• Wavefronts and modal structure of long surface and internal ring waves on a parallel shear current
• Long internal ring waves in a two-layer fluid with an upper-layer current
• Interaction between long internal waves and free surface waves in deep water
• Evolution of internal cnoidal waves with local defects in a two-layer fluid with rotation