Abstract
We study long surface and internal ring waves propagating in a stratified fluid over a parallel shear current. The far-field modal and amplitude equations for the ring waves are presented in dimensional form. We re-derive the modal equations from the formulation for plane waves tangent to the ring wave, which opens a way to obtaining important characteristics of the ring waves (group speed, wave action conservation law) and to constructing more general ‘hybrid solutions’ consisting of a part of a ring wave and two tangent plane waves. The modal equations constitute a new spectral problem, and are analysed for a number of examples of surface ring waves in a homogeneous fluid and internal ring waves in a stratified fluid. Detailed analysis is developed for the case of a two-layered fluid with a linear shear current where we study their wavefronts and two-dimensional modal structure. Comparisons are made between the modal functions (i.e. eigenfunctions of the relevant spectral problems) for the surface waves in homogeneous and two-layered fluids, as well as the interfacial waves described exactly and in the rigid-lid approximation. We also analyse the wavefronts of surface and interfacial waves for a large family of power-law upper-layer currents, which can be used to model wind generated currents, river inflows and exchange flows in straits. Global and local measures of the deformation of wavefronts are introduced and evaluated.
Highlights
The Korteweg–de Vries (KdV) equation and its generalisations such as the Gardner, Ostrovsky and Kadomtsev–Petviashvili (KP) equations are well known as good weakly nonlinear models describing long surface and internal waves that are commonly observed in the oceans, see, for example, Grimshaw et al (1998), Helfrich & Melville (2006), Grimshaw et al (2010), Ablowitz & Baldwin (2012) and Grimshaw, Helfrich & Johnson (2013)
In this study we linked the description of the ring waves in a cylindrical geometry with the description of the plane waves tangent to the ring wave and propagating at various angles to the shear flow
The general solution of this nonlinear first-order differential equation corresponds to the plane waves tangent to the ring wave, while its singular solution describes the ring wave
Summary
The Korteweg–de Vries (KdV) equation and its generalisations such as the Gardner, Ostrovsky and Kadomtsev–Petviashvili (KP) equations are well known as good weakly nonlinear models describing long surface and internal waves that are commonly observed in the oceans, see, for example, Grimshaw et al (1998), Helfrich & Melville (2006), Grimshaw et al (2010), Ablowitz & Baldwin (2012) and Grimshaw, Helfrich & Johnson (2013). The developed linear formulation provided, in particular, a description of the distortion of the shape of the wavefronts of surface and internal ring waves in a two-layered fluid by the piecewise-constant current. The wavefronts of surface and interfacial ring waves were described in terms of two branches of the envelope of the general solution of the derived nonlinear first-order differential equation, constituting further generalisation of the well-known Burns (Burns 1953) and generalised Burns (Johnson 1990) conditions. Significant squeezing of the wavefronts of interfacial ring waves, similar to that described for a piecewise-constant current, can take place for some currents in the family Such currents are close to river inflows and exchange flows in straits, while for wind-generated-type currents the wavefronts appear to be elongated in the direction of the current.
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