Wavefronts and modal structure of long surface and internal ring waves on a parallel shear current

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.

Linearizable Abel equations and the Gurevich–Pitaevskii problem

“…Here, are a few final comments: An interesting class of exactly solvable first‐order ODEs (with nonlinear dependence on the derivative) whose singular solutions can be parameterized by hypergeometric functions has appeared in Ref. 34 in the context of ring waves in stratified fluids (the so‐called directional adjustment equations). In this connection, one should mention that algebraic separatrix solutions of the equations $$A_{c_1, c_2}$$ constructed in our paper can be viewed as singular solutions. The algebra $$\mathfrak {g}=\langle \partial _z,z \partial _z-g \partial _g,z^2 \partial _z-(2zg+1) \partial _g\rangle$$ is one of four inequivalent realizations of the Lie algebra $$\mathfrak {sl}_2(\mathbb {R})$$, but it is the only one that leads to Abel equations as symmetry reductions of $$\mathfrak {sl}_2(\mathbb {R})$$‐invariant third‐order ODEs.…”

Internal ring waves in a three-layer fluid on a current with a constant vertical shear

“…However, on satellite images, internal waves generated in narrow straits [32] or by river plums [43], in particular, appear to have curvilinear wavefronts that resemble part of a ring wave and propagate over various currents, which motivates the present study. To describe ring waves and their relatives [24], we require the development of akin models in cylindrical geometry. Such models were developed and analysed in the absence of a background current [19,25,28,37,41,42,46,49,50], and extended to study the propagation of internal ring waves over a parallel shear current [24,[31][32][33], generalising the work by Johnson on surface ring waves in a homogeneous fluid over a parallel shear current [29,30].…”

“…To describe ring waves and their relatives [24], we require the development of akin models in cylindrical geometry. Such models were developed and analysed in the absence of a background current [19,25,28,37,41,42,46,49,50], and extended to study the propagation of internal ring waves over a parallel shear current [24,[31][32][33], generalising the work by Johnson on surface ring waves in a homogeneous fluid over a parallel shear current [29,30].…”

“…The simplest stratified configuration amenable to analytical studies is a two-layer system (see [24,[31][32][33]). For example, in [32,33], ring waves in a two-layer flow with piecewise-constant current, bounded above by a free surface, were considered.…”

Linearisable Abel equations and the Gurevich-Pitaevskii problem

“…• An interesting class of exactly solvable first-order ODEs (with nonlinear dependence on the derivative) whose singular solutions can be parametrised by hypergeometric functions has appeared in [10] in the context of ring waves in stratified fluids (the so-called directional adjustment equations). In this connection, one should mention that algebraic separatrix solutions of the equations A c 1 ,c 2 constructed in our paper can be viewed as singular solutions.…”