Long internal ring waves in a two-layer fluid with an upper-layer current

TL;DR

This paper analytically investigates long interfacial ring waves in a two-layer fluid with depth-dependent currents, revealing how currents influence wave structure, stability, and wavefront deformation.

Contribution

It provides an analytical framework for describing wavefronts and stability of interfacial ring waves in variable currents, extending previous models to more realistic flow profiles.

Findings

Wavefronts depend on current profiles and density jump.

Strong currents can induce instability and wavefront squeezing.

Analytical solutions involve hypergeometric functions.

Abstract

We consider a two-layer fluid with a depth-dependent upper-layer current (e.g. a river inflow, an exchange flow in a strait, or a wind-generated current). In the rigid-lid approximation, we find the necessary singular solution of the nonlinear first-order ordinary differential equation responsible for the adjustment of the speed of the long interfacial ring wave in different directions in terms of the hypergeometric function. This allows us to obtain an analytical description of the wavefronts and vertical structure of the ring waves for a large family of the current profiles and to illustrate their dependence on the density jump and the type and the strength of the current. In the limiting case of a constant upper-layer current we obtain a 2D ring waves' analogue of the long-wave instability criterion for plane interfacial waves. On physical level, the presence of instability for a sufficiently strong current manifests itself already in the stable regime in the squeezing of the wavefront of the interfacial ring wave in the direction of the current. We show that similar phenomenon can also take place for other, depth-dependent currents in the family.