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

Karima Khusnutdinova

doi:10.2205/2020es000734

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.

Highlights

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
We note that for the same surface strength U (1) = 0.0045 there exist currents in this family which appear to have wavefronts squeezed in the direction of the shear flow, in contrast to the behaviour illustrated in previous plots

Summary

AF φz

Where the function φ = φ(z, θ) satisfies the following set of modal equations: The modal equations are obtained by looking for a solution of the problem in the form of asymptotic multiple-scales expansions of the form ζ = ζ1 + εζ2+. Where we define s to be the wave speed in the absence of a shear flow (with k(θ) = 1). When a shear flow is present the function k(θ) is responsible for the adjustment of the wave speed in a particular direction, and is to be determined. The wavefront at any fixed moment of time t is described by the equation rk(θ) = constant, and for the sake of definiteness we consider outward propagating ring waves, requiring that the function k = k(θ) > 0. To leading order, assuming that perturbations of the basic state are caused only by the propagating wave, the motion is described by the solution μ1AR

Aθ R

When γ

The deformation of the wavefront of a ring wave

Conclusion

The α