Near-integrable models for long surface and internal ring waves in stratified shear flows

Abstract: In this talk I will first overview some general results concerning the effects of the parallel shear flow on long weakly-nonlinear surface and internal ring waves in a stratified fluid (e.g., oceanic internal waves generated in narrow straits and river-sea interaction zones), generalising the results for surface waves in a homogeneous fluid [1]. We showed that despite the clashing geometries of the waves and the shear flow, there exists a linear modal decomposition (separation of variables) in the far-field set of Euler equations describing the waves in a stratified fluid, more complicated than the known decomposition for plane waves [2,3]. We used it to describe the wavefronts of surface and internal waves, and to derive a 2D cylindrical Korteweg - de Vries (cKdV)-type model for the amplitudes of the waves. The distortion of the wavefronts is described explicitly by constructing the singular solution (envelope of the general solution) of a respective nonlinear first-order differential equation.

Next, we consider a two-layer fluid with a rather general depth-dependent upper-layer current (e.g. a river inflow, 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 [4]. 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 will also discuss a 2D generalisation of the long-wave instability criterion for plane interfacial waves on a piecewise-constant current [4], which on physical level manifests itself in the counter-intuitive squeezing of the wavefront of the interfacial ring wave.

REFERENCES

1. R.S. Johnson, Ring waves on the surface of shear flows: a linear and nonlinear theory, J. Fluid Mech., 215, 1638-1660 (1990).

2. K.R. Khusnutdinova, X. Zhang, Long ring waves in a stratified fluid over a shear flow, J. Fluid Mech., 794, 17-44 (2016).

3. K.R. Khusnutdinova, X. Zhang, Nonlinear ring waves in a two-layer fluid, Physica D, 333, 208-221 (2016).

4. K.R. Khusnutdinova, Long internal ring waves in a two-layer fluid with an upper-layer current, submitted (2020).

5. L.V. Ovsyannikov, Two-layer 'shallow water' model, J. Appl. Math. Tech. Phys. 20, 127-135 (1979).

Zoom limk: us04web.zoom.us/j/75476385312?pwd=dTU0U0VQMTN5VlFOMVVHNmhaS1pCZz09

machine learningmathematical softwaresymbolic computationmathematical physicsanalysis of PDEsclassical analysis and ODEsdynamical systemsnumerical analysisexactly solvable and integrable systemscomputational physicsdata analysis, statistics and probability

Audience: researchers in the topic

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Mathematical models and integration methods

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