Weakly nonlinear waves in stratified shear flows

Anna Geyer; Ronald Quirchmayr

doi:10.3934/cpaa.2022061

Article Contents

2022, Volume 21, Issue 7: 2309-2325. Doi: 10.3934/cpaa.2022061

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Weakly nonlinear waves in stratified shear flows

Anna Geyer^1, and
Ronald Quirchmayr^2, ,

1.
Delft University of Technology, Delft Institute of Applied Mathematics, Faculty of EEMCS, Mekelweg 4, 2628 CD Delft, The Netherlands
2.
University of Vienna, Faculty of Mathematics, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria

^*Corresponding author
^*Corresponding author

Received: December 2021

Revised: February 2022

Early access: March 21, 2022

Published online: March 21, 2022

The second author is supported by the Austrian Science Fund (FWF), Erwin Schrödinger fellowship J 4339-N32.

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Abstract

We develop a Korteweg–De Vries (KdV) theory for weakly nonlinear waves in discontinuously stratified two-layer fluids with a generally prescribed rotational steady current. With the help of a classical asymptotic power series approach, these models are directly derived from the divergence-free incompressible Euler equations for unidirectional free surface and internal waves over a flat bed. Moreover, we derive a Burns condition for the determination of wave propagation speeds. Several examples of currents are given; explicit calculations of the corresponding propagation speeds and KdV coefficients are provided as well.

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Mathematics Subject Classification: 35R35, 76B55, 76B70.

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Figure 1. Fig. 1a shows a sketch of the stratified fluid domain bounded by a free surface at $ \bar z = \bar\eta(\bar x, \bar t) $ and a fixed bottom at $ \bar z = -\bar d $ with an interface at $ \bar z = -\bar h + \bar H(\bar x, \bar t) $ separating the upper fluid with density $ \bar\rho = \bar\rho_0 $ from the denser lower one, where $ \bar\rho = \bar\rho_0(1+r) $. Fig. 1b illustrates an example of a background current $ \bar U(\bar z) $

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