 Award ID(s):
 1816934
 Publication Date:
 NSFPAR ID:
 10177608
 Journal Name:
 Journal of Fluid Mechanics
 Volume:
 883
 ISSN:
 00221120
 Sponsoring Org:
 National Science Foundation
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The interaction of localised solitary waves with largescale, timevarying dispersive mean flows subject to nonconvex flux is studied in the framework of the modified Korteweg–de Vries (mKdV) equation, a canonical model for internal gravity wave propagation and potential vorticity fronts in stratified fluids. The effect of large amplitude, dynamically evolving mean flows on the propagation of localised waves – essentially ‘soliton steering’ by the mean flow – is considered. A recent theoretical and experimental study of this new type of dynamic soliton–mean flow interaction for convex flux has revealed two scenarios where the soliton either transmits through the varying mean flow or remains trapped inside it. In this paper, it is demonstrated that the presence of a nonconvex cubic hydrodynamic flux introduces significant modifications to the scenarios for transmission and trapping. A reduced set of Whitham modulation equations is used to formulate a general mathematical framework for soliton–mean flow interaction with nonconvex flux. Solitary wave trapping is stated in terms of crossing modulation characteristics. Nonconvexity and positive dispersion – common for stratified fluids – imply the existence of localised, sharp transition fronts (kinks). Kinks play dual roles as a mean flow and a wave, imparting polarity reversal to solitons andmore »

We consider a strongly nonlinear long wave model for large amplitude internal waves in a threelayer flow between two rigid boundaries. The model extends the twolayer Miyata–Choi–Camassa (MCC) model (Miyata, Proceedings of the IUTAM Symposium on Nonlinear Water Waves , eds. H. Horikawa & H. Maruo, 1988, pp. 399–406; Choi & Camassa, J. Fluid Mech. , vol. 396, 1999, pp. 1–36) and is able to describe the propagation of long internal waves of both the first and second baroclinic modes. Solitarywave solutions of the model are shown to be governed by a Hamiltonian system with two degrees of freedom. Emphasis is given to the solitary waves of the second baroclinic mode (mode 2) and their strongly nonlinear characteristics that fail to be captured by weakly nonlinear models. In certain asymptotic limits relevant to oceanic applications and previous laboratory experiments, it is shown that large amplitude mode2 waves with singlehump profiles can be described by the solitarywave solutions of the MCC model, originally developed for mode1 waves in a twolayer system. In other cases, however, e.g. when the density stratification is weak and the density transition layer is thin, the richness of the dynamical system with two degrees of freedom becomes apparent and new classesmore »

Abstract This paper considers twodimensional steady solitary waves with constant vorticity propagating under the influence of gravity over an impermeable flat bed. Unlike in previous works on solitary waves, we allow for both internal stagnation points and overhanging wave profiles. Using analytic global bifurcation theory, we construct continuous curves of largeamplitude solutions. Along these curves, either the wave amplitude approaches the maximum possible value, the dimensionless wave speed becomes unbounded, or a singularity develops in a conformal map describing the fluid domain. This is stronger than what one would expect from a straightforward generalization of existing results for periodic waves. We also show that an arbitrary solitary wave of elevation with constant vorticity must be supercritical. The existence proof relies on a novel reformulation of the problem as an elliptic system for two scalar functions in a fixed domain, one describing the conformal map of the fluid region and the other the flow beneath the wave.

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