Rossby solitary waves in the presence of a critical layer
P. Caillol,Roger Grimshaw +1 more
TLDR
In this paper, the evolution of long nonlinear Rossby waves in a sheared zonal current in the regime where a competition sets in between weak nonlinearity and weak dispersion is considered.Abstract:
This study considers the evolution of long nonlinear Rossby waves in a sheared zonal current in the regime where a competition sets in between weak nonlinearity and weak dispersion. The traditional assumption of a weak amplitude breaks down when the wave speed equals the mean-flow velocity at a certain latitude, due to the appearance of a singularity in the leading order equation, which strongly modifies the flow in the critical layer. Here, nonlinear effects are invoked to resolve this singularity, since the relevant geophysical flows have high Reynolds numbers. Viscosity is introduced in order to render the nonlinear critical-layer solution unique, but the inviscid limit is eventually taken. By the method of matched asymptotic expansions, this inner flow is matched at the edges of the critical layer with the outer flow. This theory is relevant for the phenomenon of Rossby wave breaking and eventual saturation into a nonlinear wave. Spatially localized solutions are described by a Korteweg-de Vries equation, modified by new nonlinear terms, depending on the critical-layer shape. These lead to depression or elevation solitary waves.read more
Citations
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Time-Space Fractional Coupled Generalized Zakharov-Kuznetsov Equations Set for Rossby Solitary Waves in Two-Layer Fluids
Lei Fu,Yaodeng Chen,Hongwei Yang +2 more
TL;DR: In this article, a generalized Zakharov-Kuznetsov (gZK) equations set was derived to describe two kinds of weakly nonlinear waves interaction by multiple coupling terms.
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Rossby Elevation Waves in the Presence of a Critical Layer
P. Caillol,Roger Grimshaw +1 more
TL;DR: In this article, the authors investigated the evolution of solitary-wave-like development of small-amplitude Rossby waves propagating in a zonal shear current, for the particular case when the Rossby wave speed equals the mean flow velocity at a certain latitude in the β-plane.
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The Small Vorticity Nonlinear Critical Layer for Kelvin Modes on a Vortex
P. Caillol,S. A. Maslowe +1 more
TL;DR: In this article, the authors considered the propagation of neutral modes along a vortex with velocity profile being the radial coordinate and formulated a nonlinear critical layer theory valid when the critical point occurs far enough from the center of the vortex so that the vorticity there is small.
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Internal solitary waves with a weakly stratified critical layer
P. Caillol,Roger Grimshaw +1 more
TL;DR: In this paper, the authors considered the evolution of long weakly nonlinear internal waves in an incompressible Boussinesq fluid, where the motion is restricted to the vertical plane.
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A singular vortex Rossby wave packet within a rapidly rotating vortex
TL;DR: In this paper, the authors describe the quasi-steady regime attained by a rapidly rotating vortex after a wave packet has interacted with it, and derive the leading-order equations of the system of nonlinear coupled partial differential equations that govern the slowly evolving amplitudes of the wave packet and induced mean flow a long time after this interaction started.
References
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TL;DR: In this paper, it was shown that the Stewartson-warn-warn (SWW) solution for the time evolution of an inviscid, nonlinear Rossby-wave critical layer, which predicts that the critical layer will alternate between absorbing and over-reflecting states as time goes on, is hydrodynamically unstable.