Solitary waves propagating over variable topography

Combined Effect of Rotation and Topography on Shoaling Oceanic Internal Solitary Waves

Journal of Physical Oceanography ◽

10.1175/jpo-d-13-0194.1 ◽

2014 ◽

Vol 44 (4) ◽

pp. 1116-1132 ◽

Cited By ~ 49

Author(s):

Roger Grimshaw ◽

Chuncheng Guo ◽

Karl Helfrich ◽

Vasiliy Vlasenko

Keyword(s):

Solitary Wave ◽

Wave Packet ◽

Numerical Simulations ◽

Solitary Waves ◽

Combined Effect ◽

Internal Solitary Wave ◽

Internal Solitary Waves ◽

De Vries ◽

Ostrovsky Equation ◽

Variable Topography

Abstract Internal solitary waves commonly observed in the coastal ocean are often modeled by a nonlinear evolution equation of the Korteweg–de Vries type. Because these waves often propagate for long distances over several inertial periods, the effect of Earth’s background rotation is potentially significant. The relevant extension of the Kortweg–de Vries is then the Ostrovsky equation, which for internal waves does not support a steady solitary wave solution. Recent studies using a combination of asymptotic theory, numerical simulations, and laboratory experiments have shown that the long time effect of rotation is the destruction of the initial internal solitary wave by the radiation of small-amplitude inertia–gravity waves, and the eventual emergence of a coherent, steadily propagating, nonlinear wave packet. However, in the ocean, internal solitary waves are often propagating over variable topography, and this alone can cause quite dramatic deformation and transformation of an internal solitary wave. Hence, the combined effects of background rotation and variable topography are examined. Then the Ostrovsky equation is replaced by a variable coefficient Ostrovsky equation whose coefficients depend explicitly on the spatial coordinate. Some numerical simulations of this equation, together with analogous simulations using the Massachusetts Institute of Technology General Circulation Model (MITgcm), for a certain cross section of the South China Sea are presented. These demonstrate that the combined effect of shoaling and rotation is to induce a secondary trailing wave packet, induced by enhanced radiation from the leading wave.

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Evolution of solitary waves and undular bores in shallow-water flows over a gradual slope with bottom friction

Journal of Fluid Mechanics ◽

10.1017/s0022112007006817 ◽

2007 ◽

Vol 585 ◽

pp. 213-244 ◽

Cited By ~ 45

Author(s):

G. A. EL ◽

R. H. J. GRIMSHAW ◽

A. M. KAMCHATNOV

Keyword(s):

Solitary Wave ◽

Shallow Water ◽

Solitary Waves ◽

Periodic Wave ◽

Local Effect ◽

Bottom Friction ◽

Undular Bore ◽

Shallow Water Flows ◽

Adiabatic Approach ◽

Variable Topography

This paper considers the propagation of shallow-water solitary and nonlinear periodic waves over a gradual slope with bottom friction in the framework of a variable-coefficient Korteweg–de Vries equation. We use the Whitham averaging method, using a recent development of this theory for perturbed integrable equations. This general approach enables us not only to improve known results on the adiabatic evolution of isolated solitary waves and periodic wave trains in the presence of variable topography and bottom friction, modelled by the Chezy law, but also, importantly, to study the effects of these factors on the propagation of undular bores, which are essentially unsteady in the system under consideration. In particular, it is shown that the combined action of variable topography and bottom friction generally imposes certain global restrictions on the undular bore propagation so that the evolution of the leading solitary wave can be substantially different from that of an isolated solitary wave with the same initial amplitude. This non-local effect is due to nonlinear wave interactions within the undular bore and can lead to an additional solitary wave amplitude growth, which cannot be predicted in the framework of the traditional adiabatic approach to the propagation of solitary waves in slowly varying media.

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