The propagation of nonlinear waves in layered and stratified fluids
Abstract of the thesis entitled
THE PROPAGATION OF NONLINEAR WAVES IN LAYERED AND STRATIFIED FLUIDS
Derek Wing-Chiu Lai
for the degree of Doctor of Philosophy
at the University of Hong Kong
in April 2001
In this thesis the propagation of nonlinear waves in layered and stratified fluids is investigated. In the first part of this research, ?nconventional?solitary waves are obtained and their interactions are investigated by the Hirota bilinear transformation. Such solitary waves are ?nconventional?because they can be expressed analytically as some mixed exponential - algebraic expressions. Furthermore, the separation of the crests goes like a logarithm, rather than a linear function, in the time scale. In a proper frame of reference these unconventional solitary waves are usually counterpropagating waves.
These counterpropagating waves and their interactions are investigated for several nonlinear evolution equations which are of fluid dynamical interests. Firstly, 2- and 3-soliton expansions are obtained for the Manakov system, a coupled set of nonlinear Schr?inger equations arising from the propagation of multiphase modes when the group velocity projections overlap. A pair of counterpropagating waves is observed if the technique of ?erger?of the wavenumbers is performed for a 2-soliton expansion, and the separation of the crests goes like a
logarithm in time. Furthermore, temporal modulation of the amplitude is observed if the same technique is applied to a 3-soliton expansion.
A similar procedure is then applied to the (2+1)-dimensional (2 spatial and 1 temporal dimensions) long wave?hort wave resonance interaction equations in a two-layer fluid. Such long?hort resonance interactions can be considered as a degenerate case of triad resonance. The required condition is that the phase velocity of the long wave matches the group velocity of the short wave. The ?erger?technique can also be extended to the dromion solutions. Dromions are exact, localized solutions of (2 + 1) (2 spatial and 1 temporal) dimensions that decay exponentially in all directions.
In a two-layer fluid the modified Korteweg-de Vries (mKdV) systems will be the governing equation if the quadratic nonlinearity vanishes. The required condition for the case of irrotational flow is that the density ratio is approximately equal to the square of the depth ratio. Under the irrotational flow assumption only the mKdV systems with the cubic nonlinear and the dispersive terms of opposite signs (mKdV-) exist. Our contribution here is to investigate the wave propagation in a two-layer fluid with shear flows in order to demonstrate the existence of mKdV systems with the cubic nonlinear and the dispersive terms of the same sign (mKdV+). A class of counterpropagating waves and their interactions are studied for the mKdV+.
From the perspective of fluid dynamics the propagation of nonlinear waves in the first part of this research is considered in the
weakly nonlinear regime. In the second part of this research fully nonlinear internal solitary waves in stratified fluids are calculated. Such internal waves for the exponential and linear density profiles are obtained by computing the higher order terms in an asymptotic expansion where the Boussinesq and long wave parameters are comparably small. With increasing amplitude the wavelength of the solitary waves generally decreases and recirculation zones will develop. The Boussinesq approximation commonly seen in literature is thus removed or relaxed.
School:The University of Hong Kong
School Location:China - Hong Kong SAR
Source Type:Master's Thesis
Keywords:water waves mathematical models fluid dynamics
Date of Publication:01/01/2001