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Weakly Nonlinear Evolution of a Wave Packet in a Zonal Mixing Layer
Author(s) -
Mallier R.,
Maslowe S. A.
Publication year - 1999
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/1467-9590.00104
Subject(s) - wave packet , inviscid flow , instability , amplitude , physics , nonlinear system , mathematical analysis , mathematics , classical mechanics , quantum mechanics
A nonlinear integrodifferential equation governing the amplitude evolution of a wavepacket near the critical value of the beta parameter is derived. The basic velocity profile is a hyperbolic tangent shear layer and although the neutral eigensolution is regular, all higher‐order terms in the expansion of the stream function are singular at the critical point. The analysis is inviscid and in the critical layer both wave packet effects and nonlinearity are present, but the former are taken to be slightly larger. Unlike the Stuart–Watson theory, the critical layer analysis dictates the form of the amplitude equation, the outer expansion being relatively passive. A secondary instability analysis shows that the packet is unstable to sideband perturbations, but the instability is weak so its main consequence would be to produce some modulation of the packet without destroying its coherence.