DECOUPLED AND UNIDIRECTIONAL ASYMPTOTIC MODELS FOR THE PROPAGATION OF INTERNAL WAVES

2012/08/31 by Vincent Duchêne, Vincent Duchene · 26 citations
Earth and Planetary Sciences · Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Applied mathematics #Classical mechanics #Consistency (knowledge bases) #Geometry #Internal wave #Inviscid flow #Korteweg–de Vries equation #Mathematical analysis #Mathematics #Mechanics #Nonlinear Waves and Solitons #Nonlinear system #Ocean Waves and Remote Sensing #Physics #Scalar (mathematics) #Type (biology) #math.AP

paper · pdf · doi:10.1142/s0218202513500462

published in Mathematical Models and Methods in Applied Sciences 24(01), 1-65 (World Scientific)

arxiv created 2012/12/18 · openalex publication_date 2013/04/30 · arxiv updated 2021/11/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We study the relevance of various scalar equations, such as inviscid Burgers', Korteweg–de Vries (KdV), extended KdV, and higher order equations, as asymptotic models for the propagation of internal waves in a two-fluid system. These scalar evolution equations may be justified in two ways. The first method consists in approximating the flow by two uncoupled, counterpropagating waves, each one satisfying such an equation. One also recovers these equations when focusing on a given direction of propagation, and seeking unidirectional approximate solutions. This second justification is more restrictive as for the admissible initial data, but yields greater accuracy. Additionally, we present several new coupled asymptotic models: a Green–Naghdi type model, its simplified version in the so-called Camassa–Holm regime, and a weakly decoupled model. All of the models are rigorously justified in the sense of consistency.

Citations

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Cited by

• Shallow water asymptotic models for the propagation of internal waves