Statistical theory of gravity and capillary waves on the surface of a finite-depth fluid
type: Item
instance of: scholarly article
title: Statistical theory of gravity and capillary waves on the surface of a finite-depth fluid
zbMATH Open document ID: 0960.76014
DOI: 10.1016/S0997-7546(99)80031-4
published in: European Journal of Mechanics. B. Fluids
publication date: 2001-05-01T00:00:00Z
review text: Starting from general equations describing potential surface waves in inviscid fluids of arbitrary depth, the author develops an asymptotic analysis for a small-amplitude case. The resultant equations of motion are obtained in the canonical Hamilton's forms for properly defined complex amplitudes expressed in terms of two-dimensional spatial Fourier transforms of surface perturbations. To simplify the equations, the author uses a canonical transformation aimed at the elimination of quadratic nonlinearities. The final equation has the form of an integral first-order evolutional equation with linear and cubic terms (the integrations are performed in the space of the Fourier-transform wave-vectors, which are continuous parameters of complex canonical variables). Various special cases, which are of interest for applications, are considered in detail, including deep- and shallow-water limits. In the latter case, the kernel of the nonlinear integral term is formally singular, but it is possible to prove that all the singularities mutually cancel. It is stressed that the canonical transformation which eliminates the quadratic terms changes the definition of physically relevant action-type variables, which may have important consequences in the application to the shallow-water limit. The author also develops a statistical description of the weak wave turbulence, based on a combination of the integral evolution equation and Gaussian statistical correlations imposed on physically correct action-type variables. This analysis produces an effective kinetic equation, which, in particular, allows one to find a nonlinear correction to the dispersion equation for weakly nonlinear plane waves. Stationary analytical solutions to this kinetic equation are found in the form of Kolmogorov's spectra. Finally, the author demonstrates that the applicability of the above weakly nonlinear theory to the shallow-water case is strongly limited, especially in the limit of long surface waves.
reviewed by: B. A. Malomed
Mathematics Subject Classification ID: 76B15
Mathematics Subject Classification ID: 76F55
Mathematics Subject Classification ID: 76M35
Mathematics Subject Classification ID: 76M45
zbMATH DE Number: 1357998
zbMATH Keywords: Hamilton's equations
zbMATH Keywords: potential surface waves
zbMATH Keywords: turbulence spectrum
zbMATH Keywords: inviscid fluids
zbMATH Keywords: asymptotic analysis
zbMATH Keywords: small-amplitude case
zbMATH Keywords: complex amplitudes
zbMATH Keywords: two-dimensional spatial Fourier transforms
zbMATH Keywords: canonical transformation
zbMATH Keywords: integral first-order evolutional equation
zbMATH Keywords: shallow-water limit
zbMATH Keywords: weak wave turbulence
zbMATH Keywords: Gaussian statistical correlations
zbMATH Keywords: kinematic equation
zbMATH Keywords: Kolmogorov's spectra
author: V. E. Zakharov
MaRDI profile type: Publication
cites work: Nonlinear modulation of gravity waves: a rigorous approach
cites work: Is free-surface hydrodynamics an integrable system?
cites work: On the non-linear energy transfer in a gravity-wave spectrum Part 1. General theory
cites work: Q4380014
Wikidata QID: Q126298449
full work available at URL: s0997-7546(99)80031-4
OpenAlex ID: W4379614869
description: scientific article; zbMATH DE number 1357998