Asymptotics of surface waves over random bathymetry

Abstract

This paper addresses the propagation of free surface water waves over a variable seabed in the long wavelength scaling regime. We consider the situation in which the bathymetry is given by a stationary random process which has a correlation length substantially shorter than the wavelength of the principal surface wave components. An asymptotic description shows that the water waves problem is modeled by an effective system of equations that is related to the KdV, however in a reference frame given in terms of random characteristic coordinates, and in addition with a random amplitude modulation and random scattered component. The resulting random processes are strongly correlated and have canonical limits due to the Donsker invariance principle. Our analysis is based on the Hamiltonian description of water waves and long wave perturbation theory and a new criterion for asymptotic expansions of partial differential equations with rapidly varying coefficients. In this paper we give a detailed analysis of the transformation to random characteristic coordinates and the asymptotic form of the resulting transformed partial differential equations. A companion paper (de Bouard, A., Craig, W., Díaz-Espinosa, O., Guyenne, P., Sulem, C., Long wave expansions for water waves over random topography, Nonlinearity 21 (2008), 2143-2178) analyses in detail the asymptotic behavior of the resulting expression for solutions, and their consistency with the derivation of the effective model equations.

Using random walks to establish wavelike behavior in a linear FPUT system with random coefficients

“…In this paper we consider the random case, but we restrict the randomness in such a way that we can prove a fully rigorous KdV approximation. We believe that this is the first example of such a result involving randomness and nonlinear dispersive systems, though there are several earlier results which carefully derive-but do not fully justify-KdV as an effective equation for the evolution of long water waves over randomly varying topography [22,4,6,5].…”

“…Some preliminary simulations indicate this condition, which we call the translucent random mass FPUT lattice, should have a valid KdV approximation. (5) The transparency condition is hardly natural or obvious. This raises the question: are there conditions that one can place on m(j) that are less rigid than the transparency (or translucency) conditions?…”

Travelling waves over an arbitrary bathymetry: a local stability result

“…The literature on this topic is remarkably vast and a review goes far beyond the purpose of this paper, see e.g. [Van03], [Cha07], [CS10], [CLS12] and references therein. The paper [CLS12], in particular, contains an extensive analysis from the Hamiltonian point of view, in which relevant cases of periodic and O(1) bottoms are studied.…”

“…The paper [CLS12], in particular, contains an extensive analysis from the Hamiltonian point of view, in which relevant cases of periodic and O(1) bottoms are studied. The case of a random bottom, already investigated in [RP83], has been developed in [CS10]. More closely to the travelling wave problem, a remarkable attention has been devoted to the Rossby waves, either oceanic or atmospheric.…”

Convergence analysis of a variational data assimilation scheme for bathymetry detection from surface wave observations

“…Subsequently, the implementation of a variational adjoint-based scheme as presented in this study can be challenging, especially with the absence of analytical solutions of the governing nonlinear shallow water system with non-zero bathymetry. Despite this, the effect of bathymetry is unarguably significant when it comes to predicting rogue waves or tsunamis, capable of effectively modifying wave speed as well as stability of the propagating wave [5].…”