Asymptotic nonlinear wave modeling through the Dirichlet-to-Neumann operator

Abstract

Abstract. New nonlinear evolution equations are derived that generalize the system by Matsuno [16] and a terrain-following Boussinesq system by Nachbin [23]. The regime considers finite-amplitude surface gravity waves on a two-dimensional incompressible and inviscid fluid of, highly variable, finite depth. The asymptotic simplification of the nonlinear potential theory equations is performed through a perturbation anaylsis of the Dirichlet-to-Neumann operator on a highly corrugated strip. This is achieved through the use of a curvilinear coordinate system. Rather than doing a long wave expansion for the velocity potential, a Fourier-type operator is expanded in a wave steepness parameter. The novelty is that the topography can vary on a broad range of scales. It can also have a complex profile including that of a multiply-valued function. The resulting evolution equations are variable coefficient Boussinesq-type equations. These equations represent a fully dispersive system in the sense that the original (hyperbolic tangent) dispersion relation is not truncated. The formulation is done over a periodically extended domain so that, as an application, it produces efficient Fourier (FFT) solvers. A preliminary communication of this work has been published in the Physical Review Letters [1].

A Numerical Study of Linear Long Water Waves over Variable Topographies using a Conformal Mapping

“…The numerical method is implemented on MATLAB and its validation is performed by comparing numerical solutions with exact ones when the bottom is flat, and with theoretical results for uneven topographies. A similar result on the topic of this article is presented by Artiles and Nachbin [1]. Although we study the same set of equations of these authors there are remarkable differences between the two studies.…”

“…Although we study the same set of equations of these authors there are remarkable differences between the two studies. First, the mathematical formulation considered by Artiles and Nachbin [1] depends on the computation of an explicit formula of the Dirichlet-to-Neumann operator, whereas in this work this requirement is removed. Second, Artiles and Nachbin [1] considered an implicit scheme to solve the initial value problem, while here we use the classical Runge-Kutta forth-order method.…”

“…First, the mathematical formulation considered by Artiles and Nachbin [1] depends on the computation of an explicit formula of the Dirichlet-to-Neumann operator, whereas in this work this requirement is removed. Second, Artiles and Nachbin [1] considered an implicit scheme to solve the initial value problem, while here we use the classical Runge-Kutta forth-order method. In addition, our formulation allows to conclude easily that in the shallow water limit the mapping Jacobian's along the free surface carries the topographic effects.…”

Long wave expansions for water waves over random topography

“…The earlier work of Howe (1971) [15] and the paper of Rosales & Papanicolaou (1983) [24] give an asymptotic analysis of nonlinear equations of water waves. Nonlinear problems over variable topography are addressed in Nachbin (2003) [20] and Artiles & Nachbin (2004) [2]. More recent contributions which take into account the combined effect of randomness and nonlinearity include the series of papers by Mei & Hancock (2003) [17] and Grataloup and Mei (2003) [14] on the modulational scaling regime, and its extensions to the three dimensional case in Pihl, Mei & Hancock (2002) [23].…”

On weakly nonlinear gravity–capillary solitary waves

“…This method was later used to compute hexagonal periodic traveling surface wave structures (Nicholls 2001, Craig & Nicholls 2002. Studies on variable bottom topographies with this formulation also have followed (Artiles & Nachbin 2004, Guyenne & Nicholls 2005, Nicholls & Taber 2009.…”