January 14, 2008
Using Operational Calculus for modeling Water Waves
Prof. Yehuda Agnon, Department of Civil and Environmental Engineering, Technion, Haifa, Israel
Abstract
The study of linear and nonlinear diffraction and shoaling of irrotational water waves by uneven bottom topography has received considerable attention. Phenomena such as edge waves, harbour resonance and surf-beats are a few of its manifestations. Spatio-temporal evolution equations are powerful in the study of focusing and of rogue waves. In the case of slow spatial evolution, one can eliminate the vertical coordinate, z. The solution of a three-dimensional (3D) flow field is replaced by a simpler, (approximate) 2D problem. In spatially homogeneous problems (e.g. rectangular basins with an even bottom), the waves are conveniently represented in the wavenumber domain, leading to the formulation of the Zakharov equation. Non-uniformity in space due to non-periodic spatial evolution or to bottom slope, and the study of more general basins, can be accounted for through representation in the frequency, rather than wavenumber, domain. For linear shoaling, fully dispersive models include the mild slope equation (MSE), the extended, and the modified MSE (Chamberlain and Porter, 1995). Most of the work on nonlinear shoaling has been based on the assumption of weak dispersion, either by using Boussinesq type models or by assuming narrow spectra in Schrödinger equation models. This is because the full dispersion operator is an integral operator in space and its pseudo-differential representation involves spatial derivatives to infinite order.
Agnon et al. (1993, 1998) and Kaihatu and Kirby (1995) have studied a nonlinear MSE - in which the linear dispersion is exact, but the nonlinear interaction is only accurate for resonant interaction. This, apparently, limited the model's applicability to near resonance quadratic interaction, since it introduced an error in the bound waves that was too large to allow extension of the model to include cubic (four-wave) interaction. Agnon and Sheremet (1997) have derived a stochastic nonlinear MSE. Mei (1999) has derived a nonlinear MSE for long waves induced by short waves. Shemer et al. (2001) have derived a spatial Zakharov equation. Rasmussen and Stiassnie (2001) have derived a discretized spatio-temporal model on an even bottom.
Accounting for the dispersive nature of water waves in spatial models complicates the solution and requires consideration of high order derivatives (essentially to any order) of the depth and of the potential. The inherent difficulty of the problem stems from the need to account for non-local interaction. A new model for linear and nonlinear shoaling has been recently derived (Agnon, 1999; Agnon and Pelinovsky, 2001), using operational calculus. It systematically accounts for the effect of terms in high order derivatives of the bathymetry without calculating them. This is done by utilizing the fact that the dynamics of the system is governed by resonant interaction. After deriving a pseudo-differential equation, we expand the nonlocal operator in a small parameter - the detuning from resonance. The result is that the nonlocal operator is approximated by low order differential operators. Here we extend this approach to obtain more accurate nonlinear shoaling .The present work differs from the previous derivations of MSE and its linear and nonlinear extensions: no assumption is made regarding the vertical structure of the wave field (except for the lack of 'global' evanescent modes). This results in a model of very general validity. The derivation provides insight regarding in which regimes the use of existing and new models is appropriate.
We study an augmented nonlinear MSE which accurately accounts for the bound waves (subharmonics and superharmonics). This enables the enhancement of the nonlinear MSE to include cubic interaction. Cubic interaction is important in the evolution of ocean waves and is known to play a central role in subharmonic resonance of edge waves. We derive the waves which are induced by the bottom variations, and show that there exists no general MSE-type equation for describing Class II Bragg resonance (i.e. linear in the wave amplitude and quadratic in the bottom variation). We study Class III Bragg resonance (i.e. quadratic in the wave amplitude and linear in the bottom variation) over bathymetries that are not gentle, and derive an extended nonlinear MSE, and a modified nonlinear MSE. The performance of the different models in calculating resonant double-frequency harmonic generation on an oscillatory bottom variation is compared. Spatio-temporal equations are obtained directly from the spatial evolution equations by discretization of the frequency spectrum.