Uni-directional waves over slowly varying bottom. Part I: Derivation of a KdV-type of equation (original) (raw)

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N ov 2 01 8 Surface waves over currents and uneven bottom

2018

The propagation of surface water waves interacting with a current and an uneven bottom is studied. Such a situation is typical for ocean waves where the winds generate currents in the top layer of the ocean. The role of the bottom topography is taken into account since it also influences the local wave and current patterns. Specific scaling of the variables is selected which leads to approximations of Boussinesq and KdV types. The arising KdV equation with variable coefficients, dependent on the bottom topography, is studied numerically when the initial condition is in the form of the one soliton solution for the initial depth. Emergence of new solitons is observed as a result of the wave interaction with the uneven bottom. *Corresponding author. E-mail address: rossen.ivanov@dit.ie (R. I. Ivanov)

A Whitham–Boussinesq long-wave model for variable topography.

• Present bidirectional nonlocal shallow water wave model for variable depth. • Nonlocality introduced through simplified variable depth Dirichlet–Neumann operator. • Study variable depth effects in linear normal modes, examples of steepening and modulation. • Show qualitative agreement with some higher order constant depth nonlocal models. • Model exhibits combination of nonlinear and variable depth effects. a b s t r a c t We study the propagation of water waves in a channel of variable depth using the long-wave asymptotic regime. We use the Hamiltonian formulation of the problem in which the non-local Dirichlet–Neumann operator appears explicitly in the Hamiltonian, and propose a Hamiltonian model for bidirectional wave propagation in shallow water that involves pseudo-differential operators that simplify the variable-depth Dirichlet–Neumann operator. The model generalizes the Boussinesq system, as it includes the exact dispersion relation in the case of constant depth. Analogous models were proposed by Whitham for unidirectional wave propagation. We first present results for the normal modes and eigen-frequencies of the linearized problem. We see that variable depth introduces effects such as a steepening of the normal modes with the increase in depth variation, and a modulation of the normal mode amplitude. Numerical integration also suggests that the constant depth nonlocal Boussinesq model can capture qualitative features of the evolution obtained with higher order approximations of the Dirichlet–Neumann operator. In the case of variable depth we observe that wave-crests have variable speeds that depend on the depth. We also study the evolutions of Stokes waves initial conditions and observe certain oscillations in width of the crest and also some interesting textures and details in the evolution of wave-crests during the passage over obstacles.

Long-time dynamics of KdV solitary waves over a variable bottom

Communications on Pure and Applied Mathematics, 2006

We study the variable bottom generalized Korteweg-de Vries (bKdV) equation ∂tu = −∂x ∂ 2 x u + f (u) − b(t, x)u , where f is a nonlinearity and b is a small, bounded and slowly varying function related to the varying depth of a channel of water. Many variable coefficient KdV-type equations, including the variable coefficient, variable bottom KdV equation, can be rescaled into the bKdV. We study the long time behaviour of solutions with initial conditions close to a stable, b = 0 solitary wave. We prove that for long time intervals, such solutions have the form of the solitary wave, whose centre and scale evolve according to a certain dynamical law involving the function b(t, x), plus an H 1 (R)-small fluctuation. * This paper is part of the first author's Ph.D. thesis. † Supported by NSERC under grant NA7901 and Ontario Graduate Scholarships. ‡ Supported by NSF under grant DMS-0400526.

Surface waves over currents and uneven bottom

Deep Sea Research Part II: Topical Studies in Oceanography

The propagation of surface water waves interacting with a current and an uneven bottom is studied. Such a situation is typical for ocean waves where the winds generate currents in the top layer of the ocean. The role of the bottom topography is taken into account since it also influences the local wave and current patterns. Specific scaling of the variables is selected which leads to approximations of Boussinesq and KdV types. The arising KdV equation with variable coefficients, dependent on the bottom topography, is studied numerically when the initial condition is in the form of the one soliton solution for the initial depth. Emergence of new solitons is observed as a result of the wave interaction with the uneven bottom.

Generalized KdV-type equations versus Boussinesq's equations for uneven bottom -- numerical study

arXiv: Pattern Formation and Solitons, 2020

The paper's main goal is to compare the motion of solitary surface waves resulting from two similar but slightly different approaches. In the first approach, the numerical evolution of soliton surface waves moving over the uneven bottom is obtained using single wave equations. In the second approach, the numerical evolution of the same initial conditions is obtained by the solution of a coupled set of the Boussinesq equations for the same Euler equations system. We discuss four physically relevant cases of relationships between small parameters alpha,beta,delta\alpha,\beta,\deltaalpha,beta,delta. For the flat bottom, these cases imply the Korteweg-de Vries equation (KdV), the extended KdV (KdV2), fifth-order KdV (KdV5), and the Gardner equation (GE). In all studied cases, the influence of the bottom variations on the amplitude and velocity of a surface wave calculated from the Boussinesq equations is substantially more significant than that obtained from single wave equations.

Accurate modelling of uni-directional surface waves

Journal of Computational and Applied Mathematics, 2010

This paper shows the use of consistent variational modelling to obtain and verify an accurate model for uni-directional surface water waves. Starting from Luke's variational principle for inviscid irrotational water flow, Zakharov's Hamiltonian formulation is derived to obtain a description in surface variables only. Keeping the exact dispersion properties of infinitesimal waves, the kinetic energy is approximated. Invoking a unidirectionalization constraint leads to the recently proposed AB-equation, a KdV-type of equation that is also valid on infinitely deep water, that is exact in dispersion for infinitesimal waves, and that is second order accurate in the wave height. The accuracy of the model is illustrated for two different cases. One concerns the formulation of steady periodic waves as relative equilibria; the resulting wave profiles and the speed are good approximations of Stokes waves, even for the Highest Stokes Wave on deep water. A second case shows simulations of severely distorting downstream running bi-chromatic wave groups; comparison with laboratory measurements show good agreement of propagation speeds and of wave and envelope distortions.

Hamiltonian approach to modelling interfacial internal waves over variable bottom

Physica D: Nonlinear Phenomena, 2022

We study the effects of an uneven bottom on the internal wave propagation in the presence of stratification and underlying non-uniform currents. Thus, the presented models incorporate vorticity (wave-current interactions), geophysical effects (Coriolis force) and a variable bathymetry. An example of the physical situation described above is well illustrated by the equatorial internal waves in the presence of the Equatorial Undercurrent (EUC). We find that the interface (physically coinciding with the thermocline and the pycnocline) satisfies in the long wave approximation a KdV-mKdV type equation with variable coefficients. The soliton propagation over variable depth leads to effects such as soliton fission, which is analysed and studied numerically as well.