Bore formation, evolution and disintegration into solitons in shallow inhomogeneous channels

Abstract

Abstract. The propagation of nonlinear surface waves in channels of smoothly variable in space cross section is studied theoretically and by means of numerical computations. The mathematical model describing wave evolution is based on the generalized Korteweg-de Vries equation with additional terms due to spatial inhomogeneity and energy dissipation. Specifically we consider channels of variable depth and width. The breaking of Riemann waves and the disintegration of hydraulic jumps into trains of solitons have been examined. The results obtained can be useful in particular for the understanding some peculiarities of bore (mascaret) formation, viscous evolution and disintegration into solitons in inhomogeneous channels or rivers.

Steepness and spectrum of nonlinear deformed shallow water wave

“…The same processes are observed when sea waves entry in river mouths (Pelinovsky, 1982;Tsuji et al, 1991) and straits or channels (Pelinovsky and Troshina, 1994;Wu and Tian, 2000;Caputo and Stepanyants, 2003). Meanwhile, we do not know publications where the characteristics of the nonlinear deformed wave such as steepness, spectrum and location of breaking point have been analyzed in details.…”

Nonlinear long-wave deformation and runup in a basin of varying depth

“…(17) describes a simple or Riemann wave, which is well known in nonlinear acoustics (Rudenko and Soluyan, 1977;Engelbrecht et al, 1988;Gurbatov et al, 1991). The same solution with different modifications has been obtained for water waves (Burger, 1967;Varley et al, 1971;Gurtin, 1975;Pelinovsky, 1982;Caputo and Stepanyants, 2003). Equation (17) allows the description of the deformation of weakly nonlinear wave "velocity" or wave of water displacement (with the use of Eq.…”

Runup of Tsunami Waves in U-Shaped Bays

“…Even for uniform in longitudinal direction channels, it is not easy to find the solution for the nonlinear traveling waves (solitons, cnoidal waves) analytically. This problem requires solving the 2D Laplace equation in cross-section domain with curvilinear boundary (Peregrine, 1968(Peregrine, , 1969Fenton, 1973;Shen and Zhong, 1981;Das, 1985;Mathew and Akylas, 1990;Teng and Wu, 1992, 1997Caputo and Stepanyants, 2003). At the same time approximation of a rectangular cross-section does not correspond to natural narrow bays, which bottom configuration varies in both longitudinal and transversal directions.…”