Nonlinear Processes in Geophysics www.nonlin-processes-geophys.net 1023-5809 1607-7946 16 1 2009 10.5194/npg-16-111-2009 http://www.nonlin-processes-geophys.net/16/111/2009/ http://www.nonlin-processes-geophys.net/16/111/2009/npg-16-111-2009.html http://www.nonlin-processes-geophys.net/16/111/2009/npg-16-111-2009.pdf 111 122 2009-02-17 Head-on collision of two solitary waves and residual falling jet formation J. Chambarel chambarel@irphe.univ-mrs.fr C. Kharif J. Touboul Institut de Recherche sur les Phénomènes Hors Equilibre, Marseille, France The head-on collision of two equal and two unequal steep solitary waves is investigated numerically. The former case is equivalent to the reflection of one solitary wave by a vertical wall when viscosity is neglected. We have performed a series of numerical simulations based on a Boundary Integral Equation Method (BIEM) on finite depth to investigate during the collision the maximum runup, phase shift, wall residence time and acceleration field for arbitrary values of the non-linearity parameter <i>a/h</i>, where <i>a</i> is the amplitude of initial solitary waves and <i>h</i> the constant water depth. The initial solitary waves are calculated numerically from the fully nonlinear equations. The present work extends previous results on the runup and wall residence time calculation to the collision of very steep counter propagating solitary waves. Furthermore, a new phenomenon corresponding to the occurrence of a residual jet is found for wave amplitudes larger than a threshold value. Bona, J L. and Chen, M.: A Boussinesq system for two-way propagation of nonlinear dispersive waves, Physica D., 116, 191–224, 1998. Byatt-Smith, J. G B.: An integral equation for unsteady surface waves and a comment on the Boussinesq equation., J. Fluid Mech., 49, 625–633, 1971. Byatt-Smith, J. G B.: The reflection of a solitary wave by a vertical wall, J. Fluid. Mech., 197, 503–521, 1988. Chan, R. K C. and Street, R L.: A computer study of finite-amplitude water waves., J. Computat. Phys., 6, 68–94, 1970. Chen, G., Kharif, C., Zaleski, S., and Li, J.: Two-dimensional Navier–Stokes simulation of breaking waves., Phys. Fluids, 11, 121–133, 1999. Cooker, M J., Weidman, P D., and Bale, D S.: Reflection of a high-amplitude solitary wave at a vertical wall, J. Fluid Mech., 342, 141–158, 1997. Craig, W., Guyenne, P., Hammack, J., Henderson, D., and Sulem, C.: Solitary wave interactions., Phys. Fluids, 18, 1–25, 2006. Fenton, J D. and Rienecker, M M.: A Fourier method for solving nonlinear water-wave problems : application to solitary-wave interactions., J. Fluid Mech., 118, 411–443, 1982. Maxworthy, T.: Experiments on collisions between solitary waves., J. Fluid Mech., 76, 177–185, 1976. Mirie, R M. and Su, C H.: Collisions between two solitary waves. Part 2. A numerical study., J. Fluid. Mech., 115, 475–492, 1982. Oikawa, M. and Yajima, N.: Interactions of solitary waves - a perturbation approach to nonlinear systems., J. Phys. Soc. Japan, 34, 1093–1099, 1973. Pelinovsky, E., Troshina, E., Golinko, V., Osipenko, N., and Petrukhin, N.: Runup of tsunami waves on a vertical wall in a basin of complex topography., Phys. Chem. Earth (B), 24, 431–436, 1999. Power, H. and Chwang, A T.: On reflection of a planar at a vertical wall., Wave Motion, 6, 183–195, 1984. Su, C H. and Gardner, C S.: Korteweg-de Vries equation and generalizations. III. Derivation of the Korteweg-de vries equation and Burgers equation., J. Math. Phys., 10, 536–539, 1969. Su, C H. and Mirie, R M.: On head-on collisions between two solitary waves., J. Fluid Mech., 98, 509–525, 1980. Tanaka, M.: The stability of solitary waves., Phys. Fluids, 29, 650–655, 1986. Taylor, S G.: The instability of liquid surfaces when accelerated in a direction perpendicular to their planes, Proc. Roy. Soc. London, 201, 192–196, 1950. Temperville, A.: Interaction of solitary waves in shallow water theory., Arch. Mech., 31, 177–184, 1979. Touboul, J. and Kharif, C.: Two-dimensional direct numerical simulations of the dynamics of rogue waves under wind action (Ed Q W Ma), The world Scientific Publishing Co, 2009. Touboul, J., Giovanangeli, J P., Kharif, C., and Pelinovsky, E.: Freak waves under the action of wind: Experiments and simulations, Eur. J. Mech. B/Fluids, 25(5), 662–676, 2006.