Water Wave Kinematics pp 351-365 | Cite as

# Mathematical Modelling of Short Waves in Surf Zone

## Abstract

A numerical model for the propagation of breaking waves is developed. Using an apropriate F.D. scheme in the solution of BOUSSINESQ type of equations, a third-order accuracy is obtained, without the need of including the additional SERRE terms.

By providing the above equations with a suitable dissipative mechanism by introducing a dispersion term (using the Boussinesq eddy viscosity concept), we are able to simulate breaking waves. In this way it is possible to compute both the dissipation of the wave height and set-up, and in the 2-D case, the longshore currents. In the above cases the radiation stresses are genarated automatically.

## Keywords

Wave Height Eddy Viscosity Breaking Wave Hydraulic Jump Surf Zone## Preview

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## References

- Basco, D.R. (1985). ‘A qualitative description of wave breaking’
*J.Waterw.Fort Coastal Eng. ASCE*111: 171–188.CrossRefGoogle Scholar - Battjes,J.A. (1975) ‘Modeling of turbulence in the surf zone’,
*Proc. Symp. Modeling Techniques, ASCE*pp 1050–1061.Google Scholar - Battjes, J.A. and Janssen, J.P.F.M. (1978). ‘Energy loss and set-up due to breaking of random waves’ Proc. 16th Int. Conf. Coastal Eng., ASCE pp 569–587.Google Scholar
- Battjes, J.A. and Sakai T. (1980) ‘Velocity field in a steady breaker’, Proc. 17th Int. Conf. Coastal Eng., ASCE pp 498–511.Google Scholar
- Battjes, J.A. (1986) Energy dissipation in breaking solitary and periodic waves, Communications on hydraulic and geotechnical engineering, TU Delft Report nr 86–6.Google Scholar
- Battjes, J.A. (1988) ‘Surf-zone dynamics’,
*Ann. Rev. Fluid Mech.*20: 257–293.CrossRefGoogle Scholar - Dally, W.R., Dean R.G. and Dalrymple R.A. (1984) ‘A model for breaker decay on beaches’ Proc. 19th Int. Conf. Coastal Eng., ASCE pp 82–98.Google Scholar
- Goda, U. (1970), A synthesis of breaker indices,
*Trans. Jap. Soc. Civil Eng.*, vol 180, pp 39–49 (in Japanesse)Google Scholar - Horikawa, K. and Kuo C.T. (1966), ‘A study on wave transformation inside surf zone’, Proc. 10th Int. Conf. Coastal Eng., ASCE pp 69–81.Google Scholar
- Johns, B (1980) ‘tThe modelling of the approach of bores to a shoreline’,
*Coastal Engineering*, 3: 207–219.CrossRefGoogle Scholar - Koutitas C., (1988), Mathematical Models in Coastal Engineering Pentech Press.Google Scholar
- Madsen, P.A, and Svendsen, I.A. (1983) ‘Turbulent bores and hydraulic jumps’,
*J. Fluid Mech*. vol. 129, pp 1–25.CrossRefzbMATHGoogle Scholar - Mizugushi, M. (1980) ‘An heuristic model of wave height distribution in surf zone’. Proc. 17th Int. Conf. Coastal Eng., ASCE pp 278–289.Google Scholar
- Mizugushi, M. (1986) ‘Experimental study on kinematics and dynamics of wave-breaking’ Proc. 20th Int. conf. Coastal Eng., ASCE pp 589–603.Google Scholar
- Peregrine, E.H. (1972) ‘Equations for waters waves and approximations behind them’ Waves on Beaches and Resulting Sediment Transport’ (ed R.E. Meyer) Academic PressGoogle Scholar
- Peregrine, D.H. and Svendsen, I.A. (1978) ‘Spilling breakers, bores and hydraulic jumps’, Proc. 16th Int. conf. Coastal Eng., ASCE pp 540–550.Google Scholar
- Rodi W., (1980) Turbulence models and their application in hydraulics, IAHR.Google Scholar
- Rosenberg, D.U. (1969) Methods for the Solution of Differential Equations, Elsevier N.Y.zbMATHGoogle Scholar
- Sakai, S., Hiyamizu, K., Saeki, H. (1986), ‘wave height decay model within a surf zone’, Proc. 20th Int. Conf. Coastal Eng., ASCE pp 686–696Google Scholar
- Stive, M.J.F. (1980) ‘Velocity and pressure field of spilling breakers’,
*Proc. 17th Int. conf. Coastal Eng.*, ASCE pp 547–566.Google Scholar - Stive, M.J.F. and Wind, H.G. (1982) ‘A study of radiation stress and set-up in the nearshore region’,
*Coastal Engineering*, 6: 1–25.CrossRefGoogle Scholar - Stive, M.J.F. (1984) ‘Energy dissipation in waves breaking on gentle slopes’,
*Coastal Engineering*, 8: 99–127.CrossRefGoogle Scholar - Svendsen, I.A., Madsen, P.A. and Hansen J.B. (1978) ‘Wave characteristics in the surf zone’, Proc. 16th Int. Conf. Coastal Eng., ASCE pp 520–539.Google Scholar
- Svendsen, I.A., Madsen, P.A (1984) ‘A turbulent bore on a beach’
*J. Fluid Mech*. vol. 148, pp 73–96CrossRefzbMATHGoogle Scholar - Svendsen, I.A. (1984) ‘Wave heights and set-up in a surf zone’
*Coastal Engineering*, 8: 303–329.CrossRefGoogle Scholar - Svendsen, I.A. (1987) ‘Analysis of surf zone turbulence’,
*J. of Geophysical Research*, vol. 92, no C5, pp 5115–5124.CrossRefGoogle Scholar - Tennekes, H. and Lumley, J.L. (1972), A First Course in Turbulence, The MIT Press, pp 104–145.Google Scholar
- Uasuda, T., Goto, Sh. and Tsuchiya, I. (1982), ‘0n the relation between changes in intergral quantities of shoaling waves and breaking inception’, Proc. 18th Int. Conf. Coastal Eng., ASCE pp 23–37.Google Scholar
- Yasuda, T., Yamashita, T., Goto, Sh. and Tsuchiya, Y. (1982), ‘Numerical calculations for wave shoaling on a sloping bottom by K-dV equation’, Coastal Eng. in Japan, JSCE, Vol. 25.Google Scholar