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LOCAL TSUNAMI WARNING AND MITIGATION
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AMPLITUDE EVOLUTION AND RUNUP OF SOLITARY WAVES ON A SLOPING PLANE
Ahmet . Yalciner 1; Ertan Demirbas 1; Efim N. Pelinovsky 2; Fumihiko Imamura 3; Costas E. Synolakis 4
1

3

Middle East Technical University, Civil Engineering Department, Ocean Engineering Research Center, 06531, Ankara, Turkey 2 Department of Applied Mathematics, Nizhny Novgorod State, Technical University, Nizhny Novgorod, Russia Disaster Control Research Center, Graduate School of Eng., Tohoku University, Aoba 06, Sendai 980-8579, Japan 4 University of Southern California, Civil Engineering Department, Los Angeles, CA, USA E-mail: yalciner@metu.edu.tr

ABSTRACT
The runup of long waves on the sloping planes is described by the analytical solutions of the long wave equations with special initial conditions, proper approximations and boundary conditions. These studies are also verified by experimental data. It is convenient to test the numerical methods by comparing with analytical results. In this paper, the propagation and coastal amplification of solitary wave on a sloping plane is investigated numerically. The computed shape and amplitude evolution on the plane slope are compared with the existing analytical and experimental results. The performance of the numerical method is also discussed.

1. I

NTRODUCTION

The motions of long waves at shallower depths near the shoreline, run-up and the following inundation have been studied using theoretical, experimental and numerical approaches. Various analytical solutions for runup of nonlinear waves on plane slopes have been given by Shuto (1967); Gjevik & Pedersen [1981]; Pedersen and Gjevik, [1983]; Kim et. al., [1983]; Synolakis [1987]; Pelinovsky and Mazova [1992]; Synolakis and Skjelbreia, [1993]; Pelinovsky et al., [1996]; Kanoglu, [1996]; Kanoglu and Synolakis, [1997]; Lin et. al., [1999]; Carrier and Yeh, [2002]. In analytical approaches the runup problem is studied either by using empirical formulae or by solving the governing equations for specific initial and boundary conditions. Experimental data on runup of solitary waves are given among others by Hall and Watts, [1953], Pedersen and Gjevik, [1983] and Synolakis [1987], Shankar and Jayaratne, [2002], Lee and Raichlen, [2002]. A detailed analytical and experimental study on the runup and amplitude evolution of solitary waves on plane beaches is given in Synolakis, [1987]. An exact solution to an approximate theory for non-breaking solitary waves was introduced to derive the maximum runup asymptotically. Laboratory experiments had been performed to support the theory and the satisfactory prediction of the climb of the wave on the slope and maximum runup by linear theory has been determined. Pelinovsky and Mazova [1992] investigated tsunami runup on a beach with two different parameters; the angle of bottom slope and the breaking parameter. Titov, Synolakis, [1995a, 1995b, 1998], Imamura [1995], Yalciner et al., [2001[, Hubbard, Dodd, [2002[, Lynett et. al., [2002[, Lee and Raichlen, [2002[, Maiti and Sen, [1999[ are some references of numerical studies on long wave runup. The computer program, TUNAMIN2, used for the simulation of the propagation of long waves is developed by Prof. Imamura in Disaster Control Research Center in Tohoku University, Japan. TUNAMI-N2 is one of the
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LOCAL TSUNAMI WARNING AND MITIGATION ___________________________________________________________________

key tools for developing studies for propagation and coastal amplification of tsunamis in relation to different initial conditions. It solves the nonlinear form of long-wave equations and depth averaged velocities with bottom friction by finite difference technique for the basins of irregular shape and bathymetry and provides us a very convenient tool to simulate tsunamis. Shuto, Goto and Imamura (1990), Goto and Ogawa, [1992], Imamura, (1995), Goto et. al. [1997], Yalciner et. al., [2001], Yalciner et. al., [2002] are some of the studies used TUNAMI-N2. In this study particularly, the behavior of solitary wave on a sloping beach and the runup phenomenon by the non-linear numerical modeling (TUNAMI-N2) is studied. The shape of the solitary wave on the plane slope, the maximum positive amplitudes near the coastline are computed and compared with the analytical and experimental results [Demirbas, 2002]. 2. NUM
ERICAL

APPL

ICATION WITH SOLITARY

W

AVE

A solitary wave centered at a location x = X1 when t = 0 has the following surface profile:

( x ,0 ) =
=(

H sec h 2 ( x - X 1 ) d
1/ 2

(1) (2)

3H ) 4d

Where H is the amplitude of solitary wave, d is the water depth at the toe of the sloping plane, X1 is the distance from the specified location. The linearized long wave equations for the canonical problem are solved by Synolakis, (1987) and the runup law is derived for the non-breaking solitary waves.

=(

3H ) 4d

1/2

(3)

where R is the runup of solitary waves, is the angle of sloping plane with horizontal. The breaking condition of solitary waves on a sloping plane is presented by Gjevik & Pedersen [1981]: H > 0.479(cot ) d
-10 9

(4)

This criterion has been reported to be in excellent agreement with laboratory data for solitary waves by Synolakis, [1987]. The canonical problem named by Tadepalli and Synolakis [1994], in wave runup is the determination of the runup of a long wave propagating over a constant depth region and then climbing up a sloping beach of constant slope. There are a few numbers of studies for different wave profiles on the canonical problem. We selected the canonical problem with a regular shaped basin of 10 km length and width. The water depth of the horizontal bottom is chosen as 30 m. On one side of the basin the plane beach is located with bottom slope of 1/20. The other boundaries are selected as open boundaries. The cross section of the basin along x direction is shown in Fig. 1. The initial wave is inputted near the center of the basin where the wave crest is parallel to the shoreline (along z axis) and thus the wave propagation is forced along x direction towards shore without dispersion.
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Figure 1. Cross Section of the Basin, Location of the Initial Solitary Wave and the Gauge Locations where the Water Surface Elevations are Computed.

The location of the calculated maximum water elevation near the shoreline obtained by the numerical model with finite difference method does not coincide with the location of the actual runup. In the numerical model, the elevation of the water is computed at the fixed locations of grid points. Obviously the smaller grid sizes result nearer maximum elevations to actual runup. In this application the smallest possible grid size is selected to obtain optimum run time and to obtain best possible comparison between experimental/ analytical and numerical results. The grid size and time step are selected as 20 m and 0.25 seconds respectively in order to satisfy stability in computation. The time histories of water surface elevations at different locations, the sea state at different time steps, the snapshots of the surface profile along the axis of wave propagation at specified time step, the maximum water elevation reached at every grid point throughout the domain during the simulation are computed and stored. By using the stored data the shape of the wave at different locations on the slope and the water surface along the axis of wave direction at specified time steps are presented in the following. The results are compared with the analytical and experimental data of Synolakis [1987]. There are two cases presented in Synolakis [1987]. They are also selected in this application. In these cases, the normalized height of incoming wave, (H/d) is 0.019 (non-breaking) and 0.040 (breaking). The normalized water surface elevation () representing the climb of solitary wave at the toe of the slope, and at the shoreline for the non-breaking case on the 1:19.85 slope are shown in Fig. 2 as function of the dimensionless time. As seen from this Figure that the numerical model computes fairly consistent water surface fluctuation with experimental and analytical data. The comparison is extended to check the water surface profile along the axis of propagation at different dimensionless time steps. The water surface profiles at different dimensionless time steps are given in Fig. 3 and 4. As seen from these figures that the numerical model provides fully consistent shape of the wave and amplitude evolution on the plane slope with the analytical and experimental results, especially on the wet part of the slope for the both breaking and non-breaking conditions of incoming solitary wave.
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Figure 2. The normalized water surface elevation representing the climb of solitary wave at x = 19.85 (at the toe of the slope), and at x = 0.25 (at the shoreline) with H/d = 0.019 up a 1:19.85 slope as function of the dimensionless time. (--------experimental [Synolakis, 1987]; -----, analytical [Synolakis, 1987]; -----------, numerical ­ this study).

The numerical experiments are repeated by using different incoming solitary waves. The normalized maximum positive wave amplitudes near the shoreline are computed for each experiment. The comparison of numerical data with the runup law and experimental data of Synolakis [1987] is given in Fig. 5. This figure shows that the distribution of the data points of numerical results show similar trend with the analytical and experimental data, but the numerical results stay below the others. The underestimation of maximum amplitude in numerical results comes from fixed grid size of the numerical solution.

Figure 3. The normalized water surface profile representing the climb of solitary wave along the wave direction with H/d = 0.019 up a 1:19.85 slope as function of the normalized distance at different dimensionless time steps, (a) t = 25, (d) t = 40, (g) t = 55, (i) t = 65. (...., experimental [Synolakis, 1987]; ____-, analytical [Synolakis, 1987]; --------, numerical ­ this study). 176

Figure 4. The normalized water surface profile representing the climb of solitary wave along the wave direction with H/d = 0.040 up a 1:19.85 slope as function of the normalized distance at different dimensionless time steps, (a) t=20, (c) t=32, (e) t=44, (g) t=56. [...., experimental [Synolakis, 1987]; _____-, analytical [Synolakis, 1987]; ----------- , numerical ­ this study).

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3. DISCUSSIONS

OF

RESULTS

The analytical results do not cover the nonlinear terms of the long wave equations. Lin et al., [1999] states that the numerical results of the depth averaged equations models predict smaller value of runup tongue. However analytical approach is consistent with the experimental results. Therefore it is shown that the numerical approach computes satisfactory agreement of water motion when the wave climbs on the slope. But the computation gives smaller runup on the slope at land. ACKNOWLEDGE
MENTS

Figure 5. The Comparison of Numerically Computed Maximum Positive Wave Amplitudes near the Shoreline with the Runup law and Experimental Data Given in Synolakis, [1987]

This study has been partly supported by the basic research projects of METU-AFP- 2001-0303-06, TUBITAK YDABCAG-60 and INTAG-827. REFERENCES
Carrier G., Yeh H, 2002: Exact Long Wave Runup Solution For Arbitrary Offshore Disturbance. 27th General Assembly of European Geophysical Society (EGS), April 22-26, Nice France, Abstract No: EGS02-01939 Demirba E., 2002: Comparison of Analytical and Numerical Approaches for Long Wave Runup. M. Sc. Thesis, Middle East Technical University, Civil Engineering Department, Ocean Engineering Research Center, May 2002. Gjevik B., and Pedersen G., 1981: Runup of Long Waves on an Inclined Plane. Preprint Ser. No 2, Dept of Maths, University of Oslo. ISB 82-553-0453-3. Goto C. and Ogawa Y., 1992: Numerical Method of Tsunami Simulation with the Leap-Frog Scheme. Translated for the Time Project by Shuto N., Disaster Control Research Center, Faculty of Engineering, Tohoku University, 1992. Goto C., Ogawa Y., Shuto N., and Imamura F., 1997: Numerical method of tsunami simulation with the leap-frog scheme. (IUGG/IOC Time Project), IOC Manual, UNESCO, 1997, No. 35. Hall J. V., and Watts J. W., 1953: Laboratory Investigation of the Vertical Rise of the Solitary Waves on Impermeable Slopes. Tech. Memo. 33, Beach Erosion Board, US Army Corps of Engineers, 14 pp. Hubbard M. E., Dodd N., 2002: A 2d Numerical Model of Wave Runup and Overtopping. Coastal Engineering, 47, 1-26 Imamura F., 1996: Review of Tsunami Simulation with a Finite Difference Method. Long Wave Runup Models (Eds. H. Yeh, P. Liu, C. Synolakis), World Scientific. Imamura F., and Goto C., 1988: `Truncation Error in Numerical Tsunami Simulation by the Finite Difference Method. Coastal Engineering in Japan, 31, No. 2. Kanolu U., 1996: The Runup of Long Waves Around Piecewise Linear Bathymetries. PhD Thesis, University of Southern California, Los Angeles. Kanolu U., Synolakis C. E., 1998; Long Wave Runup on Piecewise linear topographies. Journal of Fluid Mechanics, 374, 1-28. PETROPAVLOVSK-KAMCHATSKY TSUNAMI WORKSHOP, SEPTEMBER 10-15, 2002 177


LOCAL TSUNAMI WARNING AND MITIGATION ___________________________________________________________________ Kim S. K., Liu P. L-F and Liggett J. A., 1983: Boundary Integral Equation Solutions for Solitary Wave Generation, Propagation and Runup. Coastal Eng., 7, 299-317 Maiti S., and Sen D., 1999: Computation of Solitary Waves during Propagation and Runup on a Slope. Ocean Engineering, 26, 1063-1083 Li Y., and Raichlen F., 2002: Non-breaking and Breaking Solitary Wave Runup. Journal of Fluid Mechanics, 456, 295-318 Lin P., Chang K. A., Liu P. L.­F., 1999: Runup and Rundown of Solitary Waves on Sloping Beaches. Journal of Waterway, Port, Coastal, and Ocean Engineering, 125, 5, 247-255. Lynett P. J., Wu T. R., Liu P. L. F., 2002: Modeling Wave Runup with Depth Integrated Equations. Coastal Engineering, 46, 89-197 Pedersen G., Gjevik B., 1983: Runup of Solitary Waves. J. Fluid Mechanics, 142, 283-299. Pelinovsky E., Kozyrev O., Troshina E., 1996: Tsunami runup in a sloping channel. In: Long-Wave Runup (Eds. H. Yeh, P. Liu, C. Synolakis), World Sci., 332 - 339. Pelinovsky E. N., Mazova R. Kh., 1992: Exact Analytical Solutions of Nonlinear Problems of Tsunami Wave Runup on Slopes with Different Profiles. Natural Hazards, 6, 227-249. Shankar N. J., and Jayaratne M. P. R., 2002: Wave Runup and Overtopping on Smooth and Rough Slopes of Coastal Structures. Ocean Engineering, 30, 221-238. Shuto N., Goto C., Imamura F.: 1990: Numerical Simulation as a Means of Warning for Near-Field Tsunami. Coastal Engineering in Japan, 33, 2, 173-193. Shuto N., 1974: Nonlinear waves in a channel of variable section. Coastal Engineering in Japan, 17, 1-12. Synolakis C. E., 1987: The Runup of Solitary Waves. J. Fluid Mechanics, 185, 523-545. Synolakis C. E., and Skjelbreia J. E., 1993: Evolution of Maximum Amplitude of Solitary Waves on Plane Beaches. Journal of Waterways, Port, Coastal, and Ocean Engineering, 119, 3, 323-342. Tadepalli S., and Synolakis C., 1994: The runup of N-waves on sloping beaches. Proc. Royal Society, London, A445, 99-112. Titov V., and Synolakis C. E., 1995a: Modeling of Breaking and Nonbreaking Long Wave Evolution and Runup Using VTSC-2. Journal of Waterways, Port, Coastal and Ocean Engineering, 121, 308-316. Titov V., and Synolakis C. E., 1996: Numerical Modelling of 3-D Long Wave Runup Using VTCS. Long Wave Runup Models, (Eds.: H. Yeh, P. Liu, and C. Synolakis), World Scientific, 242-248. Titov V. V., and Synolakis C. E., 1998: Numerical Modeling of Tidal Wave Runup. Journal of Waterways, Port, Coastal and Ocean Engineering, 124, 157-171 YalÃiner A. C., æzbay I., Imamura F., 2001: A Comparison of The Tsunami Set-up with Relation to the Dimensions of Underwater Landslide. NATO ARW, Underwater Ground Failures on Tsunami Generation, Modeling, Risk and Mitigation Vol. 1, Advanced Research Workshop, NATO Science Program, ISBN: 975-93455-0-1 (Editor Ahmet Cevdet YalÃiner), May 23-26, 2001, Istanbul, Turkey, 60-66. YalÃiner A. C., Alpar B., Altinok Y., æzbay I., Imamura F., 2002: Tsunamis in the Sea of Marmara: Historical Documents for the Past, Models for Future. Marine Geology, 190 (1-2), 445-463.

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