COMPARISON OF ANALYTICAL AND NUMERICAL APPROACHES FOR LONG WAVE RUNUP by ERTAN DEMİRBAŞ MAY, 2002

The runup phenomena is one of the important subject for coastal development in coastal engineering. The hazard of long waves generated by earthquakes have in many cases causes deaths and extensive destructions near the coastal regions. On this basis many studies on long wave runup phenomena have been presented numerically and analytically.

In this thesis, the runup of long waves is investigated using a numerical model. The results are presented and discussed with analytical and experimental studies. For different wave profiles;  Solitary wave and  N-wave the runup characteristics have been investigated.

Different from wind generated waves, the length of long waves are longer comparing to water depth. Wind waves show orbital motion, on the other hand long waves show translatory motion. It losses very little energy while it is propagating in deep water. The velocity is directly proportional to the square root of the depth. C = √(g x d) INTRODUCTION

As the water depth decreases, the speed of the long wave starts to decrease. However the change of the total energy remains constant. Therefore while the speed is decreasing, the wave height grows enormously.

The Study of Long Wave Runup Phenomena The study of long wave runup has direct consequence to tsunami hazard assessment and mitigation in coastal region. Generally the long waves have been modeled as Solitary Waves. Some examples are Carrier & Greenspan (1958), Shuto (1967), Pedersen & Gjevik (1983), Synolakis (1987). Recently N-waves have been modeled to describe the long wave characteristics (Tadepalli and Synolakis, 1994).

The Necessity of Numerical Studies The earlier studies on long wave runup relied largely on analytical approaches. Although the analytical studies provide simple analytical solutions, their applications are limited due to  Complex beach geometry,  Different generation parameters, and  Different wave parameters

Therefore the numerical studies are necessary to simulate propagation and coastal amplification of long waves in irregular topographies. This would enable us to evaluate the risks near coastal regions and mitigate the possible hazards on coastal regions.

HOWEVERHOWEVERHOWEVERHOWEVER The problem is to develop an adequate numerical model to describe the physical phenomena accurately.

LITERATURE SURVEY When studying long wave countermeasures, much attention is paid to runup and inundation of the beach. The runup of long waves have been studied using analytical and numerical approaches as well as experimental studies.

Analytical Approaches The runup of long waves have been studied analytically by,  Synolakis (1987)  Pelinovsky and Mazova (1991)  Tadepalli and Synolakis (1994)  Pelinovsky, Kozyrev and Troshina (1995)  Kanoğlu and Synolakis (1998)

Numerical Approaches ë Lin, Chang and Liu studied a combined experimental and numerical effort on solitary wave runup and rundown on sloping beaches (1999). ë Titov and Synolakis (1995) developed a finite difference model using Godunov method to simulate the long wave runup of breaking and non-breaking solitary waves. ë Also Zelt (1991), Kobayashi (1987) and Liu (1995) studied the same problem.

Our Numerical Model In this study the numerical model TUNAMI-N2 is used to simulate different cases. TUNAMI-N2 is one of the key tools incorporating the shallow water theory consisting of non-linear wave equations for developing studies with different initial conditions.

Governing Equations The basic equations used in the model are the nonlinear form of long wave equations as follows.

Those equations above sometimes do not satisfy the conservation of mass principle. Therefore in the model the equations below satisfying both the conservation of mass and momentum principles are used.

ANALYTICAL APPROACHES FOR SOLITARY WAVE RUNUP The key goal in analytical approaches is to introduce a relation between Runup (R) and Wave Height (H). Analytical studies provide simple solutions however their applications are generally limited to idealized cases.

Runup of Solitary Waves Synolakis (1987) presented an empirical relationship between the normalized runup and the normalized wave height. Runup Law

Obviously, the runup variation is different for breaking and non-breaking solitary waves as shown in figure (Synolakis, 1987). The normalized maximum runup of Solitary Waves up a 1:19.85 beach versus the normalized wave height

The breaking criterion of Solitary Waves derived by Gjevik & Pedersen (1981)

Runup of N-waves Tadepalli and Synolakis (1994) introduced N-waves firstly in the modeling of long waves. They discussed relatively general N- wave profile and two special cases of N-waves as the isosceles N-wave and the double N-wave

Generalized N-wave The maximum runup of the generalized N-wave is

Isosceles N-wave The maximum runup of the isosceles N-wave is

Numerical Applications For linear basins, more than 300 different simulations were carried out. The aim is to discuss the non-linear numerical results with the linear and also a few non-linear analytical approaches and experimental studies.

Selected Basins Three different basins are used to simulate different initial conditions. The slopes are selected as 1:10, 1:20 and 1:30. The grid size and time step is selected as 20 m and 0.25 seconds respectively in order to satisfy stabilities.

Initial Wave Solitary WaveN-Wave

Climbing of Solitary Wave The climb of a solitary wave with H/d=0.019 up a 1:19.85 beach (at the toe of the slope)

Climbing of Solitary Wave The climb of a solitary wave with H/d=0.019 up a 1:19.85 beach

Climbing of Solitary Wave The climb of a solitary wave with H/d=0.019 up a 1:19.85 beach

Runup of Solitary Waves

Runup of Generalized N-waves

Runup of Isosceles N-wave

Discussion In overall approach the numerical results show the same trend with analytical and experimental approaches. Especially the climb of the solitary wave up a 1:19.85 slope beach shows that the numerical model results almost similar values according to the available experimental study.

For the runup calculations, the numerical model results lower runup values compared with analytical studies in both Solitary Waves and N-waves. The trend of the relation between the normalized runup and initial wave amplitude at the toe of the slope is consistent for slopes steeper than 1:30 for non-breaking solitary wave.

The underestimation observed in numerical results is  believed to be the effect of the difference between the actual runup and calculated numerical runup  thought to be the result of higher reflection

For N-waves, the analytical approach gives an approximate upper limit. Furthermore the analytical results have not been supported by experimental studies yet. The numerical results are found below the upper limit as expected.

One important discussion for N-waves is also the effect of wave form on wave runup. Surprisingly, the maximum computed runup of leading depression N-wave is calculated as higher than the runup of leading elevation N-wave.

 The numerical model can be improved for the calculation of wave runup.  This application can be extended to milder slopes.  This study can be extended to investigate the runup of the long waves for piecewise linear topographies.

THANK YOU… Ertan Demirbas