Experimental and numerical determination of the shoreline motion during the run-up of tsunami waves on a plain beach

Contents

List of Tables

List of Figures

List of Symbols

Abstract

1 Introduction
1.1 Motivation
1.2 General Information on Tsunamis
1.3 Scope of the Thesis
1.4 Outline of the Thesis

2 Literature Review
2.1 Long Wave Generation Techniques.
2.1.1 Piston-type Wave Generation
2.1.2 Dam Break Analogy
2.1.3 Vertical Sea Bed Motion
2.1.4 Pneumatic Wave Generation
2.2 The Brier Score
2.3 The Shoreline Motion of Long Sinusoidal Waves
2.4 Breaking Criterion for Sinusoidal Waves
2.5 Optical Measurements of Long Wave Run-up

3 Experimental Setup
3.1 Description of Instruments
3.1.1 The Wave Flume
3.1.2 The Wave Gauges
3.1.3 The Velocity Meter
3.1.4 The Pressure Sensors
3.1.5 The High Speed Cameras
3.2 Performance of Physical Experiments
3.3 Data Processing
3.3.1 Processing of Time Series
3.3.2 Image Processing
3.3.3 Derived Quantities: Position, Velocity and Acceleration of the Shoreline

4 Wave Generation 4 Wave Generation
4.1 Introduction
4.2 The Pump-driven Wave Generator
4.3 The Controller Scheme
4.4 Summary

5 Results Of Physical Experiments 5 Results Of Physical Experiments
5.1 Time Series
5.1.1 Comparison Reference and Actual Curve, Brier Score
5.1.2 Time Series of Wave Gauges
5.1.3 Time Series of EMS Sensor
5.1.4 Time Series of Pressure Sensors
5.2 Run-up and Run-down of Long Waves
5.3 Maximum Run-up and Run-down
5.4 Shoreline Velocity During Run-up and Run-down
5.5 Maximum Shoreline Velocity During Run-up and Run-down
5.6 Wave Acceleration During Run-up and Run-down
5.7 Summary

6 Numerical Model Test
6.1 Introduction
6.2 Description of the Numerical Model
6.3 Results of the Numerical Model Study
6.3.1 Evolution of SSH During Propagation
6.3.2 Numerical Shoreline Location and Maximum Run-Up
6.4 Summary

7 Summary and Outlook

References

Acknowledgements

Statement

Appendix A: Calibration Curves

Appendix B: Tables for Rmax and Vmax 82

List of Tables

0.1 List of symbols

3.1 Technical specifications of the wave gauges

3.2 Technical specifications of the EMS probe

3.3 Overview over all waves generated in the wave flume

3.4 Camera image acquisition times for all experiments..

4.1 Chosen parameter of the PID controller

5.1 Brier score and standard deviation of all actual curves

5.2 Time shift of analytical shoreline location for wave T30_H2

7.1 Calibration formulas of all measuring instruments.

7.2 Maximum run-up and run-down for all experiments.

7.3 Maximum shoreline velocities for all experiments.

List of Figures

1.1 Schematic drawing for the run-up height of long waves.

3.1 Schematic drawing of wave flume

3.2 Wave gauge 1 and velocity meter

3.3 Electrodes on the sensor of the velocity meter

3.4 Detailed arrangement of pressure sensors

3.5 Controller scheme for wave generation and data acquisition

3.6 Camera raw images and assembled image after processing

4.1 The pump station

4.2 Flow calming features

4.3 Basic principle of a PID controller

5.1 Control curve and actual curve of T30_H2

5.2 Control curve and actual curve of T100_H4

5.3 Surface profiles of T30_H2

5.4 Current velocity of T30_H2

5.5 Time series of all pressure sensors for T30_H2

5.6 Shoreline location of T30_H2

5.7 Experimental and theoretical maximum run-up of long waves

5.8 Experimental and theoretical maximum run-down of long waves

5.9 Shoreline velocity of T30_H2

5.10 Shoreline velocity of T58_H3

5.11 Experimental and theoretical maximum shoreline velocities during un-up

5.12 Experimental and theoretical maximum shoreline velocities during run-down

5.13 Shoreline acceleration of T30_H2

6.1 Numerical grid data: gather and scatter strategy

6.2 Initial numerical ssh and shoreline location for T30_H2

6.3 Numerical evolution of ssh for T30_H2

6.4 Comparison of theoretical, experimental and numerical shoreline lo­cation for T30_H2

6.5 Numerical shoreline location in 2D for T30_H2 after 73.5s

6.6 Evolution of 2D numerical shoreline

7.1 Linear regression of wave gauge 1

List of Symbols

illustration not visible in this excerpt

Abstract

Within this master thesis the behaviour of long periodic waves during their run-up on a plain beach was investigated via physical and numerical modelling. In the experimental part, for seven leading depression, non-breaking sine waves with surf similarity parameters between 3.1 ≤ ξ ≤ 15.6 the wave velocity, wave height and run-up on a plain beach were determined. In addition, the motion of the initially still shoreline, i.e. run-up/ run-down process, run-up/ run-down velocity, wave ac­celeration, maximum run-up and maximum run-up velocity, was determined via two high-speed cameras. Comparison of the aforementioned characteristics with the theory revealed good agreement; deviations can mostly be attributed to experimen­tal performance.

For wave generation a new volume driven wave generator was used. Long waves are generated by a pair of high capacity pumps under control of a proportional-integral­-derivative controller (PID-controller). While rotating clockwise or counterclockwise water is pumped into the propagation section or extracted from it. Thereby, waves of arbitrary length can be generated. Using the relatively new strategy of observing the shoreline motion via optical mea­surements gave a comparatively exact shoreline position during wave run-up. In contrast, determination of the shoreline position during run-down was less exact due to missing evidence indicating the distinct position of the shoreline. In general, the experimentally determined shoreline position agreed with the theoretical approach. The maximum run-up/ run-down occurred for waves with (interval in which transition from breaking to non-breaking occurs). The magnitude of the theo­retical breaking point ξbreak increased for decreasing wave non-linearity ε = A0/h. For ξ ≥ ξbreak and for increasing ξ the normalized run-up and magnitude of nor­malized run-down decreased. The maximum run-up/ run-down was proportional to ε, i.e. the normalized run-up and magnitude of normalized run-down decreased with increasing wave amplitude as predicted by theory. The agreement of experi­mental and theoretical maximum run-down/ run-up depended on tuning of the PID controller and the resulting actual curve. For longer waves (ξ ≥ 10.5) suboptimal tuning of the PID controller resulted in riding waves. The experimental run-up was mostly underestimated compared to theory. There were two reasons for long waves: firstly, the riding wave overtook the main wave before the main wave could reach true run-up height and secondly, the tracers deferred the run-up process. An under­estimated maximum run-up of the shorter waves was due to a low shoreline velocity or a deviation of the water surface profile from the intended surface profile. The maximum run-up of intermediate waves agreed well with theory due to a well-tuned PID controller. Due to the limited field of camera view only maximum run-down heights of the two smallest waves could be determined: the maximum run-down of the sine wave exhibiting a wave period of 20s and a wave height of 2cm (T20_H2) was too low due to a decreased wave trough of the generated wave. The maximum run-down for T30_H2 agreed well with theory.

The experimental and theoretical shoreline velocity agreed well. Yet, the tracers decreased the run-up velocity of the shoreline resulting in a decreased maximum run-up height. The riding waves gave several strong peaks in shoreline velocity. The asymmetric behaviour of shoreline velocity during initial run-down restored the correct phase correlation between the curve of shoreline velocity and the curve of shoreline position. The maximum shoreline velocity during run-up showed low de­viations from theory for short and intermediate waves. For the longer waves the experimental velocity was too high.

For the seldom investigated maximum shoreline velocity during run-down the theo­retical correlations between surf similarity ξ, magnitude of the shoreline velocity, wave non-linearity ε and theoretical breaking point ξbreak are the same correlations as for the maximum run-up velocities. For short waves the experimental maximum shoreline velocities were significantly lower than predicted by theory. For inter­mediate waves the maximum run-down velocity agreed well. For longer waves the maximum velocities were higher than the theoretical values.

For the shoreline acceleration no resilient theory was found in literature. The experi­mental acceleration exhibited the highest variability in the vicinity of the zero water line where the highest velocities occurred. The tracers did not only decrease the run-up velocity, but also seemed to have a calming effect on the shoreline motion. Performing half of the experiments with white tracers was meant to give a distinct maximum run-up, but turned out to have a negative influence on the results. Non­floating tracers obstructed the pressure sensors installed on the beach bottom giving offsets in the time series. The tracers remained behind the moving shoreline leading to possible underestimation of the shoreline position during run-up, if numerical image analyses are considered in future studies. Arranging the tracers along the still shoreline reduced the velocity of the incoming wave and caused wave refection. However, the tracers were used as indicators during run-down, which did not give the exact shoreline position, yet at least an approximation of the run-down position and run-down velocity.

The numerical analysis was performed with the 2D shallow water model named TAM (T sunami model with A daptive M esh) which is a finite element model. To increase the accuracy of the results a nested triangular grid was applied. Thereby, the re­solution of the region around the zero water line was four times higher than the resolution of the remaining wave channel. The numerical model was tested with a leading depression, non-breaking sine wave from the experimental data set obtained from the measurements. The sine wave with a period of 30s, a wave height of 2cm and a surf similarity of ξ = 6.6 was chosen as test wave since this wave was repro­duced best in the physical wave flume. Post-processing of the numerical shoreline location revealed problems with the numerical inundation algorithm. After the ma­ximum run-up height was achieved a local upper beach section remained wet. The water in this section behaved independent of the main wave as it either remained on the beach or in some cases even ran further up. Consequently, the results of the numerical run-up deferred significantly from the physical and theoretical results. Furthermore, the numerical wave height decreased by 35% during propagation in the region of constant depth, which was probably due to too much numerical fric­tion. The decreased wave height resulted in a weaker numerical shoreline motion and in maximum run-down/ run-up heights which were significantly lower than the expe­rimental values. Therefore, the numerical inundation algorithm needs improvement before a validation of the numerical model can be performed.

1 Introduction

1.1 Motivation

On December 26, 2004 at 01:01:09 UTC a submarine earthquake of magnitude M = 9.0 hit the western coast of northern Sumatra, Indonesia (Wang & Liu, 2006). The main shock had its epicenter at 3.09°N and 94.26°E, i.e. on a fault line exten­ding from Indonesia northward to the Andaman Islands. Here, the Indo -Australian plate subducts beneath Sunda plate and Burma sub-plates at a celerity of approxi­mately 60mm/year in northeastern direction (Wang & Liu, 2006). Along the fault line and on a section extending 1200km to 1300km northward from the epicenter the seafloor was uplifted by up to 10m. As a consequence of the earthquake several tsunami waves were generated that spread at a velocity of approximately 700km/h radial from the epicenter. The first wave arrived at Sri Lanka two hours after the main shock. Three hours after the shock the waves arrived at the Maldives. Along the Indonesian coast wave heights of up to 30m were observed. In Phuket, Thailand wave heights of 10m were reported (Wang & Liu, 2006). In total, 283, 100 people lost their life and 14, 100 people are still missing. In 10 countries located around the Indian Ocean 1, 126, 900 people lost their home. The tsunami waves even crossed the southern Atlantic Ocean to arrive at the east coast of Africa where they caused enormous material damage and killed many people (Wang & Liu, 2006).

On March 11, 2011 at 05:46 UTC (14:46 Japanese time) an earthquake of magni­tude M = 9.0 struck the northeastern coast of Japan (Ozawa et al., 2011; Song et al., 2012). The epicenter was located at 38.1°N and 142.9°E in a depth of 24km, about 160km off Japanese Tohoku coast (Suppasri et al., 2011; Song et al., 2012). Here, the Pacific plate subducts beneath the Okhtosk plate (Ozawa et al., 2011). The shock displaced the sea floor in places up to 50m horizontally and about 10m vertically (Fujiwara et al., 2011). As far as 400km in length and 180km in width along the plate boundary of the Japan trench the sea floor was ruptured (Yagi & Fukahata, 2011). The earthquake generated tsunami waves that were amplified in the open ocean due to topographic features like ridges and seamounts (Song et al., 2012). Therefore, tsunami heights between 25cm and 70cm in the open ocean were observed by three different satellites (Song et al., 2012). In total, about 19, 508 people lost their life (Song et al., 2012).

Within the light of these two and other disastrous tsunamis such as the February 27, 2010 tsunami in Chile major efforts have been established to improve the under­standing of generation, propagation and run-up of shallow water waves. The run-up is the height above sea level that the water comes up to as it moves onshore after a tsunami wave has arrived at the beach. Fig.1.1 illustrates the run-up height of a solitary wave. After the maximum run-up height has been achieved the water starts to retreat. This process is called run-down. For the Sumatra tsunami a maximum run-up of 10m was measured at Sri Lanka (Wang & Liu, 2006). The maximum run-up of the Tohoku-Oki tsunami amounted to 40.4 m at some places (Mori et al., 2011).

There are three main tools to investigate the behaviour and characteristics of water waves: analytical models, numerical models and physical models. Each of these models owns specific advantages and disadvantages. Analytical models consist of a set of equations to calculate the desired wave characteristics like amplitude or run-up. The basis for analytical models is a derivation of the final equations based on state-of-the-art mathematical theories. In deriving the desired equations every

illustration not visible in this excerpt

Figure 1.1: Schematic drawing for the run-up height R of long waves. SWL: Still water level. From: Madsen & Schäffer (2009)

important step in the calculations is documented and can be reproduced by the reader. The advantage of analytical models is their physical simplicity which allows for basic considerations of the problem. Considering a problem in its most simple case offers the possibility to discover physical dependencies and correlations. An example is tsunami hazard assessment where analytical models can shed light on the parameters the run-up of a tsunami wave depends on, e.g. beach slope or height of the incident wave. Thus, with analytical models scientists have the possibility to vary the problem determining parameters and observe the effects on the result. So, the benefit of considering a problem theoretically is that physical dependencies can be identified in a comprehensive and accurate way. For this reason mathema­tical examinations lend themselves for sensitivity analyses. On the other hand the physical simplicity of analytical models leaves only limited space for considering the result under the influence of additional parameters than the basic ones.

A numerical model in its simplest form is a sequence of instructions that are executed by a computer. Since it is not possible to solve the original differential equations of motion that describe the propagation of a tsunami wave the derivatives are approxi­mated by differences. One advantage of numerical models is that many experiments can be performed in relatively short time. The consequence is a huge amount of data the programmer has to deal with. Due to the growing computational capac­ity it is possible to investigate the behaviour of tsunami waves under much more complex assumptions like a realistic topography or friction. The disadvantage of numerical models is that they usually have an own internal dynamic showing itself in e.g. numerical diffusion or growing instabilities, which influence the results. Since it is diffcult to find and isolate the dynamical effects the programmer cannot be sure whether the results exhibit errors and, if they do, of which magnitude the compu­tational error is. Furthermore, the results of numerical models strongly depend on the choice of boundary conditions and often there is uncertainty of what is the most realistic boundary condition.

A physical model is the downscaled representation of an ocean wave generated in a laboratory wave tank. In such wave tanks ocean waves of different kinds can be generated and their behaviour under idealized circumstances can be investigated. The waves investigated range from short wind generated waves to long gravitational waves. Depending on the waves of interest the wave tank has to meet specific condi­tions and accordingly, there are different designs of wave tanks. For example, in the wind-wave tank of the University of Hamburg the interaction between atmosphere and the ocean surface is investigated. For this purpose the wave tank incorporates a wind generator. With the aid of this wind engine the air column above the water is set to motion. As a consequence of the moving air waves are generated at the water surface. Furthermore, the wave tank needs to have a roof and the experimenter needs equipment to perform measurements above, on and eventually under the wa­ter surface. In a wind-wave tank small waves like capillary waves or ordinary surface waves are generated with a wave length ranging from centimeters to one meter. In contrast, when the aim is to examine long gravitational waves, such as tsunami-like waves, then the wave length of the generated wave may amount from several me­ters to decameters, depending on the length scale of the wave flume. Furthermore, no wind generator is necessary since long waves are hardly influenced by the wind. Accordingly, the wave tank does not need to have a roof. Moreover, the mechanism of generation differs significantly since in a laboratory a wave length of 10m cannot be generated by surging air. Long waves are generated in a hydrodynamic way as discussed later in this section. The shape of a wave tank can differ as well, depending on the waves of interest and focus of investigations. While the wind-wave tank in Hamburg is linear, the wind-wave tank of the University of Heidelberg is circular. A laboratory tank for long waves can be a rectangular flume or a channel exhibiting an oval shape. Both designs of a wave tank can be found at the Franzius Institute in Hanover. Concerning the investigation of long waves physical models own the advantage that the run-up of long waves can be observed under even more complex circumstances than in a numerical model. Examples involve wave breaking and the interaction of the flow with model buildings as investigated by Goseberg (2011). Experimental data serve as possible boundary conditions for numerical models, which decreases the level of uncertainty of the numerical model. Using the experimental data as numerical boundary conditions and comparing the numerical results with the labo­ratory results is called validation of the numerical model. Furthermore, if unrealistic results occur, usually the error source can be identified fast since often it is up to the experimental setup. For example the results of physical experiments often are influ­enced by friction occurring at the side walls and the bottom. Here, boundary layers develop that decelerate the flow, e.g. in regions of constant depth, which in reality would not be the case since the boundaries are that far away that an influence of friction can be neglected. The influence of friction in the wave tank can be decreased by using smoothed material. If all error sources are eliminated as far as possible, the physical model approximates the natural original, often called prototype, in a realistic and reliable way.

In the field of physical modelling it is the process of generating a wave and the resulting wave type that determine quality and reliability of the experimental data. In general, there are different types of waves that could be used to model tsunami­-like waves. The simplest one is a single sine wave which consists of one elevation followed by one depression of equal size. The second wave type is a solitary wave which consists of one elevation. The third wave is the N-wave which consists of one elevation followed by one depression or vice versa. Depending on the shape one distinguishes leading depression or leading elevation N-waves. In contrast to sinusoidal waves elevation and depression of an N-wave are not necessarily of the same size, but can be different. In the special case of equal size the wave is called an isosceles wave. In the past decades many studies made use of solitary waves to model the behaviour of tsunami waves. One reason is probably that solitary waves can be generated relatively easily. As a result a kind of solitary wave paradigm de­veloped teaching that a solitary wave is a good model to study tsunami waves. By now, scientists recognized that a solitary wave in fact is an inappropriate model of a tsunami wave since it its temporal and spatial scales are significantly shorter than those of the prototype. Instead, the state-of-the-art model now is the N-wave in its general form, e.g. elevation and depression are of different size. Although it does not make a significant difference whether leading elevation or leading depression waves are used, usually leading depression waves are generated since most tsunami waves propagate with a depression ahead. This is the reason why the water on a beach retreats in the forefront of an approaching tsunami (Rosetto et al., 2011).

Beside these standard wave types there are more types reported in literature, e.g. in Didenkulova et al. (2007). Like the understanding for the different wave types the mechanisms and techniques to generate a wave have improved as well over the past decades. Very early experiments were carried out in 1844 by Scott Russell who used a sinking box to generate a solitary wave. Monaghan & Kos (2000) adopted this technique and investigated the flow pattern around the box as it sank. In the 1980’s Synolakis conducted experiments in a rectangular wave flume using a piston type wave generator (Synolakis, 1986, 2006). Using this technique a vertical rectangular wall is hydraulically driven forward and backward to transfer momentum into the water and finally to set the water in motion. In this way one or more solitary waves are generated. Another hydraulic approach is the wave generation with a vertically moveable bottom as used by Hammack (1973). In that way it was intended to model a submarine earthquake. According to Goseberg (2012) this technique is not suitable for modelling waves in the vicinity of a coast since a distinction between a genera­tion section and a downstream section has to be made. A third approach to transfer momentum into a water body is to release a huge amount of water from above the water surface (Chanson et al., 2003). Although a wave is generated this dam break like mechanism has the disadvantage that a huge amount of turbulence is induced in the water. Furthermore, there is hardly control over the wave characteristics like amplitude and period (Goseberg, 2012). Recently, a new technique to generate long waves has been developed at the Franzius-Institute for Hydraulics, Waterway and Coastal Engineering in Hanover, Germany. Here, a pump-driven wave generator was developed. Via four pumps water is pumped into the wave tank at a high ve­locity. Thereby, the still water surface slopes and a horizontal velocity component is transferred into the water. Both effects combine to result in a wave that starts propagating along the wave flume. The overall advantage of this technique is that it allows precise control over the generated wave, which means that features like amplitude, wave length and period can be chosen in advance and are realized with great accuracy (Goseberg, 2012). Furthermore, different kinds of waves can be ge­nerated including single sinusoidal waves, solitary waves and N-waves.

Against the predominant public opinion it is not the enormous amplitude a tsunami comes up to when it jumps up on a beach, but it is the high run-up and the high current velocities that cause the severe material damage and claim the life of many people (Madsen & Fuhrman, 2008). Accordingly, the main emphasis of today’s tsunami research is given to the run-up process and run-up height of tsunami waves. Both, run-up height and current velocities are of interest for coastal engineering to calculate location and construction of coastal buildings. For example, if oceanogra­phers calculate a maximum run-up of 10m, engineers will not think of placing a civil building at 5m run-up. The knowledge of current velocities that occur during the inundation phase is essential since the velocities provide information about the forces that the surging water will exert on the coastal building. The magnitude of force exerted on the building has an influence on the choice of construction material. Knowing the area where the forces are strongest, the shape of the building can be adapted to minimize the force, e.g. by reducing the contact surface. An optimal form and the correct construction material are important factors determining the stability and safety of the building.

Beside the onshore construction another aspect of interest is the submarine con­struction, which means the installation of solid structures on the submerged part of the beach or on the continental shelf. These structures include for example the basement of offshore wind parks or submarine turbines for electricity generation. In the context of coastal hazard management submarine elements could be installed on the sloping beach. A study performed by Fernando et al. (2005) in the southwest of Sri Lanka proves the weakening effect of coral reefs and rock reefs on tsunamis. In a similar way the presence of submarine macro-roughness elements might influence the behaviour of tsunami waves, e.g. by reducing their amplitude and/ or velocity, causing the breaking of the wave before it hits the beach or by dissipating a part of the energy carried by the wave. So, instead of releasing the energy on the beach where it causes destruction most of the energy would dissipate offshore. Since there is reason to believe that submarine macro-elements have an influence on currents and on the marine ecosystem studies and investigations have to be performed in preparation of a possible installation. Furthermore, it is urgent to ensure that the presence of the macro-roughness elements really decreases the destructive potential of tsunami waves and does not increase it. For those kinds of investigation numerical and physical experiments are essential. Regarding the fact that today most of the global population is located on the coasts it is of special interest to know what potential damage a tsunami could cause in these regions. To estimate the possible destruction run-up and current velocity are calculated usually with the aid of numerical models. These models need to be va­lidated before one can trust them. According to Synolakis et al. (2008) validation is “the process of ensuring that the model accurately solves the parent equations of motion”. If the model is to be used for realistic predictions e.g. for a particular region (operational use) it needs to be verified in addition to the validation. By verifica­tion it is ensured that the model considers scaling effects and thus provides realistic results even when applied to realistic conditions such as topography or measured wave height as boundary condition (Synolakis et al., 2008). Validation is usually done by data from laboratory experiments. Measurements of the wave height in the laboratory region of constant depth are used as boundary conditions for the model. If the difference between model and experimental data is within a predefined error range, the model is validated.

Despite the benefit of numerical simulations there is no numerical model that is completely free of errors. Especially in the phase before any validation has taken place there is reason to mistrust the model’s results since there is no guarantee for the accuracy of the outputted data. To date the only way to answer the question whether a model is “good” or “bad” is to compare the model results to other nume­rical models or to compare it with data from laboratory investigations. Of course the model can also be compared to mathematical results. But, unfortunately mathe­matical investigations quickly find their limitations when it comes to more complex conditions like the consideration of friction or breaking waves. Thus, even in the age of super computers any numerical model depends on a validation from data provided by a series of physical experiments.

1.2 General Information on Tsunamis

The word “tsunami” originates from the Japanese language from “tsu” -“harbor” and “nami” -“wave”. The expression was formed by former Japanese fishermen who left their village in the morning to go fishing, but found their homes destroyed when they returned in the evening. Tsunami waves can be generated by aerial/ submarine earthquakes, aerial/ submarine volcano eruptions, aerial/ submarine landslides or impacts of meteorites. In any case momentum is transferred into the water column. This momentum is stored in the whole water column with the consequence that the wave is in contact with the sea floor at any time. As long as the water remains deep the energy stored in the water column can be distributed over a wide vertical space. Therefore, the amplitude is relatively small, being in the order of 1m. Accordingly, the tsunami wave is called a shallow water wave since W << L, i.e. the vertical scale, i.e. water depth, is much smaller than the horizontal scale, i.e. wave length. Thus, compared to the large wave length the ocean is shallow water. Consequently, the wave height is much smaller than the wave length (H << L0) and much smaller than the water depth (H << h). So, in the open ocean tsunami waves cannot be distinguished from ordinary surface waves (which is why the fishermen did not notice the destruction of their village until they returned). But tsunami waves travel much faster than any other kind of waves. Since tsunamis are shallow water waves their phase speed is [illustration not visible in this excerpt], where g is the gravitational constant and h is the undisturbed water depth. So, in an ocean of approximately 4000m depth tsunami waves travel at 200m/s or 720km/h. Simultaneously, the wave length is in the order of 100 km and the period ranges from minutes to one hour. As soon as the tsunami wave enters regions of shallow water, e.g. a beach, the velocity decreases dramatically while the amplitude increases considerably. The reason for this behaviour is that the energy stored in the water column has to be distributed over a much narrower vertical space and now needs to be converted. A part of the energy is converted into frictional energy. The only way to convert the rest of the energy is an increase of the wave amplitude. If the wave breaks, further energy is dissipated. If the wave does not break, the energy is dissipated while the wave propagates onshore. As soon as all kinetic energy is converted into frictional energy and dissipation energy the maximum run-up is accomplished and the water starts to retreat (run-down).

1.3 Scope of the Thesis

Within this master thesis the behaviour of long waves running up a plain beach of uniform slope is investigated. There are two main parts of the thesis: the first one is the experimental part where experiments in the wave flume are performed. In the second part a numerical model is validated with the data set gained from the first part.

The wave type chosen for all experiments is the leading depression, non-breaking single sinusoidal wave. The primary aim is to investigate the correlation between wave length/ wave period and shoreline motion as well as run-up. The expression shoreline motion here means velocity and location of the initially motionless water shoreline during the run-up and run-down phase of a long wave. A suffciently wide range of wave periods and amplitudes is chosen such that the typical temporal and spatial range of tsunami-like waves modelled by single sinusoidal waves is covered. For all investigations reported herein the experimenters make use of long non­- breaking waves. The reason is that breaking waves involve a significant amount of dissipation and turbulence. The measurement and consideration of these phe­nomena complicate the experiments and detract from the original objective, which is the investigation of shoreline motion and run-up.

For all experiments performed the beach slope remains the same so that no sensi­tivity analysis regarding the beach slope is conducted. Furthermore, this work does not include interactions with macro-roughness elements, neither on dry land nor installed on the submerged beach section.
In the second part of the thesis the data set gained from the first part is used to validate a numerical model. The numerical model setup resembles the conditions of the physical wave flume to assure comparability. No realistic topography will be im­plemented and, like in the experimental part, no interactions with macro-roughness elements are investigated. Furthermore, the forcing will remain the same which is the restoring force of gravity and the varying depth. No meteorological forcing like wind stress or atmospheric pressure, e.g. inverse barometric effect, is considered. Likewise the interaction between the tsunami wave and a tidal wave is not investi­gated.

With the experimental and the numerical part two of the three main tools of sci­entific investigations are used. The third one which is the mathematical tool is not implemented in this thesis. Therefore, the reader will neither find profound theo­retical analyses nor detailed derivations in this thesis. The mathematics used here are limited to the functions describing the shape of the wave types and most fun­damental calculations to derive e.g. run-up and shoreline motion, which in many cases has already been done by other authors. The reader interested in theoretical investigations is referred to e.g. Madsen & Schäffer (2009), Madsen & Fuhrman (2008), Carrier et al. (2003) and Carrier & Greenspan (1958).

In the past decades numerous authors examined the behaviour of tsunami-like waves running up a uniformly sloping beach, e.g. Briggs et al. (1993, 1995); Synolakis (1986, 2006); Synolakis & Deb (1988); Synolakis & Kong (2006); Gedik et al. (2005); Chanson et al. (2003); Jensen et al. (2003). While the focus of these studies is mainly given to the run-up of the waves there are only few studies like Apotsos et al. (2011, 2012) and Synolakis et al. (2008) investigating the current velocities associated with the run-up and run-down phase. There are even less studies observing the position of the shoreline. Thus, there is a significant gap of data concerning the shoreline motion induced by a tsunami. Exactly at this point the present thesis offers up: motivated by the lack of data on shoreline motion, the intention of this thesis is firstly to partially fill this gap of missing data and secondly to provide one more laboratory data set for the validation of numerical models.

1.4 Outline of the Thesis

This thesis is organized as follows: there will be a literature review in chapter 2 to review the history of physical modelling as well as to present past and current methods in wave generation. Current state-of-the-art generation techniques, possi­ble wave types and analytical solutions of the shoreline motion are presented. In chapter 3 the experimental setup is described. This involves description of the wave flume and the controller scheme to drive the pumps. Furthermore, the presentation, explanation and location of the measuring instruments in the setup are given. De­scription of the experiments performed and data acquisition terminate this section. In chapter 5 the results of the physical experiments are presented. At the end of each subsection the results are compared to literature and differences are discussed. The numerical part of the thesis commences in chapter 6 where the numerical model used herein is described. In the following, the experimental data set is used to calibrate and then to validate the numerical model. A decision is made whether the numerical model is capable of representing reality with suffcient accuracy. Consequently, it is judged whether the model is suitable for eventual future verification and eventual predictions. A summary of all results and an outlook is given in chapter 7.

2 Literature Review

2.1 Long Wave Generation Techniques

In the past decades major progress has been made in the field of physical long wave generation. Early experiments to generate waves in a wave flume were conducted by Russel in 1844. Russel used a sinking box to generate a solitary wave propagating in a rectangular wave flume. A comprehensive review of existing long wave generation techniques is given in Liu (2008), Goseberg (2011) and Goseberg (2012). In the following the most commonly applied techniques to generate long waves in a wave flume are reviewed. The pump-driven wave generator which was applied within this study is not part of the review since it is described in detail in section 4.
In the context of physical long wave generation and description of experimentally generated waves scientists and engineers often use the surf similarity parameter ξ. This parameter is utilized to characterize the behaviour of breaking waves on beaches in dependence of beach slope, wave period and wave height (Madsen & Fuhrman, 2008). The surf similarity parameter has proven its usefulness in several analytical and experimental examinations, e.g. Madsen & Schäffer (2009), Galvin (1968), Battjes (1974). It can be used to distinguish breaking and non-breaking waves as well as to define breaker type classifications as done by Galvin (1968). The surf similarity parameter, also called Iribarren number, was first introduced by Iribarren & Nogales (1949) as:

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where γ is the beach slope, H is the wave height, L0 is the offshore wave length and T is the wave period.

2.1.1 Piston-type Wave Generation

The piston-type generator is probably the most frequently applied wave generator. A hydraulically driven and horizontally moving wave board releases energy into the water column. Thereby, a solitary wave is generated that propagates away from the wave generator. The wave length of the wave generated depends on the stroke length. Fundamental work on the run-up of solitary waves was carried out by Syno­lakis (1986). On the basis of his experimental results Synolakis derived a theoretical approach for the run-up of linear and non-linear, breaking and non-breaking soli­tary waves. Synolakis found that the highest run-up heights occurred for breaking solitary waves. Briggs et al. (1995) investigated the run-up of solitary waves on a circular island. Interestingly, they found that in some cases the run-up on the back side of the island can be higher than on the island front facing the incoming wave. The explanation is that wave refraction and diffraction cause the wave to “wrap evenly around the island” resulting in the large run-up (Briggs et al., 1995). Galvin (1968) used solitary waves to perform a classification of breaker types based on the surf similarity parameter ξ. He found the following breaker types:

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2.1 Long Wave Generation Techniques 19

In addition, Galvin (1968) defined intermediate wave types forming a fluent transi­tion between these three main types. Baldock et al. (2009) issued an open access data set comprising data on wave kinematics including wave elevation, wave velo­city, acceleration and run-up. The authors performed experiments in 48m long and 26.6m wide basin using breaking and non-breaking solitary waves. The beach in their study exhibited two different slopes. The first beach section owned a slope of 1:15 and was completely inundated for a system at rest. The second beach section comprised a milder slope of 1:30. The waves generated covered a range of surf simi­larity between 0.26 ≤ ξ ≤ 5.6. The upper range limit was most likely determined by the maximum stroke length of 2m. The limited stroke length resulting in a limited wave length is one disadvantage of the piston-type wave generator (Goseberg, 2012). In his article Goseberg (2012) calculated the theoretical stroke length a piston-type wave generator should have to generate a long wave exhibiting a tsunami-like wave length. To model a tsunami-like down-scaled long wave Goseberg assumes a sinu­soidal model wave of wave period T = 90s, a wave height of H = 0.1m and a water depth of 0.3m. With a physical scale of 1:100 these assumptions result in a proto­type tsunami of wave period T = 15min, a wave height of H = 10m and a water depth of 30m (Goseberg, 2012). Based on the idea by Galvin (1964) of comparing the water volume displaced by the stroke with the water volume contained in the wave crest Goseberg calculates a theoretical stroke length of 8.19m. A piston-type wave generator exhibiting such a large stroke length would be an enormous engi­neering and financial challenge. To the author’s knowledge a piston-type generator with such large stroke length does not exist.

For the past decades the predominant wave form used in physical wave modelling has been the solitary wave due to limited wave generation techniques. However, during the last years several authors have questioned the existing solitary wave paradigm. For example Madsen & Schäffer (2009) state that the length and time scales of geophysically scaled solitary waves do not resemble the length and time scales of a prototype tsunami wave. Goseberg et al. (2012) agree with Madsen & Schäffer (2009) and state that the deviation between the wave length of a natural tsunami wave and the wave length of a solitary model wave is at least in the order of one magnitude. The only exception where solitary waves can be applied seems to be the investigation of landslide induced tsunami waves. Within a corresponding study performed by Fritz et al. (2009) solitary waves were found to model landslide induced tsunami events and their maximum run-up height well. For the mostly applied piston-type wave generator the attainable wave period is around T = 15s ­20s according to Goseberg et al. (2012). This corresponds to time and space scales which are significantly below the time and space scales of prototype tsunami waves so that characteristics like near-shore impacts, surge flow and wave run-up cannot be modelled adequately (Goseberg et al., 2012; Goseberg, 2012).

2.1.2 Dam Break Analogy

Another possibility to generate a physical water wave is to use a strategy resembling a dam break. Here, a huge amount of water is released from an overhead water reservoir by suddenly opening a gate so that the water plunges into a wave tank located beneath the water reservoir. Thereby, a wave is generated that evolves into a bore during propagation. Corresponding experiments were conducted by Chanson et al. (2003) to investigate the run-up of bores on a 1:5.92 beach. The authors observed a higher wave velocity compared to other dam break waves due to a high initial momentum. After having propagated a distinct distance the bore decelerated due to bottom friction and dissipation of turbulent energy (Chanson et al., 2003). By the sudden water release a huge amount of turbulence is introduced into the initially static water body, which represents a disadvantage of this generation technique (Goseberg, 2012). According to Goseberg (2012) a second disadvantage is the limited control over the generated wave. The volume of water released from the reservoir and the initial height of the released water body determine the shape and length of the final bore (Goseberg, 2012). Although some control over the wave shape is given by the opening velocity of the reservoir gate there is no possibility to control the entire wave generation mechanism resulting in insuffcient control over the final wave shape.

2.1.3 Vertical Sea Bed Motion

As a third strategy to generate long waves scientists use a bottom plate that moves in vertical direction. This wave generation technique is meant to model an earthquake­-triggered impulse wave. Early experiments with a moving sea bed were conducted by Hammack (1973). Hammack investigated the propagation of solitary waves in an ocean of constant depth experimentally and analytically. For his analysis he distinguished two sections, a generation section and a downstream section. In the downstream section non-linear effects occurring due to frequency dispersion were studied analytically. From the explicit distinction of two section Goseberg (2012) concludes that the vertical sea bed motion is an inadequate strategy to investigate near-shore wave effects as well as wave run-down and run-up.

Raichlen (1970) combined analytical calculations, field observations and examina­tions of experimental waves generated by a moving bottom to investigate wave pro­pagation by analyzing the wave frequency spectrum. Energy spectra were calculated from wave gauges that recorded time series of the wave height during the Chilean earthquake tsunami 1960 and the Alaskan earthquake tsunami 1964. Raichlen (1970) found that the shape of the lead wave changed significantly since additional waves were generated from the lead wave. Furthermore, due to frequency dispersion groups of waves were left behind as the waves departed further from the generation area (Raichlen, 1970). Another dispersion effect was that the front slope of the main wave decreased. Raichlen (1970) also addressed the resonant behaviour of harbours to the frequency of long waves. He found that a harbour which is responsive to a specific frequency within the wave energy spectra tends to mask the wave frequencies it is not responsive to. That means, if a wave train containing different wave frequen­cies arrives at a harbour with resonant character, then the harbour reinforces the frequency it is responsive to while the remaining wave frequencies and the energy contained at these frequencies are damped.

Using a pneumatic moveable bottom plate Gedik et al. (2005) generated non-breaking solitary waves and investigated their run-up on a 1:5 beach slope. In addition the authors covered the artificial beach with different types of armor units to examine the run-up height in dependence of different roughness intensities. Gedik et al. (2005) found that the armor units decreased the maximum run-up height by up to 50%. Furthermore, the maximum run-up decreased for an increasing diameter of the armor units which increased the area covered by a rough surface.

2.1.4 Pneumatic Wave Generation

A new approach in the generation of long waves was presented by Rosetto et al. (2011). The authors used a pneumatic wave generator that is capable of generating arbitrary long wave shapes on a scale of 1/100 or 1/50. Their wave generator con­sists of a wave flume and a water tank. From the tank top a fan extracts the air from the tank and thereby raises the water level in the tank. Through a submerged opening of 0.45 m height water is extracted from the wave flume generating a wave trough. A valve installed on the tank roof “releases air to generate a wave from the tank” (Rosetto et al., 2011). By controlling the position of the valve a pre-defined water surface profile is generated. The valve position is controlled by a proportional-­integral feedback control loop.

Within first test experiments Rosetto et al. (2011) generated sine waves, solitary waves, and N-waves. In addition, the 2004 Indian Ocean tsunami was reproduced using the surface elevation profile of the tsunami recorded by the “Mercator” yacht. The yacht was located 1.6km off the coast of South Phuket, Thailand as the tsunami approached the coast. An echo-sounder installed on the yacht recorded the water depth with a sampling interval of 1min assuring a high temporal resolution of the near-shore tsunami wave. For their experiments Rosetto et al. (2011) generated sine waves with periods between T = 50s to 200s and a maximum wave height of H = 4.5cm. The generated solitary waves exhibited an amplitude-to-depth ratio of 0.014 ≤ a/h ≤ 0.2 and periods between T = 4.2s to 17.1s. According to the authors the wave height, and therefore the range of the amplitude-to-depth ratio, were limi­ted by the tank height. The period of the N-waves ranged from 7.0s to 17.8s, the wave height varied between 2.2cm and 9.3 cm. The amplitude-to-depth ration was stated to be 0.011 ≤ a/h ≤ 0.16. During the experimental study the water depth in the flume was varied and amounted to 0.56m on average.

As stated by Rosetto et al. (2011) the wave flume exhibits an effective length of approximately 30m, depending on the operating depth. The wave tank is located at the flume beginning and measures 4.80m × 1.80m × 1.15m. The wave flume comprises a beach of a constant 1:20 slope located at the opposite flume site of the wave tank. The distance between tank and beach toe amounts to 15.2m.

Rosetto et al. (2011) investigated wave run-up, the repeatability of the wave gene­rator and they compared the surface profiles of their solitary waves to the waves generated by Synolakis (1986). For the solitary waves and the N-waves Rosetto et al. (2011) found very good repeatability as well as for the wave run-up. A com­parison of the water surface profiles of the solitary waves with the profiles generated by Synolakis (1986) yielded a discrepancy since the waves generated by the pneu­matic wave maker exhibited a lower steepness. The authors explain this deviation with the wave generation technique. For each position of the valve a specific time is needed to establish a static equilibrium of the air pressure inside the tank. As a consequence the water exchange between tank and wave flume is slowed down which results in a decreased wave steepness (Rosetto et al., 2011). As stated by the authors another disadvantage of the experimental setup is that major wave reflection occurs on the beach. Therefore, especially waves with a length exceeding 15.2m cannot be generated adequately since their surface profile is disturbed by reflected waves. This was also observed for the reproduced 2004 Indian Ocean tsunami. While the leading wave depression showed suffcient agreement with the prototype wave trough the subsequent reproduced wave crest was too short due to waves reflected from the beach.

2.2 The Brier Score

In the previous sections different possibilities to generate long waves were described. The long waves that were investigated in this study were generated by neither of these techniques. Instead, a new pump-driven wave generator was utilized which is described in section 4. To judge the quality of the generated long waves the Brier Score was used. This statistical quantity measures the accuracy of probabilistic predictions. It ranges from 0 to unity where 0 is a perfect forecast and unity means no agreement. In general the Brier score for probabilistic processes reads (Holle, 1995):

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with: n: total number of events, p: probability for the occurrence of an event and z the actual event variable being 0, if the event does not occur and 1, if it occurs. It is possible to modify the formula such that pi = 1. This option is chosen, if the observer is certain that the event occurs.

2.3 The Shoreline Motion of Long Sinusoidal Waves

In the following the run-up/ run-down of long waves as calculated by Madsen & Schäffer (2009) is shortly reviewed. Since in the present study only sinusoidal waves were applied the theoretical review solely addresses the shoreline motion of long sine waves. The reader interested in shoreline characteristics of additional wave types like solitary waves or N-waves is referred to Madsen & Schäffer (2009). To determine the run-up/ run-down of long waves Madsen & Schäffer (2009) chose a coordinate system with the origin at the still shoreline, the x-axis pointing offshore and the z-axis pointing upward. Therefore, the offshore water depth is [illustration not visible in this excerpt] or [illustration not visible in this excerpt], where γ: beach slope and x: distance from still water shoreline. For long waves travelling in water of constant depth the velocity is: [illustration not visible in this excerpt], with g: gravitational constant. Therefore, the travel distance is: [illustration not visible in this excerpt] or [illustration not visible in this excerpt]. According to Madsen & Schäffer (2009) the maximum run-up of long waves is:

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with A0: offshore wave amplitude and [illustration not visible in this excerpt]: wave duration with T being the wave period. Madsen & Schäffer (2009) then calculated the shoreline velocity during run-up/ run-down as:

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where θ was given as:

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Here, λ corresponds to time and tshift is an arbitrary phase shift. The shoreline location of long waves then finally amounts to:

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We now proceed with the theoretical determination of the maximum run-up/ run­down height of long waves. For this purpose Madsen & Schäffer (2009) modified the original definition of the surf similarity parameter [illustration not visible in this excerpt]. By interpreting T as a duration rather than a wave period the authors found

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With this definition Madsen & Schäffer (2009) expressed the maximum run-up/ run-down of long waves in terms of surf similarity parameter as:

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with: R: Run-up/ run-down, A0: offshore wave amplitude, [illustration not visible in this excerpt]: shoreline ele­vation, h0: offshore water depth and ξ: surf similarity parameter. In their paper Madsen & Schäffer (2009) specified [illustration not visible in this excerpt] for sinusoidal waves.

According to Madsen & Schäffer (2009) the maximum shoreline velocity during run-down/ run-up in terms of surf similarity parameter reads:

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where Madsen & Schäffer (2009) specified [illustration not visible in this excerpt] for sinusoidal waves de­pending on whether run-up or run-down is examined.

2.4 Breaking Criterion for Sinusoidal Waves

From theory one can determine whether a wave will break or not. There is a breaking criterion for the maximum run-down/ run-up as well as for the maximum shoreline velocity during run-up. Both criteria were derived by Madsen & Schäffer (2009). Their theoretical breaking criterion for the maximum run-down/ run-up reads:

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Here, [illustration not visible in this excerpt] is the shoreline elevation and [illustration not visible in this excerpt] is not further named in Madsen & Schäffer (2009), but seems to be a kind of breaking criteria number since it occurs in another expression for the breaking criterion (s. Madsen & Schäffer (2009), their equation (5.8)). To calculate the breaking criterion for a specific wave type [illustration not visible in this excerpt] and [illustration not visible in this excerpt] have to be chosen according to the wave at question. In their paper Madsen & Schäffer (2009) specified both quantities with [illustration not visible in this excerpt] and [illustration not visible in this excerpt] for sinusoidal waves. The sign of [illustration not visible in this excerpt] depends on whether the run-up or run-down is regarded.

Concerning the maximum shoreline velocity during run-up Madsen & Fuhrman (2008) derived the theoretical breaking criterion as:

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