High-Fidelity Numerical Simulation of Shallow Water Waves

Home , Cnoidal wave, Shallow water equations

Amir Zainali

Dissertation submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulﬁllment of the requirements for the degree of

Doctor of Philosophy in Geosciences

Robert Weiss, Chair Scott D. King Jennifer L. Irish Heng Xiao Nina Stark

December 2, 2016 Blacksburg, Virginia

Amir Zainali

ABSTRACT

Tsunamis impose significant threat to human life and coastal infrastructure. The goal of my dissertation is to develop a robust, accurate, and computationally efficient numerical model for quantitative hazard assessment of tsunamis. The length scale of the physical domain of interest ranges from hundreds of kilometers, in the case of landslide-generated tsunamis, to thousands of kilometers, in the case of far-field tsunamis, while the water depth varies from couple of kilometers, in deep ocean, to few centimeters, in the vicinity of shoreline. The large multi-scale computational domain leads to challenging and expensive numerical simulations. I present and compare the numerical results for different important problems — such as tsunami hazard mitigation due to presence of coastal vegetation, boulder dislodgement and displacement by long waves, and tsunamis generated by an asteroid impact — in risk assessment of tsunamis. I employ depth-integrated shallow water equations and Serre-Green-Naghdi equations for solving the problems and compare them to available three- dimensional results obtained by mesh-free smoothed particle hydrodynamics and volume of fluid methods. My results suggest that depth-integrated equations, given the current hardware computational capacities and the large scales of the problems in hand, can produce results as accurate as three-dimensional schemes while being computationally more efficient by at least an order of a magnitude. High-Fidelity Numerical Simulation of Shallow Water Waves

Amir Zainali

GENERAL AUDIENCE ABSTRACT

A tsunami is a series of long waves that can travel for hundreds of kilometers. They can be initiated by an earthquake, a landslide, a volcanic eruption, a meteorological source, or even an asteroid impact. They impose significant threat to human life and coastal infrastructure. This dissertation presents numerical simulations of tsunamis. The length scale of the physical domain of interest ranges from hundreds of kilometers, in the case of landslide-generated tsunamis, to thousands of kilometers, in the case of far-field tsunamis, while the water depth varies from couple of kilometers, in deep ocean, to few centimeters, in the vicinity of shoreline. The large multi-scale computational domain leads to challenging and expensive numerical simulations. I present and compare the numerical results for different important problems — such as tsunami hazard mitigation due to presence of coastal vegetation, boulder dislodgement and displacement by long waves, and tsunamis generated by an asteroid impact — in risk assessment of tsunamis. I employ two-dimensional governing equations for solving the problems and compare them to available three-dimensional results obtained by mesh-free smoothed particle hydrodynamics and volume of fluid methods. My results suggest that two- dimensional equations, given the current hardware computational capacities and the large scales of the problems in hand, can produce results as accurate as three-dimensional schemes while being computationally more efficient by at least an order of a magnitude. Acknowledgments

I would like to express my great appreciation to Dr. Robert Weiss, as my adviser, for his guidance, encouragement, and patience during the course of this work. I would also like to thank my dissertation committee members, Drs. Scott King, Jennifer L. Irish, Nina Stark, and Heng Xiao for their helpful comments on the draft of this thesis. I would also like to thank Kannikha Kolandaivelu and Roberto Marivela for reviewing the earlier versions of this dissertation.

The work presented in here is based upon work partially supported by the National Science Foundation under Grants No. NSF-CMMI-1208147 and NSF-CMMI-1206271.

iv Contents

Acknowledgement iv

List of Figures ix

List of Tables xvii

Nomenclature xix

1 Introduction 1

1.1 Theoretical Background ...... 6

1.2 Contributions and Outline of the Dissertation ...... 10

2 Boulder Dislodgement and Transport by Solitary Waves: Insights from

Three-Dimensional Numerical Simulations† 14

2.1 Introduction ...... 15

v Contents vi

2.2 Theoretical Background ...... 17

2.2.1 Governing Equations and Numerical Method ...... 17

2.2.2 Model Setup ...... 19

2.2.3 Non-Dimensional Parameters ...... 20

2.3 Results ...... 23

2.3.1 Validation of Numerical Results ...... 23

2.3.2 Boulder Transport by Solitary Waves ...... 25

2.4 Discussion and Conclusions ...... 28

3 High-Fidelity Depth-Integrated Numerical Simulations in Comparison to

Three-Dimensional Simulations 32

3.1 Introduction ...... 33

3.1.1 Three-Dimensional Methods ...... 34

3.1.2 Depth-Integrated Methods ...... 38

3.2 Results ...... 39

3.2.1 Non-Breaking Solitary Wave Interaction with a Group of Cylinders . . 39 Contents vii

3.2.2 Breaking Solitary Wave Run-Up on a Sloping Beach ...... 42

3.2.3 Breaking Solitary Type Wave Run-Up on a Sloping Beach ...... 42

4 Numerical Simulation of Nonlinear Long Waves in the Presence of Discon-

tinuous Coastal Vegetation † 45

4.1 Introduction ...... 46

4.2 Theoretical Background ...... 49

4.2.1 Initial and Boundary Conditions ...... 50

4.2.2 Wave Breaking ...... 51

4.3 Results ...... 52

4.3.1 Validation of Numerical Results ...... 52

4.3.2 Breaking Solitary-Type Transient Wave Run-Up in the Presence of

Macro-Roughness ...... 54

4.3.3 Eﬀects of Macro-Roughness on the Local Maximum Local Water Depth 56

4.3.4 Eﬀects of Macro-Roughness on the Local Maximum Momentum Flux . 57

4.3.5 Maximum Run-Up ...... 58

4.4 Discussion and Conclusion ...... 61 Contents viii

5 Some Examples for Which Dispersive Eﬀects Can Change the Results

Signiﬁcantly† 64

5.1 Numerical Simulation of Hazard Assessment Generated by Asteroid Impacts

on Earth ...... 65

5.1.1 Numerical Simulation of Tsunami Waves Generated by an Asteroid

Explosion near the Ocean Surface ...... 67

5.1.2 Numerical Simulation of Tsunami Waves Generated by an Asteroid

Impact into the Ocean ...... 69

5.2 Non-Breaking Cnoidal Wave Interaction with Oﬀshore Cylinders ...... 73

5.3 Hazard Assessment Along the Coastline from the Gaza Strip to the Caesarea,

Israel ...... 76

6 Future Work 79

Bibliography 81 List of Figures

1.1 Location of tsunamis that have happened since 1900 to present day; ( ):

caused by a volcanic activity; ( ) caused by a landslide; ( ) caused by an

unknown source; ( ) caused by an earthquake. Color codes are as following:

dark-red represents the tsunamis that caused more than 1000 casualties or

more than 1 billion dollars damage in total; red represents the tsunamis that

caused more than 100 casualties or more than 100 million dollars damage in

total; orange represents the tsunamis that caused more than 10 casualties or

more than 10 million dollars damage in total; yellow represents the tsunamis

that caused less than 10 casualties or less than 10 million dollars damage in

total ...... 2

ix List of Figures x

1.2 Histograms of the number of earthquakes versus the death and damage toll.

I: less than 10 casualties; II casualties between 10 and 100; III casualties

between 100 and 1000; IV casualties between 1000 and 10000; casualties more

than 10000. A: damage less than 10 million dollars; B: damage between 10 and

100 million dollars; C: damage between 0.1 and 1 billion dollars; D: damage

between 1 and 10 billion dollars; E: damage between 10 and 100 billion dollars;

F: damage more than 100 billion dollars; The y-axis is in logarithmic scale. .5

1.3 Casualties and economic loss due to I: unknown source; II: earthquake; III:

landslide; IV: volcano; V: meteorological source; The y-axis is in logarithmic

scale...... 6

1.4 Schematic of a long wave...... 7

1.5 Tsunami wave propagation caused by a landslide. The initial ratio between the

wavelength and water depth is more than 10. The ﬁgure shows the comparison

of the three diﬀerent simulations. (blue) SGN high resolution; (red) SGN low

resolution; (green) SWE high resolution...... 11

2.1 Sketch of dam-break scenario at t = 0. Water behind the gate starts to ﬂow

after sudden gate removal at t = 0. To compare the numerical results with

experiments presented by Imamura et al. (2008), we applied: lx = 10 m, ly =

0.45 m, lz = 0.3 m, lxh = 5.5 m, lw = 3 m and hw = 0.15, 0.20, 0.25, 0.3 m. . . . . 19 List of Figures xi

2.2 Comparison of numerical and experimental data presented. Experimental

−3 data are from Imamura et al. (2008). The case ρb = 1550 kg m is colored in

−3 black and case with ρb = 2710 kg m is colored in blue; ( ) experimental

results and ( ) SPH simulations...... 22

2.3 Time snapshots for scenarios CA, CB and CC with α = 0, (left column) and

scenarios CA, CB and CC with α = 1, (right column) with wave height of

0.15m. Please note that diﬀerent cases are superimposed on the same domain

just for illustration purposes...... 24

2.4 Contour plots of boulder maximum displacement, db, as a function of non-

dimensional parameters. a: aspect ratio in logarithmic scale versus Froude

number; (a-1) β = −8~30 (a-2) β = 0 and (a-3) β = 4~30. b: submergence

factor versus Froude number; (b-1) α = −1, (b-2) α = 0 and (b-3) α = 1. c:

submergence factor versus aspect ratio in logarithmic scale; (c-1) F r = 1.08,

(c-2) F r = 1.15 and (c-3) F r = 1.22. White regions indicate no signiﬁcant

−1 boulder movement, i.e. dblbz < 0.1. For gray regions, the distance boulder

−1 travels is in between 0.1 < dblbz < 2.0. Color contours indicate boulders moved

−1 signiﬁcantly, i.e. dblbz > 2.0 (colormap is in logarithmic scale) ...... 30 List of Figures xii

3.1 A schematic sketch of the solitary wave passing through and around cylinders.

The coordinate of the domain is located at the center of the cylinder on the

right hand side of the domain. Wave propagates along the x-axis. The red dots

represent the location of the center of the cylinders. The blue dots represent

the location of the wave gauges. Gauge 1: (-3.04 m, -0.14 m); Gauge 2: (-1.82

m, -0.14 m); Gauge 3: (-0.83 m, 0.00 m); Gauge 4: ( 0.00 m, -0.88 m); Gauge

5: ( 0.85 m, 0.00 m); Gauge 6: ( 1.82 m, 0.00 m); ...... 34

3.2 Comparison of numerical and experimental data presented. Experimental and » three-dimensional simulations are from Mo (2010); t∗ = t~ h~g, and ζ∗ = ζ~H;

( ) present simulation (2D-SGN), ( ) experimental results and ( ) three-

dimensional simulation (3D-VOF)...... 36

3.3 Comparison of numerical and experimental data presented. Experimental and » three-dimensional simulations are from Mo (2010); t∗ = t~ h~g, and ζ∗ = ζ~H;

( ) present simulation (2D-SGN), ( ) experimental results and ( ) three-

dimensional simulation (3D-VOF)...... 38

3.4 Schematic of the solitary wave run-up on a sloping beach with the slope of

1:19.85...... 40

3.5 Breaking solitary wave run-up compared to experimental results of Synolakis

(1987) and three-dimensional GPUSPH simulation of Marivela et al. (Under

review). ( ) present simulation (2D-SGN), ( ) experimental results and

( ) GPUSPH...... 41 List of Figures xiii

3.6 Free-surface elevation at (a) gauges 1-2, and (b) gauges 3-4. Experimental

results are summarized in Yang et al. (2016); ( ) present simulation (2D-

SGN), ( ) experimental results and ( ) GPUSPH...... 43

3.7 Local water depth at gauges 5-16. Experimental results are summarized in

Yang et al. (2016); ( ) present simulation (2D-SGN), ( ) experimental re-

sults and ( ) GPUSPH...... 44

4.1 Free surface elevation of the solitary-type transient wave over constant water

depth at (a) t = 5 s, (b) t = 15 s, and (c) t = 25 s; ( ) 1D-SGN, and

( ) 1D-NSW. The following parameters are used: h0 = 0.73 m, H = 0.50 m,

−1 k0 = 0.54 m ...... 48

4.2 Schematic of the breaking process. The vertical dashed lines indicate the

boundary of subdomains. The governing equations in the left subdomain are

SGN and NSW equations elsewhere. The boundary follows the leading wave;

(I) t < t1 ∶ SGN in the whole domain, (II) t1 < t < t2 ∶ SGN in the left

subdomain and SW in the right subdomain, and (III) t > t2 ∶ NSW in the

whole domain...... 50

4.3 Free-surface elevation at (a) gauges 1-2, and (b) gauges 3-4. Experimental

and COULWAVE results are summarized in Yang et al. (2016); ( ) present

simulation (2D-SGN), ( ) experimental results and ( ) COULWAVE. . . . . 52 List of Figures xiv

4.4 Local water depth at gauges 5-16. Experimental and COULWAVE results

are summarized in Yang et al. (2016); ( ) present simulation (2D-SGN), ( )

experimental results and ( ) COULWAVE...... 53

4.5 Sketch of the macro-roughness patches...... 54

4.6 Local water depth at gauges 5-16 for Scenario 3. Experimental results are

summarized in Irish et al. (2014); ( ) present simulation (2D-SGN), and ( )

experimental results...... 56

∗ 4.7 Maximum local water depth hmax for (a) Scenario 1, (b) Scenario 2 (c) Scenario

3, and (d) Scenario 3 in which all the patches are removed except the ﬁrst

patch. The maximum water depth for each scenario is normalized with the

reference values in the absence of macro-roughness patches...... 58

∗ 4.8 Maximum momentum ﬂux Fmax for (a) Scenario 1, (b) Scenario 2, and (c)

Scenario 3. The momentum ﬂux for each scenario is normalized with the

reference values in the absence of macro-roughness patches...... 59

4.9 Propagation of bore-lines in the presence of macro-roughness patches (Sce-

nario 2). Irish et al. (2014); (a) experimental results, and (b) present simula-

tion (2D-SGN)...... 60

4.10 Propagation of bore-lines for (a) Scenario with no macro-roughness patches

(b) Scenario 1, (c) Scenario 2 (d) Scenario 3...... 61 List of Figures xv

5.1 Comparison of water surface elevation between our SWE model and GeoClaw

results. Waves are generated by an asteroid with a diameter of 140 m explod-

ing at the altitude of 10 km. Wave gauges are located at ( ) 0.05LD,( )

0.2LD,( ) 0.5LD, and ( ) 0.8LD, where LD = 111 km...... 66

5.2 Comparison of water surface elevation obtained using SWE equations ( ),

and SGN equations ( ) at (a) t = 400 s, (b) t = 800 s, and (c) t = 1200 s.

Waves are generated by an asteroid with a diameter of 140 m exploding at

the altitude of 10 km...... 67

5.3 Water surface elevation obtained using SWE equations ( ), and SGN equa-

tions ( ) at (a) t = 1000 s, (b) t = 2000 s, (c) t = 3000 s, and (d) t = 4000 s.

Left y-axis shows the topographical variation in logarithmic scale...... 68

5.4 Water surface elevation of a tsunami wave generated by an impact into water

obtained using SGN equations at (a) t = 1000 s, (b) t = 5000 s, (c) t = 10000

s, and (d) t = 15000 s...... 70

5.5 Maximum wave height as a function of distance from the impact center. ( )

SWE ( )SGN...... 71

5.6 Schematic of a cnoidal wave interacting with a cylinder...... 73

5.7 Contour plots of the maximum water elevation for (a) Case1, (b) Case2, (c)

Case3. Blue lines denote the vertical cross sections at 1: (x − x0)~h0 = 9.83,

2: (x − x0)~h0 = 17.65, 3: (x − x0)~h0 = 41.25...... 74 List of Figures xvi

5.8 Maximum momentum ﬂux along the vertical cross sections at top: (x −

x0)~h0 = 9.83, middle: (x − x0)~h0 = 17.65, bottom: (x − x0)~h0 = 41.25. . . . . 75

5.9 Tsunami run-up along the coastline from the Gaza Strip to the Caesarea,

Israel...... 78 List of Tables

2.1 Physical and numerical simulation parameters (hw = 0.3 m. hb represents the

depth of the water at the location of boulder. c is the solitary wave celerity). √ Submergence factor: β = hb (y axis); Froude number: F r = c( gh )−1; boul- hw w

−1 der aspect ratio in logarithmic scale: α = log2(lbylbx ) [x − y plane; boulder

−1 −3 normalized width: Wb = lbyhw ; boulder normalized volume: Vb = lbxlbylbzhw ;

−1 density ratio: ρbρw ; friction coeﬃcient: µ;...... 25

3.1 Running time for three-dimensional SPH simulation, and one-dimensional

SGN simulation of solitary wave run-up on a sloping beach...... 40

4.1 Wave gauge coordinates...... 51

4.2 Geometrical parameters of macro-roughness patches. dr ∶ distance between

two horizontally (or vertically) aligned cylinders inside a patch; Ncp ∶ total

number of cylinders inside a patch; dp ∶ distance between two horizontally (or

vertically) aligned patches; Cfp ∶ coordinate of the center of the ﬁrst patch;

Dc ∶ diameter of the cylinders; Dp ∶ diameter of the patches...... 55

xvii List of Tables xviii

5.1 Simulation parameters of cnoidal waves interacting with a cylinder...... 73 Nomenclature

Abbreviations

SGN Serre-Green-Naghdi equations

SPH Smoothed particle hydrodynamics

SWE Nonlinear shallow water equations

VOF Volume of ﬂuid

Latin symbols

u Velocity vector

ζ Free surface elevation b Topographical variation g Gravitational acceleration h Local water depth p Pressure t Time

xix Nomenclature xx

Non-dimensional numbers

α Aspect ratio

β Submergence factor

µ Shallowness parameter

Re Reynolds number

ε Nonlinearity parameter

F r Froude Number Chapter 1

Introduction

Majority of the world’s mega-cities are located near oceans. More than 50% of the world’s population lives within the 60 km of ocean1. In the United States, according to NOAA2, more than 39% of the population lived in the counties next to an ocean. This number is expected to increase to 47% by 2020. Tsunami waves are one of the most common forms of natural disaster that aﬀects the life of human beings living close to coastal areas. They impose signiﬁcant threat to human life and coastal infrastructure.

A tsunami is a series of long waves that can travel for hundreds of kilometers through the ocean. Figure 1.1 shows the locations that are impacted by tsunamis since 1900. As we can see from Figure 1.1, tsunamis are usually initiated by a sudden displacement of large amount of water in the ocean or large lakes due to one or combination of the following reasons:

(a) Earthquakes are responsible for more than 80% of the tsunamis. Two of the most recent

deadliest tsunamis, i.e. 2004 Indian Ocean tsunami and 2011 Tohoku tsunami, were 1http://www.unep.org/urban environment/issues/coastal zones.asp 2National Ocean and Atmospheric Administration; http://oceanservice.noaa.gov/facts/population.html

1 Introduction 2

Figure 1.1: Location of tsunamis that have happened since 1900 to present day; ( ): caused by a volcanic activity; ( ) caused by a landslide; ( ) caused by an unknown source; ( ) caused by an earthquake. Color codes are as following: dark-red represents the tsunamis that caused more than 1000 casualties or more than 1 billion dollars damage in total; red represents the tsunamis that caused more than 100 casualties or more than 100 million dollars damage in total; orange represents the tsunamis that caused more than 10 casualties or more than 10 million dollars damage in total; yellow represents the tsunamis that caused less than 10 casualties or less than 10 million dollars damage in total Introduction 3

caused by earthquakes. The 2011 Tohuku tsunami occurred on March 11, 2011 caused

by a magnitude 9.0 earthquake. It resulted in more than 15, 000 deaths and a total

economic loss of over 220 billion dollars. Indian Ocean tsunami occurred on December

26, 2004 and killed more than 230, 000 people. The estimated total economic losses due

to this tsunami is about 15 billion dollars. Maximum recorded wave height for Indian

Ocean and Tohoku tsunamis is more than 50 and 40 meters, respectively.

(b) Landslides are the second most probable cause of tsunamis. Tsunamis caused by land-

slides are usually generated by a solo landslide or a landslide following an earthquake or

a landslide initiated from a structural failure of a volcano. The wavelength of a tsunami

due to a landslide is usually shorter (it will only aﬀect the areas at the proximity of the

origin). However the amplitude of these tsunamis can be much larger than the amplitude

of the tsunamis caused by earthquakes. In fact the maximum recorded wave-height in

history is more than 500 meters and belongs to 1958 Lituya Bay tsunami which was

caused by a landslide following an earthquake. Another notable tsunami that was due

to a landslide initiated by an earthquake is the 1964 Alsaka tsunami which killed more

than 100 people.

aﬀect only the local areas. One of the most notable tsunamis caused by a volcanic

eruption is the 1741’s tsunami in the Oshima-Oshima region in Japan. This tsunami

caused lots of damages to coastal infrastructure and the total number of casualties, for

this event, is estimated to be about 2000 (Satake, 2007).

(d) A tsunami can also be generated by a meteorological source or an asteroid impact. An-

other historically important tsunami in the South Paciﬁc ocean was generated by Eltanin Introduction 4

impact which occurred about 2.5 million years ago. Even though the estimated initial

wave amplitude, according to the numerical simulations, was about 1 km, due to the

dispersive nature of the generated tsunami it is thought to have dissipated very fast with

an estimated wave height, close to Chilean coast, of around 10 meters. This makes the

tsunami as dangerous as Indian Ocean tsunami or Sumutra tsunami (Weiss et al., 2015;

W¨unnemannand Weiss, 2015). The most notable recorded historical tsunami caused

by a meteorological source is the Hooghly River disaster that happened on October 11,

1737. This natural disaster is one of the deadliest recorded hazards in the history. The

cyclone caused a tsunami with a wave height in the range of 10 to 15 meters. The es-

timated casualties due to the tsunami and cyclone caused by this event is estimated to

be around 300, 000 people. Some researchers (e.g., Dominey-Howes et al., 2007) argued

only 10% of the overall deaths were due to the tsunami and the remaining 90% were

caused by the initial storm surge and cyclone.

Overall more than 2500 tsunamis have been recorded from 2000 B.C. through present day, according to the Global Historical Tsunami Database provided by NGDC3/WDS4. More than 900,000 people have died because of these natural disasters. The total economic loss due to these events is estimated to be around 550 billion dollars. Figure 1.2 shows the histograms of the number of earthquakes versus the death and damage toll caused by them.

Less than 10% of the tsunamis have caused casualties more than 10 or total economical loss of more than 10 million dollars. Only two recorded event caused more than 100, 000 deaths or more than 100 billion dollars damage. We note here that there is no direct correlation

3National Geophysical Data Center 4World Data Service Introduction 5 between the number of deaths and total economical loss. In other words, tsunamis with the largest amount of casualties do not necessarily correspond to the largest amount of property damage. Figure 1.3 shows the plot bar representing the casualties and deaths caused by tsunamis. Earthquakes are responsible for more than 65% of casualties and more than 85% of the economical loss. About 30% of deaths are caused by tsunamis with meteorological sources and landslides, and volcanic activities cause the remaining losses/damage due to tsunamis.

104

103

102

101

0 Number of10 Earthquakes I II III IV V VI 104

103

102

101

0 Number of10 Earthquakes A B C D F G

Figure 1.2: Histograms of the number of earthquakes versus the death and damage toll. I: less than 10 casualties; II casualties between 10 and 100; III casualties between 100 and 1000; IV casualties between 1000 and 10000; casualties more than 10000. A: damage less than 10 million dollars; B: damage between 10 and 100 million dollars; C: damage between 0.1 and 1 billion dollars; D: damage between 1 and 10 billion dollars; E: damage between 10 and 100 billion dollars; F: damage more than 100 billion dollars; The y-axis is in logarithmic scale. Introduction 6

Total number of casualties Damage in million dollars 106

105

104

103

102

101

100

10 1 − I II III IV V

Figure 1.3: Casualties and economic loss due to I: unknown source; II: earthquake; III: landslide; IV: volcano; V: meteorological source; The y-axis is in logarithmic scale.

1.1 Theoretical Background

Figure 1.4 shows the schematic of a typical long wave. Assuming a ﬂat bathymetry, two non-dimensional parameters that represent the problem are the nonlinearity parameter

a ε = , (1.1) h0 and the shallowness parameter h2 µ = 0 . (1.2) λ2

Here a is the nominal wave amplitude, h0 is the water depth when at rest, and λ is the wave length. Introduction 7

a h2 Nonlinearity: ε = µ = 0 Shallowness: 2 h0 λ

Figure 1.4: Schematic of a long wave.

ε is typically in the order of 10−4 for far ﬁeld tsunamis. For example, the 2004 Indian

Ocean tsunami’s wave height, while traveling in deep ocean (h0 ∼ 5000 m) was about 1 m.

However as the tsunami approaches the coast the wave height increases and the nonlinearity parameter can rise up to ε ∼ 0.5. The nonlinearity parameter for storm waves will be in the order of 10−2 to 10−1.

A typical shallowness parameter for far ﬁeld tsunamis will range from 10−5 < µ < 10−4. How- ever, as the tsunami wave approaches the shore and the wave height increases, the wave speed and consequently the wavelength decreases. This results in an increased nonlinearity parameter in the proximity of the coast. For coastal waves, the wavelength will be signiﬁcantly shorter which usually results in shallowness parameter of µ > 10−2.

Figure 1.5 shows a tsunami wave propagation and run-up. The initial wavelength is about Introduction 8

6 km and the initial wave amplitude is 4.5 m. The water depth at the origin of the wave is about 400 m. Thus the initial nonlinearity and shallowness parameters are ε ∼ 10−2, and

µ ∼ 10−2. We solved this problem using the nonlinear shallow water equations and weakly- dispersive fully-nonlinear Boussinesq-type equations. As we can see in Figure 1.5 the results obtained from different equations significantly differs from each other. We will explain the importance of choosing an appropriate model for the specific problem we want to solve in the following paragraphs.

The nonlinear shallow water equations (SWE) are the most common form of equations used in the numerical simulation of long waves:

(1.3) ∂th + ∇ ⋅ (hu) = 0, 1 ∂ (hu) + ∇ ⋅ hu ⊗ u + gh2I = −gh∇b + O(µ), (1.4) t 2 where u is the depth-integrated velocity vector, h is the water depth, b represents the bottom variations, and I is the identity tensor. To derive the nonlinear shallow water equations from the Euler equations, one has to assume hydrostatic pressure distribution in the vertical direction. This assumption implies that there is no acceleration in the vertical direction, i.e. velocity is constant along the z−axis. SWE are hyperbolic partial diﬀerential equations and thus they have shock capturing capability in their conservative form. Furthermore, there exists many mature numerical schemes for solving these equations which makes them computationally a good and eﬃcient choice.

However, while SWE can produce satisfactory results for far field tsunamis (µ ∼ 10−4), they become very inaccurate for waves with shorter wavelengths (such as landslide generated tsunamis, or storm waves). To overcome this drawback, different classes of higher-order Introduction 9 depth-integrated equations, based on Boussinesq wave theory, have been derived and presented in the literature on simulating shallow water flows: Wei et al. (1995); Liu (1994);

Lannes and Bonneton (2009). In this study, we employ fully nonlinear weakly dispersive

Boussinesq equations, also known as Serre-Green-Naghdi equations (SGN, Lannes and Bon- neton, 2009): (1.5) ∂th + ∇ ⋅ (hu) = 0, 1 (1.6) ∂ (hu) + ∇ ⋅ hu ⊗ u + gh2I = −gh∇b + D + O(µ2), t 2

and ⎛ ⎞ ⎜ −1 ⎟ ⎜ g 1 1 1 ⎟ D = h ⎜ ∇ζ − I + αhT gh∇ζ + hQ(u)⎟ , (1.7) ⎜α h h α ⎟ ⎝ ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶⎠ B

or 1 (I + αT ) B = g∇ζ + Q(u), (1.8) α

where

ζ = h + b. (1.9)

Here ζ denotes the free-surface elevation. T is an operator acting on a scalar ﬁeld given by (1.10) T [h, b](s) = R1[h, b](∇ ⋅ s) + R2[h, b](∇b ⋅ s) ,

Q is an operator acting on a vector ﬁeld given by

⊥ 2 (1.11) Q[h, b](s) = −2R1[h, b] ∂1s ⋅ ∂2s + (∇ ⋅ s) + R2[h, b](s ⋅ (s ⋅ ∇) ∇b) ,

T where s = (−sx, sy) . R1 and R2 are diﬀerential operators acting on scalar ﬁelds:

1 h R [h, b]w = − ∇ h3w − w∇b, (1.12) 1 3 2 Introduction 10

and

1 R [h, b]w = ∇ h2w − w∇b. (1.13) 2 2

These equations have two advantages compared to others: (a) They only have spatial derivatives up to second order while other equations usually include spatial derivatives of third order, and (b) they do not have any additional source term in the conservation of mass.

These properties improve the computational stability and robustness of the model. While using the SGN equations we can simulate waves with shorter wavelengths, solving these equations is associated with finding a solution to a linear system with 4 × nx × ny unknowns which makes them computationally more expensive in comparison to SWE. This computational cost can become significant with further refining the grid in order to properly resolve the effects of shorter waves on longer waves.

1.2 Contributions and Outline of the Dissertation

The remaining of this dissertation is organized as follows:

• In chapter two, a study on the motion and dislodgement of boulders by solitary waves

is presented. The results are obtained using a three-dimensional SPH method (a mesh-

free method). This is the ﬁrst time that three-dimensional numerical simulations of

boulder transport were made. The results show that nonlinear processes are important.

It is important to note that these nonlinear processes can only be captured with numer-

ical models. Proposed analytical approaches simplify the boulder transport problem. Introduction 11

20 20 0 0

200 0 a-I a-II

Elevation (m) 400 20 20 0 0

200 0 b-I b-II

Elevation (m) 400 20 20 0 0

200 0 c-I c-II

Elevation (m) 400 20 20 0 0

200 0 d-I d-II

Elevation (m) 400 20 20 0 0

200 0 e-I e-II

Elevation (m) 400 20 20 0 0

200 0 f-I f-II

Elevation (m) 400 20 20 0 0

200 0 g-I g-II

Elevation (m) 400 10 0 10 20 30 40 Distance (km)

Figure 1.5: Tsunami wave propagation caused by a landslide. The initial ratio between the wavelength and water depth is more than 10. The ﬁgure shows the comparison of the three diﬀerent simulations. (blue) SGN high resolution; (red) SGN low resolution; (green) SWE high resolution. Introduction 12

From a physical point of view, waves transform into a three-dimensional ﬂow ﬁeld in

the vicinity of the shoreline by breaking or by the interaction with boulders. The main

conclusion of our study is that, there is no linear trend in the parameter space other

than the boulders move larger distances for larger values of the parameters. Further-

more, this project showed that while it is important to simulate the problem using a

three-dimensional scheme, computational cost of a three-dimensional simulation make

it impractical for a large-scale real-world problem.

• In chapter three, we present an extensive comparison of the results obtained using

depth-integrated equations to the existing three-dimensional and experimental data.

Three-dimensional simulation relies on less number of assumptions. In addition, they

can fully resolve the coherent turbulent structures of the ﬂow. However, given the

domain size of the numerical geophysical problems we are interested in solving, i.e.

lx × ly ∼ 1000 km × 1000 km, in addition to the limits imposed on grid resolution by the

current computational facilities, we argue that it is not feasible to simulate long waves,

such as tsunamis and storms, in three dimensions. We demonstrate that using depth-

integrated equations, we can simulate the problems using ﬁner grids which results in

more accurate results.

• In chapter four, we present numerical simulation of nonlinear long waves interacting

with arrays of emergent cylinders. Our model employs the fully nonlinear and weakly

dispersive Serre-Green-Naghdi equations (SGN) until the breaking process starts, while

we changed the governing equations to nonlinear shallow water equations (NSW) at the

vicinity of the breaking-wave peak and during the run-up stage. Using this approach,

we avoid the numerical oscillations that can be caused by dispersive terms in SGN in the Introduction 13

vicinity of the shoreline and the coastal areas. In the literature, coastal vegetation (here

cylinders) is usually approximated as macro-roughness friction. We model the cylinders

as physical boundaries. This eliminates the approximation associated with deﬁning the

friction coeﬃcient. In addition, the presented numerical results are vigorously validated

against existing experimental and numerical data. Given the summary above, we think

our ﬁndings merit a publication in the Marine Geology.

• In chapter ﬁve, we present the risk assessment on the coast along the Gaza strip

imposed by possible earthquakes, landslides, and volcanic eruptions. In addition we will

present some preliminary results on numerical simulation of tsunamis by meteorological

impacts. Chapter 2

Boulder Dislodgement and Transport by Solitary Waves: Insights from Three-Dimensional Numerical Simulations†

†Citation: A. Zainali, and R. Weiss (2015), Boulder dislodgement andtranspor t by solitary waves: Insights from three-dimensional numerical simulations, Geophys. Res. Lett., 42,

44904497, doi:10.1002/2015GL063712.

14 3D Simulation of Boulder Dislodgement 15

Abstract

The analysis of boulder motion and dislodgement provides important insights into the physics

of the causative processes, i.e., whether or not a boulder was moved during a storm or tsunami

and the magnitude of the respective event. Previous studies were mainly based on simpliﬁed

models and threshold considerations. We employ three dimensional numerical simulation of

the hydrodynamics coupled with rigid-body dynamics to study boulder dislodgement and

transport by solitary waves. We explore the eﬀects of three important non-dimensional

parameters on the boulder transport problem, the Froude number F r, the aspect ratio in

logarithmic scale α and the submergence factor, β. Our results indicate that boulder motion and dislodgement is complex, and small changes in one of the non-dimensional parameters result in significantly different behavior during the transport process and the final resting place of boulders. More studies are need to determine the role of boulders in tsunamis and storms hazard assessments.

2.1 Introduction

Boulders can be found along the coastlines of our planet. In general it is assumed only high-energy events, such as storm and tsunami waves, can move these boulders because of their large mass that can exceed 50 tons. Other more common marine or coastal processes are unable to move these boulders (Paris et al., 2010; Barbano et al., 2010; Goﬀ et al., 2006;

Scheﬀers and Scheﬀers, 2006).

Boulders are, therefore, important event deposits that record the frequency and characteris- 3D Simulation of Boulder Dislodgement 16 tics of the storm or tsunami waves moved them. The analysis of boulder movement can be an important component of robust and defensible hazard assessments for future storms and tsunamis. One major problem, however, is that it is very diﬃcult to distinguish boulders moved during storms from those moved during tsunamis, especially in areas where storms and tsunamis are competing processes for boulder movement (Barbano et al., 2010; Noormets et al., 2004). The reason is that boulder displacement depends on many environmental and wave parameters. Among them are wave height and period, shape of the boulder, their initial location, as well as other parameters such as presence of macro-roughness elements Goto et al. (2010b,a). If coastal boulders are included in the statistics describing past tsunamis and storms, to predict future ones, we must be able to determine which event type moved which boulder.

Simplified theories and models has been developed in the past to model boulder transport, i.e. Nott (2003); Weiss (2012); Imamura et al. (2008). From a physical point of view, boulder motion is complex due to the boulder’s complicated geometry, potentially heterogeneous density distribution, and the non-linear interactions between the turbulent flow field and the boulder itself. Numerical models that are capable of handling this complex transport situation have to contain state-of-the art representations of turbulent flow, incorporate cutting-edge numerical solution techniques, and require significant computational resources.

Therefore, there is a significant benefit in simplifying boulder motion to develop simple theories. However, for these theories to be useful, simplifications made of the transport problem must be physically robust and lead to reliable and consistent results. This can be done through rigorous testing and evaluation of these models by more rigorous three-dimensional models. 3D Simulation of Boulder Dislodgement 17

Previous studies commonly assumed that the main cause of boulder movements are large tsunamis (Imamura et al., 2008). In this study, we design a broader framework to inves- tigate boulder dislodgement and transport by solitary waves with a large amplitude range.

Different representations of long waves have been discussed (Madsen et al., 2008; Madsen and Schaeffer, 2010; Tadepalli and Synolakis, 1994). However, we note that solitary wave theory still remains the most common (Tadepalli and Synolakis, 1996; Liu et al., 1995). The interaction between boulders and solitary waves causes boulder dislodgement and movement, and is classified in three different categories: no motion, motion but no dislodgement, and dislodgement (for more details see, Weiss and Diplas, 2015). To solve our problem in a three- dimensional domain, we employ GPUSPH (Héraultet al., 2010; Herault et al., 2006-2014).

GPUSPH is a numerical model that uses Smoothed Particle Hydrodynamics to solve the governing equations of ﬂuid motion.

2.2 Theoretical Background

2.2.1 Governing Equations and Numerical Method

We employ Smoothed Particle Hydrodynamics (SPH) to simulate the ﬂow of water around boulders. The respective conservation of mass and momentum in their weakly compressible form can be written as (Yoshizawa, 1986)

Dρ¯ ∂u˜ + ρ¯ i = 0 (2.1) Dt ∂xi

2 ∂τ ∗ Du˜i 1 ∂p¯ ∂ u˜i 1 ij (2.2) = − − gδi3 + ν + , Dt ρ¯∂xi ∂xj∂xj ρ¯ ∂xj 3D Simulation of Boulder Dislodgement 18

where “∼” refers to the Favre-ﬁltering operator φ˜ = ρφ(ρ)−1. The Large Eddy Simulation

approximation (LES) is employed to consider sub-particle scales. Dalrymple and Rogers

(2006) suggested the following representation of the LES approximation in SPH for the

sub-particle stress tensor: 2 2 τ ∗ = ρ¯2ν S˜ − S˜ δ − ρC¯ ∆2δ (2.3) ij t ij 3 kk ij 3 I ij

in which S˜ = − 1 ∂u˜i + ∂u˜j and C = 1~1500. The turbulent viscosity is calculated using the ij 2 ∂xj ∂xi I

2 standard Smagorinsky model: νt = (Cs∆) SS¯S (Smagorinsky, 1963, Smagorinsky constant is

Cs = 0.12). The parameter ∆ is the initial spacing of the SPH particles. u denotes the

velocity vector, p represents pressure, ρ is the density, ν is the kinematic viscosity, and g

refers to the gravitational acceleration. δ denotes the Kronecker delta.

As mentioned earlier, we employ SPH to solve the aforementioned equations of ﬂuid motion.

Because of its Lagrangian nature, the SPH method is an appropriate method for simulating complicated ﬂow situations. Refer to Dalrymple and Rogers (2006); Weiss et al. (2011);

Shadloo et al. (2015) for more information about water waves modeling using SPH and to

Monaghan (1992, 2005a, 2012) for a more complete description of the SPH method and a comprehensive list of its applications.

To employ the SPH method for our ﬂow situation, we employ GPUSPH. GPUSPH solves the governing equations with the aid of graphical processing units (H´eraultet al., 2010;

Herault et al., 2006-2014). To enforce incompressibility of the flow, GPUSPH employs an artificial speed of sound of c = 10umax such that the density variations in the fluid are limited to about 1%. For all boundaries between fluid and solid, we use the Lennard-Jones type repulsive force (Monaghan, 1994a). GPUSPH is coupled with the Open Dynamics Engine 3D Simulation of Boulder Dislodgement 19

(ODE; Smith, 2006) to simulate the interaction between a rigid body, such as a boulder,

and the ﬂuid. Aside from computing the motion of boulders, the ODE library allows the

inclusion of, for example, Coulomb’s friction law between the boulders and the underlying

boundary surface and solves an equation of motion for rigid body.

2.2.2 Model Setup

The geometrical setup of the computational domain consists of a box with length of lx = 10m,

height of ly = 0.45 m and width of lz = 0.3 m (Figure 2.1). We included a sloping beach

(S=1:10) whose toe is located at lxh = 5.5 m. The water depth is denoted with hw. We

consider two diﬀerent scenarios: A dam-break problem for validation, and solitary wave

runup for long waves. The gate that is lifted to create the dam break is located at lw.

y lz Gate Boulder

ly hw . x z lw lxh lx

Figure 2.1: Sketch of dam-break scenario at t = 0. Water behind the gate starts to ﬂow after sudden gate removal at t = 0. To compare the numerical results with experiments presented by Imamura et al. (2008), we applied: lx = 10m, ly = 0.45m, lz = 0.3m, lxh = 5.5m, lw = 3 m and hw = 0.15, 0.20, 0.25, 0.3 m.

To create solitary waves, we employ the wavemaker function as deﬁned by Goring (1978), » » for which the wavenumber is κ = (3η)(4h3)−1 and the phase speed is c = g(h + η) where

1 3D Simulation of Boulder Dislodgement 20

η is the wave amplitude.

2.2.3 Non-Dimensional Parameters

The one-dimensional shallow water equations in the absence of topographical variations

are known to predict the wave runup accurately (Li and Raichlen, 2002; Synolakis, 1987).

However, in the presence of topographical variations, ﬂow ﬁeld in its vicinity becomes three-

dimensional violating the one-dimensional ﬂow assumption.

For a brief theoretical analysis, we will continue with the one-dimensional shallow water

equations to drive the important non-dimensional parameters controlling the main wave

characteristics. Later on, we will introduce other important parameters that can character-

ize the three-dimensional eﬀects due to the presence of topographical variations, namely a

boulder in our study. One-dimensional shallow water equations can be written in the form

of: (h)t + (hu)x = 0, 1 (2.4) (hu) + hu2 + gh2 = −ghb . t x 2 x x

in which h is the water depth, u represents the velocity, and the bottom topography is

denoted by b. The variables and parameters x, t, h, u and b are normalized with xref , tref , href , uref and bref . With the help of these normalizations, Eq. (2.4) can be rewritten to:

∗ ∗ ∗ St (h )t∗ + (h u )x∗ = 0, ∗ ∗ ∗ ∗2 1 ∗2 β ∗ ∗ (2.5) St (h u ) ∗ + h u + h = − h b ∗ . t x∗ 2F r2 x∗ F r2 x where x u b St = ref , F r = » ref , β = ref (2.6) tref uref ghref href 3D Simulation of Boulder Dislodgement 21 are the Strouhal number, Froude number, and submergence factor which is the ratio between reference height and reference topographical variations. Superscript ∗ refers to normalized

−1 −1 parameters. Here we choose the following reference values: xref = κ , tref = (cκ) , href = hw, uref = c and bref = lby, where lby denotes boulder height and c and κ denote phase speed and wave number.

For simplicity, we will only consider boulders with constant width, i.e., lbz = constant.

−1 Keeping the volume of the boulder constant, we deﬁne a parameter α = log2(lbylbx ), the aspect ratio in logarithmic scale of the boulder length in the x-y plane, as another important non-dimensional parameter. Boulder density can alter dislodgement process signiﬁcantly.

−1 As a result, the ratio between the boulder density and water density, ρbρw , will play an important role in this problem as well.

Due to our simplifying assumptions compared to the real ﬂow situation, it is impossible to match every possible non-dimensional parameter with real-world data. For our analysis, we will continue with the Froude number, submergence factor and aspect ratio in logarithmic scale. We note that the Froude number is an estimate of the wave energy. The potential

2 energy of a wave is given by Ep = 0.5ρwgη where η is the wave surface elevation and overbar denotes the averaged value over the wave period. Assuming that the wave motion is under quasi equilibrium condition, using the equipartition theory we can assume that kinetic energy

Ek = Ep. Total energy becomes equal to Et = 2Ep. From the Froude number deﬁnition and

2 λ solitary wave theory we can approximate Et = ζρwg(F r − 1) where ζ and λ are constants.

The α and β parameters provide us with necessary information regarding the location of the boulder with respect to incoming water wave. Due to simplicity of the geometrical shape we 3D Simulation of Boulder Dislodgement 22

have chosen here, α parameter is intended to capture three-dimensional eﬀects. However, for more complicated geometrical shapes, other non-dimensional parameters may be needed to fully characterize shape-dependent processes. One example would be the orientation of the boulder with respect to incident wave direction.

2 ) max ( b d 1

0 0.0 0.1 0.2 0.3 0.4 h

Figure 2.2: Comparison of numerical and experimental data presented. Experimental data −3 are from Imamura et al. (2008). The case ρb = 1550 kg m is colored in black and case with −3 ρb = 2710 kg m is colored in blue; ( ) experimental results and ( ) SPH simulations. 3D Simulation of Boulder Dislodgement 23

2.3 Results

2.3.1 Validation of Numerical Results

We validate the employed numerical model with experiments carried out by Imamura et al.

(2008). These experiments are based on a dam-break scenario. The dam break is located

at lw = 3 m, and the water depth varies from hw = 0.15 m to hw = 0.30 m in steps of ﬁve

centimeters. The boulders have a constant size with 0.032×0.032×0.032m3, but two diﬀerent

1 −3 2 −3 densities, ρb = 1550 kg m and ρb = 2710 kg m , are applied.

Figure 2.2 shows comparisons for the distance, db, the boulder traveled from its original

1 location between experimental and simulated data. For a density of ρb , the comparison is

2 excellent (black color). However, for the density of ρb , there is a noticeable difference between experimental and numerical results. This difference is due to the constant friction coefficient

in the numerical model, while in experiments, diﬀerent scenarios come with diﬀerent surface

roughness values associated with diﬀerent materials, however, the exact value of friction

coeﬃcient is not reported for the scenarios. Despite this diﬀerence, we argue that GPUSPH

coupled with the ODE library capture all processes important for boulder transport. 3D Simulation of Boulder Dislodgement 24

(a) t = 0 (s) (b) t = 0 (s)

Case CA(α=0) Case CA(α=1)

Case CB(α=0) (c) t = 4.7 (s) Case CB(α=1) (d) t = 4.7 (s) Case CC(α=0) Case CC(α=1)

(e) t = 4.9 (s) (f) t = 4.9 (s)

(g) t = 5.1 (s) (h) t = 5.1 (s)

1 Velocity magnetiude [ms− ]

Figure 2.3: Time snapshots for scenarios CA, CB and CC with α = 0, (left column) and scenarios CA, CB and CC with α = 1, (right column) with wave height of 0.15 m. Please note that diﬀerent cases are superimposed on the same domain just for illustration purposes. 3D Simulation of Boulder Dislodgement 25

2.3.2 Boulder Transport by Solitary Waves

For our numerical experiments, we assume a water depth of hw = 0.3m. Although the selected waves are not a rigorous representation for the entire wave spectrum, in terms of wave shape and period, they provide an appropriate wave energy range to study the primary factors of the dislodgement process. We first created a series of subsequent solitary waves in order to see if a boulder can be moved by two or three subsequent waves for which the amplitude of each individual wave is too small to cause boulder transport. It turns out, however, as long as the first wave does not cause any movement, the effects of subsequent waves can be ignored. Therefore, boulder dislodgement can be compared on a wave by wave basis.

For more detailed analysis, we varied the aspect ratio in logarithmic scale of the boulder, submergence of the boulder and the Froude number, but all boulders have the same mass

(constant density, ρ = 2000 kg m−3.) Table 2.1 summarizes the different parameters and defines the different scenarios that are considered in our study.

Table 2.1: Physical and numerical simulation parameters (hw = 0.3 m. hb represents the depth of the water at the location of boulder. c is√ the solitary wave celerity). Submergence factor: β = hb (y axis); Froude number: F r = c( gh )−1; boulder aspect ratio in logarith- hw w −1 −1 mic scale: α = log2(lbylbx ) [x − y plane; boulder normalized width: Wb = lbyhw ; boulder −3 −1 normalized volume: Vb = lbxlbylbzhw ; density ratio: ρbρw ; friction coeﬃcient: µ;

Scenario: CA CB CC CD Submergence factor: −8~30 −4~30 0 4~30 Froude number: 1.08, 1.15, 1.22 1.08, 1.15, 1.22 1.08, 1.15, 1.22 1.08, 1.15, 1.22 boulder aspect ratio: −1, 0, 1 −1, 0, 1 −1, 0, 1 −1, 0, 1 boulder normalized width: 4~30 4~30 4~30 4~30 boulder normalized volume: 43~303 43~303 43~303 43~303 density ratio: 2 2 2 2 friction coeﬃcient: 0.6 0.6 0.6 0.6 3D Simulation of Boulder Dislodgement 26

For the numerical experiments, we define four different scenarios that vary the initial location of the boulder with regard to the still water shoreline. We employ the submergence factor to define the location from the shoreline. For scenario CA, the boulder is farthest in the water and will remain fully submerged for all aspect ratios in logarithmic scale. For scenario CB, the boulder is submerged for an aspect ratio in logarithmic scale of -1. However, for α = 1, the boulder is only partially submerged in water. The boulder in scenario CC is located directly at the shoreline, and CD defines a scenario in which boulder is initially located on dry land. In our numerical experiments, we apply three aspect ratios in logarithmic scale,

−1, 0, and 1, as well as three diﬀerent Froude numbers, 1.08, 1.15, and 1.22, to each scenario.

The Froude numbers presented here are consistent with ﬁeld observations (Etienne et al.,

2011; Jaﬀe et al., 2011; Fritz et al., 2006; Spiske et al., 2010). In terms of scaling up to the prototype scale, assuming a coastal water level of 30 m, our experimental scale represents approaching wave heights between 3.3 and 10 m resulting in a boulder with the volume of

18 m3 and a boulder mass of approximately 36 × 103 kg.

Figure 2.3a shows the initial position of the boulders for an aspect ratio in logarithmic scale of α = 0, and Figure 2.3b for an aspect ratio in logarithmic scale of α = 1. Diﬀerent cases are plotted together just for better illustration. The subsequent subplots show the impact of a wave on the diﬀerent boulders (Figure 2.3c,e,g for α = 0 and Figure 2.3d,f,h for α = 1).

We see that for α = 0 only the boulder CC moves visibly. Boulders CA and CB did not.

For the aspect ratio in logarithmic scale α = 1, all boulders, initially, ﬂip over due to their unstable position; but again, only boulder CC moves shoreward from its original location. It should be noted that for an aspect ratio in logarithmic scale of α = −1, none of the boulders moved. Boulders can saltate, rotate, slide, or a combination of all three (Nandasena et al., 3D Simulation of Boulder Dislodgement 27

2011b,a; Imamura et al., 2008). Nandasena and Tanaka (2013), however, noted that the dominant transport mode is sliding. Our numerical modeling confirms this observation. We also find that rotation can play an important role in boulder transport in the very early stage of the transport process. It should be noted that if rotation occurs even in the early stages of boulder transport, transport paths of the boulders are very different especially for the cases with α >> 0, as highlighted by red and green circles. Boulders highlighted by the red circles start to move but do not dislodge, while boulders highlighted by green circles dislodge completely.

Boulder movement can be characterized in three diﬀerent distinct modes: no motion, motion but no dislodgement, and dislodgement. Figure 2.4a-c contains the information on these modes for F r, α and β plotted against each other (Figure 2.4a: F r vs α, b: F r vs β, and c:

α vs β). The white areas in Figure 2.4a-c mark no motion in which the normalized distance that boulder travels is dblbz < 0.1, the gray color indicates motion but no dislodgement

(0.1 < dblbz < 2), and the contours represent dislodgement. For the latter, it should be noted that the darker the color, the larger the distance a boulder travels from its original position.

Boulder motion and dislodgement is achieved easily if all three non-dimensional parameters are large.

With increasing F r number, the total energy the wave carries increases. As a result, the distances boulders travel increase with increasing F r number. In our numerical experiments, the waves break in the vicinity of shoreline. After breaking, the ﬂow in the wavefront becomes supercritical with signiﬁcantly larger local Froude numbers, compared to average

Froude numbers of the wave before breaking (1.0 < F r < 3.0; Matsutomi et al., 2001). After 3D Simulation of Boulder Dislodgement 28 wave breaking and as the broken wave propagates shore-ward, the flow becomes shallower and starts to accelerate. Due to this acceleration, boulders located at β = 4~30 travel larger distances compared to boulders located at β = 0, and boulders located at β = 0 travel larger distances compared to boulders located at β = −4~30. However, with increasing β, gravity and friction forces in the opposite direction of flow will start to dominate the flow after a critical value for β is exceeded. Thus the flow will start to decelerate, and as a result, the distance boulders travel will decrease. Finally, as we mentioned earlier, rotational movement becomes very important for boulders with α >> 0 which, generally, results in larger travel distance.

2.4 Discussion and Conclusions

In this contribution, we studied the motion and dislodgement of boulders by solitary waves with the state-of-the-art hydrodynamic model GPUSPH. This is the first time that three dimensional numerical simulations of boulder transport were made. The results show that nonlinear processes are important. It is important to note that these nonlinear processes can only be captured with numerical models. Proposed analytical approaches simplify the boulder transport problem. From a physical point of view, waves transform into a three- dimensional flow field in the vicinity of the shoreline by breaking or by the interaction with boulders.

The solitary waves in our numerical experiment can be interpreted as storm and tsunami waves, based on the total energy they carry. Without loss of generality, we can assume that small solitary waves represent storm waves, and large solitary waves represent tsunami 3D Simulation of Boulder Dislodgement 29 waves. Because the ﬂow around boulders is three dimensional, we focus on the local energy and not, for example, solely on the wave speed or amplitude. We employed the Equipartition

Theory to argue for a spectrum of energy that storm and tsunami waves can have and not discrete sets of wave amplitudes and periods.

We considered boulders with very simple geometries and a homogeneous density distribution.

With the help of dimensional analysis, we identiﬁed the aspect ratio in logarithmic scale, the submergence factor and the Froude number as crucial parameters of boulder transport.

Figure 2.4a-c depicts the result of this parameter study. From these subfigures, we can see that there is no consistent trend other than boulders are transported larger distances for larger values of the different parameters. However, there is no systematic trend in the way contours exist in this parameter space. We note here that considering more realistic scenarios, predicting the distance a boulder can travel is even more difficult. A typical example of such a scenario would be a boulder with heterogeneous rather than homogeneous density distribution or a scenario in which boulder is oriented at an angle to the incident wave direction. Including these effects will result in the identification of more critical non- dimensional parameters coming from more complete theoretical analysis. More research is needed to identify the structure of these additional parameters. 3D Simulation of Boulder Dislodgement 30

moving (a) 0.8 a-I a-II a-III transition 0.4

α 0.0 at rest 0.4 − 0.8 − 1.09 1.13 1.17 1.21 1.09 1.13 1.17 1.21 1.09 1.13 1.17 1.21 F r F r F r

(b) 0.1 b-I b-II b-III

0.0 β 0.1 −

0.2 − 1.09 1.13 1.17 1.21 1.09 1.13 1.17 1.21 1.09 1.13 1.17 1.21 F r F r F r

0.0 β 0.1 −

0.2 − 0.8 0.4 0.0 0.4 0.8 0.8 0.4 0.0 0.4 0.8 0.8 0.4 0.0 0.4 0.8 − − − − − − α α α Normalized maximum boulder displacement

2 4 6 8 10 16 22 28 34

Figure 2.4: Contour plots of boulder maximum displacement, db, as a function of non- dimensional parameters. a: aspect ratio in logarithmic scale versus Froude number; (a-1) β = −8~30 (a-2) β = 0 and (a-3) β = 4~30. b: submergence factor versus Froude number; (b-1) α = −1, (b-2) α = 0 and (b-3) α = 1. c: submergence factor versus aspect ratio in logarithmic scale; (c-1) F r = 1.08, (c-2) F r = 1.15 and (c-3) F r = 1.22. White regions indicate no −1 signiﬁcant boulder movement, i.e. dblbz < 0.1. For gray regions, the distance boulder travels −1 is in between 0.1 < dblbz < 2.0. Color contours indicate boulders moved signiﬁcantly, i.e. −1 dblbz > 2.0 (colormap is in logarithmic scale)

More ﬁeld studies are needed to see how the submergence, for example, can be better esti-

mated. Therefore, three-dimensional simulations, such as ours, help to identify important

gaps in understanding boulder transport and eventually will help to narrow the gap be-

tween field and theory based research. The main conclusions of our study are that firstly, 3D Simulation of Boulder Dislodgement 31 the flow around a boulder needs to be considered in a three-dimensional fashion for which numerical simulations are needed. Second, there is no linear trend in the parameter space other than the boulders move larger distances for larger values of the parameters. The contours in the parameter space have complex and nonlinear shapes. These two conclusions reveal shortcomings of the presently used standard methods and show that more work with three-dimensional simulations is needed to successfully employ boulders as event deposits in storm and tsunami hazard assessments. Perhaps the inevitable conclusion is that such three-dimensional simulations, including variables such as angle of incident wave or heterogeneous mass distribution within the boulder, are the only credible and reliable approach to improve our understanding of boulder dislodgement. Given the model complexities as we observed earlier, it remains to be seen what role boulders can play in storms and tsunamis hazard assessment. Chapter 3

High-Fidelity Depth-Integrated Numerical Simulations in Comparison to Three-Dimensional Simulations

32 2D vs 3D 33

Abstract

In this study, we present the numerical simulation of the breaking solitary wave run-up and non-breaking wave interaction with oﬀshore cylinders using Serre-Green-Naghdi equations.

We compared the numerical results of our model to those obtained by a three-dimensional volume of ﬂuid method (VOF), and a mesh-free three-dimensional smoothed particle hydrodynamics (SPH) method and existing experimental data. Our results suggest that depth- integrated equations can produce results as accurate as three-dimensional schemes while being computationally more eﬃcient.

3.1 Introduction

Multiphase flow, where two or more fluid have inter-facial surfaces, is one of the challenging and difficult areas in the field of Computational Fluid Dynamics (CFD), which plays an important role in many industrial and natural systems. A free-surface flow can be considered as a subclass of multiphase flows for which we can neglect the stresses at the interface of the two interacting-fluid flows. The density and viscosity ratio between water and air are large and the surface tension coefficient is small between the two fluids. In addition, the

Reynolds number of the oceanic waves are large. Thus, free-surface ﬂow assumption is a good representation of ﬂow behavior in tsunami and storm waves.

In the following paragraphs, we will describe the diﬀerent numerical schemes used for numerical simulation of free-surface ﬂows. 6

2D vs 3D 34 4

2 R=0.61 (0.0,0.0)

Wave0 Direction Gauge 1 Gauge 2 Gauge 6 2R Gauge 4 2 − Gauge 3 Gauge 5

4 2.43 m −

6 − 10 5 0 5 Figure 3.1: A schematic− sketch of the solitary− wave passing through and around cylinders. The coordinate of the domain is located at the center of the cylinder on the right hand side of the domain. Wave propagates along the x-axis. The red dots represent the location of the center of the cylinders. The blue dots represent the location of the wave gauges. Gauge 1: (-3.04 m, -0.14 m); Gauge 2: (-1.82 m, -0.14 m); Gauge 3: (-0.83 m, 0.00 m); Gauge 4: ( 0.00 m, -0.88 m); Gauge 5: ( 0.85 m, 0.00 m); Gauge 6: ( 1.82 m, 0.00 m);

3.1.1 Three-Dimensional Methods

Mesh-Dependent Methods

Multiphase flow problems, so far, have been studied widely using mesh dependent techniques. Nevertheless, because of the complexity of these problems mainly associated with the necessity of tracking interface evolution, most of the current works have not gone beyond the simple problems. As can be inferred, the interface evolution is crucial to modeling of multiphase flows and thus, needs to be modeled correctly and carefully in order to obtain reliable simulation results. In mesh-dependent methods, an additional set of equations has to be solved to track inter-facial surfaces, and furthermore, depending on the problem in 2D vs 3D 35 hand (i.e., if the the topology of the flows deforms significantly), mesh-refinement might be required. In mesh-dependent methods, different techniques have been proposed and studied in literature to capture the interface evolution on a regular grid. Among them, Volume of

ﬂuid (VOF), (e.g., Hirt and Nichols, 1981), and Level set methods (LS), (e.g., Sethian and

Smereka, 2003) are Eulerian methods widely used in computational ﬂuid dynamics. These methods are generally associated with the diﬃculties in handling large topological deforma- tion. Another category belongs to Lagrangian-Eulerian hybrid methods, (e.g., Tryggvason et al., 2001; Unverdi and Tryggvason, 1992) in which external elements or markers are used to track the interface explicitly. The main advantage of these methods over the Eulerian coun- terparts is their ability of tracking interfaces in a sharper manner. However, front tracking methods are in general computationally more expensive than the Eulerian methods.

Mesh-Free Methods

An alternative to the above-mentioned methods can be purely Lagrangian methods. The

Lagrangian nature makes them potentially better candidates for tracking the interfaces with large deformations. Being a well-advanced member of Lagrangian methods, the SPH technique due to its Lagrangian nature in particular is an excellent candidate to model complex multiphase flow problems. It offers a simplified approach for tracking the interface evolution as well as incorporating the surface tension force into the linear momentum balance equations. 2D vs 3D 36

1.5 3D-VOF (a) Gauge 1 (d) Gauge 4 2D-SGN Experiment ζ∗ 0.5

0.5 −1.5 (b) Gauge 2 (e) Gauge 5

ζ∗ 0.5

0.5 −1.5 (c) Gauge 3 (f) Gauge 6

ζ∗ 0.5

0.5 − 5.0 2.5 10.0 5.0 2.5 10.0 − − t∗ t∗

Figure 3.2: Comparison of numerical and experimental» data presented. Experimental and three-dimensional simulations are from Mo (2010); t∗ = t~ h~g, and ζ∗ = ζ~H;( ) present simulation (2D-SGN), ( ) experimental results and ( ) three-dimensional simulation (3D- VOF).

Smoothed Particle Hydrodynamics (SPH) is one of the members of meshless Lagrangian particle methods used to solve partial diﬀerential equations widely encountered in scientiﬁc and engineering problems (e.g., Fang et al., 2006; Melean et al., 2004; Tartakovsky and Meakin,

2005; Cleary et al., 2002). Unlike Eulerian (mesh-dependent) computational techniques such as finite difference, finite volume and finite element methods, the SPH method does not re- 2D vs 3D 37 quire a grid, as field derivatives are approximated analytically using a kernel function. In this technique, the continuum or the global computational domain is represented by a set of discrete particles. Here, it should be noted that the term particle refers to a macroscopic part (geometrical position) in the continuum. Each particle carries mass, momentum, energy and other relevant hydrodynamic properties. These sets of particles are able to describe the physical behavior of the continuum, and also have the ability to move under the influence of the internal/external forces applied due to the Lagrangian nature of SPH. Although orig- inally proposed to handle cosmological simulations (Gingold and Monaghan, 1977; Lucy,

1977), SPH has become increasingly generalized to handle many types of ﬂuid and solid mechanics problems (e.g., Monaghan, 2005b; Sigalotti et al., 2003; Raﬁee and Thiagarajan,

2009; Liu et al., 2003; Hosseini et al., 2007; Rook et al., 2007; Tartakovsky et al., 2007). The

SPH method has recently received a great deal of attention for modeling multiphase ﬂow problems (e.g., Monaghan, 1994b; Monaghan and Kocharyan, 1995; Monaghan et al., 1999;

Landrini et al., 2007; Ferrari et al., 2010; Tartakovsky et al., 2009; Fang et al., 2006, 2009;

Zhang, 2010; Colagrossi and Landrini, 2003) owing to its obvious advantages such that it notably facilitates the tracking of multiphase interfaces and the incorporation of inter-facial forces into governing equations, allows for modeling large topological deformations in ﬂow, and does not require connected grid points for calculating partial diﬀerential terms in governing equations. Out of the above cited excellent multiphase SPH studies, works on 2D simulations of splashing processes and near-shore bore propagation (Landrini et al., 2007), and a 3D dam breaking simulations (Ferrari et al., 2010) deserve particular mention in that they have attempted to illustrated the true power and the reliability of the multiphase SPH method as a viable CFD approach by validating their results with experiments. 2D vs 3D 38

1.5 3D-VOF (a) Gauge 1 (d) Gauge 4 2D-SGN Experiment ζ∗ 0.5

0.5 − 1.5 (b) Gauge 2 (e) Gauge 5

ζ∗ 0.5

0.5 − 1.5 (c) Gauge 3 (f) Gauge 6

ζ∗ 0.5

0.5 − 5.0 2.5 10.0 5.0 2.5 10.0 − − t∗ t∗

Figure 3.3: Comparison of numerical and experimental» data presented. Experimental and three-dimensional simulations are from Mo (2010); t∗ = t~ h~g, and ζ∗ = ζ~H;( ) present simulation (2D-SGN), ( ) experimental results and ( ) three-dimensional simulation (3D- VOF).

3.1.2 Depth-Integrated Methods

Three-dimensional schemes can represent the ﬂow behavior very accurately. However, the scale of the geophysical problems we are interested to solve is very large. In addition, tracking the interface accurately is a challenging problem and requires a very ﬁne resolution, 2D vs 3D 39

especially around the interface, in order to make sure that the conservation of mass is properly

preserved. Depth-integrated equations are a very popular alternative to solving the Euler

equations since these equations increase the computational eﬃciency signiﬁcantly. These

class of equations reduce the dimension of the governing equations from three to two by

integrating the velocity along the vertical axis and assuming that the ﬂow acceleration is

zero or negligible along z-axis.

The nonlinear shallow water equations are the most common form of the depth-integrated

equations used in the literature for simulating the long waves such as tsunamis and storms.

These equations assume that the ﬂow velocity is constant along the vertical direction. This

assumption makes them only suitable for waves with very long wavelengths (i.e. λ~h0 >>

1). Diﬀerent classes of higher-order depth-integrated equations, based on Boussinesq wave theory, have been derived and presented in literature on simulating shallow water ﬂows:

Wei et al. (1995); Liu (1994); Lannes and Bonneton (2009); Dutykh and Dias (2007); Kirby

(2003); Yamazaki et al. (2009); Ma et al. (2012). These equations take into account the small variations in velocity along the vertical axis.

3.2 Results

3.2.1 Non-Breaking Solitary Wave Interaction with a Group of Cylinders

Solitary wave interaction with a group of three vertical cylinders is investigated by Mo and

Liu (2009) in which they performed three-dimensional numerical simulations using volume of 2D vs 3D 40

ﬂuid method (VOF) and compared their numerical results to experiments. We compare our

results from two-dimensional SGN with the experiments and the three-dimensional VOF » 3 simulations by Mo and Liu (2009). A solitary wave with H = 0.3h0, k0 = 3H~4h0, and x0 = 15 m inside a rectangular domain of (48.8 m, 26.5 m) interacts with some cylinders.

The diameter of cylinders is 1.22 m and they are located at (27.7 m, 0 m), (30.1 m, 1.2 m)

and (30.1 m, −1.2 m). The still water depth is h0 = 0.75 m. Uniform rectangular grid with

∆x~L0 = 1024 is used where L0 = 48.8 m is the length of the computational domain.

z x

Figure 3.4: Schematic of the solitary wave run-up on a sloping beach with the slope of 1:19.85.

Table 3.1: Running time for three-dimensional SPH simulation, and one-dimensional SGN simulation of solitary wave run-up on a sloping beach.

3D-SPH 1D-SGN 2D-SGN Processing time: ≈ 3 day ≈ 10 seconds ≈ 10 minutes Hardware cost: 1 Tesla GPU(≈5000$) 1 Xeon CPU(≈500$) 1 Xeon CPU(≈500$)

Figures 3.2, and 3.3 depict the comparison of the wave elevation at diﬀerent wave gauges

obtained using SGN with those of Mo (2010) for one (only kept the cylinder in (0.0, 0.0) in

Figure 3.1) and three cylinders, respectively. The results are in excellent agreement except

for the wave gauges immediately in front and back of the ﬁrst cylinder (see Figures 3.2(c-e) a) a)

a) a)

2D vs 3D 41 and 3.3(c-e)). Our model overestimates the leading wave height in front and underpredicts the wave height in back of the first cylinder. Similar errors are reported between experiments and Boussinesq models in Zhong and Wang (2009). These differences are due to the fact that the three-dimensional effects of the flow become more relevant in close proximity to the cylinders. Using SGN, secondary wave elevations are closer to the experimental data than the three dimensional simulation. This demonstrates the capability of SGN to simulate the effects of shorter waves that can become very important in the near shore regions.

0.4 t∗ = 10 t∗ = 15

ζ∗0.2

0.0

0.4 t∗ = 20 t∗ = 25

ζ∗0.2

0.0

0.4 t∗ = 35 t∗ = 50

ζ∗0.2

0.0 30 20 10 0 10 30 20 10 0 10 − − − − − − x∗ x∗

Figure 3.5: Breaking solitary wave run-up compared to experimental results of Synolakis (1987) and three-dimensional GPUSPH simulation of Marivela et al. (Under review). ( ) present simulation (2D-SGN), ( ) experimental results and ( ) GPUSPH. 2D vs 3D 42

3.2.2 Breaking Solitary Wave Run-Up on a Sloping Beach

In this section, we compared the run-up of a breaking solitary wave on a sloping beach to

the experimental results of Synolakis (1987), and to the three-dimensional SPH simulation

presented in Marivela et al. (Under review).

The schematic of the problem is shown in Figure 3.4. The sloping beach’s toe, with the » 3 slope of 1:19.85, is located at 3.64 m. A solitary wave with H = 0.28h0, k0 = 3H~4h0 starts

propagating toward the beach at t = 0 s.

Figure 3.5 compares the obtained results using SGN equations to the experimental data

and the three-dimensional SPH simulations. We can see that our results are closer to the

experimental data when compared to the SPH simulations. Table 3.1 summarizes the com-

putational time obtained using one-dimensional SGN equations to three-dimensional SPH

simulation. We can see that the one-dimensional simulation is, approximately, four orders

of magnitude faster. We note here that two-dimensional SGN simulation will be faster by

only two orders of magnitude.

3.2.3 Breaking Solitary Type Wave Run-Up on a Sloping Beach

We investigated solitary-type transient wave run-up on a steep sloping beach (1:10) following

the experimental setup in Irish et al. (2014). We discretized the computational domain (52

m, 4.4 m) using a rectangular uniform grid with ∆x = ∆y = 52~1024 m. The toe of the sloping beach is located at x = 32 m. The still water depth is h0 = 0.73 m. The transient

−1 solitary wave is generated by setting H = 0.50 m, k0 = 0.54 m , and x0 = 10 m at t = 0 s. A 2D vs 3D 43

symmetry boundary condition is imposed on bottom and top boundaries.

0.6 (a) Gauges 1-2 (b) Gauges 3-4

0.4 ζ

0.2

0.0

2 6 10 14 2 6 10 14 t t

Figure 3.6: Free-surface elevation at (a) gauges 1-2, and (b) gauges 3-4. Experimen- tal results are summarized in Yang et al. (2016); ( ) present simulation (2D-SGN), ( ) experimental results and ( ) GPUSPH.

Yang et al. (2016) studied this problem experimentally. We compared the free surface el-

evation (wave gauges 1-4) and the local water depth (wave gauges 5-16) with the results

presented in Yang et al. (2016) as well as to the three-dimensional simulations obtained

using GPUSPH code. The wave gauge coordinates are summarized in Table 4.1. Figure 4.3

shows the free-surface elevation ζ at gauges 1-4. While both methods capture the evolution

of the free surface accurately, three-dimensional simulations are closer to the experimental

data for the oﬀshore wave gauges.

Figure 4.4 depicts the local water depth at onshore gauges 5-16. Our results are systemati-

cally closer to the experimental data. While GPU simulations were closer to the experimental

data, the three-dimensional simulations fail to provide more accurate predictions during the

run-up stage. This is mainly due to the insuﬃcient resolution at the run-up stage imposed 2D vs 3D 44

by the hardware and computational time restrictions.

0.150 (a) Gauge 5 (b) Gauge 6 (c) Gauge 7 (d) Gauge 8

h0.075

0.000 0.150 (e) Gauge 9 (f) Gauge 10 (g) Gauge 11 (h) Gauge 12

h0.075

0.000 0.150 (i) Gauge 13 (j) Gauge 14 (k) Gauge 15 (l) Gauge 16

h0.075

0.000 11 13 15 17 19 11 13 15 17 19 11 13 15 17 19 11 13 15 17 19 t t t t

Figure 3.7: Local water depth at gauges 5-16. Experimental results are summarized in Yang et al. (2016); ( ) present simulation (2D-SGN), ( ) experimental results and ( ) GPUSPH. Chapter 4

Numerical Simulation of Nonlinear Long Waves in the Presence of Discontinuous Coastal Vegetation †

†Citation: A. Zainali, R. Marivela, R. Weiss, J. L. Irish, Y. Yang, Numerical simulation of nonlinear long waves interacting with arrays of emergent cylinders; arXiv:1610.00687

[physics.geo-ph].

45 Macro-Roughness Vegetation 46

Abstract

We presented numerical simulation of long waves, interacting with arrays of emergent cylinders inside regularly spaced patches, representing discontinues patchy coastal vegetation. We employed the fully nonlinear and weakly dispersive Serre-Green-Naghdi equations (SGN) until the breaking process starts, while we changed the governing equations to nonlinear shallow water equations (NSW) at the vicinity of the breaking-wave peak and during the runup stage. We modeled the cylinders as physical boundaries rather than approximating them as macro-roughness friction. We showed that the cylinders provide protection for the areas behind them. However they might also cause ampliﬁcation in local water depth in those areas. The presented results are extensively validated against the existing numerical and experimental data. Our results demonstrate the capability and reliability of our model in simulating wave interaction with emergent cylinders.

4.1 Introduction

Tsunamis, caused by different events such as earthquakes and landslides, and storms pose significant threat to human life and offshore and coastal infrastructure. The effects of coastal vegetation on long waves have been investigated extensively in the past (e.g. Mei et al., 2011;

Anderson and Smith, 2014). It has been considered that continuous vegetation provides protection for the areas behind them (e.g. Irtem et al., 2009; Tanaka et al., 2007). However, limited studies focused on the propagation and run-up of long waves in the presence of discontinuous vegetation (see Irish et al., 2014). The following question arises: Do discontinuous Macro-Roughness Vegetation 47 arrays of cylinders, representing coastal vegetation such as mangroves, coastal forests, and man-made infrastructure, act as barriers or as ampliﬁers for energy coming ashore during coastal ﬂooding events?

Three dimensional numerical simulation is the superior choice when computational accuracy is concerned. However, three dimensional models are computationally very expensive, and the large scale of real-world problems limits their applications in practice. Furthermore, the vertical component of the ﬂow acceleration is small compared to the horizontal components.

Thus, high ﬁdelity depth integrated formulations, such as nonlinear shallow water equations

(NSW), can be an attractive alternative for practical problems.

NSW have been extensively used for simulating long waves. Due to their conservative and shock-capturing properties, they represent a suitable approximation of the wave breaking as well as inundation. However, these equations are only valid for very long waves, and they cannot properly resolve dispersive effects before wave breaking. Thus, they become inaccurate in predicting the effects of shorter wavelength which is important for simulating the nearshore wave characteristics. On the other hand Boussinesq-type equations take dispersive effects into account and can be used to simulate the nearshore wave propagation until the breaking point more accurately. The importance of dispersive effects, until the breaking process starts, is demonstrated in Figure 4.1. Using SGN the transient leading wave almost maintains its shape, during propagation over constant water depth. However, if we ignore the effects of dispersive terms (i.e. by using NSW), after traveling a sufficiently large distance

(for this scenario 80 m) the wave height decreases by the factor of 3.

Diﬀerent classes of higher-order depth-integrated equations, based on Boussinesq wave the- Macro-Roughness Vegetation 48

(a) t = 5 s (b) t = 15 s (c) t = 25 s 0.6

0.4 ζ

0.2

0.0 0 5 10 15 20 25 30 35 40 30 35 40 45 50 55 60 65 70 60 65 70 75 80 85 90 95 100 x x x

Figure 4.1: Free surface elevation of the solitary-type transient wave over constant water depth at (a) t = 5 s, (b) t = 15 s, and (c) t = 25 s; ( ) 1D-SGN, and ( ) 1D-NSW. The −1 following parameters are used: h0 = 0.73 m, H = 0.50 m, k0 = 0.54 m . ory, have been derived and presented in literature on simulating shallow water ﬂows: Wei et al. (1995); Liu (1994); Lannes and Bonneton (2009). In this study, we employ fully nonlinear weakly dispersive Boussinesq equations, also known as Serre-Green-Naghdi equations

(SGN, Lannes and Bonneton, 2009). These equations have two advantages compared to others: (a) They only have spatial derivatives up to second order while other equations usually include spatial derivatives of third order, and (b) they do not have any additional source therm in the conservation of mass. These properties improve the computational stability and robustness of the model. Furthermore, unlike earlier studies in which vegetation eﬀects are approximated by an ad-hoc bottom friction coeﬃcient (e.g. Yang et al., 2016; Mei et al.,

2011), we model the cylinders as physical boundaries. This enables us to simulate the wave propagation through discontinuous arrays of cylinders without making any assumptions. We demonstrate the capability of the SGN equations in simulating of the waves interacting with both oﬀshore and coastal structures while sustaining the computational performance. Macro-Roughness Vegetation 49

4.2 Theoretical Background

2 We are interested in simulating long waves; thus µ = (h0~λ) << 1, where λ denotes the wavelength, and h0 is the still water depth. Assuming an inviscid and irrotationl ﬂow (in the vertical direction), expanding the Euler equations into an asymptotic series and keeping terms up to O(µ2), SGN equations can be written in the form of

(4.1a) ∂th + ∇ ⋅ (hu) = 0, 1 (4.1b) ∂ (hu) + ∇ ⋅ hu ⊗ u + gh2I = −gh∇b + D, t 2 where D is a nonlinear function of the free-surface elevation ζ, the depth-integrated velocity vector u, and their spacial derivatives; h is the water depth; b represents the bottom variations; and I is the identity tensor. See Bonneton et al. (2011) and Lannes and Bonneton

(2009) for the complete form of the equations and their derivation. Note that by setting

D = 0, Eqs. (4.1a)-(4.1b) will reduce to the NSW equations (accurate to O(µ)).

We employ the Basilisk code to solve the governing equations (Basilisk, URL: basilisk.fr,

Popinet, 2015). The computational domain is discretized into a rectangular grid and solved using the second order accurate scheme in time and space. We refer to Popinet (2015) for more information about the numerical scheme. Friction eﬀects become important as the wave approaches the coast. Following Bonneton et al. (2011), we added a quadratic friction

1 term f = cf h SuSu to Eq. (4.1b) where the friction coeﬃcient is cf = 0.0034. Macro-Roughness Vegetation 50

(III) t2 (II) t1 t (I)

z t0 t1 t2 x

Figure 4.2: Schematic of the breaking process. The vertical dashed lines indicate the boundary of subdomains. The governing equations in the left subdomain are SGN and NSW equations elsewhere. The boundary follows the leading wave; (I) t < t1 ∶ SGN in the whole domain, (II) t1 < t < t2 ∶ SGN in the left subdomain and SW in the right subdomain, and (III) t > t2 ∶ NSW in the whole domain.

4.2.1 Initial and Boundary Conditions

We imposed the initial conditions

2 cζ (4.2) ζ (x)St=0 = H sech [k0(x − x0)] , u(x)St=0 = , ζ + h0

to generate solitary-type transient long waves (Madsen et al., 2008; Madsen and Schaeﬀer, » 2010), in which c = g(h0 + H) is the phase speed. H represents the initial wave height, c

denotes the phase speed, and k0 is the characteristic wave number. This wave type removes

the constraint that exists between the wave height and the wave length and will result in

a more realistic representation of tsunami waves (e.g. Baldock et al., 2009; Rueben et al., » 3 2011; Irish et al., 2014) By choosing k0 = 3H~4h0, Eq. (4.2) will reduce to the free-surface

elevation of a solitary wave with a permanent shape. Macro-Roughness Vegetation 51

Table 4.1: Wave gauge coordinates.

Location (m) Location (m) Location (m) Gauge 1: (18.24, 0.00) 2: (24.43, 0.00) 3: (29.28, 0.00) Gauge 4: (36.42, 0.00) 5: (46.75, 0.55) 6: (45.10, 2.18) Gauge 7: (44.34, 0.83) 8: (48.40, 2.18) 9: (45.65, 0.60) Gauge 10: (49.52, 1.10) 11: (47.30, 2.18) 12: (47.30, 0.00) Gauge 13: (42.90, 0.00) 14: (45.10, 0.00) 15: (47.85, 1.65) Gauge 16: (46.20, 1.10)

4.2.2 Wave Breaking

We utilize the conditions suggested in Tissier et al. (2012) to estimate the start time of the breaking process. As noted earlier, NSW are employed as governing equations in the vicinity of the breaking-wave peak while modeling the rest of the domain using SGN. Here, we divide the computational domain into two sub-domains. The boundary that separates these two sub-domains follows the leading breaking wave (see Figure 4.2). We solve the domain behind the wave peak using SGN and the rest of the domain using NSW (t1 < t < t2 in Figure 4.2).

The distance between the wave peak and the boundary is given by xb − xw = 2hSxw where xb is the coordinate of the boundary and xw is the coordinate of the wave peak.

Note that in the very shallow areas, i.e., in the vicinity of the shoreline and the coastal areas, we always model the ﬂow using NSW equations to avoid the numerical dispersions that can be caused by dispersive terms in SGN. In this study, we considered the run-up of a single wave. After the wave breaks completely (t > t2 in Fig. 4.2), to increase the computational eﬃciency, we model the entire domain using NSW. Macro-Roughness Vegetation 52

0.6 (a) Gauges 1-2 (b) Gauges 3-4

0.4 ζ

0.2

0.0

2 6 10 14 2 6 10 14 t t

Figure 4.3: Free-surface elevation at (a) gauges 1-2, and (b) gauges 3-4. Experimental and COULWAVE results are summarized in Yang et al. (2016); ( ) present simulation (2D-SGN), ( ) experimental results and ( ) COULWAVE.

4.3 Results

4.3.1 Validation of Numerical Results

Breaking Solitary-Type Transient Wave Run-Up

We investigated solitary-type transient wave run-up on a steep sloping beach (1:10) following

the experimental setup in Irish et al. (2014). We discretized the computational domain (52

m, 4.4 m) using a rectangular uniform grid with ∆x = ∆y = 52~1024 m. The toe of the sloping beach is located at x = 32 m. The still water depth is h0 = 0.73 m. The transient

−1 solitary wave is generated by setting H = 0.50 m, k0 = 0.54 m , and x0 = 10 m at t = 0 s. A symmetry boundary condition is imposed on bottom and top boundaries. Macro-Roughness Vegetation 53

0.150 (a) Gauge 5 (b) Gauge 6 (c) Gauge 7 (d) Gauge 8

h0.075

0.000 0.150 (e) Gauge 9 (f) Gauge 10 (g) Gauge 11 (h) Gauge 12

h0.075

0.000 0.150 (i) Gauge 13 (j) Gauge 14 (k) Gauge 15 (l) Gauge 16

h0.075

0.000 11 13 15 17 19 11 13 15 17 19 11 13 15 17 19 11 13 15 17 19 t t t t

Figure 4.4: Local water depth at gauges 5-16. Experimental and COULWAVE results are summarized in Yang et al. (2016); ( ) present simulation (2D-SGN), ( ) experimental results and ( ) COULWAVE.

Yang et al. (2016) studied this problem experimentally as well as numerically using COUL-

WAVE (Cornell University Long and Intermediate Wave Modeling Package, Lynett et al.,

2002) which solves the Boussinesq equations described in Liu (1994) and Lynett et al. (2002).

We compared the free surface elevation (wave gauges 1-4) and the local water depth (wave gauges 5-16) with the results presented in Yang et al. (2016). The wave gauge coordinates are summarized in Table 4.1. Figure 4.3 shows the free-surface elevation ζ at gauges 1-4.

While both methods capture the evolution of the free surface accurately, we observe that

COULWAVE captures the breaking process slightly better. This can be due to the diﬀer- Macro-Roughness Vegetation 54 ences in the how wave breaking is handled. COULWAVE employs an ad-hoc viscosity model in the breaking process. In contrast, we solve NSW in the breaking zone and incorporate a shock-capturing scheme in the breaking process. Consequently, our model develops a steeper wave proﬁle.

dp dc dp y x dc

Figure 4.5: Sketch of the macro-roughness patches.

Figure 4.4 depicts the local water depth at onshore gauges 5-16. Our results are systemati- cally closer to the experimental data. (Park et al., 2013) also reported about overpredicting the experimental results using COULWAVE. Yang et al. (2016) argued that the diﬀerence between the simulation and the experiment can be due to the existence of an slight leak- age from the sloping beach among some other possible reasons. This can explain the slight overpredictions we observed using our model.

4.3.2 Breaking Solitary-Type Transient Wave Run-Up in the Pres- ence of Macro-Roughness

A schematic sketch of the problem is shown in Figure 4.5. The computational domain, boundary conditions, and the initial conditions are the same as the ones in section 4.3.1.

Three diﬀerent scenarios are considered here. Each macro-roughness patch consists of regu- Macro-Roughness Vegetation 55

Table 4.2: Geometrical parameters of macro-roughness patches. dr ∶ distance between two horizontally (or vertically) aligned cylinders inside a patch; Ncp ∶ total number of cylinders inside a patch; dp ∶ distance between two horizontally (or vertically) aligned patches; Cfp ∶ coordinate of the center of the ﬁrst patch; Dc ∶ diameter of the cylinders; Dp ∶ diameter of the patches.

dr (m) Ncp dp (m) Cfp (m) Dc (m) Dp (m) Scenario 1 0.1885 21 2.2 (34, 0) 0.01333 0.6 Scenario 2 0.0943 69 2.2 (34, 0) 0.01333 0.6 Scenario 3 0.1885 129 2.2 (34, 0) 0.01333 0.6

larly spaced vertical cylinders. The geometrical parameters of the patches for each scenario

are summarized in Table 4.2. We used a nested mesh with

∆x = ∆y = 52~1024 m x < 41 m, ∆x = ∆y = 52~8192 m x > 41 m.

Local water depth at gauges 5-16 is shown in Figure 4.6. The simulations are in good agreement with the experimental data. Some diﬀerences in transient wave peaks can be due to the ensemble-averaged experimental data. Ensemble averaging can smooth out some of the sharp transitions as suggested by Yang et al. (2016) and Baldock et al. (2009) Macro-Roughness Vegetation 56

0.150 (a) Gauge 5 (b) Gauge 6 (c) Gauge 7 (d) Gauge 8

h0.075

0.000 0.150 (e) Gauge 9 (f) Gauge 10 (g) Gauge 11 (h) Gauge 12

h0.075

0.000 0.150 (i) Gauge 13 (j) Gauge 14 (k) Gauge 15 (l) Gauge 16

h0.075

0.000 11 13 15 17 19 11 13 15 17 19 11 13 15 17 19 11 13 15 17 19 t t t t

Figure 4.6: Local water depth at gauges 5-16 for Scenario 3. Experimental results are summarized in Irish et al. (2014); ( ) present simulation (2D-SGN), and ( ) experimental results.

4.3.3 Eﬀects of Macro-Roughness on the Local Maximum Local Water Depth

The maximum local water depth is given by

max(h) − max(h)Sref (4.3) hmax = , max(h)Sref

where max(h)Sref is the maximum water depth in the absence of the macro-roughness patches. Unlike momentum flux we do observe water depth amplification up to 1.7 times Macro-Roughness Vegetation 57 behind the first patch in the presence of the macro-roughness patches. When the flow reaches the first patch the flow refracts away from the center of the patch toward the other patches.

However, the patches located in the second row refract and reﬂect the water toward the cen- terline. This process causes the water to amplify behind the ﬁrst patch (red area in Figure

4.7). As we can see in Figure 4.7(d), no ampliﬁcation in local water depth occurs behind the patch in the case were all other patches are removed.

4.3.4 Eﬀects of Macro-Roughness on the Local Maximum Mo- mentum Flux

The momentum flux represents the destructive forces of the incident wave. To study the effects of the macro-roughness patches on momentum flux we defined the maximum normalized momentum flux as

2 2 max(hSuS ) − max(hSuS )Sref (4.4) Fmax = 2 , max(hSuS )Sref

2 where max(hSuS )Sref is the maximum momentum ﬂux in the absence of the macro-roughness patches. Figure 4.8 shows the Fmax for Scenarios 1-3. We can see that the patches provide protection for the areas behind them for Scenario 1 (Figure 4.8(a)). With increasing density of the cylinders in patches, Fmax is decreased. Thus the level of protection against incident waves increases. However we did not observe any signiﬁcant changes with further increasing the density of the cylinders (see Figure 4.8). Macro-Roughness Vegetation 58

∗ Figure 4.7: Maximum local water depth hmax for (a) Scenario 1, (b) Scenario 2 (c) Scenario 3, and (d) Scenario 3 in which all the patches are removed except the ﬁrst patch. The maximum water depth for each scenario is normalized with the reference values in the absence of macro-roughness patches.

4.3.5 Maximum Run-Up

Maximum run-up is another important criteria in determining the eﬀectiveness of the macro- roughness patches in mitigating tsunami hazard risks. Figure 4.9 shows the comparison of the simulation to the experimental results of the bore line propagation for the Scenario 2.

The results are in good agreement. The maximum deviation from the experimental results Macro-Roughness Vegetation 59 is less than 4%.

∗ Figure 4.8: Maximum momentum ﬂux Fmax for (a) Scenario 1, (b) Scenario 2, and (c) Scenario 3. The momentum ﬂux for each scenario is normalized with the reference values in the absence of macro-roughness patches.

Figure 4.10 shows the bore-line propagation for diﬀerent Scenarios. The maximum run-up decreases with increasing the vegetation density inside the macro-roughness patches. For the very low density vegetations (Fig. 4.10(b)) this reduction is more or less uniform along the shore. However with increasing the vegetation density (Figure 4.10(c)) we observe more reduction behind the patches. With further increasing the vegetation density (Figure 4.10(d) in comparison to Figure 4.10(c)) while the maximum run-up behind the patches continues to decrease, it increases within the channel between the patches slightly. The reason for this increase is the level by which the ﬂow is channelized between the patches. Macro-Roughness Vegetation 60

(a)

2.0 (b) 1.5

y 1.0 0.5 0.0 43 44 45 46 47 48 49 50 x

Figure 4.9: Propagation of bore-lines in the presence of macro-roughness patches (Scenario 2). Irish et al. (2014); (a) experimental results, and (b) present simulation (2D-SGN). Macro-Roughness Vegetation 61

2.0 (a) 1.5

y 1.0 0.5 0.0 2.0 (b) 1.5

y 1.0 0.5 0.0 2.0 (c) 1.5

y 1.0 0.5 0.0 2.0 (d) 1.5

y 1.0 0.5 0.0 43 44 45 46 47 48 49 50 x

Figure 4.10: Propagation of bore-lines for (a) Scenario with no macro-roughness patches (b) Scenario 1, (c) Scenario 2 (d) Scenario 3.

4.4 Discussion and Conclusion

We presented numerical simulations of long water waves interacting with emergent cylinders.

We demonstrated that higher order depth integrated equations, such as SGN, are a suitable Macro-Roughness Vegetation 62 tool to simulate the wave interaction with emergent cylinders accurately except in very close vicinity of the cylinders. Three-dimensional eﬀects cannot be ignored at the proximity of the cylinders and we argue that they are the main reason for the existing diﬀerences between results from our model and the experimental data.

Cylinders, representing coastal vegetation, are usually approximated as macro-roughness friction (e.g. Yang et al., 2016; Mei et al., 2011). However there is no analytical solution for a correlation between macro-roughness patterns and the implemented friction coeﬃcient.

Thus these models need to be calibrated against available experimental data. However for scenarios with no experimental prototype these models can become inaccurate. Here, with further reﬁning the grid in the coastal areas, we modeled these macro-roughness patches as physical boundaries.

We observed that discontinuous coastal vegetation can provide protection for the areas behind them. Friction forces become dominant when the flow becomes shallower. In addition, for the areas located on the onshore slope we also have the gravitational force acting in the negative flow direction. Macro-roughness patches elongate the path of the incident wave, causing a longer local inundation period. Thus the flow will be subjected to the negative gravitational and friction force for a longer time and the maximum recorded velocity will decline in comparison to the scenario where no macro-roughness patches exist. This will lead to a reduced local momentum flux and increased protection against the destructive wave force. However, in terms of local maximum water depth this conclusion does not apply.

Even though the decreased local velocity will result in more protection against the wave, we observed for the studied macro-roughness the maximum local water depth actually increases Macro-Roughness Vegetation 63 behind the patches. We observed ampliﬁcations up to 1.6 times in local water depth. This will increase the chance of that area being ﬂooded. We note here that the main reason the

Fukushima Daiichi nuclear disaster happened was the fact that the water overtopped the protecting wall and reached the electric generators (Synolakis and Kˆano˘glu,2015).

We demonstrated the capability of the model in analyzing the wave interaction with coastal structures. However, our study was limited to the emergent cylinders and simple macro- roughness patch patterns. Studying submerged coastal vegetations and diﬀerent patch patterns will be the direction of our future studies. Chapter 5

Some Examples for Which Dispersive Eﬀects Can Change the Results Signiﬁcantly†

Contribution

The pressure boundary condition for the airburst obtained through personal communication

with Micheal J. Aftosmis, Advanced Supercomputing Division, NASA.

†Citation: A part of this chapter is in consideration for publication in PNAS:

N. Hoﬀman, et. al., Novel Integrative Approach Reveals past Mediterranean tsunami events aﬀecting Gaza to Tel Aviv.

64 Some Applications 65

5.1 Numerical Simulation of Hazard Assessment Gen- erated by Asteroid Impacts on Earth

Probably the collision of Comet Shoemaker-Levy 9, the ﬁrst direct observation of a comet

collision, with the Jupiter which happened in July 1994, was one of the most important

events that motivated scientists to try and understand the physics behind the collision of

asteroids with the Earth and to ﬁnd the possible ways that can be used to mitigate the

hazards associated with them. NASA’s Near Earth Object Program1 monitors the asteroids that orbit close to earth and publishes the probability of them hitting the Earth at SENTERY web-page ( An Automatic Near-Earth Asteroid Collision Monitoring System)2.

Assuming an unbiased asteroid impact, given the fact that 70% of the Earth is covered with

water, the probability of an asteroid impact in the ocean is 70%. Thus it is important to

study the propagation of waves generated by such an impact. Most of the asteroids will

explode before reaching the Earth’s surface. However some of the larger ones can have a

physical impact on the Earth’s surface. Smaller asteroids will explode before reaching the

Earth’s surface, while larger asteroids with a diameter approximately equal to 500 m, or

larger, will hit the surface.

For simplicity, in this section we will solve the SGN equations (Eqs. 1.5-1.6) in cylindrical

form which are given by

1 ∂ h + ∇ ⋅ (rhu) = 0, (5.1) t r 1 1 ∂ (hu) + ∇ ⋅ (rhu ⊗ u) + ∇ gh2 = −gh∇b + D + F + O(µ2), (5.2) t r 2 p 1http://neo.jpl.nasa.gov/ 2http://neo.jpl.nasa.gov/risk/ Some Applications 66

6 Geoclaw

4 NSW

0 [m] ζ 2 −

4 −

6 −

8 − 0 100 200 300 400 500 600 700 t [sec]

Figure 5.1: Comparison of water surface elevation between our SWE model and GeoClaw results. Waves are generated by an asteroid with a diameter of 140 m exploding at the altitude of 10 km. Wave gauges are located at ( ) 0.05LD,( ) 0.2LD,( ) 0.5LD, and ( ) 0.8LD, where LD = 111 km.

where the Fp is given by h F = ∇p. (5.3) p ρ

Here p represents the pressure distribution at the surface of the water. Assuming the wave

propagation is axisymmetric, Eqs. 1.5-1.6 can be further simpliﬁed to:

∂h ∂ (hu) + = −hu, (5.4) ∂t ∂r ∂hu ∂ (hu2) 1 ∂ (gh2) ∂b + + = −gh − hu2 + D + O(µ2), (5.5) ∂t ∂r 2 ∂r ∂r Some Applications 67

5.1.1 Numerical Simulation of Tsunami Waves Generated by an Asteroid Explosion near the Ocean Surface

An asteroid with a diameter of 140 m, and density of 2000 kg/m3, approaching the Earth with a velocity of 18 km/sec, which explodes at the altitude of 10 km can be approximated with an air-burst with a total energy equal to 100 Mt. Assuming that about 10% of the energy goes into radiation this will generate a pressure distribution on the water surface given by R − r R − r P (r) = G (1 − 6 )exp−3.8 , (5.6) 2W 2W

where R 2 R 2 G = 96 exp −4 + 96 exp −2 C 1.8C (5.7) R 8 R + 5 exp −2 + 34exp−2 , 5C 12C

R = 0.3915 t, C = 12.74 and W = 30.

1 (a) (b) (c)

0 [m] ζ

1 −

2 − 0 200 400 0 200 400 0 200 400 Distance [km] Distance [km] Distance [km]

Figure 5.2: Comparison of water surface elevation obtained using SWE equations ( ), and SGN equations ( ) at (a) t = 400 s, (b) t = 800 s, and (c) t = 1200 s. Waves are generated by an asteroid with a diameter of 140 m exploding at the altitude of 10 km. Some Applications 68

0.5 100 −

101 0.5 − ]

m 2

[ 10 1.5 ζ −

(a) 103 2.5 −

0.5 100 −

101 0.5 − ]

m 2

[ 10 1.5 ζ −

(b) 103 2.5 −

0.5 100 −

101 0.5 − ]

m 2

[ 10 1.5 ζ −

0.5 100 −

101 0.5 − ]

m 2

[ 10 1.5 ζ −

(d) 103 2.5 −

50 100 150 200 250 300 Distance [km]

Figure 5.3: Water surface elevation obtained using SWE equations ( ), and SGN equations ( ) at (a) t = 1000 s, (b) t = 2000 s, (c) t = 3000 s, and (d) t = 4000 s. Left y-axis shows the topographical variation in logarithmic scale. Some Applications 69

Figure 5.1 shows the water surface elevation at different wave gauges caused by the air-burst in comparison to the results obtained using GeoClaw 3 showing an excellent agreement. Fig- ure 5.2 compares the numerical results obtained using SGN and SWE. There is a significant difference between the results obtained by these two sets of equations. While one might argue that SWE produce more conservative results by generating waves with larger amplitude. However, as shown in Figure 5.3, while the wave profile obtained by SWE decays as it approaches the coast, we observe formation of an undular bore using SGN. In other words, the wave amplitude obtained by SGN becomes comparable to the wave amplitude obtained by SWE. We note here that using SWE, we usually observe one leading wave followed by smaller waves with negligible wave amplitudes. However using SGN, the predicted wave pack consists of 3-4 leading waves with amplitude of the same order of magnitude as the leading wave of SWE. This can lead to a significantly larger wave run-up.

5.1.2 Numerical Simulation of Tsunami Waves Generated by an Asteroid Impact into the Ocean

To model a tsunami wave generated by an asteroid impact into the ocean, we calculate the water surface elevation at a speciﬁc distance from impact center (in this study 20.0 km) using three-dimensional hydro-code, iSALE (impact-SALE)4. To generate a wave, we impose the obtained time series on the left boundary and simulate the wave propagation using SGN equations. Note that here, we set Fp = 0

3http://www.clawpack.org/geoclaw 4Simpliﬁed Arbitrary Lagrangian Eulerian: http://www.isale-code.de/redmine/projects/isale/wiki/ISALE/ Some Applications 70

200 (a) 100

[m] 0 ζ 100 − 200 − 20 (b) 10

[m] 0 ζ 10 − 20 − 20 (c) 10

[m] 0 ζ 10 − 20 − 20 (d) 10

[m] 0 ζ 10 − 20 − 0 1000 2000 3000 4000 Distance [km]

Figure 5.4: Water surface elevation of a tsunami wave generated by an impact into water obtained using SGN equations at (a) t = 1000 s, (b) t = 5000 s, (c) t = 10000 s, and (d) t = 15000 s.

We modeled the Eltanin impact which occurred approximately 2.15 million years ago. The

Eltanin impact occurred due to an impact of an asteroid with a diameter approximately equal to 750 m (Weiss et al., 2015; W¨unnemann and Lange, 2002; Mader, 1998; Shuvalov Some Applications 71 and Gersonde, 2014). Following Weiss et al. (2015), we assumed the rock is made of basalt

ANEOS with the density of 2700 kg/m3 and hits the water surface with the velocity of u = 12 km/s. Here we assumed a perpendicular impact for simplicity. We validated our results by comparing the wave elevation at wave gauges located at 25, 30, 35, 40 km from the center of the impact to the results obtained using iSALE. The depth of the ocean is assumed as 5000 m at the location of impact and the bottom of the ocean is assumed to be covered with 250 m thick sediment layer made of basalt ANEOS.

103

1 1.0 ζ x 102 ∝

101 Maximum Elevation [m]

1.3 ζ 1 ∝ x 100 102 103 Distance [km]

Figure 5.5: Maximum wave height as a function of distance from the impact center. ( ) SWE ( ) SGN. Some Applications 72

The leading wave of an earthquake generated tsunami usually has larger amplitude. This is

because according to shallow water theory, the phase speed of a long wave is given by c = » g(H + a) which means that the waves with larger amplitude will travel among the leading

waves in the wave packet. Figure 5.4 shows the propagation of the tsunami wave generated

by the Eltanin impact. However, we see that, unlike the earthquake generated tsunamis, the

waves with larger wave amplitudes travel slower compared to waves with smaller amplitudes.

This indicates that the wavelength associated with the asteroid generated waves, as expected,

are shorter than the earthquake generated tsunamis.

Figure 5.5 shows the decaying of the maximum wave height versus distance from the center

of impact. Neglecting friction forces for shallow water waves, the total wave energy will

1 decay proportional to geometrical spreading, i.e. E ∝ f r . Since the wave height is related √ to wave energy by ζ ∝ E. Thus the wave height decay for shallow water waves are given by 1 ζ ∝ f √ (5.8) r

However, in reality, earthquake generated waves decay at faster rates. The decay rate for

asteroid generated tsunamis while using SWE as governing equations is given by

1 ζ ∝ f (5.9) r

while, as expected, it decays faster while replacing SWE with SGN, i.e.

1 ζ ∝ f (5.10) r1.3 Some Applications 73

Table 5.1: Simulation parameters of cnoidal waves interacting with a cylinder.

m λ~h0 r~h0 Case1 0.9 6. 2.67 Case2 0.99 12. 2.67 Case3 0.999999 650. 2.67 Case4 1.0 ∞ 2.67

z λ a x h0

x y

Figure 5.6: Schematic of a cnoidal wave interacting with a cylinder.

5.2 Non-Breaking Cnoidal Wave Interaction with Oﬀ- shore Cylinders

It is widely believed by researchers that the oﬀshore and coastal structures dissipate wave energy and act as a barrier against the incident wave. However, recent studies suggest that these structures, sometimes, can resonate the wave passing through them rather than providing a protection zone behind them. Hu and Chan (2005) studied the inﬂuence of an array of vertical bottom-mounted cylinders on the propagation of long water waves. They Some Applications 74

observed focusing of water waves in a regions behind the cylinders with a maximum intensity

of 3.8.

Figure 5.7: Contour plots of the maximum water elevation for (a) Case1, (b) Case2, (c) Case3. Blue lines denote the vertical cross sections at 1: (x − x0)~h0 = 9.83, 2: (x − x0)~h0 = 17.65, 3: (x − x0)~h0 = 41.25.

To generate a cnoidal wave train, we impose the following conditions on the left boundary

(Wiegel, 1960): ¾ 2 3a (5.11) ζ (t − T )Sleft = ζ2 + a cn 3 c(t − T ); m , 4mh0

and

cζ (5.12) u(t − T )Sleft = , ζ + h0

where

a E(m) ζ = 1 − m − , (5.13) 2 m K(m)

and

» a 1 3 E(m) c = gh 1 + 1 − m − . (5.14) mh0 2 2 K(m) Some Applications 75

The phase lag T is calculated iteratively such that ζ(0 − T ) = 0. Here K(m) denotes the complete elliptic integral of the ﬁrst kind and E(m) represents the complete elliptic integral of the second kind.

The schematic of the problem is shown in Figure 5.6. The computational domain of 160 m×

160 m is discretized into a 2048×2048 uniform grid. The cylinder of 2 m diameter is located at the center of the domain. The water depth at rest is h0 = 0.75 m. Simulation parameters for the cases considered here are summarized in Table 5.1.

1.75

1.25 ∗ ) 2 hu

( 0.75

0.25

1.75

1.25 ∗ ) 2 hu

( 0.75

0.25

1.75

1.25 ∗ ) 2 hu ( 0.75

0.25 60 0 60 60 0 60 60 0 60 60 0 60 − − − − y∗ y∗ y∗ y∗

Figure 5.8: Maximum momentum ﬂux along the vertical cross sections at top: (x−x0)~h0 = 9.83, middle: (x − x0)~h0 = 17.65, bottom: (x − x0)~h0 = 41.25. Some Applications 76

Figure 5.7 shows the contour plots of the maximum water elevation. We can see that as soon

as the leading wave reaches the cylinder, it generates secondary reﬂected waves spreading

in a radial direction. This generates elliptically shaped maximum wave height contour plots

spreading out from the cylinder (see the contour pattern outside the region bounded by

the white boundary in Figure 5.7(a-c)). However, inside the region bounded by the white

curves, we observe very complicated patterns. We can distinguish three main mechanisms

that causes the propagation of the wave in this region. The leading wave that tries to retrieve

the original travel path in x−axis direction. The tailing waves with smaller amplitudes and smaller phase velocities mainly traveling in a radial direction. We also observe the formation of vorticities behind the cylinder. This causes a very complicated maximum wave height contours behind the cylinder as we can see in Figure 5.8, where we observe both reduction and increase in the maximum momentum ﬂux behind the cylinder.

5.3 Hazard Assessment Along the Coastline from the Gaza Strip to the Caesarea, Israel

In this section, we assess the risk associated with the tsunamis at the Gaza Strip. The possible triggering sources of the tsunami can be by an earthquake, volcanic eruption or a landslide in the Mediterranean Sea. Here, we assess the risk associated with all three possible scenarios. For an earthquake generated tsunami, we assume an earthquake with an epicenter located at the south-east portion of the Hellenic arc (35.58N, 28.14E) with the following parameters used in the Okada model: Some Applications 77

Depth = 45 km Strike = 225o Dip = 30o Rake = 90o Length = 140 km Width = 60 km

For the volcanic eruption, we model the tsunami caused by Thera eruption (Novikova et al.,

2011; McCoy and Heiken, 2000). Here, we simply approximate the Thera eruption by a cylindrical dam break centered at (36.5N, 26.E) and

2 2 2 h = 0 if (x − x0) + (y − y0) < 0.02 .

Here, x and y represent the longitude and latitude in degrees. And ﬁnally we approximate a landslide with the total volume of 3 km3 with

2π ζ = 10.8 sin √ . 55 gh0

Figure 5.9 summarizes the tsunami run-up along the Gaza Strip. As we can see the tsunamis generated by an earthquake impose the most significant threat at the coast followed by tsunamis generated by a landslide. However, because of the large distance between the possible origin of the volcanic eruption close to Gaza Strip, the eruption-generated tsunami imposes no significant threat on the coast along the Gaza Strip. Here we note while we observed smaller run-up for the tsunami generated by landslide, the observed maximum wave height was larger for it compared to the tsunami generated by the earthquake. This can potentially make this tsunami more hazardous for the fishing boats and ships near the

Strip. Some Applications 78

Earthquake Landslide Volcanic Eruption

A A0 Ashkelon Tel Aviv Caesaria 10 Ashdod

8 Gaza City

4 Run up [m]

2 Gaza

0 0 50 100 150 Distance along coast [km]

Figure 5.9: Tsunami run-up along the coastline from the Gaza Strip to the Caesarea, Israel. Chapter 6

Future Work

Computational power is increasing very rapidly, thanks to the fast evolving computing hardwares. However, harnessing the underlying hardware potential is becoming diﬃcult as well.

Codes, commonly used in industry, need to be altered signiﬁcantly in order to make it com- patible with the newer computing devices such as Intel MIC due to their design patterns.

While GPU computing can potentially speed-up the simulation by an order of magnitude, the current limitations with the GPU memories and the necessity to communicate between the CPU and GPU is the current bottleneck to reach the optimum speed-up. Higher order numerical schemes can achieve the same numerical accuracy on the coarser meshes. This makes them a suitable candidate for numerical codes targeting GPUs.

We are developing a software program based on higher order ﬁnite diﬀerence WENO schemes

(Xing and Shu, 2005; Vukovic and Sopta, 2002) for solving common problems in water waves and sediment transport that can be easily modiﬁed to harness the power of any new hardware that emerges in the market, while, at the same time, keeping the program easy to maintain

79 Future Work 80 and develop. Python is chosen to create a user-friendly interface. On the other hand, computationally expensive parts of the algorithm will be translated into low level native codes targeting clusters of CPUs, NVIDIA GPUs, and Intel MIC. This is accomplished by leveraging a domain speciﬁc language (DSL) derived from the Mako templating engine which is also based on Python. Using DSL language decouples the code development part from the hardware speciﬁc parts of it, enabling us to easily modify the code according to other possible hardwares that might emerge in the future. Bibliography

Anderson, M. E., Smith, J., 2014. Wave attenuation by ﬂexible, idealized salt marsh vege-

tation. Coastal Engineering 83, 82–92.

Baldock, T., Cox, D., Maddux, T., Killian, J., Fayler, L., 2009. Kinematics of breaking

tsunami wavefronts: A data set from large scale laboratory experiments. Coastal Engi-

neering 56 (5), 506–516.

Barbano, M., Pirrotta, C., Gerardi, F., 2010. Large boulders along the south-eastern Ionian

coast of Sicily: storm or tsunami deposits? Marine Geology 275 (1), 140–154.

Bonneton, P., Chazel, F., Lannes, D., Marche, F., Tissier, M., 2011. A splitting approach for

the fully nonlinear and weakly dispersive Green–Naghdi model. Journal of Computational

Physics 230 (4), 1479–1498.

Cleary, P., Ha, J., Alguine, V., Nguyen, T., 2002. Flow modelling in casting processes.

Applied Mathematical Modelling 26 (2), 171–190.

Colagrossi, A., Landrini, M., 2003. Numerical simulation of interfacial ﬂows by smoothed

particle hydrodynamics* 1. Journal of Computational Physics 191 (2), 448–475.

81 Bibliography 82

Dalrymple, R. A., Rogers, B. D., 2006. Numerical modeling of water waves with the SPH

method. Coastal engineering 53 (2), 141–147.

Dominey-Howes, D., Cummins, P., Burbidge, D., 2007. Historic records of teletsunami in

the Indian Ocean and insights from numerical modelling. Natural Hazards 42 (1), 1–17.

Dutykh, D., Dias, F., 2007. Dissipative boussinesq equations. Comptes Rendus Mecanique

335 (9), 559–583.

Etienne, S., Buckley, M., Paris, R., Nandasena, A. K., Clark, K., Strotz, L., Chagu´e-Goﬀ,C.,

Goﬀ, J., Richmond, B., 2011. The use of boulders for characterising past tsunamis: lessons

from the 2004 Indian Ocean and 2009 South Paciﬁc tsunamis. Earth-Science Reviews

107 (1), 76–90.

Fang, H. S., Bao, K., Wei, J. A., Zhang, H., Wu, E. H., Zheng, L. L., 2009. Simulations

of Droplet Spreading and Solidiﬁcation Using an Improved SPH Model. Numerical Heat

Transfer, Part A: Applications 55 (2), 124–143.

Fang, J., Owens, R. G., Tacher, L., Parriaux, A., 2006. A numerical study of the SPH

method for simulating transient viscoelastic free surface ﬂows. Journal of non-newtonian

ﬂuid mechanics 139 (1-2), 68–84.

Ferrari, A., Fraccarollo, L., Dumbser, M., Toro, E. F., Armanini, A., 2010. Three-dimensional

ﬂow evolution after a dam break. Journal of Fluid Mechanics 663, 456–477.

Fritz, H. M., Borrero, J. C., Synolakis, C. E., Yoo, J., 2006. 2004 Indian Ocean tsunami ﬂow

velocity measurements from survivor videos. Geophysical Research Letters 33 (24). Bibliography 83

Gingold, R. A., Monaghan, J. J., 1977. Smoothed particle hydrodynamics-theory and ap-

plication to non-spherical stars. Monthly Notices of the Royal Astronomical Society 181,

375–389.

Goﬀ, J., Dudley, W., Demaintenon, M., Cain, G., Coney, J., 2006. The largest local tsunami

in 20th century Hawaii. Marine Geology 226 (1), 65–79.

Goring, D. G., 1978. Tsunamis–the propagation of long waves onto a shelf. Ph.D. thesis,

California Institute of Technology.

Goto, K., Kawana, T., Imamura, F., 2010a. Historical and geological evidence of boulders

deposited by tsunamis, southern Ryukyu Islands, Japan. Earth-Science Reviews 102 (1),

77–99.

Goto, K., Okada, K., Imamura, F., 2010b. Numerical analysis of boulder transport by the

2004 Indian Ocean tsunami at Pakarang Cape, Thailand. Marine Geology 268 (1), 97–105.

H´erault,A., Bilotta, G., Dalrymple, R. A., 2010. SPH on GPU with CUDA. Journal of

Hydraulic Research 48 (S1), 74–79.

Herault, A., Bilotta, G., Dalrymple, R. A., Rustico, E., Del Negro, C., 2006-2014. GPUSPH.

Hirt, C. W., Nichols, B. D., Jan. 1981. Volume of ﬂuid /VOF/ method for the dynamics of

free boundaries. Journal of Computational Physics 39, 201–225.

Hosseini, S. M., Manzari, M. T., Hannani, S. K., 2007. A fully explicit three-step SPH

algorithm for simulation of non-Newtonian ﬂuid ﬂow. International Journal of Numerical

Methods for Heat & Fluid Flow 17 (7), 715–735. Bibliography 84

Hu, X., Chan, C., 2005. Refraction of water waves by periodic cylinder arrays. Physical

review letters 95 (15), 154501.

Imamura, F., Goto, K., Ohkubo, S., 2008. A numerical model for the transport of a boulder

by tsunami. Journal of Geophysical Research: Oceans (1978–2012) 113 (C1).

Irish, J. L., Weiss, R., Yang, Y., Song, Y. K., Zainali, A., Marivela-Colmenarejo, R., 2014.

Laboratory experiments of tsunami run-up and withdrawal in patchy coastal forest on a

steep beach. Natural Hazards 74 (3), 1933–1949.

Irtem, E., Gedik, N., Kabdasli, M. S., Yasa, N. E., 2009. Coastal forest eﬀects on tsunami

run-up heights. Ocean Engineering 36 (3), 313–320.

Jaﬀe, B., Buckley, M., Richmond, B., Strotz, L., Etienne, S., Clark, K., Watt, S., Gelfen-

baum, G., Goﬀ, J., 2011. Flow speed estimated by inverse modeling of sandy sediment

deposited by the 29 September 2009 tsunami near Satitoa, east Upolu, Samoa. Earth-

Science Reviews 107 (1), 23–37.

Kirby, J. T., 2003. Boussinesq models and applications to nearshore wave propagation, surf

zone processes and wave-induced currents. Elsevier Oceanography Series 67, 1–41.

Landrini, M., Colagrossi, A., Greco, M., Tulin, M. P., 2007. Gridless simulations of splashing

processes and near-shore bore propagation. Journal of Fluid Mechanics 591, 183–213.

Lannes, D., Bonneton, P., 2009. Derivation of asymptotic two-dimensional time-dependent

equations for surface water wave propagation. Physics of Fluids (1994-present) 21 (1),

016601. Bibliography 85

Li, Y., Raichlen, F., 2002. Non-breaking and breaking solitary wave run-up. Journal of Fluid

Mechanics 456, 295–318.

Liu, M. B., Liu, G. R., Lam, K. Y., 2003. A one-dimensional meshfree particle formulation

for simulating shock waves. Shock Waves 13 (3), 201–211.

Liu, P. L.-F., 1994. Model equations for wave propagations from deep to shallow water.

Advances in coastal and ocean engineering 1, 125–157.

Liu, P. L.-F., Cho, Y.-S., Briggs, M. J., Kanoglu, U., Synolakis, C. E., 1995. Runup of

solitary waves on a circular island. Journal of Fluid Mechanics 302, 259–285.

Lucy, L. B., 1977. A numerical approach to the testing of the ﬁssion hypothesis. The Astro-

nomical Journal 82, 1013–1024.

Lynett, P. J., Wu, T.-R., Liu, P. L.-F., 2002. Modeling wave runup with depth-integrated

equations. Coastal Engineering 46 (2), 89–107.

Ma, G., Shi, F., Kirby, J. T., 2012. Shock-capturing non-hydrostatic model for fully dispersive

surface wave processes. Ocean Modelling 43, 22–35.

Mader, C. L., 1998. Modeling the Eltanin asteroid tsunami. Science of Tsunami Hazards

16 (5), 17–20.

Madsen, P. A., Fuhrman, D. R., Sch¨aﬀer,H. A., 2008. On the solitary wave paradigm for

tsunamis. Journal of Geophysical Research: Oceans (1978–2012) 113 (C12).

Madsen, P. A., Schaeﬀer, H. A., 2010. Analytical solutions for tsunami runup on a plane

beach: single waves, N-waves and transient waves. Journal of Fluid Mechanics 645, 27–57. Bibliography 86

Marivela, R., Weiss, R., Synolakis, C. E., Under review. The temporal and spatial evolution

of momentum, kinetic energy and force in tsunami waves during breaking and inundation.

Journal of Waterway, Port, Coastal, and Ocean Engineering.

Matsutomi, H., Shuto, N., Imamura, F., Takahashi, T., 2001. Field survey of the 1996 Irian

Jaya earthquake tsunami in Biak Island. Natural Hazards 24 (3), 199–212.

McCoy, F. W., Heiken, G., 2000. Tsunami generated by the Late Bronze age eruption of

Thera (Santorini), Greece. Pure and Applied Geophysics 157 (6-8), 1227–1256.

Mei, C. C., Chan, I.-C., Liu, P. L.-F., Huang, Z., Zhang, W., 2011. Long waves through

emergent coastal vegetation. Journal of Fluid Mechanics 687, 461–491.

Melean, Y., Sigalotti, L. D. G., Hasmy, A., 2004. On the SPH tensile instability in forming

viscous liquid drops. Computer physics communications 157 (3), 191–200.

Mo, W., 2010. Numerical investigation of solitary wave interaction with group of cylinders.

[PhD dissertation], Cornell University.

Mo, W., Liu, P. L.-F., 2009. Three dimensional numerical simulations for non-breaking

solitary wave interacting with a group of slender vertical cylinders. International Journal

of Naval Architecture and Ocean Engineering 1 (1), 20–28.

Monaghan, J. J., 1992. Smoothed particle hydrodynamics. Annual review of astronomy and

astrophysics 30, 543–574.

Monaghan, J. J., 1994a. Simulating free surface ﬂows with SPH. Journal of computational

physics 110 (2), 399–406. Bibliography 87

Monaghan, J. J., 1994b. Simulating free surface ﬂows with SPH. Journal of computational

physics 110, 399–399.

Monaghan, J. J., 2005a. Smoothed particle hydrodynamics. Reports on progress in physics

68 (8), 1703.

Monaghan, J. J., 2005b. Smoothed particle hydrodynamics. Reports on Progress in Physics

68, 1703.

Monaghan, J. J., 2012. Smoothed particle hydrodynamics and its diverse applications. An-

nual Review of Fluid Mechanics 44, 323–346.

Monaghan, J. J., Cas, R. A. F., Kos, A. M., Hallworth, M., 1999. Gravity currents descending

a ramp in a stratiﬁed tank. Journal of Fluid Mechanics 379 (1), 39–69.

Monaghan, J. J., Kocharyan, A., 1995. SPH simulation of multi-phase ﬂow. Computer

Physics Communications 87 (1-2), 225–235.

Nandasena, N., Paris, R., Tanaka, N., 2011a. Numerical assessment of boulder transport

by the 2004 Indian ocean tsunami in Lhok Nga, west Banda Aceh (Sumatra, Indonesia).

Computers & Geosciences 37 (9), 1391–1399.

Nandasena, N., Paris, R., Tanaka, N., 2011b. Reassessment of hydrodynamic equations: min-

imum ﬂow velocity to initiate boulder transport by high energy events (storms, tsunamis).

Marine Geology 281 (1), 70–84.

Nandasena, N., Tanaka, N., 2013. Boulder transport by high energy: Numerical model-ﬁtting

experimental observations. Ocean Engineering 57, 163–179. Bibliography 88

Noormets, R., Crook, K. A., Felton, E. A., 2004. Sedimentology of rocky shorelines: 3.: hy-

drodynamics of megaclast emplacement and transport on a shore platform, Oahu, Hawaii.

Sedimentary Geology 172 (1), 41–65.

Nott, J., 2003. Waves, coastal boulder deposits and the importance of the pre-transport

setting. Earth and Planetary Science Letters 210 (1), 269–276.

Novikova, T., Papadopoulos, G., McCoy, F., 2011. Modelling of tsunami generated by the

giant late Bronze Age eruption of Thera, South Aegean Sea, Greece. Geophysical Journal

International 186 (2), 665–680.

Paris, R., Fournier, J., Poizot, E., Etienne, S., Morin, J., Lavigne, F., Wassmer, P., 2010.

Boulder and ﬁne sediment transport and deposition by the 2004 tsunami in Lhok Nga

(western Banda Aceh, Sumatra, Indonesia): a coupled oﬀshore–onshore model. Marine

Geology 268 (1), 43–54.

Park, H., Cox, D. T., Lynett, P. J., Wiebe, D. M., Shin, S., 2013. Tsunami inundation

modeling in constructed environments: a physical and numerical comparison of free-surface

elevation, velocity, and momentum ﬂux. Coastal Engineering 79, 9–21.

Popinet, S., 2015. A quadtree-adaptive multigrid solver for the Serre–Green–Naghdi equa-

tions. Journal of Computational Physics 302, 336–358.

Raﬁee, A., Thiagarajan, K. P., 2009. An SPH projection method for simulating ﬂuid-

hypoelastic structure interaction. Computer Methods in Applied Mechanics and Engi-

neering 198 (33-36), 2785–2795. Bibliography 89

Rook, R., Yildiz, M., Dost, S., 2007. Modeling transient heat transfer using SPH and implicit

time integration. Numerical Heat Transfer Part B: Fundamentals 51 (1), 1–23.

Rueben, M., Holman, R., Cox, D., Shin, S., Killian, J., Stanley, J., 2011. Optical mea-

surements of tsunami inundation through an urban waterfront modeled in a large-scale

laboratory basin. Coastal Engineering 58 (3), 229–238.

Satake, K., 2007. Volcanic origin of the 1741 Oshima-Oshima tsunami in the Japan Sea.

Earth, planets and space 59 (5), 381–390.

Scheﬀers, A., Scheﬀers, S., 2006. Documentation of the impact of Hurricane Ivan on the

coastline of Bonaire (Netherlands Antilles). Journal of Coastal Research, 1437–1450.

Sethian, J. A., Smereka, P., 2003. Level set methods for ﬂuid interfaces. Annual Review of

Fluid Mechanics 35 (1), 341–372.

Shadloo, M. S., Weiss, R., Yildiz, M., Dalrymple, R. A., 2015. Numerical simulation of long

wave run-up for breaking and non-breaking waves. International Journal of Oﬀshore and

Polar Engineering 25 (1), 1–7.

Shuvalov, V., Gersonde, R., 2014. Constraints on interpretation of the Eltanin impact from

numerical simulations. Meteoritics & Planetary Science 49 (7), 1171–1185.

Sigalotti, L. D. G., Klapp, J., Sira, E., Mele´an, Y., Hasmy, A., 2003. SPH simulations

of time-dependent Poiseuille ﬂow at low Reynolds numbers. Journal of Computational

Physics 191 (2), 622–638.

Smagorinsky, J., 1963. General circulation experiments with the primitive equations: I. The

basic experiment*. Monthly weather review 91 (3), 99–164. Bibliography 90

Smith, R., 2006. Open Dynamics Engine.

Spiske, M., Weiss, R., Bahlburg, H., Roskosch, J., Amijaya, H., 2010. The TsuSedMod

inversion model applied to the deposits of the 2004 Sumatra and 2006 Java tsunami

and implications for estimating ﬂow parameters of palaeo-tsunami. Sedimentary Geology

224 (1), 29–37.

Synolakis, C., Kˆano˘glu,U., 2015. The Fukushima accident was preventable. Phil. Trans. R.

Soc. A 373 (2053), 20140379.

Synolakis, C. E., 1987. The runup of solitary waves. Journal of Fluid Mechanics 185, 523–545.

Tadepalli, S., Synolakis, C. E., 1994. The run-up of N-waves on sloping beaches. In: Proceed-

ings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences.

Vol. 445. The Royal Society, pp. 99–112.

Tadepalli, S., Synolakis, C. E., 1996. Model for the leading waves of tsunamis. Physical

Review Letters 77 (10), 2141.

Tanaka, N., Sasaki, Y., Mowjood, M., Jinadasa, K., Homchuen, S., 2007. Coastal vegetation

structures and their functions in tsunami protection: experience of the recent Indian Ocean

tsunami. Landscape and Ecological Engineering 3 (1), 33–45.

Tartakovsky, A. M., Ferris, K. F., Meakin, P., 2009. Lagrangian particle model for multiphase

ﬂows. Computer Physics Communications 180 (10), 1874–1881.

Tartakovsky, A. M., Meakin, P., 2005. A smoothed particle hydrodynamics model for miscible

ﬂow in three-dimensional fractures and the two-dimensional Rayleigh-Taylor instability.

Journal of Computational Physics 207 (2), 610–624. Bibliography 91

Tartakovsky, A. M., Ward, A. L., Meakin, P., 2007. Pore-scale simulations of drainage of

heterogeneous and anisotropic porous media. Physics of Fluids 19, 103301.

Tissier, M., Bonneton, P., Marche, F., Chazel, F., Lannes, D., 2012. A new approach to

handle wave breaking in fully non-linear Boussinesq models. Coastal Engineering 67, 54–

66.

Tryggvason, G., Bunner, B., Esmaeeli, A., Juric, D., Al-Rawahi, N., Tauber, W., Han, J.,

Nas, S., Jan, Y. J., 2001. A front-tracking method for the computations of multiphase

ﬂow. Journal of Computational Physics 169 (2), 708–759.

Unverdi, S. O., Tryggvason, G., 1992. A front-tracking method for viscous, incompressible,

multi-ﬂuid ﬂows. Journal of Computational Physics 100 (1), 25–37.

Vukovic, S., Sopta, L., 2002. ENO and WENO schemes with the exact conservation property

for one-dimensional shallow water equations. Journal of Computational Physics 179 (2),

593–621.

Wei, G., Kirby, J. T., Grilli, S. T., Subramanya, R., 1995. A fully nonlinear Boussinesq model

for surface waves. Part 1. Highly nonlinear unsteady waves. Journal of Fluid Mechanics

294, 71–92.

Weiss, R., 2012. The mystery of boulders moved by tsunamis and storms. Marine Geology

295, 28–33.

Weiss, R., Diplas, P., 2015. Untangling boulder dislodgement in storms and tsunamis: Is it

possible with simple theories? Geochemistry, Geophysics, Geosystems. Bibliography 92

Weiss, R., Lynett, P., W¨unnemann,K., 2015. The Eltanin impact and its tsunami along

the coast of South America: insights for potential deposits. Earth and Planetary Science

Letters 409, 175–181.

Weiss, R., Munoz, A. J., Dalrymple, R. A., Herault, A., Bilotta, G., 2011. Three-dimensional

modeling of long-wave runup: Simulation of tsunami inundation with GPU-SPHysics.

Coastal Engineering Proceedings 1 (32), currents–8.

Wiegel, R., 1960. A presentation of cnoidal wave theory for practical application. Journal of

Fluid Mechanics 7 (02), 273–286.

W¨unnemann,K., Lange, M., 2002. Numerical modeling of impact-induced modiﬁcations of

the deep-sea ﬂoor. Deep Sea Research Part II: Topical Studies in Oceanography 49 (6),

969–981.

W¨unnemann,K., Weiss, R., 2015. The meteorite impact-induced tsunami hazard. Phil.

Trans. R. Soc. A 373 (2053), 20140381.

Xing, Y., Shu, C.-W., 2005. High order ﬁnite diﬀerence WENO schemes with the exact

conservation property for the shallow water equations. Journal of Computational Physics

208 (1), 206–227.

Yamazaki, Y., Kowalik, Z., Cheung, K. F., 2009. Depth-integrated, non-hydrostatic model

for wave breaking and run-up. International Journal for Numerical Methods in Fluids

61 (5), 473–497.

Yang, Y., Irish, J. L., Weiss, R., 2016. Investigation of tsunami impact through discontinuous

vegetation. Submitted. Bibliography 93

Yoshizawa, A., 1986. Statistical theory for compressible turbulent shear ﬂows, with the

application to subgrid modeling. Physics of Fluids (1958-1988) 29 (7), 2152–2164.

Zhang, M., 2010. Simulation of surface tension in 2D and 3D with smoothed particle hydro-

dynamics method. Journal of Computational Physics 229 (19), 7238–7259.

Zhong, Z., Wang, K., 2009. Modeling fully nonlinear shallow-water waves and their interac-

tions with cylindrical structures. Computers & Fluids 38 (5), 1018–1025.