1. TSUNAMI MODELING

2. Content - Governing Equations
- Linear form of Shallow Water Equations in Spherical Coordinates
for Far Field Tsunami Modeling
- TWO-LAYER Numerical Model for Tsunami Generation
and Wave Propagation
- Comparison of Analytical and Numerical Approaches for Long Wave Runup

3. Tsunami’s approach to the shore Summary

4. Context Two scenarios need consideration:
Locally generated tsunamis For this case warning commonly comes from perceiving earthquake motion unless caused by landslide. Timescale for warning – a few minutes.
Tsunamis arriving after significant propagation Usually approaching from deep water. Maybe an hour or more available for warning.
For both cases there is need for consideration of flows at various scales, including: oceanic, regional, coastal features, and local structures, i.e. “nested” models for assessment of vulnerable areas.

5. HL h
Parameters for wave motion
Height H = 2a Length L Local water depth h Duration/period T Gravity g

7. 1. Very small, very slight slopes
The relative sizes of these three lengths determine a wave’s behaviour
and the appropriate approximate equations. h
1. Very small, very slight slopes
a << h a << L
Linear waves
L >> h L ~ h L << h Long waves intermediate depth deep water waves non-dispersive dispersive
generally appropriate for the deep ocean

8. a L h 2. not so small, very long:
The relative sizes of these three lengths determine a wave’s behaviour
and the appropriate approximate equations. h
2. not so small, very long:
The shallow water equations are appropriate for very many aspects of tsunami flows, but the steepening they describe can lead to growth of undulations and wave breaking, and thus failure of the approximation.
L >> h
a << H a ~ h a >> h Long waves shallow water waves as above wave front steepening .

9. a L h 2. not so small, long L >> h
The relative sizes of these three lengths determine a wave’s behaviour
and the appropriate approximate equations. h
2. not so small, long
L >> h
a << H a ~ h a >> h Long waves shallow water waves as above wave steepening till or L ~ h
weak nonlinearity balances weak dispersion Boussinesq’s equations solitary waves, undular bores

10. a L
The relative sizes of these three lengths determine a wave’s behaviour
and the appropriate approximate equations. h
3. If wave height, water depth and typical wave length scales become comparable then there are no useful approximations. The full Euler equations need to be solved and even then there is no good method fro dealing with wave breaking.
In such cases one usually has to make do with the approximation of bores modelled as discontinuities within the shallow water equations.

11. Shoaling
Typical change in water depth as tsunamis leave the ocean for coastal waters is from around 4km to 100m on the continental shelf to zero at the coastline.
The topography of this change is very relevant: for a steep approach there is much wave reflection and amplitudes are not greatly increased consider ordinary waves at a cliff: ´ 2
gently sloping topography, leads to large amplification if 2D, then until a ~ h

12. Approaching the shoreline
As they approach the shoreline ordinary wind generated waves break. Long waves such as tsunamis are more like tides, which only break in the special circumstances of long travel distances in shallow water. Then tsunamis are similar to tidal bores.
For example tsunamis can have periods approaching one hour, and in the River Severn near Gloucester spring tides can rise from low to high tide in one hour. The character of a bore depends strongly on the ratio
H h
Rise in height of the water depth in front of the bore =
A bore may be undular, turbulent of breaking-undular depending on the value of this ratio.

13. H h H h H h H h > 0.6 Turbulent bore > 0.3 Breaking/undular bore
> 0.6
Turbulent bore
H h
> Breaking/undular bore
0.6 > H h
H h
0.3 > Undular bore
These properties can be used to judge water depth when watching bores Peregrine, 2005.

14. Numerical Model “TUNAMI N1”
Mesh resolution and time step, grid size
Total reflection on land boundaries

15. Governing Equations Non-linear longwave equations η : water elevation
u, v : components of water velocities in x and y directions
حx, ح y : bottom shear stress components
t : time
h : basin depth
g : gravitational acceleration

16. M, N : Discharge fluxes in x&y directions
n : Manning’s roughness coefficient ,

17. Boundary Conditions Reflection Open Boundary Initial Condition
u(x,y,0) = v(x,y,0) = h(x,y,0)

18. Numerical Technique Finite Difference " Leap Frog" y x j+1 j Dy j-1 Dx

19. Convective Terms
Truncation in the order of Dx

21. Friction Term
Discretization

22. Programme TIME : Tsunami Inundation Model Exchange
Tunami-N2

23. TUNAMI – N2 “Simulation” of propagation of long waves
solves for irregular basins
computes water surface fluctuations and velocities
is applied to Several Case Studies in Several Sea and Oceans
Application to Black Sea
( for 1939 and 1966 tsunamis )

24. Erzincan Tsunami – 1939, December 26
39,000 people died .
Epicenter was far from shore .
Comparison between measurements and numerical
solution of TUNAMI-N2 was made.

25. Erzincan, Turkey, 1939, December 26. M = 8 Earthquake
Erzincan, Turkey, 1939, December 26. M = 8 Earthquake. The sea receded 50 m near Fatsa. Tide-gauge, 53 cm at Novorossyisk
1939

26. 1939 Event. Confirmation of Trans-Sea Crossing 1939 Sevastopol Yalta
Feodosiya
Mariupol
Kerch
Novorossiysk
Tuapse
Poti
Batumi
1939

27.  Length of fault line is 120 km.
 The crest and trough amplitudes are 0.3 m and -0.9 m. respectively.

28. Tsunami arrives northern coasts between 25 minutes and 2 hours.
Distribution of maximum positive tsunami amplitudes at Black Sea coasts
Tsunami arrives northern coasts between 25 minutes
and 2 hours.
Sea level oscillations are triggered in entire basin of the
Black Sea.

29. Comparison of instrumental and numerical records

30. Anapa Tsunami – 1966, July 12 Magnitude : 5,8 Intensity : 6
Epicenter was 10 km. away from shore .
Comparison between measurements and numerical
solution of TUNAMI-N2 was made.

31. Linear Form of Shallow Water Equations in Spherical Coordinates
for Far Field Tsunami Modeling
Dispersion term is considered by Boussinesq Equation.
Long waves (small relative depth)  avertical << agravitational
Velocity of water particles are vertically uniform.

32. η : water elevation
R : radius of earth
M, N : discharge fluxes along λ and Ө
f : Coriolis coefficient
g : gravitational acceleration

33. where;

35. Computation Points for Water Level and Discharge
R1 = t / (Rcosm)
R2 = g.t / (Rcosm)
R3 = 2tsinm
R4 = gt / (R)
R5 = 2tsinm+1/2
where; , , t : directions
 ,  ,  t : grid lengths
 : angular velocity

36. TWO-LAYER NUMERICAL MODEL FOR TSUNAMI GENERATION AND PROPAGATION

37. TWOLAYER
The mathematical model TWOLAYER is used as a near-field tsunami modeling version with two-layer nature and combined source mechanism of landslide and fault motion
In two-layer flow both layers interact and play a significant role in the establishment of control of the flow. The effect of the mixing or entrainment process at a front or an interface becomes important (Imamura and Imteaz, (1995)).
Two-layer flows that occur due to an underwater landslide can be modeled using a non-horizontal bottom with a hydrostatic pressure distribution, uniform density distribution, uniform velocity distribution and negligible interfacial mixing in each layer (Watts, P., Imamura, F., Stephan. G., (2000)).
Twolayer flow, which may result from density differences within the fluid, can also be applied in the case of underwater landslides accompanying underwater earthquakes which generate tsunamis.
In twolayer flow the interaction between the two layers and the resulting effect is important.
For the case of underwater landslides, the two-layer flow can be modeled using a non-horizontal bottom, a hydrostatic pressure distribution, uniform velocity distribution and negligible interfacial mixing in each layer.

38. Theoretical Approach
Conservation of mass and momentum can be integrated in each layer, with the kinetic and dynamic boundary conditions at the free surface and interface surface (Imamura and Imteaz 1995)).
η : surface elevation
h : still water depth
ρ : is the density of the fluid
1,2 : upper and lower layer respectively (Imamura and Imteaz,(1995))
The governing equations used in this model are derived by integrating the Euler equations of mass and momentum continuties in each layer ( including full nonlinearity) (and assuming a long wave approximation).
The kinetic and dynamic conditions are applied at the free surface and the interface.
(Kinetic condition states that the water particle at the water surface remains on the surface throughout the motion; therefore the vertical velocity of this water particle is equal to the ratio of the change of water surface elevation (Delta NU) to the change of time (Delta t). )
(Dynamic condition states that the pressure at the water surface is equal to the atmospheric pressure and its absolute value is assumed as zero. At the interface the pressure is equal to the formula: p=RO1XGX(NU1+H1-NU2))
The parameters used in derivation of the formulas can be seen in this figure, where ETA is the surface elevation, H is the still water depth, RO is the density of the fluid and the subscripts 1 and 2 indicate the upper and lower layers.
The derived equations can be found in the relevant chapter and the derivation procedure is summarized in Appendix A.

39. Numerical Approach
The numerical model TWO-LAYER is developed in Tohoku University, Disaster Control Research Center by Prof. Imamura.
The model computes the generation and propagation of tsunami waves generated as the result of a combined mechanism of an earthquake and an accompanying underwater landslide.
It computes the propagation of the wave by calculating the water surface elevations and water particle velocities throughout the domain, at every time step during the simulation.
The staggered leap-frog scheme (Shuto, Goto, Imamura, (1990)) is used to solve the governing equations.
The model TWOLAYER is developed in Tohoku University, Disaster Control Research Center, Japan by Fumihiko Imamura.
As I have mentioned at the introduction, this model uses a combined source mechanism of an earthquake and an accompanying underwater landslide for tsunami generation.
Then it computes the propagation of the wave by using the staggered leap frog scheme and calculates the water surface elevation and water particle velocities throughout the domain at every time step.
(Staggered LFC: is an explicit central difference scheme with the truncation error of the second order. According to this scheme, the computation point of one variable, ETA, does not coincide with the computation point of another variable DISCHARGE (M). There are half step differences, 1/2 DELTA T and 1/2 DELTA X between computation points of two variables as shown in Figure 3.2 on Page 14 of the thesis. Thus one variable, ETA, is placed at the middle of DELTA T DELTA X rectangle and the other varibles at the four corners and vice versa.)
(The Courant-Friedrichs-Lewys Condition is applied to derive a stability condition: DELTAX/DELTA <=MAXIMUM CELERITY (LAMDA/PERIOD) OR DELTAT<=DELTAX/MAX (C1,C2))

40. Numerical Approach
Points schematics of the staggered leap-frog scheme (Imamura, Imteaz (1995))

41. Test of the Model
The model TWO-LAYER is tested by using a regular shaped basin for modeling of generation and propagation of water waves due to underwater mass failure mechanisms.
In order to obtain accurate results the duration and domain of simulation as well as the characteristics of the mass failure mechanism must be chosen accurately and described very precisely. For stability the time step and grid size should also be selected properly.
In this part the model is applied to a regular shaped rectangular basin for test purposes.
The properties of the simulation domain, the simulation time as well as the properties of the underwater mass failure mechanism are very important for the accuracy of the test results.

42. Basin and Parameters Rectangular basin w= 150 km. l= 125 km.
Three boundaries of this basin (at East, North and West) are set as open boundaries to avoid wave reflection and unexpected amplification inside the basin as shown in the figure below.
The land is located at the South
Uniformly sloping bottom starting with -100m. elevation at land and deepen up to 2000 m with a slope of 1/60.
Grid spacings: 400 m. with : 375 nodes in E-W
: 313 nodes in S-N
22 stations were selected to observe the water surface fluctuations
The test basin which is shown in this figure is a rectangular basin which is 150 km. wide and 125 km. long. The grid spacing used for the calculations is 400 m, which means a total of 375 nodes in eats-west direction and 313 nodes in south-north direction.
The land is located at the south and shown in yellow colour.Other three boundaries are open boundaries in order to avoid the wave reflection.
The land is 100 m. high and the deepest point of the test basin located at the north is 2000 m. deep, which means that there is a uniform bottom slope of 1 to 60 throughout the test basin.
In the mass failure area as shown in the figure the red parts show the slided region and the blue parts show the deposited region.
Initially 22 stations are chosen in the test basin for observing the resulting water surface levels. 6 stations as shown on the figure are chosen for presenting the obtained results as you will see in the next slide.

43. TWOLAYER
- solves the generation of the tsunami wave due to the mass failure mechanism at the source area
- calculates the water surface elevations at each grid point while propagating the wave in the basin.
- obtains the time histories of the water surface elevation at all grid points and stores 22 selected stations

44.  Mass failure mechanism is generated at a smaller rectangular region
inside the basin (w: 20 km.; l: 40 km )

45. The conservation of the moved volume of sediment
Sea bottom
after mass failure
Sea bottom
before mass failure
h+ : increase of water depth in the
eroded area due to the mass failure
h- : decrease of water depth in the
accreted area due to the mass failure
L+ : length of the eroded area
L- : length of the accreted area
Initial and final profile of the sea bottom in the mass failure area
The conservation of the moved volume of sediment
before and after the failure
h+ . L+ = h- .L-

47. COMPARISON OF ANALYTICAL AND NUMERICAL APPROACHES FOR LONG WAVE RUNUP

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

65. Initial Wave
Solitary Wave
N-Wave

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

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

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

69. Runup of Solitary Waves

70. Runup of Solitary Waves

71. Runup of Solitary Waves

72. Runup of Solitary Waves

73. Runup of Solitary Waves

74. Runup of Solitary Waves

75. Runup of Solitary Waves

76. Runup of Solitary Waves

77. Runup of Solitary Waves

78. Runup of Solitary Waves

79. Runup of Generalized N-waves

80. Runup of Generalized N-waves

81. Runup of Generalized N-waves

82. Runup of Generalized N-waves

83. Runup of Generalized N-waves

84. Runup of Generalized N-waves

85. Runup of Generalized N-waves

86. Runup of Generalized N-waves

87. Runup of Generalized N-waves

88. Runup of Isosceles N-wave

89. Runup of Isosceles N-wave

90. Runup of Isosceles N-wave

91. Runup of Isosceles N-wave

92. Runup of Isosceles N-wave

93. Runup of Isosceles N-wave

94. Runup of Isosceles N-wave

95. Runup of Isosceles N-wave

96. Runup of Isosceles N-wave

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

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

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