TSUNAMI MODELING

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” modelsfor assessment of vulnerable areas.

5. HL h Parameters for wave motion Height H = 2aLength L Local water depth h Duration/period TGravity g

7. a L The relative sizes of these three lengths determine a wave’s behaviour and the appropriate approximate equations. h a << h a << L Linear waves L >> hL ~ hL << hLong waves intermediate depth deep water wavesnon-dispersive dispersive generally appropriate for the deep ocean

8. a L The relative sizes of these three lengths determine a wave’s behaviour and the appropriate approximate equations. h L >> h a << Ha ~ ha >> hLong waves shallow water waves .as above wave front steepening.

9. a L The relative sizes of these three lengths determine a wave’s behaviour and the appropriate approximate equations. h L >> h a << Ha ~ ha >> hLong waves shallow water waves .as above wave steepening tillor L ~ h weak nonlinearity balances weak dispersionBoussinesq’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

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 Hh Rise in height of the waterdepth in front of the bore = A bore may be undular, turbulent of breaking-undulardepending on the value of this ratio.

13. Hh > 0.6 Turbulent bore Hh > 0.3 Breaking/undular bore 0.6 > H h Hh 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, حy : 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 j+1 j Dy Dx j-1 i-1 i i+1 x

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 = 8Earthquake. The sea receded 50 m near Fatsa. Tide-gauge, 53 cm at Novorossyisk 1939

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

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

28. 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)).

38. 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)) Theoretical Approach

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.

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.

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

43. TWOLAYER - solves the generation of the tsunami wave due tothe mass failure mechanism at the source area - calculates the water surface elevations at each grid point while propagatingthe 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. 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.