tsunami modeling methods to understand generation and propagation

TSUNAMI MODELING TSUNAMI MODELING METHODS TO METHODS TO UNDERSTAND UNDERSTAND

GENERATION AND GENERATION AND PROPAGATIONPROPAGATION

Parameters for wave motion

Height H = 2aLength L Local water depth h Duration/period TGravity g

HL

h

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 amplificationif 2D, then until a ~ h 4/1 ha

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

Rise in height of the waterdepth in front of the bore

= Hh

A bore may be undular, turbulent of breaking-undulardepending on the value of this ratio.

TSUNAMI MODELS

• TUNAMI N2• MOST• FUNWAVE• MIKE 21• DELFT 3D• AVI-NAMI• NAMI-DANCE• TELEMAC• …

HISTORY OF TSUNAMI MODELLING

• The TUNAMI code consists of;• TUNAMI-N1 (Tohoku University’s Numerical Analysis

Model for Investigation of Near-field Tsunamis, No.1) (linear theory with constant grids),

• TUNAMI-N2 (linear theory in deep sea, shallow-water theory in shallow sea and runup on land with constant grids),

• TUNAMI-N3 (linear theory with varying grids), • TUNAMI-F1 (linear theory for propagation in the

ocean in the spherical co-ordinates) and• TUNAMI-F2 (linear theory for propagation in the

ocean and coastal waters).

TSUNAMI MODELING

• Nonlinear Shallow Water Equations (NSW),• numerical solution procedure is from Shuto, N.,

Goto, C., Imamura, F., 1990 and Goto, C. and Ogawa, Y.,1991,

• TUNAMI N2 authored by Profs. Shuto and Imamura, and developed/distributed under the

support of UNESCO TIME Project in 1990s.

Governing Equations

0

y +hv

x+hu

t

0 x

y

u v

x

u

t

u

xgu

0y

y

vv

x

v

t

v

ygu

η : water elevationu, v : components of water velocities in x and y directionsy : bottom shear stress components ح ,xحt : timeh : basin depthg : gravitational acceleration

Non-linear longwave equations

uDhuM )(

,

vDhvN )(

0y

x

t

NM

0D D

MN

y

D

M

x

t

227/3

22

NMM

gn

xgD

M

0D D

N

y

D

MN

x

t

227/3

22

NMN

gn

ygD

N

M, N : Discharge fluxes in x&y directions

n : Manning’s roughness coefficient

Numerical Model “TUNAMI N1”

0

xg

t

u 0

yg

t

v

0][][

y

vh

x

uh

t

Mesh resolution and time step, grid size

Total reflection on land boundaries

Boundary Conditions

Reflection:

0n

ght

0n

Open Boundary:

Initial Condition: u(x,y,0)

v(x,y,0)

(x,y,0)

Numerical TechniqueFinite Difference " Leap Frog"

j+1

j

i

2

1

2

1,

k

jiN

21

21

,

k

jiN

2

1

,2

1

k

jiM 2

1

,2

1

k

jiM

1

,

k

ji

y

xi-1 i+1

j-1

y

x

k

ji

k

jitt ,

1

,

1

2

1

,2

12

1

,2

1

1 k

ji

k

jiMM

xxM

2

1

2

1,

2

1

2

1,

1 k

ji

k

jiNN

yyN

2

1

2

1,

2

1

2

1,

2

1

,2

12

1

,2

1,

1

,

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji NNyt

MMxt

Difference Scheme

Terms

0

xgD

tM

k

ji

k

ji

k

ji

k

ji

k

ji xt

gDMM ,,1,

2

12

1

,2

12

1

,2

1

k

ji

k

jiji

k

jiji

k

jihhD ,,1

,2

1,

2

1,

2

1,

2

1 21

k

ji

k

jiji

k

jiji

k

jihhD ,1,

2

1,

2

1,

2

1,

2

1, 2

1

k

ji

k

jiji

k

ji

k

ji

k

ji yt

hgDNN ,1,

2

1,

2

1,

2

1

2

1,

2

1

2

1,

Direction x

Direction y

h >

Convective Terms

2

1

,2

1

2

2

1

,2

1

31

2

1

,2

1

2

2

1

,2

1

21

2

1

,2

3

2

2

1

,2

3

11

21

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

D

M

D

M

D

M

xD

M

x

2

1

11,2

1

2

1

1,2

12

1

1,2

1

31

2

1

,2

1

2

1

,2

12

1

,2

1

21

2

1

1,2

1

2

1

1,2

12

1

1,2

1

11

1k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

D

NM

D

NM

D

NM

yDMN

y

Truncation in the order of x

2

1

2

1,1

2

1

2

1,1

2

1

2

1,1

32

2

1

2

1,

2

1

2

1,

2

1

2

1,

22

2

1

2

1,1

2

1

2

1,1

2

1

2

1,1

12

1k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

D

NM

D

NM

D

NM

xDMN

x

21

21

,

2

21

21

,

3221

21

,

2

21

21

,

2221

23

,

2

21

23

,

12

21

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

D

N

D

N

D

N

yD

N

y

Friction Term

2

2

1

,2

1

2

2

1

,2

12

1

,2

12

1

,2

13/7

2

1

,2

1

2

22

3/7

2

21

k

ji

k

ji

k

ji

k

jik

ji

NMMM

D

gNMM

D

g

2

2

1

2

1,

2

2

1

2

1,

2

1

2

1,

2

1

2

1,

3/7

2

1

2

1,

2

22

3/7

2

21

k

ji

k

ji

k

ji

k

jik

ji

NMNN

D

gNMN

D

g

Discretization

Tunami-N2

Programme TIME : Tsunami Inundation Model Exchange

RECENT TREND IN TSUNAMI MODELING

• Simulation and Animation

for Visualization

INPUT PARAMETERS

• Arbitrary shape bathymetry

• Tsunami source as initial condition

RECENT TREND IN TSUNAMI MODELING

• AVI-NAMI and NAMI DANCE simulation/animation software in C++ Language

• are brothers of TUNAMI N2

• authored by Pelinovsky, Kurkin, Zaytsev, Yalciner

Pelinovsky, Kurkin, Zaytsev, Yalciner, Imamura

Pelinovsky, Kurkin, Zaytsev, Yalciner, Imamura

Wl Lmajor

Lminor

al al

ac

TERMS

• Bottom Friction

• Pressure

• Dispersion – FUNWAVE by Kirby

– Fujima

December 26, 2004

Pelinovsky, Kurkin, Zaytsev, Yalciner, Imamura

December 26, 2004

March 28, 2005

Andaman Source

1762

Pelinovsky, Kurkin, Zaytsev, Yalciner, Imamura

MACRAN FAULT

Pelinovsky, Kurkin, Zaytsev, Yalciner, Imamura

Pelinovsky, Kurkin, Zaytsev, Yalciner, Imamura

Andaman Source

Pelinovsky, Kurkin, Zaytsev, Yalciner, Imamura

Mindanao Source

Pelinovsky, Kurkin, Zaytsev, Yalciner, Imamura

Hypothetical Tsunami Source at offshore Sabah as an example simulation in South China Sea

Pelinovsky, Kurkin, Zaytsev, Yalciner, Imamura

Hypothetical Tsunami Source at offshore Sabah as an example simulation in South China Sea

ASSESMENT OF TSUNAMI HAZARD

Simulation and animation of probable/credible tsunami scenarios, and understanding coastal amplification and arrival time of tsunamis

Acknowledgements Prof. Shuto, Imamura, Synolakis, Okal, Pelinovsky, Zaytsev

UNESCO IOC, Tohoku University Japan

Ministry of Marine Affairs and Fisheries Republic of Indonesia,

UTM, DID, ATSB, Dept. of Meteorolgy, Malaysia,

Middle East Technical University, METU, Yildiz Technical University, Chambers of Geological and Civil Engineers of Turkey,

Dr. Eng. Dinar Catur Istiyanto Ir. Widjo Kongko, M. Engand, Russian Colleagues and Team, American Colleagues and Team, Japanese Colleagues and Team, Prof. Ir. Widi

Agoes Pratikto, Dr. Ir. Subandono Dipsosaptono, Dr. Gegar Sapta Prasetya, Dr. Ir. Rahman Hidayat

THANKS and APPRECIATION THANKS and APPRECIATION

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

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.

0)cos(cos

1

NM

RtfN

R

gh

t

M

cos

fMR

gh

t

N

0

coscos

cos

1 2

1

2

1,

2

1

2

1,,

2

1,

2

12

1

,2

1

,

m

n

mjm

n

mj

n

mj

n

mj

m

n

mj

n

mj

NNMM

Rt

η : water elevationR : radius of earthM, N : discharge fluxes along λ and Өf : Coriolis coefficientg : gravitational acceleration

NfR

gh

t

MM n

mj

n

mj

m

mj

n

mj

n

mj

2

1

,2

1

,1,

2

1,

2

11

,2

1

cos

MfR

gh

t

NN n

mj

n

mj

m

mj

n

mj

n

mj

2

1

,2

1

1,,

2

1,

2

11

,2

1

sin

n

mj

n

mj

n

mj

n

mjNNNNN

2

1,

2

1,

2

1,1

2

1,14

1

n

mj

n

mj

n

mj

n

mjMMMMM

,2

11,

2

1,

2

1

2

1,

2

14

1

where;

2

1

2

1,

2

1

2

1,,

2

1,

2

112

1

,2

1

, coscosm

n

mjm

n

mj

n

mj

n

mj

n

mj

n

mj NNMMR

NRhRMMn

mj

n

mjmj

n

mj

n

mj

3

2

1

,2

1

,1,

2

12,

2

11

,2

1

MRhRNNn

mj

n

mjmj

n

mj

n

mj

5

2

1

,2

1

1,

2

1,

4

2

1,

1

2

1,

Computation Points for Water Level and Discharge

R1 = t / (Rcosm)

R2 = g.t / (Rcosm)

R3 = 2tsinm

R4 = gt / (R) R5 = 2tsinm+1/2

where; , , t : directions , , t : grid lengths : angular velocity

TWO-LAYER NUMERICAL MODEL FOR TSUNAMI GENERATION AND PROPAGATION

• 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

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 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.

Numerical Approach

Numerical Approach

Points schematics of the staggered leap-frog scheme (Imamura, Imteaz (1995))

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.

• 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

Basin and Parameters

- 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

TWOLAYER

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

0.00 20.00 40.00 60.00 80.00 100.00 120.00 140.00

East-W est D irection (km .)

0.00

20.00

40.00

60.00

80.00

100.00

120.00

So

uth

-No

rth

Dir

ecti

on

(km

.)

1 2 3 4

5 6

h+ : increase of water depth in the eroded area due to the mass failureh- : decrease of water depth in the accreted area due to the mass failureL+ : length of the eroded areaL- : 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-

Sea bottom

before mass failure

Sea bottom

after mass failure