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Pujiraharj, et al – Tsunami Simulation Using Dispersive Wave Model 371

INTRODUCTION

The Indian Ocean Tsunami (IOT) on 26

December 2004 is recorded as highest tsunami.Triggered by tectonic earthquake, the tsunamiwaves are generated by a complicated bottomuplift/downlift with multiple components ofamplitude and frequency. The tsunami wasglobally propagated over the world ocean astrans-oceanic tsunami propagation. Verycomplicated structures of spatial and temporalwaves are observed by many researchers. Oneof many lessons from the IOT event is thetsunami wave is remarkably dispersive. Kulikov2005 reported the dispersion effect tsunamiwaves in the Indian Ocean from wavelet

analysis based on satellite data record. It isindicated that for trans-oceanic tsunamipropagation, dispersion effect could besignificant factor for prediction of maximumamplitude. In the coastal area tsunami wavesinteract with very complicated bathymetry sothe nonlinear combination will affect the profileof tsunami. Hence, model equations whichinclude both dispersive and nonlinear terms areneeded for better estimation.

Preliminary results of dispersive numericalmodel of IOT has been done by Watts et al.2005 using Boussinesq-type model. Then more

detail study of dispersion effect for IOT hasbeen conducted using nonlinear shallow water,nonlinear Boussinesq and the full nonlinearNavier-Stokes models by Horillo et al. 2006.Grilli et al. 2007 also discussed the dispersioneffect for IOT event. From the discussions, thedispersion effect is noticed at the south-westdirection, while at the east part tsunami isessentially nondispersive.

The aims of this study is to reproducesimulation of IOT event using two differentmodel equations i.e.: dispersive and

nondispersive wave models as tools to studythe dispersion effect of IOT at the initial stage(up to 3 hours tsunami propagation). Themodels results are compared each other andagainst observations data. Betterunderstanding and prediction of tsunamipropagation and runup are important fortsunami warning system and for evacuation ofpeoples when tsunami occurred.

TSUNAMI SIMULATION USING DIPERSIVE WAVE MODEL

Alwafi Pujiraharjo1)

and Tokuzo HOSOYAMADA2)

1)Brawijaya University, Indonesia

[email protected] 2)

Nagaoka University of Technology, Japan [email protected]

ABSTRACT

Tsunami is basically a long wave in which the wavelength is much longer compared to water depth. Hence,

modeling of tsunami is usually conducted by using nonlinear shallow water equations which not include

dispersion effect. Although propagated as a longwave, the dispersion plays a role when tsunami globally

propagated through the world ocean. While the non-linear effects are important when tsunami propagated to

the coastal area. The effects of dispersion for tsunami propagation are numerically investigated in this study.

Weakly nonlinear dispersive wave model and nondispersive wave model i.e.: nonlinear shallow water

equations are used here to simulate tsunami generation, propagation and runup. The model equations are finite differenced using predictor-corrector scheme. The numerical models are then applied to make

simulation of the Indian Ocean Tsunami on 26 December 2004. Model simulation results are compared each

other and against observations data. General features of tsunami wave patterns showed good agreement

compared with observations data. Tsunami arrival time and maximum runup are also well predicted.

Keywords: : tsunami, Boussinesq equations, dispersion, runup



372 Enhancing Disaster Prevention and Mitigation @ ICSBE2010 (UII), Indonesia, ISBN 978-979-96122-9-8

MODEL EQUATIONS AND NUMERICALSOLUTION

Model Equations

Two sets of model equations are used herei.e.: nonlinear shallow water (NLSW) equationsand extended weakly nonlinear Boussinesq-type (WNB) equations of Nwogu (1993). Time-

dependent of water depth (bottom) terms areincluded to the models based on derivation ofLynett and Liu 2002. By including the time-dependent water depth, the models could beimplemented to simulate tsunami generation bytectonic plate motions, earthquakes andunderwater landslides. The sets of modelequations contain equation for conservation ofmass and momentum conservation. The modelequations are taken in the following form

( )

( ) ( )( ) ( )

2 21 1

1 2 6

12

( ) 0

η + + ∇ ⋅ + η

+ γ ∇ ⋅ − ∇ ∇ ⋅

+ + ∇ ∇ ⋅ + =

u

u

u

ɶ

ɶɶ

ɶɶ

t t

t

h h

z h h

z h h h h

(1)

( ) ( )211 2

( )

( )

0

+ ∇η + ⋅∇

+ γ ∇ ∇ ⋅ + ∇ ∇ ⋅ +

+ − =

u u u

u u

F F

ɶ ɶ ɶ

ɶ ɶɶ ɶ

t

t t

b br

g

z z h h (2)

where h is the still water depth, η is freesurface elevation, g is the gravitational

acceleration, while h is time dependent water

depth. Subscript t denotes partial derivativewith respect to time. Two-dimensional vector

differential operator ∇ is defined by ( ), x y∂ ∂

∂ ∂∇ = .

Equations (1) and (2) are full form of model

equations used here. Setting variables γ 1 = 1and horizontal velocity vector ( , )=uɶ ɶ ɶu v as

velocity at an arbitrary level, ɶ z , reduces the

model equations to WNB equations in which ɶ z is recommended to be evaluated at

0.531= −ɶ z h (Nwogu, 1993). Then setting γ 1 = 0

and using depth averaged horizontal velocityvector ( , )=uɶ u v i.e. by taking 0=ɶ z reduces

the model equations to NLSW equations.

The Fb and Fbr terms in (2) are additionalterms to accommodate bottom friction andenergy dissipation caused by breaking waves,respectively. The bottom friction terms aregiven in quadratic formula. Although the frictioncoefficient should be a function of bottom

roughness and velocity profile but a simpleconstant friction coefficient is used in this study.Eddy viscosity formula is used to model theturbulent mixing and energy dissipation causedby breaking waves. Treatment of wavebreaking is similar to the eddy viscosity-typeformula proposed by Kennedy et al. 2000 andChen et al. 2000.

Numerical Solution

In order to eliminate the error terms to thesame form of dispersive terms in the WNBmodel equations, fourth-order accuracy ofnumerical scheme for time stepping and first-order spatial derivatives are used (Wei andKirby, 1995). High order predictor-correctorscheme is used for time stepping, employingthird order time explicit Adam-Bashforthscheme as predictor and fourth order Adam-Moulton implicit scheme as corrector step. The

corrector step must be iterated until aconvergence criterion is satisfied. The systemequations are written in a form that makesconvenient for application high-order timestepping procedure. Hence, in Cartesiancoordinate system, (1) and (2) are written in thefollowing form

[ ]

[ ]

1

1

1

( , , ) ( ) ,

( , , ) ( ) ,

( , , ) ( )

η = η +

= η +

= η +

t t

t t

t t

E u v E h

U F u v F v

V G u v G u

(3)

( ) ( )

( )

( )

3 2

1 2

3 2

1 2

( , , )

( ) ( ) ( )

( ) ( ) ( )

η = − −

− + + +

− + + +

x y

xx xy xx xy x

xy yy xy yy

E u v Hu Hv

a h u v a h hu hv

a h u v a h hu hv

(4)

( ) ( )2 2

1 2 2( ) = − − −

x y x y

E h h a h h a h h (5)

2( , , ) ( )η = − η − − − +

x x y b br F u v g u vu F F (6)

1 1 2 2( ) ( ) = + + xy xy x

F v h b hv b hv b h (7)

2( , , ) ( )η = − η − − − + y x y b br G u v g uv v G G (8)

1 1 2 2( ) ( ) = + + xy xy y

G u h b hu b hu b h (9)

2

1 2 ( )= + + xx xxU u b h u b h hu (10)




2

1 2 ( )= + + yy yyV v b h v b h hv (11)

where = + η H h is total water depth. Subscript

t denotes partial derivative with respect to time,while subscript x and y denote spatialderivatives in the x and y direction, respectively.

Variables1 2 1 2, , ,a a b b are defined as

2 21 1 1 11 2 12 6 2 2

2

, , , / 0.531

= β − = β + = β= β = = −ɶ

a a bb z h

(12)

for WNB model equations and

1 2 1 20= = = =a a b b for NLSW model equations.

Following Wei and Kirby 1995, Adam-Bashforth scheme is used for predictor stepand written as

( )1 1 1 2

, , , , ,12

1 2

1 , 1 , 1 ,

23 16 5

2( ) 3( ) ( )

+ + − −∆

− −

Ω = Ω + Φ − Φ + Φ

+ Φ − Φ + Φ

n n n n nt i j i j i j i j i j

n n n

i j i j i j

(13)

Adam-Moulton scheme is used for correctorstep and written as

( )1 1 1 1 2

, , , , , ,24

1

1 , 1 ,

9 19 5

( ) ( )

+ + + − −∆

+

Ω = Ω + Φ + Φ − Φ + Φ

+ Φ − Φ

n n n n n nt i j i j i j i j i j i j

n n

i j i j

(14)

where ( , , )Ω = η U V , ( , , )Φ = E F G and

1 1 1 1( , , )Φ = E F G .

The values of u and v at time level (n +1) couldbe obtained by solving (10) and (11) usingdouble sweep algorithm to solve tridiagonalmatrix system.

i i+1i-1

j

j-1

j+1

∆ x

∆ y

(η, h)i,j

ui,j

vi,j

Figure 1. Staggered grid system for spatialdiscretization

A staggered grid system (C grid) in space isused to discretize spatial derivatives as shownin Figure 1. The horizontal velocity vectors (u ,

v ) and sea level (η) are organized into tripletsas visualized by triangle in Figure 1. The waterdepth is defined at the same point of sea levelat the cell center, while vectors such as velocitycomponents u and v are defined at theinterfaces of the cell. At the cell interface,

scalars are obtained by linear interpolation. Forexample, the total water depth at u point can beobtained by

( ) ( ) ( )+ ++ η = + η + + η1 1, , 1, 1,2 2

,i j i j i j i j

i j h h h (15)

Spatial discretizations are required for variousorders of spatial derivatives on the right-handside of (3) and (4) which include first-order,second-order and second-order crossderivatives. The first-order derivative of

( )= + ηf h u in the x direction is discretized by

four-point finite difference method to eliminatesfifth-order physical dispersion in the governingequations as follows

− − +− + −∂ = ∂ ∆

2, 1, , 1,

,

27 27

24

i j i j i j i j

i j

f f f f f

x x (16)

The fourth-order accurate for first-order space

derivative of η at u -point is

− + +η − η + η − η∂η = ∂ ∆

1, , 1, 2,

,

27 27

24

i j i j i j i j

i j x x (17)

First-order derivative of u2

in the x direction isdiscretized by the following scheme

− − + +− + − ∂=

∂ ∆

2 2 2 222, 1, 1, 2,

,

( ) 8( ) 8( ) ( )

12

i j i j i j i j

i j

u u u u u

x x

(18)

Three-point central scheme is used for secondorder derivatives in the x direction as

− +

∂ = − +

∂ ∆

2

1, , 1,2 2

,

12i j i j i j

i j

w w w w

x x

(19)

where w = u or (hu ). Similar expressions can beobtained in the y direction for both first-orderand second-order derivatives. The cross-derivative terms in the x direction of u xy and(hu )xy are approximated by the following finitedifference scheme




− + − +

∂ = + − − ∂ ∂ ∆ ∆

2

1, , 1 1, 1 ,

,

1i j i j i j i j

i j

w w w w w

x y x y

(8)

Again, w = u or (hu ) and similar expressionscan be obtained in the y direction for v xy and(hv )xy .

MODEL SIMULATIONS AND DISCUSSIONS

Simulation Conditions

Simulations of The Indian Ocean Tsunami26 December 2004 are conducted using thenumerical models. In order to minimize the gridsize and achieves resolution but accommodategauges and satellite data, numerical domain isselected around Bay of Bengal from longitude70

0E – 100

0E and latitude 7

0S – 23

0N.

Bathymetry data is taken from ETOPO2databank and refined into one minute resolution

by linear interpolation resulting 1800 by 1800 ofgrid points with about 1.852 km x 1.852 km gridinterval in the Cartesian coordinate. Compareto spherical coordinate, the Cartesiancoordinate has embedded error of griddefinition. The maximum error in the x directionis about 237,076 km (8.58 %) at the north part(23

ON) numerical domain and 44,052 km (1.49

%) at the south part (10oS). While at the middle

computation domain (8oN) the error is about

27,661 km (0.93 %). It is decided to do notmake correction to domain error because most

part of research discussion is close to theequator line. The mean water level specified inthe model simulation did not include the effectsof tides. Coriolis effects also did not include inthe models computation.

According to the grid resolution, timeinterval was chosen to be 2 seconds due tonumerical stability of the model. Radiationboundaries are applied at the south, west andeast part of numerical domain by addingsponge layers at the corresponding boundaries.Artificial slot technique for treatment ‘wet-dry’condition for runup of Chen et al. 2000 and

Kennedy et al. 2000 is used here. Thecontinuity equation (1) must be modified toimplement the artificial slot. Detail ofimplementation of artificial slot technique forrunup treatment is referred to Kennedy et al.2000 and Chen et al. 2000. Constant bottomfriction coefficients of 0.001 and 0.0 are appliedwhen the water depth less and greater than 1km, respectively.

-4000

-3000

-2000

-4000

-500

0

92 94 96

4

6

8

10

12

+1

-1

-4000

-3000

-2000

-4000

-500

0

92 94 96

4

6

8

10

12

S1

S2

S3

S4

S5

Sumatra Sumatra

Figure 2. Locations of five rupture segment astsunami source and final form of source elevation as

combination of five Okada source determined by

Grilli et al. 2007.

Table 1. Generating time and rising time for seafloordeformation to generate tsunami.

Seg-ment

1

Seg-ment

2

Seg-ment

3

Seg-ment

4

Seg-ment

5

t 0 (s) 0 185 315 360 420

T rise (s) 60 70 90 120 150

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-7 -5 -3 -1 1 3 5 7 9 11 13 15 17 19

N-latitude (deg)N-latitude (deg)N-latitude (deg)N-latitude (deg)

s u r f a c e

e l e v a t i o n

( m )

s u r f a c e

e l e v a t i o n

( m )

s u r f a c e

e l e v a t i o n

( m )

s u r f a c e

e l e v a t i o n

( m )

Jason 1satellite data

NLSW model

WNB model

Figure 3. Comparison of surface elevation measuredby Jason 1 satellite altimetry and results of modelsimulation using NLSW model and WNB model




Tsunami Generation

Grilli et. al. 2007 has studied source modelof IOT event. The tsunami source is developedbased on rupture (seafloor deformation)parameters which estimated by seismicinversion model and other seismological andgeological data. According to rupture trench,Grilli et al. 2005 divided the rupture zone into

five segments following the trench curvature.Parameter for each segment was characterizedand defined by seismic inversion model. Thegeometry of rupture then estimated by usingstatic dislocation formulae of Okada 1985. Thearea of five segments rupture is presented inFigure 2 (left panel). Parameters to determineOkada’s formulae for each rupture segmentcan be obtained in Grilli et al. 2007. The fivesegments of ruptures then used to find the besttsunami source after it is calibrated to therecorded data of Jason 1 satellite altimetry aspublished in Gower 2005 and Kulikov 2005.

The final form of seafloor deformation geometryis superposition of the five rupture segmentsand shown in Figure 2 (right panel).

To obtain a good agreement of sea surfacealong Jason 1’s satellite transect, the fiverupture segments are generated in differentstarting time (t 0) and rising time (T rise ) of verticalseafloor movement based on average shearwave speed about 0.8 km/s from the south tothe north. Table 1 presents starting time andrising time of vertical seafloor movement ofeach rupture segment used in this simulations.

Figure 3 shows comparison of free surfaceelevations between numerical results andJason 1’s satellite altimetry data. Modelsimulation using NLSW model and WNB modelshow a good agreement with measured data ofJason 1.

Dispersion Effect

To investigate the dispersion effect oftsunami propagation, numerical results ofNLSW model and WNB model are compared.As the first check to visualize the dispersion

effect, snapshot window of sea surface patternat time 1 hour 40 minutes of tsunamipropagation is depicted as shown in Figure 4.General features of wave evolution agreed verywell by all models. However, some differencesin reproducing dispersion effect become morenoticeable as time advances and longerdistance of propagation. The figure shows thatthe main of tsunami is propagated to the south-west direction, i.e. Maldives islands. Wave

pattern at the west part simulated by WNBmodel is slightly different compare to the NLSWmodel result. The tsunami front face is shiftedand split into more then one wave yielded aseries of wave propagation in which the firstone has higher amplitude and longer periodthan the last one.

85 90 952

4

6

8

10

12

85 90 952

4

6

8

10

12

Sumatra

Sumatra

S r i L a n k a

C1

C1

N - l a t i t u d e ( d e g r e e s )

E-longitude (degrees)

(a)

(b)

Figure 4. Snap shot window of surface elevation attime 1h 40min of tsunami propagation computed by:

(a) NLSW model and (b) WNB model

74 74.5 75 75.5 76 76.5 77

-1

0

1

2

90 91 92 93 94 95-1

-0.5

0

0.5

1

75 80 85 90 95

-1

0

1

2 time=3.0hours

E-longitude (degrees)

s e a l e v e l ( m

)

Figure 5. Spatial profiles along line C1 at t = 3 hourssimulated using NLSW model (thick lines) and WNB

model (bold lines)




The dispersive effect is proportional to thewater depth, so the dispersive effect at the westpart of the source is stronger compare to theeast part. The dispersion effect at the west isalso enhanced through longer distance ofpropagation. At the east direction, thecomputation of wave pattern by NLSW modeland WNB model is not significantly different.Beside shallowness of water depth at this area,

the dispersion effect did not have enough timeto developed because of short distancepropagation.

Spatial profiles of sea surface along transectline C1 as shown in Figure 4 is presented inFigure 5 at time 3 hours tsunami propagationsto visualize more detail of the dispersion effect.Generally, agreement between the dispersiveand nondispersive model is very good but afterlong distance propagation to the south-westdirection the advantage of dispersive model isremarkable. Initially, free surface profilesproduced by WNB and NLSW models are notsignificantly different. But after long time andlong distance propagation through relativelydeep water, the front face profile is graduallydifferent. The tsunami front face is oscillatedand changed by the dispersion effect.According to Figure 5, at time t = 3 hours theWNB model yields at least three waves oftsunami front face at the south-west directionalong line C1 with the wavelength of first,second, and third waves are 141, 83, and64km, respectively, measured from trough to

trough. The average water depth is h ≈ 5 km,

therefore the corresponding values of kh for thethree waves are 0.2206, 0.3747, and 0.486,respectively. According to the value of kh , thesecond and third waves are categorized asintermediate water wave. Hence, the NLSWmodel is less accurate for this case. Althoughthe second and third waves are categorized asintermediate water wave, the approximation ofdispersive term in WNB model still giveaccurate estimation of wave speed because thevalues of kh < 1. The leading wave height atthis time is over predicted more than 20 % bythe NLSW model.

At the east part, the agreement betweenNLSW and WNB model results are quite good.When tsunami wave entered runup phase,nonlinear interaction with complex bathymetrystrongly influenced the dispersion effect. Horilloet al. 2006 pointed out that the dispersionconsideration in the numerical models isnecessary for accurate prediction for the casesof tsunami entered continental shelf, bays orharbors in which tsunami produced oscillations

-4

-2

0

2

4

6

60 80 100 120 140 160 180

S u r f a c e l e v e l ( m )

-2.5

-1.5

-0.5

0.5

1.5

2.5

180 200 220 240 260 280 300


-3

-2

-1

0

1

2

3

180 200 220 240 260 280 300


NLSW model

WNB model

measured data

time (minutes)

(a)

(b)

(c)

Figure 6. Comparisons of temporal sea levelbetween measured data (as published in Grilli et al.,2007) and model simulations at: (a) Hannimadhoo

(b) Male, and (c) Merchator yacht

through the resonance. However, there are noestablished coastal observations which clearlyrepresent dispersion mechanism.

Following Grilli et al. 2007, measured data oftsunami elevations around simulation domainare compared to the model results. Only threegauges locations are discussed here, two tidegauges at the Maldives: Hannimadhoo (73.17

0E,

6.770N) and Male (73.54

0E, 4.23

0N), and one

by a Belgian yacht ‘’Merchator’’ at Nai Harn Bay(SW of Phuket). The measured data is digitizedfrom Grilli et al. 2007. All of model simulationsoverpredicted of tsunami arrival times at alllocations. Generally, the simulated andmeasured time history of tsunami elevations




agree very well in all tide gauges as shown inFigure 6. It is noted that because of coarsenessof bathymetry data used here, the locations oftide gauges are not perfectly match betweenmodels and observations.

At the Maldives as shown in Figure 6 (a andb), it can be seen a good agreement betweenobserved and model results. General pattern oftemporal variation of sea levels are match with

observations data for at least three waves.After long distance propagation, the dispersioneffect is noticed at those gauges. However, thebathymetry effect is reduced the dispersioneffect created by WNB model, hence, similarresults are obtained by NLSW model and WNBmodel but the NLSW model over predicted ofmaximum height compare to WNB model andmeasured data.

In Figure 6(c), the NLSW and WNB modelsgive the same results. Compare to theobservations data of yacht Merchator, theprofile is not match. Local coastal topographyeffect is not resolved very well by the models,so the time shift is different between modelsand measurement. Maximum height isunderpredited by the models. Therefore, finergrid resolution at the east area is needed forbetter estimation.

CONCLUSIONS

Numerical simulation of the December 26,2004 Indian Ocean tsunami has been

performed using dispersive and non-dispersivewave models (WNB and NLSW models,respectively). Comparisons of simulation resultsusing the two models notified the dispersioneffect especially through oceanic tsunamipropagation. Simulation results using the twodifferent model equations showed that theNLSW model is quite reliable for practicalpurpose because this model gives consistentresults compare to WNB model andobservations data. However, including thedispersive terms will improve accuracy of theprediction.

ACKNOWLEDGEMENT

This research was carried out by the financialsupport of Ministry of Education, Culture, Sport,Science, and Technology of Japan andfacilitated by Hydraulics Laboratory of NagaokaUniversity of Technology. All contributions areacknowledged.

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