On the contribution of the horizontal sea-bed...

Ocean Modelling 56 (2012) 43–56

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Ocean Modelling

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On the contribution of the horizontal sea-bed displacements into the tsunamigeneration process

Denys Dutykh a,⇑, Dimitrios Mitsotakis b, Leonid B. Chubarov c, Yuri I. Shokin ca LAMA, UMR 5127 CNRS, Université de Savoie, Campus Scientifique, 73376 Le Bourget-du-Lac Cedex, Franceb IMA, University of Minnesota, 114 Lind Hall, 207 Church Street SE, Minneapolis, MN 55455, USAc Institute of Computational Technologies, Siberian Branch of the Russian Academy of Sciences, 6 Acad. Lavrentjev Avenue, 630090 Novosibirsk, Russia

a r t i c l e i n f o

Article history:Received 15 February 2012Received in revised form 4 July 2012Accepted 9 July 2012Available online 28 July 2012

Keywords:Tsunami wavesWater wavesCoseismic displacementsWave energyFinite fault inversion

1463-5003/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.ocemod.2012.07.002

⇑ Corresponding author. Tel.: +33 4 79 75 86 52; faE-mail addresses: [email protected] (

com (D. Mitsotakis), [email protected] (L.B. C(Y.I. Shokin).

URLs: http://www.lama.univ-savoie.fr/~dutykh/ (Dcom/site/dmitsot/ (D. Mitsotakis), http://www.ict.nswww.ict.nsc.ru/ (Y.I. Shokin).

a b s t r a c t

The main reason for the generation of tsunamis is the deformation of the bottom of the ocean caused byan underwater earthquake. Usually, only the vertical bottom motion is taken into account while thehorizontal co-seismic displacements are neglected in the absence of landslides. In the present studywe propose a methodology based on the well-known Okada solution to reconstruct in more details allcomponents of the bottom coseismic displacements. Then, the sea-bed motion is coupled with a three-dimensional weakly nonlinear water wave solver which allows us to simulate a tsunami wave genera-tion. We pay special attention to the evolution of kinetic and potential energies of the resulting wavewhile the contribution of the horizontal displacements into wave energy balance is also quantified. Suchcontribution of horizontal displacements to the tsunami generation has not been discussed before, and itis different from the existing approaches. The methods proposed in this study are illustrated on the July17, 2006 Java tsunami and some more recent events.

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1. Introduction

During the years 2004–2006 several interesting tsunami eventstook place in the Indian Ocean. In December 26, 2004, the GreatSumatra–Andaman earthquake (Mw ¼ 9:1, cf. Ammon et al.,2005) generated the devastating Indian Ocean tsunami referredto in the literature as the Tsunami Boxing Day 2004 (cf. Synolakisand Bernard, 2006). The local tsunami run-ups from this eventexceeded the height of 34 m at Lhoknga in the western Aceh Prov-ince. This and other observations led many researchers to askwhether this tsunami was unusually large for this specific earth-quake size. Later it was shown by Geist et al. (2006) that the GreatSumatra–Andaman earthquake is very similar in terms of localtsunami magnitude to past events of the same size. For example,the 1964 Great Alaska earthquake (Mw ¼ 9:2, cf. Kanamori, 1970)demonstrates a similar scaling.

On March 28, 2005 another earthquake occured approximately110 km to the SE from the Great Sumatra–Andaman earthquake’sepicenter. The magnitude of this earthquake was estimated to beMw ¼ 8:6–8.7 (cf. Lay et al., 2005; Bilham, 2005; Walker et al.,

ll rights reserved.

x: +33 4 79 75 81 42.D. Dutykh), [email protected]), [email protected]

. Dutykh), http://sites.google.c.ru/ (L.B. Chubarov), http://

2005). This event triggered a massive evacuation in the surround-ing Indian Ocean countries. However, the March 28, 2005 NorthernSumatra earthquake failed to generate a significant tsunami event.The survey teams reported maximum tsunami run-up of 4 m (Geistet al., 2006). This event can be compared with the 1957 Aleutianearthquake (Mw ¼ 8:6, cf. Johnson and Satake, 1993) which pro-duced a maximum tsunami run-up of 15 m (see Lander, 1996).The deficiency of the March 2005 tsunami is related to the slip con-centration in the down-dip part of the rupture zone and to the factthat a substantial part of the vertical displacement occurred inshallow waters or on the substance of the ground (Geist et al.,2006).

On the other hand, the smaller July 16, 2006 Java earthquake(Mw ¼ 7:8, cf. Ammon et al., 2006) generated an unexpectedlydestructive tsunami wave which affected over 300 km of south Javacoastline and killed more than 600 people (Fritz et al., 2007). Fieldmeasurements of run-up distributions range uniformly from 5 to7 m in most inundated areas. However, unexpectedly high run-upvalues were reported at Nusakambangan Island exceeding thevalue of 20 m. This tsunami focusing effect could be seemingly as-cribed to local site effects and/or to a local submarine landslide/slump. July 16, 2006 Java tsunami can be compared to a similarevent occured on June 2, 1994 at the East Java (Mw ¼ 7:6) (see Tsujiet al., 1995). This 1994 Java tsunami produced more than 200casualties with local run-up at Rajekwesi slightly exceeding 13 m.

All these examples of recent and historical tsunami events showthat there is a big variety of the local tsunami heights and run-up

http://dx.doi.org/10.1016/j.ocemod.2012.07.002mailto:[email protected]:dmitsot@gmail. commailto:dmitsot@gmail. commailto:[email protected]:[email protected] http://www.lama.univ-savoie.fr/~dutykh/http://sites.google. com/site/dmitsot/http://sites.google. com/site/dmitsot/http://www.ict.nsc.ru/http://www.ict.nsc.ru/http://www.ict.nsc.ru/http://dx.doi.org/10.1016/j.ocemod.2012.07.002http://www.sciencedirect.com/science/journal/14635003http://www.elsevier.com/locate/ocemod

44 D. Dutykh et al. / Ocean Modelling 56 (2012) 43–56

values with respect to the earthquake magnitude Mw. It is obviousthat the seismic moment M0 of underwater shallow earthquakes isadequate to estimate far-field tsunami amplitudes (see also Okaland Synolakis, 2003a). However, tsunami wave energy reflectsbetter the local tsunami severity, while the specific run-up distri-bution depends on bathymetric propagation paths and other site-specific effects. One of the central challenges in the tsunamiscience is to rapidly assess a local tsunami severity from very firstrough earthquake estimations. In the current state of our knowl-edge false alarms for local tsunamis appear to be unavoidable.The tsunami generation modeling attempts to improve our under-standing of tsunami behavior in the vicinity of the genesis region.

Tsunami generation modeling was initiated in the early sixtiesby the prominent work of Kajiura (1963), who proposed the useof the static vertical sea bed displacement for the initial conditionof the free surface elevation. Classically, the celebrated Okada(1985, 1992), and sometimes Mansinha and Smylie (1967, 1971)1

or Gusiakov (1978) solutions are used to compute the coseismicsea bed displacements. This approach is still widely used by the tsu-nami wave modeling community. However, some progress has beenrecently made in this direction (Ohmachi et al., 2001; Dutykh andDias, 2007, 2009b; Dutykh, 2007; Rabinovich et al., 2008; Saitoand Furumura, 2009; Dutykh et al., 2011b).

There is a consensus on the importance of the horizontal motionin landslide generated tsunamis (Harbitz, 1992; Ward, 2001; Bar-det et al., 2003; Okal and Synolakis, 2003b). However, for tsunamiwaves caused by underwater earthquakes, the horizontal displace-ments are often disregarded by the tsunami wave modeling com-munity. We can quote here a few publications devoted totsunami waves such as one by Stevenson (2005):

‘‘Although horizontal displacements are often larger, they areunimportant for tsunami generation except to the extent thatthe sloping ocean floor also forces a vertical displacement ofthe water column.’’

Or another one by Ichinose et al. (2000):

‘‘The initial lake level values were specified by assuming thatthe lake surface instantaneously conformed to the vertical dis-placement of the lake bottom while horizontal velocities wereset to zero. The effect of horizontal deformation on the initialcondition is neglected here and left for future work.’’

The authors of this article also tended to neglect the horizontal dis-placements field in previous tsunami generation studies (Dutykh,2007; Mitsotakis, 2009). Perhaps, this situation can be ascribed tothe work of Berg (1970), who showed in the case of the 1964 Alas-ka’s earthquake (Mw ¼ 9:2) that the input into the potential energyfrom the horizontal motion is less than 1.5% of that from the verticalmovement.

The attitude to horizontal displacements changed after theprominent work by Tanioka and Satake (1996). According to theircomputations for the 1994 Java earthquake, the inclusion of hori-zontal displacements contribution results in 43% increase in max-imum initial vertical ground displacement and an increase in thewave amplitudes at the shoreline by 30%. These striking resultsincited researchers to reconsider the role of the horizontal seabed motion. Hereafter, various tsunami generation techniqueswhich involve only the vertical displacement field are referred toas the incomplete scenario. On the contrary, the complete tsunamigeneration takes also into account the horizontal displacementsfield. The term complete in our study refers to a purely kinematicinteraction between the moving bottom and the ocean layer in

1 In fact, Mansinha and Smylie solution is a particular case of the more generalOkada solution.

contrast to the horizontal impulse-momentum transfer mecha-nism (Vennard and Street, 1982) applied for the first time totsunami wave generation in Song et al. (2008).

The approach proposed by Tanioka and Satake (1996) is basedon a simple physical consideration: horizontal displacements of asloping bottom will produce some amount of the vertical motiondepending on the slope magnitude. Assuming that the slope issmall, this idea can be expressed mathematically by applying oncethe first order Taylor expansion:

uh :¼ u1hx þ u2hy;

where u1;2 are the horizontal components of the displacements fieldand hðx; yÞ is the bathymetry function. The indices x and y denotethe spatial partial derivatives. Tanioka and Satake (1996) referredto the quantity uh as the vertical displacement of water due to the hor-izontal movement of the slope. Then, the quantity uh is simply addedas a correction to the vertical displacements component u3ðx; yÞ. Tocompute the co-seismic displacements u1;2;3 they used the cele-brated Okada solution (Steketee, 1958; Okada, 1985, 1992) appliedto a single rectangular fault segment. In the present study we applythe same solution to multiple sub-faults to achieve higher resolu-tion (see also Dutykh et al., 2012). More recently, an originalapproach based on the impulse-momentum principle (Vennardand Street, 1982) has been proposed in Song et al. (2008). Their gen-esis theory is based on the idea of the momentum exchangebetween the slipped continental slope and the water column dueto the earthquake. The horizontal bottom motion contributes di-rectly to the ocean kinetic energy as any moving object in watertransfers momentum to the fluid. This idea is similar to the classicaladded mass concept. According to the authors of Song et al. (2008),the continental slope is not an exception from this principle. Thisimplies some special treatment of the water parcels in the vicinityof moving boundaries. For more details we refer to the original pub-lication (Song et al., 2008). The horizontal impulse approach leadsto predictions which are somehow different from the present andseveral previous studies. The methodology described in this articleis closer to the ideas of Tanioka and Satake (1996). Namely, wereconstruct in more details the kinematics of bottom displacementsin vertical and horizontal directions, including their evolution intime. We underline that most of tsunami generation studies, includ-ing Tanioka and Satake, 1996, assume the bottom deformationprocess to be instantaneous, that in most of the cases provides afairly good approximation. Then, the bottom motion generatessome perturbations on the free surface of the water layer. This per-turbation form long waves, propagating across the oceans under thegravity force and commonly referred to as tsunami waves due totheir destructive potential fully realized only in coastal regions(Kajiura, 1963; Velichko et al., 2002; Dias and Dutykh, 2007; Syn-olakis and Bernard, 2006).

In the current study we shed some light into the energy transferprocess from an underwater earthquake to the implied tsunamiwave (in the spirit of the study by Dutykh and Dias (2009a)), whiletaking into account and quantifying the horizontal displacementscontribution into tsunami energy balance. We focus only on thegeneration stage since the propagation phase and run-up tech-niques are well understood nowadays at least in the context ofNonlinear Shallow Water equations (Imamura, 1996; Delis et al.,2008; Kim et al., 2007; Dutykh et al., 2011a,b).

The present study is organized as follows. In Section 2 we pres-ent mathematical models used in this study. Specifically, in Sec-tions 2.1 and 2.2 a description of the bottom motion and thefluid layer solution is presented respectively. Some rationale ontsunami wave energy computations is discussed in Section 2.2.1.Numerical results are presented in Section 3. Finally, some impor-tant conclusions of this study are outlined in Section 4. In Appendix

D. Dutykh et al. / Ocean Modelling 56 (2012) 43–56 45

A of this paper we present the results concerning the tsumami gen-eration of two recent events.

2. Mathematical models

Tsunami waves are caused by a huge and rapid motion of theseafloor due to an underwater earthquake over broad areas incomparison to water depth. There are some other mechanisms oftsunami genesis such as underwater landslides, for example. How-ever, in this study we focus on the purely seismic mechanismwhich occurs most frequently in nature.

Hydrodynamics and seismology are only weakly coupled in thetsunami generation problem. Namely, the released seismic energyis partially transmitted to the ocean through the sea bed deforma-tion while the ocean has obviously no influence onto the rupturingprocess. Consequently, our problem is reduced to two relativelydistinct questions:

(1) Reconstruct the sea bed deformation h ¼ hð~x; tÞ caused bythe seismic event under consideration.

(2) Compute the resulting free surface motion.

The answers to these questions are analyzed in Sections 2.1 and 2.2respectively.

Fig. 1. Surface projection of the fault’s plane and the ETOPO1 bathymetric map ofthe region under consideration. The symbol H indicates the hypocenter’s location atð107:345� ;�9:295�Þ. The local Cartesian coordinate system is centered at the pointð108�;�10�Þ. This region is located between ð106�;�8�Þ and ð110� ;�12�Þ. Thecolorbar indicates the water depth in meters below the still water level (z ¼ 0). Inthe region under consideration the depth varies from 20 to 7100 m.

2.1. Dynamic bottom displacements reconstruction

Traditionally, the free surface initial condition for various tsu-nami propagation codes (see Titov and González, 1997; Imamura,1996; Imamura et al., 2006) is assumed to be identical to the staticvertical deformation of the ocean bottom (Kajiura, 1970). Thisassumption is classically justified by the three following reasons:

(1) Tsunamis are long waves. In this regime the vertical acceler-ation is neglected with respect to the gravity force.

(2) The sea bed deformation is assumed to be instantaneous. Itis based on the comparison of gravity wave speed (200 m/sfor water depth of 4 km) and the seismic wave celerity(�3600 m/s).

(3) The effect of horizontal bottom motion is negligible for tsu-nami generation since the bathymetry has in general mildslope (�10%) (Berg, 1970).

It is worth to note that nonhydrostatic effects as well as finite timesource duration have been modeled in several recent studies (Tod-orovska and Trifunac, 2001; Dutykh and Dias, 2007; Fructus andGrue, 2007; Kervella et al., 2007; Dutykh and Dias, 2009b; Fuhr-man and Madsen, 2009). Some attempts have also been made totake into account the horizontal displacements contribution (Tan-ioka and Satake, 1996; Geist et al., 2006; Song et al., 2008).

In the present study we relax all three classical assumptions.The bottom deformation is reconstructed using the finite faultsolution as it was suggested in our previous study (Dutykh et al.,2012). However, we extend the previous construction to take intoaccount the horizontal displacements contribution as well. Thefinite fault solution is based on the multi-fault representation ofthe rupture (Bassin et al., 2000; Ji et al., 2002). The rupture com-plexity is reconstructed using the inversion of seismic data. Fault’ssurface is parametrized by multiple segments with variable localslip, rake angle, rise time and rupture velocity. The inversion is per-formed in an appropriate wavelet transform space. The objectivefunction is a weighted sum of L1; L2 norms and some correlativefunctions. With this approach seismologists are able to recoverrupture slip details (Bassin et al., 2000; Ji et al., 2002). This avail-able seismic information is exploited hereafter to compute the

sea bed displacements produced by an underwater earthquakewith better geophysical resolution. Several other multiple seg-ments sources have been used in Geist (2002), Wang et al.(2006) and McCloskey et al. (2008).

Remark 1. In a few studies an attempt has been made toreconstruct the seismic source from tsunami tide gauge records(Piatanesi et al., 2001; Pires and Miranda, 2003; Yagi, 2004; Fujiiand Satake, 2006; Ichinose et al., 2007; Rhie et al., 2007). Thisapproach seems to be very promising and in future a jointcombination of seismic and hydrodynamic inversions should beused for the successful reconstruction of appropriate initialconditions.

Let us describe the way of how the sea bed displacements arereconstructed. In this reconstruction procedure we follow the greatlines of our previous study (Dutykh et al., 2012), while adding newingredients concerning the horizontal displacements field recon-struction. We illustrate the proposed approach in the case of theJuly 17, 2006 Java tsunami for which the finite fault solution isavailable (cf. Ji, 2006; Ozguns Konca, 2006). It was a relatively slowearthquake and thus, atypical. However, we assume that the slowrupturing process was well resolved by the finite fault inversionalgorithm.

The fault is considered to be the rectangle with vertices locatedat (109:20508� (Lon), �10:37387� (Lat), 6:24795 km (Depth)),ð106:50434�;�9:45925�;6:24795 kmÞ, ð106:72382�;�8:82807�;19:79951 kmÞ, ð109:42455�;�9:74269�;19:79951 kmÞ (see Fig. 1).The fault’s plane is conventionally divided into Nx ¼ 21 subfaultsalong strike and Ny ¼ 7 subfaults down the dip angle, leading tothe total number of Nx � Ny ¼ 147 equal segments. Parameterssuch as the subfault location ðxc; ycÞ, the depth di, the slip ui andthe rake angle /i for each segment can be found in Ji (2006) (seealso Appendix II, Dutykh et al., 2012). The elastic constants andparameters such as dip and slip angles, which are common to allsubfaults, are given in Table 1. We underline that the slip angleis measured conventionally in the counter-clockwise directionfrom the North. The relations between the elastic wave celeritiescp; cs and Lamé coefficients k;l used in Okada’s solution are classi-cal and can also be found in Appendix III (Dutykh et al., 2012).

One of the main ingredients in our construction is the so-calledOkada solution (Okada, 1985, 1992) which is used in the case of an

Table 1Geophysical parameters used to model elastic properties of thesubduction zone in the region of Java.

P-wave celerity cp , m/s 6000S-wave celerity cs , m/s 3400Crust density q, kg/m3 2700Dip angle, d 10.35�Slip angle (CW from N) 288.94�

−0.5 0 0.5 1 1.5 2 2.5 3 3.5 4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t, s

T(t)

Dynamic scenarios

Trigonometric scenarioExponential scenario

Fig. 2. Trigonometric and exponential dynamic scenarios for tr ¼ 1 s (see Ham-mack, 1973).

2 In our numerical simulations presented below we use the MATLAB TriScat-teredInterp class to interpolate the static bathymetry values given by ETOPO1database.


active fault of small or intermediate size. The success of this solu-tion may be ascribed to the closed-form analytical expressionswhich can be effectively used for various modeling purposesinvolving co-seismic deformation.

Remark 2. The celebrated Okada solution (Okada, 1985, 1992) isbased on two main ingredients – the dislocation theory of Volterra(Volterra, 1907; Steketee, 1958) and Mindlin’s fundamental solu-tion for an elastic half-space (Mindlin, 1936). Particular cases ofthis solution were known before Okada’s work, for example thewell-known Mansinha and Smylie’s solution (Mansinha and Smy-lie, 1967, 1971). Usually all these particular cases differ by thechoice of the dislocation and the Burger’s vector orientation (Press,1965). We recall the basic assumptions behind this solution:

� Fault is immersed into the linear homogeneous and isotropichalf-space.� Fault is a Volterra’s type dislocation.� Dislocation has a rectangular shape.

For more information on Okada’s solution we refer to Dutykh andDias (2007), Dias and Dutykh (2007), Dutykh (2007) and the refer-ences therein.

The trace of the Okada’s solution at the sea bottom (substitutingz ¼ 0 in the geophysical coordinate system) for each subfault willbe denoted OðjÞi ð~x; d; k;l; . . .Þ, where d is the dip angle, k;l are theLamé coefficients and dots denote the dependence of the functionOðjÞi ð~xÞ on other eight parameters (cf. Dutykh and Dias, 2007). Theindex i takes values from 1 to Nx � Ny and denotes the correspond-ing subfault segment, while the superscript j is equal to 1 or 2 forthe horizontal displacements and to 3 for the vertical component.Hereafter we will adopt the short-hand notation OðjÞi ð~xÞ for the jthdisplacement component of the Okada’s solution for the ith seg-ment having in mind its dependence on various parameters.

Taking into account the dynamic characteristics of the ruptur-ing process, we make some further assumptions on the timedependence of the displacement fields. The finite fault solutionprovides us with two additional parameters concerning therupture kinematics – the rupture velocity vr and the rise time trwhich are equal to 1.1 km/s and 8 s for July 17, 2006 Java eventrespectively. The epicenter is located at the point ~xe ¼ ð107:345�;�9:295�Þ (cf. Ji, 2006). Given the origin ~xe, the rupture velocity vrand ith subfault location~xi, we define the subfault activation timesti needed for the rupture to achieve the corresponding segment i bythe formulas:

ti ¼jj~xe �~xijj

v r; i ¼ 1; . . . ;Nx � Ny:

Consequently, for the sake of simplicity in our study we assume therupture propagation along the fault to be isotropic and homoge-neous. However, some more sophisticated approaches includingpossible heterogeneities exist (see e.g. Das and Suhadolc, 1996).We will also follow the pioneering idea of Hammack (1972,1973), developed later in Todorovska and Trifunac (2001),Todorovska et al. (2002), Dutykh and Dias (2007), Dutykh et al.(2006) and Kervella et al. (2007), where the maximum bottomdeformation is achieved during some finite time (the so-called rise

time) according to an appropriately chosen dynamic scenario. Vari-ous scenarios used in practice (instantaneous, linear, trigonometric,exponential, etc) can be found in Hammack (1972, 1973), Kanamori(1972), Dutykh et al. (2006) and Dutykh and Dias (2007). In thisstudy we adopt the trigonometric scenario which is given by thefollowing formula:

TðtÞ ¼ Hðt � trÞ þ12HðtÞHðtr � tÞ 1� cosðpt=trÞð Þ; ð2:1Þ

where HðtÞ is the Heaviside step function. This scenario has theadvantage to have also the first derivative continuous at the activa-tion time t ¼ 0. However, for comparative purposes sometimes wewill use also the so-called exponential scenario (Kanamori, 1972):

TeðtÞ ¼ HðtÞ 1� e�at� �

; a :¼ logð3Þtr

:

For illustrative purposes both dynamic scenarios are represented inFig. 2.

We sum up together all the ingredients proposed above toreconstruct dynamic displacements field ~u ¼ ðu1;u2;u3Þ at the seabottom:

ujð~x; tÞ ¼XNx�Nyi¼1

Tðt � tiÞOðjÞi ð~xÞ:

Remark 3. We would like to underline here the asymptoticbehavior of the sea bed displacements. By definition of thetrigonometric scenario (2.1) we have limt!þ1TðtÞ ¼ 1. Conse-quently, the sea bed deformation will attain fast its state whichconsists of the linear superposition of subfaults contributions:

ujð~x; tÞ ¼XNx�Nyi¼1OðjÞi ð~xÞ:

Finally, we can predict the sea bed motion by taking into ac-count horizontal and vertical displacements:

hð~x; tÞ ¼ h0ð~x�~u1;2ð~x; tÞÞ � u3ð~x; tÞ; ð2:2Þ

where h0ð~xÞ is a function which interpolates2 the static bathymetryprofile given e.g. by the ETOPO1 database (see Figs. 1 and 4).

−4 −3−2 −1

0 12 3

4

−4−3

−2−1

01

23

4

−2

−1.5

−1

−0.5

0

0.5

z

x

y

O

η (x,y,t)

h(x,y,t)h0

a

Fig. 3. Sketch of the physical domain.

Fig. 4. Side view of the bathymetry, cf. also Fig. 1.


Remark 4. In some studies where horizontal displacements weretaken into account (cf. Tanioka and Satake, 1996; Baba et al., 2006;Song et al., 2008; Beisel et al., 2009), the first order Taylorexpansion was permanently applied to the bathymetry represen-tation formula (2.2) to give:

hð~x; tÞ � h0ð~xÞ �~u1;2ð~x; tÞ � r~xh0ð~xÞ � u3ð~x; tÞ ð2:3Þ

We prefer not to follow this approximation and to use the exactformula (2.2) since it is valid for all slopes (see Fig. 4 for Javabathymetry). Another difficulty lies in the estimation of thebathymetry gradient r~xh0ðxÞ required by Taylor’s formula (2.3).The application of any finite differences scheme to the measuredh0ð~xÞ leads to an ill-posed problem. Consequently, one needs toapply extensive smoothing procedures to the raw data h0ð~xÞ whichinduces an additional loss in accuracy.

In the present study we do not completely avoid the compu-tation of the static bathymetry gradient r~xh0ð~xÞ since thekinematic bottom boundary condition (2.7) involves the timederivative of the bathymetry function:

@th ¼ �r~xh0ð~xÞ � @t~u1;2ð~x; tÞ � @tu3ð~x; tÞ:

However, the last formula is exact and it is obtained by a straight-forward application of the chain differentiation rule.

Another possibility could be to consider static horizontaldisplacements ~u1;2ð~xÞ thus keeping dynamics only in the verticalcomponent u3ð~x; tÞ. However, we do not choose this option in thiswork.

In the next section we will present our approach in couplingthis dynamic deformation with the hydrodynamic problem to pre-dict waves induced on the free surface of the ocean.

2.2. The water wave problem with moving bottom

We consider the incompressible flow of an ideal fluid with con-stant density q in the domain X # R2. The horizontal independentvariables will be denoted by~x ¼ ðx; yÞ and the vertical one by z. Theorigin of the Cartesian coordinate system is traditionally chosensuch that the surface z ¼ 0 corresponds to the still water level.The fluid domain is bounded below by the bottom z ¼ �hð~x; tÞand above by the free surface z ¼ gð~x; tÞ. Usually we assume thatthe total depth Hð~x; tÞ :¼ hð~x; tÞ þ gð~x; tÞ remains positiveHð~x; tÞP h0 > 0 under the system dynamics 8t 2 ½0; T�. The sketchof the physical domain is shown in Fig. 3.


Remark 5. Classically in water wave modeling, we make theassumption that the free surface is a graph z ¼ gð~x; tÞ of a single-valued function. It means in practice that we exclude someinteresting phenomena, (e.g. wave breaking phenomena) whichare out of the scope of this modeling paradigm.

The governing equations of the classical water wave problemare the following (Lamb, 1932; Stoker, 1958; Mei, 1994; Whitham,1999):

D/ ¼ r2/þ @2zz/ ¼ 0; ð~x; zÞ 2 X� ½�h;g�; ð2:4Þ@tgþr/ � rg� @z/ ¼ 0; z ¼ gð~x; tÞ; ð2:5Þ

@t/þ12jr/j2 þ 1

2ð@z/Þ2 þ gg ¼ 0; z ¼ gð~x; tÞ; ð2:6Þ

@thþr/ � rhþ @z/ ¼ 0; z ¼ �hð~x; tÞ; ð2:7Þ

where / is the velocity potential, g the acceleration due to gravityforce and r ¼ ð@x; @yÞ denotes the gradient operator in horizontalCartesian coordinates. The fluid incompressibility and flow irrota-tionality assumptions lead to the Laplace Eq. (2.4) for the velocitypotential /ð~x; z; tÞ.

The main difficulty of the water wave problem lies on theboundary conditions. Eqs. (2.5) and (2.7) express the free surfaceand bottom impermeability, while the Bernoulli condition (2.6)expresses the free surface isobarity respectively.

Function hð~x; tÞ represents the ocean’s bathymetry (depth belowthe still water level, see Fig. 3) and is assumed to be known. Thedependence on time is included in order to take into account thebottom motion during an underwater earthquake (Dias andDutykh, 2007; Dutykh and Dias, 2007; Dutykh et al., 2006; Kervellaet al., 2007; Dutykh, 2007).

For the exposition below we will need also to compute unitaryexterior normals to the fluid domain. The normals at the free sur-face Sf and at the bottom Sb are given by the following expressionsrespectively:

n̂f ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ jrgj2q �rg

1

�� ; n̂b ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ jrhj2

q �rh�1

�� :

In 1968 Zakharov proposed a different formulation of the waterwave problem based on the trace of the velocity potential at the freesurface (Zakharov, 1968):

uð~x; tÞ :¼ /ð~x;gð~x; tÞ; tÞ:

This variable plays a role of the generalized momentum in the Ham-iltonian description of water waves (Zakharov, 1968; Dias andBridges, 2006). The second canonical variable is the free surface ele-vation g.

Another important ingredient is the normal velocity at the freesurface vn which is defined as:

vnð~x; tÞ :¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ jrgj2

q@/@n̂f

��z¼g¼ ð@z/�r/ � rgÞjz¼g: ð2:8Þ

Kinematic and dynamic boundary conditions (2.5) and (2.6) at thefree surface can be rewritten in terms of u, vn and g (Craig et al.,1992; Craig and Sulem, 1993; Fructus et al., 2005):

@tg�DgðuÞ ¼ 0;

@tuþ12jruj2 þ gg� 1

2ð1þ jrgj2ÞDgðuÞ þ ru � rg� �2 ¼ 0:

ð2:9Þ

Here we introduced the so-called Dirichlet-to-Neumann operator(D2N) (Coifman and Meyer, 1985; Craig and Sulem, 1993) whichmaps the velocity potential at the free surface u to the normalvelocity vn:

Dg :u # vn ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ jrgj2

q@/@n̂f

��z¼g

r2/þ @2zz/ ¼ 0; ð~x; zÞ 2 X� ½�h;g�;/ ¼ u; z ¼ g;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ jrhj2q

@/@n̂b¼ @th; z ¼ �h:

��The name of this operator comes from the fact that it makes a cor-respondence between Dirichlet data u and Neumann dataffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ jrgj2q

@/@n̂f

��z¼g

at the free surface.

So, the water wave problem can be reduced to a system of twoPDEs (2.9) governing the evolution of the canonical variables g andu. For the tsunami generation problem we approximate and wecompute efficiently the D2N map DgðuÞ using the Weakly Nonlin-ear (WN) model described in Dutykh et al. (2012). This relies on theapproximate solution of the 3D Laplace equation in a perturbedstrip-like domain using the Fourier transform (û :¼ F½u�,g ¼ F�1½ĝ�):

D̂gðuÞ ¼ ûj~kj tanhðj~kjHÞ þ f̂ sechðj~kjHÞ � F½F�1½i~kû� � F�1½i~kĝ��;

where ~k is the wavenumber and f is the bathymetry forcing term

f ð~x; tÞ :¼ �@th�r/jz¼�h � rh:

Several details on the time integration procedure can be also foundin Dutykh et al. (2012). The resulting method is only weakly nonlin-ear and analogous at some point to the first order approximationmodel proposed in Guyenne and Nicholls (2007). As the hydrody-namical models, Tanioka and Satake (1996) used the Linear ShallowWater equations (Satake, 1995), while Song et al. (2008) employed ahydrostatic ocean circulation model (Song andHou, 2006).

Recently, it was shown that a tsunami generation process isessentially linear (Kervella et al., 2007; Saito and Furumura,2009). However, our WN approach will take into account somenonlinear effects when they become important, for example whenrapid changes in the bathymetry are present. This is possible whenthe generation region involves a wide range of depths from deep toshallow regions (see Figs. 1 and 4).

2.2.1. Tsunami wave energyIn this study we are more particularly interested in the evolu-

tion of the generated wave energy (Dutykh and Dias, 2009a). Inthe case of free surface incompressible flows, the kinetic and po-tential energies, denoted by K and P respectively, are completelydetermined by the velocity field and the free surface elevation:

KðtÞ :¼ q2

Z g�h

Z ZXjr/j2 d~xdz; PðtÞ :¼ qg

2

Z ZXg2 d~x:

The definition of the kinetic energy KðtÞ involves an integralover the three dimensional physical domain X� ½�h;g�. We canreduce the integral dimension using the fact that the velocity poten-tial / is a harmonic function:

jr/j2 ¼ ð/r/Þ � / D/|{z}¼0

ð/r/Þ:

Consequently, the kinetic energy can be rewritten as follows:

KðtÞ ¼ q2

Z ZSfþSb

/r/ � n̂dr

¼ q2

Z ZXuDgðuÞd~x|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}ðIÞ

þ q2

Z ZX

�/@thd~x|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}ðIIÞ

;

where �/ denotes the trace of the velocity potential at the bottom/jz¼�h (see Clamond and Dutykh, 2012). In order to obtain the last

Table 2Values of physical and numerical parameters used in simulations.

Ocean water density, q, kg/m3 1027.0Gravity acceleration, g, m/s2 9.81Epicenter location (Lon, Lat) ð107:345�;�9:295�ÞRupture velocity, vr , km/s 1.1Rise time, t0, s 8.0Number of Fourier modes in x 256Number of Fourier modes in y 256

Fig. 5. Contribution of the horizontal displacements field into the sea beddeformation expressed in percentage of the maximum vertical displacement. Thegray color corresponds to negligible values while white and black zones show themost important contributions. The colorbar shows the percentage of the signeddifference going from -21% to +7%.


equality we used the free surface and the bottom kinematic bound-ary conditions (2.6) and (2.7). The first integral (I) is classical andrepresents the change of kinetic energy under the free surface mo-tion while the second one (II) is the forcing term due to the bottomdeformation. The total energy3 is defined as the sum of kinetic andpotential ones:

EðtÞ :¼ KðtÞ þPðtÞ

¼ q2

Z ZXuDgðuÞd~xþ

q2

Z ZX

�/@thd~xþqg2

Z ZXg2 d~x:

Below we will compute the evolution of the kinetic, potentialand total energies beneath moving bottom.

3. Numerical results

The proposed approach will be directly illustrated on the Java2006 event. The July 17, 2006 Java earthquake involved thrustfaulting in the Java’s trench and generated a tsunami wave thatinundated the southern coast of Java (Ammon et al., 2006; Fritzet al., 2007). The estimates of the size of the earthquake (Ammonet al., 2006) indicate a seismic moment of 6:7� 1020 N �m, whichcorresponds to the magnitude Mw ¼ 7:8. Later this estimationwas refined to Mw ¼ 7:7 (Ji, 2006). Like other events in this region,Java’s event had an unusually low rupture speed of 1.0–1.5 km/s(we take the value of 1.1 km/s according to the finite fault solution,Ji, 2006), and occurred near the up-dip edge of the subduction zonethrust fault. According to Ammon et al. (2006), most aftershocksinvolved normal faulting. The rupture propagated approximately200 km along the trench with an overall duration of approximately185 s. The fault’s surface projection along with ocean ETOPO1bathymetric map are shown in Figs. 1 and 4. We note that IndianOcean’s depth of the region considered in this study varies be-tween 7186 and 20 m in the shallowest regions which may implylocal importance of nonlinear effects.

Remark 6. We have to mention that the finite fault inversion forthis earthquake was also performed by the Caltech team (OzgunsKonca, 2006). The main differences with the USGS inversion consiston the employed dataset. To our knowledge, A. Ozgun Konca andhis collaborators include also displacements measured with GPS-based techniques. Consequently, they came to the conclusion thatthe July 17, Southern Java earthquake magnitude was Mw ¼ 7:9.The energy of a tsunami wave generated according to this solutionwill be discussed below.

The numerical solutions presented below are given by theWeakly Nonlinear (WN) model. A uniform grid of 256� 256points4 is used in all computations below. The time step Dt is chosenadaptively according to the RK(4,5) method proposed in Dormandand Prince (1980). The problem is integrated numerically duringT ¼ 255 s which is a sufficient time interval for the bottom to takeits final shape (

Fig. 6. Location of the six numerical wave gauges (indicated by the symbol r)superposed with the steady state coseismic bottom displacement (only the verticalcomponent in meters is represented here).


The importance of the nonlinear effects have already beenaddressed in several previous studies (Kervella et al., 2007; Saitoand Furumura, 2009). In the framework of the Weakly Nonlinearapproach we studied this question in a recent companion paper(Dutykh et al., 2012) using only the vertical deformation. A fairlygood agreement has been observed with the Cauchy–Poisson(linear, fully dispersive) formulation in accordance with precedingresults (cf. Kervella et al., 2007; Saito and Furumura, 2009). Conse-quently, in the present study we decided to focus on the furthercomparison between complete/incomplete approaches and expo-nential/trigonometric scenarios (Hammack, 1973; Dutykh et al.,2006; Mitsotakis, 2009). These results are discussed hereafter.

We note that the trigonometric scenario leads in general toslightly larger amplitudes than the exponential bottom motion. Itis not surprising since the exponential scenario prescribes smooth-er and less rapid change in the bottom for the same rise timeparameter value (see Fig. 2). Later we will consider the trigonomet-ric scenario unless otherwise noted.

Then, it can be seen that in most cases horizontal displacementslead to an increase in the wave amplitude but not always as it canbe observed in Fig. 7(f) (the last wave gauge located atð108:2�;�9:75�Þ). We investigated more thoroughly this question.Fig. 8 shows the relative difference between free surface elevationscomputed according to complete gcð~x; teÞ and incomplete gið~x; teÞscenarios. More precisely, we plot the following quantity (for thetrigonometric scenario):

dð~xÞ :¼ gcð~x; teÞ � gið~x; teÞ

max~xjgið~x; teÞj; te ¼ 220 s:

Time te ¼ 220 s has been chosen because at that moment the bot-tom has been stabilized and the waves enter into the free propaga-tion regime. In Fig. 8 the gray color corresponds to the zero value ofthe difference dð~xÞ, while the white color shows regions where thewave is amplified by horizontal displacements by approximately10%. On the contrary, black zones show an attenuation effect of hor-izontal sea bed motion (about �5%). Recall that all values are givenin terms of the maximum amplitude max

~xjgið~x; teÞj percentage of the

incomplete generation approach. Some connection with the resultspresented in Fig. 5 can be noticed.

Finally, we study the evolution of kinetic, potential and totalenergies during the tsunami generation process described inSection 2.2.1. Specifically we are interested in quantifying the

contribution of horizontal displacements into tsunami energy bal-ance. Fig. 9 shows the evolution of potential (Fig. 9(a)) and kinetic(Fig. 9(b)) energies for four cases already mentioned above. Hereagain blue lines refer to complete generation scenarios while blacklines – to vertical displacements only. Consecutive peaks in the ki-netic energy come from the activation of new fault segments inaccordance with the rupture propagation along the fault. The en-ergy curves vary in an analogous way with the tide gauges records(see Fig. 7). Namely, the trigonometric scenario leads to slightlyhigher energies than the exponential one. However, once the bot-tom motion stops, this difference becomes negligible. The inclusionof horizontal displacements has a much more visible effect withhigher energies.

We performed also the same simulation but using the finitefault solution obtained by the Caltech team (Ozguns Konca,2006) and the trigonometric scenario. The results concerning thekinetic and potential energies are presented in Fig. 10. It is interest-ing to note that the potential energy evolution in the Caltech sce-nario appears to be smoother especially after t ¼ 150 s. However,the peaks corresponding to subfaults activation times are equallypresent in kinetic energies. The magnitudes of kinetic and potentialenergies predicted according to the Caltech inversion are approxi-matively 10 times higher than corresponding USGS results.

We have to underline that inversions performed by the two dif-ferent finite fault algorithms lead to very different results. Ourmethod is operational with both of them. It is particularly interest-ing to compare the total energies evolution predicted by USGS andCaltech inversions. These results are presented in Fig. 11. TheCaltech version of the rupturing process generates a tsunami wavewith much higher energy (computed after the end of the bottommotion when a tsunami enters in the propagation regime):

� USGS, complete trigonometric: Ec ¼ 2:16� 1012 J,� USGS, complete exponential: Ec ¼ 2:11� 1012 J,� USGS, incomplete trigonometric: Ei ¼ 1:95� 1012 J,� USGS, incomplete exponential: Ei ¼ 1:90� 1012 J,� Caltech, complete trigonometric: Ec ¼ 2:28� 1013 J,� Caltech, incomplete trigonometric: Ec ¼ 1:83� 1013 J.

This result is not surprising since according to the USGS solutionthe Java 2006 event magnitude is Mw ¼ 7:7 and according to theCaltech team, Mw ¼ 7:9, which means a huge difference since thescale is logarithmic.

Remark 7. We would like to note an interesting property. Forlinear waves it can be rigorously shown the exact equipartitionproperty between the kinetic and potential energies (Whitham,1999). When the nonlinearities are included, this equidistributionproperty is only approximate. One can observe that at the finaltime in our simulations the kinetic and potential energies arealready of the same order of magnitude regardless the employedbottom motion (see Figs. 9 and 10). Moreover, both curves KðtÞ andPðtÞ continue to tend to the equilibrium state according to thetheoretical predictions of the water wave theory (Whitham, 1999).

Despite some local attenuation effects of horizontal displace-ments on the free surface elevation, the complete generationscenario produces a tsunami wave with more important energycontent. More precisely, our computations show that the horizon-tal displacements contribute about 10% into the total tsunamienergy balance in the USGS scenario (this value is consistent withour previous results concerning the differences in wave amplitudesin Fig. 8). This result is even more flagrant for the Caltech versionwhich ascribes 24% of the energy to horizontal displacements.The free surface amplitudes in this case should differ as well bythe same order of magnitude. As we already noted, the differencebetween trigonometric and exponential scenarios is negligible.

Fig. 7. Free surface elevations computed numerically at six wave gauges located approximately in local extrema of the static bottom displacement. The vertical axis isrepresented in meters and time is given in seconds. The black lines correspond to the wave generated only by the vertical displacements (incomplete generation) while bluelines take also into account the horizontal displacements contribution (complete scenario).


Up to now the contribution of horizontal displacements hasbeen quantified in terms of the bottom deformation (Fig. 5), freesurface elevation at a particular moment of time (t ¼ 220 s, cf.Fig. 8) and finally in terms of the wave energy (Figs. 9–11). How-ever, the wave energy cannot be directly measured in practice.

Consequently, we continue to quantify the differences betweencomplete and incomplete approaches in terms of the waveforms.Namely, we compute also the following relative measures of thewaveforms difference which act on the whole computed freesurface:

Fig. 8. Relative difference between the free surface elevation at te ¼ 220 scomputed according to the complete gcð~x; teÞ (with horizontal displacements) andincomplete gið~x; teÞ (only vertical component) tsunami generation scenarios. Thebottom moves according to the trigonometric scenario (2.1). The vertical scale isgiven in percents of the maximum amplitude of the incomplete scenario –max~xgið~x; teÞ.


d2ðtÞ :¼jjgcð~x; tÞ � gið~x; tÞjjL2ðR2Þ

jjgið~x; tÞjjL2ðR2Þ;

d1ðtÞ :¼jjgcð~x; tÞ � gið~x; tÞjjL1ðR2Þ

jjgið~x; tÞjjL1ðR2Þ: ð3:1Þ

The simulation results are presented on Fig. 12. One can see thatboth measures grow up to 15% and then oscillate around this level,at least during the simulation time. This result is in agreement withprevious measurements of the horizontal displacements contribu-tion based on the wave energy and the bottom deformation whichwere of the order of 10%. We note also that the curve d2ðtÞ has amore regular behavior than d1ðtÞ since it is based on integralcharacteristics of the difference while the latter focusses on thecharacteristics of the extreme values.

It is also interesting to compare the tsunami energy with theenergy of the underlying seismic event. The USGS Energy and

0 50 100 150 200 2500

2

4

6

8

10

12

14x 1011

t

P(t),

J

Potential energy

Complete, trig.Incomplete, trig.Complete, exp.Incomplete, exp.

Fig. 9. Energy evolution during our simulations in the complete (blue solid line) and incright images. The time t is given in seconds.

Broadband solution indicates that the radiated seismic energy ofthe Java 2006 earthquake is equal to 3:2� 1014 J. Hence, accordingto our computations with the USGS complete generation approachand the trigonometric scenario, about 0.68% of the seismic energywas transmitted to the tsunami wave (this portion raises to 7.14%for the Caltech version). The ratio of the total tsunami and seismicradiated energies can be used as a measure of the tsunami gener-ation efficiency of a specific earthquake. The seismic radiatedenergy may not be the most appropriate parameter to use, but ithas an advantage to be relatively robust and easily observable incontrast to the more relevant seismic fracture energy (Venkatar-aman and Kanamori, 2004; Kanamori and Rivera, 2006).

This parameter can also be estimated for the Great Sumatra–Andaman earthquake of December 26, 2004. The total energy ofthe Great Indian Ocean tsunami 2004 was estimated by Lay et al.(2005), to be equal to 4:2� 1015 J. According to the same USGSEnergy and Broadband solution the radiated seismic energy wasequal to 1:1� 1017 J. Thus, the tsunamigenic efficiency of the GreatSumatra–Andaman earthquake is 3.81% which is bigger than thatof Java 2006 event but remains in the same order of magnitudearound 1%. It is possible that we do not still take into account allimportant factors in Java 2006 tsunami genesis. More examplesof transmission of seismic energy to tsunami energy can be founde.g. in Løvholt et al. (2006).

4. Discussion and conclusions

In the present study the process of tsunami generation isfurther investigated. The current tsunamigenesis model relies ona combination of the finite fault solution (Ji et al., 2002) and a re-cently proposed Weakly Nonlinear (WN) solver for the water waveproblem with moving bottom (Dutykh et al., 2012). Consequently,in our model we incorporate recent advances in seismology andcomputational hydrodynamics.

This study is focused on the role of the horizontal displacementsin the real world tsunami genesis process. By our intuition weknow that a horizontal motion of the flat bottom will not causeany significant disturbance on the free surface. However, the realbathymetry is far from being flat. The question which arisesnaturally is how to quantify the effect of horizontal co-seismic dis-placements during real world events.

The primary goal of this study was to propose relatively simple,efficient and accurate procedures to model tsunami generation

0 50 100 150 200 2500

0.5

1

1.5

2

2.5

3

3.5x 1012

t

K(t),

J

Kinetic energy


omplete (black dotted line) scenarios. Note that scales are different on the left and

0 50 100 150 200 2500

2

4

6

8

10

12

14

16

18x 1012

t

P(t),

J

Potential energy

Complete, trig.Incomplete, trig.

0 50 100 150 200 2500

1

2

3

4

5

6

7

8

9

10x 1012

t

K(t),

J

Kinetic energy


Fig. 10. Energy evolution during our simulations in the complete (blue solid line) and incomplete (black dotted line) scenarios using the finite fault inversion by CaltechTectonics Observatory (Ozguns Konca, 2006). Note that the scales are different on the left and right images. The time t is given in seconds.

0 50 100 150 200 2500

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 1012

t

E(t),

J

Total energy


0 50 100 150 200 2500

0.5

1

1.5

2

2.5x 1013

t

P(t),

JTotal energy


Fig. 11. The total energy evolution predicted according to two different versions of the finite fault inversion. Note the different vertical scales on left (4:5� 1012) and right(2:5� 1013) images. The time t is given in seconds.

Fig. 12. Relative difference between free surface elevations computed according tothe complete and incomplete scenarios. For more details see Eq. (3.1). Thehorizontal axis is given in seconds, while the vertical scale is a percentage. Theblue solid line (—) corresponds to the measure d2ðtÞ. The black dashed line (� � �)represents d1ðtÞ.


process in realistic environments. Special emphasis was payed tothe role of horizontal displacements which should be also takeninto account. Thus, the kinematics of horizontal co-seismicdisplacements were reconstructed and their effect on free surfacemotion was quantified. The evolution of kinetic and potential ener-gies were also investigated in our study. In the case of July 17, 2006Java event our simulations indicate that 10% of the energy inputcan be seemingly ascribed to effects of the horizontal bottom mo-tion. This portion increases considerably if we switch to the sce-nario proposed by Caltech Tectonics Observatory (Ozguns Konca,2006).

The results presented in this study do not still explain the rea-sons for the extreme run-up values caused by July 17, 2006 Javatsunami (Fritz et al., 2007). At least we hope that the proposedmethodology illustrated on this important real world event willbe proved helpful in future studies. In our opinion a successful the-ory should incorporate also other generation mechanisms such aslocal landslides/slumps which are subject to large uncertaintiesat the current stage of our understanding. Until now there are no

0 20 40 60 80 100 1200

2

4

6

8

10

12

14

16

x 1011

t

P(t)+

K(t),

J

Total energy


0 50 100 150 200 2500

1

2

3

4

5

6

7

8

9

10x 1025

t, s

dM/d

t, J/

s

Moment release rate

Fig. 13. Total tsunami energy and seismic moment rate function for the Mentawai 2010 event. The time t is given in seconds.

0 50 100 150 200 2500

0.5

1

1.5

2

2.5

3x 1015

t

P(t)+

K(t),

J

Total energy


0 50 100 150 200 2500

1

2

3

4

5

6

7

8

9x 1027

t, s

dM/d

t, J/

sMoment release rate

Fig. 14. Total tsunami energy and seismic moment rate function for the Great East Japan Earthquake. The time t is given in seconds.


detailed images of the seabed which could support or disprove thisassumption. In this study we succeeded to quantify the signifi-cance of horizontal displacements for the tsunami generation.

6 The total released seismic moment is given by the integral of this function over therupture duration (time).

Acknowledgement

D. Dutykh acknowledges the support from French AgenceNationale de la Recherche, project MathOcean (Grant ANR-08-BLAN-0301-01) and Ulysses Program of the French Ministry ofForeign Affairs under the project 23725ZA. French-Russian collab-oration is supported by CNRS PICS Project No. 5607. L. Chubarovand Yu. Shokin acknowledge the support from the Russian Founda-tion for Basic Research under RFBR Project No. 10-05-91052 andfrom the Basic Program of Fundamental Research SB RAS ProjectNo. IV.31.2.1.

We would like to thank Professors Didier Clamond and FrédéricDias for very helpful discussions on numerical simulation of waterwaves. Special thanks go to Professor Costas Synolakis whose workon tsunami waves has also been the source of our inspiration. Fi-nally we thank two anonymous referees for the time they spentto review our manuscript and whose comments helped us a lotto improve its quality.

Appendix A. Applications to some recent tsunami events

In this Appendix we apply the techniques described in thismanuscript to two recent significant events. We compute the en-ergy transmission from the corresponding seismic event to theresulting tsunami wave with and without horizontal displace-ments. Throughout this Appendix we use the trigonometric sce-nario and the finite fault solutions produced by USGS.

Remark 8. In later versions of the finite fault solution theactivation and rise time, as well as the seismic moment arespecified for each sub-fault. Consequently, below we use thisinformation to produce more accurate bottom kinematics. It allowsus to compute also the seismic moment rate function.6

A.1. Mentawai 2010 tsunami

In October 25, 2010 a small portion of the subduction zone sea-ward of the Mentawai islands was ruptured by an earthquake of


magnitude Mw ¼ 7:7 (Lay et al., 2011). This earthquake generated atsunami wave with run-up values ranging from 3 to 9 m (with evenlarger values at some places). On Pagai islands this tsunami causedmore than 400 victims. This earthquake is characterized by 10� dipangle, a slow rupture velocity (�1.5 km/s) which propagated dur-ing about 100 s over 100 km long source region. For our simulationwe used the finite fault solution produced by USGS (Hayes, 2010).In Fig. 13 we show the evolution of the total tsunami energy to becompared with the seismic moment release rate function.

Our computations give the following estimations:

� Complete scenario: Ec ¼ 1:11� 1012 J,� Incomplete scenario: Ei ¼ 1:06� 1012 J.

Consequently, for this event only 4.5% of the total energy is due tohorizontal displacements. According to the USGS energy andbroadband solution the radiated seismic energy is Es ¼ 1:4 �1015 J. The tsunami energy constitutes 0.08% of the radiated seis-mic energy. This surprising result can be explained by the fact thata big portion of co-seismic displacements occured on the landwhich did not fortunately contribute to the tsunami generationprocess.

A.2. March 11, 2011 Japan earthquake and tsunami

This tsunami event was caused by an undersea megathrustearthquake of magnitude Mw ¼ 9:0 off the coast of Japan. Officiallythis earthquake was named the Great East Japan Earthquake. TheJapanese National Police Agency has confirmed more than 13,000deaths caused both by the earthquake and especially by thetsunami. Entire towns were devastated. The local infrastructureincluding the Fukushima Nuclear Power Plant were heavily af-fected with consequences which are widely known.

We use again the finite fault inversion performed at the USGS(Hayes, 2011). The tsunami total energy evolution along with themoment rate function are represented on Fig. 14. At the end ofthe simulation we obtain the following result:

� Complete scenario: Ec ¼ 1:64� 1015 J,� Incomplete scenario: Ei ¼ 1:50� 1015 J.

Henceforth, the contribution of horizontal displacements is esti-mated to be 9.2%. The radiated seismic energy is Es ¼ 1:9� 1017 Jaccording to the USGS energy and broadband solution. The totaltsunami wave energy represents only 0.87% of the radiated seismicenergy. This value is much lower than the corresponding value forthe Great Sumatra–Andaman earthquake of December 26, 2004.

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http://www.tectonics.caltech.edu/slip_history/2006http://www.tectonics.caltech.edu/slip_history/2006

On the contribution of the horizontal sea-bed displacements into the tsunami generation process1 Introduction2 Mathematical models2.1 Dynamic bottom displacements reconstruction2.2 The water wave problem with moving bottom2.2.1 Tsunami wave energy

3 Numerical results4 Discussion and conclusionsAcknowledgementAppendix A Applications to some recent tsunami eventsA.1 Mentawai 2010 tsunamiA.2 March 11, 2011 Japan earthquake and tsunami

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