MJe

USA

TsunamiDispersive effectCoriolis effect

weaceanonl

fourntexat t

numerical experiments in an idealized ocean using Gaussian and di-polar sources with different source

ulatinw waten wavstanda

et al. (2006) concluded that the development in time of the wavefront is strongly connected to dispersion effects. Further support-ing this conclusion, Glimsdal et al. (2006) and Grue et al. (2008)showed, in their dispersive simulations of this event, that an undu-lar bore could evolve in shallow water, in accordance with othertsunami observations (Shuto, 1985). Finally, using the dispersiveBoussinesq model FUNWAVE (Chen et al., 2000; Kennedy et al.,

et al., 1990), dispersive Boussinesq models such as FUNWAVE,which were initially developed for modeling ocean wave transfor-mations from intermediate water depths to the coast, have usuallybeen implemented in Cartesian coordinates without Corioliseffects included. Lvholt et al. (2008) recently developed a Bous-sinesq model in spherical coordinates, including Coriolis effects.Their simulations quantied the effects of earths rotation andthe importance of Coriolis forces on far-eld propagation acrossthe Atlantic Ocean of a potential tsunami originated in La Palma(Canary Islands).

Corresponding author. Tel.: +1 302 831 2438; fax: +1 302 831 1228.

Ocean Modelling 62 (2013) 3955

Contents lists available at

o

elsE-mail address: kirby@udel.edu (J.T. Kirby).satisfactory for simulating tsunamis caused by smaller scale ormore concentrated non-seismic sources, such as submarine massfailures (SMF) (e.g., Lvholt et al., 2008; Tappin et al., 2008). More-over, even for very long waves such as found in co-seismic tsuna-mis, frequency dispersion effects may become signicant in thelong distance propagation of tsunami fronts. This was evidencedby Kulikovs (2005) wavelet frequency analysis of satellite altime-try data recorded in deep water in the Bay of Bengal during the2004 Indian Ocean tsunami. Based on these, Kulikov concludedthat a dispersive long wave model should be used for this event.In their dispersive numerical simulation of this event, Horillo

Regarding effects of sphericity and earth rotation on tsunamipropagation, even for the large 2004 event, numerical resultsshowed that a Cartesian implementation of the models (neglectingCoriolis effects) is adequate for regional scale tsunami simulations,provided distances are corrected to account for earths sphericity(e.g., Grilli et al., 2007; Ioualalen et al., 2007); this is particularlyso when the main direction of propagation closely follows a greatcircle. For global tsunami propagation, however, sphericity andCoriolis effects might play a larger role in simulating tsunami wavearrival at far distant sites. While standard NSWE tsunami simula-tion models have typically included such effects (e.g., Shuto1. Introduction

Conventional models used for simare traditionally based on the shalloglect frequency dispersion effects ostudies, however, reveal that these1463-5003/$ - see front matter 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.ocemod.2012.11.009sizes. A simulation of the Tohoku 2011 tsunami is used to illustrate the effects of dispersive and Corioliseffects at large distances from the source region.

2012 Elsevier Ltd. All rights reserved.

g global-scale tsunamisr equations, which ne-e propagation. Recentrd models may not be

2000) to simulate the same event, Grilli et al. (2007) and Ioualalenet al. (2007) quantied dispersive effects by performing simula-tions with and without dispersive terms (thus solving nonlinearshallow water equations (NSWE) in the latter case). Differencesof up to 20% in surface elevations between Boussinesq and NSWEsimulations were found in deeper water.Keywords:Boussinesq wave model

source width, while the effect of the Coriolis force increases with an increase of the source width. A sen-sitivity analysis to dispersive and Coriolis effects is carried out using the numerical model in a series ofDispersive tsunami waves in the ocean:dispersion and Coriolis effects

James T. Kirby a,, Fengyan Shi a, Babak Tehranirad a,aCenter for Applied Coastal Research, University of Delaware, Newark, DE 19716, USAbDepartment of Ocean Engineering, University of Rhode Island, Narragansett, RI 02882,

a r t i c l e i n f o

Article history:Received 15 February 2012Received in revised form 12 November 2012Accepted 16 November 2012Available online 19 December 2012

a b s t r a c t

We derive fully nonlinear,a shallow, homogeneous odeveloped for the weakly ndifference method with ascheme in time. In the coscaling analysis reveals th

Ocean M

journal homepage: www.ll rights reserved.odel equations and sensitivity to

ffrey C. Harris b, Stephan T. Grilli b

kly dispersive model equations for propagation of surface gravity waves inof variable depth on the surface of a rotating sphere. A numerical model isinear version of the model based on a combined nite-volume and nite-th-order MUSCL-TVD scheme in space and a third-order SSP RungeKuttat of tsunami generation and propagation over trans-oceanic distances, ahe importance of frequency dispersion increases with a decrease of the

SciVerse ScienceDirect

delling

evier .com/locate /ocemod

Dt0 w ; z h 8

where

DDt0

t0 u0

r0 cos h/

v 0r0h 9

Note that (8) allows for an imposed motion of the ocean bottom tobe specied.

In Boussinesq or shallow water theory, it is typical to replacethe local continuity equation (1) with a depth-integrated conserva-tion equation for horizontal volume uxes. Integrating (1) overdepth and employing the kinematic boundary conditions (7) and(8) yields

r02jg0g0t0 r02jh0h0t0 1

cos h@

@/

Z g0h0

u0r0dz0( )

1cos h

@

@hcos h

Z g0h0

v 0r0dz0( )

0 10

This equation will be simplied below based on scaling arguments.

2.1. Scaling

Based on the standard procedure for shallow water scaling, weintroduce the length scales h00, a

00 and k

0, denoting a characteristic

odeBased on recent work summarized above, it appears that theBoussinesq approximation may be a more accurate tool for per-forming tsunami simulations relative to more conventional modelsbased on the shallow water theory, since frequency dispersion ef-fects are manifested in almost all cases, either at large distances inlarger scale, co-seismic events, or at much shorter distances insmaller scale SMF events. However, the computational demandsof such simulations has been a concern. As pointed out by Yoon(2002), Boussinesq models require vast computer resources dueto the implicit nature of the solution technique used to deal withdispersion terms. Some simulations may involve a wide range ofscales of interest, from propagation out of the generation region,through propagation at ocean basin scale, to runup and inundationat affected shorelines (Grilli et al., 2007). Improvement in modelefciency can be achieved by using nested grids (e.g., Ioualalenet al., 2007; Yamazaki et al., 2011; Son et al., 2011), unstructuredor curvilinear grids (Shi et al., 2001) and parallelization of the com-putational algorithms (Sitanggang and Lynett, 2005; Pophet et al.,2011; Shi et al., 2012a).

In this study, we rigorously derive and validate equations for adispersive Boussinesq model on the surface of a rotating sphere,including Coriolis effects. The numerical scheme for the weaklynonlinear case is developed following the recent work of Shiet al. (2012a), who applied a TVD Riemann solver to Boussinesqmodel equations of Chen (2006), extended to incorporate a movingreference level as in Kennedy et al. (2001). The model is imple-mented using a domain decomposition technique and uses MPIfor communication in distributed memory systems. The relativeimportance of frequency dispersion and Coriolis effects in tsunamipropagation is evaluated both theoretically and based on numeri-cal simulations of an idealized case. The basic capability of themodel for computing basin-scale tsunami propagation is thendemonstrated using the 2011 Tohoku event; more detail of thisparticular case may be found in Grilli et al. (2012a), where an ear-lier version of the present model based on depth-averaged veloci-ties is used.

2. Model equations in spherical polar coordinates

We consider the motion of a uid column with variable stillwater depth h0/; h on the surface of a sphere of radius r00 to thestill water level, where coordinates r0; h;/ denote radial distancefrom the sphere center, latitude, and longitude, with the local ver-tical coordinate dened as z0 r0 r00 (Fig. 1). In this coordinatesystem, the dimensional Euler equations describing the ow ofan incompressible, inviscid uid are given by (Pedlosky, 1979,Section 6.2),

w0z0 2w0

r0 1r0 cos h

v 0 cos hh 1

r0 cos hu0/ 0 1

du0

dt0 u

0

r0w0 v 0 tan h 2X0w0 cos h v 0 sin h 1

qr0 cos hp0/ 2

dv 0

dt0 1r0v 0w0 u02 tan h 2X0 sin hu0 1

qr0p0h 3

dw0

dt0 u

02 v 02r0

2X0 cos hu0 1qp0z0 g 4

where X0 is the spheres angular velocity around the absolute verti-cal axis, u0 and v 0 are positive velocity components in the horizontalEasterly (/) and Northerly (h) directions respectively, and w0 de-notes the local vertical velocity. In the selected coordinate system,the total time derivative operator is dened as,

0 0

40 J.T. Kirby et al. / Ocean Mddt0

t0 u

r0 cos h/

vr0h w0z0 5Boundary conditions consist of a dynamic condition specifyingpressure ps on the free surface,

p0s/; h; t0 0; z0 g0 6together with a kinematic constraints on the velocity eld at thesurface and bottom boundary. The kinematic surface boundary con-dition (KSBC) is given by

Dg0

Dt0 w0; z0 g0 7

and the (kinematic) bottom boundary condition (BBC) is given by

Dh0 0 0 0

Fig. 1. Coordinate system for model development corresponding to Eqs. (1)(4).

lling 62 (2013) 3955water depth, wave amplitude (or amplitude of bottom displace-ment), and horizontal length; with the latter remaining to be

odeexamined. Combining these scales with each other and with r00yields a family of dimensionless parameters: h00=r00 denotingthe relative depth or thickness of the ocean layer; l h00=k0, theusual parameter characterizing frequency dispersion in Boussinesqtheory; and d a00=h00, the shallow water nonlinearity parameter.The parameter takes on values of O103 at maximum, and willthus always be taken to indicate vanishingly small effects when itoccurs in isolation.

Based on this family of parameters, we scale z-direction quanti-ties by as,

h; z h0; z0h00

; g g0

a0011

We take u00 dc00 dgh00

qto denote a scale for horizontal veloci-

ties, and let w00 denote a scale for vertical velocities, so that,

u;v u0;v 0u00

; w w0

w0012

Pressure is scaled by the weight of the static reference watercolumn

p p0

qgh0013

We introduce a rescaling of the dimensionless latitude and longi-tude according to

/; h r00

k0/; h l

/; h 14

This gives horizontal coordinates which change by O1 amountsover distances of Ok0. Retaining terms to O, the nondimensionalform of the continuity equation (1) is then given by

w00u00

wz 2w l1 zcos h u/ v cos hh O

2 15

indicating that w00=u00 Ol, as is usual in a Boussinesq or shallow

water model framework. Turning to the depth-integrated mass con-servation equation (10), we introduce the total depth,

H h dg 16and obtain

1dHt 1cos h Hu/ Hv cos hh O 17

where

u;v 1H

Z dgh

u; vdz 18

are depth-averaged horizontal velocities, and where time t0 is scaledaccording to

t x0t0 gh00

qk0

t0 19

In keeping with the notion that waves which are short relative tothe basin scale (k0=r00 or =l 1) may have frequencies which arehigh relative to the earths rotation rate (x0=X0 1), we introducethe scaling

X lX0

x0 O1 20

(Comparing to other treatments of this problem, we note that thischoice leads to a nondimensional Coriolis termwith an explicit scal-ing that changes its size in the Boussinesq regime in comparison to

J.T. Kirby et al. / Ocean Mthe shallow water regime, in contrast to the model equations inLvholt et al. (2008) or Zhou et al. (2011), for example, where therelative size of local acceleration and Coriolis terms in differentscaling limits is not apparent). Turning to the Easterly / momen-tum equation (2), we obtain

ut l

fv d ucos h

u/ vuh wuz luv tan h

d1

cos hp/

O 21where the dimensionless Coriolis parameter is dened asf 2X sin h. Similarly, the Northerly h momentum equation (3)becomes,

v t l

fu d ucos h

v/ vvh wvz lu2 tan h

d1ph O 22

The dimensionless vertical momentum equation is nally given by

dl2 wt d ucos hw/ vwh wwz h i

pz 1 O 23

In the following, we consider two relations between and l:(1) the regime l O, which recovers the shallow waterequations; and (2) the regime l O1=3, which yields theBoussinesq approximation. As a further note on the apparentscaling of the Coriolis parameter, f, consider the usual denition

of the dimensional Rossby deformation radius R0 gh00

q=f 0

gh00q

=2X0 sin h. In the present scaling, we obtain f r00=R0 or

l

f k

0

R024

which shows that the shallow water regime and Boussinesq regimecan be thought of as covering waves which are O1 in length ormuch shorter than the Rossby deformation radius, respectively,the Coriolis effects being weaker in the latter case.

2.2. Shallow water equations

Most theories of transoceanic tsunami propagation are basedon either the nonlinear shallow water equations (NSWE), or theirlinearized form, in recognition of the vanishing effects of disper-sion l! 0 for very long waves. In the present discussion, thislimit is obtained when the horizontal length scale of wavemotion approaches the horizontal scale of a global-sized oceanbasin, or k0 ! r00. This implies that the ratio =l O1, whileterms proportional to l appearing alone are essentially the sizeof already neglected terms of O. In this combined limit, thelocal vertical momentum equation (23) reduces to a hydrostaticbalance, which may be integrated down from the free surfaceto yield

p dg z 25This expression is used to evaluate pressure gradient terms in thehorizontal momentum equations, yielding the nal set of shallowwater equations

1dHt 1cos h Hu/ Hv cos hh 0 26

ut fv dcos h uu/ cos hvuh sin huv 1

cos hg/ 0 27

v t f u dcos h uv/ cos hvvh sin hu2 gh 0 28

lling 62 (2013) 3955 41where, in this limit, the scaled latitude and longitude revert to theoriginal values. Eq. (26) retains the possibility of describing wave

evolution in shallow coastal margins as well as mainly linear evo-

odelution in the deep ocean basin, we will retain the mechanics ofthe fully nonlinear Boussinesq model framework, following theapproach of Chen (2006) and Shi et al. (2012a) but working inthe framework of rotating ow. As in those studies, we use hori-zontal velocity ua at a reference level za as the dependent variableas a means of providing more accurate frequency dispersioneffects as well as for connecting the model more directly tolocal-scale models in Cartesian coordinates. In developing themodel, we seek to retain terms to Ol2 without any truncationin orders of d. This is in contrast to the classical Boussinesqapproach, which would take d Ol2 and truncate terms ofOd2; dl2;l4 and higher.

In the derivation, we retain the effect of an imposed bottommotion h/; h; t. The approximation is accompanied by theassumption that l O1=3. For O103, this implies a dis-persion term l O101, which would be reasonable for theusual surface wave problems. This choice of scaling then impliesthat O=l Ol2, indicating that Coriolis terms and undifferen-tiated advective acceleration terms are the same size as the lead-ing-order deviation of the pressure term from a hydrostaticbalance.

2.3.1. Pressure and vertical momentumPressure in the system being considered will deviate from

hydrostatic by Ol2 amounts. Denoting this non-hydrostatic com-ponent by ~p, we write

p/; h; z; t ph/; h; z; t dl2~p/; h; z; t dg z dl2~p 29

Introducing (29) in (23) and integrating up to the free surface(where ~p 0) gives

~pz Z dgz

wtdz dZ dgz

ucos h

w/ vwh wwzh i

dz O 30

The weakly nonlinear approximation with d=l2 O1would retain

~pz Z 0z

wtdz Od 31

2.3.2. The vertical structure of velocitiesIn order to use (30) to evaluate horizontal pressure gradients,

we need to establish a relation between w and components u;vthrough the continuity equation (15), which simplies to

wz lcos h u/ v cos hh O 32

Integrating (32) from h to z and using the bottom boundary con-dition gives

1 @Z z 1 @ Z z 1generation through a bottom motion ht , which appears at O1=dsince h0 is scaled by h00 rather than wave amplitude. This is impor-tant for modeling time dependent tsunami sources, such as occurfor large co-seismic events (e.g., Indian Ocean in 2004 or Tohokuin 2011) or for landslide tsunamis.

2.3. The Boussinesq approximation

We now wish to retain dispersive effects to leading order inthe description of wave motion. Further, in order to provide a uni-formly valid model which can be used to describe nonlinear wave

42 J.T. Kirby et al. / Ocean Mwzcosh @/ h

udz cosh @h h

v coshdz dht 33We now follow Nwogu (1993) and Chen (2006) and express thehorizontal velocities using Taylor series expansions about a refer-ence depth za. This approach, together with the closure assumptionthat horizontal components of vorticity are zero at leading order,expressed through

uz l2 w/

cos h Ol4; ; vz l2wh Ol4; 34

leads to the following expressions for the velocity components,

uz ua l2

cos hza zA/ 12 z

2a z2B/

Ol4; 35

vz va l2 za zAh 12 z2a z2Bh

Ol4; 36

wz A Bz 37

where

A 1dht 1cos h fuah/ va cos hhh g;

B 1cos h

fua/ va cos hh g 38

The vertical vorticity associated with (35) and (36) is given by

x x0 l2x2z Ol4; 39with

x0 1cos hva/ uah 40

and

x2 1cos h fAhza/ Bhzaza/ A/zah B/zazah g

tan hcos h

A/ za z B/ z2a

2 z

2

2

dhdh

41

In contrast to the case in Cartesian coordinates, the Ol2 contribu-tion to the vorticity here is depth dependent. In the Boussinesqregime, however, we have dh=dh l2 and thus the additionaltime-dependent term is of Ol4 overall, moving it outside theapproximate equations developed here. We thus denotex2z ~x2 l2x^2z and neglect x^2 from further analysis.

2.3.3. Fully nonlinear Boussinesq equationsFully nonlinear Boussinesq equations are obtained rst by

using the expressions (35)(37) to evaluate the pressure eld(30), giving

~pz l2 DaA dABz dg DaB dB2h i z2 dg2

2

!( )42

where

Da t dua

cos h/ vah

n o43

is a total derivative following the horizontal motion in the local tan-gent plane. Substitution of Eqs. (35)(37) into the / momentumequation (21) gives the approximate horizontal momentumequation

uat l2fva 1

cos hg/ d

uaua/cos h

vauah l2uava tan h

l2V /1 V /2 V /3 z Ol4; 44

lling 62 (2013) 3955where

uaA/ 1 2

J.T. Kirby et al. / Ocean Modelling 62 (2013) 3955 43za dg cos h vaAh 2 A dgB /46

V /3 z dva ~x2 dx0 za zAh z2a z2

2

Bh

47

Similarly, the h momentum equation can be written as

vat l2fua gh duava/cos h

vavah l2u2a tan h

l2V h1 V h2 V h3 z Ol4; 48where

V h1 z2a2Bth zaAth dg

2

2

Bt

h dgAth

49

V h2 dz2a dg2

2

!uaB/cos h

vaBh

za dg uaA/

cos h vaAh

(

12A dgB 2

)h

50

V h3 z dua ~x2 dx0cos h

A/ za z B/ z2a z22

51

At this level of approximation, all z-dependence is contained in thedispersive terms V3. In order to obtain a reduced-dimension, Bous-sinesq type system, we follow Chen (2006) and average (47) and(51) over depth to obtain the expressions

V /3 dva ~x2 dx0 Ah12z2a

16dg2 dgh h2h i

Bh za 12 dg h

52

and

V h3 dua ~x2 dx0cos h

B/12z2a

16dg2 dgh h2

B/ za 12 dg h

53

To complete the set of Boussinesq equations, the continuity equa-tion (17) is written by expressing u and v in terms of ua and va,giving

1dHt 1cos h M

// Mh cos hh

n o O;l4 54

where the volume ux components in / and h are given by

M/ H ua l2

coshza12hdg

A/ z2a

2h

2hdgdg26

!B/

" #( )

Mh H val2 za12hdg

Ah z2a

2h

2hdgdg26

!Bh

" #( )55

2.3.4. Weakly nonlinear approximationThe standard weakly nonlinear Boussinesq approximation fol-

lows from the assumption that d Ol2, leading to the immediateneglect of all terms of Odl2 in Eqs. (44), (48) and (54). The result-ing set of approximate momentum equations are given byV /1 1

cos hz2a2Bt/ zaAt/ dg

2

2

Bt

/ dgAt/

( )45

V /2 d

cos hz2a dg2

2

!uaB/cos h

vaBh (

uat l2fva 1

cos hg/ d

uaua/cos h

vauah

l2 eV /1 Ol4; dl2; 56

where

eV /1 1cos h zaAt/ z2a2 Bt/

57

and

vat l2fua gh duava/cos h

vavah

l2 eV h1 Ol4; dl2; 58

where

eV h1 zaAth z2a2 Bth

59

The approximate volume uxes in (56) follow from (55) and are gi-ven by

M/ Hua l2

cos hh za 12 h

A/ z2a

2 h

2

6

!B/

" #

Mh Hva l2h za 12 h

Ah z2a

2 h

2

6

!Bh

" #60

We note that the scaling automatically eliminates the undifferenti-ated advective acceleration terms in (56) and (58). These terms arealso absent in the weakly nonlinear model of Lvholt et al. (2008)but are retained in the model of Zhou et al. (2011).

3. Numerical approach

The fully nonlinear system described here provides a compre-hensive model for studying tsunamis from the earliest stages ofgeneration to the nal stages of shoreline inundation and runup.As pointed out in Lvholt et al. (2008), there are possible advanta-ges to the fully nonlinear system particularly in the study of thedevelopment of undular bores over shelf regions, where weaklynonlinear models are known to over-predict bore undulationheights (Wei et al., 1995). However, it is our usual practice to sim-ulate nearshore propagation and inundation using the correspond-ing model system in Cartesian coordinates, as described in Shi et al.(2012a) and used recently in Grilli et al. (2012a). For simplicity, weillustrate the numerical implementation of the weakly nonlinearsystem (56)(60) here and obtain a model which is suitable forthe generation and propagation studies considered in the followingexamples.

Recent progress in the development of Boussinesq-type wavemodels using a combined nite-volume and nite-difference TVDschemes has shown robust performance of the shock-capturingmethod in simulating breaking waves and coastal inundation(Tonelli and Petti, 2009; Roeber et al., 2010; Shiach and Mingham,2009; Erduran et al., 2005, and others). In this study, we appliedthe MUSCL-TVD scheme in space and a high-order RungeKuttascheme in time with adaptive time stepping.

3.1. Conservative form of governing equations

The numerical implementation is based on dimensional formsof the weakly nonlinear governing equations, augmented by termsrepresenting bottom friction. We also neglect the direct generationof waves due to bottom motion. The model equations are given by

odeHt 1r0 cos h Hua/ Hva cos hhn

1r20 cos h

h za 12h

A/ z2a

2 h

2

6

!B/

!" #/

1r20

h cos h za 12h

Ah z2a

2 h

2

6

!Bh

!" #h

) 0 61

uat fva 1

r0 cos huaua/ 1r0 vauah

gr0 cos h

g/

1r20 cos h

zaAt/ z2a

2Bt/

Cd

Hjuajua

0 62

vat fua 1

r0 cos huava/ 1r0 vavah

gr0gh

1r20

zaAth z2a

2Bth

Cd

Hjuajua 0 63

with

A hua/ hva cos hhcos h

; B ua/ va cos hhcos h

64

and where Cd represents a drag coefcient. The system of Eqs.(61)(63) corresponds to the Nwogu-type equations used inLvholt et al. (2008). In order to apply the combined nite-volumeand nite-difference schemes, the governing equations (61)(63)are re-arranged to a conservative form following Shi et al.(2012) for the fully non-linear Boussinesq equations in Cartesiancoordinates. We dene

n1 r0 cos h0/ /0n2 r0h h0

65

where /0; h0 are the reference longitude and latitude, respectively.n1; n2 represent coordinates in the longitude and latitude direc-tions, respectively. The conservative form of (61)(63) can be writ-ten as

@W@t

r HW S 66

where W and HW are the vector of conserved variables and theux vector function, respectively, and are given by

W H

U

V

0B@1CA; H

SpPi Q jSpP2

H 12 Spgg2 2ghh i

i PQH jSpPQH i Q

2

H 12 gg2 2ghh i

j

0BBB@1CCCA 67

where Sp is a spherical coordinate correction factor given by

Sp cos h0cos h 68

P Hua hu1 and Q Hva hv1, in which u1; v1 are dened by

u1 za h2

Sp Sphuan1n1 hvan1n2 1r0

tan hhvan1

z2a

2 h

2

6

!Sp Spuan1n1 van1n2

1r0

tan hvan1

69

v1 za h2

Sphuan1n2 hvan2n2 1r0tan hhvan2

!

44 J.T. Kirby et al. / Ocean M z2a

2 h

2

6Spuan1n2 van2n2

1r0tan hvan2

70The conserved variables U and V in (67) are given by

U Hua F 71

V Hva G 72in which

F z2a

2S2puan1n1

z2a2Sp van1n2

1r0

tan hvan1

zaS2phuan1n1

zaSp hvan1n2 1r0

tan hhvan1

73

and

G z2a

2Spuan1 n2

z2a2

van2n2 1r0tan hvan2

zaSphuan1 n2 za hvan2n2

1r0tan hhvan2

74

S in (66) represents a source array given by

S

1r0tan hHva hv1

Spgg @h@n1 fHva Cduau2a v2a

p w1gg @h

@n2 fHua Cdva

u2a v2a

p w20BB@

1CCA 75where

w1 gtF u1 hSpuau1n1 vau1n2 Spu1uan1 v1uan2 76

w2 gtG v1 hSpuav1n1 vav1n2 Spu1van1 v1van2 77The surface elevation gradient term was split into

12 Spgg2 2gh; 12 gg2 2gh

in (67) and Spgghn1 ; gghn2 in (75)in order to use a well-balanced numerical scheme (Shi et al.,2012a).

Eq. (66) is solved using the MUSCL-TVD scheme and the HLLapproximate Riemann solver. A MUSCL-TVD scheme up to thefourth-order in space (Yamamoto and Daiguji, 1993) and a third-order Strong Stability-Preserving (SSP) RungeKutta (Gottliebet al., 2001) in time were adopted. Model implementation also in-cludes wave breaking and wettingdrying schemes for shallowwater, as described in Shi et al. (2012a).

3.2. Parallelization

In parallelizing the computational model, we use the domaindecomposition technique to subdivide the problem into multipleregions and assign each subdomain to a separate processor core.Each subdomain region contains an overlapping area of ghost cellsthree rows deep, as dictated by the 4th order computational stencilfor the leading order non-dispersive terms. The Message PassingInterface (MPI) with non-blocking communication is used to ex-change data in the overlapping region between neighboring pro-cessors. Velocity components are obtained from Eqs. (46) and(47) by solving tridiagonal matrices using the parallel pipeliningtridiagonal solver described in Naik et al. (1993).

To investigate the performance of the parallel program, numer-ical simulations of an idealized ocean case were rst tested withdifferent numbers of processors on the linux cluster mills.hpc.ude-l.edu using a heterogeneous set of nodes consisting of nodes with242.4 GHz cores and 64 GB of memory or 482.4 GHz cores and128 GB of memory per node. Tests here were conducted using 24core nodes. The test case uses a numerical grid of dimension5400 3600. Fig. 2 shows the model speedup versus number ofprocessors for tests with 1, 8, 16, 24, 36, 48 and 64 processors.

lling 62 (2013) 3955The effect of inter-node communication is noted as the computa-tion moves from one to two nodes (above 24 processors) and fromtwo to three nodes (above 48 processors), but overall performance

Fig. 3 illustrates variations of l2 and =l with respect to l or

Fig. 4. Geometries for (a) a dipolar source and (b) a Gaussian source used in modeltesting.

J.T. Kirby et al. / Ocean Modeh0=W , where W represents the characteristic width of the tsunamisource. Typically, for a source width of 100 km (for example, the2004 Indian Ocean tsunami), l 0:025 and the Coriolis effectis satisfactory in comparison to an ideal arithmetic speedup basedon performance using a single processor.

4. Tests of dispersion and Coriolis effects

The orders of the frequency dispersions terms and Coriolisterms in Eqs. (47) and (48) are Ol2 and O=l, respectively.The relative importance of frequency dispersion and Coriolis forcecan be evaluated using l2 and =l values in some specic cases.

Fig. 2. Variation in model performance with number of processors for a5400 3600 domain. Straight line indicates arithmetic speedup. Actual perfor-mance shown by circles.would be expected to be relatively more important than dispersionas shown in Fig. 3. This result would apply to all co-seismic sourceswith widths in the range 50 6W 6 500 km, for which0:01 6 l 6 0:1. For narrower sources with widths on the order of25 km or less, l values lie to the right of the intersection point in

Fig. 3. Relative importance of Coriolis force (=l) and frequency dispersion (l2)with varying inverse source width l h0=W .(b)(a)

lling 62 (2013) 3955 45Fig. 3, indicating that dispersive effects are as important as theCoriolis effect, and get relatively more important as the sourcewidth diminishes. Nevertheless, in the examples considered below,we nd that Coriolis effects are uniformly less important than dis-persion effects for all the cases considered, even though the scalingindicates otherwise.

4.1. Idealized tsunami sources and examples

There is a signicant lack of benchmark examples that can beused as test cases for the determination of accuracy of sphericalcoordinate ocean wave models. Shi et al. (2012b) have describedthe testing of an earlier version of the present code, based ondepth-averaged velocity, against the standard tsunami benchmarktests provided by Synolakis et al. (2007). Here, we compare modelresults to several of the detailed measurements obtained fromDART buoys during the 2011 Tohoku-oki tsunami event. Beforeturning to the realistic example, we rst illustrate the dependenceof the maximum wave height envelope on the effect of dispersiveand Coriolis effects using idealized sources which may be taken tobe indicative either of localized SMF events or of co-seismic eventswith limited lateral extent.

We utilize two idealized sources here: an initial Gaussianelevation

ode46 J.T. Kirby et al. / Ocean Mg/; h A exp r20

W2/ /c2 h hc2

78

with center at /c; hc, nominal source width W and amplitude A,and a dipolar initial displacement given by

g/; h 22

p

Wr0/ /c exp

r20W2

/ /c2 h hc2

79

where the orientation of the dipole may be changed by altering therst / /c factor. The shapes of these initial forms are shown inFig. 4. Of these two, the dipolar source is most representative of tsu-nami-like sources, and could represent either the net upward anddownward displacement of an Okada-like source, or the net effectof lateral translation or rotation of an SMF event.

We consider an idealized ocean and use a model grid in spher-ical coordinates with a at bottom bathymetry over the entireocean basin. The computational domain is in the northern hemi-sphere and covers a region from 10N to 50N in the south-northdirection and from 20W to 20E in the west-east direction. Waterdepth h 3000 m over the whole domain. The grid resolution is0:750. As is classically done in tsunami analyses, the bottom defor-mation is transferred to the spherical Boussinesq model as an

Fig. 5. (a) Dipolar initial condition (W 0:25 , Pa 2:5), (b) dipolar initial condition (Winitial condition (W 0:25 , Pa 2:5), (e) Gaussian initial condition (W 0:5 , Pa 5:0lling 62 (2013) 3955initial free surface condition with no initial uid velocities, speci-ed by either (78) or (79).

4.2. Source size and wave dispersion effect

The rst set of examples considers the effect of varying sourcewith on the rate of appearance of dispersive effects, using both ofthe source congurations considered above. Results are analyzedin light of a parameter due to Kajiura (1963), given by

Pa 6hL 1=3 W

h

80

where h is ocean depth, W is source width (as an indicator of ba-sic wavelength), and L denotes travel distance. Kajiura indicatesthat dispersive effects should begin to become apparent whenthe value of the Pa, which decreases with travel distance, dropsbelow 4.

Fig. 5 shows a snapshot at time t 5000 s of the NE quadrant ofwave trains evolving from both sources (78) and (79) with varyingsource widths. Panels (a)(c) correspond to a dipolar source ori-ented in the EW direction, while panels (d)(f) correspond toGaussian sources. Panels (a) and (d) correspond to a narrow initial

0:5 , Pa 5:0), (c) dipolar initial condition (W 1:0 , Pa 10:0), (d) Gaussian), and (f) Gaussian initial condition (W 1:0 , Pa 10:0), T 5000 s.

odeJ.T. Kirby et al. / Ocean Msource withW 0:25, corresponding to an SMF-sized source witha width of about 25 km, with parameter Pa 2:5, indicating thestrong evolution of a dispersive wave train at the time of 1.4 h afterevent initiation. In contrast the lower panels (c) and (f) represent asource with initial width W 1 or about 100 km, consistent witha larger co-seismic slip event. At this elapsed time of 1.4 h, Pa 10and dispersive effects are not apparent, indicating that dispersion(as manifested by the presence of an oscillatory wave train) isnot likely to occur for the rst several hours during the evolutionof a wave train from a classic co-seismic event.

Figs. 6 and 7 show the effect of dispersion on the spatial patternof maximum wave height for a strongly dispersive case. Fig. 6(a)and (b) shows the wave elds at t 5000 s for dispersive and non-dispersive calculations, respectively, for the dipolar source withinitial W 0:25, corresponding to Fig. 5(a). Fig. 6(c) shows thespatial pattern of the difference in wave height envelope for thesimulations with and without dispersion, and shows that there isa general tendency towards a decrease in wave height along theprinciple propagation direction when dispersion is taken into ac-count. (This tendency also occurs in the realistic simulations ofthe Tohoku-oki event shown below, although the tendency is re-versed at distances which are relatively shorter (higher Pa) than

Fig. 6. Dipolar initial condition with EW orientation and W 0:25 . (a) With dispersivdifference in wave height envelope, dispersivenondispersive. Plots are in percent of mlling 62 (2013) 3955 47the distances considered here, possibly due to complex refrac-tion/diffraction effects over the variable ocean bathymetry.) Simi-lar results occur for the Gaussian source as indicated in Fig. 7,aside from a more uniform distribution of results due to the ini-tially symmetric source.

Figs. 8 and 9 provide a more detailed picture of the evolution ofthe dispersive wave train evolving from the dipolar or Gaussiansources respectively, with W 0:25 in either case. In each panel(a)(f), a distance D

gh

pt along an EW transect through the

source center is chosen, and then a time series of the resultingwave form passing that point is constructed with the time axisshifted by the amount t, to obtain an arrival time of zero for a non-dispersive wave in Cartesian coordinates. In each gure, the dis-tance of the measurement point from the source origin along theEW direction varies from 308.6 km (at top) to 1851.8 km (at bot-tom), corresponding to values of Pa falling from 3.11 to 1.71. In thiscase, frequency dispersion is seen to be important at all displayeddistances from the source, and the evolved wave train representsan extensive packet of waves with a gradually decreasing waveperiod as the wave passes a xed point, indicting the expectedsorting of longer and shorter waves due to phase speed depen-dence on frequency.

e terms, Pa 2:5, t 5000 s; (b) without dispersive terms, Pa 2:5, t 5000 s; (c)aximum initial displacement, equal to 1 m.

ode48 J.T. Kirby et al. / Ocean M4.3. Coriolis effect on an idealized source

The examples presented in the previous section included Corio-lis effects in the computations. Corresponding runs without Corio-lis were found to have only minor effects on the outcome, in linewith the case study of the Tohoku-oki event discussed in the fol-lowing section. In this section, we consider an additional idealizedcase of a dipolar source centered at 50N with a NS orientation.This latitude corresponds to Aleutian Island sources, and is chosenso as to give a source located in a region with elevated rotationaleffects. The source considered here has a width W 0 4 or about300 km, which is still considerably smaller than an estimate ofthe Rossby radius of deformation of 1800 km for a water depthof h0 4 km and this latitude. Correspondingly, Coriolis effects onthe solution are still weak at this latitude. Fig. 10 illustrates a com-parison of the maximum wave height envelope after 10 h of simu-lation for cases with and without Coriolis included, with solidcontours corresponding to the case with Coriolis and dashed con-tours corresponding to the case without. Contour levels correspondto percentage of the initial source height. Coriolis is seen to lead toa somewhat more rapid decay of the wave height with distanceform the source center. The results also indicate that the resultswith Coriolis are somewhat asymmetrical in the EW direction,

Fig. 7. Gaussian initial condition with W 0:25 . (a) With dispersive terms, Pa 2:5, theight envelope, dispersivenondispersive. Plots are in percent of maximum initial displling 62 (2013) 3955with wave heights being larger to the East, or left, of the maindirection of propagation. This is a seemingly paradoxical result,as we may expect the tendency of Coriolis to deect motions tothe right in the Northern hemisphere to cause greater wave heightsto the right of the propagation direction, as illustrated in the nextsection for the Tohoku-oki event. In order to examine this further,we consider an idealized case of diplolar sources with NS and EW orientation with Coriolis either included or neglected. Thesource geometry and latitude are unchanged. Fig. 11(a) and (b) dis-play results with and without Coriolis force included for the samecase as in Fig. 10. Fig. 11(c) shows the difference between run withCoriois (a) and the run without Coriolis (b). The most obvious effectnoted here is that Coriolis tends to trap a portion of the initial waveclose to its original position, leading to a persistent dipolar featurenear the source center at (50N, 0E). The pattern also indicatesthat the evolving wave form with Coriolis is somewhat lower inamplitude to the right of the Southerly propagation direction, inagreement with Fig. 10. The waveform with Coriolis also has a low-er amplitude at the leading edge of the spreading wave, indicatingthat Coriolis is reducing wave propagation speed to a small extent.The dipolar source with EW orientation (Fig. 11(d)(f)) also showsa reduction in wave speed induced by Coriolis and a tendency totrap a portion of the initial wave form near the source location.

5000 s; (b) without dispersive terms, Pa 2:5, t 5000 s; (c) difference in wavelacement, equal to 1 m.

((

(

odeJ.T. Kirby et al. / Ocean MAt the propagation distances shown here, there is no clear ten-dency for the evolving waves to be deected to the right or leftof the principle propagation direction. See the following sectionfor indications of this sort of behavior for much larger propagationdistances in a realistic event.

Overall, the effect of Coriolis terms on evolving tsunami wavefronts appears to be of minimal importance, as noted in a numberof earlier investigations (Kowalik et al., 2005; Lvholt et al., 2008).

4.4. The 2011 Tohoku-oki tsunami

The ability of the new spherical Boussinesq model to simulatebasin-scale tsunami events is demonstrated in this section byapplying the model to the recent March 11th, 2011 M9 Tohoku-oki earthquake. During this event, an extremely devastating tsu-nami was generated in the near-eld while a signicant tsunamiwas observed at many far-eld locations. Grilli et al. (2012a) pro-vide a detailed account of the event, the earthquake source, thenear- and far-eld tsunami observations, and tsunami generationand propagation modeling using both the Cartesian version (Shiet al., 2012a) and an earlier spherical version of FUNWAVE-TVDbased on depth-averaged velocity. Here, we present results of sim-

(

(

(

Fig. 8. Evolution of dispersive (solid line) and nondispersive (dashed line) wave trains forand eastward propagating waves, respectively. Measurement locations are at D

gh

pt

5400 s and 925.8 km; (d) 7200 s and 1234.5 km; (e) 9000 s and 1543.1 km; and (f) 10,8a)

b)

c)

lling 62 (2013) 3955 49ulations with the weakly-nonlinear spherical Boussinesq modeldescribed above. We specically analyze effects of dispersion andCoriolis terms in the model equations on simulated maximum tsu-nami elevation in the Pacic Ocean. Comparisons are made basedon measured and modeled time series at the location of four DARTbuoys, one near Japan (21418), one off Oregon (46404), one nearHawaii (51407), and one near Panama (32411) as shown inFig. 14, and on a comparison of synoptic maps of maximum waveheight envelopes for the entire Pacic basin.

As in Grilli et al. (2012a), the computational domain covers aregion of the Pacic Ocean from 60S to 60N in the south-northdirection, and from 132E to 68W in the west-east direction(Fig. 12). In the present simulations, the grid resolution is im-proved to 20 compared to the 40 resolution used in Grilli et al.(2012a). Bathymetry is specied in the model based on the ETO-PO1 10 data base. In these simulations, we use the tsunami sourceof Grilli et al. (2012a), which is based on the 3D FEM model ofMasterlark (2003). This source, denoted UA, was derived from acombination of seismic and GPS inversion to specify the earth-quake-induced bottom uplift or subsidence as a function of time.The model simulations here do not make use of any hydrody-namic data in the determination of the source conguration.

d)

e)

f)

a dipolar source with initial widthW 0:25 . Left and right panels show westwardwith h 3000 m and t, D = (a) 1800 s and 308.6 km; (b) 3600 s and 617.3 km; (c)00 s and 1851.8 km.

ode50 J.T. Kirby et al. / Ocean MThe non-hydrostatic model NHWAVE (Ma et al., 2012) is used tosimulate the rst 5 min of tsunami generation, as in Grilli et al.(2012a), using a smaller and ner local 1 km resolution Cartesiangrid (see red rectangle in Fig. 12), based on the UA source.

(

(

(

(

(

(

Fig. 9. Gaussian source, pa

Fig. 10. (a) Comparison maximum recorded surface elevation (relative to initial source asimulated time. Dipolar source with width W 0 4:0) oriented in NorthSouth directionlling 62 (2013) 3955NHWAVE results for surface elevation and depth-averaged hori-zontal velocity at t 5 min are then interpolated over the spher-ical Boussinesq model grid, in which computations are theninitiated as a hot start. For a more detailed description of model

a)

b)

c)

d)

e)

f)

rameters as in Fig. 8.

mplitude) with Coriolis (solid lines) and without Coriolis (dashed lines) after 10 h of.

Fig. 11. Surface elevation comparison at T 7200 s for dipolar sources with initial widthW 4 and (a) NorthSouth orientation with Coriolis, (b) NorthSouth orientationwithout Coriolis, (c) (a,b), (d) EastWest orientation with Coriolis, (e) EastWest orientation without Coriolis, and (f) (d,e).

Fig. 12. Computational domain for far-eld simulations with FUNWAVE-TVD, with the marked location of all DART buoys in the region (labeled red dots used incomparisons). The smaller red box marks the location of NHWAVEs regional computational domain (Grilli et al., 2012a,b). (For interpretation of the references to color in thisgure legend, the reader is referred to the web version of this article.)

J.T. Kirby et al. / Ocean Modelling 62 (2013) 3955 51

Fig. 13. Comparison between measured surface elevation at DART buoys (black) and model simulations using full model including dispersion and Coriolis effects. Simulationsare based on the seismic/GPS UA source described in Grilli et al. (2012a). Buoy numbers and lead in model arrival time are (a) 21418, 0 min; (b) 51407, + 5 min; (c) 46404, +

ilita

52 J.T. Kirby et al. / Ocean Modelling 62 (2013) 3955setup, as well as a more comprehensive comparison of observa-tions and model results based on the depth-averaged velocity for-mulation, see Grilli et al. (2012a).

Fig. 13 shows a comparison of DART buoy measurements and

6 min, and (d) 32411, +10 min. Model results are offset by the indicated shift to facfull model predictions (retaining dispersion and Coriolis) at thefour selected locations. Timing of arrival of the main tsunami peak

Fig. 14. Model predicted surface elevations at DART buoys: (a) 21418, (b) 51407, 4640dashed line), and no dispersion/Coriolis (green dashed line). (For interpretation of the refarticle.)at the nearest buoy 21418 is accurate, and the wave form is repro-duced accurately aside from a trailing high frequency wave trainthat follows the main peak in the observations. (Grilli et al.(2012b) have recently speculated that the primary source of this

te wave form comparisons.early manifesting, shorter period (34 min) wave train is an SMFsource located to the north of the main coseismic slip.) Using an

4, and (d) 32411. Full model (blue line), no dispersion (red line), no Coriolis (blueerences to color in this gure legend, the reader is referred to the web version of this

estimated source width W 0 100 km and an average depth ofh0 4 km, an estimated travel distance of 1300 km at buoy21418 gives a value Pa 6:6, indicating that dispersive effectsshould not be apparent for waves generated by the main co-seis-mic source. At more distant buoys, the model results lead the mea-surements in arrival timing, with a progressive increase in leadtime with distance from the initial event source. Approximate leadtimes are 300 s at 51407, 360 s at 46404, and 600 s at 32411. Thistiming discrepancy could be due to a number of factors, includingdeviations from sphericity, errors in bathymetry, errors in leadingorder model dispersion, and truncation errors associated with dis-cretization. We have not done simulations at a higher resolution of1 min in order to test convergence, but note that results at 2 minresolution are signicantly improved over results at 4 min resolu-tion, where timing discrepancies are larger (Grilli et al., 2012a).

Fig. 13(b)(d) display model results with the leading shift in timeremoved in order to facilitate comparisons of the modeled waveforms. The results at the three distant buoys indicate that the mod-el accurately predicts the evolution of the leading features of thetsunami wave train, with good reproduction of the sequence, per-iod and amplitudes of arriving wave crests.

The effects of Coriolis force and frequency dispersion are illus-trated by comparing numerical results obtained with and withouteach term in the model equations in Fig. 14. Fig. 14(a) shows thatthe effect of frequency dispersion on the wave train is signicantalready at buoy 21418, where a forward steepening of the nondis-persive wave form is apparent in comparison to the wave formwith dispersion retained, although no oscillatory dispersive tailhas appeared yet. The differences between dispersive and nondis-persive calculations increase with distance from the source, and, bythe farthest buoy 32411, dispersion has created a wave train withsignicant following crests that are largely absent in the nondis-persive case, as would be expected. The parameter Pa takes onapproximate values of 3.9, 3.7 and 3.0 at buoys 51407, 46404and 32411, respectively, indicating that dispersive effects shouldonly be mildly apparent at the two intermediate buoys, and rela-tively more apparent at the most distant buoy, as is seen in themodel results. In contrast, the gure shows that the effect of Cori-olis terms on the calculation is indistinguishable, at least at theparticular buoy locations considered.

Fig. 15 summarizes the synoptic results. The center panel dis-plays the envelope of maximum water surface elevation for thecomplete model. The upper panel displayed the difference of themodel results with dispersion and the model results without dis-persion. The presence of dispersion in the calculation leads to localchanges in maximum wave height envelope of up to 20 cm even in

J.T. Kirby et al. / Ocean Modelling 62 (2013) 3955 53Fig. 15. Envelope of maximum computed wave elevation with FUNWAVE-TVD inspherical (20) Pacic grid: difference between maximumwave height envelope with

and without dispersion (upper panel); result with dispersion and Coriolis terms(center panel); and difference between maximum wave height envelope with andwithout Coriolis terms (lower panel).Fig. 16. Percent change in maximum wave height envelope for the Tohoku-okitsunami for (top) simulations with and without dispersion and (bottom) simula-tions with and without Coriolis.

odethe deep ocean, which represents a signicant deviation in modelpredictions with and without dispersion incorporated. In contrast,the lower panel in Fig. 15 indicates that the effect of Coriolis on thecalculation is relatively minor, with deviations in wave amplitudeon the order of a centimeter over the entire ocean basin. The pres-ence or absence of Coriolis effects in the calculations does not leadto a practical difference in the form or amplitude of modeled wavesfor the Tohoku-oki simulation. Coriolis force causes waves to theright of the main direction of travel to be generally higher thanthey would be without it, indicating a subtle shift of the initial pro-gressive wave pattern to the right of the rotation-free direction ofpropagation in the generation region, consistent with results for EW oriented dipolar sources in the Northern hemisphere in the pre-vious section.

Fig. 16 provides an additional view of the synoptic results, withthe absolute wave height difference plots of Fig. 15 being replacedby plots of percent change resulting from taking the ratio of thedifference plots to the full model simulation. The top panel indi-cates that dispersion has a pronounced effect on wave height dis-tribution in the far eld. There is an overall tendency for thedispersive simulation to lead to a reduction of wave height in therelative near eld down wave of the source. However, this effectis partially reversed in the far eld, where waves are often signi-cantly larger in the dispersive case than the nondispersive case.This effect is partially due to a simple spatial shifting of concen-trated wave energy in lateral directions, as evidenced by a patternof positive and negative deviations along transects perpendicularto the main propagation direction. However, there is a net overalltendency towards increased wave height in the far eld, indicatinga systematic change in wave form due to dispersive effects.

In contrast, percent changes due to Coriolis effects are on the or-der of a few percent at most, and are likely to be insignicant rel-ative to uncertainties in source conguration and other factors in arealistic simulation. Changes along the main propagation directionare on the order of 3% in the far eld. These values are consistentwith previous results for simulations of the Cumbre Vieja volcanoevent described by Lvholt et al. (2008). The maximum effect ofCoriolis is noticed along boundaries to the north and east (or tothe left of the main propagation direction) where Coriolis effectsreduce wave heights by up to 5%, and to the south and west (orto the right of the main propagation direction), where results arerelatively increased, particularly in regions that are strongly shad-owed by island chains.

5. Conclusions

We have derived fully nonlinear Boussinesq equations forweakly dispersive wave propagation on the surface of a rotatingsphere, including Coriolis effects. The model equations incorporateimproved dispersion following Nwogu (1993) and Lvholt et al.(2008). The weakly nonlinear version of the model is implementedusing a Godunov-type method with a fourth-order MUSCL-TVDscheme in time and a third-order SSP RungeKutta scheme. Themodel is implemented using a domain decomposition techniqueand optimized for parallel computer clusters using MPI. Modelspeedup tests with multiple processors show a nearly linear speed-up, suggesting that such a Boussinesq model can be efciently usedfor modeling global wave propagation.

A scaling analysis indicates that the importance of frequencydispersion should increase with a decrease in tsunami sourcewidth, and that effects of Coriolis force should increase with an in-crease of the source width. The importance of dispersive effectsboth in the far eld of large sources as well as in the near eld of

54 J.T. Kirby et al. / Ocean Mcompact, SMF-like sources is established using idealized examples.In contrast, it is seen that tsunami wave trains corresponding totypical wavelengths for co-seismic events are relatively unaffectedby rotational effects, and it is unclear that their retention in themodel is a necessary part of obtaining realistic simulations. Theseresults are in line with recent suggestions of Kowalik et al. (2005)and Lvholt et al. (2008, 2012). As the Coriolis terms do not repre-sent a difculty in developing the numerical scheme itself, though,there is little reason to argue that they should not be retained forcompleteness.

A simulation of the Tohoku-oki event and comparison to far-eld DART buoy observations provides a strong test of the accuracyof the seismic/GPS source developed by Grilli et al. (2012a), whichappears to be the most accurate available co-seismic source amongthose which are developed without input from hydrodynamic data.(For contrast, see the recent work of Lvholt et al. (2012), wherethe inclusion of hydrodynamic measurements is probably thestrongest factor guiding the choice of source conguration.) Grilliet al. (2012b) have recently hypothesized that an additional SMFcomponent of the event is crucial to an overall understanding ofthe observed tsunami properties, both in terms of modeling coastalinundation and in reproducing short period oscillations observedin GPS and closer DART buoy records. The far eld DART buoy re-cords examined here do not provide a clear picture of these addi-tional short wave effects, as they have either dispersed at thesedistances or are buried within additional under-resolved scatteringeffects from nearby shelf and coastal boundaries.

Acknowledgements

The authors wish to acknowledge support from the NationalTsunami Hazard Mitigation Program (NOAA). Harris and Grilliacknowledge support from Grant EAR-09-11499/11466 of the USNational Science Foundation (NSF) Geophysics Program. Kirbysportion of the linux cluster Mills was supported by the Ofce ofNaval Research and the University of Delaware.

References

Chen, Q., 2006. Fully nonlinear Boussinesq-type equations for waves and currentsover porous beds. J. Eng. Mech. 132, 220230.

Chen, Q., Kirby, J.T., Dalrymple, R.A., Kennedy, A.B., Chawla, A., 2000. Boussinesqmodeling of wave transformation, breaking and runup. II: two horizontaldimensions. J. Waterway, Port, Coastal Ocean Eng. 126, 4856.

Erduran, K.S., Ilic, S., Kutija, V., 2005. Hybrid nite-volume nite-difference schemefor the solution of Boussinesq equations. Int. J. Numer. Methods Fluids 49,12131232.

Glimsdal, S., Pedersen, G., Atakan, K., Harbitz, C.B., Langtangen, H., Lvholt, F., 2006.Propagation of the December 26, 2004 Indian Ocean tsunami: effects ofdispersion and source characteristics. Int. J. Fluid Mech. Res. 33 (1), 1543.

Gottlieb, S., Shu, G.-W., Tadmor, T., 2001. Strong stability-preserving high-ordertime discretization methods. SIAM Rev. 43 (1), 89112.

Grilli, S.T., Ioualalen, M., Asavanant, J., Shi, F., Kirby, J.T., Watts, P., 2007. Sourceconstraints and model simulation of the December 26, 2004 Indian Oceantsunami. J. Waterway, Port, Coastal Ocean Eng. 133, 414428.

Grilli, S.T., Harris, J.C., Tajalli Bakhsh, T., Masterlark, T.L., Kyriakopoulos, C., Kirby,J.T., Shi, F., 2012a. Numerical simulation of the 2011 Tohoku tsunami based on anew transient FEM co-seismic source: comparison to far- and near-eldobservations. Pure Appl. Geophys.. http://dx.doi.org/10.1007/s00024-012-0528-y.

Grilli, S.T., Harris, J.C., Tappin, D.R., Masterlark, T., Kirby, J.T., Shi, F., Ma, G., 2012b. Amultisource origin for the Tohoku-oki 2011 tsunami earthquake and seabedfailure. Nature Comm., submitted for publication.

Grue, J., Pelinovsky, E.N., Fructus, D., Talipova, T., Kharif, C., 2008. Formation ofundular bores and solitary waves in the Strait of Malacca caused by the 26December 2004 Indian Ocean tsunami. J. Geophys. Res. 113, C05008. http://dx.doi.org/10.1029/2007JC004343.

Horillo, J., Kowalik, Z., Shigihara, Y., 2006. Wave dispersion study in the IndianOcean-tsunami of December 26, 2004. Marine Geodesy 29, 149166.

Ioualalen, M., Asavanant, J., Kaewbanjak, N., Grilli, S.T., Kirby, J.T., Watts, P., 2007.Modeling the 26th December 2004 Indian Ocean tsunami: case study of impactin Thailand. J. Geophys. Res. 112, C07024. http://dx.doi.org/10.1029/2006JC003850.

Kajiura, K., 1963. The leading wave of a tsunami. Bull. Earthquake Res. Inst. 41, 535571.

lling 62 (2013) 3955Kennedy, A.B., Chen, Q., Kirby, J.T., Dalrymple, R.A., 2000. Boussinesq modeling ofwave transformation, breaking and runup. I: one dimension. J. Waterway, Port,Coastal Ocean Eng. 126, 3947.

Kennedy, A.B., Kirby, J.T., Chen, Q., Dalrymple, R.A., 2001. Boussinesq-type equationswith improved nonlinear performance. Wave Motion 33, 225243.

Kowalik, Z., Knight, W., Logan, T., Whitmore, P., 2005. Numerical modeling of theglobal tsunami: Indonesian tsunami of 26 December 2004. Sci. Tsunami Haz. 23,4056.

Kulikov, E., 2005. Dispersion of the Sumatra tsunami waves in the Indian Oceandetected by satellite altimetry. Rep. from P.P. Shirshov Institute of Oceanology,Russian Academy of Sciences, Moscow

Lvholt, F., Pedersen, G., Gisler, G., 2008. Oceanic propagation of a potential tsunamifrom the La Palma Island. J. Geophys. Res. 113, C09026. http://dx.doi.org/10.1029/2007JC004603.

Lvholt, F., Kaiser, G., Glimsdal, S., Scheele, L., Harbitz, C.B., Pedersen, G., 2012.Modeling propagation and inundation of the 11 March 2011 Tohoku tsunami.Nat. Haz. Earth Syst. Sci. 12, 10171028.

Ma, G., Shi, F., Kirby, J.T., 2012. Shock-capturing non-hydrostatic model for fullydispersive surface wave processes. Ocean Modell. 4344, 2235.

Masterlark, T., 2003. Finite element model predictions of static deformation fromdislocation sources in a subduction zone: sensitivities to homogeneous,isotropic, Poisson-solid, and half-space assumptions. J. Geophys. Res. 108(B11). http://dx.doi.org/10.1029/ 2002JB002296, 17pp.

Naik, N.H., Naik, V.K., Nicoules, M., 1993. Parallelization of a class of implicit nitedifference schemes in computational uid dynamics. Int. J. High Speed Comput.5, 150.

Nwogu, O., 1993. Alternative form of the Boussinesq equations for nearshore wavepropagation. J. Waterway, Port, Coastal Ocean Eng. 119, 618638.

Pedlosky, J., 1979. Geophysical Fluid Dynamics. Springer-Verlag, New York, 624pp.Pophet, N., Kaewbanjak, N., Asavanant, J., Ioualalen, M., 2011. High grid resolution

and parallelized tsunami simulation with fully nonlinear Boussinesq equations.Comput. Fluids 40, 258268.

Roeber, V., Cheung, K.F., Kobayashi, M.H., 2010. Shock-capturing Boussinesq-typemodel for nearshore wave processes. Coastal Eng. 57, 407423.

Shi, F., Dalrymple, R.A., Kirby, J.T., Chen, Q., Kennedy, A.B., 2001. A fully nonlinearBoussinesq model in generalized curvilinear coordinates. Coastal Eng. 42, 337358.

Shi, F., Kirby, J.T., Harris, J.C., Geiman, J.D., Grilli, S.T., 2012a. A high-order adaptivetime-stepping TVD solver for Boussinesq modeling of breaking waves andcoastal inundation. Ocean Modell. 4344, 3651.

Shi, F., Kirby, J.T., Tehranirad, B., 2012b. Tsunami benchmark results for sphericalcoordinate version of FUNWAVE-TVD (Version 1.1). Research Report No. CACR-12-02, Center for Applied Coastal Research, University of Delaware, Newark.

Shiach, J.B., Mingham, C.G., 2009. A temporally second-order accurate Godunov-type scheme for solving the extended Boussinesq equations. Coastal Eng. 56,3245.

Shuto, N., 1985. The Nihonkai-Chubu earthquake tsunami on the north Akita Coast.Coastal Eng. Jpn. 28, 255264.

Shuto, N., Goto, C., Imamura, F., 1990. Numerical simulation as a means of warningfor near eld tsunamis. Coastal Eng. Jpn. 33, 173193.

Sitanggang, K., Lynett, P., 2005. Parallel computation of a highly nonlinearBoussinesq equation model through domain decomposition. Int. J. Numer.Methods Fluids 49, 5774.

Son, S., Lynett, P.J., Kim, D.-H., 2011. Nested and multi-physics modeling of tsunamievolution from generation to inundation. Ocean Modell. 38, 96113.

Synolakis, C.E., Bernard, E.N., Titov, V.V., Kanoglu, U., Gonzlez, F.I., 2007. Standards,criteria, and procedures for NOAA evaluation of tsunami numerical models.NOAA Tech, Memo. OAR PMEL-135, Pacic Mar. Env. Lab., Seattle.

Tappin, D.R., Watts, P., Grilli, S.T., 2008. The Papua New Guinea tsunami of 1998:anatomy of a catastrophic event. Nat. Haz. Earth Syst. Sci. 8, 243266.

Tonelli, M., Petti, M., 2009. Hybrid nite volumenite difference scheme for 2DHimproved Boussinesq equations. Coastal Eng. 56, 609620.

Wei, G., Kirby, J.T., Grilli, S.T., Subramanya, R., 1995. A fully nonlinear Boussinesqmodel for surface waves. I. Highly nonlinear, unsteady waves. J. Fluid Mech.294, 7192.

Yamamoto, S., Daiguji, H., 1993. Higher-order-accurate upwind schemes for solvingthe compressible Euler and NavierStokes equations. Comput. Fluids 22, 259270.

Yamazaki, Y., Cheung, K.F., Kowalik, Z., 2011. Depth-integrated, non-hydrostaticmodel with grid nesting for tsunami generation, propagation, and run-up. Int. J.Numer. Methods Fluids 67, 20812107.

Yoon, S.B., 2002. Propagation of distant tsunamis over slowly varying topography. J.Geophys. Res. 107, C10. http://dx.doi.org/10.1029/2001JC000791, 3140.

Zhou, H., Moore, C.W., Wei, Y., Titov, V.V., 2011. A nested-grid Boussinesq-typeapproach to modelling dispersive propagation and runup of landslide-generated tsunamis. Nat. Haz. Earth Syst. Sci. 11, 26772697.

J.T. Kirby et al. / Ocean Modelling 62 (2013) 3955 55

Dispersive tsunami waves in the ocean: Model equations and sensitivity to dispersion and Coriolis effects1 Introduction2 Model equations in spherical polar coordinates2.1 Scaling2.2 Shallow water equations2.3 The Boussinesq approximation2.3.1 Pressure and vertical momentum2.3.2 The vertical structure of velocities2.3.3 Fully nonlinear Boussinesq equations2.3.4 Weakly nonlinear approximation

3 Numerical approach3.1 Conservative form of governing equations3.2 Parallelization

4 Tests of dispersion and Coriolis effects4.1 Idealized tsunami sources and examples4.2 Source size and wave dispersion effect4.3 Coriolis effect on an idealized source4.4 The 2011 Tohoku-oki tsunami

5 ConclusionsAcknowledgementsReferences