EGU Journal Logos (RGB) Open Access Open Access Open Access Advances in Annales Nonlinear Processes Geosciences Geophysicae in Geophysics Open Access Open Access Nat. Hazards Earth Syst. Sci., 13, 1507–1526, 2013 Natural Hazards Natural Hazards www.nat-hazards-earth-syst-sci.net/13/1507/2013/ doi:10.5194/nhess-13-1507-2013 and Earth System and Earth System © Author(s) 2013. CC Attribution 3.0 License. Sciences Sciences Discussions Open Access Open Access Atmospheric Atmospheric Chemistry Chemistry Dispersion of tsunamis: does it really matter? and Physics and Physics Discussions Open Access Open Access S. Glimsdal1,2,3, G. K. Pedersen1,3, C. B. Harbitz1,2,3, and F. Løvholt1,2,3 Atmospheric Atmospheric 1International Centre for Geohazards (ICG), Sognsveien 72, Oslo, Norway Measurement Measurement 2Norwegian Geotechnical Institute, Sognsveien 72, Oslo, Norway 3University of Oslo, Blindern, Oslo, Norway Techniques Techniques Discussions Open Access Correspondence to: S. Glimsdal ([email protected]) Open Access

Received: 30 November 2012 – Published in Nat. Hazards Earth Syst. Sci. Discuss.: – Biogeosciences Biogeosciences Revised: 5 April 2013 – Accepted: 24 April 2013 – Published: 18 June 2013 Discussions Open Access Abstract. This article focuses on the effect of dispersion in 1 Introduction Open Access the field of tsunami modeling. Frequency dispersion in the Climate linear long-wave limit is first briefly discussed from a the- Climate Most tsunami modelers rely on the shallow-water equations oretical point of view. A single parameter, denoted as “dis- of the Past for predictions of propagationof and the run-up. Past Some groups, on persion time”, for the integrated effect of frequency dis- Discussions the other hand, insist on applying dispersive wave models, persion is identified. This parameter depends on the wave- sometimes even with enhanced nonlinear properties. These Open Access length, the water depth during propagation, and the propa- Open Access models are in-house models or available as standard codes, gation distance or time. Also the role of long-time asymp- free or commercial. Some of these are fairly well suited for Earth System totes is discussed in this context. The wave generation by Earth System implementation of tsunami applications. In the examples pre- the two main tsunami sources, namely earthquakes and land- Dynamics sented herein we employ an in-houseDynamics model (Pedersen and slides, are briefly discussed with formulas for the surface Discussions Løvholt, 2008; Løvholt et al., 2008, 2010) which is designed response to the bottom sources. Dispersive effects are then for long-distance propagation of dispersive tsunamis. Us- Open Access

exemplified through a semi-idealized study of a moderate- Open Access ing this model, we may take the Japan 2011 tsunami across strength inverse thrust fault. Emphasis is put on the directiv- Geoscientific Geoscientific the Pacific Ocean on a standard desktop during some hours ity, the role of the “dispersion time”, the significance of the Instrumentation Instrumentation of CPU time. The standard models, such as COULWAVE Boussinesq model employed (dispersive effect), and the ef- (Lynett et al., 2002; Kim andMethods Lynett, 2011 and) and FUNWAVE Methods and fects of the transfer from bottom sources to initial surface el- (Kennedy et al., 2000; ShiData et al., Systems2012) are based on more- Data Systems evation. Finally, the experience from a series of case studies, demanding numerical schemes and incorporate a number of Discussions

including earthquake- and landslide-generated tsunamis, is Open Access effects that are not relevant for oceanic propagation.Open Access Hence, presented. The examples are taken from both historical (e.g. simulations of oceanic propagation on single CPUs using Geoscientific the 2011 Japan tsunami and the 2004 Indian Ocean tsunami) Geoscientific these models may therefore be too time consuming. How- and potential tsunamis (e.g. the tsunami after the potential La Model Development ever, parallel featuresModel (e.g. DevelopmentSitanggang and Lynett, 2005; Palma volcanic flank collapse). Attention is mainly given to Discussions Pophet et al., 2011; Shi et al., 2012) of the models should the role of dispersion during propagation in the deep ocean enable large-scale applications. Open Access and the way the accumulation of this effect relates to the “dis- Open Access While the employment of dispersive codes for tsunami persion time”. It turns out that this parameter is useful as a Hydrology and computation certainly boostsHydrology the CPU and times and memory first indication as to when frequency dispersion is important, requirements, the significance of the extra physical fea- Earth System even though ambiguity with respect to the definition of the Earth System tures such codes inherit are met with skepticism by many wavelength may be a problem for complex cases. Tsunamis Sciences Sciences tsunami modelers, at least for seismic tsunamis. On the other from most landslides and moderate earthquakes tend to dis- hand, a proper description of the wave generation by land- Discussions Open Access play dispersive behavior, at least in some directions. On the Open Access slide tsunamis, and subaerial landslides in particular, requires other hand, for the mega events of the last decade disper- primitive wave models, as demonstrated for the potential La sion during deep water propagation is mostly noticeable for Ocean Science Palma tsunami (GislerOcean et al., 2006 Science; Abadie et al., 2012) and transoceanic propagation. Discussions the 1998 Papua New Guinea tsunami (Grilli and Watts, 2005; Open Access Published by Copernicus Publications on behalf of the European Geosciences Union. Open Access Solid Earth Solid Earth Discussions Open Access Open Access

The Cryosphere The Cryosphere Discussions 1508 S. Glimsdal et al.: Dispersion of tsunamis

Tappin et al., 2008). For such cases the waves are also dis- also included when feasible. The main focus is on dispersive persive in the far-field. Long-term propagation of dispersive effects during oceanic propagation in a linear context, even waves may be approximated by ray (optical) methods (Ward, though nonlinearity may be present also in the generation 2001; Ward and Day, 2001). If the waves are moderately dis- and propagation for landslide tsunamis. Dispersion may also persive, Boussinesq models without the optical approxima- be important for constructive interference due to geometry tion are now capable of simulating the far-field tsunami prop- and bore formation during shoaling. Generally, we do not ad- agation over transoceanic distances (see e.g. Løvholt et al., dress these phenomena. The exception is a brief example on 2008; Zhou et al., 2011; Kirby et al., 2013). the evolution of an undular bore, which occurs in one of the The variation of landslide thickness, relative to the water case studies that is presented. We start with a basic treatise depth, may contribute to dispersion (Ward, 2001). However, on the effects of weak dispersion and identify a parameter the hydrodynamic response from the uplift filters short-scale that describes its significance. After discussing earthquake variations due to landslide volume displacements (Geist, and landslide sources we demonstrate dispersion effects for 1998b; Glimsdal et al., 2011; Kajiura, 1963; Løvholt et al., a semi-idealized tsunami. Then, we move on to a series of 2012b; Pedersen, 2001). Generally, the leading-order wave is case studies, including the mega tsunamis of the last decade, reduced due to dispersion. In the far-field, however, the trail- with an eye on the significance of the dispersion. ing wave system is expected to eventually dominate (Løvholt et al., 2008). Finally, frequency dispersion is of less impor- tance for waves generated by large and sub-critical subma- 2 Dispersion effects rine landslides with moderate acceleration and deceleration We distinguish between the dispersive effect acting during where large wavelength components dominate (Harbitz et al., deep water propagation and the first part of the shoaling, 2006). In these cases the Froude number  1. (Froude num- when the earthquake tsunamis are linear, and the dispersion ber is the ratio between the landslide velocity and the local effects that may appear in shallow water, which are linked to wave speed.) This is true for e.g. the Storegga tsunami (Har- nonlinearity and produce undular bores. The first type, which bitz, 1992b; Bondevik et al., 2005). is the main concern in the present treatise, is described in Although frequency dispersion is often considered negligi- Sect. 2.1, while the latter is presented in Sect. 2.2, somewhat ble for earthquake-induced tsunamis, it may become notice- more briefly. able and sometimes important. Løvholt et al. (2012b) showed that the seabed displacement due to heterogeneous coseismic 2.1 Linear dispersion during propagation slip gave rise to frequency dispersion, affecting the tsunami run-up. The propagation of the 2004 Indian Ocean tsunami Frequency dispersion is the spreading of energy in the direc- gave noticeable dispersion in the Bengal Bay and Andaman tion of wave advance due to different wave celerity for wave Sea (Ioualalen et al., 2007; Horrillo et al., 2006), becom- modes of different length. For plane linear gravity waves ing more distinct at transoceanic distances (Glimsdal et al., propagating in an inviscid fluid of uniform depth, we have 2006). Similarly, frequency dispersion for the long-distance solutions in the form of harmonic modes propagation of the 2011 Tohoku tsunami is clear (Løvholt η = Acos(kx − ωt), u = B(z)cos(kx − ωt), et al., 2012b; Grilli et al., 2012). For smaller earthquakes in- (1) volving shorter-length scales, dispersion is expected to be v = C(z)sin(kx − ωt), pronounced at shorter-wave propagation distances. This is where η, u and v are the surface elevation, the horizontal for instance demonstrated for the 2009 Samoa tsunami by velocity component and the vertical velocity component, re- Zhou et al. (2012). spectively. The wave number, k, is 2π divided by the wave- Frequency dispersion in combination with nonlinearity length, λ, and ω is the frequency. For Eq. (1) to be a solution may cause the formation of undular bores during shoaling. of the governing equations we must require that the wave Undular bores related to the 2004 Indian Ocean tsunami and number, k, and the frequency, ω, fulfill the dispersion rela- the 1998 Papua New Guinea tsunami are discussed by Glims- tion dal et al. (2006) and Grue et al. (2008), and by Tappin et al. (2008), respectively. Recent investigations of the shoaling ω2 g c2 = = tanh(kh), (2) from potential ocean-wide tsunamis from La Palma also ad- k2 k dress this problem (Løvholt et al., 2008; Zhou et al., 2011). In this paper we draw on the experience from a series of where c is the phase speed, g is the constant of gravity and h earthquake and landslide tsunamis to address the significance is the equilibrium depth. Details on the derivation of Eq. (2) of dispersion. To this end we need control on crucial param- are found in many textbooks, such as Mei (1989). If com- eters in the computations and availability of sufficient data. pressibility is taken into account there exist other modes in Hence, the investigation is based primarily on cases where addition to the pure gravity mode Eq. (2), which are gener- the authors have first-hand knowledge and full access to com- ated by submarine earthquakes (see, for instance, Stiassnie, putational data. However, other studies from the literature are 2010, and references therein).

Nat. Hazards Earth Syst. Sci., 13, 1507–1526, 2013 www.nat-hazards-earth-syst-sci.net/13/1507/2013/ S.S. Glimsdal Glimsdal et et al.: al.: Dispersion Dispersion of tsunamisof tsunamis 15093

ForFor long long waves waves the the dispersion dispersion relation relation may may be be expanded expanded in in 170 termsterms of ofkhkh, L = √ght   2 2  1 1 2 2 2 4 6  c c=2 =c0c2 1 1 −(kh(kh)) +2 + (kh(kh)) +4 +OO(kh)(kh) 6 , , (3)(3) 0 − 3 3 1515 λ where c = √√gh. Long wave theories may be classified ac- h where 0c = gh. Long-wave theories may be classified ac- cording to0 how much of the contents within the outer paren- cording to how much of the contents within the outer paren- theses they reproduce. Shallow water theory only yields the theses they reproduce. Shallow-water theory only yields the 175 unitary constant, Korteweg-deVries and standard Boussinesq unitary constant, Korteweg–de Vries and standard Boussi- Fig. 1. Definition sketch of the evolution of an initial elevation from equations inherit the first two terms, while optimized Boussi- nesq equations inherit the first two terms, while optimized Fig.an earthquake. 1. Definition sketch of the evolution of an initial elevation from nesq equations (such as Nwogu, 1993) also approximates Boussinesq4 equations (such as Nwogu, 1993) also approxi- an earthquake. the O(kh) term.4 Also the model used herein may take the mate4 the O(kh) term. Also the model used herein may take Othe(khO(kh)) into4 account.into account. are a translated spatial variable and a temporal variable 180 IfIf we we ignore ignore the the finite finite time time duration duration of of a a submarine submarine earth- earth- regardfor evolutionτ as a “dispersion of dispersion time”. effects, Below, respectively. we therefore Hence, con- we quake,quake, together together with with compression compression waves waves in in the the water, water, the the sequentlymay regard useτ theas term a “dispersion dispersion time”. time whenBelow, referring we therefore to τ. tsunami will evolve from an initial elevation, η0, of the ocean tsunami will evolve from an initial elevation, η0, of the ocean Shallowuse the termwater dispersion theory corresponds time when to referring neglecting to τ the. Shallow- second surface, due to the seabed displacement. In a plane model surface, due to the seabed displacement. In a plane model this215 termwater in theory the expansion corresponds for ω toin whichneglecting case the solutionsecond term imme- in thiswill will give give rise rise to twoto two wave wave systems, systems, which which propagate propagate inthe in diatelythe expansion becomes forη =ω 1inF which(ξ). case the solution immediately 1 2 185 the positive and negative x-direction, respectively. For the = positive and negative x direction, respectively. For the sys- becomesFor largeη earthquakes,2 F (ξ). for instance, the behavior for finite systemtem moving moving toward toward increasing increasingx values,x values Eq. (2) (2) implies implies the the andFor small largeτ is earthquakes, the most interesting for instance, (see the section behavior 4). for Disper- finite solutionsolution sionand smallthen modifiesτ is the most the initial interesting wave (see shape, Sect. while4). Dispersionλ is still ∞ then modifies the initial wave shape, while λ still defines Z ∞ 220 defining the length of the wave front. In such cases we 1 Z ı(kx−ω(k)t) the length of the wave front. In such cases we may explain η(x,t) = 1 ηˆ0(k)e ı(kx−ω(k)tdk,) (4) may explain the significance of τ as follows. First, most of η(x,t) =2π ηˆ0(k)e dk, (4) the significance of τ as follows. First, most of the energy in 2−∞π the energy in the spectrum is distributed on k values rang- −∞ ingthe from spectrum zero is to distributed2π/λ, say. on k Invalues time rangingt the corresponding from zero to where ηˆ0 is the Fourier transform of η0. Again we refer to components2π/λ, say. In will time bet displacedthe corresponding by an amount components∆ct, will where be where ηˆ0 is the Fourier transform of η0. Again we refer to 190 standard textbooks, such as Mei (1989) or Whitham (1974). displaced by an amount 1c t,2 where2 1c = c(0)−c(2π/λ) ≈ standard textbooks, such as Mei (1989) or Whitham (1974).225 ∆c = c(0) c(2π/λ) 6c0h /λ . Then, this displacement Near the wave front the long parts of the spectrum domi- 6c h2/λ2.− Then, this displacement≈ must be measured against Near the wave front the long parts of the spectrum dominates must0 be measured against the length of the wave, λ. We then nates and ω in (4) may be replaced by the first two terms endthe with length the of dispersion the wave, time,λ. Weτ, then on different end with forms the dispersion and ω in Eq.1 (4) may2 be replaced by the first two terms ω ∼ ω c0k(1 (kh) ). time, τ, in different forms − 1 6 2 2 2 c∼0k(1 6−(kh) ). 1 6c h 1 6h L 6ht The effect of dispersion will depend on the wavelengths, = ∆ 0 = = (7) The effect of dispersion will depend on the wavelengths, τ c t 1 6c2 h2t 1 63h2L 63ht, 195 the depth and the time available for its evolution. Long · · λ ≈ λ 0 · · λ λ gT the depth and the time available for its evolution. Long- τ = 1c · t · ≈ · t · = = , (7) term evolution may also depend qualitatively on certain other λ λ2 λ λ3 gT 3 term evolution may also depend qualitatively on certain other where L = c0t and T = λ/c0 are the propagation distance and properties of the initial condition, such as the net volume of properties of the initial condition, such as the net volume of230 thewhere overallL = period,c t and respectivelyT = λ/c (seeare the figure propagation 1). Naturally, distance the displacement (see below). To find a simple relation we regard 0 0 displacement (see below). To find a simple relation we regard effectand the of overall dispersion period, accumulates respectively in (see time Fig. and1). thus Naturally, increases the a group of initial conditions, which are of the same shape, but a group of initial conditions, which are of the same shape but witheffectt and of dispersionL. The variation accumulates is stronger in time withand thus respect increases to h. 200 have different lengths have different lengths, However,with t and theL sensitivity. The variation is strongest is stronger with with respect respect to the to ex-h. x tensionHowever, of the the source, sensitivityλ. is strongest with respect to the ex- η0(x) = F x , η0(x) = F λ , 235 tensionFor large of the times source, (τ λ. ) an asymptotic approximation λ →→ ∞ ∞ according to the value of the parameter λ. This gives ηˆ (k) = forFor the wavelarge front times is (τ found in) an textbooks asymptotic (for instance,approximation Mei, 0ˆ = λFaccordingˆ(s), where to theFˆ is value the Fourierof the parameter transformλ. of This the gives functionη0(k)F , 1989).for the In wave the normalized front is found coordinates in textbooks this becomes(for instance, Mei, λFˆ (s), where Fˆ is the Fourier transform of the function F , 1989). In the normalized coordinates this becomes and s = kλ. Using this, focusing on the wave propagating in Fˆ(0)  ξ  and s = kλ. Using this, focusing on the wave propagating in Ai (8) 205 the positive x-direction, and invoking the two term expansion η ˆ 1 1 !, ∼ 2(12F(τ0))3 (12τξ) 3 fortheω positivein (4) wex direction, obtain and invoking the two-term expansion ∼ η 1 Ai 1 , (8) for ω in Eq. (4), we obtain 3 3 ∞ where2Ai(12isτ) the Airy(12 function.τ) It is noteworthy that the initial Z 1 ∞ ı ξs+ 1 τs3 240 condition only enters (8) through ˆ(0), i.e. the initial volume ˆ (  36 )  F η = G(ξ,τ) = 1 ZF (s)e ı ξs+ 1 τs3ds (5) where Ai is the Airy function. It is noteworthy that the ini- 2π ˆ 36 per width of the disturbance divided by λ. At some distance η = G(ξ,τ) = F (s)e ds; (5) tial condition only enters Eq. (8) through F(ˆ 0), i.e. the ini- 20π behind the front (8) may be matched to the stationary phase 0 tial volume per width of the disturbance divided by λ. At = , where the normalized variables, approximation to yield a complete asymptotic expression for ds λdk some distance behind the front Eq. (8) may be matched to the ds = λdk, where the normalized2 variables, the evolution of plane waves in constant depth. The trail- x c0t 6c0h t stationary phase approximation to yield− 1 a complete asymp- = = (6) 245 ing waves attenuate proportional to t 2 and the leading crest ξ − , τ 3 ,2 totic expression for the evolution of plane waves at constant xλ− c0t λ6c0h t ξ = , τ = , (6) will eventually be dominant. From (8) we observe that− 1 the 3 depth. The trailing waves attenuate proportional to t 2 and 210 are a translatedλ spatial variable,λ and a temporal variable for length of the leading crests increases with time. As a conse- evolution of dispersion effects, respectively. Hence, we may quencethe leading the dispersive crest will effects eventually on this be partdominant. of the Fromwavesystem Eq. (8)

www.nat-hazards-earth-syst-sci.net/13/1507/2013/ Nat. Hazards Earth Syst. Sci., 13, 1507–1526, 2013 1510 S. Glimsdal et al.: Dispersion of tsunamis we observe that the length of the leading crests increases with suggest the integrated measure time. As a consequence the dispersive effects on this part of t L s the wave system diminish, which is consistent with the fact 6 Z 6 Z h that the relative attenuation rate of the wave height goes to τ = hdtˆ = dx.ˆ (9) gT 3 gT 3 g zero. Therefore, dispersion may affect the wave front most 0 0 strongly in the early parts of the propagation, while it later becomes relatively more significant in the trailing system of In principle this expression requires slow variation of h waves. It is thus stressed that while τ represents its accumu- and integration along a ray path. Assuming an idealized ge- lated effect, the significance of dispersion does by no means ometry corresponding to a uniform slope stretching from the relate linearly to τ for larger dispersion times. source region to the shoreline, we may estimate the coastal Another consequence of the stretching of the wave front value of τ from Eq. (9): is that nonlinearity becomes comparable with dispersion 4h2L (Ursell’s paradox, see Ursell, 1953). However, given the lim- τ = 0 , ited sizes of the oceans, the evolution of both dispersion and λ3 nonlinearity is too slow to reach this stage. Extra attenuation where h0 is the depth in the source region, λ is still the source will reduce the significance of nonlinearity even further in width and L is the distance from the source region to the the three-dimensional case (and remove Ursell’s paradox). shore. We observe that the reduction of dispersion due to If the net displaced volume is zero, then Eq. (8) no longer shoaling alters the constant depth estimate in Eq. (7) only applies, but the leading crest height decays in proportion to moderately. − 2 t 3 and its shape is defined by the derivative of the Airy Real disturbances may involve several length scales. For function. In this case the trailing waves will eventually be- nonuniform sources we may have scales down to a few times come dominant. Subduction earthquakes with dip angles be- the depth, as discussed by Løvholt et al. (2012) and Pedersen ◦ ◦ tween 0 and 90 will yield net elevation of the seafloor, (2001). However, the leading wave will generally be dom- implying that Eq. (8) is correct (see discussion in Kervella inated by the longest initial length scale, which then corre- et al., 2007). Submarine landslides and slumps, on the other sponds to λ. Still, the actual choice of λ may be ambiguous. hand, are volume neutral, while subaerial landslides obvi- ously yield a net positive volume. For partially subearial 2.2 Combined nonlinearity and dispersion in shallow slides the net volume may be small in comparison to the total water: the undular bores displaced volumes, and the waves may display an interme- diate asymptotic attenuation, even for transoceanic propaga- In shoaling water the length-to-depth ratio of a tsunami in- tion (see Løvholt et al., 2008). With two horizontal dimen- creases and the dispersive effects are diminished; meanwhile sions there is an additional attenuation due to geometrical the amplitude increases and nonlinear effects may become spreading of the wave energy, introducing an extra attenua- important. However, due to the nonlinearity the front of the − 1 tsunami steepens, which may lead to breaking or bring dis- tion factor of t 2 for propagation distances much larger than the source extensions, and asymptotic analysis displays more persive effects back into play. If the front width becomes diversity (see Mei, 1989). comparable to depth while the amplitude-to-depth ratio is In his profound study of tsunami generation and propa- still less than 0.3, say, an undular bore evolves. In that case, gation, Kajiura (1963) reported many important derivations long waves undergo fission into a series of individual peaks, and observations. Among these the Fourier transforms given of solitary shape, with height up to twice that of the wave above are implicit, but another parameter was used for the before fission. Such bores are known to develop from tides significance of dispersion, namely in some rivers and estuaries, such as the Severn and the Garonne/Gironde, and have been observed for tsunamis as 1 36 3 well. The phenomenon in relation to tsunamis was pointed P = . out already by Shuto (1985), while more recent analysis and τ observations are given by, for instance, Madsen et al. (2008); Furthermore, it was suggested that dispersion has to be taken Glimsdal et al. (2006); Grue et al. (2008); Arcas and Segur into account when P < 4, which corresponds to τ ∼ 0.5 (see (2012) and references therein. The further dynamics of the also Shuto, 1991, for a discussion). In view of the interpre- undular bore may be complex, including breaking of the in- tation of τ, as given above, this limit seems somewhat small dividual crest, with crucial loss of energy and even identity as in terms of P , and large in terms of τ. Herein, we prefer to separate waves as possible outcomes (Dorn et al., 1968; Ko- employ the dispersion time τ in our subsequent discussion rycansky and Lynett, 2005). It should be kept in mind that the on the influence of frequency dispersion, being a normalized evolution of undular bores requires an interaction between time scale for evolution of dispersive effects. nonlinearity and dispersion and can thus not be reproduced Our identification of τ is made for constant depth. For vari- in either nonlinear shallow water (NLSW) models or linear able depth we may exploit the invariance of the period, T , to models.

Nat. Hazards Earth Syst. Sci., 13, 1507–1526, 2013 www.nat-hazards-earth-syst-sci.net/13/1507/2013/ S. Glimsdal et al.: Dispersion of tsunamis 1511

3 Employed physical and mathematical models ∞ Z ∞ n 1 mJ0(mr) 1 X (−1) (2n + 1) G(r) = dm = . (11) 3.1 Wave generation by earthquakes 2π coshm π 3 n=0  + 2 + 2 2 0 (2n 1) r For seafloor deformation due to a coseismic single uniform The function G decays exponentially in its argument, and slip we may immediately recognize a total width, W, and a the integration intervals in Eq. (10) may be replaced by inter- length (along the fault), B, as length scales. According to vals of length 5 h centered at x0 = x and y0 = y. At |r −r0| = earthquake scaling laws (e.g. Blaser et al., 2010; Leonard, 5 h we have r G less than 10−3. For the case in Sect. 4.1 2010) B is clearly larger than W. In addition, the deformation h the application of such an integration interval leads to an may contain shorter features, depending on the depth of the error of less than 3 × 10−4 m, while the maximum uplift is earthquake (e.g. Geist, 1998b; Mai and Beroza, 2002). Math- roughly 1 m. A table for G, covering the finite integration in- ematical models for the deformation, such as Okada’s for- terval used in Eq. (10), is computed by means of the series in mula (Okada, 1985), may even predict discontinuities. The Eq. (11), and G as a general function is then made available presence of the accretionary wedge at the plate boundaries, through interpolation. etc., will replace the discontinuity by a transition of a finite We solve the equations for tsunami propagation on a reg- length, which is still small compared to W, and even to the ular grid in either geographic or Cartesian coordinates. The water depth. Splay faults or inhomogeneous fault distribu- grid is staggered (Arakawa C type, see Pedersen and Løvholt, tions may introduce yet other short features in the source 2008; Mesinger and Arakawa, 1976) with surface nodes at (e.g. Geist, 1998b). However, the short scales are not directly r = (i1x,j1y), and the volume balance is observed for conveyed to the ocean surface. This may be due to finite du- i,j cells centered at these locations and with extensions equal to ration effects of the earthquake that are difficult to assess, the grid increments. Denoting the average seabed elevation or to the hydrodynamic response to source distributions that inside a cell by D , the discrete counterpart to Eq. (10), at does not produce surface responses with extensions less than ij location r , becomes a few depths, say (see, for instance, Kajiura, 1963; Peder- nm sen, 2001; Løvholt et al., 2012b). Still, a common procedure XX 1x1y ηnm = σij Dij G(|rnm − rij |/hij ), (12) for initiation of tsunami simulations is to copy the coseis- h2 mic bottom deformation to the sea surface. This may lead i j ij to an unphysically high content of short-wave components where σij is a correction factor, explained below, 1x and 1y in the tsunami spectrum. Even though these are sometimes are the grid increments and only the contributions from a lim- dissipated numerically, the best option is to remove them in ited range in i and j need to be taken into account due to the a sound and controlled manner. Representing the coseismic exponential decay of G in the far-field. In principle Eq. (12) deformation as a source distribution at the bottom, we may is valid only for constant depth. However, if the depth varia- compute the surface response. Herein, we employ two tech- tion over a distance of a few ocean depths is small in Eq. (12), niques for this. (1) By treating the source as a composition this is a good approximation in nonuniform depth as well. of narrow strips, normal to the fault line, we may employ The factor σi,j is chosen as to preserve volume, in the sense two-dimensional models for the hydrodynamic response. For that the contribution to the initial discrete surface elevation shallow earthquakes, with uniform slip, we may then employ from cell (i,j) equals Dij 1x 1y. As long as the grid incre- an analytic expression, while a numerical integral is used oth- ments are well below the water depth in size, σij is very close erwise (Pedersen, 2001; Løvholt et al., 2012). (2) We also to unity. If the grid increments are large compared to the wa- compute the full three-dimensional response from an uplift ter depth, on the other hand, then the volume correction pro- distribution on the bottom. Assuming a rapid event, relative cedure corresponds to copying the bottom deformation onto to the time gravity waves will spend crossing the source re- the sea surface. gion, the sea surface elevation after the event will depend A similar procedure as described above may also be em- only on the final uplift distribution, D(x,y), where x and ployed at each time step for an event of finite duration, such y are the horizontal coordinates. For simplicity we employ as a landslide (see also Glimsdal et al., 2011). Cartesian coordinates; the extension to geographical coor- The techniques for the hydrodynamic response and its dinates is straightforward. According to Kajiura (1963) the smoothing effects on the initial condition is demonstrated in initial surface elevation at constant depth, h, then becomes the first example, Sect. 4.1. A consequence of the smooth- ∞ ∞ ing effect is that sources at the bottom generate waves within Z Z |r − r0| η(x,y,0) = h−2 D(x0,y0)G dx0 dy0, (10) the realm of Boussinesq equations. Naturally, waves gener- h ated in shallow water may still end outside the long-wave −∞ −∞ regime if they propagate into the deep ocean. Sources at the where r is the position vector, and the normalized Green ocean surface, on the other hand, like impacting asteroids, function is given by rock slides plunging into water, and calving icebergs, may produce waves which are short compared to the depth. www.nat-hazards-earth-syst-sci.net/13/1507/2013/ Nat. Hazards Earth Syst. Sci., 13, 1507–1526, 2013 1512 S. Glimsdal et al.: Dispersion of tsunamis

Shorter components in the spectrum (wavelengths down to bathymetry (see e.g. Harbitz et al., 2012b, and the references a few depths, but still within the realm of Boussinesq equa- therein). tions) will propagate comparatively slower and influence the Quantification of the landslide parameters is complicated rear parts of the generated wave train. Hence, the evolution of by the transformation of the landslide from solid to fluid and the leading parts will mainly be governed by the long scales, (in many cases) to a turbidity current. Another complicating namely W or B, depending on the direction of wave advance. factor is that many submarine landslides develop retrogres- However, in a shallow-water solution shorter features will, sively, i.e. they are released progressively upwards from the artificially, also stay in the former part of the wave system, at slide toe (e.g. Kvalstad et al., 2005). least with a fine grid that yields weak numerical dispersion. For the reasons above, we do not attempt to link λ directly Hence, as an indicator for the need of a dispersive model, to the landslide parameters. Instead we extract it from the τ should be based on the shortest significant length of the freshly generated wave as twice the distance, along a tran- source. For the events presented below we have not included sect, between the first crest and the point at the front where any particular information of shorter features, but generally the elevation is 10 % of the height of this crest. compute the bottom deformation from Okada’s formula ap- After the fashion described previously (Sect. 3.1) source plied to a single or a few faults. Then, for propagation normal features shorter than a few water depths may be filtered out to the fault line, which generally means toward land or off to also for landslides (Glimsdal et al., 2011). sea, we identify the parameter λ in Eq. (7) as the total width, W, of the slip region. On the other hand, for propagation in 3.3 The tsunami propagation model the direction parallel to the fault line the length, B, is the ap- propriate one. Since B generally is much larger than W we The main model employed herein is an optimized version of must expect the waves in the direction normal to the fault to the standard Boussinesq equations. The main features are display dispersion most strongly. – Enhanced linear dispersion.

3.2 Wave generation by landslides – Simpler and more efficient than FUN- WAVE/COULWAVE for tsunami propagation purposes. Considering a landslide simply as a uniform nondeformable block moving at a sub-critical speed (ignoring dispersion) re- – Geographic coordinates or Cartesian coordinates. veals that the length of the landslide and duration of motion – Rotational effects (Coriolis) included. influence both the dominant wavelength and the surface ele- vation, while the thickness and the acceleration or decelera- Denoting longitude and latitude by ψ and φ, respectively, tion of the landslide as well as the wave speed (which again we introduce dimensionless variables according to is determined by the water depth) determine the surface ele- R2 vation. (ψ,φ) = 2(x,y), t = √ √ gh0 ˆ (13) For numerical landslide tsunami models the simplest (u,ˆ v)ˆ =  gh0(u,v), h = h0h ηˆ = h0η, source model is a sink/source distribution with prescribed shape and kinematics ignoring the two-way landslide/water where the hats indicate variables with dimension, g is the interaction. Presently, the tsunamigenic landslide models constant of gravity, h0 is a characteristic depth, R is the Equa- themselves apply simple rheological functions and ignore the torial radius of the Earth, and  is an amplitude factor. The multilayer structure of a submarine landslide with a dense characteristic horizontal length (wavelength) now becomes debris flow at the bottom and a dilute turbidity current (sus- Lc = R2, which may determine 2, and the “long-wave pa- pension flow) above. Further, rock or mud type mass gravity rameter” is accordingly recognized as flows will entrain water, and produce turbulence and large h2 vortices that cannot be conveyed properly to a depth inte- µ2 = 0 . (14) grated model, while viscous drag may have a crucial influ- R222 ence on the shape and dynamics of the mudflow. For the physical constants we substitute Rock slides plunging into fjords, lakes, or reservoirs evolve as super-critical and critical during impact, transi- g = 9.81 m s−2,R = 6 378 135 m. (15) tioning to sub-critical during the later phase of motion. The build-up of the wave persists as long as the Froude num- It is emphasized that these quantities are not constant, but ber is around unity. For rock slides, nonlinear effects may their variation is neglected along with other small effects of be important in the wave generation area, but often only in rotation and the curvature of the Earth. We emphasize that the a restricted region and during a short period of time. Their scaling given above allows us to state the Boussinesq equa- tsunamigenic power is governed by the frontal area of the tions in a transparent, custom manner. However, outside the rock slide, the velocity of the rock slide when plunging into present subsection we specify quantities in terms of physical the water body, the permeability of the rock slide, and the units.

Nat. Hazards Earth Syst. Sci., 13, 1507–1526, 2013 www.nat-hazards-earth-syst-sci.net/13/1507/2013/ S. Glimsdal et al.: Dispersion of tsunamis 1513

In dimensionless variables the continuity equation reads This is an active seismic region, with the Lisbon earthquake of 1755 as the most prominent historical case. In 1969 an ∂η ∂ ∂ ∂h = cφ = − {(h + η)u} − {cφ(h + η)v} − cφ , inverse thrust fault of magnitude Mw 7.9 in the Horseshoe ∂t ∂x ∂y ∂t Abyssal Plain south of Portugal generated a moderate tsunami that was recorded at tide gauges in Portugal, Spain, where cφ = cosφ is a map factor and the rightmost term, rep- resenting temporal bottom changes, is the source distribution and Morocco (Gjevik et al., 1997). The magnitude of the from, for instance, a submarine landslide. By means of a sur- elevations from the model simulations (based on the seismic face response similar to the one described in Sect. 3.1 the data) were consistent with the observed ones, even though ∂h there were unresolved issues concerning a single time series field ∂t may be replaced by a slightly modified distribution. The momentum equations are written as at Casablanca. We assume a dip angle of 50◦, an ocean depth h = 5 km,   ∂u +  u ∂u + v ∂u = − 1 ∂η + f v−γ µ2h2 1 ∂Dη a source width W = 50 km, a length B = 100 km and a uni- ∂t cφ ∂x ∂y cφ ∂x cφ ∂x form slip of 2 m. Combined with a shear modulus of 30 GPa, 2 h i + µ h ∂ ∂ ∂u
+ ∂ ∂v
= 2 2 ∂x ∂x h ∂t ∂y cφh ∂t this yields a moment magnitude Mw 7.6, which is some- cφ 2 h i what lower than the one given above. Still, the case should − 2 1 + h ∂ ∂ ∂u
+ ∂ ∂v
µ ( 6 γ ) 2 ∂x ∂x ∂t ∂y cφ ∂t , be characteristic of moderately strong earthquakes with large cφ dip angles. In Fig. 2 we have depicted the seabed displace-   ∂v + u ∂v + ∂v = − ∂η − − 2 2 ∂Dη ment, as obtained from Okada’s formula, and the surface re- ∂t  c ∂x v ∂y ∂y f u γ µ h ∂y φ sponse modified through Eq. (12), and compared the latter to 2 h i + µ ∂ 1 ∂ ∂u
+ 1 ∂ ∂v
2 h ∂y c φ ∂x h ∂t c φ ∂y cφh ∂t the analytic expression by Pedersen (2001). The removal of h i the discontinuity and the shortest features is apparent, while −µ2( 1 + γ )h2 ∂ 1 ∂ ∂u
+ 1 ∂ c ∂v
, 6 ∂y c φ ∂x ∂t c φ ∂y φ ∂t the analytic expression and Eq. (12) are very similar in a transect through the center of the source (y = 0). The de- where f is the Coriolis parameter and some smaller con- viations are somewhat larger at outskirts of the fault line tributions to the convective acceleration terms are omitted. (results not shown). An obvious consequence of the modi- Equations valid for Cartesian grid are obtained simply by fied sea surface elevation is a reduction of the shorter-wave putting the map factor, c , to unity. The dispersion correc- φ components in the spectrum. In a dispersive model this will tion term, D , is the Laplacian of η and was first proposed by η mainly affect trailing parts of the evolving wave patterns. On = − 1 Madsen and Sørensen (1992) with the coefficient γ 15 . the other hand, in a shallow-water model the steep front, in- = We instead choose γ 0.057, which yields dispersion prop- troduced by copying the seabed deformation directly to the erties identical to those of Nwogu (1993). In Sect. 4.1, we surface, should in principle be retained. However, in numer- will refer to this version of the model as “h.o.”, because it is ical shallow-water models on coarse grids the numerical dis- of higher-order with respect to dispersion properties, while persion will remove the steep fronts and instead yield trailing = the version with γ 0 is named “disp”. The latter choice re- noise. produces the so-called standard Boussinesq equations (Pere- In the present context we focus on properties for deep grine, 1967). Dispersion or nonlinear terms may be switched ocean propagation and employ an infinite ocean of depth off independently. Further details on the model are given by 5 km. Simulations have been performed for spatial resolu- Pedersen and Løvholt (2008); Løvholt et al. (2008, 2010). tions 1x = 1y = 3.6 km, 2 km, 1 km and 0.5 km. It is noted We emphasize that the model is fairly efficient. As an exam- that the coarsest of these correspond to a 20 resolution in ge- ple we may state that the trans-Atlantic propagation of the 0 ographical coordinates. In the present subsection, however, La Palma tsunami on a 2 grid (see Sect. 5.4) requires around the resolution 1x = 0.5 km is used unless otherwise speci- 5 h of CPU time (on a single CPU) in a cheap off-the-shelf fied.√ The time step is determined by keeping the CFL num- desktop. ber, gh1t/1x, equal to, or slightly smaller than, unity for dispersive simulations, while the CFL number is kept below 4 Seismic case studies 0.63 for shallow-water computations, since omission of the dispersive effects yields a stricter stability criterion (e.g. Ped- 4.1 Portugal (1969) ersen and Løvholt, 2008). The slight variations in the CFL numbers with spatial resolution is due to the need to syn- For some particular source configurations non- chronize the simulations at intervals for comparison. After planar extensions of Eq. (8) are available 1 hour of propagation the dispersive solutions for the three (see Mei, 1989; Clarisse et al., 1995). However, it is finest grids display relative deviations of order 0.5 × 10−3 in more instructive to study sources which are more realistic the amplitudes of the leading crest, for propagation in the representations of submarine earthquakes. To this end we direction normal to the fault line. For the coarsest grid the design a semi-idealized case inspired by a true event in error in this amplitude is increased to 5 × 10−3, which is the Atlantic Ocean south of the Iberian Peninsula in 1969. still rather small. In the rear part of the wave trains, where www.nat-hazards-earth-syst-sci.net/13/1507/2013/ Nat. Hazards Earth Syst. Sci., 13, 1507–1526, 2013 S. Glimsdal et al.: Dispersion of tsunamis 9 8 S. Glimsdal et al.: Dispersion of tsunamis is anisotropic and that the errors for propagation in directions 1514 S. Glimsdal et al.: Dispersion of tsunamis obliquely to the grid axis may be larger. For the leading part 635 of the wave train the shallow water solutions converge more slowly than the dispersive ones and are strongly affected by 72 artificial dispersion for the coarser grids (results not shown). In figure 4 we have compared the surface elevation in the transect y = 0, at t = 30min, for a dispersive simulation start- 36 m 640 ing with a copy of the seabed displacement on the surface to one where (12) has been applied. While the first crest is 1.05 0 0.90 very similar for the two dispersive simulations, we clearly y(km) 0.75 observe effects of the over-representation of shorter wave 0.60 -36 0.45 components in the former solution in the trailing crests. In a 0.30 645 corresponding study Dutykh et al. (2006) found larger differ- 0.15 0.00 ences for the leading crest, since they made the comparison -72 -0.05 at a much earlier time. The dispersion time, τ, in the fig- ure is close to 0.5, the limit where dispersion effects should -64 -32 0 32 64 be taken into account according to Kajiura (1963). How- x(km) 650 ever, comparison with the hydrostatic linerar shallow water Fig. 3. The surface elevation along the y = 0 transect after 30 min. (LSW) solution reveals that both the lengths and the heights 72 ResultsFig. 3. forThe different surface elevation resolutions along (given the y = in 0 km)transect forthe after optimized30min. of the leading crest and trough have been strongly altered BoussinesqResults for model different (h.o.) resolutions and initial (given conditions in km) for obtained the optimized with the by dispersion. Moreover, a significant trailing wave system GreenBoussinesq function model of Kajiura. (h.o.) and Waves initial are conditions propagating obtained to the right. with the has already developed in the dispersive solution, even for 36 m Green function of Kajiura. Waves are propagating to the right. 655 the smoother initial condition. This indicates that a criterion 1.05 τ < 0.5 for applicability of shallow water theory is too weak. 0 0.90 For the leading part of the wave train the shallow-water solu- For instance, the error of the LSW model may strongly affect y(km) 0.75 tions converge more slowly than the dispersive ones and are an inversion of tsunami time series for the construction of a 0.60 -36 0.45 strongly affected by artificial dispersion for the coarser grids composite source. In their study of the 2009 Samoa event 0.30 0.15 (results not shown). 660 Zhou et al. (2012) employ a source which is composite, but 0.00 In Fig. 4 we compare the surface elevation in the tran- still inherits scales consistent with W = 50km. According to -72 -0.05 sect y = 0, at t = 30 min, for a dispersive simulation, start- their figure 3 showing results for τ up to 0.3, they experience Fig. 5. The surface elevation along the y = 0 transect, for the wave ing with a copy of the seabed displacement on the surface to system propagating in the positive x-direction. Results obtained a dispersive effect on the wave front comparable to the one -64 -32 0 32 64 one where Eq. (12) has been applied. While the first crest is from models with standard and enhanced dispersion properties are in our figure 4. In figure 3 we have displayed the grid depen- x(km) very similar for the two dispersive simulations, we clearly marked by ’disp’ and ’h.o,’ respectively. Waves are propagating to 665 dence for the same time as is used in figure 4. We observe the right. that the evolution of the first crests of the wave train is very observe effects of the over-representation of shorter-wave similar for all displayed resolutions, even the coarsest with components in the former solution in the trailing crests. In a corresponding study Dutykh et al. (2006) found larger differ- ∆x = 3.6km. quite different. Transect results after one hour is shown in ences for the leading crest, since they made the comparison In figure 5 the waves propagating in the positive x- figure 6 and we observe only moderate effects of dispersion. 670 direction are shown for t 7.5min and t 1h. These times at a much earlier time. The dispersion time, τ, in the fig- 690 In this case the “dispersion time”, τ, is based on λ = B = correspond to = 0 12 and≈ = 0 96, respectively,≈ where is ure is close to 0.5, the limit where dispersion effects should τ . τ . λ 100km, which gives a value τ = 0.12 corresponding to that = 50km identified with W . Already at the earliest time the be taken into account according to Kajiura (1963). However, for the upper panel in figure 5. The dispersion effects in the effect of dispersion is noticeable, while it has transformed comparison with the hydrostatic linear shallow water (LSW) two graphs also appear to be of the same magnitude. the wave train crucially at t 1h, when the only quantity solution reveals that both the lengths and the heights of the ≈ We conclude this introductory example by reporting some 675 properly reproduced by the LSW equations is the arrival leading crest and trough have been strongly altered by disper- 695 plane simulations, with the transect profile in figure 2 as ini- time. The standard and higher order dispersion representa- sion. Moreover, a significant trailing wave system has already tial condition. For the wave system propagating in the pos- tion only makes an apparent difference late in the emerging developed in the dispersive solution, even for the smoother itive x direction we observe that the leading crest is fairly wave trains. We also observe that the second crest is slightly initial condition. This indicates that a criterion τ < 0.5 for well described by the asymptotic formula (8) after one hour higher than the leading one in the dispersive solutions. This Fig. 4. The surface elevation along the y = 0 transect after 30min. applicability of shallow-water theory is too weak. For in- of propagation, while the match is nearly perfect after two 680 is a three dimensional effect that is neither observed for Fig. 2. Upper panel: seabed displacement. Mid-panel: the sur- Dispersive results with initial conditions obtained with and without Fig. 2. Upper panel: seabed displacement. Mid panel: the sur- stance, the error of the LSW model may strongly affect an in-700 hours (figure 7). This implies that the original source length the train propagating in the negative x-direction nor in the face response as obtained by Eq. (12). Lower panel: comparison in application of the Green function of Kajiura are marked h.o. and face response as obtained by (12). Lower panel: comparison in the transect y = 0 of seabed displacement (h), surface profile from versionh.o.?, respectively. of tsunami Also time the series LSWfor solution the construction is included for of compari- a com- has become irrelevant at this stage. plane simulations (below). However, non-uniformity of the the transect y = 0 of seabed displacement (h), surface profile from Eq. (12) (3-D) and surface profile from asymptotic formula of Ped- positeson. Waves source. are propagatingIn their study to the of right. the 2009 Samoa event Zhou source, constructive interference of reflections, or formation (12) (3D) and surface profile from asymptotic formula of Pedersen 4.2 Indian Ocean Tsunami (2004) of undular bores, are presumably more likely reasons for ersen (2001) (2-D). et al. (2012) employ a source which is composite, but still in- (2001) (2D). herits scales consistent with W = 50 km. According to their 685 the larger inundation of secondary waves observed in true Fig. 3, which shows results for τ up to 0.3, they experience Here we investigate the effect of dispersion for the 26th De- tsunami cases. cember 2004 Indian Ocean Tsunami. The rupture started The behavior for waves propagating in the y direction is a dispersive effect on the wave front comparable to the one the wavelengths are shorter, the grid dependence is much in our Fig. 4. In Fig. 3 we display the grid dependence for stronger. Examples of grid dependence in surface profiles the same time as is used in Fig. 4. We observe that the evolu- are shown in Fig. 3. It is also remarked that the numeri- tion of the first crests of the wave train is very similar for all cal dispersion is anisotropic and that the errors for propa- displayed resolutions, even the coarsest with 1x = 3.6 km. gation in directions oblique to the grid axis may be larger.

Nat. Hazards Earth Syst. Sci., 13, 1507–1526, 2013 www.nat-hazards-earth-syst-sci.net/13/1507/2013/ S. Glimsdal et al.: Dispersion of tsunamis 9

Fig. 3. The surface elevation along the y = 0 transect after 30min. Results for different resolutions (given in km) for the optimized Boussinesq model (h.o.) and initial conditions obtained with the Green function of Kajiura. Waves are propagating to the right.

S. Glimsdal et al.: Dispersion of tsunamis 1515 S. Glimsdal et al.: Dispersion of tsunamis 9 Fig. 5. The surface elevation along the y = 0 transect, for the wave system propagating in the positive x-direction. Results obtained from models with standard and enhanced dispersion properties are marked by ’disp’ and ’h.o,’ respectively. Waves are propagating to the right.

quite different. Transect results after one hour is shown in figure 6 and we observe only moderate effects of dispersion. 690 In this case the “dispersion time”, τ, is based on λ = B = 100km, which gives a value τ = 0.12 corresponding to that for the upper panel in figure 5. The dispersion effects in the two graphs also appear to be of the same magnitude. We conclude this introductory example by reporting some 695 plane simulations, with the transect profile in figure 2 as ini- tial condition. For the wave system propagating in the pos- itive x direction we observe that the leading crest is fairly Fig. 4. The surface elevation along the y = 0 transect after 30 min. well described by the asymptotic formula (8) after one hour DispersiveFig. 4. The results surface with elevation initial along conditions the y = obtained 0 transect with after and30min without. applicationDispersive results of the with Green initial function conditions of Kajiura obtained are with marked and without h.o. and of propagation, while the match is nearly perfect after two h.o.application∗, respectively. of the Green Also the function LSW ofsolution Kajiura isare included marked for h.o. compari- and 700 hours (figure 7). This implies that the original source length son.h.o.? Waves, respectively. are propagating Also the toLSW the solution right. is included for compari- has become irrelevant at this stage. son.Fig. Waves 3. The are surface propagating elevation to the along right. the y = 0 transect after 30min. 4.2 Indian Ocean Tsunami (2004) ResultsIn Fig. for5 differentthe waves resolutions propagating (given in in the km) positive for thex optimizeddirec- tion are shown for t ≈ 7.5 min and t ≈ 1 h. These times cor- Boussinesq model (h.o.) and initial conditions obtained with the Here we investigate the effect of dispersion for the 26th De- respondGreen function to τ = of0.12 Kajiura. and τ Waves= 0. are96, propagating respectively, to the where right.λ is identified with W = 50 km. Already at the earliest time the cember 2004 Indian Ocean Tsunami. The rupture started effect of dispersion is noticeable, while it has transformed the wave train crucially at t ≈ 1 h, when the only quantity properly reproduced by the LSW equations is the arrival time. The standard and higher order dispersion representa- tion only makes an apparent difference late in the emerging wave trains. We also observe that the second crest is slightly Fig. 5. The surface elevation along the y = 0 transect, for the wave higher than the leading one in the dispersive solutions. This systemFig. 5. propagatingThe surface elevation in the positive along thex direction.y = 0 transect, Results for the obtained wave is a three-dimensional effect that is neither observed for fromsystem models propagating with standard in the andpositive enhancedx-direction. dispersion Results properties obtained are the train propagating in the negative x direction nor in the markedfrom models by “disp” with and standard “h.o”, and respectively. enhanced Waves dispersion are propagating properties are to plane simulations (below). However, nonuniformity of the themarked right. by ’disp’ and ’h.o,’ respectively. Waves are propagating to the10 right. S. Glimsdal et al.: Dispersion of tsunamis source, constructive interference of reflections, and/or forma- tion of undular bores are presumably more likely reasons for t = 6h 0min; τ = 0.05 the larger inundation of secondary waves observed in true quite different. Transect results after one hour is shown in 0.5 0.4 tsunami cases. figure 6 and we observe only moderate effects of dispersion. The behavior for waves propagating in the y direction is 0.3 690 In this case the “dispersion time”, τ, is based on λ = B = quite different. Transect results after one hour are shown in 0.2 100km, which gives a value τ = 0.12 corresponding to that 0.1

Fig. 6 and we observe only moderate effects of dispersion. [m] for the upper panel in figure 5. The dispersion effects in the η 0.0 = = In this case the “dispersion time”, τ, is based on λ B two graphs also appear to be of the same magnitude. −0.1 100 km, which gives a value τ = 0.12 corresponding to that −0.2 We conclude this introductory example by reporting some LSW for the upper panel in Fig. 5. The dispersion effects in the 695 plane simulations, with the transect profile in figure 2 as ini- −0.3 disp two graphs also appear to be of the same magnitude. −0.4 tial condition. For the wave system propagating in the pos- 0 200 400 600 800 1000 We conclude this introductory example by reporting some itive x direction we observe that the leading crest is fairly distance [km] plane simulations, with the transect profile in Fig. 2 as ini- well described by the asymptotic formula (8) after one hour Fig. 4. The surface elevation along the y = 0 transect after 30min. tial condition. For the wave system propagating in the pos- of propagation, while the match is nearly perfect after two Dispersive results with initial conditions obtained with and without Fig. 8. The linear hydrostatic and linear dispersive solutions of itive x direction we observe that the leading crest is fairly 700 hours (figure 7). This implies that the original source length application of the Green function of Kajiura are marked h.o. and the Indian Ocean tsunami towards Africa extracted along a transect well described by the asymptotic formula (8) after one hour has become irrelevant at this stage. h.o.?, respectively. Also the LSW solution is included for compari- with length 1000km (surface elevations). Waves are propagating to ofson. propagation, Waves are propagating while the tomatch the right. is nearly perfect after two Fig. 6. The surface elevation along the x = 0 transect. Waves are the left. hours (Fig. 7). This implies that the original source length propagating4.2Fig. 6. IndianThe to surface theOcean right. elevation Tsunami along (2004) the x = 0 transect. Waves are has become irrelevant at this stage. propagating to the right. ◦ Here we investigate the effect of dispersion for the 26th De-705 around latitude 3 N and continued about 1200km northward cember 2004 Indian Ocean Tsunami. The rupture started along the Sunda trench. The width of the source was about 200km and the maximum slip was about 20m. The earth- quake had a magnitude of Mw = 9.0 and a dip angle of about www.nat-hazards-earth-syst-sci.net/13/1507/2013/ Nat. Hazards Earth Syst. Sci., 13, 1507–1526, 2013 15◦ (e.g. Bilham, 2005; Stein and Okal, 2005). 710 Glimsdal et al. (2006) found an insignificant effect of dis- persion close to the earthquake. However, for longer prop- agation distances, the effect of dispersion was found to be more apparent. In figure 8 the linear hydrostatic and linear dispersive solution is shown along a transect towards Africa 715 in a distance of 4300km from the source area. With an aver- age sea depth of 4000m, W = 200km and B = 1200km, the dispersive parameter is τ 0.05. The solutions differ mostly in shape/steepness of the∼ leading wave. The grid resolution in the computations was 20. 720 In many videos and photos taken of the tsunami, short fea- tures are evident (e.g. Arcas and Segur, 2012). A possible explanation is that the front of the tsunami in certain places evolved into an undular bore. Glimsdal et al. (2006) and Grue et al. (2008) showed through simulations towards Malaysia 725 (Malacca Strait) that undular bores may be formed.

4.3 Japan (2011)

The 11th March 2011 Tohoku tsunami devastated the east coast of Japan and caused almost 20.000 fatalities. The earthquake with a magnitude Mw = 9.0 occurred 130km 730 east of the Sendai coast, Japan. The source extensions was B = 400km and W = 150km. The average slip was reported to be 15 20m with a maximum value exceeding 60m (e.g. Lay et al.,− 2011; Ozawa et al., 2011). To model the tsunami we apply an earthquake source with a non-uniform slip dis- ◦ 735 20m 25 Fig. 7. The surface elevation for plane simulations. Comparison of tribution, a maximum slip of , and a dip angle of dispersive simulation (h.o.) and the asymptotic formula (8) for the (see Løvholt et al., 2012b). wave front. The phase of the latter is adjusted to yield coinciding The modeled surface elevations are compared to the reg- leading peaks at t = 37.5min. Waves are propagating to the right. istered data from DART buoys (http://www.ndbc.noaa.gov- /dart.shtml). In figure 9 the maximum surface elevation for 10 S. Glimsdal et al.: Dispersion of tsunamis

t = 6h 0min; τ = 0.05 0.5 0.4 0.3 0.2 0.1 [m]

η 0.0 −0.1 −0.2 LSW −0.3 disp −0.4 0 200 400 600 800 1000 distance [km]

Fig. 8. The linear hydrostatic and linear dispersive solutions of the Indian Ocean tsunami towards Africa extracted along a transect with length 1000km (surface elevations). Waves are propagating to the left. Fig. 6. The surface elevation along the x = 0 transect. Waves are propagating to the right. 1516◦ S. Glimsdal et al.: Dispersion of tsunamis 10705 around latitude 3 N and S. Glimsdal continued et about al.: Dispersion1200km northward of tsunamis along the Sunda trench. The width of the source was about 200km and the maximum slipt = 6h was 0min; aboutτ = 0.05 20m. The earth- 0.5 quake had a magnitude of Mw = 9.0 and a dip angle of about 0.4 15◦ (e.g. Bilham, 2005; Stein and Okal, 2005). 0.3 710 Glimsdal et al. (2006) found an insignificant effect of dis- 0.2 persion close to the earthquake. However, for longer prop- 0.1

agation [m] distances, the effect of dispersion was found to be η 0.0 more apparent. In figure 8 the linear hydrostatic and linear −0.1 dispersive−0.2 solution is shown along a transect towards Africa LSW 715 in a distance of 4300km from the source area. With an aver- −0.3 disp age sea−0.4 depth of 4000m, W = 200km and B = 1200km, the dispersive0 parameter 200 is τ 0 400.05. The solutions 600 differ 800 mostly 1000 in shape/steepness of the∼ leadingdistance wave. [km] The grid resolution in the computations was 20. Fig. 8. The linear hydrostatic and linear dispersive solutions of the 720 In many videos and photos taken of the tsunami, short fea- IndianFig. 8. OceanThe linear tsunami hydrostatic towards and Africa linear extracted dispersive along solutions a transect of tureswiththe Indian are length evident Ocean 1000 (e.g. tsunami km (surface Arcas towards and elevations). Africa Segur, extracted 2012). Waves along Aare possible propagating a transect explanationtowith the length left. is1000km that the(surface front of elevations). the tsunami Waves in certain are propagating places to evolvedthe left. into an undular bore. Glimsdal et al. (2006) and Grue Fig. 6. The surface elevation along the x = 0 transect. Waves are et al. (2008) showed through simulations towards Malaysia propagating to the right. 725 (Malacca Strait) that undular bores may be formed. explanation is that◦ the front of the tsunami in certain places 705 around latitude 3 N and continued about 1200km northward 4.3evolvedalong Japan the into Sunda (2011) an undular trench. bore. TheGlimsdal width of theet al. source(2006 was) and aboutGrue et200km al. (2008and) the showed maximum through slip simulations was about 20m towards. The Malaysia earth- The(Malacca 11th March Strait) 2011 that Tohokuundular tsunamibores may devastated be formed. the east quake had a magnitude of Mw = 9.0 and a dip angle of about coast15◦ (e.g. of Japan Bilham, and 2005; caused Stein almost and 20.000Okal, 2005). fatalities. The earthquake with a magnitude Mw = 9.0 occurred 130km 710 4.3Glimsdal Japan et (2011) al. (2006) found an insignificant effect of dis- 730 eastpersion of the close Sendai to coast,the earthquake. Japan. The However, source extensions for longer was prop- BTheagation= 400km 11 March distances,and W 2011= the 150km Tohoku effect. tsunamiThe ofdispersion average devastated slip was was the found reported east to coast be 15 20m 60m toofmore be Japan apparent. and causedwith In a figure maximum almost 8 the20 value000 linear fatalities. exceeding hydrostatic The earthquake and(e.g. linear Lay et al.,− 2011; Ozawa et al., 2011). To model the tsunami withdispersive a magnitude solution isM shownw = 9.0 along occurred a transect 130 kmtowards east Africa of the we apply an earthquake source with a non-uniformB slip= dis- Fig. 7. The surface elevation for plane simulations. Comparison of 715 Sendaiin a distance coast, of Japan.4300km Thefrom source the extensions source area. were With an400 aver-◦ km 735 tribution, a maximum slip of 20m, and a dip angle of 25 Fig. 7. The surface elevation for plane simulations. Comparison of andage seaW = depth150 of km.4000m The average, W = 200km slip wasand reportedB = 1200km to be, 15–the dispersive simulation (h.o.) and the asymptotic formula (8) for the (see Løvholt et al., 2012b). wavedispersive front. simulation The phase (h.o.) of the and latter the asymptotic is adjusted formula to yield (8) coinciding for the 20dispersive m with parametera maximum is valueτ 0. exceeding05. The solutions 60 m (e.g. differLay mostly et al., The modeled surface elevations are compared to the reg- leadingwave front. peaks The at t phase= 37. of5 min. the latter Waves is are adjusted propagating to yield to coinciding the right. 2011in shape/steepness; Ozawa et al. of, 2011 the∼ leading). To model wave. the The tsunami grid resolution we apply istered data from DART buoys (http://www.ndbc.noaa.gov- leading peaks at t = 37.5min. Waves are propagating to the right. anin the earthquake computations source was with20. a nonuniform slip distribution, a /dart.shtml). In figure 9 the maximum surface elevation◦ for 720 maximumIn manyslip videos of 20and m, photos and a taken dip angle of the of tsunami, 25 (see shortLøvholt fea- 4.2 Indian Ocean tsunami (2004) ettures al., are2012b evident). (e.g. Arcas and Segur, 2012). A possible explanationThe modeled is that surface the front elevations of the tsunami are compared in certain to the places reg- Here we investigate the effect of dispersion for the 26 De- isteredevolved data into from an undular DART bore. buoys Glimsdal (http://www.ndbc.noaa.gov-/ et al. (2006) and Grue cember 2004 Indian Ocean tsunami. The rupture started dart.shtmlet al. (2008)). In showed Fig. 9 throughthe maximum simulations surface towards elevation Malaysia for the ◦ around latitude 3 N and continued about 1200 km northward 725 Pacific(Malacca Ocean Strait) is shown that undular together bores with may the be locations formed. of selected along the Sunda Trench. The width of the source was about DART buoys. In the same figure the mariograms for both 200 km and the maximum slip was about 20 m. The earth- the4.3 numerical Japan (2011) model and the measurements at DART buoys quake had a magnitude of Mw = 9.0 and a dip angle of about 21401 (about 1000 km northeast of the source area), 51407 15◦ (e.g. Bilham, 2005; Stein and Okal, 2005). (Hawaii),The 11th March and 43413 2011 (west Tohoku of tsunami Guatemala) devastated are found. the east The Glimsdal et al. (2006) found an insignificant effect of dis- comparisoncoast of Japan to the and DART caused buoy almost data 20.000 shows that fatalities. the tsunami The = 9 0 130km persion close to the earthquake. However, for longer prop- wasearthquake clearly with affected a magnitude by dispersionMw at buoy. occurred 43413, but also agation distances, the effect of dispersion was found to be 730 ateast 51407. of the At Sendai DART coast, buoy Japan. 43413 The the source height extensions of the leading was = 400km and = 150km. The average slip was reported more apparent. In Fig. 8 the linear hydrostatic and linear dis- waveB for the dispersiveW solution is close to the measured one, to be 15 20m with a maximum value exceeding 60m (e.g. persive solutions are shown along a transect towards Africa while the linear hydrostatic solution overestimates the mea- Lay et al.,− 2011; Ozawa et al., 2011). To model the tsunami at a distance of 4300 km from the source area. With an aver- sured height by as much as ∼ 30 %. At this buoy we found we apply an earthquake source with a non-uniform slip dis- age sea depth of 4000 m, W = 200 km and B = 1200 km, the that τ ∼ 0.45, with an average sea depth from the source to◦ 735 tribution, a maximum slip of 20m, and a dip angle of 25 dispersiveFig. 7. The parameter surface elevation is τ ∼ for0.05. plane The simulations. solutions Comparison differ mostly of the buoy of 4.7 km. At buoy 51407 the effect of dispersion (see Løvholt et al., 2012b). indispersive shape/steepness simulation of (h.o.) the andleading the asymptotic wave. The formula grid resolution (8) for the is still clear (but slightly reduced), while there are no visible The modeled surface elevations are compared to the reg- inwave thefront. computations The phase was ofthe 20. latter is adjusted to yield coinciding effects of dispersion at 21401. Note that the results from the istered data from DART buoys (http://www.ndbc.noaa.gov- leadingIn many peaks videos at t = and 37.5min photos. Waves taken are of propagating the tsunami, to short the right. fea- DART buoys are given a shift of up to 200 s, to match the /dart.shtml). In figure 9 the maximum surface elevation for tures are evident (e.g. Arcas and Segur, 2012). A possible arrival of the leading peak in the simulations and make the

Nat. Hazards Earth Syst. Sci., 13, 1507–1526, 2013 www.nat-hazards-earth-syst-sci.net/13/1507/2013/ S. Glimsdal et al.: Dispersion of tsunamis 11 S. Glimsdal et al.: Dispersion of tsunamis 1517

740 the Pacific Ocean is shown together with the locations of se-

comparisonlected DART easier. buoys. The In measurements the same figure at DART the mariograms buoy 21418 for is coarse and influenced by noise; the maximum value of the 21415 both the numerical model and the measurements at DART 21419 leading peak is therefore somewhat uncertain. This is the rea- 21401 buoys 21401 (about 1000km north-east of the source area), 40˚ 21418 son51407 why (Hawaii), the results and from 43413 buoy (west 21401 of Guatemala)are plotted instead are found. of 21413 buoy 21418 located in front of the source area. The compar- 745 The comparison to the DART buoy data shows that the 51407 ison between the numerical solutions and the measured data 20˚ tsunami was clearly affected by dispersion at buoy 43413, but 43413 fromalsoat eight 51407. DART At DARTbuoys are buoy elaborated 43413 the in height upon Table of the1 leading. The overall picture is not entirely clear and may be influenced 0˚ wave for the dispersive solution is close to the measured one, 51425 by directivity and the nonuniform source distribution. Still, while the linear hydrostatic solution overestimates the mea- −20˚ there is a tendency for increasing importance of dispersion 750 sured height with as much as 30%. At this buoy we found 140˚ 160˚ 180˚ −160˚ −140˚ −120˚ −100˚ −80˚ for long-distance propagation.∼ The resolution of the com- m that τ 0.45, with an average sea depth from the source to putational grid was 40. Grid refinement tests at buoy 21413 0.01 0.02 0.03 0.04 0.06 0.10 0.16 0.25 0.40 0.63 1.00 the buoy∼ of 4.7km. At buoy 51407 the effect of dispersion comparing the results from a 40 grid with those from 20 and is0 still clear (but slightly reduced), while there are no visible DART 21401, τ=0.07 1 grids, covering a smaller part of the Pacific Ocean, show a 0.8 effect of dispersion at 21401. Note that the results from the difference in the height of the leading wave of less than 1 % 755 DART buoys are given a shift of up to 200 s, to match the 0.6 for LSW and 0.5 % for the linear dispersive solution. arrival of the leading peak in the simulations and make the 0.4 comparison easier. The measurements at DART buoy 21418 4.4 Potential earthquake at Lesser Antilles is coarse and influenced by noise, the maximum value of the [m] 0.2 η leading peak is therefore somewhat uncertain. This is the 0.0 NE of Guadeloupe in the Caribbean, we have modeled a po- 760 reason why the results from buoy 21401 is plotted instead of tential earthquake with magnitude Mw = 8.0 (for details, see −0.2 buoy 21418 located in front of the source area. The com- e.g. Løvholt et al., 2010; Harbitz et al., 2012a). The dip angle parison◦ between the numerical solutions0 and the measured −0.4 is 80 . The grid resolution was 0.5 . The earthquake source 2000 4000 6000 8000 10000 12000 data from eight DART buoys are elaborated in table 1. The lies along a SSE–NNW axis with a depression facing to the overall picture is not entirely clear and may be influenced DART 51407, τ=0.27 ENE, as shown in Fig. 10. The effect of dispersion is inves- 765 by directivity and the non-uniform source distribution. Still, tigated through mariograms at two locations. The first loca- 0.3 there is a tendency for increasing importance of dispersion tion is 350 km south of the earthquake (outside Bridgetown, 0.2 for long distance propagation. The resolution of the compu- Barbados) and the0 second location is about 350 km to the 0.1 4 east.tational In the grid first was case. Grid the direction refinement of tests propagation at buoy 21413 is close by comparing the results from a 40 grid with those from 20 and

[m] 0.0 η to0 the azimuth direction of the source, and we therefore set 770 1 grids, covering a smaller part of the Pacific Ocean, show a −0.1 λ = B = 150 km, and the average depth to 3 km. In the sec- difference on the height of the leading wave of less than 1% −0.2 ond case we look at propagation mainly along the dip di- rection,for LSW and and set 0.5%λ = forW the= 50 linear km, dispersive and the average solution. depth to −0.3 5 km. The mariograms show that the effect of dispersion is 26000 28000 30000 32000 34000 36000 4.4 Potential earthquake at Lesser Antilles almost absent for the laterally propagating leading waves for location 1 (τ = 0.006), while the effects of dispersion at lo- DART 43413, τ = 0.45 NE of Guadeloupe in the Caribbean, we have modeled a po- cation 2 is crucial (τ = 0.42). Hence, source orientation and 0.2 775 tential earthquake with magnitude M = 8.0 (for details, see location govern influence of frequencyw dispersion. e.g. Løvholt et al., 2010; Harbitz et al., 2012a). The dip an- 0.1 gle is 80◦. The grid resolution was 0.50. The earthquake 4.5 Potential earthquake at the Hellenic Arc 0.0 source lies along a SSE-NNW axis with a depression facing [m] η to the ENE, figure 10. The effect of dispersion is investi- −0.1 As an example from the Mediterranean, we show the tsunami 780 gated through mariograms at two locations. The first loca- DART for a potential earthquake SW of Crete (for details, see LSW tion is 350km south of the earthquake (outside Bridgetown, −0.2 e.g. Løvholt et al., 2012a). The earthquake has a magni- disp Barbados) and the second location is about 350km to the tude M = 7.8 and a dip angle of 15◦. The extensions of the east. Inw the first case the direction of propagation is close 52000 54000 56000 58000 source correspond to B = 100 km and W = 44 km. As in the Time [s] to the azimuth direction of the source and we therefore set example from the Lesser Antilles above, we evaluate the ef- 785 = = 150km, and the average depth to 3km. In the sec- fectλ ofB dispersion along transects after 10 min. The dispersive Fig. 9. The Tohoku tsunami 2011. In the upper panel the maximum ond case we look at propagation mainly along the dip direc- Fig. 9. The Tohoku tsunami 2011. In the upper panel the maximum parameter is calculated when the leading waves have propa- surface elevation for the Pacific Ocean is shown. In the lower panels tion, and set λ = W = 50km, and the average depth to 5km. thesurface mariograms elevation (for for the three Pacific of the Ocean DART is shown. buoys) In for the the lower computed panels gated a distance of 180 (h = 1.7 km) and 250 km (h = 2 km) The mariograms show that the effect of dispersion is almost surfacethe mariograms elevations (for for three the linear of the shallow DART water buoys) (“LSW”) for the and computed linear along the transects ahead (to the south) and laterally, re- dispersivesurface elevations (“disp”) for solutions the linear arecompared shallow water to the (“LSW”) measured and surface linear spectively.absent for In the Fig. laterally11 solutions propagating along leadingthese two waves transects for loca- are elevationsdispersive (“DART”). (“disp”) solutions are compared to the measured surface 790 shown.tion 1 ( Laterallyτ = 0.006 the), while effect the is effects very limited of dispersion with τ at= location0.006, elevations (“DART”). while2 is crucial ahead of (τ the= 0 fault.42). the Hence, effect source of dispersion orientation is clearer, and loca- but stilltion small, govern with influenceτ = 0.036. of frequency The grid dispersion. resolution was 0.50. www.nat-hazards-earth-syst-sci.net/13/1507/2013/ Nat. Hazards Earth Syst. Sci., 13, 1507–1526, 2013 12 S. Glimsdal et al.: Dispersion of tsunamis

Table 1. The values of τ for eight different DART buoys, together with the simulated (linear hydrostatic - “LSW”, and linear dispersive - “disp”) and1518 measured (“DART”) height of the leading wave. L is the propagation distance, S. Glimsdalt is the propagation et al.: Dispersion time, while of tsunamish is the average p sea depth (calculated by using t and the LSW propagation speed, gh). (DART 21418 - uncertain value due to coarse resolution of the data.) Table 1. The values of τ for eight different DART buoys, together with the simulated (linear hydrostatic – “LSW”; and linear dispersive – “disp”) and measured (“DART”) height of the leading wave. L is the propagation distance, t is the propagation time, while h is the average q sea depth (calculatedDART by using #t and LSW the LSW [m] propagation disp [m] speed, DARTgh). [m] (DARTL 21418[km] – uncertaint [hrs] valueh due[km] to coarseτ resolution of the data.) 21401 0.72 0.72 0.73 1000 1.1 6.3 0.07 21413DART # 0.64 LSW [m] 0.68 disp [m] DART 0.78 [m] L 1200[km] t [h] 1.3h [km] 5.8τ 0.07 21415 0.27 0.26 0.28 2700 3.1 5.9 0.17 21401 0.72 0.72 0.73∗ 1000 1.1 6.3 0.07 2141821413 1.78 0.64 1.80 0.68 1.86 0.78 1200500 1.3 0.5 5.8 6.3 0.07 0.04 2141921415 0.51 0.27 0.51 0.26 0.56 0.28 27001300 3.1 1.3 5.9 6.8 0.17 0.11 5140721418 0.25 1.78 0.21 1.80 0.17 1.86∗ 6000500 0.5 7.5 6.3 5.0 0.04 0.27 5142521419 0.07 0.51 0.07 0.51 0.06 0.56 13007000 1.3 8.1 6.8 5.7 0.11 0.40 4341351407 0.23 0.25 0.17 0.21 0.18 0.17 11500 6000 7.5 14.7 5.0 4.7 0.27 0.45 51425 0.07 0.07 0.06 7000 8.1 5.7 0.40 43413 0.23 0.17 0.18 11500 14.7 4.7 0.45

Location 1; τ = 0.006 0.3

20˚ 0.2 0.1

[m] 0.0 η −0.1 18˚ −0.2 LSW 2 disp −0.3 2000 3000 4000 5000 time [s] 16˚ Location 2; τ = 0.42 1.2 1.0 0.8 0.6 14˚ 0.4 [m]

η 0.2 1 0.0 −0.2 LSW −0.4 disp 12˚ −0.6 −64˚ −62˚ −60˚ −58˚ −56˚ 1000 2000 3000 4000 time [s] m 0.10 0.20 0.39 0.77 1.52 3.00

Fig. 10. The maximum surface elevations for the linear hydrostatic solution is shown to the left. The locations of the mariograms are indicated Fig. 10. withThe the maximum numbers 1 surface and 2, with elevations the corresponding for the linear results hydrostaticshown in the uppersolution and islower shown right to panels. the left. The locations of the mariograms are indicated with the numbers 1 and 2, with the corresponding results shown in the upper and lower right panels. 4.6 Dispersive effects on seismic tsunamis, worldwide intermediate cases need to be evaluated individually with re- spect to dispersive effects. 4.5 Potential earthquake at the Hellenic Arc 4.6 Dispersive effects on seismic tsunamis; worldwide A global study of tsunami impact is presented in Løvholt et al. (2012a). In Table 2 we present the dispersive param- 5 Landslide-generated tsunamis As an exampleeter for a from selection the Mediterranean, of the earthquake we sources show from the thistsunami study A global study of tsunami impact is presented in Løvholt 795 for a potentialin order earthquake to demonstrate SW practical of Crete examples (for details, of the dispersion see e.g. 810 5.1et al. The (2012a). Storegga In submarine table 2 we landslide have presented the dispersive Løvholtnumber et al., for 2012a). forecasting. The The earthquake parameter τ hasis estimated a magnitude by sub- parameter for a selection of the earthquake sources from this ◦ = Mw = 7stituting.8 and a a dip propagation angle 15 distance. The extensions of L 1000 of km, the an source aver- Thestudy Storegga in order Slide to on demonstrate the continental practical slope off examples Western Nor- of the dis- age depth and a source width into Eq. (7). Both the source way around 8150 yr BP is one of the largest and best-studied correspond to B = 100km and W = 44km. As in the ex- persion number for forecasting. The parameter τ is estimated ample fromwidth the and Lesser the depth Antilles are given above, in the we table. evaluate Again wethe see effect that submarine landslides on Earth (Bugge et al., 1987, 1988; the smaller earthquakes with narrow width are expected to Haflidasonby substituting et al., a2004 propagation; Bryn et al. distance, 2005; Kvalstad of L = et 1000km al., , an 800 of dispersion along transects after 10min. The dispersive be much more affected by dispersion – see e.g. the earth-815 2005average). The depth landslide and comprised a source widtha volume into of equationabout 2400 (7). km3 Both. the parameterquake is calculated along the Makran when the fault leading (Pakistan) waves with haveτ ∼ 0 propa-.9. The Today,source the width most and common the depth view is are that given the Storegga in the table.Slide was Again we gated amega-earthquakes, distance of 180 (h in= particular, 1.7km) andhave250km much smaller(h = 2kmτ and) asee continuous that the retrogressive smaller earthquakes process, and with deposits narrow of the width corre- are ex- along theare transects hence expected ahead (toto be the less south) affected and by laterally, dispersion respec- – see spondingpected to giant be much tsunami more are found affected in Norway, by dispersion, Faroe Islands, see e.g. the tively. Ine.g. figure the Mw 11= 9.4 solutions outside alongsouthern these Chile two with transectsτ ∼ 0.01. areThe Shetlandearthquake and along Scotland the ( MakranHarbitz, 1992b fault; (Pakistan)Bondevik with et al.,τ 0.9. ∼ 805 shown. Laterally the effect is very limited with τ = 0.006 820 The mega-earthquakes, in particular, have much smaller τ while aheadNat. Hazardsof the fault Earth the Syst. effect Sci., of dispersion13, 1507–1526 is clearer,, 2013 but and are www.nat-hazards-earth-syst-sci.net/13/1507/2013/ hence expected to be less affected by dispersion, see 0 still small, with τ = 0.036. The grid resolution was 0.5 . e.g. the Mw 9.4 outside Southern Chile with τ 0.01. The ∼ S. Glimsdal et al.: Dispersion of tsunamis 1519

Table 2. Values for the dispersive parameter τ for a selection of sources applied in Løvholt et al. (2012a). τ is measured at a distance of 1000 km away from the source. The earthquake source parameters: Mw – magnitude; B – length; W – width; and D – dip angle. h is the average sea depth used for estimation of τ. Abbreviations: SA – Sunda Arc; BF – Burma Fault; MF – Makran Fault; BA – Banda Arc; NGT – New Guinea Trench; PT – Philippine Trench; TT – Tonga Trench; SST – South Solomon Trench; NHT – New Hebrides Trench; PCT – Peru–Chile Trench; PRT – Puerto Rico Trench; HA – Hellenic Arc.

Location Mw B [km] W [km] D [deg] h [km] τ SA, Andaman Islands 8.50 362 100 15 5 0.150 SA, South Sumatra 9.10 527 200 15 5 0.019 BF, Myanmar - Bangladesh 8.90 655 125 10 3 0.028 MF, Pakistan coast 8.40 398 48 10 4 0.868 BA, Eastern Banda Sea 8.50 261 150 20 3 0.016 NGT, Eastern Irian Jaya 8.50 258 100 20 4 0.096 PT, South Mindanao 8.40 176 100 20 5 0.150 MT, Western Luzon 8.20 348 70 45 5 0.437 TT, Northern part 9.00 519 200 20 5 0.019 SST, Eastern Solomon Isl. 8.30 281 100 20 5 0.150 NHT, Southern Vanuatu 8.60 314 100 20 3 0.054 NHT, Northern Vanuatu 8.60 340 100 20 3 0.054 PCT, Southern Chile 9.40 853 200 20 4 0.012 PRT, North Hispaniola 8.00 200 55 80 5 0.902 HA, South of Crete 7.70 149 75 20 3 0.128

1997a,b, 2005; Dawson et al., 1988). Best agreement with S. Glimsdal et al.: Dispersion of tsunamis 13 the observations is obtained with a maximum frontal velocity of 25–30 m s−1, a run-out distance of the dense tsunamigenic t = 0h 10min; τ = 0.006 845 flowNordic of Seas 150 km, are fairly and a limited. retrogressive As a result, release the of Storegga the total Slide vol- tsunami illustrates that dispersion is insignificant for waves 0.4 ume lasting less than one hour, i.e. 15–20 s between the re- leasesgenerated of each by large individual and sub-critical landslide submarine element (Bondevik landslides et with al., 0.2 2005moderate). The acceleration Storegga Slide and tsunamideceleration is composed dominated exclusively by wave bycomponents very long much components longer than relative the water to the depth water (Haugen depth. More- et al.,

[m] 0.0 −4

η 850 2005). In terms of the dispersive parameter we find 10 over, the propagation distances in the Nordic Seasτ are fairly in a distance of 200km towards Norway. Even for a∼ propa- −0.2 limited. As a result, the Storegga Slide tsunami illustrates gation distance of 4000km towards North America the value LSW that dispersion is insignificant for waves generated by large −0.4 disp of τ is relatively small, τ = 0.04. The numerical simulations and sub-critical submarine landslides with moderate acceler- (linear hydrostatic only) is performed on a grid with a reso- 40 80 120 160 200 240 ation and deceleration dominated by wave components much 855 lution of about 2 km. distance [km] longer than the water depth (Haugen et al., 2005). In terms t = 0h10 min; τ = 0.036 − 1.0 of the dispersive parameter we find τ ∼ 10 4 at a distance 5.2 The Hinlopen submarine landslide 0.8 of 200 km towards Norway. Even for a propagation distance 0.6 of 4000 km towards North America the value of τ is rela- 0.4 For the pre-Last Glacial Maximum Hinlopen Slide at the tively small, τ = 0.04. The numerical simulations (linear hy- 0.2 mouth of the Hinlopen cross-shelf trough on the northern

[m] 0.0 drostatic only) is performed on a grid with a resolution of

η Svalbard margin bathymetric effects as well as high speed −0.2 about 2 km. 860 and huge thickness of the dislodged mass and the rafted −0.4 blocks probably implied that shorter wave components in- −0.6 LSW 5.2 The Hinlopen submarine landslide −0.8 disp troducing dispersive and nonlinear effects were more pro- −1.0 nounced than for most other tsunamis generated by subma- 0 40 80 120 160 For the pre-Last Glacial Maximum Hinlopen Slide at the rine landslides (Vanneste et al., 2011). The head-wall is distance [km] mouth of the Hinlopen cross-shelf trough on the northern 865 several hundred meters high (exceeding 1400 m). Despite Svalbard margin, bathymetric effects as well as high speed Fig. 11. Effect of dispersion for an earthquake along the Hellenic the relatively small slide scar area (about 5 % of the size and huge thickness of the dislodged mass and the rafted Arc.Fig. In 11. theEffect upper of panel dispersion the surface for an elevations earthquake extracted along along the Hellenic a side- of the Storegga Slide area), an upper estimated volume of wiseArc. transect In the upper are shown. panel the In the surface lower elevations the transect are isextracted taken ahead along of a blocksabout 1350km probably3 (about implied 55 that % of shorter-wave the Storegga components Slide volume) in- theside earthquake. wise transect is shown. In the lower the transect is taken ahead troducingwas excavated dispersive from the and northern nonlinear Svalbard effects continental were more margin pro- of the earthquake. nounced than for most other tsunamis generated by sub- 870 (Vanneste et al., 2011).Close to the slide area the simulations marineof the tsunami landslides show (Vanneste surface elevations et al., 2011 over). The130m, headwall whereas is severalthe tsunami hundred may havemeters been high several (exceeding tens of 1400 meters m). along Despite the intermediate cases need to be evaluated individually with re- coasts of Svalbard and Greenland. The dispersive effects and spect to dispersive effects. www.nat-hazards-earth-syst-sci.net/13/1507/2013/the radial Nat. spread Hazards reduce Earth the Syst. maximum Sci., 13,surface 1507– elevations1526, 2013 as 875 the tsunami propagates out from the slide area. A simula- tion along a transect towards Greenland is shown in figure 825 5 Landslide generated tsunamis 13. The effect of dispersion is here clear with τ = 0.13. 5.1 The Storegga submarine landslide 5.3 Papua New Guinea (1998) The Storegga Slide on the continental slope off western Nor- way around 8150 years B.P. is one of the largest and best- The 1998 Papua New Guinea (PNG) tsunami gave run-up studied submarine landslides on Earth (Bugge et al., 1987, 880 heights up to 15 m and affected a 20km segment of the coast, 830 1988; Haflidason et al., 2004; Bryn et al., 2005; Kvalstad killing 2200 people (Dengler and Preuss, 2003; McSaveney et al., 2005). The landslide comprised a volume of about et al., 2000). Farther away the tsunami was not a significant 2400km3. Today, the most common view is that the Storegga event (Okal and Synolakis, 2004). Slide was a continuous retrogressive process, and deposits Initially, the tsunami was believed to originate from an of the corresponding giant tsunami are found both in Nor- 885 earthquake. However, attempts to model the tsunami using 835 way, Faroe Islands, Shetland and Scotland (Harbitz, 1992b; solely an earthquake source gave too small amplitudes and Bondevik et al., 1997a,b, 2005; Dawson et al., 1988). Best too late arrival times (e.g. Geist, 1998a). It is broadly ac- agreement with the observations is obtained with a maxi- cepted that the damaging part of the tsunami was due to a mum frontal velocity of 25 30m/s, a run-out distance of slump (Bardet et al., 2003; Tappin et al., 2003; Sweet and − the dense tsunamigenic flow of 150km, and a retrogressive 890 Silver, 2003), while the earthquake was responsible for the 840 release of the total volume lasting less than one hour, i.e. far-field tsunami and played an indirect role as the slump 15 20s between the releases of each individual landslide triggering mechanism. element− (Bondevik et al., 2005). The Storegga Slide tsunami The dipole shape and the short-wave components of the is composed exclusively by very long components relative to generated waves contribute to radial spreading (Okal and the water depth. Moreover, the propagation distances in the 895 Synolakis, 2004) and frequency dispersion (Lynett et al., 14 S. Glimsdal et al.: Dispersion of tsunamis

Table 2. Values for the dispersive parameter τ for a selection of sources applied in Løvholt et al. (2012a). τ is measured at a distance of 1000km away from the source. The earthquake source parameters are: Mw - magnitude, B - length, W - width and D - dip angle. h is the average sea depth used for estimation of τ. Abbreviations: SA - Sunda Arc, BF - Burma Fault, MF - Makran Fault, BA - Banda Arc, NGT - New Guinea Trench, PT - Philippine Trench, TT - Tonga Trench, SST - South Solomon Trench, NHT - New Hebrides Trench, PCT - Peru-Chile Trench, PRT - Puerto Rico Trench, HA - Hellenic Arc.

Location Mw B [km] W [km] D [deg] h [km] τ SA, Andaman Islands 8.50 362 100 15 5 0.150 SA, South Sumatra 9.10 527 200 15 5 0.019 BF, Myanmar - Bangladesh 8.90 655 125 10 3 0.028 MF, Pakistan coast 8.40 398 48 10 4 0.868 BA, Eastern Banda Sea 8.50 261 150 20 3 0.016 NGT, Eastern Irian Jaya 8.50 258 100 20 4 0.096 PT, South Mindanao 8.40 176 100 20 5 0.150 MT, Western Luzon 8.20 348 70 45 5 0.437 TT, Northern part 9.00 519 200 20 5 0.019 SST, Eastern Solomon Isl. 8.30 281 100 20 5 0.150 NHT, Southern Vanuatu 8.60 314 100 20 3 0.054 NHT, Northern Vanuatu 8.60 340 100 20 3 0.054 PCT, Southern Chile 9.40 853 200 20 4 0.012 PRT, North Hispaniola 8.00 200 55 80 5 0.902 1520HA, South of Crete 7.70 149 75 20 S. Glimsdal 3 et al.:0.128 Dispersion of tsunamis

Fig. 12. The tsunami after the Hinlopen submarine landslide. Panel A shows the maximum surface elevation of the generated tsunami. Values aboveFig. 12. 40 mThe are tsunami colored after red. The theHinlopen bathymetry submarine is shown landslide. in (B), where Panel the A initial shows posistion the maximum of the landslidesurface elevation (yellow ofpoint) the andgenerated the transect tsunami. for Fig.Values13 (red above line) 40 are m are shown. colored The red. figure The is bathymetry modified from is shownVanneste in panel et al. B,(2011 where). the initial posistion of the landslide (yellow point) and the transectS. Glimsdal for figure et al.: 13 Dispersion (red line) are of shown. tsunamis The figure is modified from Vanneste et al. (2011). 15

Transect towards Greenland, τ=0.13 Foralong a propagation a transect towards distance Greenland of 500km isthis shown leads in to Fig. a value13. of The 2003), respectively, which reduce the surface elevation in the that the effect of dispersion was crucial. On the other hand, 940 effectclose of to dispersion unity. After is here crossing clear the with Atlanticτ = 0. Ocean13. we find 60 τ far-field. From Synolakis et al. (2001) we found λ = 10km, τit is10 noteworthy. The resolution that Lynett in the trans-oceanic et al. (2003) simulationsreported that is even20, according40 to section 3.2, and the propagation towards land, though∼ dispersion had a crucial effect on the incident wave, while5.3 much Papua finer New grids Guinea were (1998) employed in more local simula- about2020km away, gave τ 0.1 (moderate effect of disper- the flooding was rather similar to the one obtained by a ∼ tions around the Canari Islands. 900 905 sion). [m] 0 In a distance of 50km offshore τ 5, which means NLSW model. It is not clear whether this is circumstantial η ∼ TheSome 1998 results are Papua shown New in figure Guinea 14. We observe (PNG) a strong tsunami −20 945 directivitygave run-up and that heights the leading up crest to 15 is m not dominantand affected in the a time series shown. More surprisingly, in view of the large −40 LSW 20 km segment of the coast, killing 2200 people value of , in the simulated time series for buoy 2 the devi- disp (Denglerτ and Preuss, 2003; McSaveney et al., 2000). Farther −60 ation between the LSW solution and the dispersive solution away the tsunami was not a significant event (Okal and 240 280 320 is only moderate. It must then be remarked that the gen- distance along transect [km] Synolakis, 2004). 950 eration and early propagation is computed by a dispersive Initially, the tsunami was believed to originate from an and nonlinear model up to t = 900s, then the solution is con- Fig. 13. Computations of the waves generated by the Hinlopen sub- earthquake. However, attempts to model the tsunami using veyed into either a linear dispersive model or a LSW model. marineFig. 13. landslideComputations along of a transect the waves towards generated Greenland. by the Hinlopen The front sub- of solely an earthquake source gave too-small amplitudes and themarine waves landslide has traveled along 420 a transect km from towards the landslide Greenland. area overThefront an aver- of At t = 900s the leading crest is already stretched much by too-late arrival times (e.g. Geist, 1998a). It is broadly ac- agethe depth waves of has about traveled 2 km.420km Wavesfrom are propagating the landslide to area the over right an (south- aver- dispersive effects and the subsequent evolution may thus be cepted that the damaging part of the tsunami was due to a west).age depth of about 2km. Waves are propagating to the right (south- 955 slower (see discussion below eq. (8)). west). slumpIn figure (Bardet 15 we et show al., 2003 the evolution; Tappin etof al.the, second2003; Sweet incident and crestSilver as, it2003 enters), while the continental the earthquake shelf of was North responsible America. for An the undularfar-field bore tsunami rapidly and evolves played and an frontindirect ofthe role crest as the is split slump theor a relatively more general small feature. slide scar area (about 5 % of the size intotriggering a sequence mechanism. of solitary waves. The height of the individ- of the Storegga Slide area), an upper estimated volume of The dipole shape and the short-wave components of the 960 ual peaks are close to the stability limit for solitary waves 3 about5.4 1350Potential km landslide(about 55 from % of La the Palma Storegga Slide volume) andgenerated they will waves soon contribute break during to the radial following spreading shoaling. (Okal At and was excavated from the northern Svalbard continental margin thisSynolakis location, 2004 the first) and incident frequency crest displays dispersion the ( highestLynett etsur- al., (VannesteA potential et massive,al., 2011). volcanic Close to flank the slide collapse area at the the simulations La Palma face2003 elevation), respectively, (not shown), which reduce but the the second surface crest elevation has higher in the ofIsland the tsunami was first show suggested surface by elevations Ward (2001). over Assuming 130 m, whereas a slide effectivefar-field. wave From heightSynolakis due to et the al. intermediate(2001) we found trough.λ = Hence,10 km, 3 910 volume of 500km they predicted wave heights along the the tsunami may have been several tens of meters along the965 theaccording leading crestto Sect. undergoes3.2, and the the transformation propagation to towards an undular land, coastseast coast of Svalbard of North and America Greenland. in the The range dispersive10 25m effects. How- and about 20 km away, gave τ ∼ 0.1 (moderate effect of disper- − bore later, when it is closer to the coast. To resolve the un- theever, radial this spread extreme reduce scenario the has maximum remained surface controversial elevations ever as dularsion). bore At properlya distance grid of increments 50 km offshore comparableτ ∼ 5, to which the depth, means thesince. tsunami Masson propagates et al. (2002); out from Wynn the and slide Masson area. (2003); A simulation Mas- orthat preferably the effect smaller of dispersion must be employed.was crucial. Covering On the theother conti- hand, son et al. (2006); Hunt et al. (2011) questioned the geologi- nental shelf with a resolution of, say, 20m, will be a compu- 915 cal aspects, while Mader (2001); Gisler et al. (2006) obtained 970 tational challenge. Moreover, when the crests start to break Nat. Hazards Earth Syst. Sci., 13, 1507–1526, 2013 www.nat-hazards-earth-syst-sci.net/13/1507/2013/ much smaller waves in the far-field. we still need a Boussinesq type model with breaking features In Gisler et al. (2006) the slide volume was reduced to implemented (see Korycansky and Lynett, 2005). 375km3. The wave generation and early propagation were treated with the multi-material SAGE model, while the far- 5.5 Potential rockslides in Norwegian fjords, example 920 field estimates were obtained by extrapolation of local atten- from Akerneset˚ uation rates. Løvholt et al. (2008) started with the near field

solution of Gisler et al. (2006), but treated the oceanic prop- 975 A large unstable rock volume has been identified in the agation with a set of Boussinesq type equations. By this pro- Akerneset˚ rock slope in the narrow fjord, Storfjorden, Møre cedure they obtained wave heights at the American coast that & Romsdal County, Western Norway, figure 16. The site has 925 were slightly smaller than those of Ward (2001), but still dan- been subject to extensive geological investigations (Blikra, gerous. Recently, Abadie et al. (2012) applied a similar strat- 2008, 2012) and the tsunami has been studied experimentally 3 egy, with a slide volume of 500km , and obtained somewhat 980 and numerically (NGI, 2010; Harbitz et al., 2012b). We here higher waves close to the Canary Islands than Løvholt et al. focus on a comparison between the LSW and the dispersive (2008). However, they did not compute the trans-Atlantic solutions for a volume of 54 106 m3 and an impact veloc- · 930 propagation. The La Palma tsunami case is also studied by ity of 45m/s. In figure 16 we present surfaces along two Zhou et al. (2011). transects; one through the generation area and one close to We omit most of the details on the wave field in Løvholt 985 the fjord head at Geiranger. The dispersion is crucial during et al. (2008) and focus on aspects that are most relevant in the tsunami generation and the early phases of propagation, the present context. In this case both dispersive and nonlin- see mid panel of figure 16. This is also confirmed through 935 ear effects are active during the formation of the waves and laboratory experiments in both 2D (Sælevik et al., 2009) the value of τ is somewhat dependent on the time at which and 3D (NGI, 2010). The leading peak for the south going the wavelength is measured. After 500s the generation is 990 waves differ significantly both in height (more than a fac- virtually finished and we find λ = 36km at a depth of 4km. tor two), wavelength, and shape between the two solutions. 16S. Glimsdal et al.: Dispersion of tsunamis S. Glimsdal et al.: Dispersion 1521 of tsunamis

Buoy 2 60˚ 2.0

1.0

[m] 0.0 η

45˚ −1.0 disp LSW 2 1 −2.0 6.2 6.4 6.6 6.8 7.0 7.2 7.4 3 30˚ time [hrs] Buoy 4 4 3.0 15˚ 2.0 1.0 0.0 [m]

η −1.0 0˚ −2.0 disp −3.0 LSW −4.0 −90˚ −75˚ −60˚ −45˚ −30˚ −15˚ 0˚ 5.2 5.4 5.6 5.8 6.0 6.2 6.4 time [hrs] 0.1 0.2 0.4 0.8 1.6 3.2 6.3 12.6 25.1 50.1 100.0

Fig. 14. The La Palma tsunami. Left panel: maximum wave elevation and position of buoys. Right panels: time series at two buoy locations. Fig. 14. The La Palma tsunami. Left panel: maximum wave elevation and position of buoys. Right panels: time series at two buoy locations. it is noteworthy that Lynett et al. (2003) reported that even We omit most of the details on the wave field in Løvholt though dispersion had a crucial effect on the incident wave, et al. (2008) and focus on aspects that are most relevant in the flooding was rather similar to the one obtained by a thethe present effect context. of dispersion In this case is both small dispersive for τ and < 0 nonlin-.01, while it gen- NLSW model. It is not clear whether this is circumstantial earerally effects becomes are active significantduring the formation for τ > of0 the.1, waves, say. and The form of τ or a more general feature. the value of τ is somewhat dependent on the time at which theindicates wavelength that is measured. the source After width/initial 500 s the generation wave is length vir- (for land- 1010 slides) is more important= for the significance of dispersion 5.4 Potential landslide from La Palma tually finished and we find λ 36 km at a depth of 4 km. Forthan a propagation the depth distance or propagation of 500 km thisdistance. leads to Accordingly, a value of we find τthatclose moderate to unity. After magnitude crossing the earthquakes Atlantic Ocean yield we more find dispersive A potential massive volcanic flank collapse at La Palma Is- τ ∼ 10. The resolution in the transoceanic simulations is 20, land was first suggested by Ward (2001). Assuming a slide tsunamis than the huge ones, such as the 2004 Indian Ocean 3 while much finer grids were employed in more local simula- volume of 500 km , they predicted wave heights along the tionsand around the 2011 the Canary Japan Islands. tsunami. east coast of North America in the range 10–25 m. How- 1015 SomeFor results the largestare shown earthquakes in Fig. 14. We observe frequency a strong dispersion di- only ever, this extreme scenario has remained controversial ever rectivitymodify and the that the the trans-oceanic leading crest is not propagation dominant inmildly. the time Hence, dis- since. Masson et al. (2002); Wynn and Masson (2003); Mas- series shown. More surprisingly, in view of the large value of son et al. (2006); Hunt et al. (2011) questioned the geologi- τpersion, in the simulated is not needed time series for for propagation buoy 2 the deviation in the be- near-field, but cal aspects, while Mader (2001); Gisler et al. (2006) obtained tweenmay the be LSWimportant solution if and far-field the dispersive tsunami solution data are is only used for verifi- much smaller waves in the far-field. Fig. 15. Evolution of an undular bore from the second crest on the moderate.cation of It sourcemust then properties. be remarked that the generation and In Gisler et al. (2006) the slide volume was reduced to early propagation are computed by a dispersive and nonlinear continental shelf of3 North Carolina. Waves propagate to the left. y1020 For the smaller earthquakes (M 8 or less) we observe a 375 km . The wave generation and early propagation were model up to t = 900 s; then the solution isw conveyed into ei- is the surface elevation (η), x is the distance and the numbers in the ∼ treated with the multimaterial SAGE model, while the far- therstrong a linear directivity dispersive modelof the or dispersion, a LSW model. following At t = 900 the s amplitude legends correspondsfield estimates to the were propagation obtained by time extrapolation in minutes. of local atten- thedirectivity, leading crest due is alreadyto the elongated stretched a lot shapes by dispersive of the source ef- regions. uation rates. Løvholt et al. (2008) started with the near-field fects,In the and offshore the subsequent direction evolution normal may thus to the be slower fault line (see the tsunami solution of Gisler et al. (2006), but treated the oceanic prop- discussion below Eq. 8). agation with a set of Boussinesq type equations. By this pro- signal must be expected to become completely transformed Later, after 9min the leading parts of the two solutions are In Fig. 15 we show the evolution of the second incident cedure they obtained wave heights at the American coast that1025 before reaching buoys or other continents. only slightly different along the transects outside Geiranger crest as it enters the continental shelf of North America. An were slightly smaller than those of Ward (2001), but still dan- undularOn bore the rapidly other evolves hand, and most the landslidefront of the crest induced is split tsunamis are (lower panelgerous. of figure Recently, 16).Abadie This etis al. surprising(2012) applied in view a similar of both strat- into a sequence of solitary waves. The heights of the individ- egy, with a slide volume of 500 km3, and obtained somewhat strongly affected by dispersive effects. For the leading part of 995 the dispersion time τ 0.4 and the large differences in the ualthe peaks signal, are close such to effects the stability are generally limit for solitary most waves, important during early stages.higher Presumably waves∼ close geometrical to the Canary Islands effects, than inLøvholt particular et al. and they will soon break during the following shoaling. At (2008). However, they did not compute the trans-Atlantic thiswave location generation the first incident and the crest early displays stages the highestof propagation, sur- while the fjord bifurcation,propagation. may The haveLa Palma selectively tsunami case transmitted is also studied longer by 1030 facethe elevation far-field (not properties shown), but may the presumably second crest has be higher different for sub- componentsZhou into et the al. ( branch2011). leading to Geiranger (see Harb- aerial (net volume in tsunami) and submarine (no net vol- itz, 1992a; Nachbin and da Silva Simoes, 2012). This feature ume) landslides. Extremely large landslides, moving at small 1000 awaits furtherwww.nat-hazards-earth-syst-sci.net/13/1507/2013/ analysis, but may anyhow indicate that the dis- Nat. Hazards Earth Syst. Sci., 13, 1507–1526, 2013 Froude numbers, such as the Storegga Slide is the likely ex- persion time must be used with care in complex situations. ception. The oceanic propagation of such events are virtually The grid resolution in the computations was 50m. 1035 non-dispersive. From the above discussion it is tempting to conclude that 6 Conclusions a reasonable hazard assessment in the near-field for tsunamis of seismic origin may be based on shallow water theory, In the cases investigated the value of the normalized disper- which is favorable with respect to real time tsunami computa- 1005 sion time, τ, is seen to correspond reasonably well with the1040 tions for warning purposes. However, undular bores, which apparent dispersive effects. As a rule of thumb we may say are not included in shallow water theory, may evolve dur- 16 S. Glimsdal et al.: Dispersion of tsunamis

Buoy 2 60˚ 2.0

1.0

[m] 0.0 η

45˚ −1.0 disp LSW 2 1 −2.0 6.2 6.4 6.6 6.8 7.0 7.2 7.4 3 30˚ time [hrs] Buoy 4 4 3.0 15˚ 2.0 1.0 0.0 [m]

η −1.0 0˚ −2.0 disp −3.0 LSW −4.0 −90˚ −75˚ −60˚ −45˚ −30˚ −15˚ 0˚ 5.2 5.4 5.6 5.8 6.0 6.2 6.4 time [hrs] 0.1 0.2 0.4 0.8 1.6 3.2 6.3 12.6 25.1 50.1 100.0

Fig. 14. The La Palma tsunami. Left panel: maximum wave elevation and position of buoys. Right panels: time series at two buoy locations. 1522S. Glimsdal et al.: S. Dispersion Glimsdal et of al.: tsunamis Dispersion of tsunamis 17

m the effect of dispersion is small for τ < 0.01, while it gen- 400 referee are thanked for their valuable comments on the manuscript. 0 1 erally becomes significant for τ > . , say. The form of τ 350 indicates6895 that the source width/initial wave length (for land- 300 1010 slides) is more important for the significance of dispersion than the depth or propagation distance. Accordingly, we find 250 1 200 References that6890 moderate magnitude earthquakes yield more dispersive 2 tsunamis than the huge ones, such as the 2004 Indian Ocean 150 UTM 32 [km] Abadie, S. M., Harris, J. C., Grilli, S. T., and Fabre, R.: Numerical and the 2011 Japan tsunami. 100 1055 modeling of tsunami waves generated by the flank collapse of Geiranger 1015 For6885 the largest earthquakes frequency dispersion only 50 the Cumbre Vieja Volcano (La Palma, Canary Islands): Tsunami modify the the trans-oceanicHellesylt propagation mildly. Hence, dis- 0 source and near field effects., J. Geophys. Res., 117, 26 pp, persion385 is not needed 390 for 395 propagation 400 in the 405 near-field, 410 but doi:10.1029/2011JC007646, 2012. UTM 32 [km] Fig. 15. Evolution of an undular bore from the second crest on the may be important if far-field tsunami data are used for verifi- Arcas, D. and Segur, H.: Seismically generated cation of source properties.Transect 1, t = 0h 1min continentalFig. 15. Evolution shelf of Northof an undular Carolina. bore Waves from propagatethe second to crest the on left. they 1060 tsunamis, Phil. Trans. R. Soc. A, 370, 15051542, iscontinental the surface shelf elevation of North (η), Carolina.x is the distance Waves propagate and the numbers to the left. in they1020 For80 the smaller earthquakes (Mw 8 or less) we observe a doi:10.1098/rsta.2011.0457, 2012. ∼ legendsis the surface correspond elevation to the (η propagation), x is the distance time in and minutes. the numbers in the strong directivity of the dispersion, following the amplitude Bardet, J.-P., Synolakis, C. E., Davies, H. L., Imamura, F., and Okal, 40 legends corresponds to the propagation time in minutes. directivity, due to the elongated shapes of the source regions. E. A.: Landslide Tsunamis: Recent Findings and Research Di- In the offshore direction normal to the fault line the tsunami rections, Pure Appl. Geophys, 160, 1793–1809, 2003. [m] 0 η signal must be expected to become completely transformed 1065 Bilham, R.: A Flying Start, Then a Slow Slip, Science, 308, 1126– effectiveLater, after wave9min heightthe due leading to the parts intermediate of the two trough. solutions Hence, are −40 1025 before reaching buoys or other continents. 1127, 2005. theonly leading slightly crest different undergoes along the the transformation transects outside to an Geiranger undular LSW On−80 the otherdisp hand, most landslide induced tsunamis are Blaser, L., Kruger,¨ F., Ohrnberger, M., and Scherbaum, F.: Scaling bore(lower later, panel when ofit figure is closer 16).to This the is coast. surprising To resolve in view the of undu- both strongly affected by dispersive effects. For the leading part of relations of earthquake source parameter estimates with special 995 larthe bore dispersion properly, time gridτ increments0.4 and the comparable large differences to thein depth, the 0 2 4 6 8 the signal, such effects are generally most important during focus on subduction environment, Bull. Seismol. Soc. Am., 100, or preferably smaller, must∼ be employed. Covering the con- distance along transect [km] early stages. Presumably geometrical effects, in particular 1070 2914–2926, 2010. tinental shelf with a resolution of, say, 20 m will be a compu- wave generation and the early stages of propagation, while the fjord bifurcation, may have selectively transmitted longer Transect 2, t = 0h 9min, Blikra, L.: The Aknes˚ rockslide; monitoring, threshold values and 1030 the far-field properties may presumablyτ be= 0.4 different for sub- tationalcomponents challenge. into the Moreover, branch leading when theto Geiranger crests start (see to Harb- break 15 aerial (net volume in tsunami) and submarine (no net vol- early-warning., in: Landslides and Engineered Slopes. From the weitz, still 1992a; need Nachbin a Boussinesq and da type Silva model Simoes, with 2012). breaking This features feature Past to the Future. Proceedings of the 10th International Sym- ume) landslides.10 Extremely large landslides, moving at small 1000 implementedawaits further (see analysis,Korycansky but may and anyhow Lynett indicate, 2005). that the dis- Froude numbers, such as the Storegga Slide is the likely ex- posium on Landslides and Engineered Slopes, 30 June - 4 July persion time must be used with care in complex situations. 5 1075 2008, Xi’an, China, edited by Chen, Z., Zhang, J.-M., Ho, K., ception. The oceanic propagation of such events are virtually [m]

5.5The Potentialgrid resolution rockslides in the computations in Norwegian was fjords,50m. example η Wu, F.-Q., and Li, Z.-K., Taylor and Francis, ISBN: 978-0-415- 1035 non-dispersive.0 from Akerneset˚ 41196-7, 2008. From the above discussion it is tempting to conclude that −5 LSW Blikra, L. H.: The Aknes˚ rockslide, Norway, in: Landslides: Types, a reasonable hazard assessment in the near-field for tsunamis A6 large Conclusions unstable rock volume has been identified in the disp Mechanisms and Modeling, edited by Clauge, J. and Stead, D., −10 of seismic origin may be based on shallow water theory, 1080 pp. 323–335, Cambridge University Press, 2012. Akerneset˚ rock slope in the narrow fjord, Storfjorden, Møre 0 2 4 In the cases investigated the value of the normalized disper- which is favorable with respect to real time tsunami computa- Bondevik, S., Svendsen, J., Johnsen, G., Mangerud, J., and Kaland, & Romsdal County, Western Norway, Fig. 16. The site has distance along transect [km] 1005 sion time, τ, is seen to correspond reasonably well with the1040 tions for warning purposes. However, undular bores, which been subject to extensive geological investigations (Blikra, P.: The Storegga tsunami along the Norwegian coast, its age and apparent dispersive effects. As a rule of thumb we may say are not included in shallow water theory, may evolve dur- runup, Boreas, pp. 29–53, 1997a. 2008, 2012) and the tsunami has been studied experimen- Fig. 16. Upper panel: Bathymetry of the inner part of the fjord sys- ˚ Bondevik, S., Svendsen, J., and Mangerud, J.: Tsunami sedimen- tally and numerically (NGI, 2010; Harbitz et al., 2012b). We Fig.tem. 16.AkernesetUpper panel: is marked Bathymetry with a large of the yellow inner bullet. part of The the numerical fjord sys- solutions˚ are evaluated along the two transects, 1 and 2. The depth 1085 tary facies deposited by the Storegga tsunami in shallow marine here focus on a comparison between the LSW and the dis- tem. Akerneset is marked with a large yellow bullet. The numerical is given in meters. Mid-panel: LSW and linear dispersive solutions basins and coastal lakenews, western Norway, Sedimentology, persive solutions for a volume of 54 × 106 m3 and an impact solutions are evaluated along the two transects 1 and 2. The depth isat given the generation in meters. area Mid (transect panel: LSW 1, surface and linear elevation). dispersive The solutions leftmost 44, 1115–1131, 1997b. velocity of 45 m s−1. In Fig. 16 we present surfaces along atwaves the generation are propagating area (transect to the left 1, (south) surface and elevation). the rightmost The leftmost waves Bondevik, S., Løvholt, F., Harbitz, C., Mangerud, J., Dawson, A., two transects: one through the generation area and one close wavesare propagating are propagating to the right to the (north). left (south) The center and linethe rightmost of the slide waves mo- and Svendsen, J.: The Storegga slide tsunami comparing field to the fjord head at Geiranger. The dispersion is crucial dur- aretion propagating is found at about to the 3 right.5 km. (north). Lower The panel: center solutions line of along the slide transect mo-1090 observations with numerical simulations, Ormen Lange Special ing the tsunami generation and the early phases of propa- tion2 outside is found Geiranger at about (surface3.5km. elevation). Lower panel: Solutions along transect Issue, Marine and Petroleum Geology, 22, 195–208, 2005. gation – see mid-panel of Fig. 16. This is also confirmed 2 outside Geiranger (surface elevation). Bryn, P., Berg, K., Forsberg, C., Solheim, A., and Kvalstad, through laboratory experiments in both 2-D (Sælevik et al., T.: Explaining the Storegga slide, Mar. Pet. Geol., 22, 11–19, 2009) and 3-D (NGI, 2010). The leading peak for the south- awaits further analysis, but may anyhow indicate that the dis- doi:10.1016/j.marpetgeo.2004.12.003, 2005. 1095 Bugge, T., Befring, S., Belderson, R., Eidvin, T., Jansen, going waves differ significantly in height (more than a fac- ingpersion shoaling. time must Even be though used suchwith borescare in may complex double situations. the wave E., Kenyon, N., and Sejrup, H. H. H.: A giant three- tor two), wavelength, and shape between the two solutions. heightThe grid locally, resolution their in effect the computations on inundation was are 50 more m. uncertain Later, after 9 min the leading parts of the two solutions are stage submarine slide off Norway, Geo-Mar. Lett., p. 191198, because the individual crests are short and may be strongly doi:10.1007/BF02242771, 1987. only slightly different along the transects outside Geiranger 1045 affected by dissipation due to wave breaking. Bugge, T., Belderson, R., and Kenyon, N.: The Storegga slide, (lower panel of Fig. 16). This is surprising in view of both 6 Conclusions 1100 Trans. R. Soc., A 325, 357–388, 1988. the dispersion time τ ∼ 0.4 and the large differences in the Acknowledgements. This work has been supported by the Norwe- Clarisse, J.-M., Newman, J. N., and Ursell, F.: Integrals with a large early stages. Presumably geometrical effects, in particular gianIn the Research cases Council investigated under projectthe value no. 205184. of the normalized The University dis- of parameter: water waves on finite depth due to an impulse, Pro- the fjord bifurcation, may have selectively transmitted longer Oslopersion (UiO), time, theτ Norwegian, is seen to Geotechnical correspond Institute reasonably (NGI), well and with the ceedings of the Royal Society Series A, 450, 67–87, 1995. components into the branch leading to Geiranger (see Harb- Internationalthe apparent Centre dispersive for Geohazards effects. As (ICG) a rule are also of thumb thanked we for may sup- Dawson, A., Long, D., and Smith, D.: The Storegga Slides: Ev- itz, 1992a; Nachbin and da Silva Simoes, 2012). This feature1050 portingsay the the effect work of on dispersion this manuscript. is small This for is contributionτ < 0.01, while no. 421 it 1105 idence from eastern Scotland for possible tsunami, Mar. Geol., of the ICG. 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