Hydrodynamics of tsunamis generated by asteroid impact in the Black Sea

Cent. Eur. J. Phys. • 10(2) • 2012 • 429-446DOI: 10.2478/s11534-012-0012-4

Central European Journal of Physics

Hydrodynamics of tsunamis generated by asteroidimpact in the Black Sea

Research Article

Dragos Isvoranu1, Viorel Badescu2∗

1 Faculty of Aerospace Engineering, Dept. of Aerospace Sciences ’Elie Carafoli’,University Politehnica of Bucharest, Str. Gh. Polizu Nr. 1, Bucharest 011061, Romania

2 Candida Oancea Institute, Polytechnic University of Bucharest,Spl. Independentei 313, Bucharest 060042, Romania

Received 8 September 2011; accepted 30 January 2012

Abstract: Two-dimensional and one-dimensional models are used to evaluate the seashore effects of the tsunamigenerated by an asteroid hitting the deep water in the Eastern region of the Black Sea. The shallow watertheory has been used to describe tsunami propagation. The distance between the impact point and thenearest coast is about 150 km. The effects on the coastal regions depend on many factors among whichthe most important is asteroid size. The tsunami generated by a 250 m asteroid reaches the nearest dryland location in 20 minutes and needs about two hours to hit all over the Black Sea coast. The horizontalinundation length is also known as run-in or run-off distance, according to the direction of water movement.The run-up values may be up to 39 m in the Eastern basin and a more than ten times smaller in the Westernregion. The Northern part of the Black Sea coast is not affected by the tsunami. The run-in values of atsunami generated by a 1000 m diameter asteroid are sensibly larger than the similar values associatedto a 250 m diameter asteroid. The run-in strongly depends on the distance from the impact position to theshore and on coastal topographical profile. For instance, the run-in distance in case of a tsunami generatedby a 250 m size asteroid is 0.1 km (at Varna), 0.5 km (Ordu), 0.7 km (Yalta) and 1.4 km (Sochi). In caseof the 1000 m diameter asteroid the run-in distance is 0.7 km (at Varna) and 2.9 km (Yalta). The resultsaccuracy is also discussed.

PACS (2008): 91.30.Nw Tsunamis; 92.10.Hm Ocean waves and oscillations

Keywords: asteroid impact • tsunamis • wave propagation • Black Sea© Versita Sp. z o.o.

1. Introduction

Tsunamis produced in association with earthquake activ-ity have been reported several times in the Black Sea∗E-mail: [email protected]

from the antiquity up to the present time [1–13]. Most ofthese observed tsunamis were initiated close to the coast-line. Earthquakes generated tsunamis in the Black Sea in1927, 1939 and 1966. Their maximum amplitudes reached53 cm, 53 cm and 41 cm, respectively [2]. Many studiesrefer to tsunami observed on the coasts of the CrimeanPeninsula, the Caucasus, the northern beaches of Turkeyand Bulgarian coast. The Black Sea’s sea floor topog-429

Hydrodynamics of tsunamis generated by asteroid impact in the Black Sea

raphy, particularly the extent of shallow sea-bottom, isa major factor for this noteworthy geographical distribu-tion [3]; the topographic capture of tsunami energy by theBlack Sea shelf accounts for this phenomenon. Accord-ing to historical data the maximum heights of Black Seatsunamis are not remarkably impressive [4]. The largesttsunami in the 20th century was observed in 1968 withmax. wave height of c. 3 m [5]. The most comprehensivecritical review of tsunami phenomena in the Black Seawas published recently [5].Asteroid impacts in the Black Sea are apparently un-recorded by historians [1–4]. The first study concerningthis potential event is [40]. Note that two different phe-nomena might start simultaneously when an asteroid hitsthis sea. First, the initial water cavity constitutes thesource for a tsunami wave. The second phenomenon isspecific to the Black Sea. Hydrogen sulfide is releasedinto the atmosphere and creates a gaseous ’cloud’ on thesea surface. This H2S cloud moves and disperses in themean atmospheric wind field. The two phenomena havetheir own dynamics and their effects on the coastal re-gions are different. The dynamics of the poisonous gasblanket has been discussed [14]. The present paper isconcerned with the tsunami hydrodynamics.Tsunami generation and propagation significantly de-pends on local bathymetry. In the Black Sea the southernregion is deeper than the northern one. Tsunamis withdifferent characteristics are expected to occur in the tworegions due to different initial water cavity features (seeSection 4). Also, the social effects of these catastrophicevents depend on seashore population distribution, whichis not uniform.Results concerning the run-up heights, the horizontal in-undation length (also known as the run-in distance [15]),as well as the simulated wave profiles across realisticbeaches profiles, are not very often reported in literature.Most papers referred to various mathematical models ofwave-wave interaction in shallow water, wave run-up onsloping beaches, conical obstacles or vertical cylinders,respectively [16–25]. Some of these papers compare ex-perimental setup results with results obtained from numer-ical simulation for validation purposes. However, after thegreat Sumatra earthquake of 26 December 2004 a numberof papers focused on realistic simulations of the run-ups ofearthquake generated tsunamis [2, 26–28, 30]. Other ar-ticles have been devoted to the simulation of run-ups dueto landslides [31–36]. There is a limited number of papersdealing with numerical simulation of run-ups induced byasteroid impact tsunamis [37–39]. All these works refer toasteroids hitting open seas and oceans.In a previous paper [40] we reported preliminary resultsabout the seashore effects of a 70 m size asteroid hitting

the Black Sea in the proximity of the Crimea peninsula. Acrude estimate of the initial water cavity shape has beenused and a simple 1D model has been adopted there todescribe the tsunami’s propagation. In the present work amuch more detailed analysis is performed for an asteroidhitting the deep sea waters of the Eastern Black Sea. Im-proved 2D and 1D models and bathymetry data are usedto simulate both tsunami propagation and its effect on avariety of realistic beaches. Different asteroid sizes areconsidered. Also, the associated run-in distance valuesare analyzed in several areas of the eastern Black Seacoasts and the differences due to bathymetry and topog-raphy are discussed. The tsunami effects in the westernbasin of Black Sea were studied in [41]. The results re-ported here refer to impact generated tsunamis in a con-fined water body, where refraction effects are not neg-ligible (see [6, 7]), and extend existing knowledge abouttsunamis in open seas and oceans.2. Information about Black SeaThe Black Sea is an oval-shaped body of seawater situ-ated between 40.55◦N and 46.32◦N latitude and between27.27◦E and 41.42◦E longitude; it has a coastline of 4090km (≈ 1.4% of the world’s coastline) and a maximum waterdepth of 2,200 m.The distinguishing physical peculiarity of the Black Seais the presence of a ”permanent” halocline situated at adepth of 100-200 m, a stratification generated mostly byfreshwater inputs from rivers flowing into the Black Sea.Of its total volume of water-approximately 547000 km3-some 475000 km3, or 87%, is anoxic, without dissolvedoxygen and therefore lifeless except for anaerobic bacte-ria. The Black Sea is Earth’s biggest single reservoir ofhydrogen sulfide (H2S), an oceanographic fact only dis-covered circa 1891. H2S is generated by bacterial re-duction of sulfate both in the water column and in theHolocene seafloor sediments deposited since the connec-tion of the Black Sea with the Mediterranean Sea.3. Frequency of asteroid impacts inBlack SeaRough estimates give more than 30 Near-Earth objects(NEOs) larger than 5 km in diameter, 1500 NEOs largerthan 1 km and 135000 larger than 100 m [42]. About halfof NEOs are crossing the Earth orbit and there is a chanceto collide with the Earth in near or far future. The mainsources of NEOs are the asteroid belt and the Edgeworth-Kuiper Belt (EKB). Other objects crossing the Earth orbit

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Dragos Isvoranu, Viorel Badescu

are long-period comets coming from the Oort cloud. Icybodies can also migrate inside the Solar System from theregions located between the EKB and the Oort cloud.Almost all of what is known about the potential envi-ronmental and societal consequences of asteroid impactson Earth has been obtained from numerical simulations1.Some conclusions were also derived (for smaller impacts)from extrapolations of nuclear weapons tests [44] and (forlarger impacts) from inferences from the geological recordfor the Cretaceous /Tertiary (K/T) impact about 65 Myrago. The environmental consequences from asteroid im-pacts are usually classified in three size ranges [45]: (i) re-gional disasters due to impacts of multi-hundred meter ob-jects; (ii) civilization-ending impacts by multi-km objectsand (iii) K/T-like cataclysms that yield mass extinctions.Recently, a number of researchers were arguing a forthsize range should be added, namely (iv) multi-ten me-ters impactors like Tunguska (see [46, 47] and referencestherein).There is considerable uncertainty about the environmen-tal consequences of larger impacts. It is expected thatthey have diverse physical, chemical, and biological con-sequences, which dominate the Earth ecosphere in waysthat are difficult to imagine and model. Atmospheric per-turbations due to dust and aerosols lofted by impacts aresome of these effects that have been studied by usingglobal circulation and climate models. The ”asteroidalwinter” may be a consequence, deriving from a strong in-jection of dust in the atmosphere [48]. Impacts may alsoinduce chemical changes in the atmosphere, mainly by in-jection of sulphur into the stratosphere. These are relatedto the vaporization of both the impactor and a part of thetarget. Large impact events may inject enough sulphurto produce a reduction in temperature of several degreesand a major climatic shift [37]. Additional effects on atmo-spheric chemistry are the potential for the destruction ofthe ozone layer from shock heating atmospheric nitrogenand the injection of fluorides from the vaporized hittingbody [47]. The greatest danger from smaller impacts aretsunamis, which transfer the effects of a localized oceanimpact into dangerous waves on distant shores [49]. Thecurrent philosophy of impact hazard considers the dan-ger from small asteroids as negligible. However, severalfacts claim for a revision of this philosophy. The impactorsin category (iv) may have major local consequences nearground zero. Also, they could generate social effects, po-litical ramifications and fallout from the public. In fact,there are several scientists suggesting that small asteroids1 www.boulder.swri.edu/clark/neowp.html

might be even more dangerous than larger bodies [47].The impact is a random process in geographical spacebut also in time. Estimates of such impact rates basedon the number of asteroids and dynamical considerationsare rather uncertain, so it may be more robust to deter-mine them from the historical impact records [50]. On theother hand, the latter method suffers greatly from the smallnumber statistics and unknown sample completeness. Inpractice, evaluation of impact frequency is made by usingempirical or semi-empirical formulas. The scarce existingdata yield often contradictory results.A rough estimate for the impact frequency as function ofimpactor size is [43]: (i) multi-hundred meter objects hitEarth every 104 years (ii) multi-km object impacts occur ona million-year timescale; (iii) K/T-like cataclysms occur ona 100 Myr timescale. Also, tens of meters impactors col-lide with the Earth on timescales comparable to or shorterthan a human lifetime [40].A simple way to evaluate the frequency of asteroid im-pacts in the Black Sea is to multiply the above estima-tions (in years between successive collisions with sim-ilar size objects) by the ratio between the Earth andBlack Sea surface (which is 1208, for an Earth surfaceof 510,065,600 km2) [51]. However, the possible impactorsare grouped by families according to their origin, as shownbefore. One may speculate rather similar dynamical andtrajectory properties for objects of the same family andthis may decrease the randomness degree of impact spa-tial distribution. As an example, the impact crater distri-bution on Europe’s surface shows a larger number of im-pacts around Baltic Sea and (interestingly) in the northof the Black Sea2. But this interpretation must be takenwith caution because most of the terrestrial impact cratershave been obliterated by other terrestrial geological pro-cesses [53].4. Effects of asteroids hitting theBlack Sea

Very accurate numerical simulations provided valuable in-formation about the interaction between larger size aster-oids and the atmosphere, the seawater, and the underwa-ter medium [54]. Usually, less than 0.01 of the impactor’skinetic energy is dissipated during the atmospheric pas-sage. The remaining part of the kinetic energy is absorbedby the ocean and sea floor within less than one second.The water immediately surrounding the impactor is va-2 http://www.unb.ca/passc/ImpactDatabase/europe.html

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porized, and the rapid expansion of the vapor excavatesa cavity in the water. This cavity is asymmetric in caseof oblique incidence angles, and the splash, or crown, ishigher on the side opposite the incoming trajectory. Thecollapse of the crown creates a precursor tsunami thatpropagates outward. The higher part of the crown breaksup into droplets that fall back into the water. The hotvapor from the cavity expands into the atmosphere. Whenthe vapor pressure diminishes enough, water begins tofill almost symmetrically the cavity from the bottom. Thefilling water converges on the center of the cavity andgenerates a jet that rises vertically in the atmosphere toa height comparable with the initial cavity diameter. It isthe collapse of this central vertical jet that produces theprincipal tsunami.Modeling the initial water displacement by asteroid im-pact in a water body is a daunting task and various com-putational set-up scenarios are described by using shockphysics theory [54–57]. All of them predict water distur-bances of a characteristic length scale comparable withwater depth at impact point.Due to complex water movement at impact source, theusual approach consists in designing an equivalent wa-ter cavity as in modeling waves generated by underwa-ter explosions [58] or explosions of underwater volcanoes[59]. The following relation between the radius Rc and thedepth Dc of the water cavity has been suggested [60]:Dc = qRα

c , (1)where q and α are parameters depending on asteroidproperties.Central symmetry is assumed at the initial mo-ment t = 0 for the equivalent water cavity at the impactpoint. Then, the parabolic shape of the water displace-ment at distance r from the impact point is given by [61]:

η (r, t = 0) = { Dc

(1− r2Rc

) if r ≤ RD0 otherwise , (2)where η is the water displacement relative to the unper-turbed sea level and RD is the radius of the undisturbedsea level around the impact point. The energy ET of thewater cavity formation process is defined by

ET ≡ρwg2

∫γη (r, t = 0)2dr, (3)

where ρw (∼= 1.027 kg m−3) is the seawater density andg (∼= 9.81 m/s2) is the gravitational acceleration. If RD =√2Rc , usage of Eqs. (2) and (3) yields:

ET = ρwgπ3 (RcDc)2 , (4)

It is assumed that only a fraction ε of asteroid kinetic en-ergy Ei is transformed into the energy of the water cavityformation process, i.e.Ei = εEi = ε 2ρiπ3 (

R3i V 2

i), (5)

where ρi,Vi , and Ri are the density, velocity and radiusof the impactor, respectively. Usage of Eqs. (4) and (5)yields the depth Dc of the water cavity:Dc =

D′c =√ 2ρiεR3i V

2i

ρwgR2c for D′c ≤ |h||h| for D′c > |h| , (6)

where |h| is the modulus of the water depth at the impactlocation. The assumption that the depth of the water cav-ity cannot exceed the water depth |h| has been adoptedin Eq. (6). From Eqs. (1) and (6) one gets the radius Rcof the water cavity:Rc = Ri

(2εV 2i

gRi

)δ ( ρiρw

)δ ( 1qRα−1

i

)2δ, (7)

with δ = 0.5/(1 + α). Laboratory measurements suggestthat the following value may be adopted: α = 1.27 [49,61]. Figure 1 shows experimental results reported in [49]concerning the dependence of the q coefficient enteringEq. (1) on the impactor radius Ri.

Figure 1. Dependence of the q coefficient entering Eq. (1) on im-pactor radius Ri. Experimental results of [49] are shownas well as those obtained by using the best fit curve (seeEq. (8) ).

We have fitted several relationships to the data in Fig. 1.The following power law is a best fit:q = 0.2839R−0.2671

i , (8)(root mean square error RMSE=0.0006924, squared co-efficient of correlation R2=0.9991). Equation (8) is almostsimilar to the regression relationship proposed in [15].

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Figure 2. Dependence of the cavity depth Dc and cavity radius Rc ,respectively, on the impactor radius Ri, as derived fromEqs. (7) and (8).

Figure 2 shows the dependence of the water depth cav-ity Dc and radius cavity Rc , respectively, on the aster-oid radius Ri, as derived from Eqs. (7) and (8). A valueε = 0.155 has been adopted there [49, 61]. Both thedepth cavity and the radius cavity increase by increasingthe asteroid size, as expected. However, the water cavitydepth cannot exceed the sea depth (see Eq. (6)).Some consequences of an asteroid hitting the Black Seawere described in [14] and [62]. Only results concerningthe dynamics of the H2S cloud were analyzed in detail inthose papers. Here we focus on tsunami generation andpropagation and its effects on the seashore.5. Tsunami hydrodynamicsDifferent theories are used to model the hydrodynamics ofimpact-generated tsunamis. Some of these theories andtheir estimated accuracy are briefly described in the Ap-pendix A. Here we are using a model for tsunami genera-tion and propagation based on shallow water theory.The main criterion for applying the shallow water theoryis λ � |h|, where λ is the wavelength and |h| is the wa-ter depth. In deep waters, the usual wave length rangesbetween 10 and 25 km. At impact location, |h| = 2122 m(see Section 6). The tsunami source Eq. (2) employedhere comes from linear dispersive wave theory and pack-ets of waves would be expected to be generated later on,too. The shallow waters equations (SWEs) act over theinitial cavity like a high-pass filter such that only themost energetic long wave (of wavelength L) is capturedand propagated towards the shore [37]. Following [15] and[49], L ≈ 2.12Rc . For asteroids of diameter 70 m, 250 m,and 1000 m this yields L ≈ 2700, 7250, and 21500 m,respectively. The shallow water theory criterion is ratherwell fulfilled for 1000 m diameter asteroids. In the case

of 70 m and 250 m asteroids, the accuracy of the shallowwater theory is worse (see Appendix A for more details).Nonetheless, shortly after impact the wave length growsrapidly from the initial size no matter how large the as-teroid is. Computations show that 60 s after impact thelongest wavelengths range from 7 to 15 km for asteroidsof 70 m and 250 m, respectively. Hence, the SWEs maybe used to model with reasonable accuracy tsunami wavepropagation, especially when interested on the shore ef-fects.The implementation of the shallow water theory in case ofthe Black Sea is presented next. Modified Navier-Stokesequations including bottom friction effects are used. Thedispersion term and the Coriolis effect are neglected dueto the relatively small size of Black Sea. Then, the waveequations in spherical coordinates describing the temporaland spatial evolution of the sea level are [63, 64]:∂η∂t + 1

R cosφ[∂M∂λ + ∂

∂φ (N cosφ)] = 0∂M∂t + 1

R cosφ ∂∂λ

[M2D

]+ 1R

∂∂φ[MN

D]+ gD

R cosφ ∂η∂λ + τbλ

ρ = 0∂N∂t + 1

R cosφ ∂∂λ[MN

D]+ 1

R∂∂φ

[N2D

]+ gDR cosφ ∂η

∂φ + τbφρ = 0,(9)where D = h+η is the total water depth, R is the averageEarth radius, λ is the longitude, and φ is the latitude. Mand N are the depth averaged water discharges in thelongitude and latitude directions, respectively, given by:

M = ∫ η−h νλdr = ν̄λDN = ∫ η−h νφdr = ν̄φD

, (10)where νλ and νφ are appropriate sea water velocities stem-ming from the spherical Navier-Stokes equations. The lastterms in Eqs. (9) are given by:

τbλ = f MD2√M2 +N2τbφ = f N

D2√M2 +N2 . (11)They represent bottom friction terms which become impor-tant in shallow waters. The friction coefficient f enteringEqs. (11) can be computed from the Manning’s roughnesscoefficient n:

n =√ fD1/32g , (12)such that Eqs. (11) become [63]:

τbλ = 2gMD7/3√M2 +N2

τbφ = 2gND7/3√M2 +N2 , (13)

Typical values are adopted in calculations for the Manningcoefficient n (see Section 5.2).433


5.1. Numerical approachThe system of partial differential nonlinear hyperbolicnonconservative (9) is solved using TsunamiClaw code em-bodying many features and subroutines from Clawpackpackage3. The system of PDE’s is cast into a conser-vative form with source term in order to use conservativenumerical method based on the integral form of the SWEsand appropriate up-winding near shocks like that providedby solving Riemann problems. An extensive explanationof these features and numerical approach can be found in[65]. The novel augmented Riemann solver for SWEs andtsunami simulation described in [66] is based on a finitevolume technique that allows modeling the global prop-agation regime and wet-dry interfaces accurately, mak-ing it appropriate for inundation modeling and shorelinecapturing. Further, the solver is well suited for shock-capturing; in fact it is equivalent to the Roe solver nearshocks [67]. Lastly, the solver does not require an entropyfix such as the Roe solver. The source terms are evaluatedthrough a 2-stage Runge-Kutta method. The shorelinediscontinuities are capture rather than explicitly tracked,similar in philosophy to the shock-capturing property ofGodunov-type methods, where shocks are captured auto-matically by solving Riemann problems appropriately nearthese features [68]. As the tsunami enters shallow waters,the wave is compressed and focused leading to violentflow characteristics, usually shock-waves. The nonlinear-ities in the shallow water equations dominate in this flowregime, which can lead to breaking waves and turbulentbores. There are two issues at hand here: first, for mod-elers, the issue of computing a true shock-wave solutionto the shallow water equations is a difficult task, and sec-ond, it is not clear how well a shock-wave solution to theSWEs represent the physical turbulent bore or breakingwave. The resolution of the latter issue seems to be thatshock waves represent turbulent bores surprisingly well,if one considers only the macroscopic properties of thebore, such as when it forms, its propagation speed, andhow it affects inundation. Certainly the SWEs do not de-scribe the flow features right at the edge of the turbulentregion of the bore. However, if the bore is representedmathematically as a discontinuity, the SWEs with appro-priate entropy conditions provide the correct conservationprinciples for hydrostatic flow surrounding the bore. Thiswork is more concerned with the first issue, which is nu-merically converging to correct discontinuous solutions ofthe SWEs. Regardless of the crudeness of the shock-waveapproximation to a true physical bore, certainly a correctsolution to the SWEs is a better approximation to the borethan a spurious one.3 http://www.amath.washington.edu/∼claw/clawpack.org/

Note that the Riemann solutions become unphysical atwet/dry interfaces where solutions of Rankine-Hugoniotjump conditions are affected by vanishing water depth ateither left or right states. However, in [66] it has beenshown that Riemann solutions comprise one rarefactionwave, a middle dry state, and the shock wave is nonex-istent. The rarefaction wave is associated with eitherthe first or second characteristic eigenvalue dependingon which side of the wet/dry interface the water depthvanishes.Free boundary conditions (zero order extrapolation) areused when solving the SWEs such that to capture flood-ing phenomena on the shores. The computational gridcontains both water and land domain and is spaced in E-W and N-S directions in equal steps of hλ = R cosφ∆λin longitude and hφ = R∆φ in latitude.5.2. General features of tsunami claw

The code allows modeling tsunamis and land inundationon either spherical or Cartesian coordinates with a diverserange of temporal and spatial scales. This is accomplishedby using up to two coarse level grids for entire domain andevolving rectangular sub-grids of higher refinement levelthat track moving waves and flooding around shoreline[69, 70]. The first two coarse grids track water displace-ment and wave propagation in deep waters (water depthsgreater than the pre-established value of shallow waterdepth, usually 100 m). Imposing shallow water depthone can indicate which areas are to be refined close tocoastal lines. Finer grids capture the wave behavior inshallow waters and wave run-up on shore. All grids areconstructed by interpolating geometric data directly fromthe bathymetry file as long as the characteristic cell sizeis larger than the resolution of the bathymetry file. Inshallow waters and on shores, the combined bathymetry-topography file is itself interpolated in order to providethe necessary geometric information for finer grids. Forexample, suppose that the rectangular domain coveringBlack Sea area, ranging from 27.30◦E to 42.00◦E andfrom 40.50◦N to 47.00◦N, is discretized at the coarsestlevel on a 420 × 190 grid that ensures a resolution forthe deep waters of 3.5 × 3.5 km. Next, suppose that oncertain beach areas we want to track flooding on a ≈100× 100 m grid. We can specify the number of refined gridsand refinement ratios in order to achieve our goal. Forexample, the grid refinement in either direction is 1:2 onthe second level, 1:6 for the third and 1:30 for the fourth,compared to the coarsest first level. At any given timein the computation, a particular level of refinement mayhave numerous disjoint grids associated with it that trackspecific areas of interest.

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One of the benefits of using finite volume methods for theintegral form of the SWEs, is that the formal assump-tions still hold when applied to regions surrounding wetand dry interfaces. Therefore, we can avoid special dis-cretization and specialized tracking or ad-hoc treatmentsof the shorelines. In fact, we can accurately model inunda-tion by simply solving Riemann problems between wet anddry cells-using the exact same formalism as for the rest ofthe domain. The approach we take in this situation is tosolve a preliminary Riemann problem, where we replacethe values in the dry cell with ghost cell values-valuesused simply to determine the Riemann solution. Con-sider Dleft > Dright (here D denotes water column height),but assume that Riemann velocity solution may have anyvalue. We first solve the homogeneous Riemann problemwith ghost cell values in the right cell that approximate awall boundary condition. The solution to such a Riemannproblem is easy to determine, since it must be symmet-rical about wet-dry interface. Clearly, depending on thesign of uleft, then we have either a double rarefaction,with D∗ < Dleft or a double shock, with shocks propa-gating away at the same speeds with opposite signs, withD∗ > Dleft. Here the quantity D∗ represents water columnheight associated with constant middle state of the Rie-mann problem. Due to the symmetry of either case, theoutput generated consists of only two waves with equalstrength moving to the left and to the right. We then es-timate D∗, either by checking our approximate Riemannsolution, or by approximating the root of the Riemann fluxfunction associated. Since we are solving a homogeneousproblem, D∗ can be easily and confidently determined. Wethen determine if D∗ is large enough to inundate the ’wall’,i.e. the jump in bathymetry. That is if D∗ < hright, thenno inundation would occur if the jump in bathymetry wasactually a wall. In this case, the solution to the ghost cellproblem serves to update the solution vector [η,M,N ] , byusing the leftward moving waves from the Riemann solu-tion. However, if D∗ > hright then the jump in bathymetrydoes not form a large enough barrier to prevent inunda-tion. In this case, the ghost problem is discarded, andthe actual Riemann problem is solved with the actual val-ues from the right cell. This ensures that the full sourceterm is realized when the inundating water has enoughmomentum to overcome the entire slope. Friction may beimportant for realistic run-up heights and a typical valuefor the Manning coefficient is n = 0.025.The original code was tuned for tectonically inducedtsunamis whose initial water level perturbation is gen-erated by a vertical displacement of the sea floor. Herewe have developed the code to use the profile of the wa-ter cavity generated through the asteroid impact as initialwave source. This is a major difference taking into ac-

count the large scale separation between the characteris-tic length of the water domain and that of the perturbationsource domain. Actually, from this point of view, the per-turbation induced by the asteroid impact can be regardedas a point source. In order to obtain reliable simulationresults, we treated the initial perturbation domain on athird or fourth level of grid refinement just like any otherbeach specific area. Hence, we were able to capture eventhe shore effects of a 70 m asteroid impact.6. Results and discussionsA combined bathymetry and topography data file in stan-dard GIS format for the Black Sea4 is used throughout.The data has been obtained from satellite altimetry andship depth soundings. Data resolution is 0.01×0.0114degrees (which is equivalent to ≈ 1134 × 935 m) (Fig. 3).

Figure 3. Black Sea bathymetry (m). The asteroid impact positionis denoted I.

The impact position considered in this paper has latitude43◦N and longitude 38◦E (Fig. 3) while the sea depththere is h = −2122 m. This impact position correspondsto the impact point B3 in Ref. [62]. The sea floor bed isflat around the impact point for a radius of 125 km. Theshortest distance between impact location and the coastis about 150 km.The study of tsunami propagation provides clues regard-ing major flooding risk for the nearest beach areas. Amongthem, we point out, Yalta’s region in Ukraine, Sochiand Novorossiysk surrounding areas on Russian coast-line, Sokhumi and Batumi on Georgian shore, and Trab-zon, Ordu, and Samsun on the Turkey’s shore (Fig. 4).Information about these localities is given in Table 1.Impacts by asteroids of class (i), (ii) and (iv) will be consid-ered next. The asteroid material is assumed to be dunite(mass density ρi = 3000 kg/m3) as a mock-up for typicalstony asteroids. The impact velocity is Vi = 20000 m/s.4 http://topex.ucsd.edu/marine_topo/mar_topo.html

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Table 1. Information about main localities on Black Sea coast (seeFig. 4).

Locality Latitude (◦N) Longitude (◦E)Varna 43.20 27.55Burgas 42.50 27.47Constanta 44.18 28.65Sevastopol 44.60 33.53Yalta 44.48 34.17Kerch 45.36 36.48Sochi 43.60 39.73Novorossiysk 44.57 38.02Batumi 41.39 41.44Sokhumi 43.00 41.01Trabzon 41.05 39.72Samsun 41.28 33.33Zonguldak 41.15 31.24Ordu 40.59 37.53

Figure 4. Black Sea and main localities on the shore.

The following constant values are used during the calcu-lations: α = 1.27, ε = 0.155.6.1. Impacts by class (i) asteroidsAn asteroid of diameter Di = 250 m is considered first.Taking into account Eq. (7) and the second option of Eq.(6) we obtain the water cavity’s depth Di = |h| = 2122 mand cavity’s radius Rc = 3410 m. The peak spectral com-ponent of the generated wave occurs at 7162 m wavelength(that is, 1.05 × water cavity diameter [37]). The ratio |h| /Lbetween the sea depth and peak tsunami’s wavelength is2122/7162 = 1/3.37. This wave is not a shallow waterwave (see Appendix A). The accuracy of the results re-ported next are expected to be similar to that describedin Appendix A for |h| /L > 1.4The salient features of the tsunami propagation over thewhole Black Sea are presented in Fig. 5. Complex pattern

of waves stands out mostly due to the almost confinedcharacter of Black Sea that generates wave interference.At the impact position the water is lifted to about 2000 mabove the sea level (Fig. 5a). Due to the large ratio be-tween the initial water cavity diameter and cavity depth,no breaking wave is apparent near the source. The firstimportant wave (10 to 15 m height) reaches the Russiancoast in about 20 minutes after asteroid impact (Fig. 5b).The Northern coast of Turkey is reached between Sam-sun and Ordu a few minutes later. The 12 to 18 m heighttsunami needs 40 minutes to impact the whole coastlineof Eastern region of Black Sea (Fig. 5c) except the East-ern extremity, which is reached by a wave less than 1.5 mheight one hour after impact (Fig. 5d). Most part of theBlack Sea is affected by the tsunami in about two hoursafter impact but the wave height is small, oscillating be-tween -0.25 and 0.75 m (Figs. 5e and 5f , respectively).The Northern part of the Black Sea coast is not affectedby this catastrophic event. Note the importance of theCrimean coasts orientation (Fig. 3). Indeed, the Easterncoast is affected by a tsunami wave of about 18 m height(Fig. 5c) while the Western coast is reached by wavesless than 1 m in height (Figs. 5e and 5f). We remind thatSWEs do not take into account dispersion, and thereforeshort wavelength waves are neglected. The waves rep-resented in Fig. 5 have long wavelength. Especially atshore and during the run-in process this feature is mostprominent.It is known that the wave height increases near the shore,due to the interaction between the wave and the smalldepth sea floor. This interaction causes the most devas-tating effects on the coastline. In order to accommodatethe most important features of the tsunami impact on thecoastal regions we have performed detailed simulationsfor the Eastern and Western basins of Black Sea, withemphasis on flooding risk beaches. High coverage coarsegrids (level 1 and 2) have been adopted within the numer-ical procedure for deep sea, trying to find a compromisebetween reasonable accuracy and reasonable computingtime. Finer grids (level 3) were used in the domains withsea depth smaller than 100 m. Run-up effects have beenobtained by using the finest grids (level 4) covering beachareas of interest. 1D slices through the computational do-main in these areas allow the best estimates of tsunamieffects on beaches. These 1D wave profiles through the 2Ddata arrays are obtained by interpolation. Near the shore,the tsunami wave lays very well within the assumptionsof the shallow waters theory, which predicts non-breakingwaves behavior.The initial sea level perturbation is obtained after inter-polation procedures within the impact area (Fig. 6). Itshows a peak of about 2100 m. The water cavity is as big436


Figure 5. Time snapshots for the propagation of a tsunami generated by a 250 m diameter asteroid. (a) the asteroid impact position. (b)-(f)tsunami propagation at various times measured from the impact moment. The units on the map are: degree for latitude and longitude,and second for time. Legend’s left and right: negative and positive water vertical displacements, respectively (meter).

Figure 6. Half of the initial profile of water displacement at the impactlocation of a 250 m diameter asteroid.

as 1800 m in depth. Investigations have focused on Yaltaarea in Crimea (see Table 1). In this case the coarsest gridhad a number of 420 x 190 cells while further grids wererefined in ratios of 2, 3 and 5, respectively. The finestgrid covered the flooding risk beach. The beach topog-raphy profile on the azimuth line connecting the asteroidimpact point and Yalta is illustrated in Fig. 7. The wetand dry terrain raises almost linearly more than 250 mover 5 km distance.Figure 8 shows a slice through the propagating wave at2400 s after impact along the azimuth line associated toFig. 7. The four different curves in Fig. 8a illustrate thesolution of the PDEs system (9) corresponding to differentrefinement grid levels. Grid level 1 and 2 apply to deepseawaters while grid levels 3 and 4 apply to the beach on437


Figure 7. Bathymetry and topography profile near Yalta’s seashore.

Figure 8. (a) Tsunami wave profile at 2400 s after the impact of a250 m diameter asteroid, for different grid refinement lev-els. Grid levels 1 and 2 apply to deep waters while gridlevels 3 and 4 apply to the beach on Yalta’s area. (b) Wa-ter depth along distance.

Yalta’s area. Note that the shore line is at X = 347 km(see Fig. 7 and Fig. 8b). All grids show that before reach-ing the shore the propagating wave has a 20 m high crest,a 15 m trough and about 11.5 km wavelength. In theseashore area the results depend significantly on the gridlevel. The coarser grids 1 and 2 estimate run-ups of 20 mto 35 m, respectively, while the more finer grids 3 and4 predict more realistic run-ups of 10 m and 2 m, re-spectively. Note that lower level grids are used only forextrapolation to the higher level grids, which are of rel-evance near the seashore. All next slices are associatedwith grid level 4.At 2520 s from the asteroid impact, the main tsunami wavereaches the Yalta’s shore-line (Fig. 9a). It is about 39 m inheight. There is a water cavity behind the wave more thanone km long and about 18 m maximum deep. The steepbeach gradient diminishes the run-in distance, which isestimated to only 700 m. Eighty seconds later, waterrecedes for more than 1.5 km from initial shore line (seethe run-off distance on Fig. 9b). The beach sweepingceases only one and a half hours after the impact moment.

Figure 9. (a) Run-up height on Yalta’s shore at t=2520 s after theimpact of a 250 m diameter asteroid. The inland extent ofwater flooding is comprised between the shoreline and X-coordinate given in the right text box. (b) Run-off heightat time = 2680 s. Water recedes from shoreline to X-coordinate given in the bottom text box.

In between, beside the snapshots already presented, thereare numerous series of waves hitting the coastline of lessimportant amplitudes.Figure 10 shows the water wave profile when hitting theseashore at Sochi and Ordu and the maximum run-in dis-tances on dry land. Located on the North-East shore ofthe Black Sea, Sochi is the first important locality in Fig. 4where the tsunami arrives, at about 23 minutes after theasteroid impact (Fig. 10a). The tsunami is about 13 mhigh and the water cavity behind is more than 2 km largeand about 20 m deep. The maximum run-in distance isaround 1.4 km (Fig. 10b). Ordu is located on the South-East coastline of the Black Sea (Fig. 4) and the tsunamitravels about 27 minutes to arrive there (Fig. 10c), i.e. 4minutes later than at Sochi. The wave height at shoreline is lower at Ordu but still comparable with the waveheight at Sochi (compare Figs. 10c and 10a, respec-tively). The maximum run-in distance at Ordu is about0.5 km (Fig. 10d). This rather small value is due to thesteep beach profile, similar to that on Yalta’s seashore (seeFig. 7). The run-in distance in Ordu and Yalta is similarin size (compare Figs. 10d and 9a, respectively).Figure 11 shows the rather small effect that the tsunami438


Figure 10. Tsunami wave profile generated by the impact of a 250 msize asteroid, when arriving at the seashore at (a) Sochiand (c) Ordu. The maximum run-up distance on the dryland is also shown: (b) Sochi and (d) Ordu.

wave may have in the Western basin of the Black Sea.Varna is an important port on the Western coast (Table 1).The tsunami arrives at Varna a bit earlier than two hoursafter the asteroid impact (Fig. 11a) and the maximum waveheight at shore is 1 m. The maximum run-in distance on

Figure 11. (a) Maximum run-up distance on dry land of a tsunamigenerated by the impact of a 250 m size asteroid. (b)Profile of the bottom sea and the dry land near Varna onthe direction of tsunami propagation.

the dry land is about 0.1 km (Fig. 11a). This small valueis due to the profile of the bottom of the sea and the dryland near Varna in the direction of tsunami propagation(Fig. 11b). There is a hilly area of maximum height 37 mlocated on dry land about 3 km from the coastline whichputs an upper limit to the run-in distance.6.2. Impacts by class (ii) asteroidsAn asteroid of diameter Di = 1000 m is considered now.In this case the water cavity radius is Rc = 10122 mand the computed water cavity depth is Dc = 6693 m.According to the second option in Eq. (6) the actual cav-ity depth is equal to the sea depth, i.e. Dc = 2122 m.The peak spectral component of the generated wave oc-curs at 21256 m wavelength. The ratio |h|/L between thesea depth and peak tsunami’s wavelength is 2122/21256= 1/10.01. This wave is rather close to a shallow waterwave (for which |h|/L < 1/20; see Appendix A). Conse-quently, the accuracy of the next results is expected to besignificantly better than that described in Appendix A for|h|/L > 1.4.Figure 12 provides a broad image about the tsunami prop-agation in the whole Black Sea area. At the impact posi-tion the water is lifted about 2000 m above the sea level

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Figure 12. Same as Fig. 5 for a 1000 m diameter asteroid.

(Fig. 12a). The first important wave (about 80 m high)reaches the Russian coast in 20 minutes after asteroidimpact (Fig. 12b). The wave is about seven times higherthan in case of a 250 m diameter asteroid hitting the samelocation (compare Figs. 12b and 5b, respectively). TheNorthern coast of Turkey is reached between Samsun andOrdu a few minutes later. The tsunami wave needs 40minutes to reach all the coastlines of Eastern Black Sea(Fig. 12c). The next 20 minutes wave propagation de-creases in speed and one hour after the impact the waveentered one third of the Western basin (Fig. 12d). Also,the maximum wave height decreased to about 40 m and thenegative heights show interferences with waves returningfrom the shores. Most part of the Western Black Sea isaffected by the tsunami in about two hours (Figs. 12e and12f). The maximum wave height ranges between 10 m and

15 m while the minimum oscillates between -10 m and -11m (Figs. 12e and 12f), showing interference processes.Figures 5 and 12 allow a comparison between thetsunamis generated by asteroids of various diameters. Theextent of the perturbed sea level area depend significantlyon the diameter of the asteroid (compare Figs. 5f and 12f).Refined grids were used to evaluate the effect of thetsunami generated by the 1000 m asteroid on particularlocations on the seashore. Figure 13 shows the maximumrun-in distance on the dry land at the seashore at Ordu,Yalta and Varna (for location see Fig. 4 while for seadepth see Fig. 3). The tsunami arrives at Ordu in about27 minutes after the asteroid impact. The wave crest isabout 20 m high and has behind a 9 km long trough of43 m depth. The maximum run-in distance is around 3 km(Fig. 13a). Compare these values with the tsunami gen-440


Figure 13. The maximum run-up distance on the dry land generatedby the impact of a 1000 m size asteroid. (a) Ordu, (b)Yalta, (c) Varna.

erated by a 250 m size asteroid, whose height on Ordu’sbeach is about 8 m and has a run-in distance of 0.5 km(see Fig. 10d). At Yalta the tsunami arrives in 41 min-utes (Fig. 13b). Its height is much higher than at Ordu(≈ 90 m) while the run-in distance is about 2.9 km. Com-pare these values with the tsunami generated by a 250 msize asteroid, whose height on Yalta’s beach is 39 m andhas a run-in distance of 0.7 km (Fig. 8a). The tsunami ar-rives at Varna in about 115 minutes and has a much lowerheight than at Ordu and Yalta (i.e. 8 m), as expected (Fig.13c). Of course, the run-in distance is larger than in caseof the tsunami generated by a 250 m asteroid (i.e. about0.7 km) (see Fig. 13c).

6.3. Impacts by class (iv) asteroidsAn asteroid of diameter Di = 70 m is considered here.In this case the water cavity depth is Dc = 971 m andthe water cavity radius is Rc = 1272 m. The peak spec-tral component of the generated wave occurs at 2671 mwavelength. The ratio |h|/L between the sea depth andpeak tsunami’s wavelength is 2122/2671 = 1/1.25. Thiswave is not a shallow water wave (see Appendix A). Con-sequently, the results accuracy is expected to be similarto that described in Appendix A for |h|/L = 1.The raw combined bathymetry-topography file has a res-olution of 1134 × 935 m. Near the shore the compu-tational grid has been refined to a range varying from1:80 to 1:100. The same proportion is maintained for thebathymetry-topography file through appropriate interpo-lation. This makes possible capturing the shore effect ofthe impact by the 70 m asteroid at 150 km away from theimpact position.Figure 14 provides a broad image about the tsunami prop-agation in the Eastern basin of the Black Sea. At theimpact position the water is lifted to nearly 1000 m abovethe sea level (Fig. 14a). Ten minutes after impact the frontwave has a height of about 0.5 m (Fig. 14b). The tsunamireaches the North-East shore of Black Sea in about 20minutes, between Sochi and Novorossyisk (Fig. 14c). Itsheight is less than 0.5 m. After 30 minutes the tsunamiaffected almost all the coastlines in the Eastern basin ofBlack Sea but the wave height has considerably dimin-ished, being of the order of 0.3 m (Fig. 14d). The run-ineffect is negligible.7. ConclusionsThe effects of a tsunami generated by an asteroid hittingthe Eastern region of the Black Sea were evaluated inthis paper. Models were presented for both the impactof the asteroid with the seawater and tsunami propaga-tion. Implementation of these models was based on accu-rate bathymetry and topography data provide by satellitemeasurements. The asteroid impact location adopted inthis study is in the deep Eastern Black Sea region. Theshortest distance between impact location and the coastis about 150 km. Three different asteroid sizes were con-sidered. The effects of the impact-generated tsunami werestudied on those sites on the coast where the most impor-tant cities are placed. Also, the effects on the furthestWestern and Northern Black Sea shores were described.The results accuracy is limited by the accuracy of themodel we used to describe the initial tsunami source atthe impact location, by the accuracy of the shallow watertheory we used here and by the accuracy of the near-

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Figure 14. Time snapshots for the propagation of a tsunami gener-ated by a 70 m diameter asteroid. (a) the asteroid impactposition. (b)-(d) tsunami propagation at various timesmeasured from the impact moment. The units on themap are: degree for latitude and longitude, and secondfor time. Legend’s left and right: negative and positivewater vertical displacements, respectively (meter).

shore bathymetry and topography. Shallow water theoryoverestimate wave amplitudes by factors ranging between1 and 3.75 (see Appendix A) and this should be taken intoaccount when reading the values reported in this paper,including this section.The wave generated by a 250 m asteroid reaches the near-est dry land location in 20 minutes and needs about twohours to cover the whole Black Sea coast. In the EasternBlack Sea basin the wave height may be up to 39 m. Inthe Western Black Sea regions the wave height decreasessignificantly, being more than ten times smaller in someplaces. There are, however, important differences betweenthe maximum values of the wave crest height at differentsites, depending on local bathymetry and topography. TheNorthern part of the Black Sea coast is not affected bythis catastrophic event.The effects of a tsunami generated by an 1000 m diameterasteroid are larger on the Eastern than on the Westerncoasts. The run-up values are a few times larger than thesimilar values associated to a 250 m diameter asteroid,depending on location.An asteroid of diameter Di = 70 m has been consideredtoo. The water cavity generated by the asteroid impact(1272 m radius and 971 m depth) is much smaller thanthat of the 250 m size asteroid (3410 m and 2122 m, re-spectively) and that of the 1000 m size asteroid (10122 mand 2122 m, respectively). The consequence is a muchsmaller run-up effect.All results show that large value of the tsunami’s run-indistance is a pre-requisite of devastating effects. However,these effects are proportional to the inland extent of theflooding waters, which, in turn, is strongly dependent onthe distance from the impact position to the shore andon the coastal topographical profile. For instance, therun-in distance in case of a tsunami generated by a 250m size asteroid is 0.1 km (at Varna), 0.5 km (Ordu), 0.7km (Yalta) and 1.4 km (Sochi). In case of the 1000 mdiameter asteroid the run-in distance is 0.7 km (at Varna)and 2.9 km (Yalta). The run-in distance is negligible forthe 70 m size asteroid.The main novelty of these results is that they refer totsunami effects generated by asteroids impacting a con-fined body of water. The refraction effects on coasts aretaken into consideration. Most previous studies refer toopen seas and oceans where such refraction effects arenegligible. These results may be of interest for local au-thorities when emergency plans are prepared.

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AcknowledgmentsThe authors thank the referees for useful comments andsuggestions. Dr. Alina Mihaela Badescu helped with La-TeX editing.Appendix AThe initial conditions for any sort of tsunami calculationsare a topic of debate because the choice may substan-tially alter wave height predictions. The initial stagesof tsunami generated by asteroid impacts are particu-larly difficult to model. They might come from detailed,non-linear simulations of hydrodynamic shocks. For ex-ample, the interaction of a typical stony asteroid (densityof 3320 kg/m3 and velocity of 20 km/s) with the atmo-sphere and a 5 km deep ocean with a basalt bottom hasbeen modeled using the CTH computer code for multidi-mensional, multi-material, large deformation, strong shockwave physics [56]. Alternatively, initial conditions mightcome from experiment and observation in the form of em-pirical scaling laws. Various scenarios were proposed todescribe the water disturbance (the tsunami source) due toan asteroid impacting the ocean (see [61] and referencestherein). All of these models predict large values (compa-rable with the ocean depth) of water disturbances and acharacteristic length scale (source radius) comparable tothe asteroid diameter. Two extreme cases were consideredin [37]. In the first case all of the water within the cavityejects incoherently into space and does not contribute tothe tsunami. In the second case, all of the water within thecavity deposits coherently into a bordering lip and doescontribute to the tsunami. Based on energy arguments, in[49] it has been shown that the water cavity idealizationadopted in the last case, have solid footing. This secondcase of [37] was our choice in [72] and in this work.A realistic tsunami propagation theory is fully 3-D andneither depth-averaged nor restricted to long or shortwaves [37]. However, such a complete theory is rarelyused in practice due to limitations in available computingcapacity and time resources. Depending on specific fea-tures two approximate approaches are usually adopted,namely the linear wave theory and the shallow water the-ory (which is a simplified 2D non-linear theory), respec-tively. A few details follow.Linear theory

Due to wave dispersion, the wave height is attenuatedand far from the source the wave can be considered aslinear [61]. In linear wave theory one distinguishes be-

tween long waves and short waves. A simple classifica-tion is sometimes made as a function of wave wavelengthL and modulus of water depth |h|: long waves correspondto |h|/L < 1/20 (and are usually associated to shallowwater) while short waves correspond to |h|/L > 1/2 (andare associated to deep water)5 [74]. Wavelengths at peaktsunami amplitude correspond closely with 1.05 times thediameter of the impact cavity [37]. If the water cavity ra-dius is smaller than (or comparable to) the water depth,the deep water approximation can be applied [61]. Forsuch sources the dispersion effects become important. Im-pact tsunamis have many spectral holes. Tsunamis dis-perse normally with longer waves traveling faster thanshorter waves. Waves longer than the cavity diameter ar-rive before the peak; shorter waves arrive after [49].Shallow water theory

The crucial simplifying assumption for shallow water the-ory is that the pressure is strictly hydrostatic [66]. This isthe case for long and shallow waves (i.e. waves with wave-length much larger than water depth), in which the verti-cal acceleration of fluid elements during the wave passagestays small [73]. Note that, no assumption is made aboutthe amplitude and the smoothness of shallow water wavesrelative to the depth all such nonlinearities are retainedin shallow water theory [66] (p. 37-3). The shallow waterequations are mostly used for tsunamis induced by under-water earthquakes, when the vertical displacement of thewater column is the main tsunamigenic mechanism andtsunami waves begin as long waves [61], [66] (p. 52). Theshallow water theory has also been used in the case ofimpact induced tsunamis (see, e.g. [75–77] among others)and was our choice in [72] and in this work.The largest terms neglected in the shallow water equa-tions are dispersive terms [66] (p. 58). During the passageof high waves, significant vertical accelerations can occur,leading to deviation from the hydrostatic pressure condi-tions. The erroneous consequence is that high waves com-pared to shallow ones are slower and subject to larger dif-fusion [73]. For situations which do not deviate too muchfrom hydrostatic conditions, the classical shallow waterequations can be generalized [66] (p. 59).Accuracy of approximate theories

Impact tsunami simulations by using 3D Navier-Stokesequations require impressive computing resources. A few5 http://www.ifu.ethz.ch/GWH/education/graduate/Hydraulik_II/Vorlesungen/index/k8_EN.pdf

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such results were reported in [56] and allow a brief dis-cussion about the expected accuracy of both linear andshallow water theories when used to simulate impact gen-erated tsunamis.The linear theory was used in [37] to study a hypotheticaltsunami generated by the 1.1 km - size asteroid 1950 DAimpacting a 5 km deep ocean 600 km east of the UnitedStates coast. The quoted authors state that their modelpredict a 7 m height wave at 1000 km distance when gen-erated from a 250 m impactor. The same authors state thatthis result is about 10 times less than the result previouslyderived in [78] but much larger than the result obtained bysolving 3D Navier-Stokes equations in [56]. Since resultsin [37] do not show what ’much larger’ means, we shallspecify that Table 1 of [56] gives an amplitude of 8 m at70 km distance from the impact point of the 250 m asteroidin a 5 km deep ocean. That table does not contain resultsfor distances larger than 70 km. Reference [56] states thatat 1000 km distance the amplitude ’is negligible’ (indeed,extrapolation of data in Table 1 of [56] at 1000 km dis-tance gives an amplitude of about 0.1 m. Consequently, inthis particular case, the approximate linear theory overes-timates about 70 times the result predicted by the ’exactmodel’ based on solving the full 3D- Navier-Stokes equa-tions).Tables 1 and 2 of [56] allow a comparison between resultspredicted by the ’exact model’ and the shallow-water the-ory. These tables report the amplitude of the wave atvarious distances from the impact place for asteroids ofsize 250 m, 500 m and 1000 m, respectively. A |h| = 5 kmdeep ocean is always considered. Table 1 in [56] containsvalues of the amplitude ηNS computed by using the ZUNIcode which solves the incompressible 3D-Navier-Stokesequations. A 250 meter asteroid would result in less thana 10 meter high tsunami after 60 km of travel in a 5 kmdeep ocean. The tsunami generated by a 1 km diameterasteroid would run 80 km before the tsunami wave am-plitude was less than 100 meters and 500 km before itwas less than 10 meters. The tsunami period, wavelength,and velocity increases with run distance while the am-plitude decreases. Table 2 in [56] contains values of theamplitude ηSW computed by using the approximate shal-low water theory. Computations were performed usingthe SWAN non-linear shallow water code which includesCoriolis and frictional effects [75]. Again, the diameter ofthe initial water cavity is reported for all the three im-pactors. Since the cavity diameter is nearly equal to thepeak tsunami’s wavelength L, one may assign values forthe ratio |h|/L for all cases reported in Tables 1 and 2of [56]. Note that both tables contain two sorts of data:(i) data obtained by direct numerical integration and (ii)data obtained from extrapolation of numerical results.

Figure 15. Ratio ηSW /ηNS as a function of the distance from the im-pact place. ηSW and ηNS is the peak wave amplitudeobtained by using the shallow water theory and the 3D-Navier-Stokes equations, respectively. Data providedby Tables 1 and 2 of [56] are used. These data con-sists of: (i) data obtained by direct numerical integrationand (ii) data obtained from extrapolation of numerical re-sults. |h|/L is the ratio between the water depth and wavewavelength, respectively.

Figure 15 shows the ratio ηEW /ηNS as derived from thedata in Tables 1 and 2 of [56]. The tsunami wave ampli-tudes and velocities within the ’exact model’ are smallerthan the shallow-water wave values. The ratio ηEW /ηNSis less than 5 for all cases described in Fig. 15. Whenonly data reported for direct numerical integration in Ta-bles 1 and 2 of [56] are used, the ratio ηEW /ηNS is lessthan 3.75. The ratio ηEW /ηNS is less than 2.75 when dataassociated to |h|/L less than 1/2 but higher than 1/4 areconsidered. This agrees with [79] and [56], where it is hasbeen stated that with increasing periods and wavelengths,the discrepancy between shallow water theory and 3D-Navier-Stokes theory decreases. Figure 15 shows thatresults obtained under the hypothesis of shallow watertheory are upper limit values and as much as twice toolarge at long distances of run. The uncertainty factor de-creases in case of |h|/L values smaller than 1/4.References

[1] A. A. Nikonov, Izvest Phys. Sol. Earth 33, 77 (1997)[2] S. F. Dotsenko, A.V. Ingerov, Phys. Ocean 17, 269(2007)[3] E. I. Ignatov, The Black Sea, (Springer, Berlin, 2008),47-62[4] A. Yalciner et al., J. Geophys. Res. 109, C12023 (2004)[5] G. A. Papadopoulos, G. Diakogianni, A. Fokaefs, B.Ranguelov, Nat. Hazards Earth Syst. Sci. 11, 945(2011)[6] I. Vilibic, J. Sepic, B. Ranguelov, N. Streleh-Mahovic,444


S. Tinti, J. Geophys. Res. 115, C07006 (2010)[7] B. Ranguelov, S. Tinti, G. Pagnoni, R. Tonini, F. Zani-boni, A. Armigliato, Geophys. Res. Lett. 35, L18613(2008)[8] B. Ranguelov, Tsunami investigations for the BlackSea - empirical relationships and practical applica-tions regarding the tsunami zoning, Proc.IW TsunamiRisk Assessment., June 14-16, 2001, Moscow, Rus-sian Federation, 44[9] B. Ranguelov, Tsunami repeatability for the BlackSea., Proc. Intl. Conf. ”Port - Coast - Environment”(PCE’97)., June 30 - July 4, 1997, Varna, vol. 2, 293[10] B. Ranguelov, Compt. Rend. Acad. Sci. 50, 47 (1997)[11] B. Ranguelov, Seismicity and Tsunamis in the BlackSea, in Seismology in Europe. Papers presented inthe XXV Gen. Ass. ESC, Sept. 9-14, 1996, Reykjavik,667[12] B. Ranguelov, Natural Hazards - nonlinearities andassessment (Acad. Publ. House (BAS), Sofia, 2011)327[13] B. Ranguelov, Atlas of the tsunami risk susceptibleareas along the Northern Bulgarian Black Sea coast- Balchik site (Hristo Terziev, Sofia, 2010) 25[14] V. Badescu, Earth Interactions 11, 1 (2007)[15] B. Levin, M. Nosov, Physics of Tsunamis, (Springer,Berlin, 2009)[16] D. R. Fuhrman, P. A. Madsen, Coastal Engineering 55,139 (2008)[17] A. I. Delis, M. Kazolea, N. A. Kampanis, Int. J. Numer-ical Methods in Fluids 56, 419 (2007)[18] D. H. Kim, Y. S. Cho, H. J. Kim, J. Engng. Mech. 134,277 (2008)[19] E. Pelinovsky, E. Troshina, V. Golinko, N. Osipenko,N. Petrukhin, Physics and Chemistry of the Earth,Part B: Hydrology, Oceans and Atmosphere 24, 431(1999)[20] V. Golinko, N. Osipenko, E. N. Pelinovsky, N. Zahibo,Int. J. Fluid Mech. Res. 33, 106 (2006)[21] S. T. Grilli, J. Harris, N. Greene, In: J. Mc Kee Smith(Ed.) Proc. 31st Intl. Coastal Engng. Conf. ICCE08,Hamburg, Germany, September, 2008, (World Scien-tific Publishing Co., Singapore, 2009) 1638[22] H. Tarman, U. Kanoglu, In: Long Waves Symposium(in parallel with the XXX IAHR Congress) August 25-27, 2003, Thessaloniki, Greece[23] E. van Groesen, D. Adytia, Andonowati, Nat. HazardsEarth Syst. Sci. 8, 175 (2008)[24] A. Constantin, R. S.Johnson, J. Nonlin. Math. Phys.15, Supplement 2, 58 (2008)[25] S. R. Massel, E.N. Pelinovsky, Oceanologia 43, 61(2001)[26] S. N. Ward, S. Day, Comm. Comp. Physics 3, 222

(2008)[27] T. S.Yean, K. H. Lye, Md. A. I., Ismail, In: Proc. of the2nd IMT-GT Conference on Mathematics, Statisticsand Applications, University Sains Penang, June 13-15, 2006, Malaysia[28] C. Chinnarasri, W.Pratoomchai, In: Proc. of theEighth ISOPE Pacific/Asia Offshore Mechanics Sym-posium Bangkok, November 10-14, 2008, Thailand[29] D. Park, Y.-S. Cho, S.-M. Kim, J. Coastal Res. 50,1168 (2007)[30] E. A. Okal, H. Hebert, Geophys. J. Int. 169, 1229(2007)[31] R. Weiss, H. M. Fritz, K. Wünnemann, Geophys. Res.Lett. 36, L09602 (2009)[32] S. Abadie et al., In: J. Mc Kee Smith (Ed.) Proc. 31stIntl. Coastal Engng. Conf. ICCE08, Hamburg, Ger-many, September, 2008, (World Scientific PublishingCo., Singapore, 2009) 1, 384[33] F. Enet, S. Grilli, In: 5th Internatl. Symp. WAVES2005, Madrid, Spain, 2005, paper 88[34] I. V.Fine, A. B.Rabinovich, B. D. Bornhold, R. E. Thom-son, E. A. Kulikov, The Marine Geology 215, 45(2005)[35] P. Heinrich, J. of Waterway, Port, Coastal and OceanEngineering, ASCE 118, 249 (1992)[36] G. A. Papadopoulos et al., Marine Geodesy 30, 315(2007)[37] S. N. Ward, E. Asphaug, Geophys. J. Int. 153, F 6(2003)[38] R. Weiss, K. Wünnemann, H. Bahlburg, Geophys. J.Int. 167, 77 (2006)[39] D. G. Korycansky, P. J. Lynett, Geophys. J. Int. 170,1076 (2007)[40] R. D.Schuiling, R. B. Cathcart, V. Badescu, D. Isvo-ranu, E. Pelinovsky, Nat. Hazards 40, 327 (2007)[41] V. Badescu, D. Isvoranu, Pure and Applied Geo-physics 168, 1813 (2011)[42] D. Rabinowitz, E.Helin, K.Lawrence, S.Pravdo, Na-ture 403, 165 (2000)[43] C. R. Chapman, D.D. Durda, R. E. Gold, TheComet/Asteroid Impact Hazard: A Systems Approach,Office of Space Studies, Southwest Research Insti-tute, Boulder CO 80302, Feb. 24, 2001[44] S. Glasstone, P. J. Dolan, The Effects of NuclearWeapons, 3rd edition (U.S. Government Printing Of-fice, Washington DC, US, 1977)[45] O. B. Toon, K.Zahnle, D. Morrison, R. P. Turco, C.Covey, Rev. Geophys. 35, 41 (1997)[46] D. Jewitt, Nature 403, 145 (2000)[47] L. Foschini, Astron. Astrophys. 342, L1 (1999)[48] C. S. Cockell, M. D. Stokes , Climatic Change 41, 151(1999)445


[49] S. N. Ward, E. Asphaug, Icarus 145, 64 (2000)[50] Z. Ivezic, M. Juric, R. H. Lupton, S. Tabachnik,T. Quinn, Astron. J. 122, 2749 (2001)[51] M. P. Paine, Sci Tsunami Hazards 17, 155 (1999)[52] Earth Impact Database, Planetary and Space Sci-ence Centre, University of New Brunswick, Canada[53] R. A. F. Grieve, Scientific American 262, 66 (1990)[54] G. Gisler, R. Weaver, C. Mader, M. L. Gittings, SciTsunami Hazards 21, 119 (2003)[55] V. Shuvalov, H. Dypvik, F. Tsikalas, J. Geophys. Res.107, E7 (2002)[56] D. A. Crawford, C. L. Mader, Sci. Tsunami Hazards 16,21 (1998)[57] C. L. Malder, M.Gittings, Sci. Tsunami Hazards 21, 91(2003)[58] B. Le Mehaute, S. Wang, Water waves Generated byunderwater Explosion (World Scientific, Singapore,1996)[59] N. R. Mirchina, E. N. Pelinovsky, Nat. Hazards 1, 277(1988)[60] S. Ward, E. Aspaugh, Deep-Sea Res. Pt. II, 49, 1073(2002)[61] C. Kharif, E. Pelinovsky, C. R. Physique 6, 361 (2005)[62] V. Badescu, Environmental Toxicology 22, 510 (2007)[63] M. H. Dao, P. Tkalich, Nat. Hazards Earth Syst. Sci.7, 741 (2007)[64] Z. Kowalik, W. Knight, T. Logan, P. Whitmore, Sci.Tsunamis Hazards 23, 40 (2005)[65] R. J. LeVeque, Finite Volume Methods for Hyperbolic

Problems (Cambridge University Press, Cambridge,2002)[66] D. L. George, Ph.D. Thesis, University of Washington(Washington, USA, 2006)[67] P. L. Roe, J. Comput. Physics 43, 357 (1981)[68] J. L. Stegger, R. F. Warming, J. Comput. Physics 40,263 (1981)[69] M. J. Berger, R. J. LeVeque, SIAM J. Numerical Anal.35, 2298 (1998)[70] M. J. Berger, I. Rigoutsos, IEEE Trans. Syst. Man Cy-bern. 21, 1278 (1991)[71] W. H. F. Smith, D. T. Sandwell, Science 277, 1956(1997)[72] D. Isvoranu, V. Badescu, Journal of Cosmology 2, 419(2009)[73] C. Beffa, 2D-Shallow Water Equations Basics - So-lutions - Applications, Numerical Hydraulics, 2008[74] J. A. Liggett, Fluid Mechanics (McGraw-Hill Int. Edi-tions, Columbus, OH, USA, 1994)[75] C. L. Mader, Sci Tsunami Hazards 16, 11 (1998)[76] C. L. Mader, Sci Tsunami Hazards 16, 17 (1998)[77] J. Ormo, H.Miyamoto, Deep-Sea Res Part II 49, 983(2002)[78] J. G. Hills, I. V. Nemtchinov, S. P. Popov, A. V. Teterev,In: T. Gehrels (Ed.) Hazards due to Comets and As-teroids (Univ. Arizona Press, Tucson 1994) 779[79] C. L. Mader, Sci. Tsunami Hazards 11, 93 (1993)

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