A more realistic assessment of beach effects of 2880 March 16 asteroid impact tsunami

DRAGOS ISVORANU1 and VIOREL BADESCU2

1 Dept. of Aerospace Sciences, 2 Dept. of Thermodynamics University Politehnica of Bucharest

1, Ghe. Polizu Str., Bucharest 011061 ROMANIA

ddisvoranu@gmail.com http://www.aero.pub.ro

Abstract: - Eight years ago, some scientists computed for a 1.1-km diameter asteroid (1950 DA) a 0.0 to 0.3 per cent probability for colliding with Earth. Not long after this, a quasi-analytic model for wave propagation was used to describe the dynamics of the tsunami generated by the asteroid impact somewhere at 600 km east of the U.S. coast. Starting from the same initial conditions, our paper resumes the former calculus in the outfit of numerical simulation. To everybody's relief, the catastrophic outcome of the previous computation (beach waves well over 100 m) has been reconsidered and sensible lower values for run-up heights are predicted. Similar to former computation it takes 3 hours for waves to make landfall from Cape Cod to Cape Hatteras. Due to lower topography of the East coast, run-up distances are of the order of 20-30 Km. Key-Words: - asteroid impact, Atlantic Ocean, tsunami, numerical modeling, coastal zones, risks

1 Introduction Rough estimates give more than 30 Near-Earth objects (NEOs) larger than 5 km in diameter, 1500 NEOs larger than 1 km and 135000 larger than 100 m [1]. About half of NEOs are Earth-crossers and there is a chance to collide the Earth in near or far future. The main sources of NEOs are the asteroid belt and the Edgeworth–Kuiper Belt (EKB). Other Earth-crossers are long-period comets coming from the Oort cloud. Icy bodies can also migrate inside the Solar System from the regions located between the EKB and the Oort cloud. Almost all of what is known about the potential environmental and societal consequences of asteroids impact on Earth has been obtained from numerical simulations [2]. Some conclusions were also derived (for smaller impacts) from extrapolations of nuclear weapons tests [3] and (for larger impacts) from inferences from the geological record for the Cretaceous /Tertiary (K/T) impact about 65 Myr ago. The environmental consequences from asteroid impacts are usually classified in three size ranges [4]: (i) regional disasters due to impacts of multi-hundred meter objects; (ii) civilization-ending impacts by multi-km objects and (iii) K/T-like cataclysms that yield mass extinctions. Recently, a number of researchers are arguing a forth size range should be added, namely (iv) multi-ten meters impactors like Tunguska (see [5], [6] and reference therein). There is considerable uncertainty about the environmental consequences of larger impacts (categories (i) and (ii)). It is expected that they have

diverse physical, chemical, and biological consequences, which dominate the Earth ecosphere in ways that are difficult to imagine and model. Atmospheric perturbations due to dust and aerosols lofted by impacts are some of these effects that have been studied by using global circulation and climate models. The “asteroidal winter” may be a consequence, deriving from a strong injection of dust in the atmosphere [7]. Impacts may also induce chemical changes in the atmosphere, mainly by injection of sulphur into the stratosphere. These are related to the vaporization of both the impactor and a part of the target. Large impact events may inject enough sulphur to produce a reduction in temperature of several degrees and a major climatic shift [6]. Additional effects on atmospheric chemistry are the potential for the destruction of the ozone layer from shock heating atmospheric nitrogen and the injection of fluorides from the vaporized impacting body [6]. The greatest danger from smaller (category (iii)) impacts are tsunamis, which transfer the effects of a localized ocean impact into dangerous, breaking "tidal waves" on distant shores [8]. The current philosophy of impact hazard considers the danger from small asteroids is negligible. However, several facts claim for a revision of this philosophy. The impactors in category (iv) may have major local consequences near ground zero. Also, they could generate social effects, political ramifications and fallout from the public. In fact, there are several scientists suggesting that small asteroids might be even more dangerous than larger bodies [6].

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The impact is a random process in geographical space but also in time. Estimates of such impact rates based on the number of asteroids and dynamical considerations are rather uncertain, so it may be more robust to determine them from the historical impact records [9]. On the other hand, the latter method suffers greatly from the small number statistics and unknown sample completeness. In practice, evaluation of impact frequency is made by using empirical or semi-empirical formulas. The scarce existing data yield often contradictory results. A rough estimate for the impact frequency as function of impactor size is given in [2]: (i) multi-hundred meter objects impact Earth every 104 years (ii) multi-km objects impacts occur on a million-year timescale; (iii) K/T-like cataclysms occur on a 100 Myr timescale. Also, tens of meters impactors collide with the Earth on timescales comparable to or shorter than a human lifetime. A simple way to evaluate the frequency of asteroid impacts in a specific zone is to multiply the estimations above (in years between successive collisions with similar size objects) by the ratio between the Earth and corresponding area surface [10]. However, the possible impactors are grouped by families according to their origin, as shown before. One may speculate rather similar dynamical and trajectory properties for objects of the same family and this may decrease the randomness degree of impact spatial distribution. As an example, the impact crater distribution on Europe surface shows a larger number of impacts around Baltic Sea and (interestingly) in the north of the Black Sea. But this interpretation must be taken with caution because most of the terrestrial impact craters have been obliterated by other terrestrial geological processes [11].

2 Effects of an Asteroid Impacting the Ocean Surface Very accurate numerical simulations provided valuable information about the interaction between larger size asteroids and the atmosphere, the seawater and the underwater medium [12]. A brief review of these findings follows. Usually, less than 0.01 of the impactor’s kinetic energy is dissipated during the atmospheric passage. The remaining part of the kinetic energy is absorbed by the ocean and seafloor within less than one second. The water immediately surrounding the impactor is vaporized, and the rapid expansion of the vapor excavates a cavity in the water. This cavity is asymmetric in case of oblique incidence angles and the splash, or crown, is higher on the side opposite the incoming trajectory. The collapse of the crown creates a precursor tsunami that propagates outward. The higher

part of the crown breaks up into droplets that fall back into the water. The hot vapor from the cavity expands into the atmosphere. When the vapor pressure diminishes enough, water begins to fill almost symmetrically the cavity from the bottom. The filling water converges on the center of the cavity and generates a jet that rises vertically in the atmosphere to a height comparable with the initial cavity diameter. It is the collapse of this central vertical jet that produces the principal tsunami. Modeling the initial water displacement by asteroid impact in a water body is a daunting task and various computational set-up scenarios are described in [12], [13], [14] and [15]. All of them predict water disturbances of a characteristic length scale comparable with water depth at impact point. Due to complex water movement at impact source, the usual approach consists in designing an equivalent water cavity as in modeling waves generated by underwater explosions [16] or explosions of underwater volcanoes [17]. Ward and Aspaugh [18] suggested a relation between the radius cR and depth cD of the cavity of the form

αcc qRD = (1)

where q and α are parameters depending on asteroid properties. It is assumed that only a fraction ε of asteroid kinetic energy is consumed in the cavity formation process, hence the depth of the cavity is given by

≤

=

h

hDgR

VRD c

cw

iii

c

,2

2

23

ρερ

(2)

with iρ , iV , iR are the density, velocity and radius of

the impactor, respectively,wρ is the seawater density, h is water depth and g is the gravitational acceleration.

From Eqs. (1) and (2) one gets the diameter cd of the cavity

δ

α

δδ

ρρε

2

1

2 122

= −

iw

i

i

iic

qRgR

VRd (3)

with ( )αδ += 15.0 . From laboratory investigations 27.1=α . Assuming central symmetry for the equivalent

water cavity at impact point and initial moment 0=t , the parabolic shape of the water displacement is given in [19]:

( )

>

≤

−

==

D

Dc

c

Rr

RrR

rD

tr

,0

,10, 2

2

η (4)

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with cD RR 2= and η is the water displacement relative to the un-deformed sea level. This result implies that all water deposits on the border lip of the cavity.

3 Tsunami Dynamics In order to describe the tsunami wave propagation, modified Navier-Stokes equations, including bottom friction effects, are used. Neglecting the Coriolis effect due to relatively small size of the computational domain and the dispersion term, the wave equations in spherical coordinates describing the temporal and spatial evolution of the sea level are (see [20] and [21]):

( ) 0coscos

1 =

∂∂+

∂∂+

∂∂ φ

φλφη

NM

Rt (5)

0cos

1

cos

1 2

=+∂∂+

∂∂+

+

∂∂+

∂∂

ρτ

λη

φφ

λφ

λb

R

gD

D

MN

R

D

M

Rt

M

(6)

0cos

1

cos

1

2

=+∂∂+

∂∂+

+

∂∂+

∂∂

ρτ

φη

φφ

λφ

φb

R

gD

D

N

R

D

MN

Rt

N

(7)

where η+= hD is the total water depth, R is the average Earth radius, λ is the longitude and φ is the latitude. M and N are the depth averaged water discharges in the longitude and latitude directions.

DvdrvMh

λ

η

λ == ∫−

(8)

DvdrvNh

φ

η

φ == ∫−

(9)

where λv and φv are appropriate sea water velocities

stemming from the spherical Navier-Stokes equations. The last terms in Eqs. (6-7) are given by:

222 NM

D

Mfb +=λτ (10)

222

NMD

Nfb +=φτ (11)

They represent bottom friction terms which become important in shallow waters. The friction coefficient f entering Eqs. (10-11) is computed from Manning’s roughness coefficient

g

fDn

2

3/1

= (12)

such that Eqs. (10-11) become [20]:

223/7

2NM

D

gMb +=λτ (13)

223/7

2NM

D

gNb +=φτ (14)

Typical values for the Manning coefficient n adopted in calculations are presented in Section 3.2. 3.1 Numerical Approach The system of partial differential nonlinear hyperbolic non-conservative equations is solved using TsunamiClaw code embodying many features and subroutines from Clawpack package [22]. An extensive explanation of its features and numerical approach can be found in [23]. The code is based on a finite difference technique using second-order Godunov flux-splitting scheme [24, Roe’s approximate Riemann solver with entropy fix [25] for convective terms and 2-stage Runge Kutta method for evaluating source terms. The boundary conditions of the computational domain are free boundary conditions (zero order extrapolation) such that to capture flooding phenomena around shore lines. The computational grid containing both water and land domain and is equally spaced in E-W and N-S directions in equal steps of λφλ ∆= cosRh in longitude and

φφ ∆= Rh in latitude.

3.2 General Features of TsunamiClaw The code allows modeling tsunamis and inundation on either latitude-longitude grid or on Cartesian grid with a diverse range of temporal and spatial scales. This is accomplished by using up to two coarse levels grids for entire domain and evolving rectangular Cartesian sub-grids of higher refinement level that track moving waves and flooding around shoreline (see [26] and [27]). At any given time in the computation, a particular level of refinement may have numerous disjoint grids associated with it. User may specify the refinement ratios such that, starting with a coarse 102 km grid cell that tracks the long wavelength of the deep water tsunami he/she is able to resolve the shore line and inundation potential with a level of refinement up to 102 m or even lower. Imposing shallow water depth (typical value of 100 m) one can indicate which areas are to be refined close to coastal lines. Friction may be important for realistic run-up heights and typical value for Manning coefficient is n=0.025. The original code has been developed for tectonically induced tsunamis whose initial water level perturbation is generated by a vertical displacement of the seafloor. Here we have tuned the code to use as an initial wave source the profile of the water cavity generated through the asteroid impact. A combined bathymetry and topography file in standard GIS format for east Atlantic Ocean [28] is used throughout.

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4 Results In recent years a number of studies have attempted to assess the asteroid impact probability with Earth surface ([29], [30], [31] and [32]). One candidate, asteroid 1950 DA, stands out, being attributed a 0 to 0.3 probability of colliding with Earth in 2880, March 16, somewhere in Atlantic Ocean at 600 Km from the east coast of U.S [33]. Hence, such an event is worth simulating in order to assess the tidal wave effects on shores. To our best knowledge, a similar attempt has been made by Ward & Aspaugh [35], but rather in an analytic approach. Assuming a diameter m, density kg/m3 and impact velocity m/s for the asteroid, they computed that the initial cavity would have a radius Km and a depth Km. Considering the ocean depth of 4998 m corresponding to an impact point situated at latitude N°00.35 , longitude W°00.70 , the resulting depth of the cavity follows from Eq. 2, that is Km. The shape of the initial cavity is given by Eq. 4. Different results are obtained if applying equations 1-3. For example, cavity radius would be of only Km. Anyway, for comparison reasons, we agreed in simulating the tsunami wave propagation starting from the same initial cavity features as presented in [35] and quoted above. 4.1 Grid Independence Tests The best way to validate a theoretical model is by comparing its predictions with experimental data. There is no experimental data available in the present case. The accuracy of the numerical discretization has been studied by performing a series of tests on different grid sizes. Comparison between the results of these tests gives information about the intrinsic performance of the numerical method. The computational domain (25 − 45°N and 50 − 82°W) has been first mapped onto a coarse grid having 170x170 cells on the latitudinal and longitudinal directions, respectively. This specific number of cells represents the minimum number necessary to capture the initial water level disturbance induced by the asteroid impact. Starting from this first level grid, two subsequent more refined grids have been overlaid, both in the deep water domain and shallow water domain with depths less than 100m. The highest refined grid (level 4) has been considered in order to cover inundation risk areas on the coast with cells of dimensions ~ 100x100 m. Two different refinement ratios have been considered between consecutive grids levels: (4:4:4) and (5:5:4) (this means that, in the first case, for the second level grid and each direction the number of cells is 170x4, for the third 170x4x4 and so on).

Fig. 1: Sea level displacement for 3 different discreti-zation grids. Left corner and center: deep water, right corner: shallow waters and flooding beach. Another coarse grid with 230x230 cells and subsequent refining ratios 4:4:4 has been taken into account in order to assess the sensitivity of solution against the size of the initial discretization grid. Figure 1 presents a slice through the field of water displacement along the azimuth connecting the impact position and a location defined by 37.00 N and 75.50 W for the above three discretization grids. On average, all profiles show quite the same traits but the smaller the ratio between the first level grid and the second one the larger the differences between displacements are (blue line versus red and green at the left and centre part of Fig. 1). On the other hand, in shallow waters and flooding areas (right side of Fig. 1) displacements are almost equal even if the number of cells on the first level is small but we use larger refinement ratios (red line versus green line). In these areas, the worst results are obtain on the 170 - (4:4:4) grid, on which, we noticed, literally, an unrealistic disappearance of the tsunami by numerical diffusion. Choosing between 230-(4:4:4) and 170-(5:5:4) grids is only a matter of computing power and time expenditure. We chose to work on 170-(5:5:4) grid. 4.2 Wave propagation: Run-up and Run-in The salient features of the tsunami propagation are presented in Fig. 2.

a)

d)

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b)

e)

c)

f)

Fig. 2: Tsunami propagation. (a) impact position, (b)-(f) time snapshots of the wave amplitude. Legend in meters, time in seconds. The most striking characteristic of the snapshots array presented in Fig. 2 is the wave interference between direct and reflected waves especially by the continental plateau. This feature is clearly illustrated in snapshot 2b. Other reflected wave originates from Bermuda Island. Although after only 1 hour the tidal wave is situated at 50 km from Cape Hatteras it will take another 0.5 h to the tidal wave to jump over the natural barrier surrounding Cape Hatteras and reach Fisherman’s Island. This behavior is related to tsunami slowing down in shallow waters. The almost spherical propagation of the reflected wave around Bermuda Island is apparent from snapshots 2b-f and is concurrent with other numerical simulation around conical obstacles [34]. The East Coast is flooded for more than 20 km in average from the coastal line from New-York to Charleston in 3 hours. The tsunami propagation from our simulation follows quite remarkable the timetable from [35], although major differences must be pointed out especially regarding wave amplitude. It seems that that the analytic approach of [35] is much more energy conservative, under dispersive and insensitive to bathymetry than expected. The maximum wave amplitude, in our simulation, is 5 to 10 times lower than in the above mentioned paper depending on the temporal development. Four sites on the Eastern US coast are considered next (see Table 1 for geographical coordinates and Fig. 3 for position). Maximum run-ups are presented in Fig. 3

while run-in distances and beach wave profiles are illustrated in Fig. 4. Table 1: Geographical coordinates of points on coastal areas.

Fig. 3: Maximum run-up values for various sites on US East Coast. (red tag- impact position).

Fig. 4: Wave profiles and run-in distances at location C. a) maximum wave amplitude; b) run-in distance.

Site Longitude W[°] Latitude N[°] A 73.50 39.50 B 74.60 38.10 C 75.50 37.20 D 76.10 34.20

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5 Conclusions In this paper the consequences of the asteroid 1950 DA (1.1. km diameter) impacting the Atlantic Ocean are studied and briefly reported. The adopted impact position is about 600 km far from the Eastern U.S. coast. It takes about three hours for the tsunami generated by the impactor to make landfall from Cape Cod to Cape Hatteras. The values predicted by this study are lower than previously estimated [35]. However, the maximum run-up may be as high as 16.1 m (see Fig.3) in some particular places on the shore. References: [1] Rabinowitz D., Helin E., Lawrence K., Pravdo S. A

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