1. INTRODUCTION

Asteroids and comets pose a significant danger to life on Earth (Barbee and Nuth, 2009; Cambier et al., 2009; Crowther, 2009), particularly those identified as Neart Earth Objects (NEOs). Currently, over 135,000 NEOs have been detected, and these include more than 30 Near-Earth objects (NEOs) larger than 5 km in diameter, 1,500 NEOs larger than 1 km and 135,000 larger than 100 m. About half of NEOs are Earthcrossers and there is a chance some of these may intersect the orbit of our planet and collide with 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 catastrophic environmental and societal consequences of asteroids impact on Earth has been obtained from numerical simulations (Chapman et al., 2001). Some estimates were also derived (for smaller impacts) from extrapolations of nuclear weapons tests (Glasstone and Dolan, 1977) 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 (Toon et al., 1997): (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. Also, a number of researchers are arguing a forth size range should be added, namely (iv) multi-ten meters impactors like Tunguska (see Jewitt, 2000; Foschini, 1999) and reference therein).

There is considerable uncertainty about the environmental consequences of larger impacts: categories (i) and (ii). It is expected that they will have diverse and catastrophic 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 (Cockell and Stokes, 1999). Sunlight will be blocked, temperatures will drop, plants and photosynthesizing organisms will die, plant-eating animals sill succumb, carnivores would follow, and the entire food chain lead to humans would be severely effected.

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 (Foschini, 1999). 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 (Foschini, 1999).

Much of the Earth is covered with water. Therefore, most asteroids would more likely strike the ocean rather than land. Hence, one of the greatest dangers posed by even smaller category impacts are tsunamis, which transfer the effects of a localized ocean impact into dangerous, breaking "tidal waves" on distant shores (Ward and Asphaug, 2000).

The current philosophy of impact hazard does not view the danger from small asteroids as significant. 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 extreme public reaction and hysteria. In fact, there are several scientists suggesting that small asteroids, in some respects, might be even more dangerous than larger bodies (Foschini, 1999).

The impact is a random process in geographical space but also in time (Barbee and Nuth, 2009; Cambier et al., 2009; Crowther, 2009). 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 (Ivezic et al., 2001). 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 (Barbee and Nuth, 2009; Cambier et al., 2009; Crowther, 2009). The scarce existing data yield often contradictory results.

A rough estimate (Chapman et al., 2001) for the impact frequency as function of impactor size is: (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 surface area (Paine, 1999). 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. But this interpretation must be taken with caution because most of the terrestrial impact craters have been obliterated by other terrestrial geological processes (Grieve, 1990).

2. EFFECTS OF AN ASTEROID IMPACTING THE OCEAN SURFACE

In this paper we develope a scenario where an asteroid of category (i) (i.e. multihundred meters diameter) strikes the Atlantic Ocean near the North American coast. The tsunami generation and propagation is studied and the effects of the waves on the shore are predicted. The results presented here can be seen as an illustrative example of what could take place in any other part of the world. However, this study may be representative for asteroid impacting oceans and open seas only. Different results are expected in case the impactor strikes smaller bodies of water such as confined seas.

Very accurate numerical simulations provided valuable information about the interaction between larger size asteroids and the atmosphere, the seawater and the underwater medium (Gisler et al., 2003). 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 by Crawford and Malder (1998), Shuvalov et al. (2002), Malder and Gittings (2003), and Gisler et al. (2003). All 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 (Le Mehaute and Wang, 1996) or explosions of underwater volcanoes (Mirchina and Pelinovsky, 1988). Ward and Aspaugh (2002) suggested a relation between the radius R_c and depth D_c of the cavity of the form

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

with ρ_i , V_i , R_i are the density, velocity and radius of the impactor, respectively, ρ_w is the seawater density, h is water depth and g is the gravitational acceleration. From Eqs. (1) and (2) one gets the diameter d_c of the cavity

with δ = 0.5 /(1+ α). From laboratory investigations, α = 1.27.

Assuming central symmetry for the equivalent water cavity at impact point and initial moment t = 0 , the parabolic shape of the water displacement is given by Kharif and Pelinovsky (2005):

with 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 area covered by the numerical simulation and the dispersion term, the wave equations in spherical coordinates describing the temporal and spatial evolution of the sea level are (Kowalik et al., 2005; Dao and Tkalich, 2007):

where D = h + η 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.

where v₂ and v_φ are appropriate sea water velocities stemming from the spherical Navier- Stokes equations. The last terms in Eqs. (4) are given by:

They represent bottom friction terms which become important in shallow waters. The friction coefficient f entering Eqs. (6) is computed from Manning’s roughness coefficient n

such that Eqs. (6) become (Dao and Tkalich, 2007):

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 available at (Clawpack, 2009). An extensive explanation of its features and numerical approach can be found in (LeVeque, 2002). The code is based on a finite difference technique using second-order Godunov flux-splitting scheme (Stegger and Warming, 1981) and Roe’s approximate Riemann solver with entropy fix (Roe, 1981) 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 in longitude and 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 (Berger and LeVeque, 1998; Berger and Rigoutsos, 1991). 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.

4. RESULTS

In the last twenty years a number of studies have attempted to assess the Earth impact probability for specific asteroids (Morrison, 1992; Chapman & Morrison, 1994; Gehrels, 1994; Atkinson, 2000). 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 (Giorgini et al., 2002). Studying this particular possible event is of interest, taking into account the expected societal consequences. A first attempt has been made by Ward & Aspaugh (2003) (referred to as WA2003 in the following) by using an analytic approach. Assuming an asteroid of diameter D = 1100 m, density p = 2200 kg/m³ and impact velocity V = 17800 m/s, the quoted authors estimated a radius, R = 9.5 Km and a depth D = 6.5 Km for the initial impact cavity.

Here we reconsider the tsunami generated by 1950 DA and its effects on the shores. Two are the main novelties in respect to WA2003. First, we are using numerical techniques to describe tsunami propagation and interaction with the seashore. Second, an accurate combined bathymetry and topography file in standard GIS format for Eastern Atlantic Ocean (Smith and Sandwell, 1997) is used throughout.

The geometry of the impact cavity created by 1950 DA is evaluated next from the model presented in Section 2. Considering the ocean depth of 4998 m corresponding to an impact point situated at latitude 35.00°N , longitude 70.00°W , the resulting depth of the cavity follows from Eq. (2), that is D = 4.998 Km. The shape of the initial impact cavity is given by Eq. (4). Using Eqs. (1)-(3) we obtain values which are different from those adopted in WA2003. For example, the cavity radius predicted by Eq. (3) is R = 5.577 km, which is obviously smaller than R = 9.5 km estimated by the quoted authors. However, in simulations we are using the impact cavity features given in WA2003 (i.e. D_c = 4.998 km and R_c = 9.5 km). This allows proper comparison between our results and those reported in WA2003.

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 overlayed, 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 is for the second level grid and each direction the number of cells is 170x4, for the third 170x4x4 and so on). 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 azymuth connecting the impact position and a location defined by 37.000° N and 75.500°W for the above three discretization grids.

Figure 1: Vertical water surface displacement profiles for three different discretization grids. Left corner and centre of the figure correspond to deep water displacements while the right corner refer to shallow waters and flooding areas.

The interesting conclusions that one can easily draw from the vertical water displacement profiles of Fig. 1 are as follows. 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 dissapearence 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 chosed to work on 170-(5:5:4) grid.

4.1. Wave propagation, Run-up and Run-in.

The salient features of the tsunami propagation are presented in Fig. 2.

Figure 2: Time snapshots for the propagation of a tsunami generated by a 1100 m diameter asteroid. (a) the asteroid impact position. (b)-(f) tsunami propagation at various times measured from the impact moment. The length unit on the legend below each sub-figure is meter and the time unit is seconds. η - vertical water surface level displacement.

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 Fig. 2b. Other reflected wave originates from Bermuda Island (see Fig. 3 for position).

Although after only one hour the wave arrives at 50 km from Cape Hatteras (see Fig. 3 for geographical position) it will take another hour for the wave to jump over the natural barrier surrounding Cape Hatteras and reach Fisherman’s Island (see Fig. 3 for geographical position). This behavior is related to the known effect of tsunami slowing down in shallow waters. The almost radial symmetric propagation of the reflected wave around Bermuda Island is apparent from snapshots Figs. 2b-f. This is in good agreement with results predicted by other numerical simulation of tsunami propagation around conical obstacles (Repalle et al., 2007). The US East Coast is flooded between New York and Charleston in about 4 hours (see Fig. 3 for geographical position). The maximum run-in may exceed 15 km, but on the average it stays around 5 km and lower (see for example Fig. 4).

The tsunami propagation, as described by the present simulation, follows quite remarkably the timetable from WA2003. However, major differences exist between our results and those in WA2003 regarding wave amplitude. It seems that the semi-analytic approach of WA2003 is excessively energy conservative, less dispersive and almost insensitive to bathymetry. The maximum wave amplitude in our simulation is five times lower (on average) than that reported in WA2003, depending on the temporal development (compare our Fig. 2 with Fig. 2 from WA2003).

Four sites on the Eastern U.S. coast are considered next (see Fig. 3 for position and Table 1 for geographical coordinates). Figure 3 shows the maximum run-up values at these sites while Fig. 4 presents the associated run-in distance and the wave profile on beach.

Table 1: Geographical coordinates of four sites on the U.S. coast (see also Fig. 3) Site Longitude west [°] Latitude north [°]

Figure 3: Run-up values for various sites on the U.S. East Coast. For geographical coordinates see Table 1.

The farther we move south of Cape Haterras, the run-up values are smaller and smaller (decreasing below 2 m). Hence, we decided not to display them in Fig. 3 The same trend is reflected in Fig. 3 of WA2003, but the intrinsic values are overestimated in that paper by almost an order of magnitude.

Figure 4: Wave profiles at different moments in time and run-in distances at location C (see Table 1 and Fig. 3 for geographical coordinates and position, respectively. a): maximum wave amplitute at shore line; b): run-in distance.

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 to make landfall from Cape Cod to Cape Hatteras. The values predicted by this study are lower than previously estimated (see Ward and Asphaug (2003)). However, the maximum run-up may be as high as 16.1 m (see Fig.3) in some particular places on the shore.

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