1 Introduction

This talk will review some of the basics behind the simulation of asteroid-generated tsunamis, and how this piece of the Asteroid Threat Assessment Program (ATAP) got its start.

In 1994, the United States Congress asked NASA to identify 90% of asteroids larger than 1 km in diameter that could pose a threat to Earth. This led to the Near Earth Observing (NEO) program, which catalogued the objects and tried to determine their characteristics. In 2005, NASA’s mission was expanded to track near Earth objects greater than 140 m in diameters. Obviously the largest dinosaur-killing asteroids are the most dangerous. However, the question arises, how small does an asteroid have to be before we don’t have to worry about it? Little is known about asteroids smaller than 140 m in diameter, and whether they are safe to ignore. What if one exploded over an ocean. Could it generate a tsunami that would change it from a regional to a more global hazard that would threaten coastal populations far away?

As it turns out, in February, 2013 an approximately 20-m asteroid exploded about 15 miles above the ground over Chelyabinsk, Russia. This airburst provided an unprecedented opportunity for data collection. Teams of scientists visited, collected samples of the meteor to determine its composition, analyzed web cams from Russian cars to determine the trajectory and energy deposition, canvassed the region to see how far away windows broke (evidence of the blast overpressure), etc. [15]. In other words, data was collected that could be used for model validation. The ATAP project started shortly thereafter.

A reader might wonder how often such airbursts really occur. Figure 1 shows that in fact airbursts happens quite regularly. Since most of the world’s surface is water, an investigation into airburst-generated tsunamis seems warranted.

Fig. 1
figure 1

Figure taken from https://cneos.jpl.nasa.gov/fireballs

Airbursts reports from April, 1988 to Dec, 2019.

In this talk I will focus only on simulations of smaller asteroids that explode before hitting the ground. There is very little literature on the effects of these airbursts. There is some literature on simulations of larger asteroids that do reach the ocean, and sometimes reach the ocean floor [4, 17, 18]. Impact simulations are generally performed using hydrocodes that simulate material deformation and failure, multimaterial phase changes (e.g. water turns into vapor and rises through the atmosphere), sediment excavation from the ocean floor, shock waves traveling through water, etc. A nice discussion can be found in the chapter by Gisler in [6]. These are very expensive calculations, so they tend to be axisymmetric to reduce cost, including the bathymetry.Footnote 1 Asteroid impact simulations is a dynamic area that is receiving a lot of recent attention [12, 13, 16].

In the next section we will present our simulations using the shallow water equations modeled with the GeoClaw software package, and describe how GeoClaw was adapted to model asteroid airbursts. We will review our analysis of a model problem that helps understand the simulations results. However, it turns out that airburst-generated tsunamis have smaller length scales that earthquake-generated tsunamis. Hence we will turn to the linearized Euler equations to bring in the effects of compressibility and dispersion. It will turn out that dispersion is a much more important factor at the length scales and pressures of interest, and luckily the shallow water equations seem to overestimate the effect. We will conclude that airburst-generated tsunamis do not pose a global threat. This was the conclusion reached by all participants in the joint NASA-NOAA tsunami workshop in 2016 using a variety of codes and test problems, summarized in [11].

2 Simulations of Airburst-Generated Tsunamis

2.1 Background

The simulations we first present use the open-source software package GeoClaw [9]. GeoClaw solves the depth-averaged shallow water equations on bathymetry. It uses a second order finite volume scheme with a robust Riemann solver to deal with wetting and drying [5]. Very important for trans-oceanic wave propagation where coastal inundation is also important is the use of adaptive mesh refinement. GeoClaw uses patch-based mesh refinement, allowing resolution in deep water with grid cells the size of kilometers, and on land on the order of meters. Other issues such as well-balancing (an ocean at rest on non-flat bathymetry stays at rest), and a well-balanced and conservative algorithm for adding and removing patches, are also part of GeoClaw. Desktop-level parallelism using OpenMP has also been implemented. There is no data from asteroid-generated tsunamis to use for benchmarking. We mention however that GeoClaw has had many benchmarking studies performed for earthquake-generated tsunamis, especially extensively in 2011 in [7]. This set of benchmarks was performed to allow GeoClaw to be used in hazard assessment work funded by the U.S. National Tsunami Hazard Mitigation Program.

The shallow water equations can be derived from the incompressible irrotational Euler equation using the long wavelength scaling, by assuming the ratio \(\epsilon = h/L \ll 1\). Here, h is the depth of the water and L is the length scale of interest. This scaling leads to the conclusion that the velocity of the water in the z direction only enters at \(O(\epsilon )\), and the horizontal velocities are constant in the vertical direction to \(O(\epsilon ^2)\). Eliminating the need to compute the vertical velocity reduces the three-dimensional simulation to a much more affordable calculation using only the horizontal velocities u and v.

Ordinarily the pressure only appears as a gradient in the shallow water equations, allowing the value for the pressure itself to be set arbitrarily. In our simulations however we will need to match the pressure at the top of the water column with the atmospheric pressure produced by the asteroid blast wave. Re-deriving the shallow water equations and retaining the pressure produces the following set of equations for simulation:

$$\begin{aligned} \begin{aligned} h_t + (hu)_x + (hu)_y =&\, 0 \\ (hu)_t + \left( hu^2 + \frac{1}{2}g h^2\right) _x + (huv)_y =&\, -g h B_x - \frac{h \, {p_e}_x}{\rho _w} - Du \\ (hv)_t + (huv)_x + \left( hv^2 + \frac{1}{2}g h^2\right) _y =&\,-g h B_y - \frac{h \, {p_e}_y}{\rho _w} -Dv\\ \end{aligned} \end{aligned}$$

The other terms in (1) are g, gravity, \(p_e\), the external atmospheric pressure at the water surface, and \(\rho _w = 1025 \, \text {kg/m}^3\) is the density of salt water. B(xy) is the bathymetry (underwater topography, or depth of the ocean floor). Note that the pressure forcing appears in a non-conservative form, as does the bathymetry. In these equations, a flat ocean would have \(h(x,y) = -B(x,y)\). This is often described using the water elevation \(\eta (x,y) = h+B\), where sealevel is \(\eta (x,y) = 0\). In these equations we have neglected the Coriolis force (often considered unimportant for tsunami propagation). The term \(D = \frac{gM^2 \sqrt{(u^2+v^2)}}{h^{1/3}}\) is the drag, which is important in numerical simulations that include inundation. \(M = 0.025\) is the Manning coefficient which we take to be constant.

To simulate the equation set (1), the external pressure must be known. This is obtained from detailed simulations of an asteroid entering the earth’s atmosphere at a given speed, angle, and material composition, performed by others in the ATAP project [1]. The asteroid deposits its energy in the atmosphere, causing a blast wave. The simulations extract the ground pressure \(p_e(x,y)\), and the width and amplitude of a Friedlander profile, an idealized blast wave profile, is fit to the data. This functional form is then used in the simulations for the pressure forcing. For simplicity we use a radially symmetric source term corresponding to a vertical entry angle for the asteroid. (In other simulations we have performed anisotropic simulations, with no change to our conclusions.) The blast wave in these simulations travels at 391.5 m/s, which we take to be constant. This is somewhat faster than the speed of sound in air.

Fig. 2
figure 2

A typical blast wave profile is drawn at two times. The amplitude is fit with a sum of decaying exponentials and the profile is scaled to get the pressure forcing at a given time. This functional form is then used in numerical simulations

Figure 2 shows a typical profile. A Friedlander profile has a characteristic width that describes the distance from the leading shock to the ensuing underpressure. Figure 2 is used in the simulations as follows: At a given time t in the simulation, each grid point needs to evaluate the atmospheric pressure. If the leading blast wave travels at speed \(s=391.5\) m/s, then at time t it has travelled a distance \(d = 391.5 \times t\) meters. If the grid point is farther than d from the initial location of the blast wave there is no change to the ambient pressure. If it is less, the pressure profile is evaluated at that distance away and fed to the solver. The blue curve in Fig. 2 shows the profile at 50 s. The amplitude of the overpressure at that time is approximately 100% of ambient pressure. It is zero ahead of the blast, and decays as it gets closer to blast center. These values are used in Eq. (1).

The simulation in Fig. 2 resulted from a 250 MT asteroid. This roughly corresponds to a meteor with a 200 m diameter entering the atmosphere with a speed of 20 km/s. Note that the maximum overpressure of the airburst is approximately 450%. (Explosions are measured in terms of MT (megatons) of TNT, relating the equivalent destructive power to the uses of dynamite; this is also used to quantify nuclear bombs). For comparison, the explosion of Mount Saint Helens was estimated to be 25–35 MT. The largest volcanic explosion ever records was Mount Tamboura, which was approximately 10–20 Gt, and caused global climate change and mass destruction. The airburst over Chelyabinsk was approximately 520 KT. The Tunguska event, the largest airburst of the previous century, is now thought to be about 15–20 MT.

We point out that the length scale of the Friedlander profiles are significantly shorter than those of earthquake-generated tsunamis, which are typically on the order of 50–100 km. We will come back to this point in Sect. 3.

2.2 Analytical and Computational Results for Shallow Water Equations

In [2], we propose and analyze a one-dimensional model problem that helps describe the results seen in our simulations. The model problem first assumes that the pressure disturbance is a traveling wave and then builds on this to solve the problem where the pressure disturbance starts impulsively at time zero. Of course the actual pressure disturbance is a decaying function that will generate further waves as it changes amplitude, but the initial waves are the strongest and most important.

When the pressure pulse from the airburst hits the water, it causes two distinct waves with two different wave speeds. One will be related to the pressure pulse with speed \(s_b\), and the other is the gravity wave, moving with speed \(s_g\). What we call the response wave is an instantaneous disturbance of the sea surface that is in direct response to the amplitude of the moving pressure pulse and that propagates at the same speed, \(s_b = 391.5\) m/s (this is called \(\eta \) above, but we change notation here to indicate it is a response to the pressure forcing).

Our analysis shows the following relationship between the response wave and the pressure disturbance \(p_e\):

$$\begin{aligned} h_r = \frac{h_0 p_e}{\rho _w (s_b^2 - s_g^2)} \end{aligned}$$

In (2), \(h_0\) is the undisturbed height of the water (i.e. when \(\eta = 0\)). This shows that the response wave is stronger is deeper water, (almost linearly, since \(s_g\) depends on \(h_0\) too). For 4.5 times atmospheric pressure, at a depth of 3 km, the response wave would have an initial height of approximately 10.8 m. This amplitude would decay rapidly with the strength of the blast wave. Note that this response wave has positive amplitude, since \(p_e > 0\) and \(s_b > s_g\). This is counterintuitive, since one would think that pushing on water would have lower its height. With hurricanes, the air pressure disturbance is negative, and hurricane travel slower than water waves, so again the water height increases, but this is more intuitive.

There are also gravity waves which move at the slower speed \(s_g = \sqrt{gh}\) m/s. When \(h = 3000\) m, this gravity wave moves at slightly less than 171 m/s, less than half the speed of the response wave. The initial gravity waves generated can also be estimated by linearizing the model problem and solving the homogeneous equation to get:

$$\begin{aligned} h(x,t) = h_r(x-s_b t) - \left( \frac{s_b}{s_g} + 1 \right) \frac{h_r(x-s_g t)}{2} + \left( \frac{s_b}{s_g} - 1 \right) \frac{h_r(x+s_g t)}{2} \end{aligned}$$

The first term in (3) is the response wave traveling at blast wave speed \(s_b\), and the next two are the gravity waves moving to the right and left with speed \(s_g\). We see that their amplitude is also a function of the amplitude of the response wave.

We next show results from two simulations at different distances from shore and ocean depths. More details on these particular simulations are in [2]. The first set of simulations are located off the coast of Westport, Washington. This area has been well-studied because of its proximity to the earthquake-prone M9 Cascadia subduction zone. The blast was located 180 km from shore, about 30 km from the continental shelf, and the ocean was 2575 m deep underneath the blast. Figure 3 shows the region of interest.

Fig. 3
figure 3

The first set of simulations has the blast located 180 km offshore from Westport, in 2575 m deep water, indicated by the purple star. The zoom shows the region of interest studied for inundation

Figure 4 shows 3 snapshots at intervals of 25 s after the blast wave. A black circle is drawn indicating the location of the blast, the red just inside the circle is the response wave, and further interior to the circle is the gravity waves. Note that the leading gravity is a depression (negative amplitude). Contours of the bathymetry from −1000 to −100 are drawn to show the location of the continental shelf. Although the colorbar scale is from −1 to 1, the response wave height near the blast is over 10 m.

Figure 5 shows a zoom of the waves approaching shore (2000 s), about to hit the peninsula (3000 s), and mostly reflecting (4000 s), with some smaller waves entering Grays harbor. Note that the landscape is better resolved as the waves approach, indicating that the refinement level has increased. The wave amplitudes have greatly decreased, and no inundation is observed. Note that the colorbar scale (in units of meters) has been reduced by a factor of 5 in these later plots.

Fig. 4
figure 4

Westport simulations at intervals of 25 s after the blast. The waves are spreading symmetrically around the blast center. The largest wave is over 10 m at the start

Fig. 5
figure 5

Selected times as gravity waves approach Westport coastline. The zooms cover a changing region closer and closer to shore. No inundation is observed. Note the colorbar scale is a factor of 5 smaller than in the figure above

Since the first set of results did not show any inundation despite such a large blast, the second set puts the blast much closer to shore. We locate the blast 30 km off the coast of Long Beach, California, an area with a lot of important infrastructure. Figure 6 shows the topography. The water at the center of the blast is 797 m deep.

Figure 7 shows 3 snapshots at intervals of 25 s after the blast wave. Several features are evident. The black circle, which indicates the location of the blast wave at that time, no longer coincides with the leading elevation of the response wave (the red contours). This is because the topography becomes more shallow at the blast wave approaches Catalina Island, so its instantaneous amplitude has decreased, as expected from Eq. (2). Also notice that that atmospheric blast wave in the atmosphere jumps over the island, and the response wave reappears when the blast is again over water. Once again we see that the gravity waves are mostly a depression.

Fig. 6
figure 6

The second set of simulations has the blast located 30 km from Long Beach, in 797 m deep water, indicated with the red dot. The zoom shows the region of interest studied for inundation

Fig. 7
figure 7

First row shows computed solution for Long Beach simulation at intervals of 25 s after the blast. The black circle indicates the location of the blast wave in air. Bottom row shows zooms near shore at two later times

With this proximity to shore, the blast wave has not greatly decayed before it hits shore. The blast wave will be the more important cause of casualties and damages, and not the ensuing tsunami. The zooms in Fig. 7 have more refinement than the early times. The breakwater is now resolved, and water only approaches shore through the breakwater gaps or around the edge. But since the port infrastructure is two meters high, there is still no flooding. A very tiny bit of flooding is seen along the river (not visible in these plots).

We performed a number of additional simulations in a variety of locations, bathymetries, and asteroid strengths, including one with one Gt of energy. We have not found any examples where airbursts have caused significant onshore inundation. However, in the next section we examine whether the shallow water equations is an appropriate model for airburst-generated tsunamis, and compare the previous results with similar analyses and computations using the linearized Euler equations.

3 The Linearized Euler Equations

As reviewed earlier, the shallow water equations are a long wavelength approximation to the full 3D equations. Since the length scales of the Friedlander profile are on the order of 10 km, the ratio of water depth to length scale is not that small in a 4 km ocean. Closer to shore the shallow water equations may be more appropriate. The length scales are also important in determining the effect of dispersion, which is not present in the shallow water equations.

To examine this more closely, we compare the results from the previous section using the shallow water equations with those from the linearized Euler equations. This brings in the effects of both compressibility and dispersion. The latter equations have the advantage that the free surface boundary condition of the full Euler equations becomes a simple boundary condition when linearized, so the free water surface and the atmosphere do not have to be tracked or computed. Unfortunately it does require that the vertical direction be discretized along with the two horizontal directions, and so is much more expensive than a depth-averaged equation set.

3.1 Analytical and Computational Results for Linearized Euler

Again, we first review the results from [2] for our model traveling wave problem but for the linearized Euler equations (which are also derived there). Unlike the shallow water equations, which do not have any dependence on wave length, there is such a dependence in the Euler equations. We first present results for a single frequency k, where the length scale \(L = 2 \pi /k\). We then apply our results to a function with many frequencies. Finally we show some preliminary results of radially symmetric simulations confirming the model problem conclusions.

If we denote the external pressure forcing \(p_e(m) = A_k e^{ikm}\), where \(m = x-s_b t\) is the traveling wave variable in our model problem, we can compute the response coefficients as a function of wave number, i.e. \(h_r(m) = \widehat{h_r} e ^{ikm}\) and amplitude \(A_k\), and similarly for the velocities u and now the vertical velocity w too. The traveling wave problem can no longer be solved exactly, but can be evaluated numerically. In Fig. 8, we evaluate the solution to the model problem using an ocean depth of 4 km, and an amplitude of 1 atmosphere for the overpressure. We take the speed of sound in water \(c_w = 1500\,\text {m/s}\), and density \(\rho _w = 1025\, \text {kg/m}^3\). Figure 8 also evaluates the results for an artificially faster speed \(c_w = 10^8\), in order to approach the incompressible limit.

Fig. 8
figure 8

Comparison of response wave amplitudes as a function of length scale for the shallow water and linearized Euler equations. These were evaluated for a 4 km deep ocean, and 1 atm overpressure. At smaller length scales the dominant difference is due to dispersion, not to compressibility

The green curve in Fig. 8 is the shallow water amplitude of the response wave. It is constant, since as expected there is no dependence on wave number. We can also compute the nonlinear response, which is done in [2], and overlays the linearized response. The blue curve is the linearized Euler result using the real sound speed of water. This does not appear to approach the shallow water curve. The red curve uses the artificially larger sound speed \(c_w=10^8\), which approaches the incompressible limit and does approach the shallow water curve, giving us more confidence in the results. The difference between the linearized Euler curve and the shallow water curve is roughly 10%. We are calling this the effect due to compressibility. However, at the length scale of interest for airburst-generated tsunamis, the difference between the curves is over a factor of 2. We conclude that dispersion is a much more important effect.

Figure 8 showed the amplitude response due to a single frequency pressure perturbation. In Fig. 9 we evaluate the response to a Gaussian pressure pulse \(p_e(m) = \exp (-0.5 (m/5)^2)\) that includes all frequencies. We take the Fourier transform, multiply each frequency by the Fourier multiplier shown in Fig. 8 and transform back, so this is still a static response. The left figure shows results in 4 km deep water, and the right in 1 km deep water. Again we see that compressibility accounts for a smaller portion of the height difference between shallow water and linearized Euler results than dispersion. Note also that the Euler results have broadened, an indication of dispersion. The results in shallower water match better, as expected. Luckily, in all cases the shallow water results overestimate the response including compressibility and dispersion.

Fig. 9
figure 9

Comparison of responses to a Gaussian pressure pulse in 4 km deep water (left) and 1 km deep water (right)

Finally, in Figs. 10 and 11 taken from [3] we show snapshots from time dependent simulations with the 250 Mt airburst and compare linearized Euler (denoted AG for acoustic with gravity in the legends), shallow water, and two different BoussinesqFootnote 2 models [8, 14]. We thank Popinet for the use of Basilisk in simulations using the Serre-Green-Naghdi (SGN) set of equations, and Jiwan Kim for the use of BoussClaw, which uses the Madsen Sørensen equation set [10].

We first show results in a 4 km deep flat ocean, then 1 km deep. Note that the scales are not the same in the two figures. Also, since the tsunami travels more slowly in shallower water, we only show those results every 100 s. Note that the leading shallow water gravity wave is a depression in both simulations. Also note that the two Boussinesq simulations agree with each other better than with the linearized Euler runs. The SGN simulation is in two space dimensions, and plotted as a function of radius, hence is much noisier than the other simulations which were one-dimensional radially symmetric computations. We point out that Boussinesq waves decay inversely proportional to distance traveled, whereas shallow water waves decay inversely to the square root of distance. Finally, all 4 codes show the same response wave behavior as an elevation in sealevel, albeit with different magnitudes.

Fig. 10
figure 10

Comparison of initial generation of airburst tsunami using all 4 models in a 4 km deep ocean. Selected frames every 50 s. After 300 s, the SGN and BoussClaw resuls match linearized Euler in the leading gravity wave, but not (yet) the rest. The SWE model does not generate gravity waves that match at any of the times

Fig. 11
figure 11

Comparison of airburst generated tsunamis using all 4 models in a 1 km deep ocean. Selected frames every 100 s. After about 200 s, SGN and BoussClaw match the linearized Euler results in the leading gravity wave, and by 400 s, the next few waves are very similar, though the amplitude is not quite right. The shallow water model still has very different waves

We do not think that the depth-averaged equations are suitable for simulating the initiation of gravity waves, since there is significant variation in the vertical velocity. It does seem that depth-averaged equations can be used to propagate the waves, once initiated by a higher fidelity simulation. This has been demonstrated in [3]. We do not yet know how this translates into shoreline inundation. Preliminary evidence indicates that the shallow water model provides an overestimate of run-in due to airbursts, as it did in predicting wave height for the response wave, but we need more evidence for this hypothesis.

4 Conclusions

We have presented several numerical simulations of the shallow water equations in response to a 250 Mt airburst. The results are further explained using a traveling wave model problem, for both the shallow water and linearized Euler equations. All results show that there is no significant water response (in either the response wave or the gravity wave) to the airburst. The most serious danger from an airburst would be from the blast itself if close enough to the blast center, rather than from water waves it generated.

We also found that because of the shorter wave-lengths of an airburst, the shallow water equations do not provide an accurate simulation of propagation for these waves, compared to simulations using Boussinesq or linearized Euler models. However it may be possible to use the shallow water equations to give an estimate of shoreline inundation. This is a matter for future study.