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Numerical Modeling of the 2013 Meteorite Entry in Chebarkul Lake

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Numerical Modeling of the 2013 Meteorite Entry in Chebarkul Lake Numerical modeling of the 2013 meteorite entry in Chebarkul Lake, Russia Andrey Kozelkov 1,2, Andrey Kurkin2, Efim Pelinovsky2,3, Vadim Kurulin1, Elena Tyatyushkina1 1Russian Federal Nuclear Center, All-Russian Research Institute of Experimental Physics, Sarov, 607189, Russia 5 2Nizhny Novgorod State Technical University n.a. R.E. Alekseev, Nizhny Novgorod, 603950, Russia 3Institute of Applied Physics, Nizhny Novgorod, 603950, Russia Correspondence to: Andrey Kurkin ([email protected]) Abstract. The results of the numerical simulation of possible hydrodynamic perturbations in Lake Chebarkul (Russia) as a consequence of the meteorite fall of 2013 (Feb. 15) are presented. The numerical modeling is based on the Navier-Stokes 10 equations for a two-phase fluid. The results of the simulation of a meteorite entering the water at an angle of 20 degrees are given. Numerical experiments are carried out both when the lake is covered with ice and when it isn’t. The estimation of size of the destructed ice cover is made. It is shown that the size of the observed ice-hole at the place of the meteorite fall is in good agreement with the theoretical predictions, as well as with other estimates. The heights of tsunami waves generated by a small meteorite entering the lake are small enough (a few centimeters) according to the estimations. However, the danger 15 of a tsunami of meteorite or asteroid origin should not be underestimated. 1 Introduction February 15, 2013 at 9:20 a.m. local time, in the vicinity of the city of Chelyabinsk, Russia (Fig. 1) a meteorite exploded and collapsed in the earth's atmosphere as a result of inhibition. Its fall was accompanied by a series of atmospheric explosions and propagation of shock waves over the territory of the Chelyabinsk region. Small fragments of the meteorite came down 20 on the region. It is the largest of the known celestial bodies which fell to the ground after the Tunguska meteorite in 1908. The following characteristics of the Chelyabinsk meteorite are given in recent publications [Zetser, 2014, Ionov, 2013, Kopeikin et al., 2013, Emel'yanenko et al., 2013, Popova et al., 2013, Berngardt et al., 2013, Gokhberg et al., 2013, Krasnov et al., 2014, Seleznev et al., 2013, De Groot-Hedlin and Hedlin, 2014]: . the meteorite with the diameter of 16-19 meters flew into the earth's atmosphere at about 20 degrees to the horizon with 25 the velocity ~ 17-22 km/s; . the meteorite was destroyed in several stages, and the main explosion occurred at an altitude of about 23 km. The analysis of the meteorite fall led to the estimation of the explosion energy from 380 to 1,000 kilotons of TNT; . surviving fragments of the Chelyabinsk meteorite supposedly flew at velocities up to 150 - 300 m/s, and according to the updated data, the velocity was 156 m/s; 1 . the largest meteorite fragment, weighing about 550 kg, was recovered from Lake Chebarkul near the Krutik Peninsula and it had an irregular oval shape with the average outer diameter of about 1 m; . the celestial body was exploded when it hit the ice and water. Its fragments flew more than 100 m and formed an ice- hole of the round shape with the diameter of about 8 m (Fig. 1). 5 The GPR survey of the crash site showed that the crater, which was formed by the meteorite impact on the bottom, was observed 30 m away from the ice-hole [Kopeikin et al., 2013]. The results of the numerical modeling of the processes that began in the water after the meteorite entering Lake Chebarkul are presented in this paper. We consider two cases – the occurrence of a meteorite in the ice-covered lake (as it was in February 2013), and in the lake without ice to estimate the amplitude of the wave in the case of a possible fall in the summer. 10 First of all, in section 2 is used a well-known parametric model of the tsunami source of meteoric origin [Ward and Asphaug, 2000] which estimates the disturbances in the water at the site of the meteorite entry. It is shown that the size of the source is twice the size of the observed ice-hole; that is why this model cannot explain the observed phenomenon. In section 3 we describe the hydrodynamic model of wave generation based on the Navier-Stokes equations as well as the parameters of its numerical discretization. In section 4 we present the results of numerical experiments on the tsunami wave generation in 15 open water. The results of numerical experiments on disturbance generation in the lake covered with ice are presented in section 5. The zones of maximum and minimum pressure, which could potentially lead to the destruction of the ice cover are detected. In section 6 are shown the results of estimations of the ice cover destruction, based on the calculation of the stresses. The area of ice destruction is estimated according to the famous semi-empirical formulas and pressure values obtained in the numerical experiments. This estimation is in good agreement with the observed data, as well as with the 20 estimates made earlier in [Ivanov, 2014]. The results are summarized in section 7. 2 Preliminary estimates of wave heights in the lake with a free surface To estimate the possible parameters of the water surface displacement when a meteorite falls, we use a simplified model in which the generation of waves by a meteorite entering the water is parameterized by certain initial conditions [Ward and Asphaug, 2000; Mirchina and Pelinovsky, 1988; Kharif and Pelinovsky, 2005; Levin and Nosov, 2009; Torsvick et al, 25 2010]. It is suggested that in the initial stage of a crater formation a meteorite, vertically entering the water, creates a radially symmetrical cavity on the water surface, which can be described by a simple function: imp 2 2 η (r) DC (1 r / RC ), r RD , (1) imp η (r) 0, r RD , where Dc is the depth of the cavity, Rc and RD are inner and outer radii of the cavity, respectively. In the case of RD 2RC the water ejected from the cavity, forms the outer splash – ring structure typical of the fall of the object in the water and the 30 volume of which corresponds exactly to the volume of water discharged from the cavity. 2 Considering that the meteorite kinetic energy is converted into the potential energy of the water level displacement, the following simple analytical formulas are derived to calculate the radius and depth of the cavity; see for instance, [Ward and Asphaug, 2000]: δ 1 1 δ 2δ 2ερ R3V 2 V 2 ρ 3 ρ 3 1 D I I I , R R 2ε I I w , (2) C 2 C I α1 ρwgRC gRI ρw ρI qRI 5 where w is density of water, g is gravitational acceleration, is the proportion of the kinetic energy of the meteorite, converted into the tsunami energy, I, RI, VI are density, radius and velocity of the meteorite q and are the coefficients, associated with the properties of the meteorite and the water layer. According to [Levin and Nosov, 2009] about 16% of the kinetic energy of the falling body is converted into the energy of tsunami waves. The parameter is 1.27 and the value of q and is calculated as follows: 0,26 1 ρ 1 10 δ , q 0,39 w . (3) 0,27 2α 2 ρI RI If the diameter of a meteorite is 1 meter, the density is 3.3 g/cm3 and the water entering velocity is 156 m/s, the inner radius of the cavity is equal to RC = 5.6 m, the outer radius is RD = 7.8 m and the depth of the cavity formed on the surface of Lake Chebarkul is DC = 3.2 m approximately. The resulting cavity is schematically depicted in Fig. 2. The obtained estimation of the cavity diameter (15.5 m) exceeds twice the size of the ice-hole (7-8 m) formed by the fall of a 15 Chelyabinsk meteorite fragment (Fig. 1). The estimation of the cavity size is made for the conditions of open water. If we take into account the ice cover, which prevents the water from forming vertical upward discharges at the edges of the resulting cavities, and consider only part of the underwater cavity (indicated in Fig. 2 by dashed lines), we will obtain the diameter of a little more than 10 meters, which is closer to the observed size of the ice-hole. Another reason for the differences in the estimation of the source size, in our opinion, might be the wrong extrapolation of empirical formulas for 20 the small size meteorites, especially since it does not enter the water vertically. 3 The hydrodynamic model of the tsunami source, based on the Navier-Stokes equations The models based on the numerical solution of the Navier-Stokes equations [Ferziger and Peric, 2002], which have become popular in solving the problems of asteroid tsunami, allow simulating a real meteorite entering the water [Kozelkov et al., 2015a, 2016a]. In this paper we solve the Navier-Stokes equations for a two-component (water and air) incompressible fluid 25 in a gravitational field, and both air and water are considered to be incompressible. The incompressibility / compressibility of air will mainly influence the rate of change of the wave parameters in the source during the collapse of the cavity. Qualitatively, the process will be described correctly, but the quantitative difference can be significant. This is directly related to the Mach number, determined by the characteristic velocity of the meteorite, which in our calculations is 156 m/s, 3 which is approximately 0.5 Mach number.
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