RUSSIAN JOURNAL OF EARTH SCIENCES VOL. 8, ES5003, doi:10.2205/2006ES000210, 2006

Numerical Simulation and Results

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Figure 2
[9]  The model used for the simulation of the wave of 1963 is based on the assumption of layered sediments resting on a relatively rigid base (Figure 2). The parameters involved in the calculations were the density of a layer rs, shear modulus G, bulk modulus K, cohesion c, maximum friction angle js, and tensile strength s [Garagash et al., 2003]. It is supposed that the volume of the landslide body does not depend on the disturbance magnitude and is rather controlled by mechanical properties of its components, for example, the angle of internal friction. An additional dynamic effect can be a triggering mechanism exerting a pushing action on the landslide mass on a slope until the critical stress is exceeded.

[10]  After the situation becomes unstable, the sliding process continues due to the accumulated potential energy. We introduced an initial dynamic action on the landslide process equivalent to a moderate earthquake and lasting for 6 s (Figure 2b) [Garagash et al., 2003].

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Figure 3
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Figure 4
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Figure 5
[11]  The stability of submarine slopes was analyzed using the FLAC code [Garagash et al., 2003] implementing an explicit 3-D finite difference scheme applied in geomechanical modeling. In contrast to the finite element method, the explicit difference scheme enables the modeling of nonlinear behavior of pore-saturated sedimentary masses under the condition of plastic flow above the yield strength [Garagash and Ermakov, 2001; Garagash et al., 2003]. The modeling was performed in two stages. At the first stage, we modeled the initial prestressed state of the sedimentary slope formed due to the gravity force and the seawater saturation under the sea pressure conditions. We considered layer-by-layer sliding of an upper part of the elastoplastic sedimentary layer on the slope surface that forms during the landslide process; a distinct interface is supposed to exist between water and the landslide body. The water density rw(x, z) is constant and the landslide density is a function of coordinates: rs = rs(x, z). The sliding process of sedimentary masses is essentially controlled by the friction angle. In this work, the numerical simulation was performed for the maximum friction angle fs = 20o. Assuming that the process stops whenever the velocity of the leading front of landslide masses becomes equal to zero, we found that the landsliding process lasts for 45 s. The maximum velocity of the leading front of the landslide is about 6 m s-1 and is attained at about 40 s after the beginning of the process. Figures 3, 4, and 5 show the evolution of particle velocities in the landslide body during the sliding of the sedimentary layer, the plastic strain of the landslide body during the sliding process, and the distribution of the shear strain for the angle fs = 20o and 32o. (The computations were performed for four time moments (10, 25, 40, and 55 s; the sliding process virtually stopped at the latter time moment.)

[12]  At the second stage, we performed a numerical simulation of landslide-generated surface water waves on the basis of the nonlinear system of shallow water equations with the use of a specially modified explicit difference scheme including the check for stability conditions (see below).

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Figure 6
[13]  In the computations, the origin of coordinates was chosen on the shoreline; the seaward x axis coincided with the level of the undisturbed water surface; and the z axis was directed upward (Figure 6), where z = -h(x, t) is the variable water depth and z = -h(x, 0) = hs(x) is the profile of the submarine slope that existed before the disturbance onset ( t = 0). D (x, t) is the thickness of the landslide body: h (x, t) = hs (x, t) - D (x, t). The shape of the free water surface is denoted as z = h(x, t) with h(x, 0) = 0. Figure 5 shows the successive positions of the landslide surface at a time step of 10 s. It is well seen that the landslide moves downward on the slope, continuously changes its shape and, shifting to the right, reaches the final position at a distance of about 1300 m from the shoreline ( x = 0). It is noteworthy that the upper part of the sliding body seems to move leftward as well, involving the dry coast. As a result, the immobile shoreline point ( x = 0) becomes mobile, moving leftward and reaching the position x = -200 m at the final moment of the landsliding process ( t = 45 s). The same effect is observed in a purely viscous model of a landslide. However, a landslide in the viscoplastic model moves only downward on the slope [Jiang and LeBlond, 1993, 1994].

[14]  The above modeling results obtained for the landslide movement were used for numerical simulation of the landslide-generated surface water wave. The wave generation was computated using the ordinary nonlinear system of shallow water equations, which can be written as

eq001.gif(1)

where x is the horizontal coordinate and u is the horizontal component of the water particle velocity in the wave. The coupling between the landslide masses and seawater during the surface water wave generation was described through the related continuity equation:

eq002.gif(2)

We should note here that, in contrast to the existing models [Fine et al., 1998; Iwasaki, 1997; Jiang and LeBlond, 1993, 1994], the second term on the right-hand side of (2) does not vanish because the landslide in the model under consideration moves on a surface formed by the sliding process rather than on a stationary slope surface, as is generally supposed [Fine et al., 1998; Iwasaki, 1997; Jiang and LeBlond, 1993, 1994].

[15]  To solve numerically the system of equations (1) and (2), we used an explicit two-layer scheme of the first order in time and the second order in spatial coordinates:

eq003.gif(3)

where

eq004.gif

eq005.gif

and l1 and l2 are the coefficients of artificial viscosity subjected to the condition

eq006.gif(4)

Using the perturbation method, we obtained the following stability condition for the given difference scheme:

eq007.gif(5)

where l min = min{l1, l2}, n is the time index of the point, and i is the index in the spatial variable.

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Figure 7
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Figure 8
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Figure 9
[16]  Figure 7 illustrates the evolution of the surface water wave form above the moving landslide. As is well seen, the landslide generates a dipolar wave with a trough facing the coast and a crest moving seaward. This appears to be evident because the landslide movement on the seafloor is known to generate an elevation wave above the leading front of the landslide and a depression wave above its rear front [Mazova, 2003; Iwasaki, 1997]. The situation does not change if the landslide moves on an inclined surface: an elevation wave arises above the leading front of the landslide, and a depression wave arises above the landslide "tail", thereby forming a run-down wave from the slope where landslide occurs. The run-down wave increases until the time moment t = 15 s. As is well seen, the length of this depression wave is three times as short as the wavelength of the elevation wave (the first wave crest). Beginning from t = 15 s, an additional crest (the second one in the generated wave) forms at the back front of the elevation wave; with the further movement of the landslide, it reaches the free water surface ( t = 30 s) and continues to build up. At the same time, the receding distance of the depression wave decreases and the trough disappears at t = 40 s, implying that the elevation wave approached the shoreline. At t = 45 s, the height of the crest moving toward the shoreline attains 6 m, whereas the height of the first, seaward crest is no more than 4 m (Figure 8). Figure 9 plots the velocity characteristics of the generated wave at various time moments. It is well seen that the maximum horizontal velocity of water particles in the wave receding from the dry shore does not exceed 5 m s-1, while this velocity is about 3 m s-1 in the crest region of the seaward elevation wave; in the newly formed second crest, the horizontal velocity is directed toward the shoreline and, at the final generation stage, when (the landslide stops), attains 6 m s-1. More detailed distributions of the velocity on surface water wave profiles at time moments in the interval t
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Figure 10
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Figure 11
= 15-45 s are presented in Figure 10. As is well seen at t = 15 s, the leading and trailing fronts of the first trough (the run-down wave) move toward each other. This pattern results in the formation of the second crest at the back front of the trough (at t = 25 s). As clearly seen from Figure 10, the velocities of both the back front of the smaller crest and the trough following it are directed toward the shoreline, while the first, larger crest moves seaward (the arrows in the figure). It is seen that, during the immobilization of the landslide ( t = 45 s), the surface water wave is discomposed into two wave trains: the first moves toward the open sea, while the second train consistings of the two depression waves and the wave crest moves toward the dry shore ( t = 55 s). The second wave train yields a vertical run-up of about 8 m at the isobath x = -200 m. Figure 11 shows the evolution of the shapes of the first and the second crests for time moments in the interval t = 45-65 s (after the landslide movement stops). The wave traveling into the sea retains its form and a height of about 4 m.

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Figure 12
[17]  Figure 12 presents 10 time moments of the generation of a tsunami wave due to underwater landslide movement, its propagation and run-up on the coast approximated by a plane slope. Computations showed that the heights of wave run-up of landslide origin are up to 6 m and the heights of the run-up at the opposite shoreline of the bay is of the order of 4 m; these values are well consistent with available observational data.


RJES

Citation: Lobkovsky, L. I., R. Kh. Mazova, I. A. Garagash, and L. Yu. Kataeva (2006), Numerical simulation of the 7 February 1963 tsunami in the Bay of Corinth, Greece, Russ. J. Earth Sci., 8, ES5003, doi:10.2205/2006ES000210.

Copyright 2006 by the Russian Journal of Earth Sciences

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