Nonlinear Mechanism of Tsunami Wave Generation by Atmospheric Disturbances E
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Nonlinear mechanism of tsunami wave generation by atmospheric disturbances E. Pelinovsky, T. Talipova, A. Kurkin, C. Kharif To cite this version: E. Pelinovsky, T. Talipova, A. Kurkin, C. Kharif. Nonlinear mechanism of tsunami wave generation by atmospheric disturbances. Natural Hazards and Earth System Sciences, Copernicus Publ. / European Geosciences Union, 2001, 1 (4), pp.243-250. hal-00330899 HAL Id: hal-00330899 https://hal.archives-ouvertes.fr/hal-00330899 Submitted on 1 Jan 2001 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Natural Hazards and Earth System Sciences (2001) 1: 243–250 c European Geophysical Society 2001 Natural Hazards and Earth System Sciences Nonlinear mechanism of tsunami wave generation by atmospheric disturbances E. Pelinovsky1, 2, T. Talipova1, A. Kurkin2, and C. Kharif3 1Laboratory of Hydrophysics and Nonlinear Acoustics, Institute of Applied Physics, Russian Academy of Science, Nizhny Novgorod, Russia 2Department of Applied Mathematics, Nizhny Novgorod State Technical University, Nizhny Novgorod, Russia 3Ecole Superieure´ de Mecanique,´ Universite´ de la Mediterran´ ee,´ Marseilles, France Received: 24 July 2001 – Revised: 29 October 2001 – Accepted: 30 October 2001 Abstract. The problem of tsunami wave generation by vari- of long waves of 31 m/s (Rabinovich and Monserrat, 1998), able meteo-conditions is discussed. The simpliﬁed linear and so the resonance effects should be very important. nonlinear shallow water models are derived, and their ana- lytical solutions for a basin of constant depth are discussed. The given paper analyzes the resonance effects of tsunami The shallow-water model describes well the properties of the generation by moving atmospheric disturbances which lead generated tsunami waves for all regimes, except the reso- to the appearance of anomalous large waves. The paper is nance case. The nonlinear-dispersive model based on the organized as following. The nonlinear and corresponding forced Korteweg-de Vries equation is developed to describe linear shallow-water theory of tsunami generation by the at- the resonant mechanism of the tsunami wave generation by mospheric disturbances is brieﬂy discussed in Sect. 2. The the atmospheric disturbances moving with near-critical speed resonance character of tsunami generation is investigated in (long wave speed). Some analytical solutions of the nonlin- Sect. 3. Detailed calculations are given for 1D case, where all ear dispersive model are obtained. They illustrate the dif- analytical expressions can be obtained in the explicit form. ferent regimes of soliton generation and the focusing of fre- The similarity of the considered problem to the problem of quency modulated wave packets. tsunami generation by moving landslides is mentioned here (Pelinovsky, 1996; Pelinovsky and Poplavsky, 1996; Tinti and Bortolucci, 2000a, b; Tinti et al., 2001). A nonlinear- dispersive model of the resonant generated tsunami waves is 1 Introduction brieﬂy reproduced in Sect. 4. This model is based on the famous forced Korteweg-de Vries equation derived ﬁrst by Tsunami waves may be generated by underwater earth- Akylas (1984). The solitary wave generation and interaction quakes, submarine landslides, rockslides, volcano explosions with moving forcing is studied in Sect. 5 based on the re- and rapid anomalous changes in the atmospheric pressure sults by Grimshaw et al. (1994). There is a lot of various over the sea (Murty, 1977; Pelinovsky, 1996). For a tsunami scenaria of wave dynamics depending on the speed and sign to arise, it is necessary that the water surface deviate from of the atmospheric disturbance. It is shown that the maxi- its equilibrium on a sufﬁciently large area. In these cases the mal variation of wave amplitude in the process of interaction shallow water theory or long wave theory is the good theo- can reach two to three times. The effect of wave focusing retical and numerical model to describe the properties of the of nonlinear dispersive wave packets is analyzed in Sect. 6. tsunami waves. In many cases, the tsunami source moves Such an effect is well-known for wind-generated waves and with variable speed and direction. Such a situation is typical leads to the generation of freak or rogue waves (Osborne et for meteorological tsunamis and this mechanism is investi- al., 2000; Kharif et al., 2001). The same process of wave gated by Eﬁmov et al. (1985) and Rabinovich (1993). Mete- focusing with formation of the large-amplitude waves is pos- orological tsunamis have occurred in the Mediterranean Sea sible for tsunami waves. Obtained results are summarized in and in the Okhotsk Sea (Rabinovich and Monserrat, 1998). the conclusion. The main goal of this paper is to demonstrate In particular, near the Balearic Islands, the estimated atmo- the use of possible analytical tests that characterize the role spheric wave speed of 29 m/s is very close to the phase speed of nonlinearity, dispersion and forcing in the processes of the tsunami generation by atmospheric disturbances; such solu- Correspondence to: E. Pelinovsky tions are important for interpretation of ﬁeld data and results ([email protected]) of the numerical simulations. 244 E. Pelinovsky et al.: Nonlinear mechanism of tsunami wave generation wavelength. Here a simpler algorithm is used, which consists of neglecting vertical acceleration, dw/dt in Eq. (2). In this case Eq. (2) is integrated and, with the dynamic boundary condition (Eq. 6) taken into account, determines the hydro- static pressure p = patm + ρg(η − z). (7) Substituting Eq. (7) into Eq. (1) and neglecting the vertical velocity once again, we obtain the ﬁrst equation of the long- Fig. 1. Problem geometry. wave theory ∂η ∇p + (u∇)u + g∇η = − atm . (8) 2 Shallow water model of the tsunami generation by at- ∂t ρ mospheric disturbances The second equation is yielded by integration of Eq. (3) over the depth from the bottom z = h(x, y) to the surface The basic hydrodynamic model of tsunami generation by the z = η(x, y, t), taking into account boundary conditions atmospheric disturbances is based on the well-known Euler (Eq. 4 and 5), as well as the fact that horizontal velocity does equations for ideal ﬂuid on the non-rotated Earth not depend on vertical coordinate, z ∂u ∂u 1 + (u∇)u + w + ∇p = 0, (1) ∂η h i ∂t ∂z ρ + ∇ (h + η)u = 0. (9) ∂t ∂w ∂w 1 ∂p + (u∇)w + w + = −g, (2) η ∂t ∂z ρ ∂z Equations (8 and 9) are closed as related to functions and u. They are nonlinear (the so-called nonlinear shallow- ∂w ∇u + = 0 (3) water theory), inhomogeneous (the right-hand part is non- ∂z zero), and with variable coefﬁcients due to h(x, y). with corresponding boundary conditions at the bottom and The linear version of the shallow-water theory is most gen- ocean surface (geometry of the problem is shown in Fig. 1). erally used within the tsunami problem. In this case varia- At the solid bottom z = −h(x, y), tions of water depth are assumed weak, as well as the velocity of the ﬂuid. As a result, we obtain a linear set of equations w − (u∇)h = 0. (4) ∂u ∇p + g∇η = − atm , (10) At the free surface z = η(x, y, t) , the kinematic condition ∂t ρ is dη ∂η ∂η w = = + (u∇)η, (5) + ∇[hu] = 0. (11) dt ∂t ∂t and the dynamic condition, It is convenient to exclude u and pass over to the wave equa- tion for the surface elevation p = p (x, y, t). (6) atm ! ∂2η h Here η(x, y, t) is the elevation of water surface, u and w − ∇c2∇η = ∇ ∇p , (12) ∂t2 ρ atm are horizontal u = (u, v) and vertical components of the velocity ﬁeld, x and y are coordinates in the horizontal where plane; z-axis is directed upwards vertically, ρ is water den- p sity, p is pressure and patm is the variable atmospheric pres- c(x, y) = gh(x, y) (13) sure, g is gravity acceleration, h(x, y) is the variable ocean depth. Differential operator, ∇ acts in the horizontal plane is the speed of long wave propagation. Equation (12) is the ∇ = {∂/∂x, ∂/∂y}. basic one within the linear theory of tsunami generation and Usually tsunami waves are long (as compared with the must be supplemented by the initial conditions. It is natural ocean depth). Therefore, it is natural ﬁrst to consider the to believe that at the initial moment the ocean is quiet, i.e. long-wave (or shallow-water) approximation of the model η = 0, u = 0, v = 0, or ∂η/∂t = 0, (14) of tsunami generation, and then to estimate conditions of its applicability. The theory of long waves is based on the although due to linearity of Eq. (12), a more general case can main assumption that the vertical velocity and acceleration also be considered. From the point of view of mathematical are low as compared to the horizontal ones and can be calcu- physics, the wave (Eq. 12) is too well studied to discuss the lated from the initial system using the asymptotic procedure. details of its solution here. The governing and long-wave As a small parameter it uses the ratio of the vertical velocity systems presented here are the basic hydrodynamic models to the horizontal one or the ocean depth to the characteristic of tsunami generation by the atmospheric disturbances.