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Register a new userEnergy of tsunami waves generated by bottom motion
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Authors
Denys Dutykh
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In the vast literature on tsunami research, few articles have been devoted to energy issues. A theoretical investigation on the energy of waves generated by bottom motion is performed here. We start with the full incompressible Euler equations in the presence of a free surface and derive both dispersive and non-dispersive shallow-water equations with an energy equation. It is shown that dispersive effects only appear at higher order in the energy budget. Then we solve the Cauchy-Poisson problem of tsunami generation for the linearized water wave equations. Exchanges between potential and kinetic energies are clearly revealed.
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The generation of a tsunami wave by an aerial landslide is investigated through model laboratory experiments. We examine the collapse of an initially dry column of grains into a shallow water layer and the subsequent generation of waves. The experiments show that the collective entry of the granular material into water governs the wave generation process. We observe that the amplitude of the wave relative to the water height scales linearly with the Froude number based on the horizontal velocity of the moving granular front relative to the wave velocity. For all the different parameters considered here, the aspect ratio and the volume of the column, the diameter and density of the grains, and the height of the water, the granular collapse acts like a moving piston displacing the water. We also highlight that the density of the falling grains has a negligible influence on the wave amplitude, which suggests that the volume of grains entering the water is the relevant parameter in the wave generation.
The problem of tsunami wave run-up on a beach is discussed in the framework of the rigorous solutions of the nonlinear shallow-water theory. We present an analysis of the run-up characteristics for various shapes of the incoming symmetrical solitary tsunami waves. It will be demonstrated that the extreme (maximal) wave characteristics on a beach (run-up and draw-down heights, run-up and draw-down velocities and breaking parameter) are weakly dependent on the shape of incident wave if the definition of the significant wave length determined on the 2/3 level of the maximum height is used. The universal analytical expressions for the extreme wave characteristics are derived for the run-up of the solitary pulses. They can be directly applicable for tsunami warning because in many case the shape of the incident tsunami wave is unknown.
The purpose of this article is numerical verification of the thory of weak turbulence. We performed numerical simulation of an ensemble of nonlinearly interacting free gravity waves (swell) by two different methods: solution of primordial dynamical equations describing potential flow of the ideal fluid with a free surface and, solution of the kinetic Hasselmann equation, describing the wave ensemble in the framework of the theory of weak turbulence. Comparison of the results demonstrates pretty good applicability of the weak turbulent approach.
Myths and legends across the world contain many stories of deluges and floods. Some of these have been attributed to tsunami events. Doggerland in the southern North Sea is a submerged landscape thought to have been heavily affected by a tsunami such that it was abandoned by Mesolithic human populations at the time of the event. The tsunami was generated by the Storegga submarine landslide off the Norwegian coast which failed around 8150 years ago. At this time there were also rapid changes in sea level associated with deglaciation of the Laurentide ice sheet and drainage of its large proglacial lakes, with the largest sea level jumps occurring just prior to the Storegga event. The tsunami affected a large area of the North Atlantic leaving sedimentary deposits across the region, from Greenland, through the Faroes, the UK, Norway and Denmark. From these sediments, run-up heights of up to 20 metres have been estimated in the Shetland Isles and several metres on mainland Scotland. However, sediments are not preserved everywhere and so reconstructing how the tsunami propagated across the North Atlantic before inundating the landscape must be performed using numerical models. These models can also be used to recreate the tsunami interactions with now submerged landscapes, such as Doggerland. Here, the Storegga submarine slide is simulated, generating a tsunami which is then propagated across the North Atlantic and used to reconstruct the inundation on the Shetlands, Moray Firth and Doggerland. The uncertainty in reconstructing palaeobathymetry and the Storegga slide itself results in lower inundation levels than the sediment deposits suggest. Despite these uncertainties, these results suggest Doggerland was not as severely affected as previous studies implied. It is suggested therefore that the abandonment of Doggerland was primarily caused by rapid sea level rise prior to the tsunami event.
A new method for the determination of the Alfven wave energy generated during magnetic reconnection is introduced and used to analyze the results from two-dimensional MHD simulations. It is found that the regions with strong Alfven wave perturbations almost coincide with that where both magnetic-field lines and flow-stream lines are bent, suggesting that this method is reliable for identifying Alfven waves. The magnetic energy during magnetic reconnection is mainly transformed into the thermal energy. The conversion rate to Alfven wave energy from the magnetic energy is strongly correlated to the magnetic reconnection rate. The maximum conversion rate at the time with the peak reconnection rate is found to be only about 4% for the cases with the plasma beta=0.01,0.1, and 1.0.
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