Abstract
Abstract. Run-up of long waves on a beach consisting of three pieces of constant but different slopes is studied. Linear shallow-water theory is used for incoming impulse evolution, and nonlinear corrections are obtained for the run-up stage. It is demonstrated that bottom profile influences the run-up characteristics and can lead to resonance effects: increase of wave height, particle velocity, and number of oscillations. Simple parameterization of tsunami source through an earthquake magnitude is used to calculate the run-up height versus earthquake magnitude. It is shown that resonance effects lead to the sufficient increase of run-up heights for the weakest earthquakes, and a tsunami wave does not break on chosen bottom relief if the earthquake magnitude does not exceed 7.8.
Highlights
Resonance phenomena play a significant role in the run-up amplification and lead to different physical effects for waves in coastal zones: long duration of water oscillations, later arrival of waves with maximal amplitude compared with leading waves, and group structure of waves
It should be noted that different formulas for maximum run-up of solitary waves of various shapes can be provided in terms of wave amplitude and significant wave length describing practically important cases with good accuracy (Didenkulova et al, 2008; Didenkulova and Pelinovsky, 2008; Antuono and Brocchini, 2010a, b)
The run-up of tsunami waves on the coast is studied for the following bottom geometry: ocean of constant depth, steep continental slope, and beach of gentle constant slope
Summary
Resonance phenomena play a significant role in the run-up amplification and lead to different physical effects for waves in coastal zones: long duration of water oscillations, later arrival of waves with maximal amplitude compared with leading waves, and group structure of waves. In all studies mentioned above, the rigorous analytical solutions are obtained if the wave propagates on a plane beach of constant slope Such a plane can approximate the face-shore bathymetry only, and it has to be matched with a horizontal bottom profile. If the incident wave has a bell shape, the water oscillations on the shore repeat its shape if the bottom slope is big (limiting case is a vertical wall), and accompanied by the negative second oscillation if the bottom slope is small Such behavior is explained by the resonance effects, which are weak for such geometry – from a physical point of view, it is an open resonator (Pelinovsky, 1996, 2006; Madsen and Fuhrman, 2008).
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