Analytical evolution of tsunamis induced by near-shore earthquakes on a constant-slope ocean

Abstract

Strong near-shore earthquakes are the most frequent sources of tsunamis in many oceans of the world. In the framework of the nonlinear shallow-water theory, the initial sea-surface tsunami elevation is assumed to equal the sea-floor co-seismic displacement produced by the seismic event. This is quantified by means of the analytical formulas due to Okada (1985, 1992), dealing with seismic faults buried in an elastic medium. In this work the propagation of tsunamis is studied along two-dimensional profiles on an idealized constant-slope sea bed, an approximation that allows one to reduce the governing nonlinear equations to a linear problem by means of the classical Carrier & Greenspan (1958) approach. We introduce an analytical solution that is sufficiently general to account for initial conditions associated with paradigmatic cases of sea-bottom deformations produced by near-shore earthquakes, such as subsidence or uplift of the coastal area, and can be also used to treat more complex deformations. The main result is that the amplification of the tsunami height at the coast is found to range between approximately 1 and 2. The amplification is around 1 for tsunamis induced by earthquakes with their epicentre inland and tends to grow as the fault moves seaward. We restrict our analysis to earthquakes that dislocate the shore region. Within the class of sources that we consider, the tsunamis that are most amplified are the ones having initial profiles with a crest–trough–crest system or conversely with a trough–crest–trough system. The bottom slope is found to have no effect on tsunami run-ups and run-downs, but to influence tsunami periods and tsunami speed remarkably. Breaking analysis shows that wave breaking does not occur if the initial wave height is less than 8–9 m, and that the simplest sea-level profiles, which are associated with earthquakes with their epicentre on land, are not expected to break even if their initial height exceeds 19 m.

Runup of Tsunami Waves in U-Shaped Bays

“…In the nonlinear theory the same effects can be found for a particular case of the linearly inclined bay with a parabolic cross-section m = 2 ). The solution of the nonlinear problem in this case can be obtained with the use of the Legendre (hodograph) transformation, which has been very popular for long wave runup on a plane beach (Carrier and Greenspan, 1958;Pedersen and Gjevik, 1983;Synolakis, 1987;Tadepalli and Synolakis, 1996;Li, 2000;Li and Raichlen, 2001;Carrier et al, 2003;Kânoğlu, 2004;Tinti and Tonini, 2005;Kânoğlu and Synolakis, 2006;Didenkulova et al, 2006;2008a;Antuono and Brocchini, 2007;Pritchard and Dickinson, 2007) and is valid for non-breaking waves. In this case the nonlinear system (3) can be reduced to the linear equation (Choi et al, 2008; …”

Two- and three-dimensional computation of solitary wave runup on non-plane beach

“…The main advantage of this form is that the moving (unknown) shoreline now corresponds to σ =0 (since the total depth H =0) and therefore, solution (20) is determined in the halfline −∞<σ <0 with fixed boundary. Such transformation generalizes the Carrier-Greenspan transformation (Carrier and Greenspan, 1958) actively used in the theory of long wave runup on plane beaches (Spielfogel, 1976;Pedersen and Gjevik, 1983;Synolakis, 1987;Pelinovsky and Mazova, 1992;Pelinovsky, 1996;Carrier et al, 2003;Kânoglu, 2004;Tinti and Tonini, 2005;Kânoglu and Synolakis, 2006;Didenkulova et al, 2006;Antuono and Brocchini, 2007). It is important to mention that in the case of the "parabolic" crossslope we have the simplified analytical solution to compare with the classical case of a plane slope beach when the general solution can be expressed in the integral form only.…”

The UBO-TSUFD tsunami inundation model: validation and application to a tsunami case study focused on the city of Catania, Italy

“…Their paper can be considered as a milestone in tsunami analytical studies and inspired several works in the following years (e.g. Peregrine, 1967;Thacker, 1981;Synolakis, 1987;Tadepalli and Synolakis, 1994;Li and Raichlen, 2001;Carrier et al, 2003;Kanoglu et al, 2004;Tinti and Tonini, 2005) that had also the purpose to provide analytical reference solutions for numerical models. An interesting and exhaustive review of analytical methods and solutions that have been developed in the frame of tsunami science was prepared some years ago by Synolakis and Bernard (2006).…”