Gian Mario Beltrami

Internal generation of waves in 2D fully elliptic mild-slope equation FEM models

Abstract This paper looks at an alternative approach for the treatment of open boundaries in linear wave field simulations by means of fully elliptic mild-slope equation (MSE) finite elements models. In these kinds of models, the domain of investigation is traditionally contoured both by reflecting–absorbing boundaries, which simulate the coastline or the structures that emerge from the sea, and by an ‘open’ or ‘artificial’ boundary, which separates the sea region included in the domain from the semi-infinite region that extends outward to infinity. The approach presented here assumes the domain to be completely contoured by reflecting–absorbing boundaries. A total absorbing boundary is, in particular, assumed to separate the inner (finite) from the outer (semi-infinite) sea region. Sources of energy, which generate waves of specified height and period, are located within the domain along a line in the proximity of the inner–outer sea region boundary. Reflected and scattered waves can propagate over the generation line and are absorbed at the open boundary. Numerical tests have been carried out to simulate progressive and stationary waves in a channel and long waves around a fully reflecting circular island on a parabolic shoal, and to evaluate the amplification factors of a long and narrow bay. All these validation tests show a very good agreement with the available analytical solutions. A discussion is finally carried out on the advantages and disadvantages of the presented approach with respect to traditional ones.

Forecasting Landslide Generated Tsunamis: a Review

When earthquakes occur, suddenly bottom deformations will induce water response that is almost instantaneous. Then, bottom displacements affect wide areas and generate transient waves (tsunamis) usually characterized by a relatively small heightwith respect to their length, i.e. by a small steepness. Indeed, just after the earthquake, the water free surface mimics the final bottom deformations, that can be of the order of some meters, and small amplitude long waves start to propagate resulting in a potential transoceanic devastation (Synolakis et al., 2002). When either subaerial or submerged landslides occur, thedisplacementswill takeplace onboth larger temporal-, and smaller spatial-scale. The deformations are of the order of hundreds of meters and the generated waves are quite different from those induced by submarine earthquakes. Hence, landslide tsunamis tend to be a local phenomenon, although extreme. Since two centuries ago, submarine failures were related to “sea waves without earthquake” when great waves were observed without any account of earthquakes (e.g. Mallet & Mallet, 1858; Milne, 1898; Montessus de Ballore, 1907). In an illuminating reasoning Mallet & Mallet (1858) argued that “great underwater slippage takes place [...] the effect upon the surface of the sea is at the same moment to originate a positive and a negative wave. [...] The magnitude of the wave raised is dependent upon that of the mass of solid material that has suddenly changed its place, upon the depth of water in which the slippage has occurred, upon the rapidity of the transposition, and in minor degree upon the form and material of the portion of the bank that has slipped.” One of the more meaningful event occurred at Lituya Bay (Alaska, July 9, 1958), when a subaerial landslide triggered by an earthquake entered the water at the Gilbert Inlet and generated an impulse wave (Miller, 1960). The induced run-up was the highest known: the water rose up to 524m at the slope in front of the landslide, then thewaves propagated into the Bay and eventually radiated offshore through the Bay inlet with minimum effects outside the Bay itself. Another example, related to submarine failures, is the well documented tsunami generated by an underwater slump triggered by a magnitude 7 earthquake on July 17, 1998 close to the Sissano Lagoon, Sandaun Province, Papua NewGuinea (see Synolakis et al., 2002). The tsunami struck about 30 km wide area, i.e. rather small, by inducing a maximum run-up of about 15 m and killing over 2100 people. Landslide generated impulse waves can occur not only at the sea boundaries, but also in enclosed basins (i.e. reservoirs and lakes). The event that took place at the Vajont Valley (Italy) on October 9, 1963 is perhaps the more sorrowful one. A subaerial landslide of about 6

Gravitational mode calculation of basins discretized by orthogonal curvilinear grids

Abstract This paper presents a simple and straightforward method for carrying out the direct numerical solution of the eigenvalue problem associated to the homogeneous linear shallow-water equations expressed using orthogonal curvilinear coordinates, when ‘adiabatic’ boundary conditions apply. These equations, together with the boundary conditions, define a self-adjoint problem in the continuum. The method presented here, which is thought for calculating the 2-D theoretical gravity modes of both natural and artificial basins, relies on a change of basis of the dependent variable vector. This preliminary transformation makes it, in fact, possible to formulate two different numerical approaches which guarantee the self-adjoint property of the discrete form of the system consisting of the governing equations and the boundary conditions. The method is tested using a square and a fully circular domain, both of which allow comparisons with well-known analytical and numerical solutions. Discretizing the physical domain of a fully circular basin by a cylindrical coordinate grid makes it possible to show the actual efficiency of the method in calculating the theoretical gravity modes of basins discretized by a boundary-following coordinate grid which allows laterally variable resolution.

Algorithms for Automatic, Real-Time Tsunami Detection in Sea Level Measurements

Automatic, real-time tsunami detection in sea-level measurements is a main component of a tsunami early warning system (TEWS). Although a great effort has been recently undertaken by the scientific and engineering community in developing new technologies (e.g. satellite altimetry, detectors of low-frequency elastic oscillations associated to a tsunami) capable of increasing the awareness of potential tsunamis in the minimum amount of time, at present direct detection in sea level measurements is still the main mean to confirm their actual generation and propagation. Clearly, the device used to collect these measurements has to be chosen between those equippedwith sensors capable of detecting sea-level oscillationswithin the tsunami frequency band. At present, the main devices that can be actually used should belong to the following classes: