Linear Long-wave Equations Research Articles

Real-Time Tsunami Inundation Forecast for a Recurrence of 17thCentury Great Hokkaido Earthquake in Japan

Paleotsunami studies have shown that several large tsunamis hit the Pacific coast. Many tsunami deposit data were available for the 17thcentury tsunami. The most recent tsunami deposit study in 2013 indicated that the large slip of about 25 m along the plate interface near the Kurile trench would be necessary and the seismic moment of this 17thcentury earthquake was 1.7 × 1022Nm. If a great earthquake like the 17thcentury earthquake occurs off the Pacific coast of Hokkaido, the devastating disaster along the coast is expected. To minimize the tsunami disaster, a development of the real-time forecast of a tsunami inundation area is necessary. Estimating a tsunami inundation area requires tsunami numerical simulation with a very fine grid system of less than 1 arcsecond. There is not enough time to compute the tsunami inundation area after a large earthquake occurs. In this study, we develop a real-time tsunami inundation forecast method using a database including many tsunami inundation areas previously computed using various fault models. After great earthquakes, tsunamis are computed using linear long-wave equations for fault models estimated in real time. Simulating such tsunamis takes only 1-3 minutes on a typical PC, so it is potentially useful for forecasting tsunamis. Tsunami inundation areas computed numerically using various fault models and tsunami waveforms at several locations near the inundation area are stored in a database. Those computed tsunami waveforms are used to choose the most appropriate tsunami inundation area by comparing them to the tsunami waveforms computed in real time. This method is tested at Kushiro, a city in Hokkaido. We found that the method worked well enough to forecast the Kushiro’s tsunami inundation area.

An Extended Analytic Solution of Combined Refraction andDiffraction of Long Waves Propagating over Circular Island

An analytic solution of long waves scattering by a cylindrical island mounted on a permeable circular shoal was obtained by solving the linear long wave equation (LWE). The solution is in terms of the Bessel function expressed by complex variables. The present solution is suitable for arbitrary bottom configurations described by a power function with two independent parameters. For the case of the paraboloidal shoal, there exists a singular point (α = 2) which can be removed using Frobenius series, where α is a real constant. The present solution is reduced to Yu and Zhang’s (2003) solution for impermeable circular shoal. The numerical results show some special features of the combined effect of wave refraction and diffraction caused by a porous circular island. The effect of key parameters of the island dimension, the shoal slope, and permeability on wave scattering was discussed based on the analytic solution.

Retrieval of long-wave tsunami Green’s function from the cross-correlation of continuous ocean waves excited by far-field random noise sources on the basis of a first-order Born approximation

We investigate the theoretical background for the retrieval of the tsunami Green’s function from the cross-correlation of continuous ocean waves. Considering that a tsunami is a long-wavelength ocean wave described by 2-D linear long-wave equations, and that the sea-bottom topography acts as a set of point-like scatterers, we use a first-order Born approximation in deriving the tsunami Green’s function having coda waves. The scattering pattern is non-isotropic and symmetrical with respect to the forward and backward directions. We indicate a retrieval process which shows that the derivative of the cross-correlation function of wavefields at two receivers with respect to the lag time gives the tsunami Green’s function when point noise sources generating continuous ocean waves are distributed far from, and surrounding, the two receivers. Note that this relation between the cross-correlation and the Green’s function is different from the case in which uncorrelated plane-wave incidence from all directions is assumed to be continuous ocean waves. The Green’s function retrieved from continuous ocean waves will be used as a reference to examine the validity of the Green’s function obtained by numerical simulations.

Validation and Joint Inversion of Teleseismic Waveforms for Earthquake Source Models Using Deep Ocean Bottom Pressure Records: A Case Study of the 2006 Kuril Megathrust Earthquake

The importance of accurate tsunami simulation has increased since the 2004 Sumatra-Andaman earthquake and the Indian Ocean tsunami that followed it, because it is an important tool for inundation mapping and, potentially, tsunami warning. An important source of uncertainty in tsunami simulations is the source model, which is often estimated from some combination of seismic, geodetic or geological data. A magnitude 8.3 earthquake that occurred in the Kuril subduction zone on 15 November, 2006 resulted in the first teletsunami to be widely recorded by bottom pressure recorders deployed in the northern Pacific Ocean. Because these recordings were unaffected by shallow complicated bathymetry near the coast, this provides a unique opportunity to investigate whether seismic rupture models can be inferred from teleseismic waves with sufficient accuracy to be used to forecast teletsunami. In this study, we estimated the rupture model of the 2006 Kuril earthquake by inverting the teleseimic waves and used that to model the tsunami source. The tsunami propagation was then calculated by solving the linear long-wave equations. We found that the simulated 2006 Kuril tsunami compared very well to the ocean bottom recordings when simultaneously using P and long-period surface waves in the earthquake source process inversion.

A perturbation DRBEM model for weakly nonlinear wave run-ups around islands

In this paper, the dual reciprocity boundary element method (DRBEM) based on the perturbation method is presented for calculating run-ups of weakly nonlinear long waves scattered by islands. Under the assumption that the incident waves are harmonic, the time-dependent nonlinear Boussinesq equations are transformed into three time-independent linear equations by using the perturbation method, where, besides nonlinearity ε , the dispersion μ 2 is also included in the perturbed expansion. The first-order solution η 0 is found by using the linear long-wave equations. Then η 0 is used in two second-order governing equations related to the dispersion and nonlinearity, respectively. Since no any omission and approximation for the seabed slope ∇ h and its derivatives is made, there are the third- and fourth-order partial derivatives of η 0 appeared in the right-hand sides of two governing equations of the second-order. By employing a transformation, those third- and fourth-order partial derivatives are removed therefore large errors in approximating these derivatives are eliminated. To validate the new model, wave diffractions around a large vertical cylinder for 13 cases are first considered. It is found that the nonlinear contributions to the new model are significant for weakly nonlinear waves with a much better comparison with experimental results obtained than for the linear diffraction theory. It is also found that the dispersive effects play an important role in improving the accuracy of the new model as numerical results obtained from the Boussinesq equations (with dispersion terms) are more accurate than those from the Airy's equations (without dispersion term). Then the combined wave diffraction and refraction by a conical island is also modelled and discussed. Our model is not only accurate as the dispersive effects have been included but also computationally efficient since the domain integrals are merely evaluated by distributing collocation points over that surface.

Source parameters controlling the generation and propagation of potential local tsunamis along the Cascadia margin

The largest uncertainty in assessing hazards from local tsunamis along the Cascadia margin is estimating the possible earthquake source parameters. We investigate which source parameters exert the largest influence on tsunami generation and determine how each parameter affects the amplitude of the local tsunami. The following source parameters were analyzed: (1) type of faulting characteristic of the Cascadia subduction zone, (2) amount of slip during rupture, (3) slip orientation, (4) duration of rupture, (5) physical properties of the accretionary wedge, and (6) influence of secondary faulting. The effect of each of these source parameters on the quasi-static displacement of the ocean floor is determined by using elastic three-dimensional, finite-element models. The propagation of the resulting tsunami is modeled both near the coastline using the two-dimensional (x-t) Peregrine equations that includes the effects of dispersion and near the source using the three-dimensional (x-y-t) linear long-wave equations. The source parameters that have the largest influence on local tsunami excitation are the shallowness of rupture and the amount of slip. In addition, the orientation of slip has a large effect on the directivity of the tsunami, especially for shallow dipping faults, which consequently has a direct influence on the length of coastline inundated by the tsunami. Duration of rupture, physical properties of the accretionary wedge, and secondary faulting all affect the excitation of tsunamis but to a lesser extent than the shallowness of rupture and the amount and orientation of slip. Assessment of the severity of the local tsunami hazard should take into account that relatively large tsunamis can be generated from anomalous ‘tsunami earthquakes’ that rupture within the accretionary wedge in comparison to interplate thrust earthquakes of similar magnitude.

Effects of Friction on a Localized Structure in a Baroclinic Current

Abstract In this paper, the effects of bottom and interfacial friction on localized baroclinic instability are discussed in the weakly nonlinear, long-wave limit. Using a quasigeostrophic two-layer model in which the lower layer is assumed to be deep, we have derived a coupled evolution equation set that consists of a KdV-type equation for the upper layer and a linear long-wave equation for the lower layer. A perturbation theory reveals that there are multiple equilibria in this system, where baroclinic energy conversion and frictional dissipation are in balance; the flow is not forced externally, and multiplicity here refers to the presence or absence of solitary waves propagating steadily on a zonal flow. Further, direct numerical calculations show a rich variety of behavior of solitary waves, including steady, periodic, and complicated interacting evolutions. For a two-layer model to have multiple steady or oscillatory states, both bottom and interfacial friction should be included because if one of th...

Effects of depth discontinuity on harbor oscillations

The effects of water depth discontinuity near the harbor mouth on harbor oscillations are examined. Linear long-wave equations are used as the basis of the present study. For simplicity, only the normal incident waves are considered. Assuming that the harbor mouth is small in comparison with wavelength, the method of matched asymptotic expansion is employed to obtain the ocean impedance and the harbor responses. It is found that the incident waves can be trapped over the depth discontinuity which causes large oscillations near the harbor mouth. The radiation damping also decreases because of the appearance of the depth discontinuity, which leads to large amplifications at the lowest mode.

Longshore motion due to an obliquely incident wave group

We consider longshore motion generated within the surf zone by obliquely incident breaking waves, and seek to describe the effect on such motion of variations, caused by wave grouping, in the incident longshore momentum flux. The effects of associated variations in set-up are not considered.We use the linear long-wave equations to describe the motion resulting from the longshore momentum contained in a wave group. This consists of a succession of edge waves which disperse along the beach, and, for the example considered, an eventual steady circulation cell at the position of the wave group. We suggest that such a cell is always likely to be formed if the wave group is sufficiently localized, and that higher-modenumber edge waves are more likely to be excited.We find timescales for the dispersal of the edge waves, and for the decay, due to bottom friction, of the circulation cell: we suggest that the latter may more generally be used, as a timescale for the effect of friction on longshore motion.

Application of Improved Numerical Techniques to the Tsunami Response of Island Systems

A finite-difference numerical model based on the classical linear long-wave equations has been developed to study the tsunami response of multiple-island systems. The first application of this model was to the Hawaiian Islands. Early results were satisfactory. However, one drawback to the model was the use, at the outer open boundaries of the basin, of a radiation condition which allowed only wave energy approaching the boundary at a normal angle to pass out of the basin without reflection. Recently, an improved boundary condition has been employed which takes into account the radial propagation of scattered waves near the boundary. The earlier model tests were repeated using this new condition. The major features of the results remained relatively unchanged. Certain secondary features of the response. which had made analysis of model results quite complicated, were either eliminated or greatly reduced.

Analytic solution of the linearized shallow-water wave equations for certain continuous depth variations

AbstractThe linear long-wave equations with (and without) small ground motion are considered. The governing equations are represented in a matrix from and transformations are sought which reduce the system to (for example) a form associated with the conventional wave equation. Integration of the system is then immediate. It is shown that such a reduction may be acheived provided the variation in water depth is specified by certain multi-parameter forms.

Long wave generation on a sloping beach

A general solution of the linear long-wave equation is obtained for arbitrary ground motion on a uniformly sloping beach. Numerical results are presented for specific shapes and time histories of ground motion. Near-shore large amplitude waves are also investigated using non-linear theory.