RUSSIAN JOURNAL OF EARTH SCIENCES VOL. 8, ES6004, doi:10.2205/2006ES000216, 2006

[35]  The choice of the adequate mathematical model depends on the specific properties of the solved problem, on the specific features of the basin, in which the problem is solved, and what is not less important, on the stage of the development of the phenomenon within one problem and one basin.

[36]  Application of the most complete models is not always reasonable not only owing to a serious increase in the required computational resources and complications of the needed algorithms, but also due to the impossibility of adequate determination of all input parameters of such models. The correct choice can be made only on the basis of preliminary calculations close to the content of the applied problems using different models and different algorithms at different grids. Such work should be done during the preliminary stage using agreed test problems and "real" input data. Finally, zoning of the protected territories of the basin and coast can be made on the basis of mathematical models capable to provide adequate results for each type of the problems.

[37]  Zoning by the available types of input data (determination of the grid size for providing the needed accuracy and degree of modification of the boundaries) should be done simultaneously with the above described work as well as zoning by other hydrodynamic parameters: roughness, characteristic wind stress, etc.

[38]  The experience of the authors of this article based on the solution of research and applied problems of tsunami allows us to state that even the simplest mathematical models make possible obtaining adequate estimates in the initial stage of tsunami development, while the subsequent effects require a thorough work for their reproduction.

2006ES000216-fig01
Figure 1
[39]  An example of such kind is the investigation of tsunami transformation in "wash-tub" (see Figure 1) model region, which was introduced for the first time by [Gusyakov and Chubarov, 1985, 1987]. The convenient features of this basin is explained, in particular by the possibilities to study the peculiarities of different mathematical models and algorithms to reproduce the transformation of waves in the regions with different depths and wave interaction with different types of boundaries (open and rigid reflecting boundaries). Regardless its simplicity, the bottom topography in the "wash-tab" region agrees well with the distribution of depths along the Kuril-Kamchatka Trough. The bottom topography is specified by piecewise linear function, which varies from 10 m to 9000 m and depends only on coordinate y (it is assumed that coordinate system XOY is located so that OX axis is directed along rectilinear coast, and OY is directed normal to the coast in the direction of depth increase). The length of the basin in OX direction is 555,000 m and in OY direction is 320,000 m.

[40]  Test problems solved in this model basin should facilitate answering the questions about sensitivity of real problems to the computational accuracy of the algorithms and to hydrodynamic accuracy of mathematical models: the two main problems of numerical modeling. Two problems were considered with this goal in mind:

[41]  The first of them is propagation of a solitary wave homogeneous in OX direction, and the second, is the transformation of initial perturbation of the free surface finite in both directions. Below, all linear sizes are given in meters and time is in seconds.

[42]  In the first problem, the form of the initial perturbation was specified by a well known relation:

h = acdot cosh-2 (Y),

eq001.gif

y0 = 175,000, a = 1, u (t = 0) = 0. Conditions of full reflection were specified at the coastline ( y = 0 ) and at lateral walls ( x = 0 and x = 555,000); boundary y = 320,000 was free. At time moment t = 0, the wave decomposed into two, one of which propagated in the direction of increasing coordinate, and the second propagated in the shallow water region.

2006ES000216-fig02
Figure 2
[43]  Nine calculated gauges were located along line x = 277,500 for detailed study of wave regimes. The following values of the y coordinate were chosen: yM1 = 0, yM2 = 1250, yM3 = 2500, yM4 = 5000, yM5 = 10,000, yM6 = 47,500, yM7 = 112,500, yM8 = y0 = 175,000, yM9 = 230,000 (see Figure 2).

[44]  Modeling was performed within classical linear and nonlinear (NL) models of shallow water [Stoker, 1959], weakly nonlinear model of Nwogu [Nwogu, 1993] and fully nonlinear dispersion (NLD) model [Wei and Kirby, 1995] ("one-layer" model of Liu-Lynett [Lynett and Liu, 2004]). Owing to the fact that the results obtained using the two latter models appeared almost identical, the graphs presented bellow represent the Nwogu model. This fact allows us to state that the effects of nonlinear dispersion terms are of low importance in the solved problems.

[45]  Multi-parametric versions of differential schemes of MacCormac [Fedotova, 2006] and Adams [Wei and Liu, 1995; Lynett and Liu, 2004] were used as computational algorithms realizing the first two models. The Adams scheme was used for approximating the Nwogu model.

2006ES000216-fig03
Figure 3
[46]  The calculations using different hydrodynamic models were performed using the Adams scheme (with numbers of nodes in the direction OX - Nx = 51 and in the OY - Ny = 513 direction). Records of gauges at the ninth (closest to the source) and fourth (most distant from the source) points calculated from linear and non-linear equations of shallow water and weakly non-linear dispersion Nwogu model are shown in Figure 3.

[47]  The graphs in deep water part of the region (Figure 3a) show that all models give practically the same results. Nonlinear effects begin to manifest themselves when the wave enters the shallow water zone (Figures 3d, 3e), in which the linear model differs from the other models in the description of the reflected wave. Pressure gauge records calculated in the shallow water region, beginning from the fourth demonstrate increasing difference of the linear model results also in the description of the leading incident wave. Almost complete coincidence of the results of nonlinear equations of shallow water in both NLD models at all points of gauge calculations points to the fact that dispersion effects are insignificant in the problem with such initial conditions.

[48]  In the second problem, initial perturbation of the free surface was specified in the form of exponential "caps" of different length l using relation

eq002.gif

where L = 16,000 for l = 50,000, L = 24,000 for l = 75,000 and L = 32,000 for l = 100,000; a = 2 (the length is determined on the basis of 10% section of the initial amplitude a from the foot of the wave). The second problem was solved within classical nonlinear model of shallow water [Stoker, 1959] and weakly nonlinear dispersion Nwogu model [Nwogu, 1993].

2006ES000216-fig04
Figure 4
[49]  At time moment t = 0, the center of perturbation was located at point with coordinates (x0, y0) = (465,000; 230,000). Condition of full reflection was specified at boundary y = 0; the other boundaries were assumed free. Twenty three gauges were located to record the results of calculation. A scheme of their location is shown in Figure 4.

2006ES000216-fig05
Figure 5
[50]  Spatial images of wave pattern (from above) for three time moments are shown in Figure 5. Black horizontal lines denote breaks of bottom topography lines.

[51]  Waves propagate faster in the deepest ("upper") part of the basin; break lines are characterized by the changes in the curvature of fronts (reflection). The wave reflected from a break line with the greatest angle ( aapprox 14o ) is clearly seen.

2006ES000216-fig06
Figure 6
[52]  Dispersion effects manifest themselves stronger in the case of "short" initial perturbation demonstrating the generation of not only one wave as in the case of nonlinear equations of shallow water but the appearance of wave packets propagating in different directions from the source. This effect manifests itself to a lesser degree for the perturbations of medium length. Finally, for the "longest" perturbation the dispersion trace appears practically invisible. This is confirmed by the records of gauges (see Figure 6). Wave packets, which follow the leading wave, are clearly seen in Figures 6a, 6b. If the length of the initial perturbation increases, these wave packets gradually disappear.

[53]  The character of free surface oscillations, which is seen in the gauge records in Figure 6, allows us to note the differences in the wave regimes at coastal points (11th gauge) and in the offshore direction (16th gauge). A simple signal is observed at the coast consisting of the leading wave and an elevation behind the wave (dispersion decomposes this fragment into a series of oscillations). At a distance from the coast, the leading wave is followed by an elevation (wave packet in dispersion approximation) and a wave reflected from the boundary of the underwater slope.

2006ES000216-fig07
Figure 7
[54]  A calculation for a longer period of time was performed to study the reflection process in the case of l=50,000 m. Pressure gauge records calculated at the coastal (9th) point, at a point close to the coast (11th), at a point close to the source (12th), and remote point (16th) are shown in Figure 7. These graphs demonstrate (9th point) that the existence of a lengthy shallow water interval in the pathway of the wave propagation leads to the appearance of a peculiar filter, which removes small scale dispersion oscillations and allows us to reproduce the coastal wave processes within the equations of the shallow water theory with the same efficiency. At the same time, the gauges located in the zones of greater depth (11th, 12th, and 16th gauges) demonstrate the necessity of the account for the dispersion in reproducing the initial phases of the wave regime already, whereas its final phases determined by the waves reflected from the break lines of the bottom located in deep water zones do not require application of NLD models. It is noteworthy that the amplitudes of the waves propagating over maximal depth (16th gauge) are almost five times smaller than the amplitudes of the waves propagating in the region of decreasing depth.

[55]  The observation of the peculiarities of wave propagation along different pathways characteristic of the basin studied here is very informative. In this case, the material for such analysis is observation of the wave transformation when it propagates from the source (21st gauge) to the coast (9th gauge). This analysis characterizes the capabilities of the mathematical models and algorithms to reproduce the transformation of waves over the peculiarities of bottom topography, whose depth varies from 9000 m in the region adjacent to a depth of 10 m near the coast. A comparison of gauge records showed that the form of the leading wave does not change practically, its amplitude decreases strongly after it leaves the source but in the course of the further propagation it conserves its values, while the form of the wave packet that follows the leading wave becomes more complicated due to multiple interactions with the bottom topography and break lines of the bottom. The amplitudes of the waves in the wave packet increase as the depth decreases. We note that "hydrodynamic" dispersion caused by the presence of the corresponding terms in the equations of the mathematical model manifests itself only in a slight decrease of the amplitude of the leading wave and has no effect on its phase characteristics. Thus, nonlinear and nonlinear-dispersion models lead the similar results.

[56]  The amplitude of the leading wave decreases strongly during the propagation over the deep water part of the basin. Its form is strongly determined by the influence of dispersion, which is practically not seen during the subsequent stage of the process. This part of the results characterizes the capabilities of mathematical models and algorithms to describe the wave transformation in the deep regions located far from the boundaries.

[57]  Wave regimes calculated at coastal points are very complicated. The displacements of the free surface at these points are calculated using the algorithm approximating the corresponding boundary conditions. They are determined from the assumption of the coast as a vertical wall. The analysis of calculations near the "reflecting" boundaries showed that "hydrodynamic" dispersion does not manifest itself, and the amplitude of the waves has a tendency to decrease, but not to such degree as over the "marine" pathway. The character of oscillations and their frequency properties do not practically change from one point to another.

2006ES000216-fig08
Figure 8
2006ES000216-fig09
Figure 9

2006ES000216-fig10
Figure 10
[58]  The calculation of one of the most important characteristics, the distribution of maximal amplitudes along the coast as seen in Figure 8, does not require enhanced mathematical models, at least for the specific set of "hydrodynamic" parameters.

[59]  The provisions formulated above are confirmed by a series of spatial images shown in Figure 9 and Figure 10. The first of them demonstrates the results obtained using the Nwogu model, and the second using nonlinear model of shallow water. The figures are shown in traditional lexicographic sequence (from left to right and from top to bottom) beginning from time step 500 with an interval of 500 steps. Significant differences are found only in the first four figures, the next four differ significantly less, and beginning with the ninth figure the states of the surface in the basin are practically not distinguishable. Cell structures appearing during the interaction between waves propagating to the coast and waves reflected from the coast and from the break lines of bathymetry are worth attention. The account for the dispersion strengthens this effect especially in the zone of the flat shelf and adjacent regions of the basin.


RJES

Citation: Shokin, Yu. I., L. B. Chubarov, Z. I. Fedotova, S. A. Beizel, and S. V. Eletsky (2006), Principles of numerical modeling applied to the tsunami problem, Russ. J. Earth Sci., 8, ES6004, doi:10.2205/2006ES000216.

Copyright 2006 by the Russian Journal of Earth Sciences

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