# Comparison of linear and nonlinear shallow wave water equations applied to tsunami waves over the China Sea

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DOI: 10.1007/s11440-008-0073-0

- Cite this article as:
- Liu, Y., Shi, Y., Yuen, D.A. et al. Acta Geotech. (2009) 4: 129. doi:10.1007/s11440-008-0073-0

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## Abstract

This paper discusses the applications of linear and nonlinear shallow water wave equations in practical tsunami simulations. We verify which hydrodynamic theory would be most appropriate for different ocean depths. The linear and nonlinear shallow water wave equations in describing tsunami wave propagation are compared for the China Sea. There is a critical zone between 400 and 500 m depth for employing linear and nonlinear models. Furthermore, the bottom frictional term exerts a noticeable influence on the propagation of the nonlinear waves in shallow water. We also apply different models based on these characteristics for forecasting potential seismogenic tsunamis along the Chinese coast. Our results indicate that tsunami waves can be modeled with linear theory with enough accuracy in South China Sea, but the nonlinear terms should not be neglected in the eastern China Sea region.

### Keywords

China SeaNonlinear shallow-water equationsNumerical computationTsunami waves## 1 Introduction

Models of shallow water wave equations are widely used in tsunami simulations [1, 4, 5]. The shallow water wave equations describe the evolution of incompressible flow, neglecting density change along the depth. Shallow water wave equations are applicable to cases where the horizontal scale of the flow is much bigger than the depth of the fluid. Therefore, tsunami waves can be described by shallow water models. A simple yet practical numerical model describing the propagation of tsunamis is given by the linear shallow water wave equations. They are the simplest form of the equations of tsunami propagation, which does not contain the nonlinear convective terms. In recent years, many destructive tsunamis have served as a reminder that it is important to develop a well-coordinated strategy for issuing tsunami warnings. To make a timely prediction of tsunami wave propagation in the open deep oceans, numerical simulations based on linear theory with the linear shallow-water equations are desirable because they involve a short amount of computation. In the presence of sharply varying bathymetry of the China Sea, it is important to carry out a detailed comparison between the linear and nonlinear theory for different values of bottom friction and ocean depths. Tsunami simulations are often carried out using linear shallow water modeling because it is computationally faster and easier to perform without the need to specify the seabed boundary condition. In this paper, we find the boundary condition specification to be very important for obtaining accurate results on the coastal area in tsunami propagation process. In addition, the difference in computing times for linear and nonlinear models is very substantial. The nonlinear model typically would take about four to five times computing time than the linear model for high-fidelity simulations.

## 2 Comparison of linear and nonlinear shallow water models

Seismogenic tsunami generation is a very complicated dynamical problem. Certain factors affecting tsunami sources include the duration period of earthquake rupture, geometric shape of rupture, bottom topography near the epicenter of earthquake, seismic focal mechanism, and rock physical properties [3]. Ward [18] studied tsunamis as long-period, free oscillations of a self-gravitating earth, with an outer layer of water representing a constant depth ocean. The tsunami displacement field can be constructed by summing the normal modes of the spherical harmonics. Comer and Robert [2] regarded the tsunami source excitation in the flat Earth by a point source. He emphasized that a source problem in the flat Earth differs substantially from the corresponding problem for the spherical Earth. Yamashita and Sato [20], using the fully coupled ocean–solid Earth model, analyzed the influence of the parameters of seismic focal mechanism, such as dip angle, fault length, focal depth, and the rise time of the source time function on tsunami. They took wave forms of tsunami as long period gravity wave and Rayleigh waves.

We assume one tsunami source model as a function which depends on time, the geometry of bottom topography, and other factors. The influence of different tsunami sources with the assumed model or normal elastic bottom displacement on wave propagation is that wave dispersion can be significantly influenced in nonlinear shallow water models for amplitude estimation in tsunami propagation with wave train generated in time-dependent rupture model [8, 10]. In fact, a more realistic time-dependent source function can be used. For a seismogenic tsunami, the time duration for the earthquake, in the order of minutes, could be neglected when compared with the total time of tsunami propagation, which is from a few hours to more than 10 h. Therefore, the model of tsunamogenic coseismic earthquake can be treated as an elastic model. Many previous studies have used an elastic dislocation model constrained by the seismic focal mechanism for an earthquake excitation tsunamic source.

*L*, width

*W*, and the average slip

*D*) are derived from theoretical and empirical relationships [19] that have been widely applied. The fault dips and strikes from the composite fault plane solutions come from the average dip of the fault segments according to the Harvard catalog (http://www.globalcmt.org/CMTsearch.html). Since the shallow water region in South China Sea is relatively narrow, the linear model describing the tsunami wave propagation for this area is considered first. Here, the bottom friction is ignored. We apply the linear shallow water theory for a Cartesian system. Due to the low latitude of the South China Sea, the Coriolis effect can be safely neglected. The following linear shallow-water Eq. (1) are employed.

*z*is the instantaneous water height,

*t*is time,

*x*and

*y*are the horizontal coordinates,

*M*and

*N*are the discharge fluxes in the horizontal plane along

*x*and

*y*coordinates,

*h*(

*x*,

*y*) is the undisturbed basin depth,

*D*=

*h*(

*x*,

*y*) +

*z*is the total water depth, ρ is density of water,

*g*is gravity acceleration and

*f*is bottom friction coefficient.

*τ*

_{x}and

*τ*

_{y}are the tangential shear stresses in

*x*and

*y*direction. In the nonlinear model, the effect of friction on tsunami wave propagation is included. The bottom friction is generally expressed as follows [5]:

*f*here. Rather, we will use the Manning roughness

*n*, which is familiar to civil engineers. The friction coefficient

*f*and Manning’s roughness

*n*are related by \(n=\sqrt{fD^{\frac{1}{3}}/2g}.\) This relationship holds true when the value of total depth

*D*is small. Under this condition,

*f*becomes rather large and makes

*n*nearly a constant value. Thus, the bottom friction terms can be expressed by

In our simulations we have employed the linear tsunami propagation model Tunami-N1, and nonlinear model Tunami-N2, developed in Tohoku University (Japan) and provided through the Tsunami Inundation Modeling Exchange (TIME) program [5]. This tsunami code ensures the numerical stability of linear and nonlinear shallow water wave equations with centered spatial and leapfrog time difference [5]. We use the open boundaries conditions in these models which permits free outward passage of the wave at the open sea boundaries. The computational stability depends on the relationship between the time step and spatial grid-size. Furthermore, the computational stability is also constrained by the physical process in the models. Here, the model physics must satisfy the CFL criterion, that is \(\Updelta{s}/\Updelta{t} > |\sqrt{gh}|,\) where Δ*s* is spatial grid size, Δ*t* is time step, *g* is acceleration of gravity, and *h* is water depth. The bathymetry of the South China Sea was obtained from the Smith and Sandwell global seafloor topography (Etopo2) with grid resolution of near 3.7 km. The total number of grid points in the computational domain is 361,201, which is 601 × 601 points. The time step, Δ*t*, in both models is selected to be 1.0 s to satisfy the temporal stability condition. In our simulation, since the bottom friction coefficient is larger than zero, we have *D* = *h* + *z* > 0, where *D* is total water depth, *h* is water depth, and *z* is wave height. This means that shallow water wave equations can maintain computational stability only within a computational domain filled with fluid [17]. Since the wave height of tsunami wave is only a few meters in the propagation process, we have set the smallest computational depth as the order of 10 m along coastal area in both linear and nonlinear models. This way, all of the computation domain satisfies this condition.

As mentioned in the introduction, there exists a significant difference in the computing time and computational circumstance between the two models. Linear models can run on a single PC, whereas nonlinear models need more computing power. We performed linear and nonlinear modeling using our group computer with four Central Processing Units of an Opteron-based system. The run-time for the nonlinear model is 180 min, or 4.5 times longer than the linear models, which is 40 min for 6.0 h wave propagation computing.

*n*= 0.025, recommended by Imamura, as the bottom friction coefficient in this situation [5]. The value

*n*= 0.025 is suitable for the natural channels in good condition which is valid for the South China Sea regions. We illustrate the comparison of water heights in time histories with linear and nonlinear models in various water depths, from 12 to 3,792 m. (Figs. 3, 4). These waves are taken at the receivers located on a straight line between the epicenter (marked in Fig. 2) and Hong Kong. Because the friction force is 1/

*D*in Eq. 4, when the water depth is very deep, the friction influence can be neglected. In other words, the deeper the water level, the less the frictional influence present. On the other hand, close to the shallow water region, the seabed friction tends to dominate. The comparison figures show that there is one critical zone between 400 and 500 m depth. With the ratio of wave height to water depth smaller than 0.01, wave propagation can be modeled by the linear theory with reasonable accuracy. Otherwise, the nonlinear model is necessary for making accurate assessments. If water depth is lower than the critical range, the absolute value of wave height for nonlinear models is bigger than that of the linear models with the convection terms dominating in nonlinear shallow water wave equations. Above this depth, both the linear and nonlinear models generate similar wave shapes and wave magnitudes.

*n*= 0.025 is for the natural channels in good condition,

*n*= 0.060 is for very poor natural channels, and

*n*= 0.125 is a hypothetical number for better modeling effects of sea bottom friction on the wave height. Here, we only consider the water regions with depth lower than 500 m, with tsunamis having dominating nonlinear properties. In Fig. 6, the effects of three different Manning roughness on the wave have been compared for points at depths of 12, 152, 255, 342, and 487 m, respectively, that also located on at the straight line from the hypothetical epicenter to Hong Kong (Fig. 2). The figure shows the wave height of the tsunami to be very sensitive to the friction term. With different Manning roughness, the effect of the frictional term on the initial waves is marginal. This is because that the energy of the initial waves is very strong and not sensitive at all to the Manning roughness. However, for the subsequent waves developed, with a much lower energy after over 8,000 s of propagation, the frictional term exerts a much larger effect on the wave height. In shallow water regions (e.g., 12–75 m), the dynamical effect from the bottom friction is strong.

## 3 Probabilities of potential tsunami hazard along China Sea coast

The characteristics of Chinese tsunami hazard have a long cycle with 1,000-year-period. We devise a new method, called the Probabilistic Forecast of Tsunami Hazards, in order to determine potential tsunami hazard probability distribution along Chinese coast [11]. In this method, we first locate the potential seismic zone by analyzing the detail of the geological and geophysical background, the seismic activities by Gutenberg-Richter relationship [7, 15]. Then, we simulate the tsunamis excited by potential earthquakes and we compute the heights of waves hit the coast are computed. Finally, Probabilistic Forecast of Tsunami Hazards is computed based on Probabilistic Forecast of Seismic Hazards [11], and potential tsunami hazard distribution along Chinese coast is mapped. Historical seismogenic tsunami data can provide the most reliable basis for the study of tsunami hazard. Unfortunately, in China, there are few scientific papers that deal with the analysis of tsunamis. The reliable numerical tsunami simulation generated from the potential earthquake can make up for the inadequate historical information of tsunamis hazard.

There are significant differences in the bottom bathymetry between the South China Sea bordering the southern province of Guangdong and the East China Sea and Yellow Sea adjacent to the provinces of Zhejiang, Jiangsu, and Shandong. For the two ocean regions, we will compute the probabilities of the tsunami hazard in China Sea area by using both linear and nonlinear models. In the nonlinear model, the Manning roughness is 0.025 for the natural channels in good condition which is suitable for the China Sea.

Due to the prevalence of deep region in South China Sea, the linear model is expected to perform well. Our probability for tsunami wave occurrence is computed from the probability of earthquake occurrence and the probability of maximum wave height of all seismic tsunami induced by potential earthquakes [11]. The tsunami waves with heights of (1.0, 2.0 m) and heights over 2.0 m are considered for hazard evaluation. We consider both economic and scientific factors for wave scales in our tsunami simulation. In our project we only carry out generation and wave propagation of whole tsunami process. Two meters in propagation can be amplified a few times, up to ten times [7, 9], based on local ocean topographical conditions after run-up process computation. Therefore, this wave height size could cause economic hazard after run-up for China coastal area because continental altitude of main Chinese cities only couple meters over the sea. Actually, the wave height of Chinese historical tsunami records is around from half meters to 7.5 m (Keelung, 1867) [16]. Most of Chinese tsunamis are around 1–2 m. We forecast that the probability for tsunami wave with more than 2.0 m to hit within this century is 10.12% for Hong Kong and Macau, 3.40% for Kaohsiung, and 13.34% for Shantou with the linear model. With the nonlinear model, the probability is the same for the same tsunami wave height at Hong Kong and Macau, and Kaohsiung, while a lower probability at Shantou of 10.12% is found. In general, the probabilities for most coastal cities do not change with the usage of nonlinear theory. Both results indicate that tsunami hazard can be induced on Hong Kong coastal area with more than 2.0 m tsunami hazard with around probability of 1% every decade. These results are similar to the same as the frequency of destructive seismic event in South China Sea and adjacent region.

Based on previous analysis, we can predict the potential tsunami hazard with the linear shallow water equation (Eq. 1) in South China Sea with its natural bottom boundary condition due to its bathymetry (Fig. 1). However, we have to employ the nonlinear model (Eq. 2) in the eastern China Sea region, for which the bottom frictional effect must also be considered.

## 4 Conclusion

We investigated the economic disruption caused by a pair of Taiwan earthquakes on 26 December 2006. A larger earthquake along the Luzon Trench would have much more severe global consequences because of Internet connectivity of the cables at the bottom of the South China Sea, which were damaged by the submarine landslide. We have reacted rapidly by carrying out this comparison of linear and nonlinear predictions of tsunami wave propagation across the South China Sea. From our analysis, we concluded that one can apply linear theory to a good accuracy for this critical region (Manning roughness *n* = 0.025). This would allow a much earlier warning to be issued, since the linear calculations can be carried out on laptops in nearly real time. In addition, we found that the bottom frictional properties of the seafloor due to sediments can play an important role in quenching the magnitude of incipient tsunami waves. The same statement concerning the applicability of the linear theory will not be true for eastern China Sea region. Because of its much shallower seafloor, the nonlinear theory must be carried out in this region.

## Acknowledgments

We would like to thank Professor Fumihiko Imamura for providing computational codes TUNAMI_N1 and TUNAMI_N2, and his kind guidance on tsunami numerical modeling. This research is supported by National Science Foundation of China (NSFC-40574021, 40728004) and the EAR program of the US National Science Foundation.