Abstract
Tsunamis can propagate thousands of kilometres across the ocean. Precise calculations of arrival times are essential for reliable early warning systems, determination of source and earth properties using the inverse problem, and time series modulation due to frequency dependency of phase speed. Far field observatories show a systematic discrepancy between observed and calculated arrival times. Models in present use and based on incompressible hydrodynamics and interaction with a rigid ocean floor overestimate the phase speed of tsunamis, leading to arrival time differences exceeding tens of minutes. These models neglect the simultaneous effects of the slight compressibility of water, seabed elasticity, and static compression of the ocean under gravity, hereinafter gravity. Here, we show that taking these effects into account results in more accurate phase speeds and travel times that agree with observations. Moreover, the semianalytical model that we propose can be employed near realtime, which is essential for early warning inverse models and mitigation systems that rely on accurate phase speed calculations.
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Introduction
The vast majority of conventional ocean circulation models treat the sea as an incompressible medium (ignoring the compression of the atrest water column due to its own weight as well as acoustic contributions to motion) overlying a rigid seafloor^{1,2,3}. These assumptions are valid when the time periods are relatively short, and the corresponding phase speeds are low. It has been argued in various studies that the contributions of water compressibility and earth elasticity are vital not only for proper description of surface gravity waves^{4,5,6,7,8}, but even more for waves travelling in the earth crust, such as P, S, Rayleigh and Love waves^{9}, and acousticgravity waves that propagate in the water column^{10,11,12,13,14,15,16}, or couple with the seafloor^{17}. The latter travel at speeds that far exceed the maximum tsunami phase speed carrying information on the fault geometry and dynamics^{15}, and thus could be employed for early tsunami warning systems^{18} and inverse models^{19}, if analysed realtime.
Tsunami waves have been extensively studied in recent years, with attention focused on various physical features including thermal or salinitybased density stratification^{6,20}, compressibility of the water column^{4,6,8,15,20,21,22,23,24}, elastic deformation of the underlying solid earth^{17,20,25} or combined effect of compressibility, stratification, and elasticity^{26}. These studies confirmed, independently, that the time lag between model output and observations is sensitive to many of the physical features virtually neglected. Consequently, the time series at far field observatories are shifted in order to match arrival of waveforms extracted from these models and observations (see Figs 1d and 2 of ref.^{5} and Fig. 3 of ref.^{27}). In addition, it has been observed that the arrival delay is taking place in deep ocean where nonlinear effects are negligible^{25} and the aforementioned parameters are dominant.
In this study, we use potential theory to evaluate the effects of compressibility, elasticity and gravity on the calculated phase speed of tsunami. More specifically, water is treated as an inviscid, barotropic fluid with constant sound speed, \({c}_{l}\), and fluid motion is assumed to be irrotational. On the other hand, the solid layer is treated as an elastic half space that undergoes rotation and compression with constant pressure and shear wave speeds, \({c}_{p}\) and \({c}_{s}\). The problem at hand can be expressed by three wave equations governing a velocity potential \({\phi }_{l}\) in water, and a dilatation potential \({\phi }_{s}\) and rotation potential \({\psi }_{s}\) in the solid seafloor, given in dimensionless form by
where \({\gamma }_{i}=gh/2{c}_{i}^{2}\), (with \(i=l,p,s\)) are small dimensionless parameters representing the squares of the ratios of surface gravity waves to sound, pressure and shear wave speeds. respectively. All quantities in Eq. (1) are normalized using the water depth \(h\) as a length scale, \(\sqrt{h/g}\) as a timescale, and densities are normalized by the water density \({\rho }_{l}\), (see supplementary file for dimensional analysis). The solution for linearized motion of plane progressive waves in a horizontally uniform domain then follows from applying kinematic and dynamic boundary conditions at the free surface, matching normal stress and displacements at the oceanseabed interface, and requiring the dilatation and rotation potentials vanish at \(z\to \,\infty \) (see supplementary material for details). The full dispersion relation for the prescribed frequency \(\omega \) is then given by
where \({r}^{2}={k}^{2}2{\gamma }_{l}{\omega }^{2}+{{\gamma }_{l}}^{2}\) is the eigenvalue for the vertical structure of \({\phi }_{l}\) in the water, \(k\) is the wavenumber in the horizontal direction, and \({\varepsilon }_{1}\), \({\varepsilon }_{2}\) and \(\beta \) represent the elasticity and gravity effects defined in Eqs (3)–(5). The phase speed is then computed by \({c}_{0}=\omega /k\), where \(k\) dictates the eigenvalues \(r\) (see Eq. 6). Note that \({k}^{2} > 0\) and \({r}^{2} > 0\) corresponds to the progressive gravity mode (\(n=0\)) which decays exponentially with depth. Propagating nonevanescent acousticgravity waves can rise under the special condition where \(r\) is imaginary, yet \(k\) is real. Unlike the discrete spectrum of the progressive gravity waves, the spectrum of trapped modes, exponentially decaying in \((x,y)\), is continuous where \({k}^{2} < 0\). However, there is no interaction between progressive and trapped modes in a spatially uniform domain. The general dispersion relation (2) accounts for the effects of compressibility, elasticity, and gravity. It turns out that ignoring any combination of these effects manipulates the phase speed of the propagating surface wave (Fig. 2 and Table 1). Specifically, ignoring elasticity or compressibility results in an overestimate of the phase speed, whereas ignoring gravity, both within water and seafloor, results in an underestimate. Note that neglecting \({\gamma }_{i}\) the dispersion relation Eq. (17) of ref.^{17} is retrieved, whereas if elasticity is ignored, retaining water compressibility and gravity terms, the dispersion relation Eq. (3.2) of ref.^{28} or Eq. (11) of ref.^{4} are retrieved (\({\varepsilon }_{1}=1;{\varepsilon }_{2}=0\)). Neglecting elasticity and gravity leads to the standard dispersion relation \({\omega }^{2}=r\tanh (r)\). The phase speeds corresponding to all possible combinations of compressibility, elasticity and gravity are depicted in Table 1 and Fig. 2. It is worth noting that the surface gravity wave (mode \(n=0\)) has a cutoff frequency of \(0\ Hz\), which allows propagation of the plane wave at any excitation frequency. Thus, for tsunamis within the frequency range of a few minutes to a few hours, waves are progressive with dominance of the discrete spectrum. Alternatively, a continuous spectrum should be considered when \({k}^{2} < 0\), where the modes become evanescent decaying exponentially in depth and horizontal plane. This case is relevant only in the near field, within a range of several water depths^{29,30}.
The analysis carried out in this study confirms that the standard solution, i.e. neglecting the effects of compressibility, elasticity and gravity, may still result in satisfactory calculations of the tsunami arrival times for shallow water (\(h < 2\) km) or short waves (less than 10 min period). However, for longer waves, the full solution, which considers compressibility, elasticity and gravity, becomes essential where a deeper portion of the sea bottom interacts with the ocean. For the rigid bottom case, water incompressibility is responsible for an increase in the phase speed by \({\gamma }_{l}\) while neglecting gravity decreases the phase speed by \({\gamma }_{l}/2\) of full solution in shallow water limit, \(kh\ll 1\). Unlike compressible ocean over rigid bottom assumption where the phase speed approaches to an asymptote for shallow water limit, the role of bottom elasticity is proportional to the frequency of waves. Thus, at the limit of large wavelength waves the contribution of elasticity overcomes that of compressibility. The analysis depicted in Fig. 2c shows that elasticity is negligible for waves with periods smaller than 5 min, whereas it overtakes compressibility for wave periods longer than 60 minutes. On the other hand, ignoring gravity leads to an underestimate of the phase speed. Note that changes within the range of the Preliminary Reference Earth Model (PREM) parameters, taken from ref.^{31,32} for the crust and ocean, result in variations in the phase speed that are less than \(0.2 \% \).
As a case study, we consider the Tohoku Oki 2011 tsunami^{26,27,33,34}. Analyzing the frequency spectrum of the insitu measurements reveals a range of wave period (10–90 min) for a 4 hr window near 1\({}^{st}\) peak with a mean of \( \sim \)35 min at DART buoys (Fig. 1a,b). In the present study, the mean of peak periods is considered when calculating the arrival times of tsunami front based on the solution of Eq. (1) where water compressibility, bottom elasticity and gravity are all considered. In contrast and similar to the conventional tsunami models where the ocean is treated as an incompressible medium on a rigid bottom, the arrival time is calculated and compared to the full solution as shown in Fig. 3 (see supplementary file). Calculations of the phase speed reported in literature (i.e. refs^{5,27} as shown in Fig. 1d) overestimate observations, with a discrepancy reaching 30 minutes for waves with 90 min period in some regions \( \sim \)15000 km away from the source and close to South America (i.e. see Fig. 4d). However, considering compressibility, elasticity, and gravity results in a noticeable improvement of the arrival times at all observation locations (Figs 1c and 3). Note that the tsunami wave is a superposition of waves with different frequencies, travelling with corresponding phase speeds. As a result, the modulation would change if the correct phase speed is considered, leading to a more reliable model output.
The results of this study outline the contribution of ocean compressibility, bottom elasticity and static compression of the ocean under gravity on the propagation speed of waves, generated by tsunamigenic events ranging from 10–180 minutes periods. Parametric analysis, comparing existing ocean circulation models with the proposed full solution, reveals up to 1.3% and 4\( \% \) increase in phase speed due to neglecting water compressibility and sea bottom elasticity, respectively, and up to 1\( \% \) reduction in speed due to neglecting the effects of field gravitational potential. The lower the frequency of the wave, the higher the discrepancy between the full solution and standard models become, leading to earlier arrival of signals at farfield observatories, as well as changes in the signal modulation. Proper consideration of these parameters would lead to a better understanding of an interactive environment comprising a compressible ocean and an elastic earth system for a variety of waves travelling at the sea surface such as tsunami, or propagate in deep water, e.g. ocean acoustics, or in the earth crust, such as P, S, Rayleigh and Love.
There has been an increased requirement for a multicomponents early warning system due to the severity of impact of such frequent events in recent decades. A reliable system should have the capability to correlate precursors and destructive tail via accurate dynamic interactions between media. The findings here not only give a better measure for the phase speed that is essential for reliable warning systems, but also construct the basic pillars of the next era of research on the propagation of waves in the ocean where water compressibility, seabottom elasticity, and field gravity should all be considered to better understand the physical processes involved. These should have a direct impact on major fields within geosciences, physical oceanography, and natural hazards.
Coefficients of full dispersion relation (Dimensionless)
and
where \(\lambda \) and \(\mu \) are the Lame’s constants, \(r\), \(q\), and \(s\) are the eigenvalues of the three differential equations in Eq. (1), respectively.
Calculations of tsunami travel time
The tsunami travel time is calculated from the epicentre for a given frequency using dispersion relations presented in Table 1 (\(\#1\) and \(\#6\)). Via a time marching scheme, the furthest points the front wave can reach within \(\Delta t\) is calculated considering depth dependent phase speed and distance from front wave. The computation starting point is the epicentre, at \({t}_{0}=0\), corresponding to rupture time and calculating the coverage of tsunami for the succeeding time steps until either the tsunami covers the whole domain or reaches the coast. The Haversine method is used to convert spacing between WGS84 degrees and planar meters. For Tohoku event, we have considered \({t}_{0}=0\) on March 11, 2011 at 14:46 (JST) and \(\Delta t=60\) s for 24 hrs. Employing a tsunami travel time estimator, the front wave of a long wavelength gravity wave with the same average period, \( \sim 35\) min, of the Tohoku 2011 tsunami (see Fig. 1) is shown in Fig. 3.
Data availability
Correspondence and requests for materials should be addressed to Ali Abdolali. (email: ali.abdolali@noaa.gov).
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Acknowledgements
J. T. Kirby acknowledges support from NSF Engineering for Natural Hazards program, grant CMMI1537232. The author wish to thank two anonymous reviewers for useful comments on the text.
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A.A. postulated the idea, derived the solutions, performed insitu data analysis, developed the the numerical calculations of tsunami travel time and prepared all figures and contributed in text preparation; U.K. contributed in the derivation of the solutions, nondimensionalizing and writing main text; J.T.K. contributed in data analysis and text preparation.
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Abdolali, A., Kadri, U. & Kirby, J.T. Effect of Water Compressibility, Seafloor Elasticity, and Field Gravitational Potential on Tsunami Phase Speed. Sci Rep 9, 16874 (2019). https://doi.org/10.1038/s41598019524750
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DOI: https://doi.org/10.1038/s41598019524750
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