New insights on tsunami genesis and energy source
Abstract
Conventional tsunami theories suggest that earthquakes with significant vertical motions are more likely to generate tsunamis. In tsunami models, the vertical seafloor elevation is directly transferred to the sea-surface as the only initial condition. However, evidence from the 2011 Tohoku earthquake indicates otherwise; the vertical seafloor uplift was only 3–5 m, too small to account for the resultant tsunami. Surprisingly, the horizontal displacement was undeniably larger than anyone's expectation; about 60 m at the frontal wedge of the fault plate, the largest slip ever recorded by in situ instruments. The question is whether the horizontal motion of seafloor slopes had enhanced the tsunami to become as destructive as observed. In this study, we provide proof: (1) combining various measurements from the 2011 Tohoku event, we show that the earthquake transferred a total energy of 3.1e + 15 joule to the ocean, in which the potential energy (PE) due to the vertical seafloor elevation (including seafloor uplift/subsidence plus the contribution from the horizontal displacement) was less than a half, while the kinetic energy (KE) due to the horizontal displacement velocity of the continental slope contributed a majority portion; (2) using two modern state-of-the-art wave flumes and a three-dimensional tsunami model, we have reproduced the source energy and tsunamis consistent with observations, including the 2004 Sumatra event. Based on the unified source energy formulation, we offer a competing theory to explain why some earthquakes generate destructive tsunamis, while others do not.
Key Points
- A set of large-scale laboratory experiment with a piston wavemaker is conducted to replicate the tsunami energy ratio associated with earthquake-induced seafloor slope displacements
- The laboratory experiment reveals a three-dimensional nature of tsunami generation mechanism, differing from the vertically forced tsunami theory
- A new source energy formulation with both potential and kinetic energy for tsunami generation is proposed and applied to scale tsunamis
1 Introduction
It is known that throwing a rock into a pond generates waves, while throwing it harder makes bigger waves. Basic physics tells us that the rock has two energy components: its size gives the potential energy (PE), while its speed generates the kinetic energy (KE) [Vennard and Street, 1982]. Likewise, when a submarine earthquake causes a large section of the continental slope to slip into the ocean, the slip generates both potential and kinetic energies. Both types of energy are equally important to generate and propagate tsunamis [Voit, 1987; Song et al., 2008; Dutykh and Dias, 2009]. However, the potential and kinetic energies have different characteristics: the PE depends on the volume added into the ocean (the continuity principle), whereas the KE is the result of the momentum imparted into the ocean (the impulse-momentum principle). The division of the total energy into these two types of source is the key to resolve the tsunami genesis mystery and helps to predict the power and directivity of a tsunami because the ocean has to receive sufficient energy from an earthquake to generate destructive waves.
To focus on the tsunami formation mechanism, we need to recognize the full life cycle of a tsunami. A tsunami cycle can be divided into three stages: formation of initial waves due to earthquake forcing (hydrodynamic process), propagation of the waves in the open ocean (long-wave theory), and transformation of the waves though the shallow seas and runup on to beaches (nonlinear waves). Most observations and modeling studies have been made mainly on the second and third stages through wave measurements and two-dimensional (2-D) tsunami models. Tsunami propagations in the open ocean, runup, and inundation on beaches have been studies extensively [e.g., Satake, 1995; Merrifield et al., 2005; Liu, 2009]. However, the first stage of tsunami generation by earthquakes at the ocean bottom—the interface between earthquake and ocean water—has not been well-understood [Song et al., 2008; Song and Han, 2011; Titov et al., 2016]. This study focuses on how earthquakes transfer energy into the ocean to trigger tsunamis instead of how tsunamis propagate.
For a long-time, it is believed that the uplift of seafloor, more precisely, the total seafloor elevation change (hereafter, “seafloor elevation” includes the vertical displacement of seafloor and the vertical component of the horizontal displacement of the sloping seafloor) induced by the earthquake, is the main source of tsunami [Abe, 1973; Tanioka and Satake, 1996]. Most tsunami models, based on the 2-D shallow-water equations, treat the ocean as one-layer flow or like “a piece of paper” in the whole tsunami life cycle (including the excitation period). The earthquake-induced seafloor elevation is often assumed to be instantaneously transferred to the sea-surface and thus serves as the initial condition [Satake, 1995]. This simplification has been shown successful for modeling tsunami (long-wave) propagations in large scale, but clearly ignored or bypassed the hydrodynamic process in the excitation period. However, to date, the lack of coseismic measurements at the fault area provides challenges to verify the conventional theory. Although significant progress has been made on tsunami research, particularly in the last decade on far-field tsunami propagation, our understanding of the tsunami genesis is still incomplete.
Historically, laboratory wavemaker experiments were used to establish the tsunami genesis theory. For example, Hammack [1973] demonstrated that the vertical uplift of a wavemaker bottom could generate tsunami-like waves. This is true because ocean water is practically incompressible; therefore, the seafloor uplift can be essentially translated to the sea-surface for tsunami generation study. However, this does not mean the seafloor uplift is the only major cause of tsunami, because no other forcing mechanisms are considered in Hammack's experiments. In reality, no major recorded tsunami has been generated at a flat ocean bottom or in an ocean of uniform depth. Instead, most tsunamis have been generated near the continental edges because giant earthquakes often occur where large oceanic plates underthrust continental margins and these earthquakes also involve significant lateral slope displacements. To include the lateral displacement of slopes, Iwasaki [1982] conducted pioneering wavemaker experiments based on a movable tank with a sloping bottom to reproduce the horizontal ground motion. He concluded from the experiments that the kinetic energy due to the horizontal motion was negligible. However, his experiments maybe fundamentally flawed because the key nondimensional parameter in his experiments did not represent the scale of realistic earthquakes. Specifically, his laboratory parameter, the ratio of the water-tank depth (D = 6 cm) to the slip distance (L = 3–40 cm), represents an unphysical ratio of ocean depth (6 km) to slip distance (3–40 km), thus creating excessive potential energy (PE) while diminishing the slip-induced kinetic energy (KE). Therefore, the experimental results cannot be extrapolated to explain the real physics of tsunami genesis. Since then, to the authors' knowledge, no laboratory experiment has been conducted to confirm or refute Iwasaki's conclusion on the tsunami formation mechanism. Clearly, a systematic study on the tsunami genesis, based on recent field observations and modern wavemaker facilities, is needed to shed light on the puzzle why some earthquakes generate tsunamis, while others do not.
In this study, we use two state-of-the-art wave basins and a three-dimensional numerical tsunami model to quantify the earthquake-induced oceanic energy. Different from previous studies, our laboratory experiments allow for a wide range of nondimensional parameters in quantifying the ratio between the vertical displacements that generate potential energy (PE) and the horizontal displacements that induce the ocean kinetic energy (KE). We use nondimensional and normalized approaches to interpret and analyze the laboratory results instead of the dimensional approach of Iwasaki [1982]. In addition, our three-dimensional tsunami model takes into consideration the two energy components and the three-dimensional hydrodynamic process in the tsunami excitation period. The laboratory experiments and the three-dimensional tsunami simulations are guided by the near-field in situ measurements from the 2011 Tohoku earthquake and tsunami, including coseismic measurements from real-time GPS stations [Simons et al., 2011] and long-term acoustic measurements on the seafloor and at the front wedge of the fault [Sato et al., 2011; Ito et al., 2011]. In this novel comprehensive study, we take advantage of the information from all these data in the design of the laboratory experiments and the development of the tsunami models, and use the tsunami observations to validate the model simulations.
This article is organized as follows. In section 2, we analyze in situ measurements from the 2011 Tohoku earthquake. Based on the in situ data, we estimate the potential and kinetic oceanic energy that powered the devastating tsunami. In section 3, we present results from our laboratory experiments based on the state-of-the-art large-scale wave basins, which allows us to confirm and contrast Iwasaki's conclusions. In section 4, we use a three-dimensional tsunami model to demonstrate that, by tuning the input parameters, either the vertical displacement (potential energy) or the horizontal displacement (kinetic energy) can be used equally effective in producing the observed tsunami, suggesting that the vertically forced tsunami theory is not ubiquitous. However, based on the proposed source energy formulation, considering both potential and kinetic energies, the model simulations explain the tsunami height and propagation pattern better than just using one of the two energy components alone as in most current practice. Omitting either of these energy components could significantly underestimate the tsunami's destructive power. A summary and discussions are given in the final section.
2 Tsunami Energy From In Situ Data
The 2011 Tohoku earthquake and tsunami was one of the most measured and observed disaster by far. An unprecedented amount of data was collected. Here, we combine all those data in a systematic way, as schematically shown in Figure 1. Specifically, five sets of in situ data from the near-field coastal land to the deep trench are analyzed here:
- On the coastal land, real-time GPS measurements of the ground motions at about 1200 sites with an average spacing of 20 km over Japan became available days after the event [Simons et al., 2011]. These land-based GPS stations, operated by the Geospatial Information Authority of Japan (GSI), provided precise information on the coseismic slip continuously in every second. For example, the closest coastal station at the Oshika Peninsula gave a coseismic displacement of about 1 m vertical subsidence and 5 m horizontal displacement.
- The second set of data was given by Sato et al. [2011], who surveyed the seafloor displacements from ocean bottom GPS/acoustic stations on the shallow slope, about 1500 m below the ocean surface. Their data consist of five stations at MYGI, KAMS, KAMN, MYGW, and FUKU with vertical and horizontal displacements of (3 m, 24 m), (1.5 m, 23 m), (1.6 m, 15 m), (–0.8, 15), and (0.9, 5), respectively.
- A third set of measurements on the deeper continental slope came from Fujiwara et al. [2011], who compared repeated multibeam surveys from 1999, 2004, and 2011, and calculated average displacements along a strip from the trench axis. Using a time differential approach, we obtained an uplift of 4 m and a horizontal displacement of 41 m at the location for the same period.
- The measurements at the frontal wedge of the fault plate were provided by Ito et al. [2011], who reported an uplift of 5 m and a horizontal displacement of more than 60 m by differing the ocean-bottom pressure gauge and acoustic ranging measurements before and after the 2011 Tohoku-Oki earthquake.
- In addition, seismic reflection and refraction images from Tsuji et al. [2011] provided information on the plate boundary geometry. They estimated a slip of 80 m along the boundary, which is consistent with the measurements of Ito et al. [2011] at the frontal wedge deformation after consideration of the slip angle.
It should be noted that among these five data sets, only the land-based real-time GPS data provides the coseismic event of the earthquake. Other data sets, processed after the event, contain the effects of postseismic slip activities, which should be adjusted for tsunami initials. Previous studies on the 2004 Sumatra earthquake by Chlieh et al. [2007] and Hsu et al. [2006] reported a rate of 30–35% of postseismic slip within months of the main shock. The tsunami was induced only by the main shock, roughly in the period of a few tens of minutes. Therefore, the postseismic effect did not contribute to the tsunami forcing, which was considered carefully in this study.
Despite the differences in the surveying methodologies, all of the data sets show a consistent slip distribution of seafloor: the whole continental slope slipped into the ocean from 5 m at the coast, to about 20 m at the middle slope of 1500 m, and to 50–60 m at the front edge of the deep trench of 7800 m below the ocean surface. In addition, the fault plate tilted along the hanging line at the middle slope with a subsidence of 1 m at the coast and a larger uplift of about 5 m at the trench. Figure 1 shows these data sets in a color-coded format.
Based on the land-based GPS data, we have derived the seafloor displacements due to the main shock of the 2011 Tohoku earthquake, as shown in Figure 2. The derivation of the three-dimensional seafloor displacements is based on the empirical profile method of Song [2007], which extrapolates the coseismic slip from the coastland to the ocean floor up to the frontal wedge of the fault plate. Figure 2a gives the horizontal displacement and Figure 2b gives the vertical uplift/subsidence of the seafloor. The modeled coseismic displacements (black arrows) are compared with those after-event measurements (magenta and red arrows). It can be seen that the coseismic slip is slightly less than the after-event measurements, but agrees within the margin of 30%, consistent with previous studies on the postseismic slip [Chlieh et al., 2007; Hsu et al., 2006; Ozawa et al., 2011]. Other GPS-inversion methods have also been used in deriving the seafloor displacements, for instance, Liu et al. [2014], showing similar results.
As mentioned before, the ocean has to receive energy from an earthquake to generate tsunamis. The slip of the large-scale continental slope into the ocean not only adds volume, but also transfers a huge amount of momentum to the ocean [Song et al., 2008]. The added volume (leading to raised sea-surface) increases the ocean's potential energy (PE), which is proportional to the slip distance, while the added momentum (leading to motions within the layer of bottom water) imparts the ocean kinetic energy (KE), which is proportional to the slip speed and distance, depending on local topography. The resultant tsunami energy has to be generated by the three-dimensional seafloor (land surface) motions.
It should be noted that previous studies referred to the seafloor uplift/subsidence as the vertical effect and the rest in equation 1 as the horizontal effect of seafloor motions [e.g., Satake et al., 2013]. We think they should be referred as one “vertical component” because both disturb the sea-surface vertically and simultaneously, independently to time (static), and generate the ocean potential energy without any distinction.
Also, note that equation 3 actually gives the maximum potential energy by the seafloor elevation. An early study by Kajiura [1970] suggested that if the vertical movement was completed in a few seconds or shorter, the energy transferred to the ocean might be larger than the tsunami energy. Therefore, the conventional tsunami models may have prescribed the maximum source energy by transferring the seafloor elevation to the sea-surface instantaneously.
and are the bottom layer thickness, h is the ocean bathymetry, and h_{x} and h_{y} are the eastward and northward slopes of the subfault surface, respectively. In the formulation, L_{H} is the effective scale of the horizontal motion. Also z is the vertical coordinate at the undisturbed ocean surface, and Δu_{b}(z) and Δv_{b}(z) are the near-bottom water velocity at a subfault or a grid cell within the range of effective scale, respectively. According to Cooker and Peregrine [1995], the effective scale is proportional to the depth, i.e., L_{H} = h. The coefficient is about 0.542 based on their laboratory experiments. Note that it is difficult to validate the coefficient by earthquake measurements in deep oceans. We have found that the tsunami source calculation is sensitive to the coefficient value. Therefore, the coefficient should be used as a reference. Also, note that the horizontal displacement does not generate any energy on a flat ocean bottom (i.e., h_{x} = h_{y} = 0) or in the direction parallel to the slope (i.e., h_{x} =0 or h_{y} = 0).
In this study, we define the rise-time as the seafloor displacement period for a grid cell by . To determine the seafloor rise-time , we have used high-rate (1 Hz) GPS data with 1 s sampling interval. Figure 3a gives the selected GPS sites and Figure 3b gives the rise-time for each of the selected GPS sites. It can be seen that the rise-time decreases roughly 20 s in 100 km from the far-field GPS (green) to the near-field GPS (blue) sites. Extrapolating the two groups to the epicenter and the trench, which is about 200 km from the near-field GPS site, we obtain a maximum rise-time of about 30 s, indicated by the dashed line in Figure 3b. Noticing the two-stage nature of the near-field GPS displacement, we expect an event shorter rise-time for some patches at the epicenter and near the trench. For this reason, we have used a rise-time of 20 s in the calculation.
Note that the seafloor rise-time is not the earthquake rupture time. The latter is for the whole fault area and can take a few minutes (e.g., the 2011 Tohoku case) or over 10 min (e.g., 2004 Sumatra case) to complete, depending on the rupture speed and the size of the fault area. Also, the seafloor rise-time should not be confused with the tsunami excitation time or sea-surface rise-time. The seafloor rise-time is the seafloor responding time to the earthquake slip, while the sea-surface rise-time is the sea-surface responding time to the seafloor motions, which involves the oceanic hydrodynamics and depends on the ocean depth.
Based on the 2011 Tohoku event measurements (Figure 1), the derived three-dimensional seafloor motions (Figure 2), and equations 1-7, we obtain a total energy transferred by the earthquake of 3.1 × 10^{15} joule to the ocean, in which the potential energy due to the vertical seafloor elevation (including seafloor uplift/subsidence plus the contribution from the horizontal slip of the slope) is only 1.5 × 10^{15} joule, while the kinetic energy due to the horizontal displacement velocity of the continental slope is 1.6 × 10^{15} joule. Recently, Titov et al. [2016] compared the tsunami source energy by the GPS approach with those estimates by tsunami-wave inversion from DART buoys, and found the results to be consistent for all of the cases examined, particularly for the 2011 Tohoku event, where more data were available.
3 Laboratory Experiment
- The large wave flume has a single piston wavemaker and a channel of 104 m long, 3.7 m wide, 4.6 m deep, and a long slope for mimicking sand beaches; and
- The multidirectional wave basin has a snake piston wavemaker with 30 actuators and 29 paddles in a wave basin of 48.8 m long, 26.5 m wide, and 2.1 m deep.
The wavemakers are controlled electronically using state-of-the-art wavemaker theories. The laboratory facilities allow a wide range of model parameters (e.g., Reynolds and Froude numbers) for simulating complex problems in fluid dynamics.
To demonstrate kinetic energy generation from horizontal displacement velocity, we divide our experiment into several sets of test runs—each set has a fixed and constant stroke distance (L), but with increasing wavemaker stroke velocity (V = L/ ) by varying the rise-time , as shown in Table 1. This design allows us to quantify the increasing kinetic energy (KE) determined by the velocity of the stroke (V), in comparison with a reference potential energy (PE) determined by a constant stroke distance (L) or a fixed volume added into the water. We conduct 13 sets of test runs with each set consists of 1 to 20 test runs, as summarized in Table 1.
Stoke Distance (L) Water Depth (h) | Stroke Time: (s) | ||||||
---|---|---|---|---|---|---|---|
L = 0.5 m h = 2.0 m |
0.5 | 0.75 | 1 | 1.625 | 1.75 | 1.875 | |
L = 0.5 m h = 1.35 m |
0.5 | 0.75 | 1 | 1.25 | 1.5 | ||
L = 0.5 m h = 1.0 m |
0.5 | 0.75 | 1 | 1.25 | 1.5 | ||
L = 0.5 m h = 0.65 m |
0.5 | 0.75 | 1 | 1.25 | 1.5 | ||
L = 1.0 m h = 2.0 m |
0.75 | 1.25 | 1.75 | ∼ | 3.25 | 3.5 | 3.75 |
L = 1.0 m h = 1.35 m |
1.25 | 1.75 | 2.25 | 2.75 | |||
L = 1.0 m h = 1.0 m |
0.75 | 1.25 | 1.75 | 2.25 | 2.75 | ||
L = 1.0 m h = 0.65 m |
0.75 | 1.25 | 1.75 | 2.25 | 2.75 | ||
L = 2.0 m h = 2.0 m |
1 | 1.5 | 2 | 3.75 | 4 | 4.25 | |
L = 2.0 m h = 1.35 m |
3 | ||||||
L = 2.0 m h = 1.0 m |
0.5 | 1 | 1.5 | 2 | |||
L = 2.0 m h = 0.65 m |
2.0 | 2.5 | 3.0 | ||||
L = 3.0 m h = 2.0 m |
2.5 | 2.75 | 3 | 3.25 | 3.5 | 3.75 | 4.25 |
- ^{a} L is the stroke distance, h is the depth, and is the stroke time. In each set (row) of the test runs, the stroke distance and water depth are constant.
In each test run, the total energy inputted from the wavemaker to the fluid is computed and its kinetic energy component is calculated based on the stroke velocity and measurements of wave height and celerity. In the LWF, the wave height and the fluid velocity are measured along the channel at six locations, i.e., at 1, 3, 5, 6, 8, and 12 m from the neutral wavemaker position. A sampling frequency of 50 Hz is used in all test cases. In the DWB, only the leading wave height is measured. The resulting leading-wave height is a function of the total energy. An important assumption is that the theoretical energy of the wavemaker imparting to the water is completely transferred to the fluid with no energy loss. From physical intuition, it can be seen that for a set of experiments with a constant stroke distance, the potential energy transferred from wavemaker to the fluid is constant because the volume added into the water is the same, while the kinetic energy differs depending on the stroke velocity, as schematically shown in Figure 4.
The approximation in equation 9 is for cases when a constant stroke velocity V = L/ is used. Strictly speaking, it is difficult to have a constant stroke velocity for wavemaker starting from zero and ending in zero, as this would require infinite acceleration and deceleration, respectively. An approximate constant velocity curve is achieved by a short duration maximum physically allowable acceleration at start and a corresponding short duration maximum physically allowable deceleration at end. This short duration acceleration and deceleration has also to be sufficient long to minimize possible generation of breaking waves at the vicinity of the wavemaker surface.
Figure 5a shows the leading wave heights as a function of the kinetic to potential energy ratio for each set of the experiments. Since within each set of the test runs, the stroke distance is constant, thus the corresponding potential energy is also constant. Hence all the measurements within the set with increasing stroke velocity can be categorized according to stroke distance. It can be seen that in each set of the test runs, the kinetic energy increases with the stroke speed, which is consistent with our theoretical formulation of energy equations 8-9, suggesting that the kinetic energy plays an increasing role in determining the magnitude of the wave height.
In which is the height of the leading wave, equals the still water depth, and KE and PE are the inputted kinetic and potential energy per unit width, respectively. All the parameters are grouped nondimensionally. The relationship in nondimensional scales clearly suggests that the kinetic energy can contribute up to a normalized wave height of 0.6 in some extreme cases.
Based on the experimentally obtained equation above, we calculate the 2011 Tohoku and the 2004 Sumatra cases in the same content, and place them in Figures 5a and 5b. First, we notice that from Figure 5a, even our modern wavemakers do not represent the realistic length scale because the tsunami height ( ) relative to the ocean depth (h) is too small to be quantified by the wavemakers. However, after normalizing the experimental results, as shown in Figure 5b, the analytical equation 10 represents the two real cases well, projecting a maximum wave height of 12 m for the 2011 Tohoku case and 10 m for the 2004 Sumatra case near the fault, consisting with previous studies [Satake et al., 2013; Titov et al., 2016; Song et al., 2005].
Now we explain how our experiments is conducted differently from the one presented in Iwasaki [1982], who concluded that the contribution by the horizontal motion of the slope was negligible. Table 2 compares the parameters used in Iwasaki's experiment with those of ours, and the realistic parameters for the 2004 Sumatra earthquake and the 2011 Tohoku earthquakes. Three nondimensional parameters are highlighted: the length scale R = L/h that determines the ratio of the slip distance (L) relative to the water depth (h); the Froude umber , which is a widely used parameter in fluid dynamics; and the newly introduced energy scale Er = KE/PE, representing the ratio of the kinetic energy to the potential energy.
Length Scale R = L/h |
Froude Number , Where V = L/ |
Energy Ratio Er = KE/PE |
|||||||
---|---|---|---|---|---|---|---|---|---|
Depth h (m) | Slip L (m) | R | Rise-Time (s) | Velocity V (m/s) | Fr | PE (10^{15} J) | KE (10^{15} J) | Er | |
2011 Tohoku | 5000 | 20–60 | 0.004–0.012 | ∼10 | 2–6 | 0.008–0.024 | 1.5 | 1.6 | 1.1 |
2004 Sumatra | 4000 | 10–40 | 0.0025–0.01 | ∼10 | 1–4 | 0.004–0.018 | 2.2 | 3.8 | 1.7 |
Iwasaki [1982] | 0.06 | 0.03–0.4 | 0.5–6.7 | 0.2–2.8 | 0.004–0.8 | 0.005–1 | ∼L^{2} | ∼V^{2}h | |
This study | 2 | 0.5–2.99 | 0.25–1.5 | 0.12–4.2 | 0.12–12 | 0.027–2.7 |
- ^{a} Three nondimensional parameters (scales) are presented: length scale (R), Froude number (Fr), and energy ratio (Er). Note: Assuming the same fault area, the table indicates as a rule of thumb that the potential energy (PE) is proportional to the square of the initial surface perturbation (L^{2}), independent of the water depth. Differently, the kinetic energy (KE) is proportional to the square of the stoke velocity (V^{2}) and the water depth (h), because more volume of water would be excited. The kinetic energy explains why destructive tsunamis are often generated at deep trenches associated with large horizontal displacements of continental slopes.
First, we can see that both Iwasaki's and our experiments represent the Froude number reasonably well in comparison with the real earthquake-tsunami events of 2004 Sumatra and 2011 Tohoku. Therefore, the experimental results can be used to explain the relationship between the stroke velocity and gravity wave propagation. However, neither Iwasaki's experiment nor our modern wave basin facility could match reality in the length scale (R = L/h). The problem is that the slip distance (or laboratory wave height) relative to the water depth of the wavemaker experiment is too large, about 100 times of the realistic earthquake-tsunami length scales. Since the potential energy is proportional to the slip distance (L), such an unrealistic length scale would result in an exaggerated potential energy. Such an exaggeration in the length scale, but not in Froude number, led to the flawed conclusion in Iwasaki [1982] who ignored the stroke velocity or kinetic energy in the tsunami source energy. This flawed conclusion from Iwasaki's experiment led to the tsunami initial of Tanioka and Satake [1996] in citing Iwasaki's conclusion. It should be noted that Tanioka and Satake is the first to include the vertical component of sloping faults in the tsunami initial conditions, which is needed in the tsunami source.
Different from Iwasaki [1982], our experiment was designed to examine the contribution of kinetic energy in tsunami generation relative to potential energy using the nondimensional parameter of energy ratio (Er = KE/PE). Within each set of our test runs, kinetic energies are compared with each other on the same reference value of the potential energy, as shown in Figure 5a. Notice that the KE is only a fraction (<30% in this study) of the PE in most of the experimental cases. In particular, Iwasaki's experimental data fall practically all less than 5%. This could easily draw a misleading conclusion that the KE is negligible. By comparing results in each set of the experiment, we found that the wave height increases rapidly with the KE under the same PE reference value; therefore, we suggest that the movement of bathymetry slopes can contribute significantly to tsunami generation. Unfortunately, even our modern laboratory facility does not have the capability to reasonably match the physical length scale (deep enough) to directly determine the energy quantitatively, as shown in Figure 5a. Despite of the limitations of the laboratory wave basins, the experimental energy ratios examined in this study, for the first time, clearly illustrate the importance of the kinetic energy in determining the resultant wave height. Indeed, our experimentally determined normalized wave height to energy ratio function (equation 10), as shown in Figure 5b, can be used to predict realistic earthquake tsunami parameters. Because large earthquakes involve significant continental slope displacements, the kinetic energy generated by the impulsive displacements of the continental slopes at the trench should not be ignored.
Equations 4a and 4b have been used in the KE equation for a rectangular fault section with width of W and depth of h. It can be seen that the simplified source energy formations (11)–(12) are equivalent to the laboratory formulations of equations 8-9 as and , respectively. Also, in the kinetic energy formulation, the steepness of the continental slope does not play a role in the calculation. In fact, the height of the slope h, multiplied by the effective scale L_{H}, is the key factor in generating tsunami kinetic energy.
4 Three-Dimensional Tsunami Modeling
The idea of including the earthquake-induced momentum perturbation in this tsunami model is based on the impulse-momentum principle of fluid mechanics [Vennard and Street, 1982], and recently formulated by Song et al. [2008]. When a large-scale continental section slips into the ocean due to earthquakes, a transfer of momentum occurs and a three-dimensional force is exerted on the fluid through a distance besides the volume added into the ocean. In that sense, it is just like throwing a rock into a pond, the resulting wave height not only depends on the size or volume (potential energy) of the rock, but also the speed and shape (kinetic energy) of the rock. Both energies are equally important in generating waves (the energy principle of fluid mechanics), as demonstrated in the results of our wave basin experiment. However, to validate the tsunami source energy by observations, we need numerical models to link tsunami observations.
By including the seafloor-induced velocity (horizontally) along with the seafloor-deformed volume (vertically) from the GPS data (Figure 2) in the three-dimensional ocean model, we reproduce the 2011 Japanese tsunami.
Figure 6 shows four snapshots of the simulated 2011 Japanese tsunami at 1, 5, 10, and 25 min after the earthquake, respectively. Note that after 1 min of the quake (Figure 6a), only a narrow sea-level elevation is formed along the front wedge of faulting plate. Within 5 min (Figure 6b), the narrow sea-level elevation splits into two leading waves toward the Japan coast and to the open ocean, respectively. Since the horizontal flow induced by the seafloor horizontal motion is mainly eastward, thus it only increases the wave height to the east of the source, as clearly seen in this period. At 10 min of the quake (Figure 6c), the two leading waves are still dominant with very weak tertiary waves formed perpendicular to the seafloor slip direction. After 25 min of the quake (Figure 6d), the leading wave toward the coast runs up and reaches the Sendai coast, while the leading wave to the open ocean has not reached the closest DART buoy yet.
We also compared the results from the full three-dimensional tsunami source (kinetic energy included) with that from the vertical seafloor elevation only. Figure 7a shows the selected comparison locations. Figure 7b compares the waveforms at the coastal locations, while Figure 7c compares the waveforms at the DART buoys in the deep ocean. It can be seen that the simulation with the full source energy (red) agrees well with observations in terms of the arrival times, the largest amplitude (the primary wave), and the overall patterns of the time series, although the agreements for the secondary waves are not as good as for the primary ones. However, the simulation with vertical seafloor elevation only (green) does not match the field observations as well, particularly in maximum amplitude and along the axis of the seafloor slip direction.
To further demonstrate the three-dimensional nature of the earthquake forcing, we use previously simulations for the 2004 Sumatra event [Song et al., 2008; Song and Han, 2011]. Figure 8 compares the simulated tsunamis with each component of the tsunami source. The tsunami initial conditions have been derived from GPS measurements in Song [2007]. Although DART buoy was not present then in the Indian Ocean, both satellite altimetry and tide gauges have observed the tsunami. Again, it can be seen that each of individual initial conditions (vertical and horizontal) can generate the tsunami, almost like that by the total forcing, but with lower amplitude (see Figure 8d). In this simulation, we have recalculated the tsunami source energy and obtained a KE to PE ratio of 2.7 (Table 2).
Besides the lower amplitude, the vertically and horizontally forced tsunamis have different effects on the tsunami pattern. Figure 9 shows the asymmetric tsunami pattern recorded by tide gauges showing leading-elevation waves toward Sri Lanka (Figure 9b) and leading-depression waves toward Thailand (Figure 9c). Such an asymmetric tsunami pattern, or “N-waves” [Tadepalli and Synolakis, 1994], is best explained by the horizontally forced component. In contrast, vertical forced tsunamis usually propagate around the source without enhanced directionality. The commonly observed “N-waves” pattern, traveling along the seafloor slip direction or perpendicular to the trench, is most likely forced by the inclining slope of the fault plate, rather than by the vertical seafloor elevation. This particular observation confirms the role played by the horizontal forcing toward the deep oceans.
5 Summary and Discussions
In this study, we have provided unprecedented amount of evidence from the 2011 Tohoku and the 2004 Sumatra events, and used three different approaches—in situ data analysis, wave-generation experiment, and three-dimensional hydrodynamic modeling, to shed lights on the long-puzzled and poorly understood tsunami genesis. Our key strategy is to formulate tsunami source energy by considering both potential energy due to the vertical seafloor elevation (i.e., the vertical displacement and the vertical component of the horizontal displacement of the sloping seafloor, both are static) and the kinetic energy due to the horizontal displacement velocity of the continental slope (hydrodynamic component). For the first time, in situ measurements from the 2011 Tohoku earthquake provided a complete set of data about the seafloor displacement along a section of the continental margin from the coast to the front edge of the fault plate. The data allow us to estimate the tsunami source energy. Estimate from the in situ data gives the total tsunami source energy of 3.1 × 10^{15} joules, in which the potential energy was slightly less than a half, while the kinetic energy contributed the majority. The estimate is found to be consistent with the values independently inverted from DART buoy measurements [Tang et al., 2012; Titov et al., 2016].
Laboratory experiment is a simplification of reality, but represents the true physical process of solid objectives forcing on the fluid to generate tsunami-like waves. Such a physical process is difficult to be prescribed in tsunami models. We have conducted laboratory experiments using a fixed stroke distance with increasing stroke velocity to quantify the ratio of kinetic energy to potential energy. Both nondimensional and normalized approaches have been used in analyzing the results and have shown consistency with the 2011 Tohoku and the 2004 Sumatra cases. Our wavemaker experiments provided a new means to investigate the tsunami formation mechanism in the laboratory. The experimental results, analyzed by a normalized nondimensional approach, conclude that the kinetic energy cannot be ignored in the resulting wave height or tsunami source energy—a contradiction to Iwasaki's conclusion.
Numerical models have been widely used in producing observed tsunamis. Particularly, most of the tsunami models are based on 2-D shallow-water equations. A sea-surface perturbation can be easily tuned to match observed tsunamis after the event. However, the initial sea-surface perturbation is not the seafloor deformation of the earthquake. Our study shows that the earthquake forcing at the seafloor is three-dimensional and cannot be simply represented by a 2-D tsunami model without considering the hydrodynamic process. Here we have used a 3-D tsunami model, not only for simulating the tsunami propagation waves but also for including the formation period during the three-dimensional earthquake forcing to the ocean water and to the free surface. Our model clearly demonstrated that both the vertical seafloor elevation and the associated horizontal velocity of the continental slope are of equal importance in generating the tsunami. In fact, inclusion of the horizontal effect in the tsunami model best replicates tsunami observations in wave amplitude and pattern, such as N-waves, as shown in Figures 7 and 9.
- Seafloor displacement: Immediately after the December 2004 Indian Ocean tsunami, the Royal Navy survey vessel HMS Scott conducted bathymetric mapping of the Sumatra subduction zone. Surprisingly, only a small area of seafloor uplift was found, leading many to wonder how the enormous amount of tsunami energy was transformed from the earthquake [Moran et al., 2005]. Observations from the 2011 Tohoku event seems to shed some light on the puzzle: an unprecedented horizontal displacement of 58 m, but only an uplift of 5 m at the frontal wedge of the fault plate [Sato et al., 2011; Ito et al., 2011].
- Gravity evidence: Submarine earthquakes often generate gravity signals, which can be measured by gravity satellites, such as the Gravity Recovery and Climate Experiments (GRACE) mission. This is because the seafloor elevation has to displace the volume of ocean water above and the resulted density difference creates a gravity anomaly (note that the horizontal motion does not). Data from GRACE measurements on the 2004 Sumatra earthquake do not support seafloor elevation large enough to match the resultant tsunamis [Song and Han, 2011]. Tectonic motion is largely horizontal in accordance with continental drift theory. The vertical elevation of a large scale of seafloor has to overcome Earth's gravity. The vertically forced tsunami theory seems against the odds of gravitational principle of physics. We are confident that GRACE follow-on and other gravity missions will provide better measurements to improve our understanding of this matter.
- Tsunami observation and model: In tsunami models, the initial sea-surface perturbation can be tuned to match observed tsunamis after event. However, the tuned sea-surface perturbation often does not explain the seafloor elevation of the earthquake. For example, using tsunami observations from the 2011 Tohoku event, Satake et al. [2013] obtained a rise-time of 3 min over the fault. However, our high-rate GPS data (Figure 3) suggest a seafloor rise-time of 30 s or less. What is the reason for the discrepancy or the delayed response by the sea-surface? The rupture time can be one factor, but the delay due to the hydrodynamic process in transferring the horizontal momentum to the sea-surface should not be ignored. Clearly, the seafloor elevation should not take minutes to affect the sea-surface above. However, the horizontal momentum would need time for converting the kinetic energy to potential energy, and then to reach the surface. The hydrodynamic process in the tsunami formation stage should be considered to explain the discrepancy. Furthermore, tsunami directivity can be another clue of the source mechanism. It is known that tsunami propagations often do not symmetric to the source. Bathymetry variation can be one explanation. However, the kinetic energy clearly enhances the tsunami height mostly along the axis of the slip direction or perpendicular to the trench (Figures 6 and 8), where large casualties often occurred. In addition, the widely observed asymmetric tsunami waves, or N-waves, with leading-elevation toward offshore and leading-depression toward onshore, can also be explained by the proposed three-dimensional tsunami forcing mechanism, as shown by Figure 9.
- Submarine landslide: Currently we do not have evidence or sufficient data to support the hypothesis of major landslides in the 2011 Tohoku tsunami or in the 2004 Sumatra tsunami. To our knowledge, submarine landslides are local phenomenon, driven by gravity, and should move very slowly at ocean bottom in broken pieces. Therefore, the gravity-driven landslides should be greatly different from the horizontal displacements of a continental slope section driven by the enormous earthquake force in the tsunami formation process.
- Tsunami waveform inversion: Tsunami measurements and waveforms have often been used to invert the earthquake source with 2-D shallow-water equations models [e.g., Fujii and Satake, 2006; Satake et al., 2013]. It should be realized that the inverted results over the fault are actually the sea-surface initial, rather than the seafloor elevation or the earthquake deformation. For example, using tsunami waveforms, many studies, including Satake et al. [2013] and Tang et al. [2012], inverted a tsunami initial of 10 m or higher, but only 5 m of vertical seafloor displacement is measured near the trench (Figure 1). This seems problematic by assuming the sea-surface initial as the seafloor deformation to constrain seismic inversions. The three-dimensional tsunami initials proposed here do not require overprescribing the vertical displacement because the horizontal momentum of sloping fault contributes the rest, suggesting that the horizontal processes of tsunami excitation should be considered in the waveform inversion for earthquake studies.
- Tsunamigenic earthquakes: Finally, we would like to provide a competing theory to explain why some earthquakes generate tsunamis, while others do not. Different from the conventional theory that the vertical force is the only cause of tsunamis, our study based on in situ measurements, laboratory experiments, numerical formulations of tsunami source energy suggests that earthquakes generate three-dimensional force to the ocean. The vertical component (the vertical displacement and the vertical component of the horizontal displacement of the sloping seafloor) gives the ocean potential energy, which is proportional to the square of the seafloor elevation (∼Δh^{2}), as theorized by our equation 11. This component is an important contributor to the tsunami source, but it is not the only cause as previously perceived. In fact, the horizontal force creates kinetic energy, which is proportional to the depth of the continental slope and the square of its velocity (∼h*Δ ), as theorized by our equation 12. The nature of the kinetic energy—depending on the slip distance and water-depth—is the key to explaining why earthquakes near deep ocean trenches are more likely to generate large tsunamis. For examples, earthquakes with large displacements at deep trenches are likely to generate powerful tsunamis, as in the 2011 Tohoku and the 2004 Sumatra cases. While earthquakes at shallow seas, even close to the coast, like the 2005 Nias Islands M8.8 earthquake of the fourth largest earthquake ever recorded by modern instruments, do not necessarily generate destructive tsunamis.
This study aims to explain how earthquakes transfer energies to the ocean to generate tsunamis. Clearly, without precisely knowing how tsunamis are formed from earthquakes and without accounting for all the energy components that creates the tsunami, it would be difficult to develop a reliable warning system for saving lives and property in tsunami emergencies. There has been a knowledge gap between the earthquake magnitude and the tsunami energy [Titov et al., 2016]. The proposed tsunami source energy formulation provides a framework to bridge the gap. Our previous publications [e.g., Song, 2007; Song et al., 2012], as well as Figure 2 in this article, have shown that real-time GPS stations along coastlines can be used to detect seafloor motions due to large earthquakes. Therefore, the proposed source energy formulation with kinetic energy can be used to convert the GPS-derived earthquake information into tsunami energy and to detect tsunami scales for early warnings in practice. In fact, such a GPS approach has been tested in a NASA pilot program—the GPS-aided Real-Time Earthquake and Tsunami (GREAT) Alert System, as reported by Naranjo [2013], and its real-time performance will be reported in a future paper.
Acknowledgements
The research described here was conducted at the Jet Propulsion Laboratory (JPL), California Institute of Technology, under contracts with the National Aeronautics and Space Administration (NASA). The work of Mohtat and Yim was funded by the NASA Physical Oceanography program through subcontract with JPL. We thank the Geospatial Information Authority of Japan (GSI) for providing the GPS data. Seafloor measurements are available from the corresponding references. Tsunami observations are available at http://near-goos1.jodc.go.jp/index.html and http://www.ndbc.noaa.gov/dart.shtml.