The open ocean energy decay of three recent trans-Pacific tsunamis



[1] The 2009 Samoa (Mw 8.1), 2010 Chile (8.8), and 2011 Tohoku (9.0) earthquakes generated destructive tsunamis recorded by a large number of DART stations in the Pacific Ocean. High-resolution (15 s) DART records yield mean energy decay times for these events of 17.3, 24.7, and 24.6 h, respectively. We attribute these differences to the frequency content of the tsunamis. Specifically, the Samoa tsunami was a “high-frequency” event with periods of 2–30 min whereas the Chile and Tohoku tsunamis were “broad-band” events with periods of 2–180 min. Differences in frequency content are linked to differences in the source parameters: Samoa was a relatively small deep-water earthquake while Chile and Tohoku were extensive shallow-water earthquakes. Frequency-dependent analysis of the Chile and Tohoku tsunamis indicates that shorter period waves attenuate much faster than longer-period waves (decay times range from 15 h for 2–6 min waves to 29 h for 60–180 min waves).

1 Introduction

[2] Understanding tsunami energy decay in time and space is of primary scientific importance and critical for effective tsunami warning and mitigation. Munk [1963] suggested that tsunami energy in the ocean decays much like sound intensity does in an enclosed room. Van Dorn [1984, 1987] used this “acoustic analogy” to examine the attenuation of five major tsunamis and concluded, based on coastal measurements only, that the energy, E(t), for all tsunamis decays as E(t) = E0e− δt, where E0 is the tsunami energy index, δ is the energy decay (attenuation) coefficient, and t0 =  δ− 1  is the e-folding “decay time.” He found this decay to be event independent and nearly uniform for each ocean basin with  t0 ≈ 22 h for the Pacific, 14.6 h for the Indian, and 13.3 h for the Atlantic oceans. Both Munk [1963] and Van Dorn [1984, 1987] postulated that the main energy losses are associated with absorption during multiple reflections from the mainland coasts at a rate of about e-1 per reflection. Thus, the decay time for each ocean was assumed to be of the order of mean “reflection” time defined as tr = L*/c, where math formula is the long-wave speed in mid-ocean, g is the gravitational acceleration, H is the mean water depth, and L* is the mean travel path of tsunami waves.

[3] Early studies of tsunami decay were hampered by the small number and low quality of the analog pen-and-paper records available from coastal tide gauges at the time. The extensive high-quality data collected during the global 2004 Sumatra tsunami enabled Rabinovich et al. [2011] to use 173 coastal tsunami records to examine the tsunami energy decay in the Indian, Atlantic, and Pacific oceans. These results revealed that the decay time, t0, within a given oceanic basin is not uniform but depends on the absorption properties of the shelf adjacent to the observation site and on the travel time from the source region. Decay times for the 2004 Sumatra tsunami ranged from about 13 h for islands in the Indian Ocean near the source region to 40–45 h for remote mainland stations in the North Pacific, which is roughly twice the decay time of 22 h estimated by Van Dorn [1984, 1987] for the Pacific Ocean. The reasons for this difference in decay time were unclear and suggested the need for further investigation.

[4] Wave records from the 2009 Samoa, 2010 Chile, and 2011 Tohoku trans-Pacific tsunamis provide an excellent opportunity to re-visit the energy decay problem. In addition to originating from widely separated regions in the same ocean basin, the three tsunamis were the first to have been recorded by a large number of open-ocean Deep-ocean Assessment and Reporting of Tsunamis (DART) stations distributed throughout the entire Pacific Ocean. Wave records from these stations are free from regionally distorting topographic effects, making it possible to evaluate “pristine” tsunami decay times and to compare these times to the more data-limited estimates of Munk [1963], Van Dorn [1984, 1987] and Rabinovich et al. [2011].

2 Observations

[5] DART buoys operated by the U.S. National Atmospheric and Atmospheric Administration (NOAA) record seafloor bottom pressure at a sampling interval Δt =15 s. In the absence of a tsunami event, these data are averaged internally to a default sampling interval Δt =15 min and then transmitted every hour via satellite to the U.S. National Data Buoy Center. Upon sensing a tsunami event, DART begins to transmit the “raw” 15 s data directly for several minutes before the instrument switches to 1 min averages until the end of the event mode [Mofjeld, 2009]. The 15 s tsunami data are stored in the instrument package and downloaded following instrument retrieval [Mungov et al., 2013]. It is these retrieved 15 s datasets—consisting of 24 records for the 2009 Samoa, 23 for the 2010 Chile, and 18 for the 2011 Tohoku tsunamis—that we use for the present study. For the 2011 tsunami, we also made use of 11 “event” DART records, which were long enough to reliably estimate E0 and t0. Figure 1 shows the specific DART stations used in our analyses. We first calculated the variance of each wave record based on 6 h data segments with 3 h overlaps. As illustrated by the examples in Figure S1 (see supporting information), the tsunami energy decayed exponentially with time. To evaluate the decay parameters E0 and t0, we used least squares analysis to derive the best fit to the variance values. The record durations available to estimate these parameters ranged from 2.5 to 5 days and were dependent on how long the “ringing” of the tsunami signal was detectable above the background noise.

Figure 1.

Map of the Pacific Ocean showing the location of the Mw = 8.1 2009 Samoa, Mw = 8.8 2010 Chile, and Mw = 9.0 2011 Tohoku earthquake epicenters (stars) and positions of the DART open-ocean stations (circles).

3 The 2009 Samoa Tsunami

[6] The Mw = 8.1 Samoa earthquake occurred at 17:48:10 UTC on 29 September 2009 [Lay et al., 2010]. The ensuing tsunami had a maximum runup on the Samoa coast of 17.6 m and was responsible for 189 fatalities [Okal et al., 2010]. The high-resolution (15 s) DART records from this event were used to estimate the tsunami parameters and to compare these parameters with numerical model results [Thomson et al., 2011]. Figure 2a shows maximum computed amplitudes for the 2009 tsunami waves and estimated E0 values (see Table S1 of the supporting information for numerical values of the computed E0 and t0 estimates). According to these computations, the main “beam” of the tsunami energy was directed northeastward, toward the coast of Mexico. For other directions, the computed energy flux was approximately isotropic and yields E0 values that are comparable with those obtained from the data for different DART sites. Minimum E0 values of 0.15–0.20 cm2 were observed at remote sheltered DARTs 52404, 21415, 21416, and 46408, while maximum values of 2.07–2.10 cm2 were observed at more exposed DART sites 52401 and 51426 (see Figure 1 for DART locations); the mean value of E0, averaged over all DART records, was 0.80 cm2.

Figure 2.

Simulated tsunami wave heights along with the tsunami wave energy indices, E0, for the (a) 2009 Samoa, (b) 2010 Chile, and (c) 2011 Tohoku tsunamis. Positions of the DART buoys are indicated by white circles, with the size of the circles proportional to E0. Red stars indicate the earthquake epicenters. Solid thin blue lines are hourly computed isochrones of tsunami travel time from the source areas.

[7] The decay times derived from DART records for this event vary from t0 = 13.0–14.0 h at DARTs 21414, 46410, 51425, and 51426 to t0 = 22.8–24.2 h at DARTs 46408 and 52405. There appears to be a large-scale structure in the energy decay time. Specifically, the two DARTs (51425 and 51426) located in the vicinity of the source have t0 < 15 h, while the “eastern DARTs” (located along the coast of America) have t0 = 15–20 h, and the “western DARTs” (located in the western Pacific) have mainly t0 = 20–25 h (Figure 3a). The mean value for all records was t0 = 17.3 ± 0.7 h.

Figure 3.

As in Figure 2, but for the energy decay times, t0, of the three tsunamis. Positions of DARTs are indicated by color circles; colors denote the tsunami decay time. Circles with red contours in Figure 3c indicate “event mode” DARTs with variable samplings.

4 The 2010 Chile Tsunami

[8] The 2010 tsunami event was generated by a magnitude Mw = 8.8 thrust-fault earthquake at 06:34 UTC on 27 February 2010 near the coast of Central Chile. The earthquake was the largest in the Southern Hemisphere since 1960 [Delouis et al., 2010]. The resulting tsunami claimed 124 victims in coastal areas of Chile where the maximum observed tsunami runup was 29 m [Fritz et al., 2011]. The tsunami was recorded by more than 200 coastal tide gauges and by many DART stations. Rabinovich et al. [2013] formulated a numerical model for this event that is in good agreement with the DART and tide gauge data. According to this model, the main beam of tsunami energy was directed northwestward toward the Marquesas and Hawaiian islands and Japan (Figure 2b); a substantial fraction of the tsunami energy was also directed northward to the coast of Mexico and California, in good agreement with observations [cf. Reymond et al., 2013; Borrero and Greer, 2013]. The maximum estimated E0 value of 44.0 cm2 was obtained for DART 51406 located in the core of the main tsunami energy beam (Figure 2b); significant E0 values of 12.8–17.6 cm2 were also obtained for other DART stations located to the northwest of the source (DARTs 32412, 52401, and 21413). In contrast, E0 values were minimum (2.1–2.9 cm2) at DART sites 46411, 46407, and 46410, which were outside the beams of the propagating wave field (Table S2, supporting information). The mean E0 averaged over all 23 available DART records was 9.0 cm2, which is significantly larger than that for the 2009 Samoa tsunami, reflecting the fact that the 2010 earthquake and associated tsunami were much larger than for the 2009 event (note that the mean E0 value for this event, as well as for the 2009 Samoa and 2011 Tohoku events, was strongly dependent on the availability of DART stations in the vicinity of the source).

[9] The estimated decay times for the 2010 Chile tsunami records vary from 28.6–28.1 h at DARTs 46419 and 32411 to 20.8–21.8 h at DARTs 46402 and 51406. In general, the more “western” DARTs have slightly shorter decay times than the “eastern” DARTs (Figure 3b). The mean value for the 23 DART records is t0 = 24.7 ± 0.4 h.

5 The 2011 Tohoku Tsunami

[10] At 05:46 UTC 11 March 2011, a giant thrust fault earthquake of magnitude Mw 9.0 occurred off the coast of Tohoku District, northeastern Honshu, Japan. The earthquake was the strongest in Japan's history and one of the strongest ever instrumentally recorded [Simons et al., 2011; Saito et al., 2011]. Tsunami runup heights for this event were up to 40 m along the coast of Japan [cf. Mori et al., 2011] and were responsible for almost 20,000 deaths. The 2011 tsunami was recorded by approximately 250 coastal tide gauges throughout the Pacific Ocean and by numerous bottom pressure gauges [cf. Song et al., 2012; Saito et al., 2011; Borrero and Greer, 2013]. The recorded data were used by Fine et al. [2013] and Tang et al. [2012] to examine tsunami energy propagation and transformation in the Pacific Ocean.

[11] We have estimated the energy decay parameters for the 2011 tsunami based on the 15 s records downloaded from 18 DART stations and “event mode” data obtained from 11 additional DARTs with various sampling intervals (Table S3, supporting information). Results from this analysis are compared with the model results of Fine et al. [2013] (Figure 2c). According to the model, the tsunami energy flux was spread over a wide array of ray paths. The two main energy beams were directed eastward toward California and Mexico, and southeastward toward Peru and Chile. Two additional branches radiated northeastward toward the Aleutian Islands and southward to New Guinea and New Zealand. In general, there is good agreement between simulated and measured tsunami waves, as indicated by the strong coincidence between the computed beams of maximum wave amplitude and sites with high recorded E0. The maximum tsunami energy was observed at DART 21418 (E0 =161 cm2) located close to the source area, at two nearby DART sites 21413 (127 cm2) and 21419 (73 cm2), and at DART site 51407 (93 cm2) located within the main branch of the propagating tsunami energy (Figure 2c). The minimum value of E0 (6.0–7.0 cm2) was found at DARTs 46409, 46410, and 51426 situated in remote corners of the DART network relative to the tsunami source. These results match well coastal observations of the 2011 tsunami [cf. Reymond et al., 2013; Borrero and Greer, 2013]. The mean E0 value for all 29 DART records was 31.9 cm2, which is roughly 3.5 times greater than for the 2010 Chile tsunami, consistent with the greater magnitude of the 2011 Tohoku earthquake.

[12] The decay times estimated for the 2011 Tohoku tsunami are shown in Figure 3c. These times are mutually consistent, with t0 values varying from 29.3 h at DART 46409 to 20.9 h at DART 46419. As with the 2010 Chile tsunami, the “western” group of DART stations yields slightly shorter decay times than the “eastern” group (Figure 3c). The mean t0 value for the 29 records was 24.6 ± 0.4 h, in close agreement with estimates by Tang et al. [2012] for three selected coastal stations: Kahului, Hawaii 21.2 h, Crescent City, California 23.5 h, and Adak, Aleutian Islands 20.9 h.

6 Discussion and Conclusions

[13] Our DART-based estimates of t0 for the 2011 Tohoku and 2010 Chile tsunamis (24.6 and 24.7 h, respectively) are almost identical. However, these values differ significantly from the value t0 = 17.3 ± 0.7 h for the 2009 Samoa tsunami. Two basic questions arise from our analysis: (1) Why is the decay time t0 for the 2009 Samoa tsunami considerably less than those for the 2010 Chile and 2011 Tohoku tsunamis? and (2) why is t0 for the three recent trans-Pacific tsunamis much shorter than the corresponding value of t0 for tsunami waves observed within the Pacific Ocean following the 2004 Sumatra tsunami [Rabinovich et al., 2011]?

[14] The second question is straightforward to answer. All three tsunamis that we examined in the present paper originated in the Pacific Ocean and essentially remained in the Pacific where they dissipated due to shelf/coastal absorption processes. In this sense, the acoustic analogy of sound intensity in an enclosed room [Munk, 1963] is valid. In contrast, the 2004 Sumatra mega-tsunami originated in the Indian Ocean and then subsequently entered the Pacific Ocean through a variety of connecting passages [Rabinovich et al., 2011]. The wave energy continued to enter the Pacific for several days, where it maintained a high energy level and effectively reduced the “natural” rate of tsunami decay.

[15] The answer to the first question is likely related to differences in the frequency content of the tsunami wave fields. To examine this assumption, we conducted spectral analyses of the 15 s tsunami records collected by the DART buoys and compared the resulting tsunami spectra, Stsu(ω), with the background spectra, Sbg(ω), at each of the corresponding sites. For analysis of background signals, we used the 5 day period immediately preceding the tsunami arrival times, yielding ν = 64 degrees of freedom; for the tsunami waves, we used 21.3 h periods immediately following the wave arrivals, for which ν = 14. Following Rabinovich [1997], we calculated spectral ratios, Rs(ω) = Stsu(ω)/Sbg(ω), which give the amplification of the long-wave spectrum during the tsunami event relative to the background conditions. We began by estimating the mean regional values, math formula, for various groups of DARTs and then determining the mean ratios, math formula, averaged over all available DART records (Figure 4).

Figure 4.

Estimated spectral ratios, Rs(ω) (tsunami/background), for the (a) 2009 Samoa, (b) 2010 Chile, and (c) 2011 Tohoku tsunamis. Individual color curves in plots are related to the mean value of math formula for the jth group of DARTs; thick black curves are the mean ratios, math formula, averaged over all available DART ratios (24 values for Figure 4a, 23 for Figure 4b, and 18 for Figure 4c). Shaded areas denote the mean tsunami response showing amplification of the tsunami wave spectra relative to the background wave spectra.

[16] Our findings show that the 2009 Samoa tsunami was a pronounced “high-frequency” event with periods ranging from 2 to 30 min and a distinct spectral peak at periods of 7–8 min. In contrast, the 2010 Chile and 2011 Tohoku tsunamis were “broad-band” events with periods ranging from 2 to 180 min and predominant low-frequency energy with periods in the range of 10–100 min (Figure 4). The differences in the wave frequency content appear to be related to the markedly different source parameters. In particular, the Samoa event was characterized by a relatively small-size deep-water source [Lay et al., 2010] whereas the Chile and Tohoku events were characterized by extensive shallow-water sources [cf. Delouis et al., 2010; Simons et al., 2011]. Rabinovich et al. [2011] argue that relatively small-scale, high-frequency tsunami wave components are absorbed more actively and decay more rapidly than larger-scale lower-frequency wave components.

[17] Following the suggestion of one of our reviewers, we examined the dependence of the time decay on wave frequency and the position of given DART stations with respect to tsunami energy beams. For the major events in 2010 and 2011, we selected two groups of DART stations: those “in-beam” and those “off-beam.” Each tsunami record was band-pass filtered to isolate wave oscillations in period ranges of 2–6, 6–20, 20–60, and 60–180 min (examples of the band-passed records are presented in Figure S2 of the supporting information). The filtered records were then used to estimate E0 and t0 for each range of periods (Figure 5; also Table S4 of the supporting information). Although we find no apparent difference between the in-beam and off-beam responses and, hence, no dependence of decay time, t0, on wave amplitude, there is a clear dependence of the decay time on wave period (Figure S3, supporting information). In particular, the estimates of t0 ~ 22 h by Munk [1963] and Van Dorn [1984] closely correspond to the mean value t0 = 22.4 h we obtain for the 20–60 min band. Longer period motions decay more slowly (t0 ~ 29.1 h) than these historical estimates, while shorter (higher frequency) waves decay more rapidly (t0 ~ 15.1 h, 2–6 min periods and ~18.6 h, 6–20 min periods). The latter estimates are in close agreement with our estimate t0 = 17.3 h for the Samoa tsunami which had typical wave periods of 2–20 min. The fundamental dependency of the decay times on wave frequency accounts for the markedly different rates of energy decay for the 2009 and 2010–2011 tsunamis.

Figure 5.

The tsunami decay time, t0, as function of wave period for the (a) 2010 Chile and (b) 2011 Tohoku tsunamis. The calculations are for two groups of stations: “in-beam” stations are those within the main beams of the tsunami energy propagation; “off-beam” stations are those outside the main beams.

[18] It is safe to assume that the effects of irregular bottom topography and coastal geometry cause tsunamis originating from different source regions in the Pacific Ocean to undergo different rates of energy dissipation. It is also apparent that extensive shallow-water shelves serve as energy “sinks,” actively absorbing and dissipating tsunami energy [Rabinovich et al., 2011]. Shelves and coastal resonance might also influence energy decay in offshore regions. However, regardless of the particular subtleties associated with each particular event, the results of the present study suggest that there is an underlying, universal decay rate linked to the frequency content of the original tsunami wave field in the Pacific Ocean.


[19] We thank Isaac Fine (Institute of Ocean Sciences, Sidney, Canada) for making available the results of his numerical models and for his helpful discussions, and George Mungov (NOAA, NGDC, Boulder, CO) for providing us with the retrieved high-resolution DART data. We thank Dominique Reymond for his valuable comments and one anonymous reviewer for his/her insightful suggestions regarding frequency-dependent energy decay. Work on this by ABR was partly supported by RFBR grants 12-05-00733-a and 12-05-00757-а.