ISSN 00014338, Izvestiya, Atmospheric and Oceanic Physics, 2014, Vol. 50, No. 5, pp. 498–507. © Pleiades Publishing, Ltd., 2014. Original Russian Text © E.A. Kulikov, I.V. Fine, O.I. Yakovenko, 2014, published in Izvestiya AN. Fizika Atmosfery i Okeana, 2014, Vol. 50, No. 5, pp. 567–577.
Numerical Modeling of the Long Surface Waves Scattering for the 2011 Japan Tsunami: Case Study E. A. Kulikova, I. V. Fineb, and O. I. Yakovenkoa a
Shirshov Institute of Oceanology, Russian Academy of Sciences, Nakhimovskii pr. 36, Moscow, 117997 Russia b Institute of Ocean Sciences, Sidney, B.C., Canada email:
[email protected] Received December 19, 2013; in final form, February 27, 2014
Abstract—The March 11, 2011, megaquake caused a catastrophic tsunami recorded throughout the Pacific. This paper presents an analysis of the sealevel records obtained from deepwater tsunami meters (DART and NEPTUNE). To evaluate the effect of the sealevel oscillations’ decay, a statistical analysis of observations and numerical modeling of tsunami generation and propagation have been conducted. The main goal is to uncover physical mechanisms of the tsunami wave field formation and evolution at scales up to tens of thousands of kilometers in space and a few days in time. It is shown that the tsunami lifetime is related to the waveenergy diffusion and dissipation processes. The decay time of the variance of the tsunamigenerated level oscillations is about 1 day. Multiple reflections and scattering by irregularities of the bottom topography make the field of the secondary tsunami waves stochastic and incoherent: the distribution of the wave energy in the ocean reaches a statistical equilibrium in accordance with the Rayleigh–Jeans law of equipartition of the wave energy per degree of freedom. After the tsunami front has passed, the secondarywave energy density turns out to be inversely proportional to the water depth. Keywords: tsunami, numerical model, wave scattering, wave energy, statistical equilibrium DOI: 10.1134/S0001433814050053
1. INTRODUCTION On March 11, 2011, at 05:46 UTC, a megaquake with a moment magnitude Mw = 9.1 and hypocenter at a depth of 32 km struck offshore eastern Honshu, Japan. This earthquake and its resulting tsunami killed more than 20000 people and caused massive destruc tion. This unique event is among the six largest earth quakes in the past 60 years (1952 to 2011) accounting for more than half of the seismic energy released over this period. The modern sealevel monitoring system includes not only coastal tide gages, but also deepwater level sensors deployed in the open ocean. The system is cur rently operating 59 Deepocean Assessment and Reporting of Tsunamis (DART) stations (http://nctr.pmel.noaa.gov/Dart/) and several cable stations measuring bottom hydrostatic pressure oscil lations in real time. NEPTUNE cables are available in Japan and Canada (http://www.neptunecanada.ca/). The location of DART and NEPTUNE stations in the Pacific Ocean is shown in Fig. 1. The tsunami of March 11, 2011, was recorded by 23 DARTs and sev eral NEPTUNE stations. Almost all phases of the event, from the formation of the tsunami wave to its propagation and decay, were traced across the entire Pacific Ocean.
In this work the available sealevel records are used to study the propagation of the 2011 Japanese tsunami and the evolution of the wave field in time and space. The aim was to analyze the transformation processes of the waves at periods exceeding a characteristic travel time of the wave across the Pacific Ocean. The effects arising from multiple wave reflections are poorly understood. In [1] the tsunami propagation in the ocean was considered in the framework of acoustic analogy with wave reverberation and waveenergy absorption effects. The timescale of the waveenergy decay was estimated using simple qualitative consider ations and tsunami data available at that time. This acoustic analogy is used in the present paper to analyze tsunami records and model simulations. To study the tsunami transformation and decay, it is suggested that the tsunami wave trains be divided into two parts: the leading frontal (head) zone and the trail ing (tail) zone. Evidently, the head of the signal is more related to the initial form of the signal in the source and to the direction of waveenergy radiation. Most tsunami hypocenters have a relatively simple form and, therefore, the initial form of the ocean surface displacements has the shape of a dipole consisting of a crest and a trough. However, the hypocenter is most often located close to the shore, and the resulting wave traveling into the open ocean is almost inseparable
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N 60° 46410 21415 46409 889 FP 21416 46403 C 46402 BC 46404 21414 46408 21419 46407 21401 46411 21418 46412 21413
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North America
20° 43412 43413
52402 52403
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PACIFIC OCEAN
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South America
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20°
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Australia 54401
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S E 120°
140°
160°
180°
160°
140°
120°
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Fig. 1. Map of the Pacific Ocean and location of deepwater tsunami gages: circles are DART stations and squares are NEPTUNE (Canada) bottom cables. The star shows the epicenter of the March 11, 2011, earthquake near Japan.
from a secondary wave reflected from the shore (or continental slope). The partition into a “head” and a “tail” is therefore quite conventional. It is reasonable to assume that the trailing (tail) zone of the signal begins about 1 h after the arrival of the tsunami front. When the tsunami propagates in the open ocean, the head signal holds the initial shape well, decaying with distance. In the tail zone the wave field is con stantly modified and fed by multiple wave scattering by the seafloor irregularities and by reflections from islands. The role of wave scattering at long travel dis tances is quite important. It turns out that the decay of the wavefront amplitude is due not only to a geomet ric discrepancy, but also to the energy loss from scat tering. The problem of tsunami propagation in a basin with irregular bottom topography was considered in [2]. The decay of the mean wave field was found to be determined by the height of irregularities and by the radius of spatial correlation of the bottom topography. One important practical question arising when studying the tsunami is to determine the ringing time of the anomalous level oscillations. In case of the remotesource locations, the amplitude of the first wave may be not the largest one. An extreme wave is sometimes the third, fourth, or even a later wave, which is able to cause significant damage. For exam ple, the maximum wave height at SeveroKurilsk on IZVESTIYA, ATMOSPHERIC AND OCEANIC PHYSICS
March 11, 2011, was about 1 m, but it arrived 5.5 h after the arrival of the first wave, or 1.5 h after the tsu nami alert was canceled at this location. For the oper ational tsunami forecasting services, the problem of determining the duration of the tsunami manifestation is quite important. The aim of the present study is to describe the evolution of the tsunami wave field and, in particular, estimate the lifetime of the tsunami in the Pacific Ocean. 2. NUMERICAL MODEL OF THE 2011 JAPANESE TSUNAMI WAVES The propagation of the 2011 Japanese tsunami waves was simulated using a numerical hydrodynamic model [3], which is based on the finitedifference approximation of shallowwater equations of motion. The model in spherical coordinates is used for the Pacific Ocean area (56° S to 63° N, 120° E to 70° W; Fig. 1); the grid spacing is 2⬘ in latitude and longitude and the time step is 2 s. The dataset of depths with 2min space resolution was extracted from ETOPO2v2 (http://www.ngdc.noaa.gov/mgg/glo bal/etopo2.html). We used a numerical scheme, a modification of the wellknown TUNAMI program [4], in which a finite Vol. 50
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difference approximation of shallowwater linear equations was resolved in spherical coordinates:
⎡ ∂ζ ∂(V cos ϕ)⎤ + 1 ⎢∂U + = 0, ∂t R cos ϕ ⎣ ∂λ ∂ϕ ⎥⎦
(1)
∂U + gh ∂ζ = 0, ∂t R cos ϕ ∂λ
(2)
∂V + gh ∂ζ = 0, (3) ∂t R ∂ϕ where ϕ and λ are the current latitude and longitude, R is the Earth’s radius, U and V are the components of the horizontal fluidflux vector integrated over depth (h), ζ is the freesurface elevation, g is the gravitational acceleration, and t is time. We solve the Cauchy prob lem, i.e., it is assumed that, at the initial time t = 0, the fluid is in at rest and the initial freesurface elevation ζ 0 ( x, y ) is given. At the outer boundary Γ, the radia tion condition is specified, which relates the normal component U n and the level ζ:
Vertical displacements of the seafloor were calcu lated by the Okada formula [5] for a fault plane con sisting of 325 segments 25 × 20 km in size each. Such displacements have a typical dipole pattern oriented along the coast of Honshu with uplift on the ocean side and subsidence on the island side. The range of the ini tial vertical displacement was –1.3 to 10.1 m. The classical shallowwater approximation assumes a “pis tonlike mechanism” of the formation of the tsunami source, when the sealevel rise follows the shape of the vertical bottom displacement (see [7]). However, this approximation is incorrect for small sources located in deep water [8]. For this reason we used the solution of the full threedimensional Laplace equation [9, 10] to reconstruct the initial form of the tsunami source. Such a reconstruction of the seasurface displacement is smoother than that of the seafloor displacement. The variance (energy) of the seasurface displacement amounts to 90% of the variance of the level displace ments in a classical approximation with the same expelled volume of fluid.
(4) U n = ζ gh. The impermeability condition is assumed at the coastal boundary G specified in the form of a vertical wall
3. DISSIPATION OF TSUNAMI ENERGY IN THE OCEAN AND WAVEDECAY MODELING
(5) U n = 0 at G. In this model, the coastal boundary (wall) was set up at the 10m isobath. To eliminate artificial numerical dissipation in the simulation, we used in the model a linearized version of shallowwater equations: the corresponding finite difference approximation is conservative. Although such linearization somewhat restricts the accuracy of the model in the coastal zones, it permits the wave field to be modeled for a long time, thus providing more control over the energy dissipation in the model. The most important part of the modeling is formula tion of initial conditions. To estimate source parame ters, we used a seismotectonic finitesized fault model which is able to determine displacements along the fault plane at the earthquake hypocenter. In turn, these displacements can be recalculated into bottom displacements on the basis of the dislocation theory in an isotropic, homogeneous, and elastic halfspace [5]. In our research, the finite fault model [6] was used as a source. It gives detailed information on spatial dis placements in the area and minimizes errors in evalu ating the source sizes. The fault plane was defined using the CMT magnitude (Mw = 9.0), the USGS hypocenter location (38.32° N, 142.37° E; depth 32 km), and the Global CMT moment tensor solution (http://www.globalcmt.org). The dip was 10° and the strike was 194.4°, with a distance 650 km along the strike and 260 km along the dip. The release of the seismic moment in this model was 4.9 × 1022 N m, slightly less than in the Global CMT solution (5.3 × 1022 N m).
Numerical simulation of tsunami propagation is usually conducted for estimating wave amplitudes on the shore, runup, inundation zone, etc. The travel time of waves in modeling rarely exceeds 10–20 h, even for remote sources. The aim of our research is to estimate the decay effects of a tsunami wave, so the physical time necessary for the model simulation of the wavefield evolution must be comparable to the tsunami lifetime and equal to several days. In Munk’s work [1] the evolution of the tsunami wave field in the Pacific Ocean is considered in the framework of the acoustic analogy. Multiple wave reflections (reverberation) from shores, islands, and bottom irregularities play a primary role in the redistri bution of wave energy across the water area. From geo metric consideration, Munk estimates the average number of reflections to be 1.7 a day, and the reverber ation time (tsunami lifetime), defined as the time of the waveenergy decrease by 106 times, is estimated to be about a week. The leveloscillation intensity decay (variance) is determined by three main factors: (1) dif fusion, (2) dissipation (absorption) of wave energy, and (3) leakage into other oceans. The diffusion pro cess is understood as the redistribution of tsunami energy across the water area, with the total wave energy in the ocean kept constant. In other words, the decrease in the intensity of level oscillations as a result of wave propagation is due to a geometric discrepancy, dispersion, and wave refraction and scattering. Fur ther, the wave energy continues to decrease because of absorption: the wave energy is absorbed due to turbu lent friction (mainly in shallow water) and ultimately turns into heat. The leakage of tsunami energy into the
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43413
(а)
Sea level, m
0.1 0 –0.1 –0.2 τ = 0.5 day
Level variance, m2
Indian and Atlantic oceans is estimated by Munk to be 15 and 5%, respectively. The 6day record of level oscillations at DART 43413 located at a depth of 3380 m off the coast of Mexico at 12000 km from the source is shown in Fig. 2. Two stages can be clearly seen in the pattern of the change in the variance of tsunamigenerated level oscillations: the first stage lasts 1 or 1.5 days, during which the energy decreases by about one order. Then this stage is followed by a slower change in variance. It is easily seen that the range of level oscillations reaches a background level within 6 days. It is evident that the initial stage of energy decay is caused by tsunami wave propagation across the water area, i.e., by diffusion, whereas a further variance decrease is determined by absorption, i.e., by gradual absorption of wave energy in marginal seas and in shallow water. The estimate of the characteristic time of energy decay (efold) for the diffusion process is close to 12 h, which agrees with the value indicated by Munk in [1]. Having lost about 90% of energy, level oscillations come to another slower regime (absorption) with a characteristic decay time of 24 h. Although most of the energy is lost by diffusion, the standard estimate of the “mean” decay time calcu lated for the entire tsunami “ringing time” is closer to 24 h. It is this estimate that is given in [11, 12]. For remote stations, the diffusion stage is much shorter than for nearsource stations, hence the effective (average) decay coefficient is also less. A comparison of the numerical modeling results and the field observations (level oscillations) rarely shows the full agreement for longterm records. Only the first two or three waves are usually reproduced well. Certainly it is difficult to expect a complete coinci dence of the leveloscillation records caused by multi ply reflected waves. In our research, the numerical modeling results are compared with deepwater bottom hydrostatic pressure records in which the effects caused by local bottom topography features (coastline, shallow water, bights, etc.) are least pronounced. The tsunami records used in the paper were obtained at stations located at depths from 100 m to 6 km along the continental borders of the Pacific Ocean (DART and NEPTUNE), and almost of them recorded the 2011 tsunami. Examples of the tsunami records obtained by nine DARTs on March 11 (tidal oscillations have been removed) are shown in Fig. 3. Most of the records show a distinct transition from a large head part (less than 1 h) to a decaying tail, which is called a “coda” in acoustics and seismology. The record at station 51407 located west of the Great Island (Hawaii) seems to be the exception: there are longterm oscillations with amplitudes comparable to that in the head part. This may be due to wave trapping by the islands. The graphs for the model calculations are also shown in the figure. They show that the simu lated level oscillations fit well the head zone of the sig
501
10–3
(b) Max
Diffusion 10%
10–4
Absorption 10–5
τ = 1 day
Noise level 10–6
0
2 4 Time after earthquake, day
6
Fig. 2. (a) Tsunami record at DART 43413 station and (b) a change of the leveloscillation variance with time. The variance was calculated over 6h intervals with a 3h lag. Solid straight lines show the exponential dependence of the vari −t τ ance change, ~e , for two values τ = 0.5 and 1.0 day (decay timescale). Horizontal dashed lines show the observed maximum, 10% of the maximum, and the back ground noise level (~2 × 10–6 m2).
nal. However, the records diverge several hours later, although the range and the character of oscillations remain similar. The discrepancy in the simulated and observed oscillations is due to the loss of accuracy because of the errors in the bottom topography data used in simulations, too coarse grid spacing in model ing the wave propagation in shallow water and the wave reflection from the shore, and other effects. The event of multiple wave reflections (reverberation) and scattering in the numerical model can, however, be reproduced quite adequately as a statistical process. It is assumed that, neglecting the shallowwater wave dissipation, we are able to reproduce changes in the amplitude of the head part of the signal that are caused by the geometric discrepancy and scattering, and to model the head of the signal, whose statistical charac teristics are adequate to the observed openocean level oscillations in the first 2 or 3 days after the earthquake. Numerical modeling of the tsunami decay induced by waveenergy absorption is much more difficult. Wave dissipation occurs in shallow water, in small bay, upon reflection from a flat shore, etc. These effects can hardly be reproduced in modern numerical models on the scale of the entire Pacific Ocean. The tsunami decay for a constantdepth ocean was estimated in Vol. 50
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(
21401 21413
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Sea level, m
21415 1 52406 0
51407 51425 46411
6
8
10 12 14 Time, h (UTC) March 2011
16
18
Fig. 3. Records of the sealevel oscillations induced by the March 11, 2011, tsunami at several deepwater DART sta tions: solid lines are model simulations and dashed lines are observations.
[13]. Given a mean depth H = 4 km, the typical dis tance at which the decay effects become apparent is ~108 km. As the authors claim, significant dissipation manifestations are possible only at shallow depths below 10 m. Since the resolving capability of the model fails to reproduce the wave decay processes in shallow water, we introduce the artificial viscosity required to ensure the energy loss by friction equiva lent to physical dissipation of the wave energy in the Pacific Ocean. For simulating the absorption, we rewrite the time derivatives in the moment equations by adding a term with the characteristic linear friction
∂ ⇒ ∂ + b, (6) ∂t ∂t where b is a constant equal to 10–5 s. Introducing this term yields the energy balance equation in the form ∂E = − 2bE − ∇F , ( ) k ∂t
(7)
)
where E k = 1 ρh u 2 + v 2 is the density of kinetic 2 energy and E = 1 ρg ζ 2 + E k is the total energy density 2 per unit sea surface, F = ρghζ ⋅ u is the specific energy flux (per unit length of the wave front), ∇ = ∂ i + ∂ j, ∂x ∂y ρ is the water density, h is the height of the water col umn, u and v are the depthintegrated velocity com ponents, ∇ = ∂ i + ∂ j, x and y are horizontal coordi ∂x ∂y nates. Equation (7) is valid for the shallowwater approx imation. It is also valid for a linear finitedifference analog. Parameter b corresponds to the timescale of the exponential decay equal to 27 h; this value is close to the Van Dorn estimate [14] for the Pacific Ocean (22 h). The decay timescales calculated according to the analysis of 29 records of the 2011 Tohoku tsunami are reported in a recent paper [11]. They give 24.6 ± 0.4 h on average. It should be noted that the observed decrease in the leveloscillation energy in the Pacific Ocean includes nondissipative effects, such as the leakage of the wave energy across the open boundaries and into shallow seas (for example, the Bering Sea and the Sea of Okhotsk). The leakage of the wave energy into the Atlantic and Indian oceans can be easily taken into account in the numerical model through the radiation condition at the outer boundaries of the computa tional domain. 4. TSUNAMI WAVE SCATTERING, ENERGY DECAY, AND STATISTICAL EQUILIBRIUM OF SECONDARY WAVES The propagation of long waves in the ocean is accompanied by the refraction and scattering effects because of reflections from the shores and seafloor irregularities: underwater ranges, troughs, seamounts, islands, reefs, etc. Multiple reflections and scattering make the wave field stochastic, so level oscillations become incoherent and less predictable. As a result of stochastization, the region between the tsunami front and its source gradually fills with secondary waves and turns into a random wave field. The tsunami front propagating in the ocean is con stantly stretching; the amplitude of the head wave decreases with distance (the effect of geometric diver gence). The deepwater observations show that the frontal signal retains its initial shape fairly well even at large distances from the source. At the same time, the tail waves are continuously transformed and some times amplified by refraction and multiple reflections.
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102
20°
101
0° 100 20° 10–1 40° 10–2
S E
140°
160°
180°
160°
140°
120°
100°
80° W
Fig. 4. Frontal specific tsunami energy flux (W m–1) averaged over 1h interval after the arrival of the front. The magnitude of this energy flux is in gray shades, and its direction is marked by arrows. Solid lines show distances (in km) from the source of the tsunami.
At large distances, this leakage of energy spent on the formation of secondary waves substantially weakens the tsunami head wave. To estimate the evolution of the waveenergy field, the frontal specific energy flux per unit frontal width (W m–1) averaged over the first hour after the arrival of the tsunami front at every point was calculated from the numerical modeling results (Fig. 4). It is clearly seen that this energy flux is mainly directed to the southeast toward southern America. The specific energy flux decreases with distance from the source because of the geometric divergence and scattering. At large distances, the influence of refraction becomes more distinct: the flux divides into separate “arms” (beams); such focusing results from the scattering effect on the uneven bottom. The decay effect of the tsunami head wave was esti mated by integrating the specific energy flux over the entire length of each isoline of the distance from the source. The calculated total energy of the tsunami front, i.e., the total wave energy that passed in 1 h across the isoline of the distance from the source, is shown in Fig. 5. Should the scattering and dissipation effects have been absent, the total energy of the head wave would have remained constant. However, it is seen in Fig. 5 that the energy decreases exponentially with distance with the decrement γ ≈ 0.00021 km–1 (the decay scale L = γ –1 ≈ 4700 km). IZVESTIYA, ATMOSPHERIC AND OCEANIC PHYSICS
The spatial decay coefficient γ can be recalculated into the time decrement δ through the mean deep ocean wave velocity c = 220 m s–1 (780 km h–1): it makes up 4.6 × 10–5 s–1 (T = δ–1 ≈ 6 h). This coeffi cient is much greater than the parameter b used in the calculations of the model dissipation (6). Hence, the decrease in the frontal wave energy is largely related to the wave scattering, not to dissipation. For the amplitude of the head wave, these decay scales are, respectively, L ≈10000 km and Т ≈ 12 h. In [2] the problem of the surface wave (tsunami) propagation in the ocean with an irregular bottom was solved in a statistical formulation: the emphasis was on the decay of the mean wave field. In this paper the decay decrement values were reported for the waves with a length much greater than the horizontal scale of bottom irregularities 2 2 h 1 π k 3l 2, kl Ⰶ 1, γ= ≈− L 2 H 02
where
h
2
is the meansquare height of bottom
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Total energy, MJ
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e–x/L
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0
5000 10000 Distance, km
Fig. 5. Total frontal tsunami energy (total wave energy that passed in 1 h across the isoline corresponding to the dis tance from the source) as a function of distance. The −x L
, dashed line indicates the exponential dependence e where x is the distance from the source and L = 4700 km.
ularities whose scale does not exceed 10 km on aver age. According to [15], the correlation radius of the abyssal bottom irregularities in the Pacific Ocean is 2.5 km and the meansquare height is h 2 ≈ 125 m. According to the estimation [2], tsunami waves with λ ≤ 50 km decay on a scale of about 20 000 km. Longer waves with λ ~ 150–250 km hardly decay at all: L ~ 106 km. Such a weak decay is evidently caused by neglecting the largescale irregularities of the ocean bottom. More complete insight into the spectrum of the bottom irregularities in the Pacific Ocean can be taken from [16]. According to the analysis of bathy metric surveys obtained by echosounding profiling, the typical height of bottom irregularities is about 300 m and the horizontal scale of correlation is about 100 km. For a characteristic tsunami wavelength λ ~ 200 km, the decay scale is then on the order of 10000 km, which is close to the abovementioned modelderived estimate of the amplitude decay of the tsunami head wave. Since the scattering does not change the total wave energy, the energy of secondary waves increases as the frontal wave decays. They, in turn, also undergo scat tering and reflection. Such multiple reflections from the shores and scattering ultimately result in a new redistribution of the wavefield energy. As was shown in [17], this equilibrium state of the wave field can be described within the framework of the Rayleigh–Jeans law [18, 19]. The evolution of tsunami waves gradually leads to a statistical equilibrium of scattered waves (in about 2 or 3 days), for which the energy density is inversely proportional to the ocean depth, E ~ 1/H, where H is depth. It is noteworthy that applying the
theorem of energy equipartition per degrees of free dom to tsunami waves was first suggested by Munk in [1], where the analogy between acoustic waves in a room and tsunami waves in the Pacific Ocean was noted. In a given case, the number of degrees of free dom in a specified frequency range is calculated as the number of normal modes divided by the square of the water area. Figure 6, which shows for the March 11, 2011, tsu nami, the density of wave energy in the ocean versus depth, can be an illustration of the application of the Rayleigh–Jeans law to secondary (scattered) tsunami waves. The graph presents the average waveenergy densities calculated from DART and NEPTUNE tsu nami records (open circles) and from model mareo grams at grid points (solid circles). The waveenergy density was timeaveraged for intervals 30–36 h (Fig. 6a) and 60–66 h (Fig. 6b). It is clearly seen that the dots are grouped along the line fitting the E ~ H–1 law. In a later time interval (60–66 h), the scatter is far less: the wave field comes to the nearequilibrium state. 5. BACKGROUND SPECTRUM OF LEVEL OSCILLATIONS IN THE RANGE OF TSUNAMI WAVES An investigation of the backgroundlevel spectrum in the frequency band typical for tsunami waves is of much interest. The mean amplitude of the longwave noise is determined by the balance between the energy flux from external atmospheric forcing onto the ocean surface and waveenergy dissipation in shallow water. In the deep ocean, the noise level is rather stable in time and depends more on the location of a gage. This stability is caused by the relatively high “transparency” of the ocean for long surface waves of this type. As we have shown above, the characteristic scale of their decay by scattering is about 10000 km, which is com parable to the width of the Pacific Ocean. Hence, the distributed chaotic impact of pressure and wind fluc tuations on the ocean surface produces a statistically homogeneous wave field on the scale of the entire water basin and does not depend on the local intensity of pressure variability. In is shown in [20] that the meansquare amplitude of the noiselevel fluctuations decreases severalfold with distance from the shore. Like tsunami waves, these waves, which are of meteorological origin, natu rally undergo multiple reflection and scattering. The Rayleigh–Jeans law of the waveenergy equipartition per degree of freedom must therefore be also satisfied for such “noise” oscillations. To make sure that the assumption of the statistical equilibrium state of the waves that form the back ground noise is valid, we calculated the variance of
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(b) 60–66 h after EQ
30–36 h after EQ Average energy density, kJ/m2
505
100
Observations Model
10–1
10–2
10–3
10–4
101
102
103
104 Depth, m
101
102
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Fig. 6. Average energy density of the sealevel oscillations (E) as a function of depth (H) estimated from the level records at several DART and NEPTUNE stations (open circles) and from the model simulations at grid points (solid circles). Averaging was done over (a) 30–36 h and (b) 60–66 h after the earthquake (EQ). The dashed line shows the dependence E ~ H–1.
oscillations of bottom hydrostatic pressure (sea level) in a tsunamispecific frequency range from 10–2 to 1 cpm. For this, we used long records at several deep water stations equipped with bottom pressure meters. The table lists information on the data used in the analysis available at http://www.ngdc.noaa.gov/ nndc/struts/results?&t=102597&s=1&d=1. The dependence of the variance of background level oscillations (energy) on the depth of gage deploy ment is shown in Fig. 7. The variance was calculated
by integrating the spectrum of sea levels in a frequency band from 10–2 to 1 cpm, which is typical of tsunami waves. It is clearly seen that the energy of oscillations decreases severalfold as the depth increases from 1712 to 6669 m. Even taking into account the considerable scatter of the variance estimates, we should note that the dependence of the background noise level in the ocean on depth agrees well with the Rayleigh–Jeans law, which is characteristic of scattered waves in the state of statistical equilibrium. Like the case of the sec
Records of bottom hydrostatic pressure (sea level) oscillations at deepwater stations Depth, m
Latitude
Longitude
Start of recording
End of recording
Δt, s
ak90_1999
1712
54.29° N
158.55° E
05.14.1999
05.27.2000
15
nemo_2000
2260
45.86° N
130.00° E
07.06.2000
07.19.2002
15
nemo_2001
2262
45.86° N
130.00° E
07.01.2001
07.19.2002
15
d123_1999
3190
36.48° N
122.61° E
05.12.1999
02.25.2000
15
d128_2001
6669
45.86° N
128.77° E
06.21.2001
07.01.2002
15
d130_2001
3467
42.91° N
130.91° E
04.01.2001
06.29.2002
15
d125_2003
4500
8.49° S
125.02° E
01.29.2003
05.11.2004
15
d125_2002
6668
8.49° N
125.01° E
01.15.2002
01.27.2003
15
d125_2001
6669
8.49° N
125.01° E
09.02.2001
09.15.2002
15
d157_1999
4540
52.65° N
156.94° E
10.07.1999
08.15.2000
15
Station
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0.10 0.09 0.08 0.07 0.06
ak90 1999
nemo_2000
0.05
nemo_2001
Variance, cm2
direct simulation of the absorption effect at the Pacific Ocean scale within the framework of the modern numerical models is practically impossible. We there fore used artificial viscosity by choosing a decay coef ficient in accordance with empirical estimates. A successful effort to model the wavefield evolu tion as a statistical process can be considered as the main result of our work. The basic characteristic prop erties of the spatial and temporal tsunami wave changes can be adequately reproduced by numerical models actually in the entire interval of the tsunami “ringing time,” i.e., for several days.
d123 1999
0.04 0.03 d130_2001
d128_2001 d125_2003
0.02
d125_2002 d125_2001
d157_1999
0.01 1000
2000
4000 Depth, m
ACKNOWLEDGMENTS This work was supported by the Russian Founda tion for Basic Research, grant nos. 120500773 and 120500757.
6000 8000 10000
REFERENCES Fig. 7. Dependence of the variance of the sealevel oscilla tions on the depth of the gauge deployment (H). Variance (σ2) was calculated in the range of tsunami frequencies from 10–2 to 1 cpm. The dashed line is the dependence 2
−1
σ ~ H . Every value of the calculated variance (a circle) is marked by the name of the file used in the calculation (see table).
ondary tsunami waves, the energy of the background noise is inversely proportional to the ocean depth. 6. CONCLUSIONS The analysis of the transformation processes of a tsunami wave field on the scale of the entire Pacific Ocean and duration of up to a week was based on the acoustic analogy, which was first used for tsunami research by Munk [1]. From the analysis results of the deepwater DART tsunami records and model simu lations, the spatial and temporal scales of frontalwave energy decay were estimated to be L ≈ 10000 km and Τ ≈ 6 h. Numerical modeling was used for the first time to investigate the evolution of the statistical character istics of the secondary tsunami waves arising from multiple reflections and scattering on bottom topogra phy irregularities and shoreline. It is shown that the energy decay of sealevel oscillations can be divided into two stages: diffusion and absorption. The diffusive wave decay is due to the spatial redistribution of wave energy (reverberation and scattering), which lasts slightly less than 2 days in the Pacific Ocean. As a result, the field of secondary tsunami waves comes to a nearequilibrium state: the wave energy is distributed unevenly across the ocean and actually depends only on the depth as E ~ H–1. The process further enters the absorption stage: the dissipation of wave energy becomes a dominant factor for decay. Unfortunately, a
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NUMERICAL MODELING OF THE LONG SURFACE WAVES SCATTERING 11. A. B. Rabinovich, R. N. Candella, and R. E. Thomson, “The open ocean energy decay of three recent trans Pacific tsunamis,” Geophys. Res. Lett. 40 (2013). doi: 10.1002/grl.50625 12. T. Saito, D. Inazu, S. Tanaka, and T. Miyoshi, “Tsu nami coda across the Pacific Ocean following the 2011 TohokuOki earthquake,” Bull. Seismol. Soc. Am. 103 (2B), 1429–1443 (2013). 13. B. W. Levin and M. A. Nosov, Physics of Tsunami and Related Phenomena in the Ocean (YanusK, Moscow, 2005) [in Russian]. 14. W. G. van Dorn, “Some tsunami characteristic deduc ible from tide records,” J. Phys. Oceanogr. 14, 353–363 (1984). 15. T. H. Bell, “Statistical features of seafloor topogra phy,” DeepSea Res. 22, 883–892 (1975). 16. G. V. Shevchenko and V. N. Patrikeev, “Statistical char acteristics of abyssal roughnesses in the northwestern
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Translated by N. Tret’yakova
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