Supplementary material to “The Asian Tsunami in Sri Lanka: A personal experience”—Discussion
Chris Chapman, Schlumberger Cambridge Research, Cambridge, U.K.
Citation:
Chapman, C. (2005),
The Asian Tsunami in Sri Lanka: A personal experience,
Eos Trans. AGU, 86(2), 13.
[Full Article (pdf)]
The basic theory for the propagation of the tsunami has been given by Jeffreys and Jeffreys [1962, section 17.09] in an analysis of the dispersion of gravity water waves in a flat ocean of constant depth. In terms of a dimensionless frequency,
Ω = ω ( d / g )1/2 ,
where d is the water depth, g is the gravitational acceleration, and ω is the angular frequency, and a dimensionless wave number,
K = k d ,
where k is the wave number, and the dispersion relationship for gravity water waves is
Ω2 = K tanh K .
The dispersion curves for this function are shown in Figures 1 and 2.
Fig. 1. The dispersion curve for gravity water waves, with the short- and long-wavelength limits illustrated, and the derivation of the phase c and group v velocity from this curve.
Fig. 2. The phase and group velocities for gravity water waves, derived from the dispersion curve in Figure 1. The dashed lines show the long-wavelength approximations.
In fact, for tsunami we only need the long-wavelength limit, K << 1, when the phase and group velocities can be approximated by the quadratic term; i.e., for dimensionless velocities, these are
C = c / ( g d )1/2 = 1 − K2 / 6
V = v / ( g d )1/2 = 1 − K2 / 2 ,
respectively. Then a wave number integral can be evaluated using the Airy function to give the wave displacement in the form of the so-called Jeffreys phase [Bullen and Bolt, 1985, p. 465]
where X = x/d is the dimensionless range and T = t(g/d)1/2 is the dimensionless time. For a water depth of d = 5 km and a range of x = 2000 km, this gives a velocity of c = v = 220 m/s (C = V = 1) and an arrival time of t = 9000 s = 150 min or T = X = 400. These approximate values are consistent with our observations. Each dimensionless time unit is 0.375 min or 1 hour = 160 units. The Jeffreys phase is illustrated in Figure 3.
Fig. 3. The Jeffreys phase for X = 400.
The important features of the Jeffreys phase, caused by the stationarity of the velocity with respect to wave number/frequency, i.e., C′(0) = 0, are the slow decay with range, X −1/3, due to dispersion (this is the decay due to second-order dispersion only, i.e., one-dimensional wave propagation. Including the geometrical spreading in two dimensions, the decay rate would be increased to X −5/6), the buildup to a large initial wave, and the slow amplitude decay and decreasing periods at later times. The periods of oscillation depend on the dispersion and are only a function of time. The amplitude and sign are arbitrary depending on the excitation. Taking the inverse of this curve, the prolonged initial retreat and large wave agree with observations. The smaller precursor wave is missing, but the main disagreement is the large difference in the period of oscillations. In this figure, the periods of about 20 units correspond to 7.5 min, much shorter than our observations. The small precursor wave could probably be modeled by including a higher-order term or phase shift in the wave number integral, but the longer period will presumably need a complex (large) source mechanism.
Preliminary rupture models by Chen Ji 1 indicate that the rupture propagated north-westward for nearly 400 km (80 dimensionless units) with a speed of 2.0 km/s and a maximum vertical surface displacement of some 10 m. Normally when a source propagates toward the observer, the Doppler shift increases the frequency (decreases the pulse width), e.g., for a train approaching or for seismic waveforms (seismic signals in Sri Lanka will be of a higher frequency than in Australia). But the rupture velocity is supersonic with respect to the tsunami velocity (2000 m/s compared with 220 m/s), so the effect is different. Effectively an observer in Sri Lanka in the direction of the rupture propagation sees tsunami waves from the last point of rupture (the nearest point) first, and from the first point of rupture (the farthest point) last. In fact the rupture velocity is so high (as it is in rock), about 10 times the tsunami velocity, that the direction of rupture relative to the observation point is not very important. The tsunami from the nearest point of rupture always arrives first, and from the farthest last. For simple numerical calculations, we assume that all the rupture occurs instantaneously, i.e., an infinite rupture velocity. The pulse can be broadened by the interference of waves generated all along the rupture. This can be simulated by integrating the Jeffreys phase
where w(x) is a weighting function indicating the source strength along the rupture. Numerical experiments show that the resultant waveforms are very sensitive to the width and form of this weighting function. Trials have been made with triangular and boxcar weights of various widths. If the weighting function is narrow (a few dimensionless units), the wave shape is close to the Jeffreys phase (Figure 3), of course. With a greater width (say 20 units), the main change is that waves from the two ends of the rupture of slightly different frequencies (because the frequencies in the Jeffreys phase increase with propagation time) interfere and cause beats and a more rapid decay (see Figure 4).
Fig. 4. The Jeffreys phase for a boxcar source 20 units wide. Note the interference beats at later times and that the time axis is extended to 800 units (5 hours) compared with Figure 3.
For long source widths (40 units and greater), the wave begins to have approximately the form of the weighting function with reduced later oscillations (for large integration lengths, the Airy function in the above integral looks more like a Dirac delta function; see Figures 5 and 6).
Fig. 5. The Jeffreys phase for a boxcar source 40 units wide. Compared with Figure 4, the initial pulse begins to approximate the source, and the later oscillations are reduced.
Fig. 6. The Jeffreys phase for a triangular source 40 units wide. Again as in Figure 5, the initial pulse approximates the source. The later oscillations are much reduced due to the lack of waves from the start and end of the source.
In reality, it is unlikely that the high-frequency oscillations in Figure 5 would be propagated coherently due to spatial variations of the source and ocean. If these oscillations were removed from Figure 5, the remaining long period oscillations would begin to approximate the observations better. Although these numerical simulations are instructive, further numerical experiments with such a simple model seem pointless given the sensitivity of the final waveform to the source weighting function, and the undoubted complexity of such a large earthquake. The varying water depth of the actual ocean will cause further dispersion and focusing of energy. Full numerical simulations for realistic models have already been performed 2 including all these effects.
