Earthquake-Generated Landslides and Tsunamis

1. Introduction

The main difficulty in tsunami hazard assessment is to estimate the generation of wave energy at the source. In many cases, this is simply solved by running a CFD model and estimating the ocean surface amplitude at the source [1] from it. This estimate of the amplitudes involves many problems and can be very uncertain, when the measured amplitudes are small, but a better method is still to be found.

Sometimes it is better to estimate the initial wave itself, and when that is done, the energy transmission from the source to the point of impact can be quite accurately modeled in CFD models [2, 3, 4], if the shallow water wave equations are in the numerically stable domain.

Strong earthquakes cause large deformations of the surface of the earth, so tsunamis are more often than not strengthened by landslides triggered by the earthquake. Often the triggering earthquake does not contribute any significant amount of energy to the tsunami wave, and this seems to be the general case in the North Atlantic, where few dangerous tsunamis are reported.

There have also been speculations in the scientific community about the danger of tsunamis in the North Atlantic from gigantic glacial flood waves of volcanic origin, (jökulhlaups), emerging on the south coast of Iceland. In this case, there is a well-defined probability of occurrence [5] and clear geological evidence of the volcanic events [6] but practically no historical evidence of tsunamis. For the landslide tsunamis, we introduce a translatory wave model to estimate the initial disturbance [7]. Block slide models are popular for this purpose [8]; in them the blocks must reach very high velocities to create a serious tsunami. The translatory wave model assumes that sliding blocks break up and become debris flows when the velocity is high enough.

When the wave crest of the tsunami approaches a beach, instability of the wave fronts, wave breaking, and reflection set in. There may be little reflection on a flat beach but large energy dissipation due to wave breaking.

Few authors discuss how this problem is to be handled in practical hazard assessment, and most often this is simply done by making all coasts completely reflecting. This is on the safe side in run-up estimation. But how the correct boundary conditions should be formulated in terms of reflection and energy dissipation in the various models reported is an open question.

The following treatment on tsunamis has the emphasis on the estimation of initial disturbance and energy formation in the tsunami. The underlying theories are formulated in [8, 9] and the case studies in Chapter 7. The theory of CFD modeling of a tsunami and the associated procedures of preparing tsunami warnings are left out, as they can be found in various internet resources of the government institutions that make them.

2. Properties of a tsunami wave

Tsunamis are ocean waves that are considerably different from the best known types of ocean waves, storm waves, and ocean tides. There is much less periodicity in tsunamis, and they can run over dry land in a more or less unpredictable way. On dry land, large tsunami waves have a devastating power that resembles flood waves of the type “translatory waves” [6], such waves knock down most everything in their way. Out in the oceans (deep water), it is normally like a long wave and can be modeled using the shallow water wave theory. In this, it resembles the tidal wave.

However, there are several snags in the numerical modeling of tsunamis. Ordinary storm waves create steep wave crests that break in the shore line, but tsunamis are more like a bore, or a moving hydraulic jump; Figure 1 illustrates this difference. How the final inundation height is affected is unclear. Energy is lost in the jump as the bore moves inland, but with water behind has not, so it keeps moving. Then there is the role of the bathymetry; it has an influence on the dispersion and reflection of the wave fronts that is sometimes significant and sometimes not.

In short, the mathematics and numerical schemes are well developed in tsunami modeling, but there are pitfalls that can be difficult to avoid. Even models that have been through a scrutineer’s process of calibration and validation can fail. Still, the initial conditions in the source area are the biggest uncertainty. It has therefore been concluded that verification and validation are necessary for each case, even for models that have been through this process before [10].

There are several numerical methods available to treat such waves using, for example, the well-known St. Venant’s equations, see [11], to take an example. But the names used here, St. Venant’s equations and translatory waves, are usually not mentioned. Analytical solutions are possible for stationary flows using a wave progressing with constant velocity down an inclined plane, or in a funnel, as the translatory wave in [7]. In there it is also demonstrated that numerical solutions of the St. Venant’s equations can produce exactly this kind of flow without presuming a constant velocity wave.

However, when a wave runs upwards a mild slope with constant celerity c, as a translatory wave, c will inevitably reach the shallow water wave celerity, c = √gD (g acceleration of gravity and D the water depth) somewhere, and then the surface profile may become unstable, which can result in a breaking wave (bore) or a series of braking waves if the wave is very long. Figure 2 shows a nearshore breakup of this kind. The successive waves, resulting from the breakup, ride upon each other making the ultimate tsunami run-up very difficult to predict, even when the deep-water wave is well known. This difficulty is discussed further in Sections 5 and 7.

3. Propagation of tsunami waves

4. Causes of tsunamis

Large earthquakes usually start a tsunami. The earthquake deforms the bottom landscape and creates a surface disturbance in the source area and the associated transfer of energy to the water mass. This energy is transmitted from the source area by the tsunami wave.

An earthquake of magnitude 7 or larger on the Richter scale usually starts a tsunami. However, this is a rule of thumb only; smaller earthquakes can trigger a tsunami by starting a submarine landslide on bottom slopes. If it does, the magnitude of the tsunami depends on the size of the landslide, so the tsunami can be enormous even though the earthquake is small. Some years ago, it was discovered that huge submarine landslides on the continental shelf of the North Atlantic Ocean have caused large tsunamis. In Table 7.1 in [14], 11 submarine landslides with slide volumes 20–20,000 km³ are listed. Submarine landslides of that magnitude run as translatory waves. The deadliest tsunami attacks in the recent years have struck Indonesia and the Indian Ocean coasts; the most recent one is the Anak Krakatau volcano, where a submarine landslide of type Case 1a (see Section 5.1) caused a tsunami December 22, 2018.

If a landslide is triggered or not, it can be stated that a movement on the sea bottom creates a surface disturbance with a certain potential energy, deduced by the common methods of wave mechanics. Some kinetic energy will also be created in the boundary layer around the moving object; this is mechanical energy and can thus be converted in wave energy in the tsunami wave. But as a rule, the kinetic energy will be converted to turbulent energy that cannot be converted into wave energy and is dissipated. Thus, the total potential energy of a mass that flows from land and into the sea is not converted into wave energy; only the potential energy of the initial disturbance it causes on the sea surface contributes to the tsunami.

5. Initial disturbance and the tsunami wave energy

In estimating the mechanical wave energy generated in the source area of a tsunami, three types must be considered and separate estimates devised for each one. The different types are landslides down mountain slopes and in the water where V > c (Case 1). Totally submerged submarine landslides where V < c at least in for a part of the slide (Case 2) and bottom landscape features are moving vertically and horizontally (Case 3). The initial tsunami wave height is estimated. The energy transmitted to the water by the movement of the landslide is estimated from the moving mass. The total wave energy is estimated as the potential energy in the water mass the slide displaces from the still water surface, using the assumption that turbulent energy transmitted to the water cannot be regenerated as mechanical energy in the tsunami wave.

The wave energy transmission away from the source area can be translatory or by a solitary group of oscillatory waves. The wave transmission can be estimated in numerical models, and the shoaling also until either the point of breaking or where the numerical stability is lost in the model. In the following we will therefore estimate the wave energy generation in the source area and the corresponding wave amplitudes. In the following, the expressions for the wave energy and amplitude h depend on the variables listed here.

For further discussion and estimation of slide dimensions, see Section 7 and [6, 9, 15].

6. Studies of individual tsunamis and regional risk assessment

6.1 Tohoku tsunami in Japan

The earthquake event and the devastation caused by this tsunami is very well documented; it is the most famous tsunami event of recent years. It took place off the Pacific coast of Tohoku, Japan, on Friday, March 11, 2011 at 05:46 UTC. It was caused by a M_w 9.0 (magnitude moment) undersea megathrust earthquake with the epicenter approximately 70 km east of the Oshika Peninsula of Tohoku with the hypocenter in approximately 30 km deep water (see Figure 3 gray arrow).

From the data obtained in the exploratory drilling at the site shown in Figure 3, it was concluded in [16] that the tsunami was caused by the mass movement shown in Figure 4.

The details of the bottom deformation are estimated and pictured in Figure 3b in [18]. This picture resembles a 150 km long slide scar with L_s and B about 110 km and y₀ about 8 m using the symbols in Chapter 5.2. This would be a Case 2b slide, stopping 25–75 km from the trench, see Figure 4; here the average bottom slope is about 4/50 or 8–9%. It is interesting to note that layers of fine sediments on such a slope can easily liquify, slide down the slope, and cause a bottom deformation like the one pictured in [18] and indicated by black arrows in Figure 4. No evidence has been found in [16] to support this suggested slide event, but the information on the bottom deformation given in [18] is considered reliable, and it is supported by a coseismic slip model in Figure 4 in [17]. The suggested slide would have characteristics that can be calculated using the equations for the Case 2b slide. If the slide is due to liquefaction, the movement will start at the onset of the strong motion and stop when it stops. According to graphs in [18], this time is about 30 seconds; this is an information additional to what we have in Sections 5.2.2 and 5.2.3.

V = 2 m/s corresponds to the 9% slope and y₀ = 8 m. Together with the time 30 s, this gives a horizontal flow path of 60 m. This corresponds well to Figure 4, giving 56 m as maximum horizontal deformation.

The water depth gives c = 250 m/s so we get L_bh = 250 × 30 = 7500 m = 7.5 km for the initial disturbance. As the flow path is short the κ ∼ 0. Now we get.

L_bh = L_s + 7.5 = 110 + 7.5 = 117.5 km.

The uplift caused by the coseismic movement means that there will be just a small hole in the scar area. The volume of the heap caused by the slide will be.

W_T2b = By₀L_s = 1 × 110000 × 8 × 110000 = 9.68 × 10¹⁰ m³ = 96.8 km³.

The height of the wave with same volume as in Section 5.2:

h_2b = W_T2b/(L_hb B) = 8 × 110/117.5 = 7.5 m.

Eq. (7) must be modified due to the uplift and the small scar hole; this is done by putting κ ∼ 0 as before, and then we have for the energy in the source area:

E_W2a = ½h_2b² L_hbρ g B = ½ 7.5² × 117500 × 1025 × 9.81 × 110000 = 3.6 10¹⁵ Nm.

This result can be checked against the simulation results published by NOAA [19]; it is on Figure 5 and shows the spread of the tsunami very well. Comparing this with a ring wave spreading in an effective 90° conical channel gives a resulting average wave height 2–3 feet 900 km from the source. According to C = 250 m/s (800 km/h), this should occur after little more than 1 hour. This checks well against Figure 5.

The bottom deformations that caused the very strong Tohoku tsunami in the Pacific Ocean, simulated numerically by the Japanese and USA scientists, [17, 19], can be explained by a submarine landslide. This suggests that the coseismic slip of the earthquake triggers a sliding of the surface sediments. In combination they cause the bottom deformation. Finally, it can thus be concluded that the coseismic slip and the landslide are both responsible for the Tohoku tsunami in March 2011, not the coseismic slip alone.

This shows that in assessing the tsunami risk in the Pacific coastal regions of Japan, the landslide risk must be considered. This fact may result in that considerably larger events than the Tohoku tsunami are possible if a larger slide than this 8 m thick slide is released. This landslide is not very high compared to what has happened elsewhere. The assessment of this possibility of larger slides can be difficult; there is considerable uncertainty in estimating the height (y₀) of possible landslides.

6.2 Tsunami risks in the North Atlantic Ocean

There are many tsunami sources in the Atlantic Ocean, but in the northern regions, the tsunami risk is less than in many other places, and the source of the main threat may be unknown, both location and magnitude. A good method is presented in [9], to estimate the hazard curves for a reference point in south Iceland. This involves estimating initial wave heights at the source and their frequencies. Then the transfer functions must be applied, and the hazard curves are found by numerical integration.

6.2.1 Risk assessment methods

To assess the risk, we have to estimate event return periods for the various event magnitudes and the correlation structure of the event history. This correlation may be between time length between events and event magnitude and autocorrelation (positive or negative) in the time history.

The very long records necessary for a complete picture of the event statistics are normally not available. Certain assumptions are necessary, but we must estimate the basic statistics such as average time between events, t_a, the standard deviation associated to it, t_s and the correlation between magnitude and event interval ρ (note the different meaning of ρ in chapter 5). Now the following formula can be derived for the interval between tectonic events, it being earthquakes, submarine slides, or volcanic events:

gki=ρhki+√1−ρ2ekiE9

i	Number of event occurring at time t(i)
gk(i) = ((t(i + 1) − t(i) − ta)/ts	Dimensionless relative time between events
hk(i) = ρ(Hk(i) − Ha)/Hs	Relative magnitude, e.g., wave height
ρ = E{gk(i)hk(i)}	Magnitude time correlation (E denotes average)
ek(i)	Random function ek(0.1)
k	Series number

The time between the tectonic events i and i + 1 is known when g_k(i) is known so series for the occurrence of events in time can be simulated when h_k(i) is known. If the simulation period is a limited number of years into the future, it is necessary to simulate sufficiently many series (the number k) so the statistical distribution of H is represented.

If this statistical distribution is not known, some classification that fits available observations of H has to be assumed. Three classes, small, medium, and large events, should be considered as minimum. Then the Monte Carlo method is used to simulate the k series, and they are used to determine the statistics of interest empirically. These are various hazard curves and event probabilities for fixed periods to come, e.g., economical lifetime of structures and so on. It must be noted that the probability of a specified event of a given class happening in the next year is not a constant. This probability will increase with time.

6.2.2 Example from South Iceland

There are several methods to estimate the distribution functions we have to use as building blocks in Eq. (9) recommended in the literature. The log-normal distribution is often usable for g_k, especially when ρ is low [5].

When there is more than one source, the procedure has to be repeated for all significant sources. Then the transfer functions have to be applied and H_a and H_s calculated for the chosen reference point. H_s is much more difficult to estimate from observations than H_a, but sometimes it is possible to estimate the coefficient of variation C^v = H_a/H_s. If not it has to be included as a parameter in order to estimate the accuracy of calculated probabilities and risks.

In [9], all this is done for a reference point south of Iceland. There are eight possible sources for tsunamis in the North Atlantic, six of these are found significant for the reference point chosen. For clarification the reference point is shown and the resulting risk curve.

The only significant Icelandic source contributing to the calculated tsunami height in the reference point on Figure 6 is the Katla volcano [6]. The hazard curves for these points are in Figure 7. Here all the risk curves follow the Gumbel distribution P(H > x) = exp(−exp(−y)) where y = 3.91 + 1.12 x rather closely. Here the probability P does not denote the next event, but the maximum to be expected in the next year (annual maximum); it will be x = (y − 3.91)/1.11 in each point in Figure 7.

The probability is for the maximum to be expected in the next year (annual maximum). To take an example, the maximum to be expected in the very next year with probability P = 1% or 0.01 has the Gumbel variate 0.4, corresponding to a wave height of x = 0.1 m which is rather insignificant, but for P = 0.1%, the Gumbel variate is 2.8 giving seven times larger wave height.

7. Discussions

Tsunami attack is very difficult to predict, even though the models that simulate the propagation of the tsunami wave over deep water are very good and in most cases reliable. The mathematics of these models is very difficult, but effective, as long as the wave stays in deep water [20]. The difficulty in modeling is to predict the shoaling and the run-up. And then there is the uncertainty about the initial wave height and wave energy formation in the source area.

The importance in such analysis is to identify the sources that cause the largest tsunami threats. The methods devised in Chapter 5 can be used to estimate the initial wave and energy in the source area when the bottom deformation is known. This is demonstrated in the case study of the Tohoku tsunami, in Section 6.1; in [21] is a detailed description of this huge event and its consequences.

The cause of the bottom deformation is directly or indirectly an earthquake. It can start a submarine landslide, or the earthquake wave itself can deform the bottom so much that a large tsunami is produced, especially earthquakes above 7 in magnitude. But this very information tells us that the variability in tsunami properties will be very great in the source area. The magnitude of the average event may be possible to estimate from existing observations and geophysical data, but their standard deviation will always be difficult to estimate; usually there are not enough observations of serious tsunami events to establish a reliable value for this coefficient.

In the example taken in Section 6.2, there are eight identified tsunami sources in the North Atlantic Ocean, six of these contribute significantly to the hazard curve in the danger zone on Figure 7. This is the zone above 2 m, meaning that the tsunami has to be 2 m or higher to pose any significant threat alone, i.e., without being accompanied by a flood of different origin [9] or attacking upon a spring tide flood.

The effect of the coefficient of variation on the hazard curve Figure 7 is quiet surprising. The estimation of its value is very difficult. However, to leave it out corresponds to estimating its value to be zero, and that is totally unsatisfactory. In Figure 7 the effect of three different values, common in geophysical data, is demonstrated, and the difference is quite striking. The difference in frequency of occurrence is of one decade, up or down, from the C^v = 1/2 value.

8. Conclusion

The translatory wave theory is used to estimate the expressions for energy and wave height in the source area, and it can be used for the initial conditions in wave models.

Earthquakes, of too small a magnitude to produce dangerous tsunamis themselves, can do so by triggering submarine landslides.

Approximate analytical methods can give good results as a first approximation for the transfer functions that link the wave heights in the source area to wave heights in a reference point selected in the wave propagation path. The best position for such a point is offshore, outside the zone of nonlinear shoaling and wave breaking, but near the places where the attack of the tsunami is expected. Then a hazard assessment may produce a wave height-frequency curve for this point.

It is clearly demonstrated that it is very important to include the C^v factor for the event magnitude in the hazard assessment. Otherwise the tsunami wave heights for a given frequency may be seriously underestimated.

For high C^v values the hazard curves may be expected to follow the Gumbel distribution or possibly another distribution of the extreme value distribution family because the hazard curve is the maximum expected frequency for a given wave height.

Acknowledgments

Part of the theories herein is developed for the Básendaflóð (The Flood at Basendar Iceland in 1799) research directed by Sigurður Sigurðarson, Section Engineer, Icelandic Road and Coastal Administration, and in cooperation with Gísli Viggósson Consulting Engineer, Reykjavík Iceland. This project was supported financially by the Research Fund of the Icelandic Road and Coastal Administration who also sponsored this publication. Their support is greatly acknowledged.

The authors acknowledge Japan Agency for Marine-Earth Science and Technology and the NOAA Center for Tsunami Research, Pacific Marine Environmental Laboratory, for the use of material referred by them.

Conflict of interest

The author declares that there are no conflict of interests regarding the publication of this chapter.