Tsunami Numbers

Manuel Garcia, Jr.
30 December 2004

The magnitude 9.0 Sumatra-Andaman Islands earthquake of 26 December 2004 generated a tsunami -- one or more large gravity waves -- that ran across the Bay of Bengal devastating the coast and causing an immense loss of life especially in Sri Lanka, India, Bangladesh, Myanmar, Thailand, Malaysia and the Indonesian island of Sumatra. The tsunami even caused deaths across the Indian Ocean along the west coast of Africa between Madagascar and Somalia. On 30 December, the published estimate of lives lost stood at 125,000. (1)

Much has already been written about the poison added to an already horrible natural disaster by separatist disputes, particularly in Sri Lanka but also in Aceh Province in Sumatra. Another disheartening and embarrassing story is that of American parsimony and cold-heartedness, in the person of President Bush's obvious annoyance at having to waste time and money even if only as a gesture on behalf of the Asian victims. The Europeans are seeking to call emergency meetings of the Group of Eight (G8, the rich nations) and the Paris Club (the creditor nations) to gather billions in emergency aid, and to cancel the debt of stricken nations. Clearly, Europe will respond either as a Union or with independent national efforts.

President Bush offered $15M when he was finally compelled to emerge momentarily from his vacation in Crawford, Texas. (2) UN emergency relief coordinator Jan Egeland called this "stingy," to which a New York Times editorial concurred. Later, functionaries in the government found another $20M to toss in the bucket, so the US contribution swelled to $35M. (3) Well, at least you've got to give the man (Bush) credit for honesty -- for a change -- he's not even pretending to care. Americans will no doubt be far more generous in supporting the international relief effort than is reflected by their (sic) government.

However, it is not politics that concerns me now; what I wish to describe is a sense of the physical scale of the tsunami. Why? To help place this event in perspective, and to help motivate thinking about shared global problem solving. Also, to offer some possibly humbling reflections on the power of nature.

Data: Earthquake And Tsunami

The essential facts about the magnitude 9.0 earthquake and its many aftershocks are available from the US Geological Survey. (4) Information about tsunamis generally, and the Indonesian one in particular, is presented by the US National Oceanic and Atmospheric Administration. (5) An animation of the tsunami was produced by the National Institute of Advanced Industrial Science & Technology of Japan. (6)

The earthquake was caused by a rupture -- perhaps 1000 kilometers in length -- along fault lines in the floor of the Andaman Sea, on the eastern side of the Bay of Bengal between Sumatra and Myanmar. The Burma Microplate is a small tectonic plate underlying the seafloor of the Andaman Basin, perhaps 2000 kilometers long and several hundred wide. The India Plate moves northeast at 61 millimeters a year and subducts under the western side of the Burma Microplate along the Sunda Trench. The Australia Plate is south of the India Plate and also moves northeast subducting at the Sunda Trench. To the east of both the Burma Microplate and the Australia Plate is the large Sunda Plate, underlying the Malay Peninsula and southern Indo-China. The epicenter of the 26 December M9.0 earthquake occurred at the southern end of the Burma Microplate, a nearly quadruple junction of four tectonic plates: the India, Australia, Sunda and Burma. The Burma Microplate shifted by an upward thrust of its western edge at the Sunda Trench, and a parallel slip along its north-south fault lines separating it from the Sunda plate to the east. [7]

The impulsive vertical motion of the seafloor displaced an immense quantity of seawater, launching the tsunami.

Gaging The Tsunami

How much energy went into the tsunami?

Assume a volume of seawater 1000 km long, 32 km wide and 3.6 km deep (typical of the Bay of Bengal) is lifted 2 meters. The energy involved equals the product of the mass of seawater, the acceleration of gravity (9.81 m/s^2) and a vertical displacement of 2 m. The mass of seawater is itself the product of its density, taken to be uniform at 1100 kg/m^3, and the volume of 1.15 x 10^5 km^3, or 1.15 x 10^14 m^3. The energy is E = (1100) * (1.15 x 10^14) * (9.81) * (2) = 2.5 x 10^18 joules.

To make this quantity more easily comprehensible, let us find its equivalent in numbers of Hiroshima atomic bombs. One kiloton (kt) of TNT has an explosive energy of 4.184 x 10^12 joules, the Hiroshima bomb had a yield of about 15 kt or 6.3 x 10^13 joules. The energy E pumped into our hypothetical seafloor upthrust is equivalent to nearly 40,000 Hiroshima bombs (39,683.5, rounded up), that is 600,000 kt or 600 Mt (megatons). One could imagine trying to engineer a comparable seafloor upthrust to that of the 26 December earthquake by planting 40,000 Hiroshima scale bombs spaced at 25 m in a 1000 km line.

The Hiroshima bomb had a yield that is fairly low compared to most bombs produced since 1945. The largest nuclear bomb detonated by the United States was Castle/Bravo at 15 Mt, and the largest bomb ever exploded was a USSR device yielding over 50 Mt. These bombs could be thought of as 1/40 and 1/12 equivalents, respectively, to the hypothetical 600 Mt seafloor uplift.

The Hiroshima bomb destroyed an area 12 km^2, killed up to 80,000 people and resulted in total casualties of up to 150,000. (8) The 26 December tsunami has probably killed 125,000, and displaced millions. It is possible a land area as large 10^4 km^2 has been inundated. This estimate would be equivalent to 3200 km (2000 miles) of coastline affected up to 3 km inland. In a few months, the reports of geophysicists will begin to be published, with refined estimates of inundation based on satellite imagery and ground surveys. While the nature of the destruction from a tsunami and a nuclear blast are very different, it is still useful to reflect that the scale of destruction by the Indonesian tsunami may be hundreds of times that of Hiroshima.

It is also a somber realization that with small numbers of large nuclear bombs a few governments begin to approach the destructive capability of nature itself.

Wave Height And Speed

How does a tsunami propagate?

Imagine once more our column of seawater, with its 32 km wide, 1000 km long "footprint." This is momentarily "lifted," which is to say it is pressurized by the sea floor displacement so that water mounds up above normal sea level. Being liquid, water will not hold a shape, so as it is lifted it will slosh away. The actual mound above normal sea level may only be 10 centimeters (4 inches) in the footprint area.

Under the force of gravity the mound will seek to level out, and the gravitational energy impulsively infused into the column of water by the sea floor motion will seek to equilibrate throughout the mass of the ocean. The tsunami is a wave or series of waves that carries this initial energy from the source volume out to the surrounding ocean.

It is energy that flows to great distance, not water. The wave moves because small parcels of water slosh back and forth (in circular vertical paths, for deep water) and this local activity transfers the high pressure of a particular wave forward (think of a sequence of meshed gears). The wave may have a wavelength of many meters, even kilometers, and within that distance the wave will exhibit both a high pressure crest and a low pressure trough. Tsunamis are invisible from the air or satellites because they can have crests of less than a meter in height -- even just inches -- spaced at hundreds to thousands of meters. The water itself moves little and feels "normal," tsunamis pass unnoticed under boats on deep water.

Let L represent the wavelength of a tsunami, and h represent the depth of the sea. Then, a deep water wave will have h > (L/6.3), while a shallow water wave will have (L/6.3) > h. The speed of a deep water wave is given by V = (1.56*L)^(1/2), where V is in m/s and L is in meters. Longer waves travel faster in deep water. The speed of shallow water waves is independent of the wavelength, they all have the same speed set by the depth, V = (9.81*h)^(1/2).

For the Bay of Bengal, deep water waves will be those with wavelength L less than (6.3*h) = 23 km, and shallow water waves will have larger wavelengths. So, tsunamis with L > 23 km will travel at 188 m/s (420 mph), and tsunamis of shorter wavelength will travel at speeds between zero and 188 m/s, for wavelengths L from zero to 23 km, respectively, according to the formula (1.56*L)^(1/2). The Indonesian tsunami crossed 1400 km of the Bay of Bengal to arrive at Sri Lanka in 2 hours.

The sea floor rises as the tsunami approaches land, diminishing the depth of water, which has three effects: the waves slow down (they all become shallow water waves and h is diminishing), they bunch up and their crest heights increase dramatically.

Deep water tsunamis (those with wavelengths under 23 km for the Bay of Bengal) move at a wide range of speeds. Longer waves (under 23 km) will outspeed shorter waves, so they disperse as they move away from the source volume, which can emit a variety of wavelengths, depending on how the earthquake flapped and rattled the sea floor. As tsunamis become shallow water waves their speeds converge, and as they move along a rising sea floor they continually slow and they bunch up. As a sequence of waves bunches up, the individual waves also compress to smaller wavelength with taller crests. The wave tries to preserve the volume of its crest mound. What may have been a 10 cm high, 32 km wide mound in deep water might strike the shore as a 16 m (53 ft) high, 200 m wide wall of water moving at 13 m/s (30 mph).

The limit on cresting is set by the instability of water piled higher than the width of its base -- the wave breaks -- and by the distance from the source volume of the tsunami.

If the cresting of the previous example is doubled to 32 m (106 ft) the base would be 100 m, a still reasonably stable height-to-base ratio of 1/3.

The tsunami is just a gigantic ripple expanding from a point of disturbance. Close to the source, the ripple has more energy per unit volume because it is a small circular arc. As this arc expands, a greater volume of water contains the same amount of energy, so every meter of linear extent along the wave arc has less energy -- or pressure -- with which to rise against the force of gravity as it races up a rising sea floor.

Parting Thoughts: Clinging To Roots Against The Flood

A disaster of this magnitude makes it painfully clear that the most concentrated forms of power humans have at their disposal are still dwarfed by natural forces. That this power is in the forms of nuclear weapons, military power and overbearingly greedy economics only rubs salt into the wounds of the dispossessed. A disaster of this magnitude should make it clear to anyone with any degree of human feeling that we are all simply castaways on this island Earth, and that the sharing of burdens and the assistance of each other is not really an option but a necessity for our own survival. It is to that end that our power, and planning and resources should be aimed. We need some major realignments of our thinking. Nature is indifferent to us, and its power is such that with just a twitch any number of us could be wiped away from the world of the living. The only thing we have that approaches security is each other.

The next article describes the energy of the tsunami waves, see below (15 December 2005).

References

[all web sites active 30 December 2004]

[1] Tomi Soetjipto and Dean Yates, "Tsunami Toll Jumps to over 125,000, Fear Lingers," Reuters, 30 December 2004, http://www.reuters.com/newsArticle.jhtml;jsessionid=N4T0FTC02ODQYCRBAEZSFEY?type=topNews&storyID=7210780, also: http://www.truthout.org/docs_04/123104X.shtml (1 back)

[2] Patrick Martin, "Bush's response to South Asia disaster: indifference compounded by political incompetence," World Socialist Web Site, 30 December 2004, reference provided by John Steppling, http://www.wsws.org/articles/2004/dec2004/bush-d30.shtml, (2 back)

[3] "Are We Stingy? Yes," The New York Times, Editorial, 30 December 2004, (http://www.nytimes.com, and obnoxious "registration", or) http://www.truthout.org/docs_04/123104B.shtml (3 back)

[4] "Magnitude 9.0 - OFF THE WEST COAST OF NORTHERN SUMATRA 2004 December 26 00:58:53 UTC," Preliminary Earthquake Report, U.S. Geological Survey, National Earthquake Information Center World Data Center for Seismology, Denver, http://earthquake.usgs.gov/eqinthenews/2004/usslav/

and

"M9.0 Sumatra - Andaman Islands Earthquake of 26 December 2004," map of Andaman Islands and Aceh Province in Sumatra, http://earthquake.usgs.gov/eqinthenews/2004/usslav/tect_lg.gif (4 back)

[5] "NOAA REACTS QUICKLY TO INDONESIAN TSUNAMI," National Oceanic and Atmospheric Administration, US Department of Commerce, 26 December 2004, http://www.noaanews.noaa.gov/stories2004/s2357.htm (5 back)

[6] "2004 Sumatra Earthquake," Tsunami Animation - National Institute of Advanced Industrial Science & Technology (Japan), http://staff.aist.go.jp/kenji.satake/animation.gif (6 back)

[7] USGS, see [4], http://earthquake.usgs.gov/eqinthenews/2004/usslav/tectsetting_lg.gif (7 back)

[8] Daniel Green, "The Atomic Bomb," World War II Air Power, http://www.ww2guide.com/atombomb.shtml (8 back)


The Energy of the Indian Ocean Tsunami of 2004

Manuel Garcia, Jr.
15 December 2005

In the example described in "Tsunami Numbers" on 30 December 2004, the abstracted earthquake pressurized a lens of water 1000 km long by 32 km wide by 3.6 km deep with a force (equal to the weight of water) of 1.25 x 10^18 Newtons (N) across a seafloor area of 3.2 x 10^4 km^2 = 3.2 x 10^10 m^2. The resulting pressure (above normal) was 3.91 x 10^7 N/m^2 = 5666 psi (pound/in^2).

Every meter of depth held 6.94 x 10^14 joules = 166 kt (kilo-tons of TNT) of energy to be dissipated. This energy flowed laterally into the surrounding ocean. The top 10 meters of ocean would dissipate 1.66 Mt (mega-tons) of energy, and this can be taken as the energy of the tsunami.

Because the source volume is so thin compared to its length, the waves emanating from it will initially approximate those from a "line source", they will be fairly linear waves 1000 km long emanating from each long side of the source. Ignoring the local distortion of the waves because of islands and changes of depth to the seafloor, the waves will begin to curve and stretch into circular arcs once they are well past 1000 km distance from the source region. At a distance of 1000 km, the waves may have lengthened to 3000 km, approximating a half circular arc. Beyond this point (ignoring local distortions) the waves will expand as half-rings, their length proportion to 3.14 (pi) times their distance from the source.

Let R represent the wave distance from the source region. The energy per unit length of wave (each side) will vary, approximately, with R as follows:

0.83 kt/km @ R < 1000 km,

0.27 kt/km @ R ~ 1000 km,

270 kt/R @ R > 1000 km.

Waves striking land within several hundred km of the source region will carry energy of 0.83 kt (830 tons TNT) for every km of length. Waves arriving at Sri Lanka and India, 1400 km away, will hold 0.19 kt/km (190 tons/km). At 2000 pounds/ton, the waves at Sri Lanka were holding an energy equivalent to 385 pounds of TNT for every meter of length. Waves striking the coast of Sumatra only 100 km from the epicenter of the earthquake would carry an equivalent to 1660 pounds of TNT per meter of linear extent.

In reality, "the tsunami" was actually a series of waves spaced at intervals of minutes (about 30 minutes at Thailand). The major force was carried by the first three waves (over 1.5 hours at Thailand, the third wave being the largest, small tsunami waves continued during the rest of the day). (1)

The energy equivalent to the waves arriving at Sri Lanka was comparable to crates of TNT about the size of refrigerators, stacked in a continuous wall along the shoreline. The energy equivalent at the coast along northwest Sumatra was comparable to a continuous line of side-to-side tightly parked truck-bombs.

Obviously, the manner and rate in which energy is released in an explosion of TNT and by a tsunami (as a series of waves) are quite different. However, it is evident that the energy available to a tsunami drawn from the top 10 meters of ocean surface (in this example) is more than enough to cause major damage when it strikes the shore and proceeds inland.

In thinking of the wave energy in equivalents to TNT explosions and nuclear bombs, one begins to understand why this tsunami series of waves could produce a war-like level of catastrophic damage over a very wide area, all within the course of one to two hours.

It is interesting to consider the degree of pressurization of the water rushing ashore. Here, I am using the term "pressurization" to describe the "total pressure" as given by the Bernoulli equation (the integral of the conservation of momentum). This "total pressure" is the sum of three effects: the static pressure of a parcel of water, it's kinetic energy (sometimes called dynamic or ram pressure, which is proportional to the velocity squared and is a thrust in the direction of motion), and the effect of gravity due to the height (or depth) of the water. This sum is expressed in units of pressure (Pascals in metric units, pounds-per-square-inch (psi) in English units). At the surface of the water, the static pressure will be normal air pressure, the gravitational effect (the weight of overlaying water) is zero, and the "total pressure" is largely a horizontal thrust of water. So, the total pressure experienced at the shoreline is largely the thrust of water inland (or out) instead of a static pressurization (e.g., a super-carbonated beverage). Physicists use units of pressure in the Bernoulli equation because then the "total pressure" indicates the total energy-per-unit-volume of the fluid. Clearly, as the wave expands from the source region, the energy-per-unit-volume decreases. Given this definition of terminology, let us continue with the example.

At the source the water is pressurized to 5666 psi above its normal state (at its given depth). The pressurization will drop as the wave recedes from the source region, because for any given layer (say the top 10 m) the same total energy is distributed in a greater quantity of water. The volume of the wave increases as it moves away from the source because it becomes a longer arc.

The pressure of each wave (the entire wave series for each of 2 sides) will vary, approximately, with R as follows:

2833 psi @ R < 1000 km, (waves on each side at 1/2 the energy density of source)

922 psi @ R ~ 1000 km,

922*(1000/R) psi @ R > 1000 km.

If we assume that the tsunami series is made of three significant waves of equal size, then each would have:

944 psi @ R < 1000 km,

307 psi @ R ~ 1000 km,

307*(1000/R) psi @ R > 1000 km.

Waves striking shore would carry pressurized water that could be expected to froth, foam and bubble as it churned inland -- from a combination of motion-induced turbulence, as well as any momentary static pressurization caused by the passing of the wave, or energy pulse, through the fluid. Waves at Sri Lanka would be pressurized to about 219 psi, while those at Sumatra to 944 psi. These pressurizations are equivalent to that of still water at a depth of 15 m and 64 m, respectively. (2)

The rapid de-pressurization of surface water as the energy of the tsunami wave passes through it would be equivalent to a rapid rise of subsurface water -- or a human diver. Gases trapped during pressurization would be quickly released, causing bubbles and adding to the turbulence of the liquid. One of the warning signs of the imminence of a tsunami (a rise in sea level near shore) is the "boiling" or "bubbling" of the sea. The pictures of tsunamis sweeping inland (e.g. at the Thai holiday beaches, see 1) show frothy, turbid, highly agitated water both at wave faces and along their backs.

References

[web sites active 15 December 2005]

[1] "2004 Indian Ocean Earthquake", wikipedia,
http://en.wikipedia.org/wiki/2004_Indian_Ocean_earthquake

This reference contains data on the actual events observed on 26 December 2004, and the published scientific estimates of the physical parameters of the earthquake (e.g., length of seafloor rupture at 1200 km) and subsequent tsunami, as well as casualty figures. There are several links to other useful sources. The picture at the head of the article is a graphic illustration of the "linear TNT explosion" concept I have used here as an analogy with which to describe the energy of the tsunami.

The wikipedia article gives the energy release of the earthquake at between 250 Mt and 800 Mt; and it states that the tsunami held about 5 Mt of energy. My estimate from 30 December 2004 of earthquake energy release was 600 Mt. My estimate for tsunami energy (described above, continuing the prior work) is 1.66 Mt for the top 10 meters of ocean. If a layer 20 m thick is assumed to transmit its energy past the shoreline, then 3.32 Mt is transmitted. (1 back)

[2] "Water pressures at ocean depths",
http://newport.pmel.noaa.gov/nemo_cruise98/education/pressure.html
"We often speak of pressure in terms of atmospheres. One atmosphere is equal to the weight of the earth's atmosphere at sea level, about 14.6 pounds per square inch. If you are at sea level, each square inch of your surface is subjected to a force of 14.6 pounds. The pressure increases about one atmosphere for every 10 meters of water depth. At a depth of 5,000 meters the pressure will be approximately 500 atmospheres or 500 times greater than the pressure at sea level." (2 back)


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