Tsunami and Earthquakes: the dynamic connection

Here I would like to briefly discuss the physics connected with the Tsunami. We heard the magnitude of the earthquake was 8.9 in Richter scale. Richter scales are logarithmic as most of the other things in physics, like magnitude of stars, the sound intensities (decibels) and the wind speeds (Beaufort scale). Richter introduced the logarithmic scale to quantify the earthquakes. So suppose you know there was an earthquake of Richter 8.9, how do we estimate the energy released? The formula that is normally used is given as
log E=2.5 M-1.2.
Here E is the energy released in Joules and M is the magnitude of the earthquake.
It is a coincidence that a similar formula is also used for the luminosity scale of stars. So now we can calculate that for 8.9 it works out to be about 10^21 J. For a typical 1 Megaton nuclear weapon the energy released is about 10^17 J . So the intensity of the earthquake was equivalent to that of 10 000 large nuclear bombs.And this earthquake also triggered a tsunami. To understand this we need a little more information. The earth is made up of tectonic plates. During an underground earthquake a tectonic plate goes over another tectonic plate and the material in between is pushed up. This doesn’t happen in any other planet. Venus and Mars don’t have tectonics. There is an essential condition for the formation of plates that there should be lot of water in the planet. This water goes deep inside and mixes with the rock forming a molten mixture which is a highly viscous and elastic fluid called Magma.This phenomenon affects only the crust and the upper part of the mantle. The upper mantle goes to about 20 km. So most of the earthquakes have their origin about 10 km deep. So essentially as far as the diameter of the earth is concerned these are all surface phenomena.There is an increase of pressure and temperature with depth, So if we go a few kilometres down, the rock are molten. Originally these rocks became molten because of the radioactivity in the earth's crust coming from isotopes like 40 K, Rh, U etc. which initially provided much of the heat to keep the earth hot. The continental plates are floating on this magma. Every continent has got its plate, even other landmasses have got their plate. India has got its own plate called the Indian plate. Alfred Wegener was the first to notice that like in a jigsaw puzzle, the west coast of Africa fits nicely to the east coast of south America. Similarly the other landmasses could also be fitted together like a jigsaw puzzle . He then proposed that all the continents were part of only one landmass and that the continents are actually drifting. Initially there was only one super continent Pangaea, then two huge continents Laurentian and Gondwanaland. Apparently about 200 Myrs back India was near the Atlantic in the southern hemisphere. Then the Indian plate broke up and joined the Asian plate. Now with GPS (Global Positioning System), we know this is the reality because we are now actually able to measure the small drift in continents . The Indian plate is still going northwards and pushing against China’s plate. The Himalayas are the result of Indian plate colliding with the Asian plate about 40 Myrs ago. So Himalayas are quite young mountains. As an evidence for this we find fossils of marine animals on top of mount Everest! This shows that once upon a time it was all part of the sea. This also explains why the Himalayas are growing taller. All these mountain building and movements of continents are all connected. The earth surface is in a state of continuous flux. All the continents are moving around. Even now the Mediterranean is closing because the African plate is pushing northwards again and Europe is trying to merge with America and thereby shrinking the Atlantic! In another 100 Myrs the face of the continents will be very different. The pacific is the biggest ocean right now , but it may not be so after 100 Myrs. There is a ring of fire around Japan and all these islands in the pacific. These are highly earthquake prone regions because their pleates collide with each other every now and then and the intensity of the collision depends on how much matter is thrust up. In 2004 there was an underground earthquake in Indonesia that led to a big tsunami and death of 2 lakh people including several in the Sri-Lankan coast and Chennai. Last year in Chile 2010 there was a big earthquake and one in Haiti which also caused a tsunami. In the past year there have been three or four such events.
Tsunami literally means harbor waves in Japanese because when they break on the coast they rise to great heights. So how do we estimate the height of the tsunami wave? Waves can propagate by two means namely : gravity waves and capillary waves. For any body of water if there is a small perturbation, like throwing a pebble on water, the ripples on the surface can be treated as capillary waves. These are generated because of surface tension in the water. Now suppose we have a bigger perturbation like an explosion under water, gravity waves will be created. There is a column of water formed which is denser than the surrounding and they will be pushed up because of buoyancy. There is a formula for the frequency of these waves which depends on the wave number k (inverse of the wavelength) and g is the acceleration due to gravity. For gravity waves the formula is given as
omega^2=g k .
For capillary waves this formula depends on the surface tension gamma , the density of the liquid rho and k^3 given by:
omega^2=k^3 gamma/rho.
So the general expression for a combination of gravity waves and capillary waves.
is given by
omega2=g k +k^3 gamma/rho.
There is a transition between one form of wave to another depending on the acceleration due to gravity, the density and the surface tension. For larger values of k capillary waves will dominate and for longer wavelengths, gravity waves will dominate. Tsunami waves are nothing but the gravity waves. The velocity of the waves depends on the depth of the wave, d
v^2=gd.
The waves which are moving with a higher velocity are submerged to a greater depth. That's why tsunami waves could be quite high and they move at high velocities. We have heard that the velocities of the tsunami waves were about 1000 km/hr, comparable to a jet liner. We can roughly estimate these velocities in the following way: a substantial part of the energy released in the earthquake (10^21 J) would be transmitted to the water causing the wave motion.
We saw that the area involved was something like 1000 square km. The depth of the wave would be about 100 m which can be deduced from the epicentre where the earthquake took place (how much of the plate was broken off at the fault line). So the volume covered is roughly about 10^14 cubic meters. So we know the mass of the water was about 10^17 kg. We can now work out the velocity. The kinetic energy of the water mv^2 , is roughly of the order of 10^21 J. So, if you do this calculation so you get a v roughly about close to what they have said, about 1000 km/hr. Off course for high values of wave number k, there would be a surface wave and the transition is interesting to work out. For example, the surface tension of water is about 70 dynes/cm and density 1g/cc. So for water the transition frequency is something like 10 Hertz. For a frequency smaller than this gravity wave dominates and for a larger frequency capillary waves dominates. So we can see that for the case of tsunami surface waves are not important and the deeper gravity waves are important. So we are justified in estimating the velocity of the waves in the above fashion.
If we have different planet or satellite say like Titan which is supposed to have oceans of liquid methane (one of the findings of Cassini spacecraft was that Titan has got weather similar to earth but running on methane's cycle). Now suppose a similar thing happens there, the surface tension for alcohol is much lower (compared to water) say around 20 whereas the densities of the order of comparable about 0.8 g/cc . So again the crossover frequency can be worked out. The surface gravity on Titan is quite small compared to the earth . The frequency of waves would get affected. For a given wave of particular frequency the velocity will be different now. Its velocity will be smaller and wavelength probably 10 times longer. If you take another example another example of Europa ,one of the satellites around the Jupiter, they think there might be a lot of water below the surface (which is why some space craft are being planned to exclusively monitor the satellite Europa). This huge liquid ocean underneath the surface might be subjected to all these kind of wave phenomena. For the Europa, the surface gravity is only one seventh of the earth. So you see any velocity of any of these waves will be much less . It is easy to deduce many things from this kind of analysis . So next time you read about some earthquakes and all these things about Richter scales, you can plug in the formulae and see how much energy is released and what would be the velocities of the waves.