Modeling the Sendai Tsunami

Today, I gave a brief lesson on the Richter Scale, earthquakes, and tsunamis. I initially made a pretty poor model (2-D) of the tsunami wavefront for my students in 3^rd period physics. By 7^th period physics, I had was using a crude 3-D cylindrical model, but unfortunately most students were tired and worried about the TOEFL test. So, here it is, reproduced for anyone curious how relatively simple math and physics can model a tsunami.

The 8.9 earthquake released approximately 1.41x 10^18 J (the equivalent of 336 megatons of TNT,about  7 times larger than the largest nuclear weapon ever tested).

Imagine all of that energy going into a cylindrical ring of water surrounding the impact. (This is really unrealistic - a lot would be lost into the Earth, or due to friction, or evaporation, or melting of rocks.) The outer edge of the wave is a cylinder of radius r (in km) and height h (in km). I’m estimating the shockwave to be roughly 1 km (just because I need a number to plug in – help from fluid engineers/hydrophysicists would be welcome!), which means that the inner cylinder has a radius of  r-1 and a height of h.

The volume of the outer cylinder is

And the volume of the inner cylinder is

Therefore, the volume of water (in km^3) in the expanding cylindrical ring is

The mass of water in that cylinder is simply the density of water times that volume of water.

Finally, assuming that the entire energy of the earthquake is transmitted into kinetic energy of the  water,

And the velocity of the water is

Now, in the deep ocean, h ~ 4km. Using this rough model, the velocity of the water with r=1 (just after the explosion) is about

The epicenter occurred about 80 miles east of Sendai (~130 km). Now, at a distance of r = 130 km, assuming the ocean remains the same depth, the velocity of the water would be

But near the coast, the water is actually much shallower. Using a more realistic depth, say about 30 meters (~100 feet), the velocity of the water should be

Remember – this is a very crude model. We ignore energy losses due to friction and phase changes or transmission into the Earth. But it captures the essential features of a tsunami:

1. The energy dissipates the further you go from the epicenter.

2.  However, shallower water means that the energy is spread across less mass, meaning that the water will actually have a high energy density at the shore.