How does a Tsunami work?

Usually I talk about the weather in this blog.  The fact is, a lot of the dynamics of weather are derived from basic fluid dynamics which apply not only to air, but to water as well.  We talk about waves in the atmosphere all the time--short waves, long waves, buoyant waves, etc.--and the same mechanics describe water waves.  So today I'm going to give a brief look at some of the simple mathematics behind tsunami-like waves.

Of course, the motivation for this is the tsunami generated by the very powerful earthquake that occurred last night in Japan.  How powerful was it?  According to the USGS, a magnitude 8.9-9.0 earthquake.  According to the list of strongest earthquakes on Wikipedia, this places it as the fifth most powerful earthquake ever recorded (in terms of magnitude).

It was so strong that even short-range seismographs around the world were able to pick up on the tremor.  Here's the seismograph from a small seismomenter on the Mount Augustine volcano in southern Alaska from the past 24 hours.  The earthquake is clearly visible just before 630Z in the upper right part of the trace:

Fig 1 -- Helicorder seismograph on the Mount Augustine Volcano in Alaska from 20Z March 10, 2011 to 20Z, March 11, 2011.

In fact, almost every single volcanic seismomenter in Alaska picked up the earthquake signal.  The lower 48 seismometers picked it up to--here's a similar plot from the Black Mountain, California, seismometer.  Once again, the quake shows up just before 630Z:

Fig 2 -- Seismograph from Black Mountain, California, from 2315Z, March 10, 2011 to 2315Z, March 11, 2011.

An earthquake of this magnitude can (and did) generate a tsunami--a very powerful ocean wave that swamps inland.  There are a plethora of videos now of the tsunami impacting Japan--and the horrible destruction brought with it.  This wave had so much energy in it that it easily transversed the Pacific Ocean.  One of NOAA's DART buoy's off the coast of Hawaii showed the interruption of the normal tidal cycle as the tsunami went through:

Fig 3 -- Water Column height from NOAA DART buoy 51047 off the coast of Hawaii from 00Z, March 7, 2011 to 20Z, March 11, 2011.

Note that for the past few days, the height of the water column rose and fell very periodically--these are the normal tides we see on a daily basis.  However, there was a sudden jump early this morning (around 11-12Z) where the cycle was interrupted by a powerful wave.  The switch to green colors indicates a change from 15 minute data collection to one minute data collection.  You can see that, at least at this buoy (which is some distance out to sea), the actual change in the height of the water column wasn't even as much as the change we usually see in height due to the normal tidal cycle.  But this was way out to sea--tsunamis move and behave differently when they get close to land.  How does that work?

WARNING--MATH CONTENT AHEAD

It turns out that tsunamis can be reasonably well explained by a set of equations called the shallow-water equations.  They are some of the simplest equations to describe wave flow on the free surface of a fluid.  They basically related the change in the height of the surface with time to the slope of the surface at a given time.   If we assume something called the shallow water limit,  which means that we assume that the wavelength between wave crests is on about the same scale as the depth of the fluid (the waves are about as far apart as the water is deep), we can ignore vertical perturbations and get one simple differential equation:

Where the Greek letter eta (the lowercase 'h' looking thing) represents the height of the free surface of the water, H represents the mean depth of the water, and g is the acceleration due to gravity.  I know...I know...looks complicated.  But if you've taken some basic calculus, it's actually pretty easy to sort through this.  Basically, we know we're looking for a WAVE as an answer to this, right?  So why not just assume that our free surface height is a simple sine wave:

We'll just assume our free surface looks like a wave of amplitude A that varies in space (x) and time (t).  The wavelength of the wave is given by lambda and the frequency of the wave is given by sigma.  Just a simple wave.  Now we can plug that wave equation in for eta in the first equation--take the second derivative with respect to time and then take the second derivative with respect to position x.  After some cancelling, we're left with this expression:

This is actually something called the dispersion relation  for this wave--it tells you how frequency (sigma) and wavelength (lamda) are related.  We can rearrange this equation to get:

Now, how do we find the speed of a (non-dispersive) wave?  The speed of such a wave is given by the frequency (in units of seconds^-1 (per second)) multiplied by the wave length (given in meters).  Put those units together and you get...meters per second.  The speed of the wave.  Well, the frequency times the wavelength is just sigma times lambda:

So we can just solve our expression above for sigma times lambda:

And there we go!  We've solved for the speed of shallow water gravity waves.  It's just the square root of the mean depth of the water (H) multiplied by gravity (9.81 meters per second squared).


END OF FANCY EQUATIONS


I find that result fascinating--the speed of waves in this shallow water wave approximation just depends on the depth of the fluid and gravity--that's it.  Amazing.


So it turns out that tsunamis moving through the ocean can be well-approximated by this shallow wave limit.  That seems kind of counter-intuitive--after all, they're moving through the deep ocean--how is that "shallow water"?  Remember the definition of "shallow water" above--it says that the wavelength between the wave crests is about the same as the depth of the fluid.  The Pacific Ocean is around 4 kilometers deep on average.  So when you think about it that way, the wave crests of the tsunami only need to be about 4 kilometers apart to be considered in the "shallow water" limit.  It makes more sense that way.  It really makes sense when you think of how big the Pacific Ocean is--

It's almost 20,000 kilometers across at its widest point from east to west--but only 4 kilometers deep on average.  That's a very "shallow" pool of water given its size...


Anyhow, if we know that the average depth of the ocean is 4 km (4000m) we can compute the speed of a wave moving through it:

So a wave moving through the deep ocean (in the shallow water limit...ha!) would be moving at about 443 miles per hour, on average.  That's amazingly fast!  This explains how the earthquake could have happened late in the night on the west coast of the US and the tsunami wave was already there by the next morning.  At that speed, the wave would have covered the 4800 or so miles from Tokyo to Seattle in about 10 hours.  Which is right on the ball for the time it actually took.


But what happens when the wave reaches shore?  There's really no strong signal of the wave amplitude in the deep ocean, like in the buoy trace above where there was definitely a strong interruption to the tidal cycle, but its amplitude was much smaller than even the normal tidal cycle. But when a tsunami reaches shore, the height of the wave increases dramatically.  What's going on there?


It has to do with several things--one is the slope of the ocean floor.  As the ocean floor goes upward as we approach the coast, the mean depth of the fluid gets shallower, forcing the water to pile up higher.  At the same time, notice our equation for the speed of the wave above--if the mean depth of the water (H) is decreasing, the speed of the wave will also decrease. (Actually, the depth will decrease enough that the shallow-water limit is no longer applicable and we have to consider the deep-water limit--which is completely different and makes less sense, but still...in approximation...).  So the wave is slowing down as it approaches the coast too.  This is why, though we see a wall of water moving toward the shore, that wall of water is definitely not moving at 443 miles per hour--that would be some devastation.  No...the waves do slow down somewhat.


But then we have to consider the conservation of energy.  Remember in classical mechanics, energy comes in two forms--kinetic energy (the energy of motion) and potential energy (based generally on how high off the ground you are).  If you hold a ball high in the air, it has a lot of potential energy--it has the potential to fall due to gravity.  Once you let that ball go, it starts falling--it loses that potential to fall by converting that energy to the kinetic energy of it falling.  It's basic mechanics.


So if the wave is slowing down as it's approaching shore, it's losing kinetic energy--it's no longer moving as fast.  But total energy has to be conserved--that lost energy has to go somewhere.  What does it do?  It converts into potential energy by raising the wave higher.  So the kinetic energy of the fast-moving wave transfers into the potential energy of a much higher (taller) wave as it approaches the coast.  Here's my quick sketch describing this:

So we see that the structure of the tsunami is described by this give-and-take between its kinetic energy (its speed) and its potential energy (its height).  Over the open ocean, the kinetic energy dominates (it moves much faster).  As the wave approaches land, it loses kinetic energy as that converts to potential energy (it slows down, but its amplitude increases).  


Of course, some of that energy is also lost to friction and other dispersive forces as the wave travels.  So generally, the further you are from the source of the wave, the weaker the resulting tsunami.


Anyhow, I think that's enough for one night.  I hope you found at least some of this enlightening as to how tsunami waves actually propagate through the ocean and why they become taller as they approach land.  I don't usually throw a lot of math into these blog posts, but I like to once in a while just to show where things come from.  If you didn't follow that, that's ok.  It's the concept that really counts.