Tsunami Geometry: Calculating the Height of a Tsunami using Basic Geometry

Since we’re working on geometry this cycle, I thought it would be an interesting exercise to think about how we could use geometry to think about how the strength of tsunamis decreases with distance from the source.

Of course, we’ll have to do this using some intense simplification so we can actually apply the tools we have available. The first is to approximate the tsunami as a circular wall of water centered on the epicenter of the earthquake.

tsunami-geometry — Simplified tsunami geometry.

This lets us figure out the volume of the wave pretty easily because we know that the volume of a cylinder is:

(1) $! V_c = \pi r^2 h$

The size of the circular water wall we approximate from the reports from Japan. The maximum height of the wave at landfall was somewhere in the range of 14 m along the northern Japanese coast, which was about 80 km from the epicenter. Just as a wild guess, I’m assuming that the “effective” width of the wave is 1 km.

Typically, in deep water, a tsunami can have a wavelength greater than 500 km (Nelson, 2010; note that our width is half the wavelength), but a wave height of only 1 m (USSRTF). When they reach the shallow water the wave height increases. The Japanese tsunami’s maximum height was reportedly about 14 m.

At any rate, we can figure out the volume of our wall of water by calculating the volume of a cylinder with the middle cut out of it. The radius of our inner cylinder is 80 km, and the radius of the outer cylinder is 80 km plus the width of the wave, which we say here is 1 km.

Calculating the volume of the wave

However, for the sake of algebra, we’ll call the radius of the inner cylinder, r_i and the width of the wave as w. Therefore the inner cylinder has a volume of:

(2) $! V_i = \pi r_i^2 h$

So the radius of the outer cylinder is the radius of the inner cylinder plus the width of the wave:

(3) $! r_o = r_i + w$

which means that the volume of the outer cylinder is:

(4) $! V_o = \pi (r_i + w)^2 h$

So now we can figure out the volume of the wave, which is the volume of the outer cylinder minus the volume of the inner cylinder:

(5) $! V_w = V_o - V_i$

(6) $! V_w = \pi (r_i + w)^2 h - \pi r_i^2 h$

Now to simplify, let’s expand the first term on the right side of the equation:

(7) $! V_w = \pi (r_i ^2 + 2 r_i w + w^2) h - \pi r_i^2 h$

Now let’s collect terms:

(8) $! V_w = \pi h \left( (r_i ^2 + 2 r_i w + w^2) - r_i^2 \right)$

Take away the inner parentheses:

(9) $! V_w = \pi h (r_i ^2 + 2 r_i w + w^2 - r_i^2)$

and subtract similar terms to get the equation:

(10) $! V_w = \pi h ( 2 r_i w + w^2 )$

Volume of the wave

Now we can just plug in our estimates of width and height to get the volume of water in the wave. We’re going to assume, later on, that the volume of water does not change as the wave propagates across the ocean.

(11) $! V_w = \pi (14) ( 2 r_i (1000) + (1000)^2 )$

rearrange so the coefficients are in front of the variables:

(12) $! V_w = 14 \pi ( 2000 r_i + 1000000 )$

So, at 80 km, the volume of water in our wave is:

(13) $! V_w = 14 \pi ( 2000 (80000) + 1000000 )$

(14) $! V_w = 7081149841 m^3$

Height of the Tsunami

Okay, now we want to know what the height of the tsunami will be at any distance from the epicenter of the earthquake. We’re assuming that the volume of water in the wave remains the same, and that the width of the wave also remains the same. The radius and circumference will certainly change, however.

We take equation (10) and rearrange it to solve for h by first dividing by rearranging all the terms on the right hand side so h is at the end of the equation (this is mostly for clarity):

(15) $! V_w = \pi ( 2 r_i w + w^2 ) h$

Now we can divide by all the other terms on the right hand side to isolate h:

(16) $! \frac{V_w}{\pi ( 2 r_i w + w^2 )} = \frac{\pi ( 2 r_i w + w^2 ) h}{\pi ( 2 r_i w + w^2 )}$

so:

(17) $! \frac{V_w}{\pi ( 2 r_i w + w^2 )} = h$

which when reversed looks like:

(18) $! h = \frac{V_w}{\pi ( 2 r_i w + w^2 )}$

This is our most general equation. We can use it for any width, or radius of wave that we want, which is great. Anyone else who wants to calculate wave heights for other situations would probably start with this equation (and equation (15)).

Double checking our algebra

So we can now figure out the height of the wave at any radius from the epicenter of the earthquake. To double check our algebra, however, let’s plug in the volume we calculated, and the numbers we started off with, and see if we get the same height (14 m).

First, we’ll use all our initial approximations so we get an equation with only two variables: height (h) and radial distance (r_i). Remember our initial conditions:

w = 1000 m
r_i = 80,000 m
h_i = 14 m

we used these numbers in equation (10) to calculate the volume of water in the wave:

V_w = 7081149841 m³

Now using these same numbers in equation (18) we get:

(19) $! h = \frac{7081149841}{\pi ( 2 (r_i) (1000) + (1000)^2 )}$

which simplifies to:

(20) $! h = \frac{7081149841}{ 2000 r_i \pi + 1000000 \pi }$

So, to double-check we try the radius of 80 km (80,000 m) and we get:

h = 14 m

Aha! it works.

Across the Pacific

Now, what about Hawaii? Well it’s about 6000 km away from the earthquake, so taking that as our radius (in meters of course), in equation (20) we get:

(21) $! h = \frac{7,081,149,841}{ 2,000 (6,000,000) \pi + 1,000,000 \pi }$

which is:

h = 0.19 m

This is just 19 cm!

All the way across the Pacific, Lima, Peru, is approximately 9,000 km away, which, using equation (20) gives:

h = 0.13 m

So now I’m curious about just how fast the 14 meters drops off to less than 20 cm. So I bring up Excel and put together a spreadsheet of tsunami height at different distances. Plot on a graph we get:

So the height of the tsunami drops off relatively fast. Within 1000 km of the earthquake the height has dropped by 90%.

How good is this model

This is all very nice, a cute little exercise in algebra, but is it useful? Does it come anywhere close to reality? We can check by comparing it to actual measurements; the same ones used by NOAA to compare to their model (see here).

tsunami-model-data-honolulu — The red line is the tsunami's water height predicted by the NOAA computer models for Honolulu, Hawaii, while the black line is the actual water height, measured at a tidal gauge. Other comparisons can be found here.

680_20110311-TsunamiWaveHeight — Tsunami wave heights in the Pacific, as modeled by NOAA. Notice how the force of the tsunami is focused across the center of the Pacific.

The graph shows a maximum height of about 60 cm, which is about three times larger than our model. NOAA’s estimate is within 20% of the actual maximum heights, but they’ve spent a bit more time on this problem, so they should be a little better than us. You can find all the gruesome details on NOAA’s Center for Tsunami Research site’s Tsunami Forecasting page.

Notes

1. The maximum height of a tsunami depends on how much up-and-down motion was caused by the earthquake. ScienceDaily reports on a 2007 article that tried to figure out if you could predict the size of a tsunami based on the type of earthquake that caused it.

2. Using buoys in the area, NOAA was able to detect and warn about the Japanese earthquake in about 9 minutes. How do they know where to place the buoys? Plate tectonics.

buoys-warning — The locations of the buoys in NOAA's tsunami warning system.

Update

The equations starting with (7) did not have the 2 on the r_iw term. That has been corrected. Note that the numerical calculations were correct so they have not changed. – Thanks to Spencer and Claude for helping me catch that.

Volcanic eruption in Japan: Shinmodake

shinmoedake-eruption — Shinmodake Volcano in southern Japan (center). This picture predates the big earthquake. Image from NASA Earth Observatory: Shinmoe-dake Volcano Erupts on Kyushu..

The Shinmoedake Volcano erupted on January 19th after being dormant for two years, however, two days after the big Japanese earthquake, it began spewing ash once again. The two are not necessarily connected.

Volcanos and convergent margins go together. Typically, the plate being subducted melts as it is pushed deeper into the Earth and temperatures rise. It also helps that the water in the crust and sediment of the subducting plate makes it easier to melt, and makes the resulting magma much more volatile and explosive.

convergence-japan-volcanoes — The subducting plate melts producing volatile magma.

But although Shinmoedake is in Japan, it is not on the same tectonic boundary as the earthquake. The northern parts of Japan are where the Pacific Plate is being subducted beneath the Okhotsk Plate. This volcano is connected to the subduction of the Philippine Plate to the south.

japan-subduction-volc — The large earthquake's epicenter and the Shinmoedake volcano are on different plate margins. Image adapted from Wikimedia Commons user Sting.

This does not necessarily mean that the two occurrences are totally unrelated. Seismic waves from the big earthquake could have been enough to incite magma chambers that were just about ready to blow anyway.

The map below is centered on the series of craters in the region of the erupting volcano.

View Larger Map

Plate Tectonics and the Earthquake in Japan

The magnitude 8.9 earthquake that devastated coastal areas in Japan shows up very clearly on the United States Geologic Survey’s recent earthquake page.

earthquake-japan — The big red square marks an aftershock of the magnitude 8.9 earthquake off Japan. (Image via USGS). Note that most of the earthquakes occur around the edge of the Pacific Ocean (and the Pacific Plate).

Based on our studies of plate tectonics, we can see why Japan is so prone to earthquakes, and we can also see why the earthquake occurred exactly where it did.

View Larger Map

The obvious trench to the east and the mountains and volcanoes of the Japanese islands indicate that this is a convergent margin. The Pacific plate is moving westward and being subducted beneath the northern part of Japan, which is on the Okhotsk Plate.

japan-plates — The tectonic plates and their boundaries surrounding Japan. The epicenter of the earthquake is along the convergent margin where the Pacific Plate is being subducted beneath the Okhotsk Plate. Image adapted from Wikimedia Commons user Sting.

The epicenter of the earthquake is on the offshore shelf, and not in the trench. Earthquakes are caused by breaking and movement of rocks along the faultline where the two plates collide.

In cross-section the convergent margin would look something like this:

convergence-japan — Diagram showing the tectonic plate movement beneath Japan. Note the location of the earthquake is beneath the offshore shelf and not in the trench.

The shaking of the sea-floor from the earthquake creates the tsunamis.

So where are there similar tectonic environments (convergent margins)? You can use the Google Map above to identify trenches and mountain ranges around the world that indicate converging plates, or Wikimedia Commons user Sting’s very detailed map, which I’ve taken the liberty of highlighting the convergent margins (the blue lines with teeth are standard geologists’ markings for faults and, in this case, show the direction of subduction):

Tectonic_plates_boundaries_detailed-en-conv — Convergent plate boundaries (highlighted blue lines) shown on a world map of tectonic boundaries. The blue lines with teeth are standard geologic symbols for faults, with the teeth showing the direction of the fault underground. Image adapted from Wikimedia Commons user Sting.

The Daily Dish has a good collection of media relating to the effects of the quake, including footage of the tsunami inundating coastal areas.

Cars being washed away along city streets:

Our thoughts remain with the people of Japan.

UPDATES:

1. Alan Taylor has collected some poignant pictures of the flooding and fires caused by the tsunami and earthquake. TotallyCoolPix has two pages dedicated to the tsunami so far (here and here).

2. Emily Rauhala summarizes Japan’s history of preparing for this type of disaster. They’ve done a lot.

3. Mar 12, 2011. 2:10 GMT: I’ve updated the post to add the map of the tectonic plates surrounding Japan.

4. A CNN interview that includes video of the explosion at the Fukushima nuclear power plant (my full post here).

5. NOAA has an amazing image showing the tsunami wave heights.

They also have an excellent animation showing the tsunami moving across the Pacific Ocean. (My post with more details here).

6. The United States Geological Survey (USGS) put out a podcast on the day of the earthquake that has interviews with two specialists knowledgeable about the earthquake and the subsequent tsunami, respectively. Over 250 kilometers of coastline moved in the earthquake which is why the tsunami was so big. They also have a shakemap, that shows the area affected by the earthquake.

shake-map — USGS ShakeMap for the earthquake. Image via the USGS.

7. ABC News (Australia) and Google have before and after pictures.

8. The University of Hawaii has a page about, Why you can’t surf a tsunami.

9. A detailed article on earthquake warning systems, among which, “Japan’s system is among the most advanced”, was recently posted in Scientific American.

10. Mar 15, 2011. 9:15 GMT: I’ve added a map of tectonic boundaries highlighting convergent margins.

11. The Shinmoedake Volcano erupted two days after the earthquake, but they may be unrelated.

guardian-reactor-status — Fukushima reactor status as of March 16th, 5:00 pm GMT from the Guardian live blog.

12. The Guardian’s live blog has good, up-to-date information on the status of the nuclear reactors at Fukushima.

POV-Ray: 3d rendering

Giles Tran’s amazing rendering of glasses on the counter inspired me to check through my own POV-Ray generated library. Nothing nearly as good, but some of it is still might be useful.

Axial_tilt — Demonstrating the axial tilt of the Earth, this image shows the Earth at the northern hemisphere's summer solstice.

120px-ElipticN_000_2xAplaneClose — Rotating Earth at the northern hemisphere's winter solstice.

You build 3d models in POV-Ray and then export 2d images from whichever point of view you want, so once you have the model set up you can easily change the perspective or even move objects to create animations.

POV-Ray does not have any useful sort of user interface; you’re usually creating your models with computer code. It can therefore be challenging to use, and, as with any 3d programming language, a bit of geometry, trigonometry and algebra are needed.

However, the final results can be impressive. I’m continually amazed each year by the quality of the work added to their Hall of Fame.

For much easier, quicker and not so sophisticated 3d results, I use VPython, which is also a great way to learn programming that outputs 3d images.

City in the sandbox

sandbox-city-diagram — The City of Apolypse.

My small group that had trouble getting SimCity to behave itself on the laptop decided to go build their city in the sandbox instead.

They had just looked through all the civic buildings and zoning options before they took the outside option, so they started with SimCity’s basic introduction to urban planning concepts.

The group chose to locate their city on the ocean, with a river. Previously, when the class had looked up and down the U.S.’s eastern seaboard in Google Maps, we’d noticed that most of the bigger cities, like New York and Charleston were on or near estuaries. (We’d also noticed that most of the cities were protected by some sort of barrier from the direct influence of the oceans.)

[googleMap name=”New York City” description=”NYC on the river and ocean.” width=”480″ height=”400″ mapzoom=”8″ mousewheel=”false”]New York City[/googleMap]

This group gained some significant advantages over just playing the computer game because the sandbox model allowed them create features not built into the game.

In particular, they sculpted an earthen dam with a hydroelectric power plant, that was the centerpiece of their city.

By putting a dam across the estuary they could acquire both fresh water reservoir and hydroelectric power.

It’s probably not unfair to guess that the idea for the dam came primarily from our visit to the Pickwick Landing Hydroelectric Plant last year. I say so because the eight grader who came up with the idea was reminiscing about last year’s immersions for the rest of the day.

apocalypse-150g — The decline and fall of Apocalypse.

The group did a great job, although they did site their landfill upstream of their reservoir. This became a problem because after they presented to the class they turned on the river. We relearned the biblical lesson about not building on the sand. This was not entirely unexpected though; the students had named the city Apocalypse.

The combination of computer simulation and physical model really worked well. So much so that two years from now, when I do this again, I think I’ll require at least one group to do the physical model. But it really worked for them to have at least seen the computer game so I’ll have to build that into the project too.