# Plate Tectonics on the Eminence Immersion

#### October 13, 2014

The picture of a convergent tectonic boundary pulls together our immersion trip to Eminence, and the geology we’ve been studying this quarter. We saw granite boulders at Elephant Rocks; climbed on a rhyolite outcrop near the Current River; spelunked through limestone/dolomitic caverns; and looked at sandstone and shale outcrops on the road to and from school.

The subducting plate melts producing volatile magma.

Citing this post: Urbano, L., 2014. Plate Tectonics on the Eminence Immersion , Retrieved April 20th, 2018, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

# Plate Tectonics and the Earthquake in Japan

#### March 11, 2011

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.

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.

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.

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:

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):

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.

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.

Tsunami wave heights modeled by NOAA. Note the colors only go up to 2 meters. The maximum wave heights (shown in black in this image), near the earthquake epicenter, were over 6 meters.

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.

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.

Shinmoedake Volcano.

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

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.

Citing this post: Urbano, L., 2011. Plate Tectonics and the Earthquake in Japan, Retrieved April 20th, 2018, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

# Geothermal energy and plate tectonics

#### April 26, 2010

Major tectonic plates (from USGS).

Seafloor topography around the Hawaiian Hotspot (from NOAA)

The question came up about where are good places for geothermal energy, and the answer, of course, was to introduce plate tectonics. It was a quick introduction, and a refresher for the 8th graders, but the interest was there and it seemed impactful.

It also provided a link to talk about the Icelandic volcano that’s been disrupting air traffic in Europe. NASA has an amazing picture of the eruption on its Picture of the Day for April 19th.

Google Maps is a great tool for showing features like the mid-ocean ridges (use the satellite view), zooming in and out of the mountain ranges, tracing the Hawaii hotspot and watching East Africa split apart.

Citing this post: Urbano, L., 2010. Geothermal energy and plate tectonics, Retrieved April 20th, 2018, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

# Assessment with the Toilet Paper Timeline of Earth History

#### December 17, 2012

With a larger class, and quite a bit of space in the gym, I had more flexibility working on the toilet paper timeline compared to the last time.

Labeling the timeline in the gym.

I built in a friendly race to see which group could find a set of events first, and allowed me to highlight nine different, important, series of events along the timeline.

The adapted spreadsheet, racing sequences, and a short summative quiz are on this Toilet Paper Timeline spreadsheet.

I broke the class up into 4 groups of 4, and each group created their own timeline based on a handout.

Groups of students lay out their toilet paper timelines. Post-it notes were used to label the events.

Then, I gave each group a slip of paper with four events on it (one event per student), and they had to race to see which group would be first to get one person to each event on the list. Once each group got themselves sorted out, I took a few minutes to talk about why the events were important and how they were related.

Table 1: The series of events.

1) We’ll be talking about plate tectonics soon, so it’s good for them to start thinking about the timing of the formation and breakup of the supercontinents.
Event 1 Event 2 Event 3 Event 4
Formation of Rodinia (supercontinent) Breakup of Rodina Formation of Pangea Breakup of Pangea
2) This sequence emphasizes the fact that most free oxygen in the atmosphere comes from ocean plants (plankton especially), and that a lot of free atmospheric oxygen was needed to to form the ozone layer which protected the Earth’s surface from uv radiation, which made the land much more amenable to life. Also, trees came way after first plants and oxygen in the atmosphere.
Event 1 Event 2 Event 3 Event 4
First life (stromatolites) Oxygen buildup in atmosphere First land plants First Trees
3) Pointing out that flowering plants came after trees.
Event 1 Event 2 Event 3 Event 4
First life First land plants First trees First flowering plants
4) The Cambrian explosion, where multicellular life really took off, happened pretty late in timeline. Longer after the first life and first single-celled animals.
Event 1 Event 2 Event 3 Event 4
First life (stromatolites) First animals First multicelled organisms Rise of multicelled organisms
5) Moving down the phylogenetic tree from mammals to humans shows the relationship between the tree and evolution over time.
Event 1 Event 2 Event 3 Event 4
First mammals First Primates Homo erectus Homo sapiens
6) More tectonic events we’ll be talking about later.
Event 1 Event 2 Event 3 Event 4
Opening of the Atlantic Ocean Linking of North and South America India collides with Asia Opening of the Red Sea
7) Pointing out that life on land probably needed the magnetic field to protect from the solar wind (in addition to the ozone layer).
Event 1 Event 2 Event 3 Event 4
Formation of the Earth First life Formation of the Magnetic Field First land plants
8) Fish came before insect. This one seemed to stick in students’ minds.
Event 1 Event 2 Event 3 Event 4
First Fish First Insects First Dinosaurs First Mammals
9) Mammals came before the dinosaurs went extinct. This allowed a discussion of theories of why the dinosaurs went extinct (disease, asteroid, mammals eating the eggs, volcanic eruption in Deccan) and how paleontologists might test the theories.
Event 1 Event 2 Event 3 Event 4
First Dinosaurs First Mammals Dinosaur Extinction First Primates

The whole exercise took a few hours but I think it worked out very well. The following day I gave the quiz, posted in the excel file, where they had to figure out which of two events came first, and the students did a decent job at that as well.

Citing this post: Urbano, L., 2012. Assessment with the Toilet Paper Timeline of Earth History, Retrieved April 20th, 2018, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

# Inverse Relationships

#### October 16, 2012

Inverse relationships pop-up everywhere. They’re pretty common in physics (see Boyle’s Law for example: P ∝ 1/V), but there you sort-of expect them. You don’t quite expect to see them in the number of views of my blog posts, as are shown in the Popular Posts section of the column to the right.

Table 1: Views of the posts on the Montessori Muddle in the previous month as of October 16th, 2012.

Post Post Rank Views
Plate Tectonics and the Earthquake in Japan 1 3634
Global Atmospheric Circulation and Biomes 2 1247
Equations of a Parabola: Standard to Vertex Form and Back Again 3 744
Cells, cells, cells 4 721
Salt and Sugar Under the Microscope 5 686
Google Maps: Zooming in to the 5 themes of geography 6 500
Market vs. Socialist Economy: A simulation game 7 247
Human Evolution: A Family Tree 8 263
Osmosis under the microscope 9 219
Geography of data 10 171

You can plot these data to show the relationship.

Views of the top 10 blog posts on the Montessori Muddle in the last month (as of 10/16/2012).

And if you think about it, part of it sort of makes sense that this relationship should be inverse. After all, as you get to lower ranked (less visited) posts, the number of views should asymptotically approach zero.

## Questions

So, given this data, can my pre-Calculus students find the equation for the best-fit inverse function? That way I could estimate how many hits my 20th or 100th ranked post gets per month.

Can my Calculus students use the function they come up with to estimate the total number of hits on all of my posts over the last month? Or even the top 20 most popular posts?

Citing this post: Urbano, L., 2012. Inverse Relationships, Retrieved April 20th, 2018, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

# The Geology of St. Albans, Missouri

#### March 15, 2012

The area around the Fulton School has just two types of geology: young, floodplain sediments; and old limestone bedrock.

The geology of St. Albans consists of young floodplain sediments (yellow), and old limestone bedrock (blue). Image adapted from the USGS.

• Missouri River Flood Plain Sediments:
• The flat area next to the Missouri River that would get flooded regularly if the rivers weren’t regulated)
• Holocene (last 10,000 years)
• Clays and Silts (mud) deposited when the river floods.
• Bedrock. Mostly limestone:
• Can be found outcropping on the hills.
• Mississippian Limestones (USGS ref.) (330-360 million years old.): found on some of the hilltops.
• Ordovician Dolomites and Limestones (USGS ref.) (435-500 million years old)

The geology of St. Albans consists of young floodplain sediments (yellow), and old limestone bedrock (blue). Image adapted from the USGS.

# Geologic History

## The continents form

To reconstruct the geologic history, we can start a bit deeper, with the fact that we’re sitting in the middle of a continent, which means that if you drill deep enough you’ll get to some of the original, granitic rocks that formed just after the crust of the Earth cooled — about four and a half billion years ago.

The froth that floats on top of the boiling jam is a bit like the continental crust.

The continental crust is a bit like the froth that forms on moving water (or the top of boiling jam), and just like froth it tends not to want to sink. So there’s some pretty old continental crust beneath the continents.

However, also just like froth on water, the continental crust is pushed around on the surface of the Earth. This is called continental drift (which is part of the theory of plate tectonics). Sometimes, the continental crust can split apart, making space for seas and oceans between the drifting continents, and causing parts of the continent to subside beneath the oceans.

At other times, such as when two continents collide, they can push each other up to mountains out of areas that were once seas.

And that’s how we ended up with limestone rocks in the middle of Missouri.

## Forming Limestone Rocks (Ordovician)

Five hundred million years ago (500,000,000 years ago) the continents were in different places, and Missouri was under a shallow part of the Iapetus Ocean.

The location of Missouri 458 million years ago. Image from: "Plate tectonic maps and Continental drift animations by C. R. Scotese, PALEOMAP Project (www.scotese.com)"

Many of the micro-organisms that lived in that ocean made shells out of calcium carbonate.

100 million year old, calcium carbonate shell (from Coon Creek).

When you accumulate billions of these shells over the course of millions of years, and then bury them, compress them, and even heat them up a bit, you’ll end up with a rock made of calcium carbonate. We call that type of rock: limestone.

Limestone outcrop on St. Albans Road (Ordovician).

## Emerging from the Oceans: The Formation of Pangea.

The Mississippian limestone rocks formed in the same way, but about 360 million years ago. Why is there a gap between the Ordovician rocks (450 million years ago) and the Mississippian ones? Good question. You should look it up (I haven’t). There may have been rocks formed between the two times but they may have been eroded away.

I can make a good guess as to why there are no limestone rocks younger than about 300 million years old, however. At that time the continents, which had been slowly sidling toward each other, finally collided to form a super-continent called Pangea.

What would become North America (called Laurentia), ran into the combined South America/Africa continent (called Gondwana) pushing up the region, and creating the Ozarks and Appalachian Mountains.

Laurentia collides with Gondwana. Image from "Plate tectonic maps and Continental drift animations by C. R. Scotese, PALEOMAP Project (www.scotese.com)"

And that’s the story the geology can tell.

# References

The USGS has good, detailed, interactive maps of the geology of the states in the US.

A nice geologic map of St. Louis County can be found here.

A geologic time scale from the USGS.

The geologic time scale. From the USGS via Wikipedia.

Citing this post: Urbano, L., 2012. The Geology of St. Albans, Missouri, Retrieved April 20th, 2018, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

# The Freezing Core Keeps the Earth Warm

#### August 30, 2011

The internal structure of the Earth.

The inner core of the Earth is made of solid metal, mostly iron. The outer core is also made of metal, but it’s liquid. Since it formed from the solar nebula, our planet has been cooling down, and the outer core has been freezing onto the inner core. Somewhat counter-intuitively, the freezing process is a phase change that releases energy – after all, if you think about it, it takes energy to melt ice.

The energy released from the freezing core is transported upward through the Earth’s mantle by convection currents, much like the way water (or jam) circulates in a boiling pot. These circulating currents are powerful enough to move the tectonic plates that make up the crust of the earth, making them responsible for the shape and locations of the mountain ranges and ocean basins on the Earth’s surface, as well as the earthquakes and volcanics that occur at plate boundaries.

Conceptual drawing of assumed convection cells in the mantle. (via The Dynamic Earth from the USGS).

Eventually, the entire inside of the earth will solidify, the latent heat of fusion will stop being released, and tectonics at the surface will slow to a stop.

The topic came up when we were talking about the what heats the Earth. Although most of the energy at the surface comes from solar radiation, students often think first of the heat from volcanoes.

Note: An interesting study recently published showed that although the core outer core is mostly melting, in some places it’s freezing at the same time. Unsurprising given the convective circulation in the mantle.

Model of convection in the Earth's mantle. Notice that some areas on the mantle are hotter, creating hot plumes, and some are cooler (image from Wikipedia).

Note 2: Convection in the liquid outer core is what’s responsible for the Earth’s magnetic field, and explains why the magnetic polarity (north-south) switches occasionally. We’ll revisit this when we talk about electricity and dynamos.

Citing this post: Urbano, L., 2011. The Freezing Core Keeps the Earth Warm, Retrieved April 20th, 2018, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

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

#### March 24, 2011

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.

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, ri 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 (ri). Remember our initial conditions:

w = 1000 m
ri = 80,000 m
hi = 14 m

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

Vw = 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:

Tsunami heights with distance from earthquake, assuming a circular wall of water.

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).

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.

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.

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

# Update

The equations starting with (7) did not have the 2 on the riw 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.

Citing this post: Urbano, L., 2011. Tsunami Geometry: Calculating the Height of a Tsunami using Basic Geometry, Retrieved April 20th, 2018, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.