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Michael Lydon

F09 Midterm Paper

Introduction to and Conceptualizing the
Study of Mathematics in Vulcanism

It
was August 26th 1883. The awesome power of earth was unleashed. The
small, remote island of Krakatoawould make all the
citizens of the earth respect her. She took the lives of thousands in seconds;
those close enough to gaze upon her in that moment had their corneas cauterized
by the blinding light. The explosion according to Deborah Hopkinson
(2004), was the “loudest sound you've ever
heard. Multiplied by thousands.” “A sound
so deafening, an explosion so terrifyingly loud, people more than 2,000 miles
away recorded hearing it” (Hopkinson, 2004).
Literally 20 cubic kilometers of ash and pumice were flung into the air within
hours (Scarth, 1999). The resulting tsunamis killed
tens more thousands; in several different countries, hundreds of miles away.
“The explosive force of Krakatoa was equal to
13,000 times the nuclear yield of the Little Boy bomb that devastated
Hiroshima, Japan during World War II;  and Average global temperatures
fell by as much as 1.2 degrees Celsius in the year following the eruption.
Weather patterns continued to be chaotic for years and temperatures did not
return to normal until 1888” (Wikipedia, 2009).
Scary, huh?

Volcanoes
are one of the few natural disasters that have been able to humble mankind time
and time again. The natural phenomena they create is
absolutely fascinating. Although this essay may have first appeared to be about
the eruption of Krakatoa, I was simply giving my
audience a taste of the wonderfully violent nature of volcanoes; as well as
secretly incorporating several mathematical statistics about them. After all
title of this research paper is called Introduction to and Conceptualizing
the Study of Mathematics in Vulcanism.

The
study of volcanoes is brimming with math. Volcanoes make up just one part of
the geological processes of the earth, but they are particularly interesting to
scientists. Where else on the earth do they get to study the content from the
center of the earth; and watch the very formation of rocks? Where else do they
get to the see a major, natural instrument of climate change?  As you can imagine, all these topics
could be simplified and explained through math and physics. In this paper, my
goal is to create an expository that provides entertaining information about
volcanoes and discusses three mathematical concepts involved with vulcanism; Bernoulli's Equation, The Modified Bernoulli's
equation (that relates to volcanoes) and the Volcanic Explosive Index (V.E.I.).

Image from:
http://education.ed.pacificu.edu/sweb/gutierrez/volcano/index.html

When
we see an eruption, what do we think of? Probably most of us would think of
something like we see pictured above. In reality, there are very many types of
eruptions. Some are quite explosive like the island
of Krakatoa.
Yet some are more calm like Hawaii,
where lava is large in quantity but lacks the large explosions. Here in California,
I  live on
what is called a subduction zone; these are the areas
where the most frequent volcanic activity occurs. This area is a hotbed of
volcanic activity because we live on the verge of two tectonic plates that are
shearing against each other. When this happens magma (what lava is classified
when underground) percolates up between the gapped areas of the moving tectonic
plates. This forms a noticeable chain of what are called
“hot-spots” all down the are of the
Pacific coast. This is actually only one stretch of a larger continuation of
hotspots that exist all around the Pacific Plate. Because of the frequent
volcanic activity around this roughly circular area,  it has become cleverly known as
“The Ring of Fire.”

Many
other types of hot-spots exist that are not concentrated along the lines of
plates. Most of the islands that are in the middle of the ocean are formed by
volcanic activity. According to Burke and Wilson out of the 122  known active hot-spots, 53 exist
on the ocean basins (1976). In the ocean the Earth's crust is much more thin than on the continents and lava frequently seeps
out to form giant underground mountains. The Hawaiian Islands
were formed this way, they are part of a traveling hot
spot that (going back in time) originally formed not only the Hawaiian chain
but all of the Northwestern (Leeward) Islands. Islands
such as Midway were probably just as big and volcanically active as the big island
of Hawaii.

Speaking
of activity, let's get started on explaining how the more explosive eruptions
work with regards to mathematics. Over time, a hot-spot will build up some type
of pressure in the form of a liquid (the magma) on a solid (the crust). Andrea Steffke (2003) describes the three processes that happen
during an eruption-

An
explosive eruption involves three different processes: fragmentation of the
magma; blasting             of
the fragmented mass through the vent to the surface, and the ascent of the
eruption column.           An
explosive eruption will only take place if the pressure within the magma
exceeds the           strength
of the surrounding rock. These explosive eruptions are driven by the thermal
energy           stored
in the magma. The thermal energy is transferred into kinetic energy of the
eruption           column
through the expansion of gases into growing vesicles (air bubbles).

In this situation we are discussing
how a mass of molten fluid is ascending through the main vent of a volcano,
called a pipe; and out of the hole on top called a vent. The magma has to have
some type of pressure build up to form an explosion. This occurs when the
molten hot magma hardens as it comes up through the vent, hardens and rapidly
decompresses (fragmentation).

I
can begin explaining how this process happens, by using a differential equation
to describe
concept of fluid dynamics. This will give us understanding on a
basic level how the functions of a volcano are sometimes similar to common
household items.

Image From:
http://www.princeton.edu/~asmits/Bicycle_web/Bernoulli.html

This is called Bernoulli's
equation. It is a representation of the law describing pressure and fluid. The
law states “The pressure in a fluid decreases as the speed of the fluid
increases” (Hewitt, 1993). Let us use the analogy of a watering hose to
describe the equation. When the nozzle at the end has no valve the water, when
turned on, will squirt out normally. When a pressure washing valve is added,
the water condenses tightly. The more pressure behind the liquid, and the
narrower the valve at the end, the more quickly the water will flow out. This
is facilitated because of the rapid decrease of pressure when the water is
ejected out of a narrow opening.

The
equation that we see here is designed to represent the previous analogy. We can
translate the above formula as: Pressure + ½ of Density and
Velocity² + Density and Gravitational Acceleration and Elevation =
Constant Flow Rate. This is a very controlled way to describe any of the
variables listed in the formula. Bernoulli's equation is very practical for
many industries, and is used in urban utilities and such. But for our case this
basic equation will just not do. Volcanoes introduce some new variables that are
very different from water flowing out of a hose. Since we have conceptualized
the behavior of pressure flow in a condensed area; we can move on to a more
specialized differential equation that was designed specifically for volcanoes.

Volcanoes
are dramatically diverse, because the process of the lava being ejected can
vary radically. In the case of Krakatoa, there was a
massive build up of pressure and a very narrow vent. Like
prying out a cork out of a bottle of Champagne.

Image From:
http://ffden-2.phys.uaf.edu/212_fall2003.web.dir/Andrea_Steffke/eruptions.htm

Just like a bottle of Champagne
however, there are additional forces that act on the rapid ejection of liquid.
The equation above is an evolution of the Bernoulli's equation designed by
reputed vulcanologist Lionel Wilson. It includes
several additional variables that show up in volcanic eruptions.

US-
Represents Ejection Velocity

Pi- Represents the pressure in gas.

PS
– Represents Atmospheric Pressure

SC-
Represents Magma Density

Let
us introduce the first variable in Wilson's
Equation- gas pressure. Pressure, as in a bottle of soda or Champagne
can increase the velocity and force behind a liquid. This is the same for
volcanoes. Gas can be in dissolved form within a liquid, just like carbon
dioxide in soda. And when the pressure is released it can expand quite rapidly.
Krakatoa was an example of an eruption that had a lot
of gas as the driving force of the huge explosion.

The second part of the equation is atmospheric
pressure. Atmospheric pressure is normally defined, according to the Physics Hypertextbook, (2009) as the ratio of force applied per area covered or P=F/A
(Anonymous). So basically, the more force over the liquid magma and the more
concentrated area it is in determine how much pressure it has exerted on it. Mauna
Loa in Hawaii has a
large reservoir of lava, and is, in fact the largest volcano in the world. But,
because it's magma is not in a very concentrated area
(and pressure very low), it does not erupt nearly as violently as some of the
other smaller volcanoes. Instead it flows out very softly, layering over time
to increase the size of
Hawaii.

The
third variable in Wilson's equation
is Magma Density. This factor is very important to Vulcanologists,
as well as geologists. It is determined by the type of molten rock that the
magma is composed of. It varies radically, volcano to volcano, how this factor
will affect the equation. When the lava from a volcano cools, crystallizes and
becomes solid, it is classified as an igneous rock (Christopherson,
2006). Igneous magma before it becomes a rock is either classified as Felsic or Mafic. Felsic is lighter and less dense, Mafic has a higher density, and is more dense (Christopherson). These characteristics, in addition to the
temperature that they are; will determine what kind of an effect magma density
will have on the eruption.

So
what do all these variables add up to? The answer is the boom! They show how
fast the tephra (ash, rocks, dust, lava etc...) will
be ejected from the volcano. “So now what?”
You might ask; well we can now move on to another exciting mathematical
concept!  Vulcanologists
have conceived a way to actually describe the size of the eruption (or boom! If you prefer).

Developed by C.G. Newhall, the V.E.I. Of Volcanic Explositivity Index that measures size of an eruption.

Image From: http://www.volcano.si.edu/world/vei.jp

Quoted from the United
States Geological Survey (2009):

The
Volcanic Explosivity
Index, or VEI, was proposed in 1982 as a way to describe the
relative         size
or magnitude of explosive volcanic eruptions. It is a 0-to-8 index of
increasing explosivity.           Each increase
in number represents an increase around a factor of ten. The VEI uses several     factors to assign a number,
including volume of erupted pyroclastic material (for
example,        ash
fall, pyroclastic flows, and other ejecta), height of eruption column, duration in hours, and   qualitative descriptive terms.

This scale is mathematically
relevant, because it can be used to help develop theoretical models of
eruptions and their effects on the world. It uses statistics of past eruptions
to determine what if scenarios for future ones. Now with modern technology
scientists can develop early warning systems and evacuation plans to aid in
getting people to safety.

There
is also an algebraic formula that deals with the volume of earth an eruption
displaces. Based on the quantity (volume) of ejecta
put into the sky by a volcano we can surmise the effect it will have on the
immediate area under the ash plume. This is very useful because the ash plume
can influence many things; including effects on the environment, weather and
also the people who live around the volcano. Vulcanologists
study the cone shaped areas where past ash flows have fallen as well. These can
show us what kind of dispersion a future eruption may have.

Scientists
also use the objective visual measurements to determine the “Qualitative
Description” of the eruption event (Taken from above image). Throughout
(written) human history, there has never been an eruption that was threatening
on the scale past a number seven on the V.E.I. classification scale. However,
there has been many past events that have easily topped
that. The Toba eruption that took place around
seventy-thousand years ago was over twenty times the size as Krakatoa. This eruption had consequences that, at the time,  had threatened
the existence of humanity. At a time when the earth was already in  period of
global cooling; the resulting amount of tephra
released by“The Toba
Catastrophe” most likely increased the severity of the already taxing
conditions that threatened early humanity. “This massive environmental
change created population bottlenecks in the species that existed at the time;
this in turn accelerated differentiation of the isolated human populations,
eventually leading to the extinction of all the other human species except for
the two branches that became Neanderthals and modern humans”  (Taken From-"Toba
Catastrophe Theory").

Volcanoes
through out human history have had a wide range of effects on our world. They
never fail to captivate people, and will ultimately continue to shape our
history.  In this essay we have
discussed Bernoulli's equation, the Modified Bernoulli's equation (Volcano
Equation) and the Volcanic Expositivity Index.
Hopefully the reader has gained a new appreciation for volcanoes and has a good
idea of how mathematics translate into how they
function. We should never underestimate the effect volcanoes can have on our
world; and should always respect the fact that the next major eruption can  change how
we live.

Works Cited Page (APA)

Books:

Burke, K., & Wilson, J. (1976).
Hot spots on the earths surface. Scientific
American: volcanoes and the    earth's
interior, 3-31.

Christopherson, R.W. (2006). Geosystems, sixth edition.. Upper Saddle
River, New Jersey: Pearson          Prentice
Hall.

Hewitt, P.G. (1993). Conceptual physics.

New
York, N.Y.: HarperCollins
College Publishers.

Scarth, Alwin.
(1999). Vulcans fury: man against the
volcano.

New
Haven and London:
Yale University
Press.

Sigurdsson, H., Houghton, B., McNutt, S., Rymer,
H., & Stix, J. (2000). Encyclopedia
of volcanos.

San Diego,
United States of America:
Academic Press.

Websites:

Anonymous. (1998-2009).
Viscosity. Retrieved from

http://physics.info/viscosity/

Anonymous. (n.d.). Krakatoa. Wikipedia.org. Retrieved (2009, October 19) from     http://en.wikipedia.org/wiki/Krakatoa

Steffke, Andrea. (2003). Physics of volcanoes. Retrieved from

http://ffden-2.phys.uaf.edu/212_fall2003.web.dir/Andrea_Steffke/table.htm

U.S.G.S., . (n.d.). VHP photo glossary: vei.
Retrieved from   http://volcanoes.usgs.gov/images/pglossary/vei.php

Various authors, Initials. (n.d.). Volcano
and Hydrologic monitoring techniques. Retrieved from             http://vulcan.wr.usgs.gov/Monitoring/techniques.html

(n.d.). Toba Catastrophe
Theory. Wikipedia.org. Retrieved (2009, October 23) from             http://en.wikipedia.org/wiki/Toba_catastrophe_theory

Online Articles:

Hopkinson, D. (2004, January). Witness account. Storyworks,
11(4), Retrieved from             http://teacher.scholastic.com/activities/wwatch/volcanoes/witnesses.htm

Pictures:

Gutierrez, J. (n.d.). All about
volcanoes(image). Retrieved from             http://education.ed.pacificu.edu/sweb/gutierrez/volcano/index.html

Anonymous, Initials. (n.d.). Bernoulli's
equation (image). Retrieved from             http://www.princeton.edu/~asmits/Bicycle_web/Bernoulli.html

Brooks & Cole. (2001). Volcanic explositivity index
(image). Retrieved from             http://www.uwsp.edu/gEo/faculty/ozsvath/images/volcanic_explosivity_index.htm

(n.d.). VEI
(taken from url).
Retrieved from http://www.volcano.si.edu/world/vei.jpg