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lege Logo"/>
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</v:shape><![endif]--><![if !vml]><img width=3D32 height=3D32
src=3D"Lydon_files/image002.jpg" align=3Dleft alt=3D"Small Mission College =
Logo"
v:shapes=3D"Picture_x0020_2"><![endif]><!--[if gte vml 1]><v:shape id=3D"Pi=
cture_x0020_3"
 o:spid=3D"_x0000_s1030" type=3D"#_x0000_t75" alt=3D"Small Math Dept Logo" =
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ogo"/>
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</v:shape><![endif]--><![if !vml]><img width=3D32 height=3D32
src=3D"Lydon_files/image003.jpg" align=3Dleft alt=3D"Small Math Dept Logo" =
v:shapes=3D"Picture_x0020_3"><![endif]><b><span
style=3D'mso-bidi-theme-font: minor-bidi'>Math Department, </b><st1:place><=
st1:PlaceName><b>Mission</b></st1:PlaceName><b>
 </b><st1:PlaceType><b>College</b></st1:PlaceType></st1:place><b>, </b><st1=
:place><st1:City><b>Santa
  Clara</b></st1:City><b>, </b><st1:State><b>California</b></span><span
  style=3D'mso-bidi-theme-font: minor-bidi'></st1:State></st1:place> <o:p><=
/o:p></p>

<p><b></span>Go to <a
href=3D"http://www.missioncollege.org/depts/math/default.html">Math Dept Ma=
in
Page</a> | Go to <a href=3D"http://www.missioncollege.org/default.html">Mis=
sion
College Main Page</a></b> <span style=3D'font-family:"Arial Unicode MS"'><o=
:p></o:p></span></p>

<p><b><span style=3D'color:black'>This paper was written as an assignment f=
or Ian
Walton's Math G - Math for liberal Arts Students - at </span></b><st1:place=
><st1:PlaceName><b><span
  style=3D'color:black'>Mission</span></b></st1:PlaceName><b><span
 style=3D'color:black'> </span></b><st1:PlaceType><b><span style=3D'color:b=
lack'>College</span></b></st1:PlaceType></st1:place><b><span
style=3D'color:black'>. If you use material from this paper, please acknowl=
edge
it. </span></b><o:p></o:p></p>

<p><b><span style=3D'color:black'>To explore other such papers go to </span=
><a
href=3D"http://www.missioncollege.org/depts/math/gproject.htm">the Math G
Projects Page</a></b><o:p></o:p></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal align=3Dright style=3D'text-align:right'><o:p>&nbsp;</=
o:p></p>

<p class=3DMsoNormal align=3Dright style=3D'text-align:right'><o:p>&nbsp;</=
o:p></p>

<p class=3DMsoNormal align=3Dright style=3D'text-align:right'>Michael <span
class=3DSpellE>Lydon</span></p>

<p class=3DMsoNormal align=3Dright style=3D'text-align:right'>F09 Midterm P=
aper</p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><o:p>&nbsp;=
</o:p></p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><o:p>&nbsp;=
</o:p></p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><b style=3D=
'mso-bidi-font-weight:
normal'><span style=3D'font-size:14.0pt'>Introduction to and Conceptualizin=
g the
Study of Mathematics in <span class=3DSpellE>Vulcanism</span><o:p></o:p></s=
pan></b></p>

<p class=3DMsoNormal style=3D'line-height:200%'><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal style=3D'line-height:200%'><span style=3D'mso-tab-coun=
t:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </s=
pan>It
was August 26<sup>th</sup> 1883. The awesome power of earth was unleashed. =
The
small, remote island of <span class=3DSpellE>Krakatoawould</span> make all =
the
citizens of the earth respect her. She took the lives of thousands in secon=
ds;
those close enough to gaze upon her in that moment had their corneas cauter=
ized
by the blinding light. The explosion according to Deborah <span class=3DSpe=
llE>Hopkinson</span>
(2004<span class=3DGramE>),</span> was the &#8220;loudest sound you've ever
heard. <span class=3DGramE>Multiplied by thousands.&#8221;</span> &#8220;A =
sound
so deafening, an explosion so terrifyingly loud, people more than 2,000 mil=
es
away recorded hearing it&#8221; (<span class=3DSpellE>Hopkinson</span>, 200=
4).
Literally 20 cubic kilometers of ash and pumice were flung into the air wit=
hin
hours (<span class=3DSpellE>Scarth</span>, 1999). The resulting tsunamis ki=
lled
tens more thousands; in several different countries, hundreds of miles away.
&#8220;The explosive force of <span class=3DSpellE>Krakatoa</span> was equa=
l to
13,000 times the nuclear yield of the Little Boy bomb that devastated
Hiroshima, Japan during World War II<span class=3DGramE>;<span
style=3D'mso-spacerun:yes'>&nbsp; </span>and</span> Average global temperat=
ures
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&#8221; (<span class=3DSpellE>Wikipedia</span>, =
2009).
<span class=3DGramE>Scary, huh?</span> </p>

<p class=3DMsoNormal style=3D'line-height:200%'><span style=3D'mso-tab-coun=
t:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </s=
pan>Volcanoes
are one of the few natural disasters that have been able to humble mankind =
time
and time again. The natural phenomena they create <span class=3DGramE>is</s=
pan>
absolutely fascinating. Although this essay may have first appeared to be a=
bout
the eruption of <span class=3DSpellE>Krakatoa</span>, 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 <u>Introduction to and Conceptualizi=
ng
the Study of Mathematics in <span class=3DSpellE>Vulcanism</span></u>. </p>

<p class=3DMsoNormal style=3D'line-height:200%'><span style=3D'mso-tab-coun=
t:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </s=
pan>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 interestin=
g to
scientists. Where else on the earth do they get to study the content from t=
he
center of the earth; and watch the very formation of rocks? Where else do t=
hey
get to the see a major, natural instrument of climate change?<span
style=3D'mso-spacerun:yes'>&nbsp; </span>As you can imagine, all these topi=
cs
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 <span
class=3DSpellE>vulcanism</span>; Bernoulli's Equation, The Modified Bernoul=
li's
equation (that relates to volcanoes) and the Volcanic Explosive Index (V.E.=
I.).
</p>

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200%'><!--[if gte vml 1]><o:wrapblock><v:shape
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  <tr>
   <td width=3D16 height=3D0></td>
  </tr>
  <tr>
   <td></td>
   <td><img width=3D632 height=3D454 src=3D"Lydon_files/image005.jpg" v:sha=
pes=3D"_x0000_s1026"></td>
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style=3D'mso-ignore:vglayout' clear=3DALL>
<span style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp; </span><span
style=3D'font-size:10.0pt;line-height:200%'>Image from:
http://education.ed.pacificu.edu/sweb/gutierrez/volcano/index.html<o:p></o:=
p></span></p>

<p class=3DMsoNormal style=3D'line-height:200%'><span style=3D'mso-tab-coun=
t:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </s=
pan>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 <st1:place><st1:PlaceType>isla=
nd</st1:PlaceType>
 of <st1:PlaceName><span class=3DSpellE>Krakatoa</span></st1:PlaceName></st=
1:place>.
Yet some are <span class=3DGramE>more calm</span> like <st1:State><st1:plac=
e>Hawaii</st1:place></st1:State>,
where lava is large in quantity but lacks the large explosions. Here in <st=
1:State><st1:place>California</st1:place></st1:State>,
<span class=3DGramE>I<span style=3D'mso-spacerun:yes'>&nbsp; </span>live</s=
pan> on
what is called a <span class=3DSpellE>subduction</span> 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 classifi=
ed
when underground) percolates up between the gapped areas of the moving tect=
onic
plates. This forms a noticeable chain of what are called
&#8220;hot-spots&#8221; all down <span class=3DGramE>the are</span> 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<span class=3DGramE>,<sp=
an
style=3D'mso-spacerun:yes'>&nbsp; </span>it</span> has become cleverly know=
n as
&#8220;The Ring of Fire.&#8221; </p>

<p class=3DMsoNormal style=3D'line-height:200%'><span style=3D'mso-tab-coun=
t:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </s=
pan>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 <span class=3DG=
ramE>122<span
style=3D'mso-spacerun:yes'>&nbsp; </span>known</span> active hot-spots, 53 =
exist
on the ocean basins (1976). In the ocean the Earth's crust is much <span
class=3DGramE>more thin</span> than on the continents and lava frequently s=
eeps
out to form giant underground mountains. The <st1:place>Hawaiian Islands</s=
t1:place>
were formed this <span class=3DGramE>way,</span> they are part of a traveli=
ng hot
spot that (going back in time) originally formed not only the Hawaiian chain
but all of the Northwestern (Leeward) <st1:place>Islands</st1:place>. Islan=
ds
such as Midway were probably just as big and volcanically active as the big=
 <st1:place><st1:PlaceType>island</st1:PlaceType>
 of <st1:PlaceName>Hawaii</st1:PlaceName></st1:place>. </p>

<p class=3DMsoNormal style=3D'line-height:200%'><span style=3D'mso-tab-coun=
t:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </s=
pan>Speaking
of activity, let's get started on explaining how the more explosive eruptio=
ns
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). And=
rea <span
class=3DSpellE>Steffke</span> (2003) describes the three processes that hap=
pen
during an eruption- </p>

<p class=3DMsoNormal style=3D'line-height:200%'><span style=3D'mso-tab-coun=
t:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </s=
pan>An
explosive eruption involves three different processes: fragmentation of the
magma; blasting <span style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>of
the fragmented mass through the vent to the surface, and the ascent of the
eruption column. <span style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&n=
bsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>An
explosive eruption will only take place if the pressure within the magma
exceeds the <span style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp; </span>strength
of the surrounding rock. These explosive eruptions are driven by the thermal
energy <span style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp; </span>stored
in the magma. The thermal energy is transferred into kinetic energy of the
eruption <span style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp; </span>column
through the expansion of gases into growing vesicles (air bubbles). </p>

<p class=3DMsoNormal style=3D'line-height:200%'>In this situation we are di=
scussing
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 h=
ave
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 rapid=
ly
decompresses (fragmentation). </p>

<p class=3DMsoNormal style=3D'line-height:200%'><span style=3D'mso-tab-coun=
t:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </s=
pan>I
can begin explaining how this process happens, by using a differential equa=
tion
to <span class=3DGramE>describe<span style=3D'mso-spacerun:yes'>&nbsp;
</span>concept</span> 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.</p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center;line-height:=
200%'><!--[if gte vml 1]><o:wrapblock><v:shape
  id=3D"_x0000_s1027" type=3D"#_x0000_t75" style=3D'position:absolute;left:=
0;
  text-align:left;margin-left:0;margin-top:0;width:272.6pt;height:56.15pt;
  z-index:2;mso-wrap-distance-left:0;mso-wrap-distance-right:0;
  mso-position-horizontal:center' filled=3D"t">
  <v:fill color2=3D"black"/>
  <v:imagedata src=3D"Lydon_files/image006.png" o:title=3D""/>
  <w:wrap type=3D"topAndBottom"/>
 </v:shape><![endif]--><![if !vml]><img width=3D363 height=3D75
 src=3D"Lydon_files/image007.jpg" v:shapes=3D"_x0000_s1027"><![endif]><!--[=
if gte vml 1]></o:wrapblock><![endif]--><br
style=3D'mso-ignore:vglayout' clear=3DALL>
<span style=3D'font-size:10.0pt;line-height:200%'>Image From:
http://www.princeton.edu/~asmits/Bicycle_web/Bernoulli.html<o:p></o:p></spa=
n></p>

<p class=3DMsoNormal style=3D'line-height:200%'>This is called Bernoulli's
equation. It is a representation of the law describing pressure and fluid. =
The
law states &#8220;The pressure in a fluid decreases as the speed of the flu=
id
increases&#8221; (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, w=
hen
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. Th=
is
is facilitated because of the rapid decrease of pressure when the water is
ejected out of a narrow opening. </p>

<p class=3DMsoNormal style=3D'line-height:200%'><span style=3D'mso-tab-coun=
t:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </s=
pan>The
equation that we see here is designed to represent the previous analogy. We=
 can
translate the above formula as: Pressure + &frac12; of Density and
Velocity&sup2; + Density and Gravitational Acceleration and Elevation =3D
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 tha=
t are
very different from water flowing out of a hose. Since we have conceptualiz=
ed
the behavior of pressure flow in a condensed area; we can move on to a more
specialized differential equation that was designed specifically for volcan=
oes.</p>

<p class=3DMsoNormal style=3D'line-height:200%'><span style=3D'mso-tab-coun=
t:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </s=
pan>Volcanoes
are dramatically diverse, because the process of the lava being ejected can
vary radically. In the case of <span class=3DSpellE>Krakatoa</span>, there =
was a
massive build up of pressure and a very narrow vent. <span class=3DGramE>Li=
ke
prying out a cork out of a bottle of </span><st1:State><st1:place><span
  class=3DGramE>Champagne</span></st1:place></st1:State><span class=3DGramE=
>.</span>
</p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center;line-height:=
200%'><!--[if gte vml 1]><o:wrapblock><v:shape
  id=3D"_x0000_s1028" type=3D"#_x0000_t75" style=3D'position:absolute;left:=
0;
  text-align:left;margin-left:0;margin-top:0;width:144.1pt;height:61.75pt;
  z-index:3;mso-wrap-distance-left:0;mso-wrap-distance-right:0;
  mso-position-horizontal:center' filled=3D"t">
  <v:fill color2=3D"black"/>
  <v:imagedata src=3D"Lydon_files/image008.png" o:title=3D""/>
  <w:wrap type=3D"topAndBottom"/>
 </v:shape><![endif]--><![if !vml]><img width=3D192 height=3D82
 src=3D"Lydon_files/image009.jpg" v:shapes=3D"_x0000_s1028"><![endif]><!--[=
if gte vml 1]></o:wrapblock><![endif]--><br
style=3D'mso-ignore:vglayout' clear=3DALL>
<span style=3D'font-size:10.0pt;line-height:200%'>Image From:
http://ffden-2.phys.uaf.edu/212_fall2003.web.dir/Andrea_Steffke/eruptions.h=
tm<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'line-height:200%'>Just like a bottle of <st1:=
State><st1:place>Champagne</st1:place></st1:State>
however, there are additional forces that act on the rapid ejection of liqu=
id.
The equation above is an evolution of the Bernoulli's equation designed by
reputed <span class=3DSpellE>vulcanologist</span> Lionel Wilson. It includes
several additional variables that show up in volcanic eruptions. </p>

<p class=3DMsoNormal style=3D'line-height:200%'><st1:country-region><st1:pl=
ace><i><span
  style=3D'font-size:14.0pt;line-height:200%'>U</span></i><i><span
  style=3D'font-size:8.0pt;line-height:200%'>S</span></i>-</st1:place></st1=
:country-region>
Represents Ejection Velocity</p>

<p class=3DMsoNormal style=3D'line-height:200%'><i><span style=3D'font-size=
:14.0pt;
line-height:200%'>P</span>i- </i>Represents the pressure in gas.</p>

<p class=3DMsoNormal style=3D'line-height:200%'><i><span style=3D'font-size=
:14.0pt;
line-height:200%'>P</span></i><i><span style=3D'font-size:8.0pt;line-height=
:200%'>S
&#8211;</span> </i>Represents Atmospheric Pressure</p>

<p class=3DMsoNormal style=3D'line-height:200%'><st1:State><st1:place><i><s=
pan
  style=3D'font-size:14.0pt;line-height:200%'>S</span></i><i><span
  style=3D'font-size:8.0pt;line-height:200%'>C</span></i>-</st1:place></st1=
:State>
Represents Magma Density</p>

<p class=3DMsoNormal style=3D'line-height:200%'><span style=3D'mso-tab-coun=
t:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </s=
pan>Let
us introduce the first variable in <st1:City><st1:place>Wilson</st1:place><=
/st1:City>'s
Equation- gas pressure. Pressure, as in a bottle of soda or <st1:State><st1=
:place>Champagne</st1:place></st1:State>
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 rapi=
dly.
<span class=3DSpellE>Krakatoa</span> was an example of an eruption that had=
 a lot
of gas as the driving force of the huge explosion. </p>

<p class=3DMsoNormal style=3D'line-height:200%'><i><span style=3D'font-size=
:14.0pt;
line-height:200%'><span style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp; </span></span></i><span
style=3D'mso-spacerun:yes'>&nbsp;</span>The second part of the equation is =
atmospheric
pressure. Atmospheric pressure is normally defined, according to the Physic=
s <span
class=3DSpellE>Hypertextbook</span>, (2009) as<i><span style=3D'font-size:1=
4.0pt;
line-height:200%'> </span></i>the ratio of force applied per area covered o=
r <i>P</i>=3D<i>F/A</i>
(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=
. <st1:place>Mauna
 Loa</st1:place> in <st1:State><st1:place>Hawaii</st1:place></st1:State> ha=
s a
large reservoir of lava, and is, in fact the largest volcano in the world. =
But,
because <span class=3DGramE>it's</span> magma is not in a very concentrated=
 area
(and pressure very low), it does not erupt nearly as violently as some of t=
he
other smaller volcanoes. Instead it flows out very softly, layering over ti=
me
to increase the size <span class=3DGramE>of<span style=3D'mso-spacerun:yes'=
>&nbsp;
</span></span><st1:State><st1:place><span class=3DGramE>Hawaii</span></st1:=
place></st1:State>.</p>

<p class=3DMsoNormal style=3D'line-height:200%'><span style=3D'mso-tab-coun=
t:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </s=
pan>The
third variable in <st1:City><st1:place>Wilson</st1:place></st1:City>'s equa=
tion
is Magma Density. This factor is very important to <span class=3DSpellE>Vul=
canologists</span>,
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 fac=
tor
will affect the equation. When the lava from a volcano cools, crystallizes =
and
becomes solid, it is classified as an igneous rock (<span class=3DSpellE>Ch=
ristopherson</span>,
2006). Igneous magma before it becomes a rock is either classified as <span
class=3DSpellE>Felsic</span> or <span class=3DSpellE>Mafic</span>. <span
class=3DSpellE>Felsic</span> is lighter and less <span class=3DGramE>dense,=
</span> <span
class=3DSpellE>Mafic</span> has a higher density, and is more dense (<span
class=3DSpellE>Christopherson</span>). These characteristics, in addition t=
o the
temperature that they are; will determine what kind of an effect magma dens=
ity
will have on the eruption.<span style=3D'mso-spacerun:yes'>&nbsp;&nbsp; </s=
pan></p>

<p class=3DMsoNormal style=3D'line-height:200%'><span style=3D'mso-tab-coun=
t:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </s=
pan>So
what do all these variables add up to? The answer is the boom! They show how
fast the <span class=3DSpellE>tephra</span> (ash, rocks, dust, lava etc...)=
 will
be ejected from the volcano. <span class=3DGramE>&#8220;So now what?&#8221;=
</span>
You might ask; well we can now move on to another exciting mathematical
concept!<span style=3D'mso-spacerun:yes'>&nbsp; </span><span class=3DSpellE=
>Vulcanologists</span>
have conceived a way to actually describe the size of the eruption (or boom=
! <span
class=3DGramE>If you prefer).</span><span style=3D'mso-spacerun:yes'>&nbsp;=
 </span></p>

<p class=3DMsoNormal style=3D'line-height:200%'><span style=3D'mso-tab-coun=
t:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </s=
pan><span
class=3DGramE>Developed by C.G. Newhall, the V.E.I.</span> Of Volcanic <span
class=3DSpellE>Explositivity</span> Index that measures size of an eruption=
. </p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center;line-height:=
200%'><!--[if gte vml 1]><o:wrapblock><v:shape
  id=3D"_x0000_s1029" type=3D"#_x0000_t75" style=3D'position:absolute;left:=
0;
  text-align:left;margin-left:0;margin-top:-13.05pt;width:426.1pt;height:24=
9.7pt;
  z-index:4;mso-wrap-distance-left:0;mso-wrap-distance-right:0;
  mso-position-horizontal:center' filled=3D"t">
  <v:fill color2=3D"black"/>
  <v:imagedata src=3D"Lydon_files/image010.png" o:title=3D""/>
  <w:wrap type=3D"topAndBottom"/>
 </v:shape><![endif]--><![if !vml]><img width=3D568 height=3D333
 src=3D"Lydon_files/image011.jpg" v:shapes=3D"_x0000_s1029"><![endif]><!--[=
if gte vml 1]></o:wrapblock><![endif]--><br
style=3D'mso-ignore:vglayout' clear=3DALL>
<span style=3D'mso-spacerun:yes'>&nbsp;</span><span style=3D'font-size:10.0=
pt;
line-height:200%'>Image From: http://www.volcano.si.edu/world/vei.jp<o:p></=
o:p></span></p>

<p class=3DMsoNormal style=3D'line-height:200%'>Quoted from the <st1:countr=
y-region><st1:place>United
  States</st1:place></st1:country-region> Geological Survey (2009):</p>

<p class=3DMsoBodyText style=3D'line-height:200%'><span style=3D'mso-tab-co=
unt:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <=
/span>The
<strong><span style=3D'font-weight:normal'>Volcanic <span class=3DSpellE>Ex=
plosivity</span>
Index</span></strong>, or VEI, was proposed in 1982 as a way to describe the
relative <span style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp; </span>size
or magnitude of explosive volcanic eruptions. It is a 0-to-8 index of
increasing <span class=3DSpellE>explosivity</span>. <span style=3D'mso-tab-=
count:
1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>Each incre=
ase
in number represents an increase around a factor of ten. The VEI uses sever=
al <span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp; </span>factors to assign a num=
ber,
including volume of erupted <span class=3DSpellE>pyroclastic</span> materia=
l (for
example, <span style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p; </span>ash
fall, <span class=3DSpellE>pyroclastic</span> flows, and other <span
class=3DSpellE>ejecta</span>), height of eruption column, duration in hours=
, and <span
style=3D'mso-tab-count:1'>&nbsp; </span>qualitative descriptive terms. </p>

<p class=3DMsoNormal style=3D'line-height:200%'>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 erupti=
ons
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. </p>

<p class=3DMsoNormal style=3D'line-height:200%'><span style=3D'mso-tab-coun=
t:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </s=
pan>There
is also an algebraic formula that deals with the volume of earth an eruption
displaces. Based on the quantity (volume) of <span class=3DSpellE>ejecta</s=
pan>
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 plu=
me
can influence many things; including effects on the environment, weather and
also the people who live around the volcano. <span class=3DSpellE>Vulcanolo=
gists</span>
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. </p>

<p class=3DMsoNormal style=3D'line-height:200%'><span style=3D'mso-tab-coun=
t:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </s=
pan>Scientists
also use the objective visual measurements to determine the &#8220;Qualitat=
ive
Description&#8221; of the eruption event (Taken from above image). Througho=
ut
(written) human history, there has never been an eruption that was threaten=
ing
on the scale past a number seven on the V.E.I. classification scale. Howeve=
r,
there <span class=3DGramE>has</span> been many past events that have easily=
 topped
that. The <span class=3DSpellE>Toba</span> eruption that took place around
seventy-thousand years ago was over twenty times the size as <span
class=3DSpellE>Krakatoa</span>. This eruption had consequences that, at the=
 time<span
class=3DGramE>,<span style=3D'mso-spacerun:yes'>&nbsp; </span>had</span> th=
reatened
the existence of humanity. At a time when the earth was already <span
class=3DGramE>in<span style=3D'mso-spacerun:yes'>&nbsp; </span>period</span=
> of
global cooling; the resulting amount of <span class=3DSpellE>tephra</span>
released <span class=3DSpellE>by&#8220;The</span> <span class=3DSpellE>Toba=
</span>
Catastrophe&#8221; most likely increased the severity of the already taxing
conditions that threatened early humanity. &#8220;This massive environmental
change created population bottlenecks in the species that existed at the ti=
me;
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&#8221;<span
style=3D'mso-spacerun:yes'>&nbsp; </span>(Taken From-&quot;<span class=3DSp=
ellE>Toba</span>
Catastrophe Theory&quot;).<span style=3D'mso-spacerun:yes'>&nbsp; </span></=
p>

<p class=3DMsoNormal style=3D'line-height:200%'><span style=3D'mso-tab-coun=
t:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </s=
pan>Volcanoes
through out human history have had a wide range of effects on our world. Th=
ey
never fail to captivate people, and will ultimately continue to shape our
history.<span style=3D'mso-spacerun:yes'>&nbsp; </span>In this essay we have
discussed Bernoulli's equation, the Modified Bernoulli's equation (Volcano
Equation) and the Volcanic <span class=3DSpellE>Expositivity</span> Index.
Hopefully the reader has gained a new appreciation for volcanoes and has a =
good
idea of how mathematics <span class=3DGramE>translate</span> 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 <span
class=3DGramE>can<span style=3D'mso-spacerun:yes'>&nbsp; </span>change</spa=
n> how
we live.</p>

<p class=3DMsoNormal style=3D'line-height:200%'><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><b style=3D=
'mso-bidi-font-weight:
normal'>Works Cited Page (</b><st1:stockticker><b style=3D'mso-bidi-font-we=
ight:
 normal'>APA</b></st1:stockticker><b style=3D'mso-bidi-font-weight:normal'>=
)<o:p></o:p></b></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><b>Books:<o:p></o:p></b></p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><span class=3DGramE>Burke, K., &amp; Wilson, J. (1976)=
.</span>
Hot spots on the <span class=3DGramE>earths</span> surface. <i>Scientific
American: volcanoes and the <span style=3D'mso-tab-count:1'>&nbsp;&nbsp; </=
span>earth's
interior</i>, 3-31.</p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><span class=3DSpellE>Christopherson</span>, R.W. (2006=
). <span
class=3DSpellE><i>Geosystems</i></span><i>, sixth edition<span class=3DGram=
E>.<span
style=3D'font-style:normal'>.</span></span></i> <st1:place><st1:City>Upper =
Saddle
  River</st1:City>, <st1:State>New Jersey</st1:State></st1:place>: Pearson =
<span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; =
</span>Prentice
Hall. </p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal>Hewitt, P.G. (1993). <span class=3DGramE><i>Conceptual=
 physics</i>.</span>
</p>

<p class=3DMsoNormal><span style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span><st1:place><st1:City>New
  York</st1:City>, <st1:State>N.Y.</st1:State></st1:place>: <st1:place><st1=
:PlaceName>HarperCollins</st1:PlaceName>
 <st1:PlaceType>College</st1:PlaceType></st1:place> Publishers. </p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><span class=3DSpellE>Scarth</span>, <span class=3DSpel=
lE>Alwin</span>.
(1999). <span class=3DSpellE><i>Vulcans</i></span><i> fury: man against the
volcano</i>. </p>

<p class=3DMsoNormal><span style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span><st1:City><st1:place>New
  Haven</st1:place></st1:City> and <st1:City><st1:place>London</st1:place><=
/st1:City>:
<st1:place><st1:PlaceName>Yale</st1:PlaceName> <st1:PlaceType>University</s=
t1:PlaceType></st1:place>
Press. </p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><span class=3DSpellE><span class=3DGramE>Sigurdsson</s=
pan></span><span
class=3DGramE>, H., Houghton, B., McNutt, S., <span class=3DSpellE>Rymer</s=
pan>,
H., &amp; <span class=3DSpellE>Stix</span>, J. (2000).</span> <span class=
=3DGramE><i>Encyclopedia
of <span class=3DSpellE>volcanos</span></i>.</span></p>

<p class=3DMsoNormal><span style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span><span
style=3D'mso-spacerun:yes'>&nbsp;</span><st1:place><st1:City>San Diego</st1=
:City>,
 <st1:country-region>United States of America</st1:country-region></st1:pla=
ce>:
Academic Press. </p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><b>Websites:<o:p></o:p></b></p>

<p class=3DMsoNormal><span class=3DGramE>Anonymous.</span> <span class=3DGr=
amE>(1998-2009).
<i>Viscosity</i>.</span> Retrieved from </p>

<p class=3DMsoNormal><span style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>http://physics.info/vis=
cosity/
</p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><span class=3DGramE>Anonymous.</span> <span class=3DGr=
amE>(<span
class=3DSpellE>n.d</span>.).</span> <span class=3DSpellE><span class=3DGram=
E>Krakatoa</span></span><span
class=3DGramE>.</span> <i>Wikipedia.org</i>. Retrieved (2009, October 19) f=
rom <span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp; </span>http://en.wikipedia.org=
/wiki/Krakatoa
</p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><span class=3DSpellE>Steffke</span>, Andrea. <span
class=3DGramE>(2003). <i>Physics of volcanoes</i>.</span> Retrieved from </=
p>

<p class=3DMsoNormal><span style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>http://ffden-2.phys.uaf=
.edu/212_fall2003.web.dir/Andrea_Steffke/table.htm
</p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal>U.S.G.S.<span class=3DGramE>, .</span> <span class=3DG=
ramE>(<span
class=3DSpellE>n.d</span>.).</span> <i>VHP photo glossary: <span class=3DSp=
ellE>vei</span></i>.
Retrieved from <span style=3D'mso-tab-count:1'>&nbsp; </span>http://volcano=
es.usgs.gov/images/pglossary/vei.php
</p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><span class=3DGramE>Various authors, Initials.</span> =
<span
class=3DGramE>(<span class=3DSpellE>n.d</span>.).</span> <span class=3DGram=
E><i>Volcano
and Hydrologic monitoring techniques</i>.</span> Retrieved from <span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp; </span>http://vulcan.wr.usgs.gov/Monitoring/techniques.ht=
ml
</p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><span class=3DGramE>(<span class=3DSpellE>n.d</span>.)=
.</span> <span
class=3DSpellE><span class=3DGramE>Toba</span></span><span class=3DGramE> C=
atastrophe
Theory.</span> <i>Wikipedia.org</i>. Retrieved (2009, October 23) from <span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp; </span>http://en.wikipedia.org/wiki/Toba_catastrophe_theo=
ry
</p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><b>Online Articles:<o:p></o:p></b></p>

<p class=3DMsoNormal><span class=3DSpellE>Hopkinson</span>, D. (2004, Janua=
ry). <span
class=3DGramE>Witness account.</span> <span class=3DSpellE><i>Storyworks</i=
></span>,
11(4), <span class=3DGramE>Retrieved</span> from <span style=3D'mso-tab-cou=
nt:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </=
span>http://teacher.scholastic.com/activities/wwatch/volcanoes/witnesses.htm
</p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><b>Pictures:<o:p></o:p></b></p>

<p class=3DMsoNormal>Gutierrez, J. (<span class=3DSpellE>n.d</span>.). <i>A=
ll about
<span class=3DGramE>volcanoes(</span>image)</i>. Retrieved from <span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp; </span>http://education.ed.pacificu.edu/sweb/gutierrez/vo=
lcano/index.html
</p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><span class=3DGramE>Anonymous, Initials.</span> <span
class=3DGramE>(<span class=3DSpellE>n.d</span>.).</span> <span class=3DGram=
E><i>Bernoulli's
equation (image)</i>.</span> Retrieved from <span style=3D'mso-tab-count:1'=
>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>=
http://www.princeton.edu/~asmits/Bicycle_web/Bernoulli.html
</p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><span class=3DGramE>Brooks &amp; Cole.</span> <span
class=3DGramE>(2001). <i>Volcanic <span class=3DSpellE>explositivity</span>=
 index
(image)</i>.</span> Retrieved from <span style=3D'mso-tab-count:1'>&nbsp;&n=
bsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>http://ww=
w.uwsp.edu/gEo/faculty/ozsvath/images/volcanic_explosivity_index.htm
</p>

<p class=3DMsoNormal><o:p>&nbsp;</o:p></p>

<p class=3DMsoNormal><span class=3DGramE>(<span class=3DSpellE>n.d</span>.)=
.</span> <i>VEI
(taken from <span class=3DSpellE><span class=3DGramE>url</span></span>)</i>.
Retrieved from http://www.volcano.si.edu/world/vei.jpg </p>

<p class=3DMsoNormal style=3D'line-height:200%'><o:p>&nbsp;</o:p></p>

</div>

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