environmental mathematics 4 - thin films (earth's atmosphere and seas) (ppt)

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Mathematics that shows Earth's atmosphere and seas to be onion-skin-thin surface films, as opposed to the immense entities that they seem to be. This is the second presentation in our new Environmental Mathematics series (the first one addresses demographics and the riddle of a billion homework questions).This "PowerPoint 2" in our "Biospherics Literacy 101" collection is made available courtesy of The Wecskaop Project (What Every Citizen Should Know About Our Planet). The mathematics outlined here shows that earth's oceans are on average, only about 6/100ths of 1% as thick as the diameter of the earth itself, and, if one were to wipe a 35 cm classroom globe with a wet paper towel, the film of water left behind would, proportionally-speaking, be too thick to properly represent the average depth of the earth's oceans.Because both the atmosphere and seas are thin surface films, our societal and industrial wastes may pose a far greater potential for adverse damage than we might otherwise expect.

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Page 1: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)
Page 2: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

Razor-Thin Surface Films

Earth’s atmosphere, oceans, and seas

Page 3: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

This presentation is a courtesy of

The Wecskaop Project

What Every Citizen Should Know About Our Planet

Copyright 2011, The Wecskaop Project.All rights reserved.

It is entirely free for non-commercial use by scientists, students, and educators anywhere in the world

Page 4: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

Is our planet fragile or robust?

Page 5: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

If the functioning of natural systems is robustand if the earth's water, seas, and atmosphere are immense

Page 6: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

then pollution and other human abuses might be

viewed as relatively minor insults to an otherwise robust

and pristine system

Page 7: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

If we assume that the earth and its atmosphere and seas

are so immense

as to be beyond harm from human activities,

then we may treat these criticalsystems with complacency

Page 8: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

But is such a worldview justified?

or do earth’s atmosphere and seas simply SEEM immense as a result of our own diminutive size?

Page 9: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

As a planet, earth has several characteristics that allow life as we know it to exist

Page 10: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

First, earth has immense quantities of water, so that we could call ourselves “The Water Planet"

Page 11: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

Compared to other known planets, for example, earth has immense quantities of water

Approximately 77% of its surface is covered with some

form of water

(such as oceans, lakes,glaciers, and snow)

Page 12: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

Secondly, it orbits the sun at a distance that allows most of its water to exist in its liquid state

Page 13: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

If we were a little further from the sun, all our water would

freeze and exist as ice

Page 14: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

If we were a little closer to the sun, our

water would exist primarily in its gaseous state

Page 15: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

The Pacific Ocean alone, for example, covers more of the

earth's surface than all of our land masses combined

Page 16: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

In addition, water covers 60% of the northern hemisphere and approximately

80% of the southern hemisphere

Page 17: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

Furthermore, if we were to use all earth's mountains and land masses to fill in the deepest parts of the sea,

we would end up with no land at all

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Instead, earth would be covered with a layer of water 2.5 kilometers deep

Page 18: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

In addition to these immense reservoirs of water, earth also has hidden reservoirs of water

Its atmosphere is filled with clouds, rain, water vapor, fog, and humidity

Page 19: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

Our farms and cities rely on dwindling

underground aquifers containing fossil water

that fell as rain thousands of years ago

Page 20: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

And even the cells, tissues, and bodies of

living things constitute rich reservoirs of water

Living cells, for example, are up to 98% salt water

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Page 21: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

In these respects, then, we can think of the earth as an "Unlikely Planet”

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Page 22: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

In reality, however, this seeming enormity and abundance is simply an illusion

In one way, therefore, the numbers cited so

farunderscore an

abundance of water

Page 23: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

Because as living organisms, each of us is so tiny compared to the size of our planet

that earth's oceans only seem large if they are compared to our own diminutive body size

Page 24: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

If we assess earth's oceans, however, as simply a surface featureof our planet, an entirely different perspective emerges

Mathematically speaking, 99.94% of our planetconsists of its crust, mantle, and its molten interior

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Page 25: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

The thin layer of water that we refer to as an ocean

exists only as an inexpressibly thin and precarious surface filmthat is only 6/100 ths of 1% as thick as the earth itself

See appendix twoFor supporting mathematics

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Page 26: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

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To proportionally illustrate such a depth to scale on a classroom globe, we would need a thin film of water

approximately 12/1000 ths of one inch deep

to correctly depict the proportionaldepth of the earth's oceans

Page 27: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

If we were to wipe a wet paper towelacross a twenty-inch globe

to properly characterize the depth of earth's oceans

the film it leaves behindwould be too deep

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Page 28: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

Thus viewed from a planetary perspective, our oceans exist as a thin and precarious surface film with greater

vulnerability than we might intuitively suppose

Page 29: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

Thus, the seeming immensity of our oceans is actually an illusion

for we have seen that, in planetary terms, our oceans are THINsurface films that are just 6/100ths of 1% as thick as the earth itself

Page 30: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

See appendix two forsupporting mathematics

Page 31: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

Earth’s Atmosphereas another thin and fragile surface film

Part Two

Page 32: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

It turns out that our ocean of air, earth's atmosphere, can be viewed in a similar way

Page 33: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

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If we analyze the proportional depth of earth's atmosphere,

we find that earth’s“ocean” of air

is also little more than another thin and

fragile film

Page 34: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

Astronauts and cosmonauts, while taking photographs from

space, have likened earth’s atmosphere

to a single layer of skinon an onion

Page 35: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

And this onion-skin-thin surface film of air may exhibit far greater vulnerability

than we commonly imagine

Page 36: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

Seen from this perspective, our collective individual impacts

could contribute seriously topotentially-calamitous outcomes

Page 37: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

The fact that earth’s atmosphereand oceans are razor-thin surface films

requires us to consider the implications of our

current worldwide levels

of pollution, disruption, and environmental damage

Page 38: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

What outbreaks of dinoflagellate ‘red-tide’

in marine environments may tell us about ourselves

Part Three

Page 39: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

In the ocean, one-celled dinoflagellates such as Karenia brevis release small amounts of

toxininto their surroundings

During such outbreaks of “red-tide,” a one-liter water sample cancontain 1,000,000 or more dinoflagellate cells per liter

Page 40: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

One of the most striking characteristics of ‘red-tide’ outbreaks

is that, taken together, all one million dinoflagellate cells per

liter in a red-tide outbreak

physically-occupy less than2/1000 ths of 1%

of the one-liter samplein which they reside

Page 41: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

See appendix onefor supporting mathematics

Page 42: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

What the white dot in this image shows most dramatically

is one of earth’s classical real-world examples of population-

environment calamities

Page 43: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

that routinely take place in environments that visually

appear to be

ALMOST entirely EMPTY

What the white dot in this image shows most dramatically

is one of earth’s classical real-world examples of population-

environment calamities

Page 44: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

In addition, dinoflagellate red-tides are one of nature’s quintessential

examples of calamities

that arise from population explosions accompanied by the release of wastes

Dinoflagellate red-tide calamities, however, arise from their release of

cellular and metabolic wastesinto their surroundings

Page 45: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

Except, of course, our own species supplements

its biological and cellular wastes with a daily worldwide avalanche of

industrial and societal wastes

Because our own species also releases wastes into its surroundings,

we may be following a trajectory

that is provocatively similar to that ofan outbreak of dinoflagellate red-tide

Page 46: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

So that, from at least one point of

view, we may actually be on a trajectory that is considerably worse

than that of the dinoflagellates

and multiple orders ofmagnitude worse at that

for each dinoflagellate cellreleases ONLY its

metabolic and biological wastes into its surroundings

Page 47: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

See appendix onefor supporting mathematics

Page 48: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

Part Four

No Other Animals Do This

Page 49: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

It is provocative to consider that today our own species, surrounded by a seemingly

enormous atmosphere and seemingly

“vast amounts of open space”

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to a significant alteration ofthe entire gaseous environment in which we live

Page 50: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

As a test of this last observation, envision an individual animal of any species other than our own

In virtually all of these cases, the organism’s daily pollution of its

environment is limited to its daily production of its bodily wastes

Page 51: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

Continuing, however, envision this same human being in an automobile, backed up in crowded traffic on a busy eight-lane highway

All around in every direction are hundreds of other cars and trucks and buses, each spewing exhaust from an internal combustion engine

This indicates that each of us as individuals are contributing much more than our body wastes to our surroundings

All of the organisms below, for example, limit themselves to therelease of their biological, cellular, and metabolic wastes

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Page 52: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

Continuing, however, envision this same human being in an automobile, backed up in crowded traffic on a busy eight-lane highway

All around in every direction are hundreds of other cars and trucks and buses, each spewing exhaust from an internal combustion engine

This indicates that each of us as individuals are contributing much more than our body wastes to our surroundings

In virtually all of these cases, each organism’s daily pollution of its environment is limited to daily production of its bodily wastes

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Page 53: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

Next, however, envision an ordinary human being living in an industrialized country

One’s daily body wastes are again a factor, of course,

but humanity’s collective biological wastes are natural productions that have, in a

planetary sense, little impact on global systems

Page 54: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

Continuing, however, envision this same human being in an automobile, backed up in crowded traffic on a busy eight-lane highway. All around in every direction are hundreds of other cars and trucks and buses, each spewing exhaust from an internal combustion engine.

This indicates that each of us as individuals are

contributing much more than our body wastes to

our surroundings

Page 55: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

And we repeat this behavior every day - again and again and again – in Beijing, Los Angeles, Mumbai, Tokyo, Cairo, Karachi, Jakarta,Paris, Moscow, Rio de Janeiro, Cape Town, and New York City

releasing more multiple millions of tons of waste, without fail, relentlessly into the onion-skin-thin

layer of air that makes up earth’s atmosphere

Page 56: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

Chemist Nate Lewis

As quoted by Friedman, 2008 - HOT, FLAT, AND CROWDED -

“Imagine you are driving your car and every mile you drive you throw a pound of trash out your window.

And everyone else on the freeway in their cars and trucks are doing the exact same thing, and people driving Hummers are throwing two bags out at a time – one out the driver-side window and one out the passenger-side window.

How would you feel? Not so good. Well, that is exactly what we are doing; you just can’t see it.

Only what we are throwing out is a pound of CO2 – that’s what goes into the atmosphere, on average, every mile we drive”

Page 57: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

If world population did not grow at all, all of these impacts would likely double

as the world’s poorest nations industrialize

and seek to emulate our own standard of living

Page 58: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

toaster-ovens microwaves hair-dryers steel mills

shopping malls motor-boats televisions, computers and hot-water heaters

our heating or air-conditioning units run a dishwasher and clothes drier run our lawnmowers and weed-trimmers

Now we switch on:

our refrigerators freezers street lights fluorescent lights

and we are not evenat home or at work yet

We are the only animals on earth that do this

Page 59: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

And we still have not included

all the wastes generated by

unwanted catalogue mailings tons of disposable, throw-away containers

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Page 60: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

Every day, from all of those tailpipes on each and every bumper-to-bumper interstate, boulevard, and highway, we spew molecules of

carbon dioxide and carbon monoxide and other noxious fumes

We are the only animals on earth that do this,

and we do so during eachand every rush hour,

on every grocery run, on every holiday trip to visit family, and during every postal delivery

Page 61: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

And then we repeat these same

activities every day,

again and again and again

so that our power plants, on our behalf,release still more tons of wastes and fumes

into the onion-skin-thinlayer of air that constitutes earth’s

atmosphere

without fail, relentlesslyand endlessly,

Page 62: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

Notice also that these additional wastes do not constitute a ‘once-in-a-lifetime’ contribution by each of us

Instead, we repeat these assaults again and again and again,

day after day after day, throughout our lives

Page 63: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

How can we imagine that endless billions of us can endlessly behave in this way without calamitous repercussions?

If we intend to enjoy such extravagance, our

populations must be smaller

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We are the only animals on earth that do this

Page 64: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

No other animal species supplements its cellular and biological wastes with a planet-wide avalanche of industrial and societal wastes the way that we do

We are the only animals on earth that have EVER done this

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Page 65: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

And no population explosions of red-tide dinoflagellates(which poison their environments by the release of wastes)

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Page 66: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

And these behaviors are NOT a minimal or incidental footnote to the biology of our species

Instead, they are one of our most distinctiveand all-encompassing characteristics

Page 67: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

Knowing that earth’s atmosphere is not

responding to our assaults very well right now,

consider that the U.N.’s newest world population projections show that we are nevertheless on-track to ADD at least

our 7th, 8th, 9th and 10th BILLIONS to our numbers by 2100

Page 68: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

And their “high-fertility” projections show us

reaching

15.8 BILLION

by century’s end

Page 69: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

Because three-quarters of the earth's surface iscovered with lakes, rivers, oceans, seas, and ice,

it is both easy and descriptive topicture our home as "a water planet“

so that we could easily call ourselves "Planet Ocean“

(IOF, 1978; Anson, 1991, 1996, 2007)

On the other hand, if we consider earth's oceans and atmosphere as strictly surface features of our planet

an entirely different assessment presents itself

Page 70: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

However, it is at least provocative to consider that today our own species, surrounded by a seemingly enormous

atmosphere and seemingly “vast amounts of open space”

also appears to be well on its way, via an ongoing release ofan assortment of industrial and societal wastes,

to a significant alteration of the entire gaseous environment in which we live

(not to mention the catastrophic physicaldamage that we inflict everywhere else)

Page 71: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

Review of Key Concepts

Page 72: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

Review of Key Concepts

climb and collapse really happen and we are not immune

collapse routinely occurs in environments that can appear to be almost entirely empty

calamities can arise from wastes and damage (as opposed to “running out” of things)

earth’s atmosphere and seas as onion-skin-thin surface films

dinoflagellate red-tides as quintessential examples of population explosions that induce calamity by the release of wastes

a thin-film of water on a globe

Page 73: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

collapse with 99% mortality is a biological reality

we are not immune to collapse, and compared to any other animals or dinoflagellates that have ever lived, we are behaving comparatively badly

by our release of wastes, we exhibit a behavioral similarity with population explosions of red-tide dinoflagellates

we may well be on a trajectory that is far worse than outbreaks of dinoflagellate red-tide because we supplement our biological and metabolic wastes with daily onslaughts of industrial and societal wastes

while outbreaks of dinoflagellate red-tide can be categorized as localized events, the impacts of our own species are global in nature

Page 74: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

We are dangerously misled by ourprevailing “open-space” suppositions

for it is a misperception to presume that human population growth and overpopulation cannot be truly serious so long

as “vast amounts of open space” remain

First, earth’s atmosphere and oceansare onion-skin-thin surface films

And climb and collapse really do occur – and they do soin environments that can appear almost entirely empty

When less than 2/1000ths of one percent of seemingly-available “space” is occupied

Review of Key Concepts

Page 75: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

In addition, the suppositionthat “running out” of things

such as space, food,resources, or anything else

are necessarily the first or only factorsthat could threaten us

is an incomplete assessment of our current condition

Page 76: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

and we are the only animals that do this, or that have ever done this

and are doing so on a worldwide scale so that we are not a localized phenomenon

and our behaviors in this respects are not a minimal or incidental footnote to the biology of our species

but are instead one of our most distinguishing and all-encompassing characteristics of our

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Page 77: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

Appendix 1

Supporting math – Red-tides

Page 78: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

Supporting Math – Red-tidesSevere red-tide conditions are common when Karenia brevis populations reach concentrations ranging between 100,000 to 1,000,000 or more cells per liter. Secondly, approximate dimensions of a typical K. brevis cell:

(1) Volume of 1 liter = 61.024 cubic inches(2) The approximate dimensions of a single cell of K. brevis are:

L: ~30 um (= 0.03 mm) = ~ 0.0012 inches ** W: ~ 0.0014 inches (“a little wider than it is long") *D: ~ 10 – 15 um deep (10 um = .0004; 15 um = .0006), so average = ~ .0005 in

** Nierenberg, personal communication, 2008 ** Floridamarine.org, 2008

Using the above:

Volume of a typical cell of K. brevis = (L) x (W) x (D)

= (.0012) x (.0014) x (.0005) = ~ .000 000 000 840 cubic inches

Thus one million Karenia brevis cells occupy approximately (1,000,000) times (.000 000 000 840) or a physical volume of about 0.000 84 cubic inches.

Recalling that one liter equals 61.024 cubic inches, subtracting 00.000 84 occupied cubic inches leaves (61.024) – (00.000 84) or about 61.023 16 cubic

inches remaining unoccupied. In other words, one million dinoflagellate cells in a one-liter sample still have approximately 61.023 16 cubic inches of unoccupied volume that would appear to remain theoretically-available to them.

Percentage Unoccupied

Therefore, the percentage unoccupied equals (61.023 16) divided by (61.024 00) so that about .999 987 2 or about 99.998 72% of the available volume remains unoccupied.

This means that such a K. brevis population manages to routinely visit calamity upon itself and the environment in which it resides, even asthe cells themselves physically-occupy less than2/1000ths of 1% of the total volume that appears to remain seemingly-available.

Thus, (100%) – (99.998 72%) = .001 28 %, or less than 2/1000ths of one percent of the volume that appears to remain theoretically-available.

Thus, even though the K. brevis cells occupy a volumetrically-insignificant portion of the "open-space" that visually appears to remain almost entirely “empty,” they manage, by their combined overpopulation and production of invisible and calamitous wastes, to catastrophically-alter the aqueous surroundings in which they live.

Page 79: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

Supporting Math

The image shown left depicts the physical amount of space that constitutes two one-thousandths of one percent. Note that the dot in the image denotes two one-thousandths of one percent of the dark rectangle.

The step-by-step mathematics outlined below permits preparation of a two-dimensional illustration like the one shown here that visually depicts the proportional amount of area occupied by two one-thousandths of one percent.

(1) Use imaging software to open a rectangle 500 pixels high by 350 pixels wide = 175,000 square pixels (Here: dark rectangle without frame)(2) Thus, one percent of this area = (175,000) x (.01) equals 1750 square pixels(3) In addition, 1/1000ths of one percent = (1750) times (.001) equals1.750 square pixels(4) And two1000ths of one percent = (1750) x (.002) equals 3.5 square pixels(5) Calculating the square root of 3.5 square pixels equals1.87 pixels, so that a square of (1.87 pixels) by (1.87 pixels) equals 3.5 square pixels

Thus beginning with a rectangle of 500 x 350 pixels, a small square of 1.87 pixels by 1.87 pixels (length times width) would visually depict a physical region of two one-thousandths of one percent.

2/1000ths ofone percent

Real-world population calamitiesin nearly “empty” environments

Page 80: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

Supporting Math – Reindeer of St. Paul IslandConcerning V. B. Scheffer’s classic reindeer climb-and-collapse study on St. Paul Island, Alaska, our estimate that the reindeer of St. Paul Island, Alaska physically-occupied “less than 2/1000ths of 1%” of the island’s total area at the time of collapse is derived as follows.

L: Assume an average reindeer is approximately 44” long

(Female reindeer ~ 38” long; males ~ 46” long; . so for our purposes, assume an average of 44”)

W: Assume that the width of an average reindeer is approximately 24” wide

Girth will vary with time of year; food, pregnant . . . females, etc., so for our purposes assume 24”

Thus the area physically-occupied by an average member of the population would equal (44 inches) x (24”) or approximately 1056 square inches each

Given a peak reindeer population of St. Paul island of slightly more than 2000 animals, (2000) times (1056) equals a combined area that is physically occupied by reindeer bodies of approximately 2,112,000 square inches (by the entire herd).

One square foot = (12) x (12) = 144 square inches, so that 2,112,000 divided by 144 means that the

bodies of an entire herd of 2000 animals would physically-occupy a total of 14,667 square feet.

If the area of St. Paul Island, Alaska is about 41 square miles, then if one square mile is equal to 27,878,400 square feet, then the total square footage of the island would equal ( (27,878,400) x (41) or approximately 1,143,014,400 square feet.

Next, we can subtract the14,667 square feet that are physically-occupied by the entire herd from the total square footage of the island so that (1,143,014,400) minus (14,667) results in a total “unoccupied” square footage of 1,142,999,733 square feet. Lastly, dividing the island’s total unoccupied space (1,142,999,733) by the total area of the island (1,143,014,400) gives the percentage of total unoccupied space at the time of the peak reindeer population, which was 0.999 987 168. Notice then that the collapse (and 99% die-off) of the St. Paul Island reindeer population began at a time when 99.999% of the island’s total area appeared to remain theoretically-available. Notice, therefore, that the herd’s collapse and 99% die-off both began (and proceeded to devastation) in surroundings that visually appeared to be

almost entirely empty.

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Supporting Math – Reindeer of St. Matthew IslandWe can apply the same approach to D.R. Klein’s classic reindeer climb-and-collapse study on St. Matthew Island, Alaska (1968). Our estimate that the reindeer of St. Matthew Island physically-occu-pied “less than 2/1000ths of 1%” of the island’s total area at the time of collapse is derived as follows.

L: Assume an average reindeer is approximately 44” long

(Female reindeer ~ 38” long; males ~ 46” long; . so for our purposes, assume an average of 44”)

W: Assume that the width of an average reindeer is approximately 24” wide

Girth will vary with time of year; food, pregnant . . . females, etc., so for our purposes assume 24”

Thus the area physically-occupied by an average member of the population would equal (44 inches) x (24”) or approximately 1056 square inches each

Given a peak reindeer population of St. Matthew isl-and (1963) of slightly more than 6000 animals, (6000) times (1056) equals a combined area that is physic-ally occupied by reindeer bodies of approximately 6,336,000 square inches (by the entire herd).

One square foot = (12) x (12) = 144 square inches, so that 6,336,000 divided by 144 means that the

bodies of an entire herd of 2000 animals would physically-occupy a total of 44,000 square feet.

If the area of St. Matthew Island, Alaska is about 138 square miles, then if one square mile is equal to 27,878,400 square feet, then the total square foot-age of the island would equal ( (27,878,400) x (138) or approximately 3,847,219,200 square feet.

Next, we subtract the 44,000 square feet that are physically-occupied by the entire herd from the total square footage of the island so that 3,847,219,200 minus (44,000) results in a total “unoccupied” square footage of 3,847,175,200 square feet. Lastly, dividing the island’s total unoccupied space (3,847,175,200) by the total area of the island (3,847,219,200) gives the percentage of total unoccupied space at the time of the peak reindeer population, which was 0.999 988 Notice then that the collapse (and 99% die-off) of the St. Paul Island reindeer population began at a time when 99.999% of the island’s total area appeared , visually-speak-ing, to remain theoretically-available. Notice, therefore, that the herd’s collapse and 99% die-off both BEGAN (and proceeded to devastation) in surroundings that visually appeared to be

almost entirely empty.

Page 82: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

Appendix 2

Supporting math – Thin Films

Page 83: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

Here is the supporting mathematics:

(i) Earth's oceans are, on average, approximate-ly 3.6 kilometers deep. If we have 3.6 kilome-ters of water on one side of our planet and an-other 3.6 kilometers on the opposite side, this represents an addition of 7.2 kilometers added to earth's overall diameter.

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(ii) Earth's overall diameter (including its molt-en interior, rocky mantle, crustal plates, and covering of oceans) is approximately 12,740 kilometers.

(iii) Thus, without the 7.2 kilometers of ocean, the earth's diameter would only be 12,732.8 km.

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(iv) This means that 12,732.8 km out of 12,740 km (99.94%) of earth's diameter consists of its molten interior, rocky mantle, and crustal plates.

(v) Thus, the math shows that the average depth of the oceans accounts for only six one-hundredths of one percent of earth's diameter – an inexpressibly thin film indeed.

(12,732.8 divided by 12,740 = 0.9994) and (100 minus 0.9994 = .0006)

Page 86: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

(For a twenty-inch classroom globe, .0006 times 20 inches would equal oceans, so that the classroom scale model would need a layer of water that is 12/1000ths of an inch deep to represent the ocean's depth in proportionally correct terms.) ( 20 times .0006 = .012 )

Page 87: Environmental Mathematics 4 - Thin Films (Earth's atmosphere and seas) (ppt)

This presentation is a courtesy of

The Wecskaop Project

What Every Citizen Should Know About Our Planet

Copyright 2011, The Wecskaop Project.All rights reserved.

It is entirely free for non-commercial use by scientists, students, and educators anywhere in the world

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Population – Science - Sustainability

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aopProject

Science – Math - Environment

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Graphs and Data Setshttp://www.flickr.com/photos/pali_nalu

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What Every Citizen Should KnowAbout Our Planet

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Razor-Thin Surface Films

Earth’s atmosphere, oceans, and seas