Transcript ESP_2_Language of Science_v2x

2. THE
LANGUAGE OF
SCIENCE
In his book Planet Earth Cesare
Emiliani notes that Americans
have a singular aversion to
mathematics. He suggests that
this may be because many of us
consider ourselves to be very
religious Christians. There is
nothing much more than simple
arithmetic in the Bible and not
much of that. Certainly no
geometry, other than to state
that the Earth is flat and
rectangular. Since there is no
math in the Bible, Cesare
conjectures that God did not
invent it. If God did not invent
it, he concludes, it must be the
work of the Devil!
LOCATING THINGS IN SPACE
Renatus Cartesius a.k.a. René Descartes (1596-1650), in his Dutch finery
Legend has it that, while sick
and confined to his bed, René
Desartes conceived his ideas
for expressing position in space
by watching the movement of a
fly in the corner of his room.
First, he considered its location
on one wall – in terms of
rectangular coordinates
Then, he realized he could
locate it in space – in terms of
three coordinates
We now refer to these as ‘Cartesian Coordinates
180° E or W Longitude
90° N Latitude
For locations on the Earth we
use a sperical version of the
Cartesian coordinates – a
LATITUDE-LONGITUDE grid
Prime Meridian (‘Greenwich;
Meridian’ = 0° Longitude)
Equator (0° Latitude)
90° W Longitude
Determining longitude is easy (in the Northern hemisphere) - it
is the elevation of the pole star (Polaris) above the horizon. In
the Southern hemishere there is no star above the pole so you
have to know where to look.
Determining latitude is a wholly different problem. Longitude
is very important to seafarers - it tells how long it will take
to get somewhere. But you need to know both what time it is
in Greenwich (or Rome) and what your local time is. Galileo
(1564-1642) figured that one could use the eclipses of the
moons of Jupiter as a universal ‘clock in the sky’. But he didn’t
have an accurate enough clock here on Earth. He knew his
‘water clock’ wasn’t accurate and tried to make a pendulum
clock, but died before it was finished. The Pendulum clocks
was invented by Christian Huygens (1629-1695) in 1645. But
pendulum clocks do not travel well at sea. Not at all!
Galileo’s idea of using the moons of Jupiter as a clock was used
later to determine the locations of a few critical sites on land.
However, the problem of determining
longitude remained vexing, partuclarly
for ships at sea. In 1714 the British
government established the Board of
Longitude which offered a prize to be
awarded to the first person to
demonstrate a practical method for
determining the longitude of a ship at
sea.
The problem was eventually solve by
British clockmaker John Harrison in
1726. it is a fascinating story, told in
Dava Sobel’s book Longitude, The True
Story of a Lone Genius Who Solved the
Greatest Scientific Problem of His
Time.
TRIGONOMETRY
Trigonometry originally started out as the study of geometrical
objects with three sides — triangles. In school you certainly heard
about Pythagoras’ theorem: in a triangle, one of whose angles is a
right (90°) angle, the area of a square whose side is the hypotenuse
(the side opposite the right angle) is equal to the sum of the areas of
the squares of the two sides that meet at the right angle.
Pythagoras lived about 570 to 495 BCE (BCE = Before Common Era =
BC of Christians) and taught philosophy and mathematics in the
Greek colonial town of Croton in Calabria, southern Italy. Although
attributed to Pythagoras, this and many other mathematical ideas
may be much, much older and were apparently known to the
Babylonians who had figured all this out by about 2,000 BCE.
Therein lies another important story. Why is a right angle 90°, and
not, say, 100° ? It goes back to the Babylonians and their method of
counting. They figured the year was about 360 days long. With four
seasons, marked by the time the sun was in its lowest point in the
southern sky (the winter solstice), its mid-pint in returning north
(vernal equinox), the northernmost point in the sky (summer solstice),
and the midpoint in the return south (autumnal equinox). You divide
360 by 4 and get 90. Ninety degrees is one quarter of the way around
a circle.
The Babylonian mathematicians are also responsible for our
timekeeping system. The ‘normal’ human pulse rate is 1 per second.
The Babylonians counted in 60's, so 60 seconds is a minute, and 60
minutes is an hour. Using this system they found it took 24 hours for
the sun to return to the same place in the sky each day. Now in Earth
measure, a circle of latitude or longitude is 360 degrees. Each degree
is divided into 60 minutes, and each minute into 60 seconds. Latitude
and longitude expressed to the nearest second will locate something
on the surface of the Earth quite well; to within a square 31 meters
(=102 feet) on a side at the equator.
PYTHAGORAS’ THEOREM FOR A 30° – 60° – 90° TRIANGLE
PYTHAGORAS’ THEOREM FOR A 30° – 60° – 90° TRIANGLE
TRIGONOMETRIC FUNCTIONS
The sine (sin) is the ratio between
the side opposite an angle divided by
the hypotenuse.
The sin of 30° is ½ or 0.5; the sin of
60° is 3/2 = 1.732/2 = 0.866.
The sin of 0° is 0 over the hypotenuse,
or 0. Sin 90° is the hypotenuse over
the hypotenuse, or 1.
The cosine (cos) is the side adjacent to the angle divided by the
hypotenuse; cos 30° = 3/2 = 1.732/2 = 0.866; the cos of 60° is ½ =
0.5. The cosine is the reciprocal (1 over whatever) of the sine.
There is another common trigonometric function, the tangent (tan).
The tangent is the ratio of the side opposite the angle to the side
adjacent. It varies from 0 137 at 0° to infinity (∞) at 90°.
SINE
COSINE
TANGENT
In thinking about the Earth, it is important to realize that the distance
between lines of longitude change with latitude. How? The distance is
proportional to the cosine of the latitude. A degree of longitude at the
equator was originally intended to be 60 (nautical) miles, but it
deceases towards the poles. The length of a degree of latitude remains
constant. A minute longitude at the equator or a minute of latitude is 1
nautical mile. Nautical miles are a form of measure that goes back to
antiquity. A nautical mile is 1,852 meters or 6076 US feet.
The length of a degree of latitude (60 nautical miles) is constant
between parallels, but the length of a degree of longitude varies from
60 nautical miles at the equator to 0 at the poles. That is where
knowing the cosine of the latitude is important. The cosine is 1 at the
equator and 0 at the pole. The cosine of 30°is 0.5, so half of the area
of the Earth lies between 30° N and 30° S. The southern margin of
North America is about 30° N. The part of the northern hemisphere
poleward of 30° N
is 1/4 of the total area of the Earth.
In older climate studies it was often assumed that what happens in the
48 contiguous United States is typical of the Earth as a whole. Wrong:
the 48 states occupy only about 1.6% of the area of the Earth – the US
is not a representative sample of the planet as a whole.
MAP PROJECTIONS
Representing the surface of a sphere on a flat piece of paper is a
challenging task. For the Earth the problem is compounded by the fact
that the planet is not actually a sphere , but is flattened slightly at the
poles because of its rotation; technically, it is an oblate spheroid.
The difference between the equatorial and polar diameters is 43 km
(26.7 statute miles; a statute mile is 5,280 feet). The average
diameter of the earth is 12,742 km (7,918 statute miles; 6,880 nautical
197 miles). Multiply those by 2 π and you get a circumference of
40,000 km or 198 24,854 statute miles or 21,598 nautical miles. The
even 40,000 km is because the meter was originally defined as
1/10,000,000 of the distance from the equator to the north pole on
the meridian passing through Paris, France — one fourth
of the way around the world. The modern measure of 21,598 nautical
miles is remarkably close to the 21,600 nautical miles if there were 1
nautical mile per second of a degree as the ancients assumed.
The simplest representation of the Earth is the Equirectangular
projection, sometimes called the Cylindrical Equidistant projection. It
was invented by Marinus of Tyre in about 100 AD. The Earth is a
rectangle and the lengths of degrees of latitude and longitude are the
same everywhere. Claudius Ptolemy (90–168 AD) who lived in Alexandria,
Egypt and who produced an eight volume geography of the known world
described this projection as the worst representation of Earth he
knew of.
NASA’S IMAGE OF THE WHOLE EARTH USING THE
EQUIRECTANGULAR (CYLINDRICAL EQUIDISTANT) PROJECTION
Sadly, the results of many climate models are shown using this
projection
Gerardus Mercator’s projection (1569) was made for the
specific purpose of navigation at sea.
Lines of constant heading or course, also called rhumb lines,
are straight lines on this projection..
Another way to look at the world is from above either of the poles. These are
Polar Orthographic projections - the view you would have if you were
infinitely far away. Hipparchus (190 - 120 BCE), a Greek astronomer and
mathematician, is credited both with their invention. On this projection the
shortest distance between two points is a straight line - a segment of a ‘great
circle’ passing through the two points and encircling the globe. Long distance
air flights generally follow a great circle path, making adjustments to take
advantage of the winds.
Finally, my favorite, the Mollweide Equal Area projection. It
preserves the correct areas of the lands and oceans at the expense
of some distortion. It was invented by mathematician and
astronomer Karl Brandan Mollweide (1774 – 1825) in Leipzig in 1805.
MEASUREMENT IN SCIENCE – THE METRIC SYSTEM
In the metric system, everything is ordered to be multiples of the
number 10. No more 12 inches to the foot, 5,280 feet to the mile, 16
ounces to the pound, or 2,000 pounds to the ton.
The metric system originated in France. It had been suggested by
Louis XVI but it remained for the Revolution of 1798 to begin the real
organization of standards of weights and measures.
The originators were ambitious; the meter was intended to be one tenmillionth of the distance along the surface of the Earth between the
equator and the North Pole along the meridian passing through Paris.
However, they were wise enough to realize that they did not know that
distance exactly, so they made the length of a metal bar in Paris the
standard. (They figured that the Earth’s meridional circumference, the
circle going through both poles and, of course, passing through Paris,
was 40,000 kilometers; back in 1790 they were off by 7.86
kilometers).
Napoleon Bonapart
spread the metric
system throughout
Europe. Britain was
spared and kept its
archaic system
(recently abandoned).
Although the French
helped us in the War
of 1812, the U.S. has
kept the old British
system.
. Everyone one else has long ago gone metric (except Belize). (In fact
the U.S. standards for measurement are defined in metric terms but
no one seems to now that).
THE METRIC SYSTEM
THE STANDARD FOR LENGTH
Although the meter (metre) was originally intended ot be 1/10,000,000
the distance from the North Pole to the Equator along the meridian
passing through Paris, this turned out to be inconvenient for practical
use. Because of the uncertainty involved a physical standard meter was
constructed in 1889.
The standard meter is the distance between two line engraved on a
standard bar composed of an alloy of ninety percent platinum and ten
percent iridium, measured at the melting point of ice. It is in the Bureau
International des Poids et Mesures in Sèvres, France.
In 1960 the meter was redefined as
equal to 1,650,763.73 wavelengths of
the orange-red emission line in
the electromagnetic spectrum of
the krypton-86 atom in a vacuum.[8]
THE METRIC SYSTEM
THE STANDARD FOR WEIGHT
The standard kilogram is a
cylinder of the same 90%
platinum 10% iridium alloy,
also kept at the Bureau
International des Poids et
Mesures in Sèvres, France.
Because ‘weight’ is a
function of both mass and
gravitational attraction
there has been a problem.
The standard kilogram has
been losing weight with
time.
THE PART OF THE METRIC SYSTEM THAT
DIDN’T SURVIVE – THE CALENDAR
The French revolutionaries tried to carry their enthusiasm for ‘10 ’
to timekeeping. Their new calendar consisted of twelve 30 day
months: vendémiaire, brumaire, frimaire (autumn); nivôse, pluviôse,
ventose (winter); germinal, floréal, prairial (spring); messidor,
thermidor, fructidor (summer).
Each of the months consisted of three 10-day ‘weeks’ called décades.
The names of the days of the décade were simply numbered: primidi,
duodi, tridi, quartidi, quintidi, sextidi, septidi, octidi, nonidi, décadi.
The extra 4 to 6 days needed to fill out a year become holidays
added at the end of the year.
The day was divided into 10 hours each with 100 minutes and the
minutes divided into 100 seconds.
The world has preferred the ancient Babylonian system
using 60’s as its base.
Orders of magnitude (+)
Number
Scientific Notation
Name
SI Prefix
1
1 x 100 or 1 x 10^0
one
-
10
1 x 101 or 1 x 10^1
ten
deca-
100
1 x 102 or 1 x 10^2
hundred
hecto-
1,000
1 x 103 or 1 x 10^3
thousand
kilo-
1,000,000
1 x 106 or 1 x 10^6
million
mega-
1,000.000,000
1 x 109 or 1 x 10^9
billion
giga-
1,000,000,000,000
1 x 1012 or 1 x 10^12
trillion
tera-
1,000,000,000,000,000
1 x 1015 or 1 x 10^15
quadrilliion
peta-
1,000,000,000,000,000,000
1 x 1018 or 1 x 10^18
quintillion
exa-
1,000,000,000,000,000,000,000
1 x 1021 or 1 x 10^21
sextillion
zeta-
1,000,000,000,000,000,000,000,000
1 x 1024 or 1 x 10^24
septillion
yotta-
Orders of magnitude (-)
Number
Scientific Notation
Name
SI Prefix
1
1 x 100 or 1 x 10^0
one
-
0.001
1 x 10-3 or 1 x 10^-3
thousandth
milli-
0.000001
1 x 10-6 or 1 x 10^-6
millionth
micro-
0.000000001
1 x 10-9 or 1 x 10^-9
Billionth
nano-
0.000000000001
1 x 10-12 or 1 x 10^-12
pico-
0.000000000000001
1 x 10-15 or 1 x 10^-15
femto-
0.000000000000000001
1 x 10-18 or 1 x 10^-18
atto-
0.000000000000000000001
1 x 10-21 or 1 x 10^-21
zepto-
0.000000000000000000000001
1 x 10=24 or 1 x 10^-24
yocto-
0.0000000001 m
1 x 10-10 or 1 x 10^-10 m
Ångström or Angstrom
EXPONENTS AND LOGARITHMS
In the table you saw how exponents of the number 10 can be used to
indicate the number of 0’s before or after the decimal point. They
indicate the number of times you have to multiply a number by itself to
get a given value.
103 is read as ‘10 to 3 the third power.’ Exponents save a lot of space in
writing out very large or very small numbers and make comparisons of
numbers much easier.
There is another way of doing the same thing, using logarithms. A
logarithm is the power to which a standard number, such as 10 must be
raised to produce a given number. The standard number on which the
logarithm is based is called, of all things, the base.
Thus, in the simple equation nx = a, the logarithm of a, having n as its
base, is x. This could be written logn a = x.
103 = 1,000; or written the other way round: log10 1,000 = 3
What is not so easy to figure out is what the log10 of 42 would be.
(You may recall from Douglas Adams’ The Hitchhiker’s Guide to the
Galaxy that 42 is the answer to the question ‘what is the meaning of
life, the Universe, and everything?’)
To determine log10 42 you need to do some real mathematical work.
When I was a student, you could buy books of tables of logarithms to
the base 10, and look it up. Now most calculators and computer
spreadsheet programs, such as Excel, will do this for you.
For your edification log10 42 is 1.623249
You may be aware that the acidity or alkalinity of a solution is
described by its ‘pH’ which is a number ranging from 0 to 10. Now for
the bad news: pH is the negative logarithm to the base 10 of the
concentration of the hydrogen ion (H+) in the solution. This means
that a small number means there are lots of H+ ions in the solution,
and a large number means that a there are only a few.
How’s that for convoluted logic?
Some other very useful logarithms don’t use 10 as their base.
Computers depend on electrical circuits that are either open or closed
— two possibilities. So computers use a system with base 2, the ‘binary
logarithm.’
Then there is a magical number that is the base of a logarithm that will
allow you to calculate compound interest, or how much of a radioactive
element will be left after a given time, or anything that grows or
decays away with time. This base is an ‘irrational number;’ that is, it has
an infinite number of decimal places. It starts out 2.7182818284 . . . . .
and goes on forever; it has a special name in mathematics, it is the
Euler number, or simply ‘e ’. Logarithms with e as their base are called
‘natural logarithms.’
Although determining the value of e is attributed to Swiss
mathematician Leonhard Euler (1707-1783) , it now seems that the
Babylonians already knew about ‘e ’.
Lots of thought has been given to that four digit repetition after 2.7,
the ‘18281828’. But is unique – it never occurs again in much longer
calculations of ‘e ’.
EARTHQUAKE SCALES
The classification of earthquakes began in the 19th century with a scale
developed by Michele Stefano Conte de Rossi (1834-1898) of Italy and
François-Alphonse Forel (1841-1912) of Switzerland. Their
scale was purely descriptive ranging from I, which could be felt by only a few
people, through V, which would be felt by everyone, and might cause bells to
ring, to X, which would be total disaster.
It was modified and refined
by the Italian volcanologist
Giuseppe Mercalli (18501914) toward the end of the
19th century, and a revision
ranging from I to XII
became widely accepted in
1902, just in time for the
San Francisco earthquake;
the city had some areas
near the top of the scale.
The Mercalli scale been modified a number times since to become the
‘Modified Mercalli Scale’ with intensities designated MM I to MM XII
based on perceived movement of the ground and damage to structures.
On this scale
the 2010 Haiti
earthquake
ranged from
MM XII at the
epicenter near
the town of
Léogâne and
MM VII in
Port-au-Prince
about16 miles
(25 km) to the
east. The MM
scale is purely
descriptive of
the damage.
In
In 1935, the seismologist Charles Richter (1900-1985) developed a more
‘scientific’ scale based on the trace of the earthquake recorded in a
particular kind of seismograph, the Wood-Anderson torsion
seismometer. (A seismograph is an instrument that measures the motion
of the ground.) Originally developed for use in southern California,
Richter’s scale has become known as the ‘local magnitude scale’
designated ML. It is based on logarithms to the base 10, so that an
earthquake with ML 7 is ten times as strong as one with an intensity of
ML 6. Richter and his colleague Beno Gutenberg estimated that on this
scale the San Francisco earthquake of 1906 had a magnitude of about
ML 8. Unfortunately, the Wood-Anderson torsion seismometer doesn’t
work well for earthquakes of magnitude greater than ML 6.8.
Most journalists do not know that the Richter Scale has been replaced
by another scale that better represents the energy involved in the
earthquake
In 1979, seismologists Thomas C. Hanks and Hiroo Kanamori (1936- ) published
a new scale, based on the total amount of energy released by the earthquake.
Development of this new scale was made possible by more refined instruments
and an increasingly dense network of seismic recording stations. The new
scale, in use today, is called the ‘Moment Magnitude Scale’ (MMS) and is
designated MW. It is based on determining the size of the surface that
slipped during the earthquake, and the distance of the slip. It too is a
logarithmic scale, but its base is 31.6. That is, an earthquake of MW 7 is 31.6
times as strong as one of MW 6.
On this scale the
Haiti earthquake
was MM 7. The 2010
Chile earthquake, a
few days later, was
MM 8.8, 500 times
stronger.
From experience,
Chile was prepared
for earthquakes.
Sites of some recent major earthquakes. 1 - Valdivia, Chile, May 22, 1906, subduction, MW =
9.5, 5,700 (est.) deaths; 2 - Sumatra, December 26, 2004, subduction, MW = 9.2, 227,898
deaths, mostly by drowning from the ensuing tsunami; 3 - Eastern Sichuan, China, May 12,
2005, intra-plate, MW = 7.9, 87,587 deaths; 4 - Haiti, January 12, 2010, MW =7.0,
transform fault, 222,570 deaths; 5 -Maule, Chile, February 27, 2010, subduction, MW = 8.8,
577 deaths; 6 - Tohoku, Japan, March 11, 2011, subduction, MW = 9.0, more than 13,000
deaths, mostly by drowning from the ensuing tsunami.
HURRICANE INTENSITY SCALES
The Saffir–Simpson
hurricane scale (SSHS,
1971):
classifies Hurricanes (Western Hemisphere
tropical cyclones that
exceed the intensities
of tropical depressions
and tropical storms) –
into five categories
according to the
maximum intensities of
their sustained winds.
HURRICANE INTENSITY SCALES
In its simplicity, the
Saffir-Simpson scale is
very much like the old
Mercalli Earthquake scale.
There is more to a
hurricane than maximum
sustained wind velocity –
size, intensity, and total
energy.
Kerry Emanuel of MIT has
been working to devise a
system to categorize
tropical cyclones by their
intensity and overall
energy
HURRICANE INTENSITY
The destructive force of a tropical cyclone is an ‘exponential function’ !
EXPONENTIAL GROWTH AND DECAY
Exponential growth and decay are concepts very useful in understanding
both why we are on the verge of a climate crisis and how we know about the
ages of ancient strata. First, let us consider what these terms mean.
They can be expressed by simple algebraic equations, well worth knowing if
you are trying to keep track of your savings account and investments, and
are planning for the future. The expression is
y = a × bt
where a is the initial amount, b is the decay or growth constant, and t is the
number of time intervals that have passed. If b is less than 1, y will get
smaller with time, it is said to decay away; if b is larger than 1, y will
increase with time, or grow. If the decay or growth is a constant percent,
the equation Scan be written
y = a × (1!b)t for decay, or
y = a × (1+b)t for growth
EXPONENTIAL GROWTH
What does exponential growth mean in practical terms? If
you put $ 100.00 into a savings account with 3% interest per
year for 10 years, the equation and its solution looks like
this:
y = $100 × (1+ 0.03)10 = $100 × (1.03)10 = $100 × (1.3439) =
$134.39
The ‘1.3439' is the result of multiplying 1.03 by itself ten
times. However, you may also wish to know how long it will
take to double your money at 3% interest. This can be done
by dividing the natural logarithm of 2 (ln2), which is
666 0.693147 by the interest rate (0.03), and the answer
comes out to be 23.1 years.
The natural logarithms have as their base that ‘irrational
number,’ e, specifically 2.71828182845904523536…. (The
dots mean the number is infinitely long).
Here is a little trick you can use to be able to do these
calculations in your head. A close approximation of the natural
logarithm of 2 (ln2), 0.693147, is simply 0.70, so you can
simply divide that number by the 3% interest rate.
But most of us would have to write it all down to make sure we
get the decimal point in the right place, so here is the trick:
multiply both sides by 100 so you don’t have to convert 3% to
0.03. Then it becomes a matter of dividing 70 by 3 = 23.33
years, a close approximation to the exact answer.
You will immediately realize that if you were getting 7%
interest, the time required to double your money would be
only (about) 70/7 or 10 years.
POPULATION GROWTH
Reverend Thomas Robert Malthus (1766–1834). He was an English scholar now
remembered chiefly for his theory about population growth. In his book An
Essay on the Principle of Population he argued that sooner or later
populations get checked by famine and disease. He did not believe that humans
were smart enough to limit population growth on their own.
Then, starting with the Industrial Revolution (1775) but accelerating during
the middle of the 20th century, the spread of antibiotics to prevent disease
and extend life, and pesticides and fertilizer for protecting crops and growing
more food made it possible for the growth rate of the human population to
rise to 2.19% per year. That is a doubling time of 31.65 years.
Earth’s human population in 1961 was about 3 billion, and in 2000 it was about 6
billion. When Cesare Emiliani published his book The Scientific Companion in
1988 he calculated that at its rate of growth at the time, doomsday would
occur in 2023. Doomsday was defined as the date when there would no longer
be standing room for the human population on land. He had a second, really bad
doomsday that would be reached only a few decades later, when the Earth is
covered by human bodies expanding into space at nearly the speed of light.
Fortunately, by 2001 the global human growth rate had slowed to 1.14%, for a
doubling time of 61 years. In 2011 the global growth rate dropped to 1.1%, a
doubling time of 63 years.
Projections by the Intergovernmental Panel on Climate Change (IPCC) indicate a
global population of 8.7 to 10 billion by 2050. Most projection suggest a decline
thereafter but others show a global population of 17 billion by 2100
SRES = IPCC’s
Special Report on
Emission Scenarios
(2000)
EARTH’S HUMAN CARRYING CAPACITY
The carrying capacity of a biological species in an environment is
defined as the maximum population size of the species that the
environment can sustain indefinitely, given the food, habitat, water
and other necessities available in the environment.
T
From the Club of Rome’s
1972 ‘Limits to Growth’
The human carrying
capacity of planet Earth
is unknown, but is
estimated to be between
1 and 4 billion people,
most probably about 2
billion.
We are already past
Earth’s human carrying
capacity.
The root cause of
environmental problems
EARTH’S HUMAN CARRYING CAPACITY
The root cause of environmental problems (water and food supply,
pollution, climate change) facing us today is overpopulation.
The carrying capacity of a biological species in an environment is
defined as the maximum population size of the species that the
environment can sustain indefinitely, given the food, habitat, water
and other necessities available in the environment.
Books that discuss these problems in detail.
Donella H. Meadows, Dennis L. Meadows Jørgen Randers and
William W. Behrens III (1972): The Limits to Growth
Donella Meadows, Jorgen Randers, Dennis Meadows (2004): The
Limits to Growth - The 30 Year Update
Lester Brown (2011): World on the Edge
Anders Wijkman, Johan Rockström, (2011): Bankrupting Nature
Jorgen Randers (2012) 2052
EXPONENTIAL DECAY
Examples of exponential decay include:
1.
The decay of a radioactive isotope of an element with
time
2. The decrease of atmospheric pressure with height; it
decreases approximately exponentially at a rate of
about 12% per 1000m
3. The transfer of heat when an object at one temperature
placed in a medium of another temperature follows
exponential decay
4. The intensity of light or sound in an absorbent medium
follows an exponential decrease with distance into the
absorbing medium
THE LOGISTIC EQUATION
In 1838 Belgian mathematician Pierre François Verhulst (1804
- 1849). he published his first version of what is now known as
the ‘logistic equation:’
P 𝑡 = 1 /1 - e-t
where P(t) represents the population as a proportion of the
maximum possible number of individuals at time t and e is the
Euler Number, 2.71828183. If the population is 1, i.e. the
maximum possible number of individuals, this is said to be the
‘carrying capacity’ of the environment
THE LOGISTIC EQUATION
THE LOGISTIC EQUATION
The ‘logistic curve’ devised by
Verhulst to describe the
growth of a population through
time until it reaches stability.
The population increase
approximates exponential
growth until it reaches t = 0,
then the growth rate begins to
decline. At population 1, the
population is stable; the birth
rate equals the death rate (or
starvation rate as Malthus
would put it).
It looks great – but there is a small problem. As you will see in
10. The Climate System, if we make tiny changes to the
equation the solutions become chaotic and unpredictable.
PRECISE DEFINITIONS OF TERMS YOU ALREADY KNOW
As you will see later, only a small
dration of the energy emitted by an
incandescent light bulb is in the visible
range
MORE PRECISE DEFINITIONS OF SCIENTIFIC TERMS
F stands for Fahrenheit, the oldest temperature scale still in use, but
only in the United States and Belize.
C stands for Celsius, the metric temperature scale where water freezes
at 0° and boils at 100°. This is sometimes called the Centigrade scale.
K stands for Kelvin, the absolute temperature scale; at 0 K there is no
motion of molecules, at least in theory. There are 100 K between the
freezing and boiling points of water. Degrees Kelvin are K, no degree
symbol is used. 0 K is -273.15 ° C.
The ‘calorie’ is now obsolete, replaced by the Joule, but some of us
remember that 1 calorie is the amount of energy required to raise 1 g of
H2O 1° C. 1 calorie = 4.18 J.
BUT you also need to remember that 1 Calorie (with a capital C, still in use
to describe the energy in food) = 1000 calories.
234.18 × 103 J / kg is the specific heat of water – the amount of energy
required to raise its temperature 1° C (this changes slightly with the
temperature of the water).
0° C = 273 K = 32° F = the freezing point of water at sea level on the surface
of the Earth.
100° C = 373 K = 212° F = the boiling point of water at sea level on the surface
of the Earth.
Your normal body temperature is 37.1° C or 310.1 K or 98.6° F; each day you
perform about 10 × 106 J of work.
A Watt is 1 Joule per second (1 J/s). There are 86,400 seconds in a day, so if
you perform 10 × 106 J of work in a day, your wattage is 115.74.
Actually, your energy flow at rest is about 75 W, but can rise to 700 W
during violent exercise.
The average temperature of the earth for the period 1940-1980 is usually
taken to have been about 15° C = 288 K = 50° F. Now it is about 0.6° C = 1.1° F
warmer.
The average temperature of the surface of the sun is 6000 K = 5727° C =
10,277° F .
At 0° C, the latent heat of vaporization (or condensation) of water (L) is about
2.5 × 106 J kg -1; at 100 ° C it is 2.26 × 106 J kg -1.
The general relation for latent heat of vaporization of water is L = (2.5 0.00235 × C) x 106 J kg-1; (where C = temperature Celsius).
At 0° C, the latent heat of fusion (or freezing) of water (F) is about 0.335 ×
106 J kg-1.
If ice sublimates to vapor at 0 ° C, the latent heat of sublimation is about 2.83
× 106 J kg -1 (the sum of L and F).
The solar constant, the amount of energy received by a panel in space facing
the Sun at the average distance of the Earth to the Sun, averages 1353 watts
per square meter (= W/m2, = Wm-2) as measured by satellites, but it varies
slightly.
The 1353 is usually rounded up to 1360 Wm-2. To get the amount
received in a day by 1 m2 at the top of the atmosphere of our
almost spherical Earth , you need to divide by 4 (the surface of a
sphere is four times the area of a disc of the same diameter).
Using 1360 Wm-2 as the solar constant, that works out to be a
convenient 340 Wm-2, which everyone cites as though it were the
real value everywhere. But a lot happens between the top of the
atmosphere and Earth’s surface.
The average insolation (amount of energy received from the Sun)
at the Earth's surface in the equatorial region is about 200 W/m2.
The average insolation at the Earth’s surface at Boulder, Colorado
is about 200 Wm-2 in summer, but drops to 100 Wm-2 in winter.
The geothermal heat flux from the interior of the Earth is about
60 x 10-3 Wm-2 (60 mWm--2)
mW = milliwatt. The geothermal heat flux is about 1/22,550th of
the energy flux from the Sun.
For atmospheric pressure a variety of terms are still in use:
In Thomas Jefferson’s day atmospheric pressure was measured using the
barometer, which had been invented in the 17th century. Early barometers
expressed pressure in terms of the height of a column of mercury (Hg) it
would support. In the US it was given in inches (in) of mercury. Sea level
pressure was 29.92 inches Hg.
For the metric system, the average sea-level pressure at the bottom of the
atmosphere was supposed to be 1 Bar. In the US today, atmospheric pressure
is usually given in thousandths of a Bar, millibars.
Unfortunately, when the Bar was established it was determined based on
measurements of atmospheric pressure in Europe and North America, in a
zone of low pressure.
The global average is actually 1.01325 Bar or 1013.25 millibar.
The metric system now uses Pascals, rather than millibars, and if you are in
another part of the world the weather report will give you the pressure in
hectapascals (hundreds of Pascals) which, happily, happen to be equal to
millibars
An Avogadro, named in honor of Lorenzo Romano
Amedeo Carlo Bernadette Avogadro di Quaregna e
Cerreto (1776-1856), is 2.06 x 1023 things. You will
hear a lot more about him later.
A mole (M) is the mass of an Avogadro of things. 1
M = 2.06 x 1023 things expressed in kg.
A hectare is 10,000 square meters (a square 100 m
on each side) and is equal to 2.47 U.S. acres. An
acre is 43,560 square feet.
FEEDBACKS
Feedback is the process by which part of the output of a system is
returned to its input in order to regulate its further output.
Feedback can be either positive or negative.
You know positive feedback as the screech from the loudspeakers
when the microphone picks up the performing rock star’s electric
guitar, amplifies its sound, picks up that sound, amplifies it even
more, and on and on until everyone is holding their ears.
Unfortunately, some elements of the climate system are prone to
positive feedbacks, so that a little perturbation by one factor leads
to enhancement of that perturbation by another factor, and so on.
Negative feedback is used to control systems. Negative feedback
causes the effect of a perturbation to be reduced and may result in
its eventual elimination.