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 nogeometry, 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!

Renatus Cartesius a.k.a. René Descartes (1596-1650), in his Dutch finery

LOCATING THINGS IN SPACE

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

For locations on the Earth we use a sperical version of the Cartesian coordinates – a

LATITUDE-LONGITUDE grid

Equator (0° Latitude)

Prime Meridian (‘Greenwich; Meridian’ = 0° Longitude)

90° W Longitude

90° N Latitude180° E or 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 /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° =/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 distancebetween lines of longitude change with latitude. How? The distance isproportional 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° Nis 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 PROJECTIONSRepresenting 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 nautical197 miles). Multiply those by 2 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 fourthof 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 heknew 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 expenseof 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 thenumber 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 ten-millionth 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).

. 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).

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.

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 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 METRIC SYSTEM THE STANDARD FOR WEIGHT

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 systemusing 60’s as its base.

THE PART OF THE METRIC SYSTEM THAT DIDN’T SURVIVE – THE CALENDAR

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 SCALESThe 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. Theirscale 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 (1850-1914) 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 thenumber 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 is666 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 weget 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 laterpopulations get checked by famine and disease. He did not believe that humanswere 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)

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

EARTH’S HUMAN CARRYING CAPACITY

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 (water and food supply, pollution, climate change) facing us today is overpopulation.

Lester Brown’s book (2011). World on the Edge, outlines the problem in detail.

From the Club of Rome’s 1972 ‘Limits to Growth’

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 GrowthDonella Meadows, Jorgen Randers, Dennis Meadows (2004): The Limits to Growth - The 30 Year Update Lester Brown (2011): World on the EdgeAnders Wijkman, Johan Rockström, (2011): Bankrupting NatureJorgen Randers (2012) 2052

EARTH’S HUMAN CARRYING CAPACITY

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

In 1838 Belgian mathematician Pierre François Verhulst (1804 - 1849). he published his first version of what is now known as the ‘logistic equation:’

P1 /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 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, thepopulation 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

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.

MORE PRECISE DEFINITIONS OF SCIENTIFIC TERMS

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.