A Mathematical Perspective of Global Warming

Global warming and the environmental impact of humans is a hot topic in the news as well as in the scientific community. People, myself included, observe localized climate change. These localized observations are often projected into a global observation by those less informed or those prone to paranoia. To say "the climate is changing" is an asinine statement even to someone who doesn't have the mathematical perspective. Yes, the climate is changing. It has been changing since Earth was a fledgling planet. It will continue to change long after humankind has killed itself off.

When a normal human measures time in seconds, minutes, hours, and so forth, geologists (and I think physicist use the same term when talking about the age of the universe) use the idea of "deep time." Deep time doesn't bother with minutes or seconds or even years. This perspective considers time in chunks of millions of years. In doing so, all resolution on the relative small scale is lost. For perspective John McPhee describes deep time in the following way [1]

"Consider the earth's history as the old measure of the English yard, the distance from the King's nose to the tip of his outstretched hand. One stroke of a nail file on his middle finger erases human history."

In the 4 or 5 billion years the earth has been around, humankind has made its lasting mark only in the last 10,000 years or so, barely enough to talk about when considering deep time. Over the course of Earth's history, dramatic climate change has been caused by the motion of the continents, the instantaneous (remember "instantaneous" in deep time has a completely different meaning) atmospheric conditions, oceanic currents, and the Milankovitch cycles. Some of these effects are byproducts of others that additively support climate change, some are strong enough on their own, and some are independent but must be supported by another factor in order to produce a drastic effect.

Now that I have tried to give perspective on deep time, I will take a step back. I am not sure if the Milankovitch cycles are considered deep time. This term refers to the eccentricity (elliptical nature of Earth's orbit), axial tilt (the angle of the axis of rotation measured from the perpendicular to the plane of orbit), and precession (the wobble of Earth on its axis; like a top about to fall over) of Earth as it travels around the sun. "The Earth's axis completes one full cycle of precession approximately every 26,000 years. At the same time, the elliptical orbit rotates, more slowly, leading to a 21,000-year cycle between the seasons and the orbit. In addition, the angle between Earth's rotational axis and the normal to the plane of its orbit moves from 21.5 degrees to 24.5 degrees and back again on a 41,000-year cycle. Currently, this angle is 23.44 degrees and is decreasing [2]." Individually, these effects may not have enough strength to cause drastic climate change, but their periodic additive nature is attuned to the 100,000-year ice age cycles. Even if the individual parts of the Milankovitch cycles don't create ice ages, they certainly will cause global climate change. Whether it is a general time shift of the seasons, or regional change in climate, the effects are certainly present.

Now you may say "this is all very interesting, but so far there is no math." Don't worry my friend, the math is about to begin! The term "chaos" has become an everyday word to describe something's random or unpredictable nature. It is a horrible misnomer that is like fingernails on a chalkboard to me when I hear it used improperly. Dr. Malcolm from the Jurassic park movie sparked a public interest in chaos by using it as a tool to warn Mr. Hammond of the dangers of creating dinosaurs. He did NOTHING to help expand the general knowledge of chaos. He wasn't entirely wrong, but his perspective on chaos being completely random certainly is. Mathematically, in the nice little world we wish we lived in chaos does look random. However, if we look a little closer we see great structure and organization in chaos.

In 1963 a mathematician and meteorologist from MIT name Edward Lorenz inadvertently stumbled across true mathematical chaos. He was not the first, maybe Poincare was, but his discovery did give chaos theory its roots. The story goes Dr. Lorenz was working on a 3D model for convective roll in the atmosphere as a step toward weather prediction. The equations are a set of coupled nonlinear ODEs given as

I don't exactly know what the greek constants represent. "All σ, ρ, β > 0, but usually σ = 10, β = 8/3 and ρ is varied. The system exhibits chaotic behavior for ρ = 28 but displays knotted periodic orbits for other values of ρ. For example, with ρ = 99.96 it becomes a T(3,2) torus knot [3]." Anyway, he's solving these numerically on the old style vacuum tube computers. These vacuum tubes often broke mid-simulation and the customary way of proceeding was to replace the tube and back up in the calculations, restarting the simulation using the values at the new starting spot as the initial conditions. The solutions are mesh together at the end. He did this, and noticed that initially everything was fine. The code reproduced the same numbers it had before even though he had to restart the simulation in the middle of the interval. After a short while he watched a very small error form in the smallest decimals place. This error grew until the original numbers were nothing like the new ones calculated. This was chaos. This type of chaos occurs in nonlinear systems when small differences in initial conditions propagate into major differences in the final solution. This is referred to as the "butterfly effect." A butterfly flaps it's wings in South America and the motion of the air causes a typhoon in Indonesia. In his case, the machine calculated to more decimal places than it displayed maybe 8 to 4, I can't remember. So his initial conditions were missing 4 decimal places of accuracy. In most cases it doesn't matter, the final solution will be indistinguishable, but not in chaos! I personally have seen chaos rear its ugly head all the way to the 16^th decimal place and beyond when programming in different languages! Dr. Lorenz had discovered what is now called the Lorenz Strange Attractor. In the real world sense, his problem is unpredictable, but if we look closer, we see clear order. The first picture is a phase plot of the Lorenz attractor. The second is a Poincare map. Both show definite structure, but not necessarily predictive capabilities.

Hopefully you're beginning to see the picture I'm trying to paint. Lorenz's predictive capabilities diverged within a couple months because he couldn't account for 4 decimal places. Translating that into the global climate means the exact affect of greenhouse gases, continental drift, oceanic currents, Milankovitch cycles, volcanic eruption, electricity production, human population, ecological oxygen production, etc. must be incorporated. It is ridiculous to consider calculating for all those affects. Even if we did, chaos is still at the root of convective roll; one of the simplest parts! And yet people still try.

A major force in the development of computers 50 or 60 years ago was to predict the weather on the short and long scale. More complete weather prediction methods use the Boussinesq Equation. It is an approximate representation of the Navier-Stokes equations and the Energy equation tailored to produce weather models. I say approximation because that is exactly what it is. It makes a huge assumption about mass conservation being "incompressible" (more accurately constant density) while convection driven density changes remain in the momentum and energy equations. It seems to work alright in practice but is inherently wrong.

So what options do we have? The answer is statistical analysis. Today we have climatic records dating back several thousand years and geological records dating back much farther. Those records suggest that during the period of human civilization and more drastically since let's say the industrial revolution, the climate has been affected. I can't argue that. In fact I feel that we have had a significant and negative impact on the global climate. I'm simply pointing out the problems with blindly jumping on the band wagon. Anyway, the problem with statistical analysis goes back to the deep time concept. I think our geological records have climate predictions back 700,000 years or so, I'm not sure exactly. Statistically, an experimentalist needs something like 11 samples to begin to get an accurate standard deviation and draw viable conclusion. So when thinking about deep time, 700,000 years isn't necessarily enough. An even deeper problem is that in a lab an experimentalist controls all the environmental variables. That way every test is the same. In the context of global climate, we're back to the problem of continental drift, oceanic currents, Milankovitch cycles, geothermal characteristics, and so on and so forth. The external stimulus has never been the same over the course of the history of Earth in the context of deep time. Therefore, you have 700,000 years of single data points that cannot be applied reliably to the next 700,000 years or more from a statistical perspective. In the local scope you can make observations toward the "human element," but since 700,000 years is barely deep time, the causes and effects are mere speculation.

I'm almost done, stay with me. In this day and age we seek to find more ecofriendly ways to maintain our standard of living. Alternative fuels such as methane, ethanol, and biodiesel are common buzzwords. Fuel cell technology has been implemented for everything from space shuttle power generation to electric power plants to powertrains for cars. The push for these technologies should be handled with care. More than they are environmentally motivated, they are seen as a way to relieve the dependence on oil. For that I can't knock it. Eco-hippies and generally uninformed people alike should realize that perfect combustion produces CO2 and water vapor. What are two significant greenhouse gases? CO2 and water vapor! Sorry combustion based alternatives. What about fuel cells? I'm not sure about all of them, but I know Proton Exchange Membrane (PEM) fuel cells produce electricity through the ionization of hydrogen and recombination with oxygen. Bingo! Water vapor! Since this type of fuel cell operates at a manageable size and temperature, they are applicable to the automotive industry. Either way, a popular way to get hydrogen is through electrolysis. That typically uses combustion for electricity generation! The same goes for electric vehicles. Economically they remain viable, ecologically, maybe not so great.

I'm not saying humans haven't changed the climate, I'm saying there is more to it than what meets the eye. We need to weigh the facts before we throw money at a problem with inherent flaws. Alright, I'm done. Thanks for stayin with me!

And now, a deep thought

-What would happen to the lemmings if at the edge of the precipice one stopped and said, "Wait? This doesn't seem right!"

[1] McPhee, J., Basin and Range (1981) ISBN 0-374-10914-1. Republished in Annals of the Former World.

[2] http://en.wikipedia.org/wiki/Milankovitch_cycles

[3] http://en.wikipedia.org/wiki/Lorenz_attractor