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

Logical Morality

There was a time when I considered myself a very religious person. Right now I would consider myself to be more of a confused person. In my time as a dedicated Christian I noticed something about many of the moral stances a Christian might take. I found that generally they can all be justified on a logical basis with a root in human rights. Being the logical minded fellow that I am it was typical that I would not truly embrace a moral stance unless I could justify it rationally. That always made more sense to me than ‘the bible says so.’

As an example I will start with the hot topic of abortion. This usually gets people fired up: ‘Pro-life’ vs. ‘Pro-Choice’. I will let you know right off that I do not agree with abortion so there could be bias in my argument. However, I have done my best to eliminate that. As I mentioned before we need to declare our human rights. This is similar to our Bill of Rights; the right to life, liberty, pursuit of happiness… that kind of stuff. The major right applicable to this argument is the right to life. It is a difficult one to argue against. I am going to state it as this: All humans have the right to life. Simple.

As far as the abortion argument is concerned we need to determine if the mother is violating the right to life of the unborn child. This comes down to whether or not the unborn is considered a human or in other words, if it is granted the right to life. Since we all can agree that a child which was recently born is alive and has the right to life, we will start there. The first question is if the act of being born makes you ‘alive.’ I think it is difficult to think that a child which is moments from coming into the world is not alive and one who has been birthed is alive. Additionally there have been several murder trials were a late term unborn child was considered a murder victim. This brings us into the womb.

Now, one would need to decide at what point you become human or alive in the womb. This is a difficult question and I am not going to attempt to answer it. My thought is that we just don’t know. What determines life anyways? Because I consider the violation of the right to life to be major abuse I would prefer to avoid it at all costs (when in doubt avoid violating someone’s rights.) Thus we should define life at its least common denominator. In this case it would be after conception.

This is certainly debatable, but in the end we are just a collection of cells. Whether we are bazillions or two makes little difference to me, they are still human cells. If at some point in time someone presented a logical argument for the point which life begins I would be happy to consider it.

There are many counter arguments to the pro-life stance. One which I would like to quickly address is the woman’s right to choose. To be honest I think it is silly. Nobody has an ultimate right to choose as they will; there are other people’s rights to consider. Additionally we have other choices to make, such as the choice to have sex. This brings up the case of the raped girl caring the child of her assailant. This is a difficult topic, to be fair it is rare compared to normal situations of abortion. This is one situation where the right of life of the unborn could be weighed against other considerations. However I still have to hope that the right to life stands above other rights and that it is always given careful consideration.

In the end this become complicated and I guess my moral of the story is that we should take a more logical, thought out approach to our morality and maybe we can all come to agreements on some of these hot topics. If anything the defining of clearer questions (such as when does life begin?) produces more productive arguments, allowing for real progress.

P.S. To humor myself I may write up a few more logical thoughts for other moral decisions.

Book Review: The Thorn Birds by Colleen McCullough

A Celtic legend from the first pages…

"There is a legend about a bird which sings just once in its life, more sweetly than any other creature on the face of the earth. From the moment it leaves the nest it searches for a thorn tree, and does not rest until it has found one. Then, singing among the savage branches, it impales itself upon the longest, sharpest spine. And dying, it rises above its own agony to out carol the lark and the nightingale. One superlative song, existence the price. But the whole world stills to listen and God in His heaven smiles. For the best is only bought at the cost of great pain…Or so says the legend."

I haven't written a book report since elementary school, but I am excited to report on this one! To give a little perspective before I begin; I am not a person who enjoys reading books with exorbitant amounts of description. Books that describe the trees and the birds and the emotions and any other nic nac thing are usually tiresome. Granted I have a better appreciation for them now than I did when I muddled through the Grapes of Wrath as a required read in high school. Also, I have historically not enjoyed romance novels much either. They seem to prey on the weak minded by exploiting rudimentary, but strong, emotion without the need for character development. That makes it very difficult to identify with (or despise) the characters. I am not ashamed to admit that as I get older my notions of romance have surfaced. For that reason, it's possible that I would not have enjoyed this book as much in one of my previous lifetimes, but somehow I seriously doubt it.

If you've been paying attention so far;), you've probably already gathered that this book does indeed use a moderate amount of description to incite emotion. Also, a large portion of the novel is about forbidden romance. If you're very astute you've probably already guessed that I was deeply moved by this book. Truly, it is one of the best reads I have ever partaken. My Mom absolutely loves it. I remember watching the TV miniseries as a kid and being completely enthralled. I probably wasn't more than 7 or 8, but I still remember my parents discussing how well it followed the book.

Reader be warned, I will elude to some "spoilers," but I'll try not to elude to anything you wouldn't gleam from the book jacket.

This book is not an easy task. It is a classic, but I feel it's fallen out of the eye of my generation of readers. It sits around 700 pages of fine print with a larger vocabulary than me, and I suspect the average person has. Also, from time to time the sentence structure feels a little awkward, but definitely not enough to criticize. In reflection, there were long periods when nothing of great significance happens. Perhaps that's the beauty of it: It holds you fixed throughout.

This is a family saga. It is divided into long sections titled by the name of the character it follows most closely through a given set of years. The entire course of the book spans 3 generations from 1915-1969. The main character is by far Meggie Cleary, although as I said each section follows more or less the life of a specific character. The story begins right around her first day of school in New Zealand. She is the only daughter of several offspring of Fee and Patsy Cleary. Patsy is a poor sheep farmer who in a moment of good fortune is invited to move to his rich, estranged sister's large sheep ranch in Gillanborne (spelling?) Australia. She is old and he is to learn the business of running a large operation in order for the family to retain ownership after she passes. Soon after arriving they meet the handsome and charming young priest, Father Ralph de Bricassart. There is an instant reciprocated affection between him and Meggie. I probably don't have to say much more in the way of foreshadowing for you to guess what will happen. Mind you, she is 8 or so and he is 20ish so there is a lot of story left before we get to it so to speak. That age difference is also a major factor in the perception and development of Meggie.

Father Ralph is a very ambitious, yet devout man. He is described as a man of true purity and faith. I infer that he's a man of powerful observational and problem solving skills when applying his talents to others, but no better than a child when pointing those powers on himself. Anyway, he's instantly likeable. Even though he's a priest, surprisingly enough he is easily identified with through his all too human faults.

The first half or so of the book is about life on Drogheda (the name of the sheep ranch) and watching Meggie come to be a young woman. The author's descriptive ability allow no other way to view Meggie as anything but a beautiful, naive, and sadly fragile young woman. She becomes infatuated with Father Ralph at a very young age, which blossoms into a deep Love she doesn't seem to fully understand. Being surrounded by only men, and her mother becoming emotionally closed after the death of her father, she has no idea about her feelings, her sexuality, or herself in the eyes of Father Ralph. Ralph on the other hand, observes Meggie as she falls for him, but is almost obtuse to the idea. He still perceives her as a child and does not respect the magnitude of her feelings. Of course being a priest…and being a priest who repeatedly is described as trying to become "more than a man"…is unable to reciprocate her emotions.

It's during this time that we are first enlightened to the tragedy that is the Cleary family's story. The author has a very unique and effective way of exciting emotion in the reader. Throughout the book, great losses and gains are always very spontaneous and occur entirely over the course of a couple pages. I found myself rereading pages thinking I missed the foreshadowing leading up to significant events. In most cases, the author simply "drops a bomb" on the reader.

Anyway, Meggie is maybe 17 or so when Father Ralph is promoted for the first time. His time at Drogheda is done. Meggie of course, becomes the ultimate martyr of Love. Her advances are ignored and diverted by Father Ralph, leaving her emotionally crippled. Some time passes and she finds herself romantically inclined to one of the new hands, Luke O'Neill. They marry in spite of her feeling for Ralph and move to western Australia where Luke is determined to cut sugar cane long enough to buy his own ranch. By the way, Luke is an asshole, and I saw Meggie's agreeing to marry him a direct result of the emotional inadequacy paralleling so many women around 20 years old. My attachment to Meggie made me frustrated that she was settling for Luke rather than Ralph. Meggie lives with a nice couple as a housemaid while Luke is away with the cane. Months turn to years and Meggie's motherly instincts kick in despite her frustration with her husband. After she got pregnant she despised the baby, even once it is born. I think she was disappointed that it wasn't Father Ralph's baby…who I think by this time was Arch-Bishop…but it might have been out of disdain for Luke.

She is sent on vacation to some tropical island to relieve some of the stress her life has accrued. About this time, Ralph shows up realizing his Love for her. Through a bit of awkwardness he succumbs to his human desires. I want to make a point of this. I was impressed beyond words at how the Love scene was written. I've read scenes that range from modest to what I could only describe as deeply pornographic, but nothing like this. Emotion is the only thing described. It is at this moment that Ralph realizes that he will forever be a man. Forgive my pagan reference; it's as if this is his realization that this woman was his true path to God. This is why I can identify with Ralph. The resounding fact is clear: we are not infallible; that at the heart of it all, we are all bound to the tragedy that is our own existence. With all our passion and ambition and progress toward a better self, we are inevitably human in all its splendor and disrepute. We cannot escape it. This is the underlying theme, and source of tragedy in this book.

I haven't mentioned her first baby, Justine. She embodies everything I dislike in strong willed women. She is one of those women who are a little too big for their britches. This is shown as she makes a series of irrational decisions. She is also emotionally crippled from the start. It's not until her 30s that we see her realize the error of her ways. Fortunately, her brother Dane, and Ralph's lovechild(I bet you never saw that coming;)) is the image of Ralph himself and is instantly likeable. Meggie leaves Luke and neither child ever meets him. Of course, nobody is fooled as to the real father of Dane.

Unlike Ralph, Dane life does not follow my insight into the human tragedy. Rather his life is the tragedy. He is pure. So pure, he enters the priesthood under his unknowing father. Unfortunately Dane's story is ultimately a large part of downward spiral of the Cleary family.

As Meggie endures through the years she becomes much less fragile and much more emotionally withdrawn – much like her mother. That is a tragedy of its own. Her fall from innocence and grace weighed on me as much as any other loss to her or her family. The story ends with Meggie as a woman entering the twilight of her life. Her family is in disarray and she is lonely; reflecting on her life, her children and her business. But still, she is there, persevering till the end in such a way that can only be felt with deep sadness.

I've written more than I wanted to and I don't feel I've done the story justice so I'll leave you my deep thought taken directly from Meggie.

And now, a deep thought

"Each of us has something within us which won't be denied, even if it makes us scream aloud to die. We are what we are, that's all. Like the old Celtic legend of the bird with the thorn in its breast, singing its heart out and dying. Because it has to, it's driven to. We can know what we do wrong even before we do it, but self-knowledge can't affect or change the outcome, can it? Everyone singing his own little song, convinced it's the most wonderful song the world has ever heard. Don't you see? We create our own thorns, and never stop to count the cost. All we can do is suffer the pain, and tell ourselves it was well worth it."




Oh NASA, My NASA!

My work has made me a bit more aware of the current state of science in America and the rest of the world. So far my contributions are small in the scheme of things, but they have given me the opportunity to observe how others are contributing. When I go to conferences I'm always left with a frustrated feeling because typically only one or two presentations seem novel, pertinent, or non trivial. Students latch on to their professor's work which may or may not be important. They end up making a career out of work that no one will care about.

I'm not the best researcher out there. My abilities are always growing, but I can't say that they will ever be as great as I would like them. Like I said, my contributions are small so far, but then again, I'm still just starting out. In time I expect to be a significant contributor. I see a great need for CFD, but I also see an over-reliance on it. By using the "black box" engineering technique the researcher loses all physical concept of the problem. Even if he programs the computations himself, it is still just mindless math unless he formulated the problem as well. I'm fortunate enough to be brought up as an analytical researcher. The trade off is a gain in physical reasoning, but a serious complication in solving problems with significant complexities. Many problems can only be solve computationally. Some can be solved with a combination of the two.

The buzzword around the propulsion community is "instability." It is an ominous word to those who know what it is. The important part for propulsion is simply explained as an observed growth in pressure oscillations and mean pressure. There are a number of mechanisms to create this phenomenon. It's amazing though, every rocket designer builds a rocket with all the best specifications and are shocked when the flow is unstable. It's almost guaranteed to happen. Expect it, plan for it as best you can, but expect it.

The following poem makes me sound like I'm blasting NASA. I do mean to poke fun at them for some stupid things they have been doing for the last 60 years, but understand that I think they are showing promise. I also make some bold statements. In case Big Brother is reading, I'm well aware that the problem is not so easily solved. I also think the right people are working on the problem now.

This poem is loosely based on Oh Captain, My Captain by Walt Whitman

Oh NASA my NASA! The journey has begun,

The shuttle has failed, the funding has bailed, the ARIES project won.

The moon is right to set our site, the daunting task ahead.

Been done before, now we want more, but documents lay shred.

Oh NASA my NASA! How much you waste to save,

Projects canceled to save the hassle, took back are funds you gave.

The tools were here for 60 years, yet common sense still slips

Bald heads ponder and weak minds wander, yet no one comes to grips.

Oh NASA my NASA! What makes the rockets blow?

And eyes that jiggle and structures that wiggle, when flight command says go?

The flow's unstable but you're not able to figure out just why,

But if you read and used your head, you'd find the work you buy!

Oh NASA my NASA! Don't reinvent the wheel!

People are jeering and countries are sneering expecting you to fail!

But you can prevent your imminent death, just scan the pasts research,

Or come to me, I'll do it for free, and show you what will work!

to continue the poetic theme...

and now, a deep thought

-I was once told, the sky is blue,

-The oceans, the lakes, and some rivers too!

-But what would I see from your point of view?

-Would blue still be blue, or a different hue?