Note -- This is yet another idea by Richard Klein, but because of its length and complexity, it has been given its own tab or subsection. The topic has little to do with bicycling and getting rid of training wheels, other than the fact that both topics are rooted in mathematical philosophy. In a sense, this topic is somewhat autobiographical as I will reveal some of my inner thought processes. For five other ideas, also with an autobiographical flair, refer to the nearby tab labeled "About Ideas."

When we think back to the trials and tribulations of Galileo, he encountered fierce hostility from the establishment, the church, when he argued that the Earth was not the center of the universe. His premise was that Earth rotated around the sun, and not the converse. What Galileo had in hand was visual evidence that allowed him to make a hypothetical explanation based on observations using his newly devised telescope.

When we start to ask questions concerning something as immense in both time and space as ice ages, the time scales of our human lives are dwarfed. The conventional rules of science involving the four steps of hypothesis, experiment, validation, and publication cease to be viable options, as we run out of time and resources as mortal humans. When we deal with processes beyond the time frame of our human lives, we must resort to reasoning and hypothesis, along with aid using properly constructed computer models. With that said, let us proceed to discuss the centuries old riddle of the origins of the ice ages?

In order to set the stage, I will make an assertion, or what can be called an hypothetical explanation.

The Earth’s record of ice ages, also known as glaciations, is little more than an “Old Faithful” process wherein the deep oceans uncork and thus de-gas on a periodic basis. Moreover, the cause or “trigger” of the deep oceans de-gassing (and thus the timing of the ice ages or glaciations) is internal or self-generating to the process, and not due to the presence of some exogenous (external) trigger. In essence, the ice ages are the result of a cyclic process wherein deep oceans gradually absorb atmospheric carbon gases, thus causing the Earth’s atmosphere to gradually cool. With diminished atmospheric carbon concentrations, the Earth’s temperature cools and the Earth gradually enters into an ice age because of this atmospheric cooling (due to removal of carbon gases from the atmosphere). The upper layer of ocean water in turn cools being in contact with the cooler atmospheric air. The cooler upper layer of the ocean increases in density in large part due to cooling, as compared to the deeper lower levels. This density stratification results in a classic hydrodynamic inversion. The more dense surface waters sink at some point, and this sinking action displaces the deeper and carbon laden ocean water. As the deep ocean waters rise the hydrostatic pressure decreases, and the ocean undergoes a tumultuous upheaval due to the degassing. In terms of the champagne bottle analogy, the bottle periodically blows or pops its cork. Old Faithful in Yellowstone can serve as a metaphor to aid our mental grasp of the process. Said in other terms, the ice ages of the Earth are analogous to the periodicity of Old Faithful. We note also that when Old Faithful erupts or ‘blows its cork,’ the eruption is violent and sudden. In short, a common trait of bodies of water that degas, the degassing is a violent and sudden event.

In order to explain this hypothetical explanation to a lay audience, I elect to retreat and start with some comments on science and how the human mind goes about discovering the inner truths of the physical world. As a scholar with three earned degrees and expertise in mechanical engineering and systems theoretic principles in particular, what I write to follow will be one where I pick and choose my words with excruciating care. I assume that my reader is a lay person and as such I will hopefully restrict my words and explanations to suit my audience.

Science is an interesting and yet often misunderstood word. If we are to understand science we must first understand that science is a disciplined study of various aspects of the physical world, or the world made up of living organisms, or possibly the study of how humans and living things act and react. Hence, science can be divided into the physical sciences, the life sciences, and the social sciences. In the following I will focus on the physical sciences – where we are trying to discover and hence uncover some truths concerning how and why the world works. The scientific method is based upon what involves hypothesis, experiment, observation, and conclusion based on a four-step process related to an experimental outcome. Not all persons regarded as scientists adhere to the four-step process. As an example, few would dispute the assertion that Albert Einstein was a scientist. What is interesting about Einstein is that he studied the world using his extraordinary powers of mathematical reasoning. It was mathematical reasoning that lead Einstein to pronounce his special theory of relativity – and it was literally decades later that others came along and used experiments and observations as per the scientific method to validate or conversely invalidate Einstein’s conclusions.

The topic of global warming is in the news daily, and it can in every sense be called a hot button topic. Because of the enormity of scale of the Earth, the long time durations of the global temperature balance, and the absence of a control experiment; if we as a society are to unravel the mysteries of global warming and periodic glaciations, the application of the four-step scientific process is not a viable option. In addition, it is sad that most who are weighing in on global warming have long ago abdicated science and have entered the world of politics and political scheming. I say that because the global warming combatants seem to have become so focused on global warming as a given and as an evil. The extant body of findings regarding glacial periods and interglacial periods gets short riff. The body of peer-reviewed literature of glacial periods and interglacial periods is largely undisputed for all who care to look, but the global warming crowd can’t be bothered with looking at the evidence. In my view, the following five assertions can be made and are supported by a general consensus in the literature:

1. The Earth has undergone a succession of glacial periods.
2. The glacial periods are, moreover, interspersed with warmer periods in between called interglacials.
3. The most recent glacial period of major significance occurred approximately 18,000 years ago.
4. The glacials and interglacials exhibit a periodicity, and thus a regularity which allows them to be described in terms of specific periods.
5. The extant peer-reviewed record shows that both the magnitude and abruptness of the cycles are significant.

As many as sixty theories have been proposed in the last several centuries, each attempting to explain the causes and origins of the ice ages. None of these theories are even close to being accepted by scientists as a whole, and none come close to explaining, in particular, the last aspect – the magnitude and the abruptness of change such as reversal from a deep ice age back to a relatively warmer climate. This is the point where my attention was aroused, as I am by training a mathematician and one interested in mathematical riddles involving causality issues.

As a control systems theorist and practitioner, there are many areas of control that interest me. One interest of mine is in an area known as non-linear control. It is a proven mathematical fact that non-linear feedback systems can exhibit, given proper conditions, a behavior known as limit cycles. A limit cycle is like a self-induced chatter – a self repeating or internally generated zig-zag. One example is the side-to-side swinging of the steering mechanism on a towed farm implement, as it sways from side to side on the highway while being pulled. If one looks at long-term Earth climate swings, it becomes apparent that the ice ages tend to resemble periodic behavior. If one observes a periodic oscillation or behavior in a system, the cause might be external – or the cause might be internal, and hence a limit-cycle type of behavior. I took the approach of hypothesizing that ice-age related climate swings are a limit cycle based phenomenon and thus symptomatic of a closed-loop feedback system. Conversely, I reject arguments claiming that external driving forces are the cause of the ice ages. The reasons for my rejection of the external exogenous hypothesis will be discussed later. Now, such a closed system, if self-generating Lyapunov limit cycles are present, characteristically has feedback loops, nonlinear properties, as well as an energy source.

The longest of the recognized ice age cycles is estimated by most scientists to be of the order of 100,000 to 110,000 years. A discussion of how climate scientists recover the past temperature records of the Earth is beyond the scope of this discourse, but in short a number of techniques are used, including both land records as well as ocean core methods. There is general consensus from an array of independent sources and studies that agree in large part. One of the more well known dating/temperature methods comes from the Vostok ice cores, and is named as the Vostok record. In what follows I will refer to the longer Vostok cycles as 100,000 year cycles. Moreover, shorter cycles are extant at 25,000 years, 33,000 years, and 41,000 years as well as other shorter periods, some at 1,500 years and some even shorter such as El Nino cycles. Modern digital computing algorithms allow scientists to compute what is referred to as power spectral density charts (based on digitized Fourier Transform techniques). The resulting power spectra clearly show the presence of periodic cycles as well as the strength of each of the various cycles. As a side quirk of the Fourier series method of representing a time history with sharp changes, one often gets or sees what is referred to as alias resonances in the truncated power spectrum. Such higher frequency spectra are expected due to the mathematics of trying to represent a time history with a Fourier series composed of sinusoidal functions, but these have little physical meaning.

My research and mathematical investigations focused on the 100,000 year cycles, and for several reasons – (1) these are large amplitude cycles with abrupt characteristics, (2) the search for an internal “time constant” associated with an internal explanation to ice ages points one rather quickly to the oceans as a capacitance capable of storing dissolved carbon-dioxide and its various compounds, and (3) the abruptness of the cycles (as the Earth comes out of an ice age) suggests the action of some form of hard or sharp nonlinearity within the closed system. If one is to be serious about explaining ice ages, one has to explain, in effect, several things:

1. Why does the Earth slip or descend into an ice age (as opposed to possibly rising in temperature)?
2. What mechanism is present to cause the Earth to come back out of an ice age (as opposed to remaining in a snow-ball covered Earth outcome)?
3. Why does the cycle display such an uncanny periodicity or regularity?
4. What explains the abrupt or cataclysmic tendency of fast temperature reversal as the Earth comes out of an ice age?
5. Is it possible to get a handle on the matter of the cycle’s stability? On this last point, as mathematicians know who are versed in Lyapunov limit cycle behavior, the presence of a cycle for a limited number of cycles is not a proof of stability of that observed cycle. Mathematicians and computer experts in non-linear simulations are also keenly aware that “Simulation is not a proof.”

The fourth question or requirement is particularly baffling as the climate record shows that the Earth returns from a deep ice age or glacial back to quite normal temperatures literally in a flash – some twenty years which is next to nothing as compared to the long term base cycle of 100,000 years.

Grounds for Rejecting the Exogenous Hypothesis. Exogenous arguments such as Earth orbit changes, continental drift and respective plate changes, interstellar cloud encounters, or fluctuations due to pulsating solar output fail miserably when examined. An example might be the notion that some as yet undetected interstellar meteor cloud intervenes on a 100,000 year basis and thereby solar energy dims at that periodicity – and yet no evidence is extant to suggest the existence of such a solar dimming agent or mechanism other than its presence as an unseen hand. Instead of hypothesizing some unseen hand acting exogenously, I have every confidence that the explanation of the ice ages must be internal, and not exogenous. My conviction can be defended on one point in particular – that the ending of the great 100,000 year ice age cycle is a sudden event, as it occurs according to the best evidence available, in a miniscule instant of about ten to twenty years. I call this an “instant” or mere wink of the eye when we are dealing with time frames of a hundred thousand years, and yet the Earth’s temperature rebounds in one quick eye wink, so to speak, of the Earth. Mechanisms such as known orbit changes and even solar fluctuations would necessarily produce far more gradual temperature changes – changes that evolve over thousands of years given that the hypothesized exogenous driving mechanism has a base periodicity of 100,000 years.

Lake Nyos and other considerations. I will keep the explanations simple, and merely state – as an hypothesis -- that the Earth experiences periodic ice ages because the deep oceans act like capacitors capable of absorbing carbon-dioxide gasses – as well as violently belching back those gasses. The cooling of the Earth brings with it an ice age as the oceans absorb sufficient quantities of atmospheric carbon-dioxide. Moreover, the ability of the deep ocean to absorb and retain carbon-dioxide gases is critically dependent upon non-linear limits. A liquid can become saturated, and even over saturated. Hence, saturation is a key nonlinearity with the ability to play a role in a carbon cycle temperature control mechanism. A deep body of water can also exhibit hydrodynamic instability – in essence being subject to “flipping,” also commonly known as overturning.

Numerous lakes in the upper Midwest of the United States flip or overturn annually, usually in the late fall. Other deep lakes flip less often however the flipping can result in catastrophic consequences. As a case in point, Lake Nyos in Cameroon Africa "flipped" on August 21, 1986 and thus a violent degassing occurred. The gas cloud released killed by asphyxiation approximately 1,700 nearby persons as well as livestock. The cloud of carbon based gases as released or belched was so massive on that fateful night of August 21, 1986, that the water level of the lake fell about one meter. Carbon-dioxide gas is somewhat heavier than air, especially when cold as would be the case with a deep water degassing, and so a cloud of poisonous gas displaced the oxygen normally present – and the unfortunate residents living nearby went to sleep unsuspectingly that night and never woke. Persons seeking verification of the events of Lake Nyos can find numerous scientific reports by searching under "Lake Nyos Cameroon" using any standard Internet search engine.

To repeat the premise, the Earth slowly enters into an ice age because the oceans gradually absorb by diffusion available carbon-dioxide from the atmosphere. As carbon-dioxide is removed from the atmosphere, the Earth tends to cool due to a lessening of what is called the atmospheric “green house gas effect.” This gradual absorption of carbon-dioxide by the oceans by diffusion requires thousands or even tens of thousands of years, and thus is indeed gradual. But, when the oceans are at saturation and/or if a temperature inversion should develop – where upper layers of water become denser than the deeper water, the body of water is no longer stable. With instability comes the setting for a violent upheaval or change.

Several mechanisms appear to be plausible explanations as what might cause a body of water to de-gas in a sudden and cataclysmic fashion. In what follows please bear in mind that the density of sea water is dictated in large by three variables; salinity, temperature, and pressure.

First, the body of water can become unstable with respect to gravity as this would occur when the upper waters cool, for example as would happen given the onset of winter temperatures as is the case with annual flipping, or with an ice age in the case of a long term cooling of the Earth. In this scenario the upper layers of water would become more dense as compared to lower waters, and thus a gravitational instability situation occurs. In this case, a body of water will become prone to “flipping” as the heavier top waters sink causing the deeper waters –laden with dissolved carbon gases – to rise. We usually call the situation of more dense upper water as an hydrological inversion.

While ocean waters circulate and mix on a continual basis, the reality is that the mixing is largely in the horizontal direction. The layers of ocean water, deep ocean waters in particular, are actually subject to very little vertical mixing – except when an instability or limnic degassing or eruption might occur. Oceans are layered, with three layers in particular; the upper or mixed layer which extends from the surface down to about 200 meters. The middle layer extends from about 200 meters to 1,000 meters in depth. The deep water extends from 1,000 meters to the ocean floor, typically 5,000 meters or about five kilometers. Of course, ocean depths vary worldwide but for the purposes of our discussions the deeper oceans such as the Pacific are the focus of our interest. Temperature, pressure, and salinity control the density of seawater, which in turn, controls the layering structure of the oceans. The deep oceans are deep indeed, and exhibit very little mixing. Without sunlight and absent any significant mixing mechanism, the temperatures in deep oceans remain unchanged and are quite cold as compared to atmospheric standards. When we think of ocean currents those are currents in the mixed or surface layer which have little impact on deep layers.

The second possible mechanism responsible for a degassing mechanism would be when the carbon laden deep water becomes supersaturated. With a state of super-saturation, any small perturbation, such as stirring, or a geological event like a rock slide, undersea earthquake or volcanic eruption, can trigger a release of dissolved gases back into vapor form – and a subsequent rising of nucleating gas bubbles. In the case of Lake Nyos, once nucleating bubbles started to rise from deep towards the top, this brought up more carbon-dioxide rich water which then also started to develop bubbles. The result would be a sudden upward rush of bubbles and, in turn, an upward flow of carbon-dioxide rich water. The reduction in hydrostatic pressure as deep water rose caused additional release as saturation limits were exceeded. The end result was a sudden and catastrophic belching of poisonous gas from the lake – as if the spigots under the lake had been opened. The trigger could be minuscule – analogous to the famous flapping of a butterfly’s wings as discussed in chaos theory. The issue is not the matter of what caused or acted as the trigger, but rather the potential for an eruption based on the fact that a liquid that is laden with dissolved gasses is prone to a limnic eruption given appropriate circumstances.

The third mechanism can best be described as the semi-infinite slab problem. This is a complex topic to discuss, but the discussion is included for the sake of completeness. In what follows in this paragraph, radiation and convection are considered as negligible. This leaves us with conduction which relies on molecular excitation between adjoining molecules. The ability of a solid substance to conduct heat is described by its property known as thermal diffusivity. If we add a boundary condition to represent the periodic fluctuations in surface temperature, then heat transfer engineers can calculate the manner in which the surface fluctuations penetrate into the material, treated as a semi-infinite slab. Two characteristics are present in the semi-infinite slab problem when forced with sinusoidal surface heating. The effect of the changes in surface temperature die out or diminish in an exponential fashion with depth, and the peak of the sinusoidal oscillation lags in time what happens at the surface. As an example, considering annual heating and cooling of a surface like ordinary soil. Surface soil is normally warmest coinciding with the peak heat of the summer, but at depth the time of year of the warmest soil temperature lags. If soil temperature is measured at various depths, we find that a thermal lag phenomenon occurs, and as such there is a depth at which the lower level temperature lags the surface by 180 degrees. If you would dig down, say 1 ½ meters, in the summer, you would find soil that is coldest in its annual cycle. On farms of the past century, root cellars were constructed by digging down about five or six feet, as the temperature is coldest at that depth in the summer months.

The mathematics of the semi-infinite slab problem can be applied to the oceans, remembering that the oceans are stratified and that scant vertical mixing occurs. If we use the thermal diffusivity constant for seawater, and also the periodic forcing at 100,000 years, we get the astonishing calculated result that the ocean temperature at a depth of about 4,000 to 5,000 meters is at a peak when the surface temperature is at a minimum – and thus in an ice age given this scenario. Hence, we have the basis to argue that a third mechanism might potentially enter to suggest that the deeper oceans can exhibit cyclic and yet lagged changes in temperature – and thus a change in the saturation limit. Cold liquids have greater saturation limits than warmer liquids, as most beer lovers can attest, and so we have a possible mechanism to suggest why deep ocean water can become supersaturated – because its temperature rises in delayed response to periodic surface cooling. For those desiring a suitable reference that discusses the semi-infinite slab problem and thermal lag of transients in solids, the text “Heat Transfer,” by Jack P. Holman (McGraw-Hill 1988) will suffice.

When a density inversion occurs whatever be the reason, and we have outlined three reasons which might work independently or in some concert form yet to be determined, the more dense surface water then sinks and the less dense deep water rises. In lake terminology it is said that a lake, for example, “flips.” Lakes of moderate depth commonly flip, often in the late fall. Some lakes are known to turnover or flip several times a year, notably Lake Ida in central Minnesota. Naturalists from the Department of Natural Resources in Minnesota conjecture that the flip in late summer that occurs in Lake Ida is due to issues related to oxygen depletion which influence specific gravity (density) of the water by layer. The flip of Lake Ida in late fall, typically November, is accepted as being due to cooling of the surface water as winter approaches. Lake Okoboji in Northern Iowa is also prone to flipping annually, again usually in the late fall. When these lakes of the upper Midwest flip, it is well known that the lakes get stirred up releasing gasses and in general, bringing debris to the top. As related to larger bodies of water that flip, deep water that is laden with carbon derivatives of gas under tremendous pressure – again rises to the top. When the hydrostatic pressures are lowered as when a liquid rises from the deep, stored gasses are released suddenly causing additional lifting and circulation – an irruption and belch occurs (in this scenario) that would make a Tsunami tidal wave look like a weak puppy. Think of a carbonated beverage bottle or container that suddenly spews out bubbles and its liquid contents as it degasses. A rapid drop in pressure, as when you open the lid of a pressurized container suddenly, compounds the problem as the degassing causes mechanical agitation and stirring of the beverage as bubbles rise.

The Issue of Scale – Looking at Bodies of Water Known to “Flip.”

There is no physical or mathematical question that much larger bodies of water, even oceans, are also capable of inversions and/or super-saturation instabilities and subsequent violent flipping with enormous quantities of gasses being released in very short time intervals. In essence, I assert or hypothesize that the Earth’s oceans “burp” or belch periodically and thus de-gas or vent off gasses in a relatively short period, and the Earth’s temperature rises back to pre-ice age levels – as if in a flash. Moreover, based on issues of depth as well as the volume of water involved, the Pacific Ocean would appear to be the primary body of water central to this mechanism. The periodicity is relatively simple to explain, just as we understand why “Old Faithful” in Yellowstone Park is indeed faithful or predictable. The Old Faithful geyser is cyclic and predictable in a manner similar to how old fashioned percolator coffee pots percolate.

Recall that perhaps as much as 98 percent of the free carbon-dioxide in the air and oceans is presently in the oceans in dissolved forms – waiting to de-gas at the first instant of any super-saturation upset or density inversion. Our current international debate over greenhouse gasses should be focused on the component in the oceans, as that is where the action ultimately hinges. I see this periodic ocean belching hypothesis as a precise and logical explanation for long term ice ages. Because this epistle will be posted in a public web site for lay readers, the mathematics and physics underlying the mechanisms responsible for the inversions and the periodic behavior, the Lyapunov limit cycle, are beyond the scope of what can be presented in text format to a lay audience. Hence the results and conclusions are stated as assertions, so in the spirit of fun let us say that the postings are for your enjoyment and intellectual stimulation.

Of course, critics can challenge my ideas and say that I’m wrong, or they can say that I have no evidence to support my claims, or they can say that I lack the credentials and qualifications to render an opinion on such a weighty topic. I need to caution those who shout “foul,” as they are reacting in a knee jerk sense, and not making a case based on any reading of the literature, knowledge of Earth’s radiation heat balances, and knowledge of theoretic principles in nonlinear system dynamics, notably the theory of Lyapunov limit cycles. Such critics don’t understand science and the scientific method. Moreover, they have scant knowledge of theoretic as well as applied feedback mathematical principles and nonlinear systems in particular. I can no longer be proved right nor can I be proved wrong. Whether any scientist thinks that he/she is right, that does not change the fact that glacials and interglacials exist, and thus there is a definitive reason or causal mechanism. Discussions and even arguments among the various researchers won’t alter any of the underlying facts or mechanisms. Few of us will be alive in tens of thousands of years to see the actual result. The best that we can do is to study the physics of the Earth, to study the climate record, to invoke arguments from physics, and proceed to use mathematical models and mathematical reasoning aided with computer models so as to allow us to make a case. The heat balance of the Earth is relatively simple and thus it is also simple to describe and account for mathematically. The Earth’s heat balance is dictated by radiation in and radiation out. Moreover, the Earth’s radiation balance is dictated by processes well understood in physics and engineering. As a point of philosophy, we don’t require a massive computer with ten trillion pieces of information and super-computing processing to explain why something goes tick-tock or zig-zag in a consistent manner.

In essence, in engineering, especially control systems engineering, its practitioners strive to use low order and thus relatively simple models to explain simple and regular processes. If it was indeed true that we needed to examine ten thousand variables, and if each of the 10,000 variables or parameters played a central role, the outcome would be more like a random walk or chaotic process, not a rhythm as simple as a heart beat – or a clock going “tick-tock.” If the Earth’s temperature record was a random walk, the power spectrum would be flat or “white,” as opposed to a power spectrum with distinct spikes or peaks at specific frequencies. In contrast, the Earth’s temperature record shows strong evidence of periodicity.

The Biblical Account of the Flood and Noah

For the critics who really love to be negative and jump on ideas of which they have done no homework or critical thinking -- we have yet another thought. Genesis tells us the story of Noah and how it rained for forty days raising the ocean's level, and then receding after a period of events recorded in Genesis to expose dry land again. Few of us were around then, so we can not rely on firsthand knowledge. The story of Noah as told to us in Genesis is one ancient record to consider. Imagine if you will, the deep oceans "flipping" and thus releasing enormous volumes of dissolved gasses. The oceans would be as if they had burst their cork -- like a pot boiling over on the stove. In such a scenario, it is quite plausible that the Earth would pull itself out of an ice age in a mere heartbeat as compared to the 100,000 period of the glacial-interglacial.

Genesis tells us about the great flood and Noah. Most of us as readers look at the issues raised of forty days of rain and then the waters ultimately receding. What we need to do is to look at the Genesis passages in detail (all Biblical quotes below unless noted are from the New International Version of the Bible):

1. The Genesis account tells us that Noah was instructed by God to make the ark out of cypress wood (wood known to be strongly impervious to rot and insect infestation) as a sealed vessel, where it was covered and the ports were sealed with tar – thus not allowing outside gasses and vapors to get inside the ark. “Coat it (the ark) with pitch inside and out.” Genesis 6: 14
2. The account in Genesis tells us that waters came in part from underneath the oceans. “... on that day all the springs of the great deep burst forth, ...” Genesis 7:11
3. The King James Version also says, “And they that went in, went in male and female of each flesh, as God had commanded him; and the Lord shut him (Noah) in.” Genesis 7:16
4. The Genesis account tells us that all living things that breathed and that were outside of the ark died – and hence asphyxiated or were drowned. “Everything on the dry land that had breath of life in its nostrils died.” Genesis 7: 22
5. The account in Genesis tells us that the waters receded in the days following the flood, and that Noah opened the hatch on the ark when it was safe to do so. “after forty days Noah opened the window he had made in the ark and sent out a raven, ...” Genesis 8: 6

My sense is that if indeed the oceans flipped as per the scenario suggested, the event would be a degassing or belching of poisonous gasses of catastrophic proportions. I assert that virtually all life on Earth that breathes would die unless it was safely within an ark or enclosed vessel of some sort. I also assert that the cloud of deadly gases would in relatively short duration mix with the balance of the atmosphere, as temperatures would equalize and atmospheric mixing would occur, thus allowing living things to once again breathe outside as before.

“... and he (God) sent a wind over the earth ...” Genesis 8: 1

I submit that mathematical reasoning underlying the explanation as advanced herein for the ice ages is in conformity with the matters of periodicity, glacial periods and interglacials, and matters such as magnitude of cyclic swings as well as abruptness in temperature reversals – in particular related to the coming out of a deep glacial. The account as set forth (in my research) also takes into account an estimate of the time constant of the deep ocean – an ocean typified by the depth of the Pacific. The word “time constant” has engineering and mathematical significance, as it relates to a characteristic measure of the length of time for a capacitance to converge exponentially to a new state or condition. In the case of the Pacific Ocean, its time constant can be determined from known formulae in engineering heat transfer given the material properties of water, some simple assumptions about mixing or lack of mixing, as well as the depth dimension – and it is amazingly close to the period of 100,000 years.

Balancing Priorities in Life

As time permits in my retirement I hope to be able to devote more time to substantiating this mathematical hypothesis. At present, it would consume years of my life to write in detail each and every nuance. Moreover, providing a review of the literature would be an Herculean task at best. Quite frankly, the global warming crowd is focused on a political agenda and isn’t interested in truth, much less an account with citations in both engineering and scripture. I wish to remind my reader that I do hold an earned doctorate in mechanical engineering from Purdue University, and that in the three decades from 1968 to 1998 I held a professorial teaching and research position at a prestigious land grant institution – The University of Illinois in Urbana-Champaign. During my tenure I taught university level engineering courses in thermodynamics, heat transfer, engineering dynamical systems, control systems theory, distributed parameter systems, engineering economics, senior design courses, kinematics, and engineering design.

It is also somewhat odd to consider that what constitutes news and even a crisis in the world is subject to the whims of the media and the opinion leaders of our society. For example, a mere thirty years ago, in 1975, Newsweek trumpeted a story, characteristic of the press at that time – that an ice age was coming and (alarm) what can we do to prevent this pending catastrophe? Now, the Chicken Little’s (including Al Gore) are busy running and screaming that global warming is coming instead. The children with whom I work in the adapted bike program are at present more deserving – and that is where I envision spending my time – not arguing with Chicken Little’s who have made up their respective minds in advance about the evils of global warming and don’t have the mathematical tools or training to stay with me. I will say that I have been examining this general topic of ice age mechanism explanation in excess of 35 years, my credentials are sound, and my resolve and faith in myself and my ideas are as strong as ever. If you will examine my resume elsewhere in this web site, you will see a citation to an ice age hypothesis paper presented by myself and three former students in 1972. If you account for a year or two in research and submission of the paper, it is easy to arrive at 35 years.

Who is A. M. Lyapunov? As an aside, for those not familiar with A. M. Lyapunov (Aleksandr Mikhailovich Lyapunov, 1857-1918), he was a Russian mathematician, a student of the renowned mathematician Chebyshev. Lyapunov defended his doctoral thesis in Czarist Russia in 1892, wherein he addressed methods to permit one to ascertain stability of certain classes of nonlinear differential equations. He is perhaps best known for what is now called the Second or Direct Method of Lyapunov. His dissertation title before the University of Moscow (as best translated from the Russian) was, “The General Problem of Stability of Motion.” Moreover, in the process of translating his works from the original Russian, to the French which happened in 1947, and then the German, and finally to English, a number of variations in the spelling of his name exist. Another common English spelling variation is “Liapunov.”

The Uniqueness and Pushing Over the Edge Questions?

From a mathematical standpoint, the outcome now that man has altered the equation depends not only on what man has done and is doing, but when it happens in the sequence of when things happen as per the underlying cycle. For the sake of an illustration, think of a child swinging on a swing-set. If you would give the child a forward push or shove, the outcomes depends upon at what point in the swing cycle you apply that push. At one time a push will cause greater swing amplitudes (when you coincide the push with forward velocity), or conversely if you would push at the time of backward velocity the amplitude will decrease or even cause a dead-beat response. In short, exactly the same pushing effort will do different things to the outcome depending upon the timing of when you apply a push.

I can envision different scenarios in which the Earth, depending upon the timing of the push, becomes a snow ball Earth, and yet other scenarios when the Earth becomes warm perpetually like Venus. Both of these outcomes are possible if one looks at the underlying mechanisms and even some nearby planets and their moons. Our solar system presently has bodies that are Venus like and trapped in a hot cycle, whereas there are other bodies that are analogous to ice-covered snowballs. Venus stays hot in part because the sun’s inbound radiation is trapped, and in the case of the snowball spheres the snowball conditions remain because of the reflectivity (or albedo effect) created by the ice covering. In short, based on glimpses from mathematics, we have a situation with multiple solutions or multiple outcomes being possible.

It is critical to understand that the Earth is posibly in a presumably stable Lyapunov limit cycle of temperature where our stability is dependent upon the maintenance of the fluctuating cycle. If some coincidences or circumstances would cause Earth to either freeze solid or to boil over so to speak, the physics of radiation balances suggests that Earth could then become trapped (or attracted to, using chaos terminology) in a new level of temperature and behavior that would also be stable. Again think of an attractor state to use chaos terminology. To express all this in mathematical language, the underlying set of nonlinear differential equations (representing the Earth’s radiation balance and the conduction and convective flows that distribute heat and substances about) in all likelihood has the property of non-uniqueness of solutions. My reasoning based on an examination of the mechanisms suggests at least three possible solution scenarios – a snow ball Earth, a Venus-like hot box Earth, and an Earth with a middle range stable Lyapunov limit cycle. Quite frankly, I think that mankind is indeed fortunate to have the third option, and I personally take offense that supposed experts cry wolf and insist that mankind should start to regulate the Earth’s temperature mechanisms and even before mankind even understands what is happening in the first place.

If I gave any answer right now about the future and mankind’s present impact I would be speculating. My guess, however, is that if mankind would continue to dig up and release carbon to the air-ocean system, then we eventually risk that our temperature realm would elevate to a new (higher level) where we as a planet would go into a perpetual Venus type situation. In order to state anything with confidence, I would first want to see that a computer simulation was done once the underlying pre-man ice age cycles are explained by means of a suitable model sans mankind. It represents the height of intellectual arrogance to suggest that what man is doing, which is X, will certainly cause Y to happen – and it is arrogance because we don’t understand the physics of what is (was) happening as a baseline prior to the very recent intervention of mankind. When I say "model," I really mean a set of equations and accompanying logic to allow us to foresee what the Earth's climate system will do assuming Z and if we as humans do W, etc.

Some Commonly Asked Questions.

Why an Internet Posting? Some have a right to ask why I have elected to post this epistle on a web site, and thus give the appearance of junk science, as opposed to seeking the peer-reviewed literature means of publication? I have thought about this scientific question for almost four decades, and I remain absolutely convinced that my reasoning is on target for the long sought after mechanism to explain ice ages and the perpetual riddle they represent. Quite frankly, I am not getting any younger, and some day my days on this Earth will end. I wanted to write what I did so that these thoughts would not be lost in a distant world when I am no longer able to express them. By choice, I posted the writing on my own web site as opposed to sending it to an editor of some journal to be considered for review. I am not interested in arguing with editors and reviewers right now. I must note that as a scientist I respect and admire editors and reviewers and the role of peer review, but in my case I feel that I am in a race against time. The demands of the adapted bicycle program are so great and the rewards so significant – that I must content myself with publishing these glacial and ice age musings in web site form. The Internet is a fascinating mechanism for diffusion of information, certainly far faster than diffusion of temperature in the oceans, and of course the Internet browser has to be cautious as so much garbage is mixed in with the good. I assume that the reader must be knowledgeable enough to be able to distinguish good vs. garbage.

What triggers the de-gassing? A common question raised concerns the triggering mechanism – what triggers the de-gassing? The mechanisms of the eruption or venting are worthy, but from a stability perspective, the actual trigger is quite immaterial given the underlying situation that an unstable hydrodynamic stratification comes about. I firmly believe that deep bodies of water are susceptible to overturning and thus violent de-gassings will take place given the slightest provocations. The question of identification of the trigger is relatively immaterial given an unstable stratification, as gravity’s action does not have to wait for some trigger to start the process.

What role does undersea venting of gases (from the Earth’s mantel) play? Questions have been raised concerning how carbon related gases are added to the oceans due to underwater venting of gases through the Earth’s mantel. Similar speculations or questions abound with regard to the carbon build-up and subsequent degassing of Lake Nyos. My suspicion is that ocean floor vents releasing carbon gasses remain at constant rates over long intervals, and thus may be a factor but it isn’t a reason for the periodicity per se – and it certainly isn’t a reason to explain the cooling of the Earth which happens when the Earth slides into an ice age. On the other hand, the absorption of carbon gasses by the oceans from the air itself, especially over tens of thousands of years, is a significant factor to take into account in any model or ice age explanation. The absorption by the oceans of atmospheric gases is significant because it causes a reduction in atmospheric green house gases and thus leads to a cooling of the Earth’s atmosphere and surface temperature because it increases the radiation loss of Earth to outer space.

How much complexity do we need to include in a computer model? It is also commonly argued by some that whatever model is used to examine and model the Earth’s temperature balance must be of high order so as to include an array of variables, parameters, and frankly -- minutiae. If indeed all the tens of thousands of parameters and variables have to be accounted for – this particular ocean current, this volcano eruption, this Antarctic ice shelf, Amazon deforestation, the role of coral reefs and the destruction of such reefs by mankind, the submergence of a coastal plain, the emergence of a particular fauna, etc. -- then the time history of Earth’s atmospheric temperature would not in any reasonable likelihood be a simple tick-tock with discernable cycles – analogous to Old Faithful. Instead, the time history would be more like a chaotic or white noise outcome. It is clear that the record of Earth’s temperature over the last half-million years or more has been a fairly consistent and repeating pattern of identifiable cycles. Because of this, I feel very comfortable focusing on a few pivotal internal mechanisms and components, and most everything else just reacts and comes along for the ride so to speak. A handful of components, feedbacks, and nonlinearities are the key ingredients, and we need to focus on them, and not to get caught up in a quest to include every variable that contributes, even in minute ways, to the ice age cycle.

Closure

As time permits I do want to add some eye relief, so this would mean adding some pictures and diagrams. Eye relief will help most readers, and I would be able to make my arguments more understandable, such as diagrams of how water moves when a lake overturns. My family and I have vacationed each summer since the 1980's in central Minnesota at Lake Ida and Lake Miltona, so I have had ample opportunity to talk to the locals to get detailed accounts and technical descriptions. It is interesting that the locals accept the flipping of lakes as routine, and they don't see any particular significance as to how this is a general mechanism that occurs in many bodies of water, nor the potential for significant climate impact when the eruptions of the type at Lake Nyos occur.

I fully recognize that the recent impact of man has done much (in the last 200 years and even the last 50 years), but my premise is that I am looking for the long term (pre-mankind) solution of the ice age riddle and have discounted (for the moment) the most recent actions by mankind and industrialized society. I feel that we need to first understand and agree on ice ages and their cause (given pre-mankind conditions) and then we can later ask the questions related to how matters will evolve or happen once man has had this recent impact? I could write a complete document on that aspect alone, but again as I said in my writing, I need to truncate my time devoted to this topic. It would be possible at any moment to dive into this topic and quest for a definitive answer and spend the remainders of our lives on this topic, and quite frankly I don't want to and won't do that. My experience and training as a systems theorist gives me the confidence in knowing that I am on the right track. Yes, someday some other researchers will do exhaustive computer simulations and verify many details and nuances, but for me the fundamental mechanisms are sufficiently distinguishable to me. Even if a person sat down just to summarize the literature, the job would be Herculean as there is so much literature pertinent to the topic. One has to look at marine life, marine chemistry (actually very complex), core samples, ocean movements, build-up of coral reefs (which consume some carbon), biology, history of fossil digs, ocean chemistry, solar balances, effects of radiation changes due to reflectivity of ice vs. open land vs. forests, etc., changes in solar reflection due to clouds and humidity, dinosaurs, rise and fall of ocean levels, searches for meteor clouds, the physics of the sun, continental drift and plate changes, etc, etc.

Respectfully,

Richard E. Klein, Ph.D.
November 10, 2005, revised September 3, 2008