Real-world science is quite unlike school science. At school you learn that 2 + 2 = 4, 3 + 1 = 4, 2² = 4,  6 – 2 = 4, and so on, and by extension you can manipulate all sorts of inputs to get an output. But forensic  science, and indeed most real-world analytical science, begins with the output and asks “what is 4?” or more often “what are the most likely constituents of 4?” (Or more probably, “why did the wings fall off/ the product not sell/the patient die…?”) And as any good murder mystery or real-life medical scandal will show you, the most politically convenient suspect is not necessarily the culprit. Presumption is the enemy of truth and the starting point for self-aggrandising error.

The problem with most climate change models is that they begin with a presumption, usually that anthropogenic  carbon dioxide drives climate change, then attempt to fit recent data to that assumption. Having done so successfully, they then presume that the same driver (though clearly not of anthropic origin) was responsible for pre-human climate change.

This approach is fundamentally unscientific and generates more questions than answers. For instance why, in the ice core record, does temperature change take place before CO₂ level change? And why is prehistoric global temperature (measured by a wide spectrum of proxies) so regularly cyclic?

The word “cyclic” awakens the sleeping scientist!  Periodicity implies causality. If we can determine the fundamental period and other analytic variables we may be able to home in on a cause.

So let’s look at a long-term proxy record of mean global temperature.

http://www.seed.slb.com/uploadedImages/Science/Earth_Science/Global_Climate_Change_and_Energy/Related_Articles/global_temp2.jpg

Without presuming anything, we can see that it is roughly periodic, fairly rigidly  bounded, and obviously independent of human activity. The detail between the peaks is itself quite interesting. We seem to be looking at a chaotic oscillator.

Now whilst a truly chaotic system is inherently unpredictable, all real systems are bounded by their finite mass, finite power input, or whatever – some externality or internality imposes limits and/or quasi-periodicity over the chaotic detail.

If we don’t look too closely at the detail, we can see an underlying “sawtooth” form to the graph. Sharp rises are quickly followed by gradual, almost exponential decay towards the lower bound. The system stays near the  lower bound for a short time then undergoes another sharp rise and the cycle repeats. Fortunately, engineers and musicians know a lot about sawtooths!

We can synthesise this behaviour by adding related sinusoidal curves. Why sinusoidal? Because most things in macroscopic nature respond smoothly to whatever input they receive, and a sine wave is a nice smooth curve containing only one frequency.  Just to remind ourselves, here’s part of a graph of  y = sin(x/10)

It’s bounded and cyclic, which is a good start. We haven’t made any assumptions about what drives it, just a simple mathematical calculation.

Now let’s add some harmonics. A harmonic is a function with the same form but a multiple of the frequency of the “fundamental” that we began with.  So the second harmonic of sin(x/10) is

y = sin(2x/10)

Let’s make one assumption, that,whatever drives the fundamental, the harmonics are driven to a smaller amplitude. It’s reasonable because otherwise you would need an infinite amount of power to drive a real system from one limit to the other – bear with me on that, but it should become obvious later! So we’ll add half of the second harmonic, a third of the third harmonic, and so forth:

y = sin(x/10) + (sin(2x/10))/2 + (sin(3x/10))/3 +….(sin(nx/10))/n…..

There’s something particularly familiar about the sums to the 3^rd and 6^th harmonic – the rapid rise and a couple of “hiccups” on the downslope look very much like one cycle of the global temperature graph.

I’ve generated these curves from a spreadsheet rather than a recursive program, partly through laziness, but also because we can add a bit of “inertia” to the system model by coarsening the sampling interval. If we double the sampling interval  (so we are summing sin(x/5), etc.) we see that the behaviour over two cycles to the twelfth harmonic

is beginning to look very much like the temperature graph.

So if there is any value in this sawtooth model, where do the fundamental and the harmonics come from in the real climate? Let’s begin with the harmonics.

Suppose there is a simple feedback mechanism in climate change.  We don’t need to be prescriptive about what it is, but water vapour is an obvious example. Water is a greenhouse gas: the more there is in the atmosphere, the warmer the air and the ground underneath becomes. That much is undisputed.  It is also the case that the warmer the air, the more water vapour it can hold before the water condenses as fog, cloud, rain, or whatever. So if the sun heats the air above a source of water (and the entire planet’s surface  is, or contains, water) and the water evaporates, the temperature rises nonlinearly. Suppose for simplicity that the rate of temperature rise is roughly proportional to the amount of water vapour already present. Then if the primary driving source is d, the response varies with d², so if the fundamental temperature cycle is sin(x), the feedback adds an element sin²(x).

Now sin²(x) = ½ –(cos(2x))/2.  A cosine wave is simply a sine wave with a phase shift – i.e. delayed by a quarter cycle. But note the variable part of the  function is the cosine of 2x, not x. The simplest first-order feedback mechanism generates the second harmonic of the driving function.

The more nonlinearities  (x³, x^1.9, x⁴, etc) we add to the system, the more harmonics we generate. It’s pretty clear that if the sun shines on an inhomogeneous world, with bits of ice, snow, cloud, and complex tributary/lake/river/ocean systems, there will be a lot of interacting and phase-shifted nonlinearites due to one phenomenon alone – the melting and evaporation of water. But most importantly, they will always sum to a sawtooth.

Now this is pretty impressive. We have made no implausible assumptions, nor have we invoked any hitherto unknown geological or ecological catastrophes, sunspots, or cometary impacts, but we have made a remarkably accurate model of known past behaviour that generates  periodic and catastrophic outputs with no catastrophic inputs.

There is, of course, one big underlying assumption: the primary sinusoid. Setting aside any feedback elements for a moment, let us consider how an isolated system with a single input (i.e. the earth in sunshine) could undergo cyclic temperature changes.

Watch the sky on a sunny, windless summer day. Initially cloudless, the sun heats the ground and warm, moist air rises.  The moisture condenses and forms fluffy cumulus clouds. These gradually reduce the solar power input but the thermal inertia of the ground maintains the convection process even when the sky is completely clouded. The clouds continue to build until the sky is “overconvected” and some rain falls. The surface temperature decreases and the clouds disperse.  It’s rare to see the cycle repeated before sunset in Britain because the intensity of sunlight is insufficient, but two or three cycles per day is not unusual  in the American Midwest. Can we propose an enormously longer , global cycle?

Consider a “snowball earth” where all the water is frozen on the surface. There are no clouds and the air is dry. Ice and snow have a high albedo, that is, most of the incoming sunlight is reflected and there is very little solar heating during the day, matched by radiation at night, and the snowball is at a near-equilibrium temperature. Due to the anomalous expansion of water below 4°C, and the transparency of clear ice, even where sunlight does penetrate the surface, most of the heat is delivered to the water below and the ice always floats. We see this wherever a frozen puddle melts: thin ice melts from below when the sun shines. So the surface temperature at any point does not rise above 0°C, and very little water evaporates, until  all the ice has melted.

Now consider the latent heat of fusion of water. Heating ice from -1 to 0°C requires 2.1 joules per gram.  Heating water from 0 to 1°C requires 4.2 J/g. But melting ice to water at 0°C requires 334 J/g. Thus  an enormous input of energy produces no temperature change  as the surface temperature passes through the freezing point.  If the solar input power is constant, whether the earth is heating up or cooling down, there is  a huge hysteresis  in surface temperature change at the melting point.

As more ice melts, so the mean albedo decreases. Furthermore the energy required to increase the surface temperature at any point diminishes sharply once that point has passed 0°C: the specific heat capacity of dry rock is less than that of ice or wet rock.

We will continue to ignore feedback for the time being, and just suppose that as the global surface temperature rises, so the amount of cloud cover increases – just as we observe on a summer day in Kansas, but averaged over the entire surface and hundreds of years. Thus the rate of heating, initially slow then accelerating, begins to diminish until we have total cloud cover and a new equilibrium, determined by the balance between cloud albedo and cloud radiance. In effect, we have turned the snowball inside out.

With considerably less solar input to the surface, and an outer shell of ice around the atmosphere, the surface will begin to cool.

So the  simplest linear model is inherently oscillatory, and any nonlinearities (such as the hysteresis of melting) will tend towards the observed sawtooth behaviour. Can we calculate the amplitude or period of oscillation?

All our historic temperature proxies are derived from land-based plants, ice cores, or whatever.  We have no reliable record of the surface temperature of the oceans before the 19^th century AD, when surveying and thermometry became matters of direct observation. But we do know that dry land heats and cools much more rapidly than the mid ocean, and our evolving model of water-driven climate change must surely take account of the 75% of the earth’s surface that is permanently covered with deep water.

Watch this space (or at least the one next to it)!