Tag Archives: wind

Rotating sail ships and why your curve ball doesn’t curve.

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The Flettner-sail ship, Barbara, 1926.

Sailing ships are wonderfully economic and non-polluting. They have unlimited range because they use virtually no fuel, but they tend to be slow, about 5-12 knots, about half as fast as Diesel-powered ships, and they can be stranded for weeks if the wind dies. Classic sailing ships also require a lot of manpower: many skilled sailors to adjust the sails. What’s wanted is an easily manned, economical, hybrid ship: one that’s powered by Diesel when the wind is light, and by a simple sail system when the wind blows. Anton Flettner invented an easily manned sail and built two ships with it. The Barbara above used a 530 hp Diesel and got additional thrust, about an additional 500 hp worth, from three, rotating, cylindrical sails. The rotating sales produced thrust via the same, Magnus force that makes a curve ball curve. Barbara went at 9 knots without the wind, or about 12.5 knots when the wind blew. Einstein thought it one of the most brilliant ideas he’d seen.

Force diagram of Flettner rotor (Lele & Rao, 2017)

The source of the force can be understood with help of the figure at left and the graph below. When a simple cylinder sits in the wind, with no spin, α=0, the wind force is essentially drag, and is 1/2 the wind speed squared, times the cross-sectional area of the cylinder, Dxh, and the density of air. Add to this a drag coefficient, CD, that is about 1 for a non-spinning cylinder. More explicitly, FD= CDDhρv2/2. As the figure at right shows, there is a sort-of lift in the form of sustained vibrations at zero spin, α=0. Vibrations like this are useless for propulsion, and can be damaging to the sail. In baseball, such vibrations are the reason knuckle balls fly erratically. If you spin the cylindrical mast at α=2.1, that is at a speed where the fast surface moves with the wind, at 2.1 times the wind speed, and the other side side moves to the wind, there is more force on the side moving to the wind (see figure above) and the ship can be propelled forward (or backward if you reverse the spin direction). Significantly, at α=2.1, you get 6 times as much force as the expected drag, and you no longer get vibrations. FL= CLDhρv2/2, and CL=6 at this rotation speed

Numerical lift coefficients versus time, seconds for different ratios of surface speed to wind speed, a. (Mittal & Kumar 2003), Journal of Fluid Mechanics.

At this rotation speed, α=2.1, this force will be enough to drive a ship so long as the wind is reasonably strong, 15-30 knots, and ship does not move faster than the wind. The driving force is always at right angles to the perceived wind, called the “fair wind”, and the fair wind moves towards the front as the ship speed increases. If you spin the cylinder at 3 to 4 times the wind speed, the lift coefficient increases to between 10 and 18. This drives a ship with yet force. You need somewhat more power to turn the sails, but you are also further from vibrations. Flettner considered α=3.5. optimal. Higher rotation speeds are possible, but they require more rotation power (rotation power goes as ω2, and if you go beyond α=4.3, the vibrations return. Controlling the speed is somewhat difficult but important. Flettner sails were no longer used by the 1930s when fuel became cheaper.

In the early 1980s, the famous underwater explorer, Jacques Cousteau revived the Flettner sail for his exploratory ship, the Alcyone. He used light-weight aluminum sails, and an electric motor for rotation instead of Diesel as on the Barbara. He claimed that the ship drew more than half of its power from the wind, and claimed that, because of computer control, it could sail with no crew. This latter claim was likely bragging. Even with today’s computer systems, people are needed as soon as something goes wrong. Still the energy savings were impressive enough that other ship owners took notice. In recent years, several ship-owners have put Flettner sails on cargo ships, as a right. This is not an ideal use since cargo ships tend to go fast. Still, it’s reported that, these ships get about 20% of their propulsion from wind power, not an insignificant amount.

And this gets us to the reason your curve ball does not curve: you’re not spinning it fast enough. You want the ball to spin at a higher rate than you get just by rolling the ball off your fingers. If you do this, α = 1 and you get relatively little sideways force. To get the ball to really curve, you have to snap your wrist hard aiming for α=1.5 or so. As another approach you can aim for a knuckle ball, achieved with zero rotation. At α=0, the ball will oscillate and your pitch nearly impossible to hit, or catch. Good luck.

Robert Buxbaum, March 22, 2023. There are also various Flettner airplane designs where horizontal, cylindrical “wings” rotate to provide lift, power too in some versions. The aim is high lift with short wings and a relatively low power draw. So-far, these planes are less efficient and slower than a normal helicopter.

What drives the jet stream

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Having written on controversial, opinion things, I thought I’d take break and write about earth science: the jet stream. For those who are unfamiliar the main jet stream is a high-altitude wind blowing at about 40,000 feet (10 km) altitude at about 50° N latitude. It blows west to east at about 100 km/hr (60 mph), about 12% of the cruising of a typical jet airplane. A simple way to understand the source of the jet stream is to note that the earth spins slower (in mph) at the poles than at lower latitudes, but that the temperature difference between the poles and equator guarantees that air at high altitude is always traveling toward the poles from the lower latitudes.

Consider that the earth is about 40,000 km is circumference and turns once every 24 hours. This suggests a rotation speed of 1667 km/hr at the equator. At any higher latitude the speed is 1667 cos latitude. Thus it’s 1070 km/hr at 50° latitude, 0 km/hr at the north pole; 1667km/hr cos 50°= 1070 km/hr.

Idealize north-south circulation of air around our globe.

Idealized north-south circulation of air around our globe.

It’s generally colder at the poles than it is at lower latitudes — that is nearer the equator (here’s why). This creates a north-south wind where the air becomes more compact as it cools in northern climate (50°latitude  and further north), and this creates a vacuum at high altitudes and a high pressure zone at low altitudes. The result is a high altitude flow of air towards north, and a flow of low altitude air south, a process that is described by the idealized drawing at right.

At low altitudes in Detroit (where I am) we experience winds mostly from the north and from the east. Winds come from the east — or appear to — because of the rotation of the earth. The air that flows down from Canada is moving west to east at a slower speed than Detroit is moving west to east. We experience this as an easterly wind. At higher altitudes, the pattern is reversed. At 9 to 12 km altitudes, an airplane would experience winds mostly from the south-west. Warm air from lower latitudes is moving eastward at 1200 or more km/hr because that’s the speed of the earth. As it moves north, it discovers that the land is moving eastward at a much slower speed, and the result is the jet stream. The maximum speed of the jet stream is about 200 km/hr, the difference in the earth’s east-speed between that at 40°N and at 50°N, while the typical speed is about half of that, 100 km/hr. I’d attribute this slower speed to friction or air mixing.

One significance of the jet stream is that it speeds west-east air-traffic, e.g. flights from Japan to the US or from the US to Europe. Airlines flying west to east try to fly at the latitude and altitude of the jet stream to pick up speed. Planes flying the other way go closer to the pole and/or at different altitudes to avoid having the jet stream slowing them down, or to benefit from other prevailing winds.

I note that Hurricanes are driven by the same forces as the jet stream, just more localized. Tornados are the same, just more localized. A localized flow of this sort can pick stuff up here’s how they pick stuff up. Robert Buxbaum, May 22, 2018

What drives the gulf stream?

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I’m not much of a fan of todays’ kids’ science books because they don’t teach science IMHO. They have nice pictures and a few numbers; almost no equations, and lots of words. You can’t do science that way. On the odd occasion that they give the right answer to some problem, the lack of math means the kid has no way of understanding the reasoning, and no reason to believe the answer. Professional science articles on the web are bad in the opposite direction: too many numbers and for math, hey rely on supercomputers. No human can understand the outcome. I like to use my blog to offer science with insight, the type you’d get in an old “everyman science” book.

In previous posts, I gave answers to why the sky is blue, why it’s cold at the poles, why it’s cold on mountains, how tornadoes pick stuff up, and why hurricanes blow the way they do. In this post, we’ll try to figure out what drives the gulf-stream. The main argument will be deduction — disproving things that are not driving the gulf stream to leave us with one or two that could. Deduction is a classic method of science, well presented by Sherlock Holmes.

The gulf stream. The speed in the white area is ≥ 0.5 m/s (1.1 mph.).

The gulf stream. The speed in the white area is ≥ 0.5 m/s (1.1 mph.).

For those who don’t know, the Gulf stream is a massive river of water that runs within the Atlantic ocean. As shown at right, it starts roughly at the end of Florida, runs north to the Carolinas, and then turns dramatically east towards Spain. Flowing east, It’s about 150 miles wide, but only about 62 miles (100 km) when flowing along the US coast. According to some of the science books of my youth this massive flow was driven by temperature according to others, by salinity (whatever that means), and yet other books of my youth wind. My conclusion: they had no clue.

As a start to doing the science here, it’s important to fill in the numerical information that the science books left out. The Gulf stream is roughly 1000 meters deep, with a typical speed of 1 m/s (2.3 mph). The maximum speed is the surface water as the stream flows along the US coast. It is about 2.5 metres per second (5.6 mph), see map above.

From the size and the speed of the Gulf Stream, we conclude that land rivers are not driving the flow. The Mississippi is a big river with an outflow point near the head waters of the gulf stream, but the volume of flow is vastly too small. The volume of the gulf stream is roughly

Q=wdv = 100,000 x 1000 x .5 =  50 million m3/s = 1.5 billion cubic feet/s.

This is about 2000 times more flow than the volume flow of the Mississippi, 18,000 m3/s. The great difference in flow suggests the Mississippi could not be the driving force. The map of flow speeds (above) also suggest rivers do not drive the flow. The Gulf Stream does not flow at its maximum speed near the mouth of any river.  We now look for another driver.

Moving on to temperature. Temperature drives the whirl of hurricanes. The logic for temperature driving the gulf stream is as follows: it’s warm by the equator and cold at the poles; warm things expand and as water flows downhill, the polls will always be downhill from the equator. Lets put some math in here or my explanation will be lacking. First lets consider how much hight difference we might expect to see. The thermal expansivity of water is about 2x 10-4 m/m°C (.0002/°C) in the desired temperature range). To calculate the amount of expansion we multiply this by the depth of the stream, 1000m, and the temperature difference between two points, eg. the end of Florida to the Carolina coast. This is 5°C (9°F) I estimate. I calculate the temperature-induced seawater height as:

∆h (thermal) ≈ 5° x .0002/° x 1000m = 1 m (3.3 feet).

This is a fair amount of height. It’s only about 1/100 the height driving the Mississippi river, but it’s something. To see if 1 m is enough to drive the Gulf flow, I’ll compare it to the velocity-head. Velocity-head is a concept that’s useful in plumbing (I ran for water commissioner). It’s the potential energy height equivalent of any kinetic energy — typically of a fluid flow. The kinetic energy for any velocity v and mass of water, m is 1/2 mv2 . The potential energy equivalent is mgh. Combine the above and remove the mass terms, and we have:

∆h (velocity) = v2/2g.

Where g is the acceleration of gravity. Let’s consider  v = 1 m/s and g= 9.8 m/s2.≤ 0.05 m ≈ 2 inches. This is far less than the driving force calculated above. We have 5x more driving force than we need, but there is a problem: why isn’t the flow faster? Why does the Mississippi move so slowly when it has 100 times more head.

To answer the above questions, and to check if heat could really drive the Gulf Stream, we’ll check if the flow is turbulent — it is. A measure of how turbulent is based on something called the Reynolds number, Re#, it’s the ratio of kinetic energy and viscous loss in a fluid flow. Flows are turbulent if this ratio is more than 3000, or so;

Re# = vdρ/µ.

In the above, v is velocity, say 1 m/s, d is depth, 1000m, ρ = density, 1000 kg/m3 for water, and  0.00133 Pa∙s is the viscosity of water. Plug in these numbers, and we find a RE# = 750 million: this flow will be highly turbulent. Assuming a friction factor of 1/20 (.05), e find that we’d expect complete mixing 20 depths or 20 km. We find we need the above 0.05 m of velocity height to drive every 20 km of flow up the US coast. If the distance to the Carolina coast is 1000 km we need 1000*.05m/20 = 1 meter, that’s just about the velocity-head that the temperature difference would suggest. Temperature is thus a plausible driving force for 0.5 m/s, though not likely for the faster 2.5 m/s flow seen in the center of the stream. Turbulent flow is a big part of figuring the mpg of an automobile; it becomes rapidly more important at high speeds.

World sea salinity

World sea salinity. The maximum and minimum are in the wrong places.

What about salinity? For salinity to work, the salinity would have to be higher at the end of the flow. As a model of the flow, we might imagine that we freeze arctic seawater, and thus we concentrate salt in the seawater just below the ice. The heavy, saline water would flow down to the bottom of the sea, and then flow south to an area of low salinity and low pressure. Somewhere in the south, the salinity would be reduced by rains. If evaporation were to exceed the rains, the flow would go in the other direction. Sorry to say, I see no evidence of any of this. For one the end of the Gulf Stream is not that far north; there is no freezing, For two other problems: there are major rains in the Caribbean, and rains too in the North Atlantic. Finally, while the salinity head is too small. Each pen of salinity adds about 0.0001g/cc, and the salinity difference in this case is less than 1 ppm, lets say 0.5ppm.

h = .0001 x 0.5 x 1000 = 0.05m

I don’t see a case for northern-driven Gulf-stream flow caused by salinity.

Surface level winds in the Atlantic.

Surface level winds in the Atlantic. Trade winds in purple, 15-20 mph.

Now consider winds. The wind velocities are certainly enough to produce 5+ miles per hour flows, and the path of flows is appropriate. Consider, for example, the trade winds. In the southern Caribbean, they blow steadily from east to west slightly above the equator at 15 -20 mph. This could certainly drive a circulation flow of 4.5 mph north. Out of the Caribbean basin and along the eastern US coat the trade winds blow at 15-50 mph north and east. This too would easily drive a 4.5 mph flow.  I conclude that a combination of winds and temperature are the most likely drivers of the gulf stream flow. To quote Holmes, once you’ve eliminated the impossible, whatever remains, however improbable, must be the truth.

Robert E. Buxbaum, March 25, 2018. I used the thermal argument above to figure out how cold it had to be to freeze the balls off of a brass monkey.

Advanced windmills + 20 years = field of junk

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Everything wears out. This can be a comforting or a depressing thought, but it’s a truth. No old mistake, however egregious, lasts forever, and no bold advance avoids decay. At best, last year’s advance will pay for itself with interest, will wear out gracefully, and will be recalled fondly by aficionados after it’s replaced by something better. Water wheels, and early steamships are examples of this type of bold advance. Unfortunately, it is often the case that last years innovation turns out to be no advance at all: a technological dead end that never pays for itself, and becomes a dangerous, rotting eyesore or worse, a laughing-stock blot or a blot on the ecology. Our first two generations of advanced windmill farms seem to match this description; perhaps the next generation will be better, but here are some thoughts on lessons learned from the existing fields of rotting windmills.

The ancient design windmills of Don Quixote’s Spain (1300?) were boons. Farmers used them to grind grain or cut wood, and to to pump drinking water. Holland used similar early windmills to drain their land. So several American presidents came to believe advanced design windmills would be similar boons if used for continuous electric power generation. It didn’t work, and many of the problems could have been seen at the start. While the farmer didn’t care when his water was pumped, or when his wood is cut. When you’re generating electricity, there is a need to match the power demand exactly. Whenever the customer turns on the switch, electricity is expected to flow at the appropriate amount of Wattage; at other times any power generated is a waste or a nuisance. But electric generator-windmills do not produce power on demand, they produce power when the wind blows. The mismatch of wind and electric demand has bedeviled windmill reliability and economic return. It will likely continue to do so until we find a good way to store electric power cheaply. Until then windmills will not be able to produce electricity at competitive prices to compete with cheap coal and nuclear power.

There is also the problem of repair. The old windmills of Holland still turn a century later because they were relatively robust, and relatively easy to maintain. The modern windmills of the US stand much taller and move much faster. They are often hit, and damaged by lightning strikes, and their fast-turning gears tend to wear out fast, Once damaged, modern windmills are not readily fix, They are made of advanced fiberglass materials spun on special molds. Worse yet, they are constructed in mountainous, remote locations. Such blades can not be replaces by amateurs, and even the gears are not readily accessed to repair. More than half of the great power-windmills built in the last 35 years have worn out and are unlikely to ever get repair. Driving past, you see fields of them sitting idle; the ones still turning look like they will wear out soon. The companies that made and installed these behemoth are mostly out of the business, so there is no-one there to take them down even if there were an economic incentive to do so. Even where a company is found to fix the old windmills, no one would as there is not sufficient economic return — the electricity is worth less than the repair.

Komoa Wind Farm in Kona, Hawaii June 2010; Friends of Grand Ronde Valley.

Komoa Wind Farm in Kona, Hawaii, June 2010; A field of modern design wind-turbines already ruined by wear, wind, and lightning. — Friends of Grand Ronde Valley.

A single rusting windmill would be bad enough, but modern wind turbines were put up as wind farms with nominal power production targeted to match the output of small coal-fired generators. These wind farms require a lot of area,  covering many square miles along some of the most beautiful mountain ranges and ridges — places chosen because the wind was strong

Putting up these massive farms of windmills lead to a situation where the government had pay for construction of the project, and often where the government provided the land. This, generous spending gives the taxpayer the risk, and often a political gain — generally to a contributor. But there is very little political gain in paying for the repair or removal of the windmills. And since the electricity value is less than the repair cost, the owners (friends of the politician) generally leave the broken hulks to sit and rot. Politicians don’t like to pay to fix their past mistakes as it undermines their next boondoggle, suggesting it will someday rust apart without ever paying for itself.

So what can be done. I wish I could suggest less arrogance and political corruption, but I see no way to achieve that, as the poet wrote about Ozymandias (Ramses II) and his disastrous building projects, the leader inevitably believes: “I am Ozymandias, king of kings; look on my works ye mighty and despair.” So I’ll propose some other, less ambitious ideas. For one, smaller demonstration projects closer to the customer. First see if a single windmill pays for itself, and only then build a second. Also, electricity storage is absolutely key. I think it is worthwhile to store excess wind power as hydrogen (hydrogen storage is far cheaper than batteries), and the thermodynamics are not bad

Robert E. Buxbaum, January 3, 2016. These comments are not entirely altruistic. I own a company that makes hydrogen generators and hydrogen purifiers. If the government were to take my suggestions I would benefit.

Land use nuclear vs wind and solar

An advantage of nuclear power over solar and wind is that it uses a lot less land, see graphic below. While I am doubtful that industrial gas causes global warming, I am not a fan of pollution, and that’s why I like nuclear power. Nuclear power adds no water or air pollution when it runs right, and removes a lot less land than wind and solar. Consider the newly approved Hinkley Point C (England), see graphic below. The site covers 430 acres, 1.74 km2, and is currently the home of Hinkley Point B, a nuclear plant slated for retirement. When Hinkley Point C is built on the same site, it will add 26 trillion Watt-hr/ year (3200 MW, 93% up time), about 7% of the total UK demand. Yet more power would be provided from these 430 acres if Hinkley B is not shut down.

Nuclear land use vs solar and wind; British Gov't. regarding their latest plant

Nuclear land use vs solar and wind; British Gov’t. regarding their latest plant

A solar farm to produce 26 trillion W-hr/year would require 130,000 acres, 526 km2. This area would suggest they get the equivalent of 1.36 hours per day of full sun on every m2, not unreasonable given the space for roads and energy storage, and how cloudy England is. Solar power requires a lot energy-storage since you only get full power in the daytime, when there are no clouds.

A wind farm requires even more land than solar, 250,000 acres, or somewhat more than 1000 km2. Wind farms require less storage but that the turbines be spaced at a distance. Storage options could include hydrogen, batteries, and pumped hydro.; I make the case that hydrogen is better. While wind-farm space can be dual use — allowing farming for example, 1000 square km, is still a lot of space to carve up with roads and turbines. It’s nearly the size of greater London; the tourist area, London city is only 2.9 km2.

All these power sources produce pollution during construction and decommissioning. But nuclear produces somewhat less as the plants are less massive in total, and work for more years without the need for major rebuilds. Hinkley C will generate about 30,000 kg/year of waste assuming 35 MW-days/kg, but the cost to bury it in salt domes should not be excessive. Salt domes are needed because Hinkley waste will generate 100 kW of after-heat, even 16 years out. Nuclear fusion, when it comes, should produce 1/10,000 as much after-heat, 100W, 1 year out, but fusion isn’t here yet.

There is also the problem of accidents. In the worst nuclear disaster, Chernobyl, only 31 people died as a direct result, and now (strange to say) the people downwind are healthier than the average up wind; it seems that small amounts of radiation may be good for you. By comparison, in Iowa alone there were 317 driving fatalities in 2013. And even wind and solar have accidents, e.g. people falling from wind-turbines.

Robert Buxbaum, January 22, 2014. I’m president of REB Research, a manufacturer of hydrogen generators and purifiers — mostly membrane reactor based. I also do contract research, mostly on hydrogen, and I write this blog. My PhD research was on nuclear fusion power. I’ve also written about conservation, e.g. curtains; insulation; paint your roof white.

What’s Holding Gilroy on the Roof

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We recently put a sculpture on our roof: Gilroy, or “Mr Hydrogen.” It’s a larger version of a creepy face sculpture I’d made some moths ago. Like it, and my saber-toothed tiger, the eyes follow you. A worry about this version: is there enough keeping it from blowing down on the cars? Anyone who puts up a large structure must address this worry, but I’m a professional engineer with a PhD from Princeton, so my answer is a bit different from most.

Gilroy (Mr Hydrogen) sculpture on roof of REB Research & Consulting. The eyes follow you.

Gilroy (Mr Hydrogen) sculpture on roof of REB Research & Consulting. The eyes follow you. Aim is that it should withstand 50 mph winds.

The main force on most any structure is the wind (the pyramids are classic exceptions). Wind force is generally proportional to the exposed area and to the wind-speed squared: something called form-drag or quadratic drag. Since force is related to wind-speed, I start with some good statistics for wind speed, shown in the figure below for Detroit where we are.

The highest Detroit wind speeds are typically only 16 mph, but every few years the winds are seen to reach 23 mph. These are low relative to many locations: Detroit has does not get hurricanes and rarely gets tornadoes. Despite this, I’ve decided to brace the sculpture to withstand winds of 50 mph, or 22.3 m/s. On the unlikely chance there is a tornado, I figure there would be so much other flotsam that I would not have to answer about losing my head. (For why Detroit does not get hurricanes or tornadoes, see here. If you want to know why tornadoes lift things, see here).

The maximum area Gilroy presents is 1.5 m2. The wind force is calculated by multiplying this area by the kinetic energy loss per second 1/2ρv2, times a form factor.  F= (Area)*ƒ* 1/2ρv2, where ρ is the density of air, 1.29Kg/m3, and v is velocity, 22.3 m/s. The form factor, ƒ, is about 1.25 for this shape: ƒ is found to be 1.15 for a flat plane, and 1.1 to 1.3 a rough sphere or ski-jumper. F = 1.5*1.25* (1/2 *1.29*22.32) = 603 Nt = 134 lb.; pressure is this divided by area. Since the weight is only about 40 lbs, I find I have to tie down the sculpture. I’ve done that with a 150 lb rope, tying it to a steel vent pipe.

Wind speed for Detroit month by month. Used to calculate the force. From http://weatherspark.com/averages/30042/Detroit-Michigan-United-States

Wind speed for Detroit month by month. Used to calculate the force. From http://weatherspark.com/averages/30042/Detroit-Michigan-United-States

It is possible that there’s a viscous lift force too, but it is likely to be small given the blunt shape and the flow Reynolds number: 3190. There is also the worry that Gilroy might fall apart from vibration. Gilroy is made of 3/4″ plywood, treated for outdoor use and then painted, but the plywood is held together with 25 steel screws 4″ long x 1/4″ OD. Screws like this will easily hold 134 lbs of steady wind force, but a vibrating wind will cause fatigue in the metal (bend a wire often enough and it falls apart). I figure I can leave Gilroy up for a year or so without worry, but will then go up to replace the screws and check if I have to bring him/ it down.

In the meantime, I’ll want to add a sign under the sculpture: “REB Research, home of Mr Hydrogen” I want to keep things surreal, but want to be safe and make sales.

by Robert E. Buxbaum, June 21, 2013

The Gift of Chaos

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Many, if not most important engineering systems are chaotic to some extent, but as most college programs don’t deal with this behavior, or with this type of math, I thought I might write something on it. It was a big deal among my PhD colleagues some 30 years back as it revolutionized the way we looked at classic problems; it’s fundamental, but it’s now hardly mentioned.

Two of the first freshman engineering homework problems I had turn out to have been chaotic, though I didn’t know it at the time. One of these concerned the cooling of a cup of coffee. As presented, the coffee was in a cup at a uniform temperature of 70°C; the room was at 20°C, and some fanciful data was presented to suggest that the coffee cooled at a rate that was proportional the difference between the (changing) coffee temperature and the fixed room temperature. Based on these assumptions, we predicted exponential cooling with time, something that was (more or less) observed, but not quite in real life. The chaotic part in a real cup of coffee, is that the cup develops currents that move faster and slower. These currents accelerate heat loss, but since they are driven by the temperature differences within the cup they tend to speed up and slow down erratically. They accelerate when the cup is not well stirred, causing new stir, and slow down when it is stirred, and the temperature at any point is seen to rise and fall in an almost rhythmic fashion; that is, chaotically.

While it is impossible to predict what will happen over a short time scale, there are some general patterns. Perhaps the most remarkable of these is self-similarity: if observed over a short time scale (10 seconds or less), the behavior over 10 seconds will look like the behavior over 1 second, and this will look like the behavior over 0.1 second. The only difference being that, the smaller the time-scale, the smaller the up-down variation. You can see the same thing with stock movements, wind speed, cell-phone noise, etc. and the same self-similarity can occur in space so that the shape of clouds tends to be similar at all reasonably small length scales. The maximum average deviation is smaller over smaller time scales, of course, and larger over large time-scales, but not in any obvious way. There is no simple proportionality, but rather a fractional power dependence that results in these chaotic phenomena having fractal dependence on measure scale. Some of this is seen in the global temperature graph below.

Global temperatures measured from the antarctic ice showing stable, cyclic chaos and self-similarity.

Global temperatures measured from the antarctic ice showing stable, cyclic chaos and self-similarity.

Chaos can be stable or unstable, by the way; the cooling of a cup of coffee was stable because the temperature could not exceed 70°C or go below 20°C. Stable chaotic phenomena tend to have fixed period cycles in space or time. The world temperature seems to follow this pattern though there is no obvious reason it should. That is, there is no obvious maximum and minimum temperature for the earth, nor any obvious reason there should be cycles or that they should be 120,000 years long. I’ll probably write more about chaos in later posts, but I should mention that unstable chaos can be quite destructive, and quite hard to prevent. Some form of chaotic local heating seems to have caused battery fires aboard the Dreamliner; similarly, most riots, famines, and financial panics seem to be chaotic. Generally speaking, tight control does not prevent this sort of chaos, by the way; it just changes the period and makes the eruptions that much more violent. As two examples, consider what would happen if we tried to cap a volcano, or provided  clamp-downs on riots in Syria, Egypt or Ancient Rome.

From math, we know some alternate ways to prevent unstable chaos from getting out of hand; one is to lay off, another is to control chaotically (hard to believe, but true).