Category Archives: Weather

Why is it hot at the equator, cold at the poles?

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Here’s a somewhat mathematical look at why it is hotter at the equator that at the poles. This is high school or basic college level science, using trigonometry (pre calculus), a slight step beyond the basic statement that the sun hits down more directly at the equator than at the poles. That’s the kid’s explanation, but we can understand better if we add a little math.

Solar radiation hits Detroit at an angle, as a result less radiation power hits per square meter of Detroit.

Solar radiation hits Detroit or any other non-equator point at an angle, As a result, less radiation power hits per square meter of land.

Lets use the diagram at right and trigonometry to compare the amount of sun-energy that falls on a square meter of land at the equator (0 latitude) and in a city at 42.5 N latitude (Detroit, Boston, and Rome are at this latitude). In each case, let’s consider high-noon on March 21 or September 20. These are the two equinox days, the only days each year when the day and night are equal length, and the only times when it is easy to calculate the angle of the sun as it deviates from the vertical by exactly the latitude on the days and times.

More specifically the equator is zero latitude, so on the equator at high-noon on the equinox, the sun will shine from directly overhead, or 0° from the vertical. Since the sun’s power in space is 1050 W/m2, every square meter of equator can expect to receive 1050 W of sun-energy, less the amount reflected off clouds and dust, or scattered off or air molecules (air scattering is what makes the sky blue). Further north, Detroit, Boston, and Rome sit at 42.5 latitude. At noon on March 31 the sun will strike earth at 42.5° from the vertical as shown in the lower figure above. From trigonometry, you can see that each square meter of these cities will receive cos 42.5 as much power as a square meter at the equator, except for any difference in clouds, dust, etc. Without clouds etc. that would be 1050 cos 42.5 = 774 W. Less sun power hits per square meter because each square meter is tilted. Earlier and later in the day, each spot will get less sunlight than at noon, but the proportion is the same, at least on one of the equinox days.

To calculate the likely temperature in Detroit, Boston, or Rome, I will use a simple energy balance. Ignoring heat storage in the earth for now, we will say that the heat in equals the heat out. We now ignore heat transfer by way of winds and rain, and approximate to say that the heat out leaves by black-body radiation alone, radiating into the extreme cold of space. This is not a very bad approximation since Black body radiation is the main temperature removal mechanism in most situations where large distances are involved. I’ve discussed black body radiation previously; the amount of energy radiated is proportional to luminosity, and to T4, where T is the temperature as measured in an absolute temperature scale, Kelvin or Rankin. Based on this, and assuming that the luminosity of the earth is the same in Detroit as at the equator,

T Detroit / Tequator  = √√ cos 42.5 = .927

I’ll now calculate the actual temperatures. For American convenience, I’ll choose to calculation in the Rankin Temperature scale, the absolute Fahrenheit scale. In this scale, 100°F = 560°R, 0°F = 460°R, and the temperature of space is 0°R as a good approximation. If the average temperature of the equator = 100°F = 38°C = 560°R, we calculate that the average temperature of Detroit, Boston, or Rome will be about .927 x 560 = 519°R = 59°F (15°C). This is not a bad prediction, given the assumptions. We can expect the temperature will be somewhat lower at night as there is no light, but the temperature will not fall to zero as there is retained heat from the day. The same reason, retained heat, explains why it is warmer will be warmer in these cities on September 20 than on March 31.

In the summer, these cities will be warmer because they are in the northern hemisphere, and the north pole is tilted 23°. At the height of summer (June 21) at high noon, the sun will shine on Detroit at an angle of 42.5 – 23° = 19.5° from the vertical. The difference in angle is why these cities are warmer on that day than on March 21. The equator will be cooler on that day (June 21) than on March 21 since the sun’s rays will strike the equator at 23° from the vertical on that day. These  temperature differences are behind the formation of tornadoes and hurricanes, with a tornado season in the US centering on May to July.

When looking at the poles, we find a curious problem in guessing what the average temperature will be. At noon on the equinox, the sun comes in horizontally, or at 90°from the vertical. We thus expect there is no warming power at all this day, and none for the six months of winter either. At first glance, you’d think the temperature at the poles would be zero, at least for six months of the year. It isn’t zero because there is retained heat from the summer, but still it makes for a more-difficult calculation.

To figure an average temperature of the poles, lets remember that during the 6 month summer the sun shines for 24 hours per day, and that the angle of the sun will be as high as 23° from the horizon, or 67° from the vertical for all 24 hours. Let’s assume that the retained heat from the summer is what keeps the temperature from falling too low in the winter and calculate the temperature at an .

Let’s assume that the sun comes in at the equivalent of 25° for the sun during the 6 month “day” of the polar summer. I don’t look at equinox, but rather the solar day, and note that the heating angle stays fixed through each 24 hour day during the summer, and does not decrease in the morning or as the afternoon wears on. Based on this angle, we expect that

TPole / Tequator  = √√ cos 65° = .806

TPole = .806 x 560°R = 452°R = -8°F (-22°C).

This, as it happens is 4° colder than the average temperature at the north pole, but not bad, given the assumptions. Maybe winds and water currents account for the difference. Of course there is a large temperature difference at the pole between the fall equinox and the spring equinox, but that’s to be expected. The average is, -4°F, about the temperature at night in Detroit in the winter.

One last thing, one that might be unexpected, is that temperature at the south pole is lower than at the north pole, on average -44°F. The main reason for this is that the snow on south pole is quite deep — more than 1 1/2 miles deep, with some rock underneath. As I showed elsewhere, we expect that, temperatures are lower at high altitude. Data collected from cores through the 1 1/2 mile deep snow suggest (to me) chaotic temperature change, with long ice ages, and brief (6000 year) periods of warm. The ice ages seem far worse than global warming.

Dr. Robert Buxbaum, December 30, 2017

Global warming’s 19 year pause

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

Global temperatures measured from the antarctic ice shows stable chaos and self-similarity.

The global climate is, as best I can tell, chaotic with 100,000 year ice-age cycles punctuated by smaller cycles of 1000 years, 100 years, etc. On the ice-age time scale shown at left, the temperature rise of the last century looks insignificant and very welcome; warm seems better than cold in my eyes. But the press and academic community has focused on the evils of warmth — global warming. They point out that temperatures have risen 1 1/2 °C since the little ice age of the early 1600s, and that 1/2 °C of this has occurred since 1900. Al Gore won a Nobel prize for his assertion that the rate of rise had accelerated to 4°C per century — a “hockey-stick change” caused by industrial CO2. This change was expected to bring disaster by 2015: The arctic was supposed to be ice-free, and Manhattan was expected to sink. I’ve posted a “Good Morning America” clip from 2008 highlighting this “inconvenient truth”.

Our 19 1/2 year global warming pause; plot from Andrew Watts with Al Gore's prediction shown in red. During the time shown, the atmospheric CO2 content has gone up by about 25%, but the prediction has not come to pass.

Our 19 1/2 year global warming pause; plot from Andrew Watts with Al Gore’s prediction shown in red. So far, the prediction has not come to pass.

As it happens, not only hasn’t global warming accelerated, it seems to have paused. There have been no significant temperature changes since late 1997, as shown.  The main explanations are clouds and solar variation: variations that the Obama administration claims will end any day now. The problem, as I see it, is that climate is fundamentally chaotic, and thus unpredictable except on the very long, ice-age, timescale. It will thus always make fools of those who predict.

This is not to say that pollution is good, or that CO2 is, but it suggests our models and remedies are flawed. The CO2 content of the air has increased 25% over the past 19 years. It now mostly comes from China and India, countries that enthusiastically endorse having us reduce our output. My thinking is that lowering US production will, in no way, protect us from the dire predictions below.

Despite pressure from China and India, the US pulled out of the Paris climate accord last month. It now seems several other countries will pull out as well.

Robert Buxbaum, July 27, 2017. I’ve also written about how the global warming of the mid 1800s lead us to have the president’s Resolute desk.

Global warming and the president’s Resolute desk

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In the summer of 2016, the Crystal Serenity, a cruse ship passed through the Northwest passage, going from the Pacific to the Atlantic above the Canadian arctic circle. It was a first for a cruise ship, but the first time any modern ship made the passage, it was 162 years ago, and the ship was wooden and unmanned. It was the British Resolute; wood from that ship was used to make the President’s main desk — one used by the last four presidents. And thereby hangs a tale of good global warming, IMHO.

President Trump meets with college presidents at the Resolute desk. Originally the front had portraits of Queen Victoria and President Hayes. Those are gone; the eagle on the front is an addition, as is the bottom stand.

President Trump meets with college presidents at the Resolute desk. Originally the front had portraits of Queen Victoria and President Hayes. Those are gone; the eagle on the front is an addition, as is the bottom stand. The desk is now 2″ taller than originally. 

The world today is warmer than it was in 1900. But, what is not generally appreciated is that, in 1900 the world was warmer than In 1800; that in 1800 it was warmer than in 1700; and that, in 1640, it was so cold there were regular fairs on the frozen river Themes. By the 1840s there were enough reports of global warming that folks in England thought the northwest passage might have opened at last. In 1845 the British sent two ships, the Erebus and the Terror into the Canadian Arctic looking for the passage. They didn’t make it. They and their crews were lost and not seen again until 2014. In hopes of finding them though, the US and Britain sent other ships, including the Resolute under the command of captain Edward Belcher.

The Resolute was specially made to withstand the pressure of ice. Like the previous ships, and the modern cruise ship, it entered the passage from the Pacific during the peak summer thaw. Like the ships before, the Resolute and a partner ship got stuck in the ice — ice that was not quite stationary, but nearly so, The ships continued to move with the ice, but at an unbearably slow pace. After a year and a half captain Belcher had moved a few hundred miles, but had had enough. He abandoned his ships and walked out of Canada to face courts martial in England (English captains were supposed to “go down with the ship”). Belcher was acquitted; the ice continued to move, and the ships moved with it. One ship sank, but the Resolute, apparently unscathed, passed through to the Atlantic. Without captain or crew, she was the first ship in recorded history to make the passage, something that would not happen again till the Nautilus nuclear submarine did it under the ice, 100 years later.

 

The ghost ship Resolute was found in September 1855, five years after she set sail, by captain Buddington of the American whaler, George Henry. She was floating, unmanned, 1200 miles from where captain Belcher had left her. And according to the law of the sea, she belonged to Buddington and his crew to use as they saw fit. But the US was inching to war with Britain, an outgrowth of the Crimean war and seized Russo-American property. Franklin Pierce thought it would help to return the ship as a sign of friendship — to break the ice, as it were. A proposal for funds was presented to congress and passed; the ship was bought, towed to the Brooklyn Navy yard for refitting, and returned to Britain as a gift. The gift may have worked: war with Britain was averted, and the seized property was returned. Then again, Britain went on to supply the confederacy early in the Civil War. None-the-less, it was a notable ship, and a notable gift, and when it was broken up, Parliament decided to have two “friendship desks” made of its timbers. One desk was presented to President Hayes, the other to Queen Victoria. One of these desks sits the British Naval museum at Portsmouth; its American cousin serves Donald Trump in the Oval office as it has served many president before him. It has been used by Coolidge, Kennedy, Carter, Reagan, Clinton, Bush II, and Obama before him — a reminder that global warming can be good, in both senses. If you are interested in the other presidents’ desks, I wrote a review of them here.

As for the reason for the global warming of the mid 1800s, It seems that climate is chaotic. ON a related note, I have proposed that we make a more-permanent northwest passage by cutting thorough one of the islands in northern Canada. If you want to travel the Northwest passage in 2017, there is another cruise scheduled, but passage is sold out.

Robert Buxbaum, March 16, 2017.

A plague of combined sewers

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The major typhoid and cholera epidemics of the US, and the plague of the Al Qaeda camps, 2009 are understood to have been caused by bad sewage, in particular by the practice of combining sanitary + storm sewage. Medieval plagues too may have been caused by combined sewers.

Combined sewer system showing an rain-induced overflow, a CSO.

A combined sewer system showing an rain-induced overflow, a CSO.

A combined system is shows at right. Part of the problem is that the outfall is hard to contain, so they tend to spew sewage into the lakes and drinking water as shown. They are also more prone to back up during rain storms; separated systems can back up too, but far less often. When combined sewers back up, turds and other infected material flows into home basements. In a previous post, “follow the feces,” I showed the path that Oakland county’s combined sewers outfalls take when they drain (every other week) into Lake St. Clair just upstream of the water intake, and I detailed why, every few years we back up sewage into basements. I’d like to now talk a bit more about financial cost and what I’d like to do.

The combined sewer system shown above includes a small weir dam. During dry periods and small rain events, the dam keeps the sewage from the lake by redirecting it to the treatment plant. This protects the lakes so that sometimes the beaches are open, but there’s an operation cost: we end up treating a lot of rain water as if it were sewage. During larger rains, the dam overflows. This protects our basements (usually) but it does so at the expense of the drinking water, and of the water in Lake St. Clair and Lake Erie.

As a way to protect our lakes somewhat Oakland county has added a retention facility, the George W. Kuhn. This facility includes the weir shown above and a huge tank for sewage overflows. During dry periods, the weir holds back the flow of toilet and sink water so that it will flow down the pipe (collector) to the treatment plant in Detroit, and so it does not flow into the lake. Treatment in Detroit is expensive, but nonpolluting. During somewhat bigger rains the weir overflows to the holding tank. It is only during yet-bigger rains (currently every other week) that the mixture of rain and toilet sewage overwhelms the tank and is sent to the river and lake. The mess this makes of the lake is shown in the video following. During really big rains, like those of August 2014, the mixed sewage backs up in the pipes, and flows back into our basements. With either discharge, we run the risk of plague: Typhoid, Cholera, Legionnaires…

Some water-borne plagues are worse than others. With some plagues, you can have a carrier, a person who can infect many others without becoming deathly sick him or herself. Typhoid Mary was a famous carrier of the 1920s. She infected (and killed) hundreds in New York without herself becoming sick. A more recent drinking water plague, showed up in Milwaukee 25 years ago. Some 400,000 people were infected, and 70 or so died. Milwaukee disinfected its drinking water with chlorine and a bacteria that entered the system was chlorine tolerant. Milwaukee switched to ozone disinfection but Detroit still uses chlorine.

Combined sewers require much larger sewage treatment plants than you’d need for just sanitary sewage. Detroit’s plant is huge and its size will need to be doubled to meet new, stricter standards unless we bite the bullet and separate our sewage. Our current system doesn’t usually meet even the current, lower standards. The plant overflows and operation cost are high since you have to treat lots of rainwater. These operation costs will keep getter higher as pollution laws get tougher.

In Oakland county, MI we’ve started to build more and more big tanks to hold back and redirect the water so it doesn’t overwhelm the sewage plants. The GWK tank occupies 1 1/2 miles by about 100 feet beneath a golf course. It’s overwhelmed every other week. Just think of the tank you’d need to hold the water from 4″ of rain on 900 square mile area (Oakland county is 900 square miles). Oakland’s politicians seem happy to spend money on these tanks because it creates jobs and graft and because it suggests that something is being done. They blame politics when rain overwhelms the tank. I say it’s time to end the farce and separate our sewers. My preference is to separate the sewers through the use of French drains or bio-swales, and through the use of weir dams. I’m running for drain commissioner. Here’s something I’ve written on the chemistry of sewage, and on the joy of dams.

Dr. Robert E. Buxbaum, July 1-Sept 16, 2016.

Why are glaciers blue

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i recently returned from a cruse trip to Alaska and, as is typical for such, a highlight of the trip was a visit to Alaska’s glaciers, in our case Hubbard Glacier, Glacier bay, and Mendenhall Glacier. All were blue — bright blue, as were the small icebergs that broke off. Glacier blocks only 2 feet across were bright blue like the glaciers themselves.

Hubbard Glacier, Alaska. Note how blue the ice is

Hubbard Glacier, Alaska. My photo. Note how blue the ice is

What made this interesting/ surprising is that I’ve seen ice sculptures that are 5 foot thick or more, and they are not significantly blue. They have a very slight tinge, but are generally more colorless than glass to my ability to tell. I asked the park rangers why the glaciers were blue, but was given no satisfactory answer. The claim was that glacier ice contained small air bubbles that scattered light the same way that air did. Another park ranger claimed that water is blue by nature, so of course the glaciers were too. The “proof” to this was that the sea was blue. Neither of these seem quite true to me, though there seamed some grains of truth. Sea water, I notice, is sort of blue, but isn’t this shade of blue, certainly not in areas that I’ve lived. Instead, sea water is a rather grayish similar to mud and sea-weeds that I’d expect to find on the sea floor. What’s more, if you look through the relatively clear water of a swimming-pool water to the white-tile bottom, you see only a slight shade of blue-green, even at the 9 foot depth where the light you see has passed through 18 feet of water. This is far more water than an iceberg thickness, and the color is nowhere near as pure blue and the intensity nowhere near as strong.

Plymouth, MI Ice sculpture -- the ice is fairly clear, like swimming pool water

Plymouth, MI Ice sculpture — the ice is fairly clear, like swimming pool water

As for the bubble explanation, it doesn’t seem quite right, either. The bubble size would be non-uniform, with many quite large resulting in a mix of scattered colors — an off white– something seen with the sky of mars. Our earth sky is a purer blue, but this is not because of scattering off of ice-crystals, dust or any other small particles, but rather scattering off the air molecules themselves. The clear blue of glaciers, and of overturned icebergs, suggests (to me) a single-size scattering entity, larger than air molecules, but much smaller than the wavelength of visible light. My preferred entity would be a new compound, a clathrate structure compound, that would be formed from air and ice at high pressures.

An overturned ice-burg is remarkably blue: far bluer than an Ice sculpture. I claim clathrates are the reason.

An overturned ice-burg is remarkably blue: far bluer than an Ice sculpture. I claim clathrates are the reason.

Sea-water forms clathrate compounds with natural gas at high pressures found at great depth. My thought is that similar compounds form between ice and one or more components of air (nitrogen, oxygen, or perhaps argon). Though no compounds of this sort have been quite identified, all these gases are reasonably soluble in water so that suggestion isn’t entirely implausible. The clathrates would be spheres, bigger than air molecules and thus should have more scattering power than the original molecules. An uneven distribution would explain the observation that the blue of glaciers is not uniform, but instead has deeper and lighter blue edges and stripes. Perhaps some parts of the glacier were formed at higher pressures one could expect that these would form more clathrate compounds, and thus more blue. One sees the most intense blue in overturned icebergs — the parts that were under the most pressure.

Robert Buxbaum, October 12, 2015. By the way, some of Alaska’s glaciers are growing and others shrinking. The rangers claimed this was the bad effect of global warming: that the shrinking glaciers should be growing and the growing ones shrinking. They also worried that despite Alaska temperatures reaching 40° below reasonably regularly, it was too warm (for whom?). The lowest recorded temperature in Fairbanks was -66°F in 1961.

18 year pause in global warming

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Here is an updated version of the climate change graph. It’s now 18+ years, and as was true with last year’s version, 17+ years of no climate change, I see no significant climate change. Similar to this: Global Warming takes a 15 year rest.

18 years of Global Temperature anomaly to July 2015

18+ years of Global Temperature anomaly to July 2015. The climate seems to have stopped changing.

Though the average planetary temperature has remained constant, there is local variation. It’s been warm in California for the past 2+ years, but cold in Michigan. Before that, it was warm in Michigan and California was cold. The Antarctic ice is at record high levels while the arctic ice has shrunk enough that we should make a Northwest passage.

Climate vs weather from the blog of Steven Goddard

Climate vs weather, from the blog of Steven Goddard. It’s funny because…

Theory suggests we should see global warming because of increased CO2 trapping of atmospheric heat 2 miles up or so. The problem with the theory is that it doesn’t include clouds. A few extra clouds, e.g. from Chinese industry, could have more cooling power than a lot of CO2 has heating power. It seems that the effects cancel, and temperature 2-3 miles up is about what you’d expect from entropy.

My biggest climate fear, BTW, is global cooling: a new ice age. They come every 110,000 years or so and we seem overdue.

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

Global temperatures from the antarctic ice show ice ages every 110,000 years. cyclic chaos and self-similarity.

Robert Buxbaum, July 22, 2015. You may not have noticed, but there have been relatively few hurricanes — something that could change at any minute. Here’s a link to 1/2 hour lecture by a Nobel physicist, Ivar Giaever on the subject. Like me, he notices no change, and thinks warmer is better.

Can you spot the man-made climate change?

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As best I can tell, the only constant in climate is change, As an example, the record of northern temperatures for the last 10,000 years, below, shows nothing but major ups and downs following the end of the last ice age 9500 years ago. The only pattern, if you call it a pattern, is fractal chaos. Anti-change politicos like to concentrate on the near-recent 110 years from 1890 to 2000. This is the small up line at the right, but they ignore the previous 10000 or more, ignore the fact that the last 17 years show no change, and ignore the variation within the 100 years (they call it weather). I find I can not spot the part of the change that’s man-made.

10,000 years of climate change based on greenland ice cores. Ole Humlum – Professor, University of Oslo Department of Geosciences.

10,000 years of northern climate temperatures based on Greenland ice cores. Dr. Ole Humlum, Dept. of Geosciences, University of Oslo. Can you spot the part of the climate change that’s man-made?

Jon Stewart makes the case for man-made climate change.

Steven Colbert makes his case for belief: If you don’t believe it you’re stupid.

Steven Colbert makes the claim that man-made climate change is so absolutely apparent that all the experts agree, and that anyone who doubts is crazy, stupid, or politically motivated (he, of course is not). Freeman Dyson, one of the doubters, is normally not considered crazy or stupid. The approach reminds me of “the emperor’s new clothes.” Only the good, smart people see it. The same people used to call it “Global Warming” based on a model prediction of man-made warming. The name was changed to “climate change” since the planet isn’t warming. The model predicted strong warming in the upper atmosphere, but that isn’t happening either; ski areas are about as cold as ever (we’ve got good data from ski areas).

I note that the climate on Jupiter has changed too in the last 100 years. A visible sign of this is that the great red spot has nearly disappeared. But it’s hard to claim that’s man-made. There’s a joke here, somewhere.

Jupiter's red spot has shrunk significantly. Here it is now. NASA

Jupiter’s red spot has shrunk significantly. Here it is now. NASA

As a side issue, it seems to me that some global warming could be a good thing. The periods that were warm had peace and relative plenty, while periods of cold, like the little ice age, 500 years ago were times of mass starvation and plague. Similarly, things were a lot better during the medieval warm period (1000 AD) than during the dark ages 500-900 AD. The Roman warm period (100 BC-50 AD) was again warm and (relatively) civilized. Perhaps we owe some of the good food production of today to the warming shown on the chart above. Civilization is good. Robert E. Buxbaum January 14, 2015. (Corrected January 19; I’d originally labeled Steven Colbert as Jon Stewart)

 

17+ years of no climate change

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Much of the data underlying climate change is bad, as best I can tell, and quite a lot of the animosity surrounding climate legislation comes from the failure to acknowledge this. Our (US) government likes to show the climate increasing at 4-6°C/century, or .05°C/year, but this is based on bad data of average global temperatures, truncated conveniently to 1880, and the incorrect assumption that trends always continue — a bad idea for stock investing too. We really don’t have any good world-wide temperature going back any further the 1990s, something the Canadian ice service acknowledges (see chart below) but we do not. Worse yet, we adjust our data to correct for supposed errors.

Theory vs experiment in climate change data

Theory vs experiment in climate change data; 17 years with no change.

High quality observations begin only about 10 years ago, and since then we have seen 17+ years of no significant climate change, not the .05°C per year predicted. Our models predicted an ice-free Arctic by 2013, but we had one of the coldest winters of the century. Clearly the models are wrong. Heat can’t hide, and in particular it can’t hide in the upper atmosphere where the heat is supposed to be congregating. The predictive models were not chaotic, and weather is, but instead show regular, slow temperature rises based on predictions of past experimental data.

In Canada and Australia, the climate experts are nice enough to put confidence bars on the extrapolated data before publishing it. Some researchers are also nice enough to provide data going back further, to late Roman times when the weather was really warm, or 20,000 years ago, when we had an ice age (it’s unlikely that the ice age ended because of automobile traffic).

Canada's version of Ice coverage data. The grey part is the error bar. Canada is nice enough to admit they know relatively little of what the climate was like in the 70s and 80s. We do not.

Canada’s version of Ice coverage data. The grey part is the error bar. Canada is nice enough to admit they don’t know what it was like in the 70s and 80s. We do not.

So what’s so wrong about stopping US coal use, even if it does not cause global warming. For one, it’s bad diplomatically — it weakens us and strengthens countries that hate us (like Iran), and countries like China that burn lots of coal and really pollute the air. It also diverts the US from real air pollution and land use discussions. If you want less air pollution, perhaps nuclear is the way to go. Finally, there you have to ask, even if we could adjust the earth’s temperature at will, who would get control of the thermostat? Who would decide if this summer should be warm or cold, or who should get rains, or sun. With great power comes great headaches.

Robert Buxbaum, June 21, 2014

If hot air rises, why is it cold on mountain-tops?

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This is a child’s question that’s rarely answered to anyone’s satisfaction. To answer it well requires college level science, and by college the child has usually been dissuaded from asking anything scientific that would likely embarrass teacher — which is to say, from asking most anything. By a good answer, I mean here one that provides both a mathematical, checkable prediction of the temperature you’d expect to find on mountain tops, and one that also gives a feel for why it should be so. I’ll try to provide this here, as previously when explaining “why is the sky blue.” A word of warning: real science involves mathematics, something that’s often left behind, perhaps in an effort to build self-esteem. If I do a poor job, please text me back: “if hot air rises, what’s keeping you down?”

As a touchy-feely answer, please note that all materials have internal energy. It’s generally associated with the kinetic energy + potential energy of the molecules. It enters whenever a material is heated or has work done on it, and for gases, to good approximation, it equals the gas heat capacity of the gas times its temperature. For air, this is about 7 cal/mol°K times the temperature in degrees Kelvin. The average air at sea-level is taken to be at 1 atm, or 101,300  Pascals, and 15.02°C, or 288.15 °K; the internal energy of this are is thus 288.15 x 7 = 2017 cal/mol = 8420 J/mol. The internal energy of the air will decrease as the air rises, and the temperature drops for reasons I will explain below. Most diatomic gases have heat capacity of 7 cal/mol°K, a fact that is only explained by quantum mechanics; if not for quantum mechanics, the heat capacities of diatomic gases would be about 9 cal/mol°K.

Lets consider a volume of this air at this standard condition, and imagine that it is held within a weightless balloon, or plastic bag. As we pull that air up, by pulling up the bag, the bag starts to expand because the pressure is lower at high altitude (air pressure is just the weight of the air). No heat is exchanged with the surrounding air because our air will always be about as warm as its surroundings, or if you like you can imagine weightless balloon prevents it. In either case the molecules lose energy as the bag expands because they always collide with an outwardly moving wall. Alternately you can say that the air in the bag is doing work on the exterior air — expansion is work — but we are putting no work into the air as it takes no work to lift this air. The buoyancy of the air in our balloon is always about that of the surrounding air, or so we’ll assume for now.

A classic, difficult way to calculate the temperature change with altitude is to calculate the work being done by the air in the rising balloon. Work done is force times distance: w=  ∫f dz and this work should equal the effective cooling since heat and work are interchangeable. There’s an integral sign here to account for the fact that force is proportional to pressure and the air pressure will decrease as the balloon goes up. We now note that w =  ∫f dz = – ∫P dV because pressure, P = force per unit area. and volume, V is area times distance. The minus sign is because the work is being done by the air, not done on the air — it involves a loss of internal energy. Sorry to say, the temperature and pressure in the air keeps changing with volume and altitude, so it’s hard to solve the integral, but there is a simple approach based on entropy, S.

Les Droites Mountain, in the Alps, at the intersect of France Italy and Switzerland is 4000 m tall. The top is generally snow-covered.

Les Droites Mountain, in the Alps, at the intersect of France Italy and Switzerland is 4000 m tall. The top is generally snow-covered.

I discussed entropy last month, and showed it was a property of state, and further, that for any reversible path, ∆S= (Q/T)rev. That is, the entropy change for any reversible process equals the heat that enters divided by the temperature. Now, we expect the balloon rise is reversible, and since we’ve assumed no heat transfer, Q = 0. We thus expect that the entropy of air will be the same at all altitudes. Now entropy has two parts, a temperature part, Cp ln T2/T1 and a pressure part, R ln P2/P1. If the total ∆S=0 these two parts will exactly cancel.

Consider that at 4000m, the height of Les Droites, a mountain in the Mont Blanc range, the typical pressure is 61,660 Pa, about 60.85% of sea level pressure (101325 Pa). If the air were reduced to this pressure at constant temperature (∆S)T = -R ln P2/P1 where R is the gas constant, about 2 cal/mol°K, and P2/P1 = .6085; (∆S)T = -2 ln .6085. Since the total entropy change is zero, this part must equal Cp ln T2/T1 where Cp is the heat capacity of air at constant pressure, about 7 cal/mol°K for all diatomic gases, and T1 and T2 are the temperatures (Kelvin) of the air at sea level and 4000 m. (These equations are derived in most thermodynamics texts. The short version is that the entropy change from compression at constant T equals the work at constant temperature divided by T,  ∫P/TdV=  ∫R/V dV = R ln V2/V1= -R ln P2/P1. Similarly the entropy change at constant pressure = ∫dQ/T where dQ = Cp dT. This component of entropy is thus ∫dQ/T = Cp ∫dT/T = Cp ln T2/T1.) Setting the sum to equal zero, we can say that Cp ln T2/T1 =R ln .6085, or that 

T2 = T1 (.6085)R/Cp

T2 = T1(.6085)2/7   where 0.6065 is the pressure ratio at 4000, and because for air and most diatomic gases, R/Cp = 2/7 to very good approximation, matching the prediction from quantum mechanics.

From the above, we calculate T2 = 288.15 x .8676 = 250.0°K, or -23.15 °C. This is cold enough to provide snow  on Les Droites nearly year round, and it’s pretty accurate. The typical temperature at 4000 m is 262.17 K (-11°C). That’s 26°C colder than at sea-level, and only 12°C warmer than we’d predicted.

There are three weak assumptions behind the 11°C error in our predictions: (1) that the air that rises is no hotter than the air that does not, and (2) that the air’s not heated by radiation from the sun or earth, and (3) that there is no heat exchange with the surrounding air, e.g. from rain or snow formation. The last of these errors is thought to be the largest, but it’s still not large enough to cause serious problems.

The snow cover on Kilimanjaro, 2013. If global warming models were true, it should be gone, or mostly gone.

Snow on Kilimanjaro, Tanzania 2013. If global warming models were true, the ground should be 4°C warmer than 100 years ago, and the air at this altitude, about 7°C (12°F) warmer; and the snow should be gone.

You can use this approach, with different exponents, estimate the temperature at the center of Jupiter, or at the center of neutron stars. This iso-entropic calculation is the model that’s used here, though it’s understood that may be off by a fair percentage. You can also ask questions about global warming: increased CO2 at this level is supposed to cause extreme heating at 4000m, enough to heat the earth below by 4°C/century or more. As it happens, the temperature and snow cover on Les Droites and other Alp ski areas has been studied carefully for many decades; they are not warming as best we can tell (here’s a discussion). By all rights, Mt Blanc should be Mt Green by now; no one knows why. The earth too seems to have stopped warming. My theory: clouds. 

Robert Buxbaum, May 10, 2014. Science requires you check your theory for internal and external weakness. Here’s why the sky is blue, not green.

Amazing tornado drought of 2014

At 143 days as of April 10, 2014, the span between major tornadoes (EF3s and stronger) is the 6th longest in the last 60 years, and it isn’t over yet. Even small tornadoes are becoming more scarce. Last year saw few hurricanes and tornadoes, and so far we’ve had only 100 total tornadoes (see below); in a typical year there’d be 323. The good news has gone unreported, I think, because there’s no event, no photo-opportunity; no interviews with survivors, police, and experts.

US tornadoes: typical and year to date, January 1 to April 10 2014, NOAA

US tornadoes: typical and year to date, January 1 to April 10 2014, from NOAA, storm center

Perhaps this is a bonuses from global warming, or from the very cold winter just passed, or from the chaotic, weather; it’s hard to tell weather from climate. Whatever the reason, it’s happening and good. Here’s how tornadoes lift stuff up, with video (Einstein’s explanation). Here’s an explanation of hurricanes (my explanation).

Robert E. Buxbaum, April 11, 2014. In other good news, the ozone hole is shrinking. genetically modified foods don’t seem to cause cancer, and many bad things are good for you, like sunlight. Enjoy the good.