400 words in 30 minutes on chaos and the weather

Goal: 400 words researched and written in half and hour. For me, for practice. Corrections welcome in the comments.

∞

Why does the weather forecast only go a few days into the future, and why are they so often wrong about what will happen in a couple of days time? If they know where the weather systems - clouds and so on - are right now, and they have a good computer model for how things change over time, why can't they predict the weather a fortnight from now, or a year?

The answer is that weather is a chaotic system. In everyday language, chaos means complete disorder without pattern but in mathematics the same word chaos refers to a particular phenomenon. Chaos theory studies the behaviour of systems that are very sensitive to small changes in initial conditions. This means slight differences in the starting point of a system can give very different results in the long term.

Because the system gives very different results when you run it twice with similar starting conditions, you might think that it involves some random elements that change each time you run the process. However, this mathematical chaos does not involve random elements. If you run the process twice with exactly the same starting conditions, it will give the same results. This is not like a random process - each time you run it you get a different outcome. We call such a process, in which running the system in the same way gives the same results every time, deterministic, because if we know everything about it we can determine in advance what will happen.

What makes chaotic systems interesting is that they are deterministic but so sensitive to initial conditions that they aren't predictable. A slight difference in the initial conditions can give massively different outcomes.

For the weather forecast, the computer model is fed by initial data from weather monitoring stations. If some of that data is slightly off, or if the computer model doesn't correctly guess what is happening between stations, the model can give radically different answers to what will actually happen. These differences mount up - if you got tomorrow's weather forecast slightly wrong and used it as the initial conditions to forecast the next day, then that could be very wrong indeed.

This is why weather forecasts are continually updated and why forecasts are only made reliably for a short period in the future.

∞

Time: 39 minutes. 388 words. Performance: took too long. I started without a clear idea and started writing as I was researching. At 25 minutes I gave up on one idea and started anew. Bad practice! I should try to remember to take some time to process before I start writing.

Explanation. More 400 words.

Pedantry on Euler and masts

I listened to the second episode of A Brief History of Mathematics on Euler yesterday. I was quite taken with a quote from Euler which, to me, says something of the potential dangers of the application of mathematics to the real world. The relevant section of the programme is:

"In the year that Issac Newton died, Euler, then aged just 19, was awarded the most prestigious mathematical prize of the day, the first of many, many achievements and accolades he would receive during his lifetime. The annual mathematical challenge issued by the Paris Academy of Science in 1727 was this: "What is the best way to arrange masts on a ship?" At first sight it's a very practical problem, but the young Swiss mathematician Leonhard Euler attacked it as a purely mathematical puzzle. Despite having never set foot on a ship, he felt perfectly well qualified to calculate the optimal arrangement of masts. For him, it was a problem that could be solved by mathematics alone.
"'I did not find it necessary to confirm this theory of mine by experiment because it is derived from the surest and most secure principles of mathematics, so that no doubt whatsoever can be raised on whether or not it be true and takes place in practice.'
"Leonhard Euler had absolute faith in mathematics...."

Having copied this out and investigated a little, I have a couple of small factual problems with this passage.

Newton died in 1727 (on 20 or 31 March, depending on your calendar) and Euler was born 15 April 1707, so I'm happy Euler was 19 in the year Newton died. And Euler's report was submitted to the 1727 Grand Prize of the Paris Academy on the arrangement of masts on a ship. (Whether the Prize was awarded pre- or post-15 April I cannot find, but this seems almost irrelevant really; that, according to the Euler Archive, Euler wrote the paper in 1726 isn't really relevant since the claim made is of the award of the prize.) Euler didn't, however, win the prize. He came second. The first prize was shared between Pierre Bouguer and Charles-Étienne Camus. This is regarded as a substantial achievement in any case given Euler's age and the fact that Bouguer was an established expert on ships. The programme doesn't quite say he came first, but the wording "was awarded the most prestigious mathematical prize of the day" is strange when Euler was second to Bouguer and Camus.

My second factual issue is that I can't find the quote as quoted anywhere. I can only find it quoted online as:

"I did not find it necessary to confirm this theory of mine by experiment because it is derived from the surest and most secure principles of mechanics, so that no doubt whatsoever can be raised on whether or not it be true and takes place in practice." (emphasis added)

The quote is from Euler's submission to the Grand Prize, which was not written in English, so there are translation issues here. The Euler Archive at Dartmouth provides a scan of the original printed in Recueil des pièces qui ont remporté lex prix de l’académie royale des sciences (1732):

"Haud opus esse existimavi istam meam theoriam experientia confirmare, cum integra et ex certissimis et irrepugnabilibus principiis Mechanicis deducta, atque adeo de illa dubitari, an vera sit ac an in praxi locum habere queat, minime possit."

Ian Bruce provides an original translation where the relevant passage is given as:

I do not think that it is necessary to confirm my theory by experiment, since the whole has been deduced from both the surest and the most irrefutable principles of mechanics, and thus concerning that there cannot be the least doubt or the truth can be put to the test in practice.

In any case the latin "Mechanicis" or the Ian Bruce translation "mechanics" suggests "mechanics", not "mathematics".

This puts me in mind of another recent programme, The Beauty of Diagrams, in an episode on Newton and optics Marcus says "and, as Newton took up residence again at Trinity, over in Italy Galileo was busy working out the speed at which light reaches us from the Sun". The problem here is that Galileo died almost a year before Newton was born.

My feeling is that the Prize issue may have been a quick shorthand as this is a side point and the programme is very short overall. The essential point is "Euler made fantastic achievements from an early age and here's something of his view on the application of mathematics". Then the mathematics/mechanics mixup seems likely to me to have been a change to avoid having to explain the term mechanics.

I have a lot of sympathy with getting obscure facts wrong with this sort of thing, which I feel is an ever-present danger in everything I do and I'm sure I will sometimes land on the wrong side of correct. I also have a lot of sympathy with trying to get the core message across without getting distracted by details in a limited timeframe. This is something I haven't had a lot of experience with in my self-published podcast endeavours, in which if you need an extra 30 seconds to explain something you can just take it (although you perhaps shouldn't).

In any case, I would recommend any applied mathematicians hang a sign somewhere with the following wording, as a warning against arrogance:

"Haud opus esse existimavi istam meam theoriam experientia confirmare, cum integra et ex certissimis et irrepugnabilibus principiis Mechanicis deducta"
"I do not think that it is necessary to confirm my theory by experiment, since the whole has been deduced from both the surest and the most irrefutable principles of mechanics."
Leonhard Euler, c.1727.

Also, consider including the wording in your next grant application.

Cosmic-Ray Research Centre and Cold War Helter-Skelter in Secret Cave City

Samuel Hansen directed me to this article: "The secret cave city under Nottingham". The article is highlighting The Nottingham Caves Survey, who offer:

The Nottingham Caves Survey is in the process of recording all of Nottingham’s 450+ sandstone caves. The project is now underway and we are surveying caves even as you read this. Keep checking the website for newly- surveyed caves! You can read more about the caves, see photographs, watch fly-through videos and take virtual tours.

The survey is producing some interesting images, such as the following (credit: Trent & Peak Archaeology / The University of Nottingham; click to enlarge):

Laser-scanned image of King David’s Dungeon, below Nottingham Castle. King David II of Scotland was reputedly held captive here in 1346.

Laser scanned orthographic plan of the Goose Gate caves, Nottingham. These caves include a medieval malt kiln, 18th-century brewery cellars and a 19th-century butchery.

And some very compelling videos:

I'm not sure about the use of the phrase "secret cave city" and the talk of the Survey using techniques to "reveal elaborate cave systems under the town of Nottingham" (in the post, not used by the Survey itself). I am pleased that the Survey is producing detailed scans of these caves, it's interesting business and nice to see them being made accessible through multimedia, but the caves were not previously secret. In my experience, the existence of widespread caves under Nottingham is fairly well known. The Survey knows this and is specifically scanning those caves that are physically accessible (although perhaps not all open to the public).

Nottingham is referred to by a biographer of Alfred the Great in 868 as "Tigguo Cobauc" which you see translated as "House of Caves," "Place of Caves" and "City of Caves". A book on the history of Nottingham I have has the following:

"Entering the town at the beginning of the seventeenth century, Bishop Corbett exclaimed: 'Why, the people live not in howses but are earthed in holes!' If a man was destitute, explained one writer in 1639, 'he has only to go to Nottingham with a mattock, a shovel, a crow or an iron, a chisel or a mallet, and with such instruments he may play the mole, the coney, and work himself a hole or burrow for his family where over their heads the grass and pastures grow, beasts do feed and cows are milked.' The caves in the area of the castle have been used as mushroom farms, cockpits, gambling dens, privies, drinking cellars and burial vaults. During World War II, some factories enlarged them as air raid shelters for their workers. Today the caves house social clubs, a rifle range, store-rooms and a cosmic-ray research centre. From time to time somebody gets even more exotic ideas for them - such as linking them together for a Tube railway or turning them into atomic shelters, to be reached quickly by people sliding down on mats as if on a helter-skelter. Builders, of course, regard them as a nuisance and a hazard, and refuse to give a firm quotation for work in the centre of Nottingham. But the tours of underground Nottingham run for the 1972 festival proved so popular that the city has woken up to its subterranean history as a tourist attraction. The thirteenth-century tannery (the only underground one in Britain) revealed in the multi-million pound Broad Marsh development, has been spared"
(Portrait of Nottingham by E. Bryson, 3rd ed. 1983, p. 26).

I reckon there's a good alternative history story in those imagined subterranean Thunderbirds-esk helter-skelter cold war bunkers.

The quote above is all interesting but the phrase "cosmic-ray research centre" in particular catches my eye. There is a paper 'The Absolute Intensity of Muons at 31.6 hg cm^-2 Below Sea-Level' by Crookes & Rastin (Nuclear Physics B 58 pp. 93-109, 1973) which describes an experiment in a cave under the Castle:

"A cave in bunter sandstone of uniform composition lies below Nottingham Castle, and this provides an experimental site at 43.0 hg cm^-2 from the top of the atmosphere. The apparatus used by Crookes and Rastin to measure the absolute intensity at sea-level in both the vertical and inclined directions was therefore used to provide similar data at this underground location" (p. 96).

The first picture above shows some of the caves under the Castle and it seems some of them provided an experimental site for detection of cosmic rays for physicists from the University of Nottingham in the 1970s.