Graham Readfearn has an article in the Guardian titled:
“Scientist’s theory of climate’s Titanic moment the ‘tip of a mathematical iceberg’
Formula for climate emergency shows if ‘reaction time is longer than intervention time left’ then ‘we have lost control.’”
drawing our attention to an article in Nature by Timothy Lenton, Johan Rockström, Owen Gaffney, Stefan Rahmstorf, Katherine Richardson, Will Steffen & Hans Joachim Schellnhuber.
Redfearn summarises it as follows:
When is an emergency really an emergency? […]Rather than being something abstract and open to interpretation, Schellnhuber says the climate emergency is something with clear and calculable risks that you could put into a formula. And so he wrote one.
Emergency = R × U = p × D × τ / T
In a comment article in the journal Nature Schellnhuber and colleagues explained that to understand the climate emergency we needed to quantify the relationship between risk (R) and urgency (U). Borrowing from the insurance industry, the scientists define risk (R) as the probability of something happening (p) multiplied by damage (D). For example, how likely is it that sea levels will rise by a metre and how much damage will that cause. Urgency (U) is the time it takes you to react to an issue (τ) “divided by the intervention time left to avoid a bad outcome (T)”, they wrote.
Schellnhuber, of the Potsdam Institute for Climate Impact Research in Germany, tells Guardian Australia the work on the formula was just the “tip of a mathematical iceberg” in defining the climate emergency. […] There is a time lag between the rapid cuts to greenhouse gases and the climate system reacting. Knowing if you have enough time tells you if you’re in an emergency or not. Schellnhuber used “standard risk analysis and control theory” to come up with the formula, and he was already putting numbers to it.
“As a matter of fact, the intervention time left for limiting global warming to less than 2C is about 30 [years] at best. The reaction time – time needed for full global decarbonisation – is at least 20 [years].” As the scientists write in Nature, if the “reaction time is longer than the intervention time left” then “we have lost control”. […]
Prof Will Steffen, of the Australian National University and the Stockholm Resilience Centre, and a co-author of the article, says: “Emergency can mean many things to many people. But there are some hard numbers behind why so many people are saying we are in a climate emergency. This formula sharpens our thinking. So we have 30 years to decarbonise and to stabilise our pressure on the climate system.” […]
“There are a range of these intervention times left,” Steffen says. “How long do we have before [the Greenland ice sheet] goes? Maybe we have 20 to 25 years and then we might be committed to losing Greenland.[…] “Our reaction time has to be fast and to decarbonise by 2050 we have to really move now. That’s the point of [Schellnhuber’s] maths. “To err on the side of danger is a stupid thing to do.”
John Ridgeway, our resident expert on Risk, may have something to say on this, but there are some points about the article that seem to call for my particular expertise, which is in O-level arithmetic. I checked the Nature article to make sure that Readfearn had got the sums right, and he has. The Nature article ends:
EMERGENCY: DO THE MATHS
We define emergency (E) as the product of risk and urgency. Risk (R) is defined by insurers as probability (p) multiplied by damage (D). Urgency (U) is defined in emergency situations as reaction time to an alert (τ) divided by the intervention time left to avoid a bad outcome (T). Thus:
E = R × U = p × D × τ / T
The situation is an emergency if both risk and urgency are high. If reaction time is longer than the intervention time left (τ / T > 1), we have lost control.
We argue that the intervention time left to prevent tipping could already have shrunk towards zero, whereas the reaction time to achieve net zero emissions is 30 years at best. Hence we might already have lost control of whether tipping happens.
Well, yes, you may argue that, but not with that formula you don’t. All the formula does is assert that an emergency is a risk times urgency. Urgency is measured in time divided by Time, and therefore has no units, but is simply a ratio. Risk is a probability and Damage is presumably measured in money or some econometric equivalent, like lives lost, psychological trauma, destruction of planet etc. Ergo Emergency is also defined in £s sterling. Odd, that.
So in order to estimate how much time we have left before we need to react to a particular Emergency, all you need is the Schellnhuber formula and a mathematical trick (Geoff’s Nature trick) which I learned about sixty years ago, which involves multiplying or dividing both sides by the same thing. Following this simple rule, I derive:
τ = ET/pD
Readfearn gives two examples to which this formula might be applied: sea level rise and the Titanic. Taking the latter one first: the limiting case is when the reaction time τ = time till you hit the iceberg T. In which case the probability of disaster is 1, and E = D, or Emergency = Damage, measured in £s sterling, loss of lives, jewellery, mummies etc. If the probability is near zero, then τ is very large, and Emergency and Damage both also tend to zero and the captain has all the time in the world. In any case, he can be reassured that, whatever the odds, since E is proportional to D, and p is a fixed number (either 1, if τ≥T,or somewhere between 1 and zero if τ<T) then the time he has to save the ship is proportional to the time before they hit the iceberg. Zeno would be proud of Hans Joachim.
For sea level rise, you can consider the case of a rise of one metre as Readfearn does, or 1 foot by the end of the century, which is in line with current trends, or any rise you like, since any old level is going to be reached some time or other. So let’s consider a rise of x ft at time T and give it a probability of 1. So τ = ET/D.
Let us further make the reasonable assumption that the reaction time to alert is equivalent to the intervention time left to avoid a bad outcome. This is what people do when they see something obviously going wrong. They fix it. This was the case with sea level rise in e.g. the Netherlands where, as soon as they realised sea levels were rising, many centuries ago, they started building dikes, canals etc. at 3mm per year, or faster if conditions permitted. It follows that E = D or, in layman’s terms, the Emergency equals the Damage. And, likewise, the Damage equals the Emergency. Well done Hans.
I invite those of you who are less mathematically challenged than I am to play with this fascinating formula devised by Potsdam’s finest. Assume (as I am sure we all do) that the probability p of something horrid happening is 1, and you find that, by the arithmetical conjuring trick I performed above, E = τD/T, which means that, for an Emergency of a given seriousness, you can reduce the Danger by spending more time τ reacting to the alert.
On the other hand, for a given Danger D, the more time τ you spend reacting to an alert, the greater the Emergency which will result, and Schellnhuber has the formula to prove it.
Which is why I’m leaving my Christmas shopping until the 24th.