In June, there was only one day here in Salem that was below normal -- June 1st, at -0.5°F below average. For the month we were 7.5°F above average, with only 0.67 inches of rain, just 45% of average. Ugly.
Is this high, prolonged heat wave due to manmade global warming? Would this heat wave have happened in a world without AGW? I don't know. Cliff Mass says it's just natural variability, but others have disagreed with him in the past.
But here's something: a warmer world increases the chances of extreme temperatures exponentially.
Here's what I mean, with some gory details, but not too bad. And I get to practice my LaTeX, which I wrote my thesis in but haven't used since.
Suppose the background temperature of the world is T1, and we want to calculate the probability of an event with temperature T -- that is, the change of occurence of temperature T.
A few years ago Tamino calculated the probability spread using data for the continental US (USA48), and got the figure to the right. It's very close to a normal (Gaussian) distribution, and that's what I'll assume for the spread of probabilities. We're calculating the vertical distance between the blue and red curves, for any value of temperature along the x-axis.
As Wikipedia says,
So, if the background temperature is T1, the probability of the occurrence of a temperature T is
Now suppose the world gets warmer, and the background temperature changes from T1 to T2, both less than T. Let ΔT = T2 - T1 be the amount of warming.
There is a similar equation to the above for p(T,T2), the probability of an event with temperature T in the warmer world T2, just by replacing T1 above with T2. I'll assume the spread of the distributions, σ, is the same in both worlds.
We want to calculate the change in the probability, Δp, of the event with temperature T:
as a function of ΔT.
After some algebra, we find
where it's understood, just to make things look simplier, T, T1 and T2 are all divided by σ√2.
We can gnaw on this further to get
where I've used the fact that
(except maybe on Venus!). All the crap on the left-hand side is junk, not dependent on ΔT or T2. So we get
where f is some function we don't really care about here.
So -- for a temperature change ΔT, the probability of the occurrence of temperature T goes up exponentially.
This, it seems to me, is proof of why extreme temperatures get much more likely in a warming world, faster than the linear amount by which the world warms.