James Hansen recently published analysis showing that temperature has increased enough that the probability of extremes (in particular, extreme heat) is significantly greater than it used to be. This has consequences.



We have indeed witnessed a spate of extreme heat events recently: the 2003 European heat wave, the monster Russian heat wave of 2010, Texas/Oklahoma 2011, and this year’s heat/drought in the USA. Hansen emphasized that these extreme events are so much more common that such a concordance simply wouldn’t have happened without global warming. Extremes are 10 times more likely than they used to be. We would of course have heat waves anyway, but global warming has made them distressingly more frequent.

Cliff Mass expressed another viewpoint. He suggested that the net temperature increase is small compared to the extremety of the observed heat waves, so global warming is only responsible for a small fraction of the heat. Therefore, he suggests, global warming has had little impact on heat-related stress on humanity. We have heat waves anyway, global warming has only made them a little worse.

I think that a proper evaluation should emphasize the risk of extreme heat (or general extremes). Risk is sometimes defined as the product of probability times cost. I’ll explore what that would be in an extremely simple model of the impact of changing temperature. This is just an exercise to explore the impact of making extremes both more extreme and more common, and the model I’ll apply is simple — in the extreme. But, you’ve got to crawl before you walk.

Here’s the model. Temperature (which could represent temperature on any time scale you choose, but should not be longer than seasonal or annual averages) is a random variable which during some reference period is governed by the normal probability density f(x). We’ll be interested in the temperature in some region and its cost, or damage, to that specific region. We’ll measure temperature in standard deviations above or below the mean during the reference period (which we will take to be a time of minimum risk), so our probability function is the “standard” normal with mean zero and standard deviation one:

Because global warming will change that distribution, we’ll need to use the full form of the normal pdf, namely

,

where is the mean (which starts at zero) and is the standard deviation (which starts at one).

We’ll say that the damage due to climate is a function D(x) of that temperature. During the reference period, we’ll suppose that the damage is zero unless temperature is at least two standard deviations above or below the mean. Beyond those extremes, it will increase slowly at first, but by the time temperature reaches 10 standard deviations (either high or low) we’ll consider damage to be total. I used this damage function, which is zero up to +2 std.devs, follows a sinusoidal curve to 10 std.devs, and remains total (damage=1) beyond that, with the damage function for negative temperature being a mirror-image of that for positive temperature:

Finally, we’ll define the risk as the expected value of the damage. This is

,

where the integration limits are from minus to plus infinity.

For convenience, I’ll define a standard normal variable

,

so that

.

Then our risk integral can be written as

.

Now let’s look at how the risk changes when the mean and standard deviation change. If we simply increase the mean while holding the standard deviation constant, the risk changes dramatically. Here’s the ratio of the risk at a variety of mean temperatures to the risk when the mean is zero (the risk ratio):

Note that even modest temperature increase, a single standard deviation, increases risk substantially. At one standard deviation, risk increases by a factor of 6.5. That is indeed a sizeable increase, and argues very strongly that Hansen’s perspective is a realistic viewpoint of the impact of global warming, while Mass’s perspective is not.

Perhaps more disturbing, as the mean continues to increase risk increases ever faster. When the increase reaches two standard deviations, risk has been magnified by a factor of more than 40. If climate-related damage is 40 times greater, then suggesting that 2 standard deviations is a small fraction of the size of a massive heat wave is shown to be a fundamentally faulty perspective.

Here’s the impact of increasing the standard deviation while holding the mean constant:

Again, significant changes in standard deviation lead to large changes in risk. I will mention that for this constant-mean case, half of the risk involves the likelihood of extreme cold events.

You’re probably wondering about the combined effect of increasing mean and standard deviation.

Note that the graph shows the logarithm (base 10) of the risk ratio. Increasing either mean or standard deviation leads to large risk magnification, increasing both together amplifies the effect.

Of course this model is an oversimplification. But I think it does demonstrate the main point, that Cliff Mass’s perspective is fundamentally flawed because it fails to account for the dramatic increase in the frequency of extreme events. It also illustrates that as warming progresses, risk increases much more rapidly. We’ve already experienced “1 sigma” of warming, and paid the price. The consequences of the next sigma will be devastatingly greater.

This much is certain: dealing with climate change is not a game. I don’t think we can afford the risk.