My investigations suggest that the strongest influence on extreme heat is the increase in average temperature during summer; the shape of the distribution can change, and that has an effect, but change in the average value dominates. So I decided to look at how summertime heat has changed in each climate division of the conterminous USA (i.e. the “lower 48 states”), according to the data for high temperature from NOAA.

For each division, I fit a smooth curve (lowess smooth), then estimated the “summer warming” as the difference between the smoothed values now (i.e. in 2021) and at the start (i.e. in 1895). Some of them show considerable warming, in fact the northeast corner of Utah has warmed by a whopping 6.05°F:

Although most climate divisions show summer warming, not all of them do; in fact in Alabama there’s a division which shows cooling by -2.39°F:

Whichever divisions in the USA have warmed by the most, are most at risk for never-before-seen extreme heat. And here they are as red dots (bigger dots, bigger risk), with blue dots indication regions which have shown net summer cooling (rather than heating) since 1895:

Two regions stand out as being at greatest risk. First is the entire U.S. west, westward of longitude 100°W. Second is the northeast coast, northward of Washington D.C.

Cliff Mass likes to refer to the recent heat wave in the Pacific Northwest as a “black swan” event. The term refers to “Black Swan Theory,” developed by Nassim Nicholas Taleb, which wikipedia describes as a “metaphor that describes an event that comes as a surprise, has a major effect, and is often inappropriately rationalized after the fact with the benefit of hindsight.“

Here’s how Taleb himself defines a “black swan” event:

“What we call here a Black Swan (and capitalize it) is an event with the following three attributes.

First, it is an outlier, as it lies outside the realm of regular expectations, because nothing in the past can convincingly point to its possibility. Second, it carries an extreme ‘impact’. Third, in spite of its outlier status, human nature makes us concoct explanations for its occurrence after the fact, making it explainable and predictable.

I stop and summarize the triplet: rarity, extreme ‘impact’, and retrospective (though not prospective) predictability. A small number of Black Swans explains almost everything in our world, from the success of ideas and religions, to the dynamics of historical events, to elements of our own personal lives.“

I stop and emphasize the phrase “not prospective.”

Taleb means that there’s only retrospective explanation, there’s an absence of prospective (i.e. predictive) explanation. I’d like to remind Cliff Mass that extreme heat waves, of greater frequency and severity than seen before, has been a prediction of climate change science for decades. Therefore, according to Taleb’s own definition, the recent heat wave in the Pacific Northwest is not a black swan event.

Taleb emphasizes the folly of “explaining” an extreme event with major effect which nobody saw coming, as though we should have known all along. He’s got a point. I’ll emphasize the folly of the opposite mistake: to use “Black Swan Theory” to dismiss what scientists have been warning us about for decades, as nothing but an unpredictable outlier.

Perhaps Cliff Mass will insist that the heat wave in the Pacific Northwest was so severe, that nobody predicted a heat wave this strong. It was so much an outlier, we have to call it a “Black Swan.”

If we’re going to do that … then there are a whole lot of Black Swans popping up these days. All over the world. With a frequency, and of a blackness, far beyond what we’ve seen before, a veritable population explosion of Black Swans. The heat wave in the Pacific Northwest is far from the only example, but it is the one which Cliff Mass can’t ignore.

I decided to apply the method used in the post about the heat wave in the Pacific Northwest, to look at ERA5 data for daily high temperature, but for other regions. Having looked at a number of spots, let me show you what it looks like for two of them which represent very different recent history of extreme heat.

One is latitude 45°N, longitude 5°E, in France, not far from the city of Lyon (which is not far from the Swiss border). The other is just north of New Orleans, at latitude 30°N, longitude 90°W. We’ll start with France, and here’s the data for temperatures 90°F and hotter:

Right off the bat it “looks like” (there’s that phrase again) there have been more such days lately. I found the highest temperature for each year (excluding 2021, which isn’t over yet), and the trend (red line in the graph with pink shading for uncertainty range) is not only real (i.e. overwhelmingly statistically significant), it’s substantial, increasing by 6.4°F from 1950 to now.

To look for changes in the distribution, first I isolated the summer months (June, July, and August), then computed anomaly (the difference from the average value for the given time of year) in order to remove the seasonal cycle. Then I split the time span (1950 to now) into three segments: from 1950 to 1975, from 1975 to 2000, and from 2000 to now (this differs from the previous analysis, in which I only used two segments). Finally I estimated the pdf (probability density function) for each interval, both by constructing a histogram and by a smoothed estimate. Here they are, with interval 1 (1950 to 1975) in blue, interval 2 (1975 to 2000) in black, and interval 3 (2000 to now) in red:

Clearly the most recent time span (2000 to now) has far greater likelihood of extreme heat (which we get from very high temperature anomalies in summer). It also “looks like” the distribution got wider, making the excursion into extreme heat even more extreme.

All three time intervals have different distributions. But interestingly, if we offset each series by its own average, to get a distribution with the same shape but with average value zero, then the first two intervals show no significant evidence of a difference in their distributions. The real difference from time span 1 to time span 2, is the small increase in its mean value.

But the third interval has a pdf which is definitely of a different shape than the other two. In particular, its variance increased (the distribution got wider). More to the point at hand, the high-temerature end of the survival function tells us the likelihood of extreme heat during each span:

This region shows exactly the kind of temperature increase, not just of the average but of the extreme-heat region, which makes it vulnerable to never-before-seen heat waves.

Turning our attention to the western hemisphere, let’s look a bit north of New Orleans at latitude 30°N, longitude 90°W. Here’s the data for temperatures 92°F and hotter:

This time, the yearly high temperature series shows no significant trend:

Performing the same procedure as before to study possible changes in the distribution, I get this for this part of the USA (interval 1 in blue, interval 2 in black, interval 3 in red):

They all look similar, and when we use the Kolmogorov-Smirnov test, none of them shows any statistically significant sign of being different from the others. This overall picture, of a location not in danger of extreme heat because it has shown neither strong warming of its average nor widening of its profile, is confirmed by a close look at the high-temperature end of the distribution:

So far, all indication are that the change in a location’s susceptibility to extreme heat is dominated by the change in its summertime average temperature. I have not yet found a case where a shape change dominates, either to creates or to suppresses a significant change in extreme heat. It’s the change in average temperature (during summer, at least) which carries the day. But … I have a lot of locations yet to study.

When extreme heat gripped the Pacific northwest recently, people noticed. They noticed in Seattle, Washington, where they set an all-time record high of 104°F on June 27th, only to break it the next day at 108°F. They noticed in Portland, Oregon, where they set the all-time record high of 112°F on June 27th, only to break it the next day at 116°F. They noticed in Lytton, Canada, where they set the all-time record high for all of Canada at 121°F, only to burn to the ground the following day.

Naturally this has led to speculation about the relationship of this particular heat wave to man-made climate change (global warming). One of the reasons we expect global warming to increase extreme heat, is illustrated in this graph:

It compares two probability distributions, one (in yellow) with a lower average, another (in red) with a higher average. As the average (i.e. the mean) temperature increases, the amount of extreme heat increases dramatically — provided, of course, that the shape of the distribution remains the same.

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Does the shape of the distribution remain the same? Of course not — at least, not perfectly so. But it turns out that for summer in the Pacific Northwest, the change in shape is quite small, in fact so small it’s hard to confirm it exists at all, while change in the average sticks out like a sore thumb.

Let’s use data for daily high temperature averaged over the “Pacific Northwest” as defined by the study region of a recent attribution study of the connection between this event and climate change, shown by the box in this map:

It covers the region from latitude 45°N to 52°N, longitude 119°E to 123°E. The data are from the ERA5 re-analysis data set, which was also used in that attribution study. It’s provided in Kelvins, but I converted that to degrees Fahrenheit for a more familiar temperature scale (at least, for my American readers). And here it is, daily temperature for the study region from 1950 through the end of June 2021:

Here’s data for just temperatures over 85°F, to show where the extreme heat has happened:

Note that before this year, the average temperature throughout the region never exceeded 95°F, but this year it easily broke 100°F.

I’ll define summer as the months June, July, and August. Here’s the data for just the summer months:

It’s straightforward to average the temperature during each year, except we should leave out the year 2021 because it’s not over yet. Averages for each year (except 2021) look like this:

The red line is the linear trend (from linear regression), is strongly statistically significant (p-value < .0001), and indicates that the average summertime temperature increased 3.9°F from 1950 through 2020.

There’s no doubt that the average has changed over time — nobody in his right mind disputes that. But what about the distribution itself? Has its shape changed, and in particular, what has happened to the “extreme heat” end of the distribution?

Let’s look at the hottest temperature for each year, but just for fun let’s omit this year’s record-breaking value:

The trend line (in red) is estimated by linear regression, is strongly statistically significant (p-value 0.001), and suggests an increase in yearly maximum temperature of 4°F from 1950 through 2020. So, for summer at least, it’s not just the average that has gotten hotter, so too has the yearly maximum value.

Still, I’d like to get at the distribution itself for summertime. To that end, I’ll transform temperature into temperature anomaly by subtracting, for each value, the average for that time of year. Doing so, I remove the annual cycle from the data (which is still present because early and late summer are cooler than mid-summer). Here are the anomalies:

They aren’t too different from the temperature data itself.

Now let’s separate the anomaly data into two big intervals of time: “early” from 1950 to 1985, and “late” from 1985 to now. We can estimate the probability distribution for each by making a histogram, as well as a smooth estimate, and I’ll compare them with pre-2000 data in blue and post-2000 data in red (solid lines are smoothed estimates):

It’s rather striking how almost all of the negative anomalies are less likely lately than earlier, and almost all of the positive anomalies are more likely now than before. It certainly seems plausible that the “late” distribution has the same shape as the “early” distribution, but shifted to the right, toward higher temperatures.

Let’s focus on the extreme-heat end by looking at the survival function (which is 1 minus the cdf, or cumulative distribution function) for high temperature anomalies:

There’s no doubt, the Pacific northwest is getting more extreme heat. But is that just because the distribution shifted to the right, or is shape change part of the story?

Let’s compute probability distributions, not for the anomalies as we just did, but for the anomalies offset by a constant to make their average values the same (in particular, zero). I’ll call these “De-Trended” anomalies (a crude term). Here are the histograms and smooth estimates of the probability density function:

There may be some shape change, but perhaps not — perhaps the differences are just random fluctuations within the uncertainty range. We’ll get a better idea of what’s happening on the high end with the survival function:

Again, it’s plausible that they are the same because they are almost entirely within each other’s uncertainty range. It is possible that the distribution has changed at the high end with the appearance of de-trended anomalies not before see, but the statistics don’t confirm that yet.

Perhaps the “go-to” statistical test for whether two distributions are the same, is the Kolmogorov-Smirnov test. When I use it to compare the anomalies (not de-trended), it proves they are different (p-value 0.00000000000000022), which of course they are because they have a different mean value. But when I compare the de-trended anomalies, there is no significant evidence of a difference between the distributions. The p-value is 0.4974 — not even close to significant.

My bottom line: The evidence demonstrates that extreme heat has gotten more frequent and more severe, and we can expect it to continue, because the probability distribution has shifted to hotter values with no confirmable change in its shape. But it is possible (not yet confirmed) that the recent heat wave is so much hotter than we’ve seen before that there’s more going on — which means we can expect even worse.

Yet people continue to dispute even the simple idea, which is easily shown for the Pacific northwest, that the increase in the average has brought with it an increase in extreme heat. One such is Judith Curry (from whom I got the first graph shown), who gives references to the peer-reviewed literature which I went and read, only to make me wonder whether or not she read them. At least she delivers on the promise of her blog title: hot air.

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