Hot Days

Climate is defined as the mean and variation of weather over long periods of time (typically, 30 years). I emphasize the “and variation” part because climate change doesn’t just refer to a changes in the mean, it also refers to changes in the variation.

One aspect of that change in variation is highlighted in a new paper (Seneviratne et al. 2014, Nature Climate Change, 4, 161-163) which points out that although global average temperature may have been increasing more slowly recently than in the previous two decades or so, the frequency of hot days in land areas has been increasing faster recently than in the previous two decades or so.

To illustrate the principle, I’ll need daily data, which is perhaps most easily obtained from ECA (European Climate Assessment & Dataset). Here are monthly averages of daily high temperature anomaly from Kremsmuenster, Austria since 1950.


I’ve plotted monthly averages just so the plot won’t be so crowded, but the analysis which follows uses the daily data. The red line (a lowess smooth) shows that there is an upward trend since about 1980, but it was fastest in the 1980s and has been a bit slower since.

I transformed these temperature anomalies to normalized anomalies, by dividing each value by the standard deviation. But I used the standard deviation for the given time of year, so that the seasonal pattern in variation wouldn’t bias the result. Then, I counted how many days out of each year the normalized anomaly exceeded 2.326, which is the 99th percentile for a normal distribution. This doesn’t mean the data necessarily follow the normal distribution, but it is a reasonable cutoff point which should include about 1% of the data in a “normal” year.

And here are the counts for each year:


It too begins to rise about 1980, when the mean high temperature starts its rapid increase. But unlike the mean, this count of extreme hot days doesn’t rise more slowly after 1990 — instead it rises more quickly. In fact its increase is quite dramatic after about 2005, and the years since 2010 have been scorching hot.

We can compare the trends (the smooths) for mean temperature, and for number of hot days, by normalizing them and plotting them on the same graph:


Clearly, although the mean high temperature has risen more slowly since 1990 than it did during the 1980s, the number of hot days has risen faster.

A number of factors could bring about such a change, and one of them is an increase in the variance of daily high temperature. Such an increase is noticeable if we compute the standard deviation of high temperature anomaly, averaged over each year:


The most noticeable change is the increased standard deviation over the last five years, especially the highly deviant year 2013. We see the same thing in the standard deviation of high temperature anomaly averaged over each month:


The smoothed version reveals, among other things, an increase since about 2010:


So, not only has the mean high temperature at this location changed recently, so too has the variation — and both of those are examples of climate change. The increased variation is one of the factors leading to more rapid increase in the number of hot days per year.

I did a similar analysis for all stations in the ECA which had sufficient data for at least 60 years out of the 64-year period from the beginning of 1950 to the end of 2013. Then I averaged the number of hot days over all stations. This is an extremely crude way to estimate the regional effect since it has no area-weighting, but at least it’s a first pass at exploring the phenomenon. Here’s the result:


At first I was a bit surprised that the 2003 European heat wave and the 2010 Moscow heat wave don’t stand out more prominently, in fact the single most prominent “standout” year is 1990. But then I remembered that not only is this a pan-European average, it also includes as “hot days” those days in winter (and spring and fall) which are much hotter than average for those times of year. Although parts of Europe suffered greatly from the heat waves in 2003 and 2010, overall the number of hot days relative to the norm for a given time of year, has shown a more steady increase (excepting 1990) and a more recent increase (there’s no evidence of a slowdown since whatever year the denialists want to cherry-pick).

Seneviratne et al. used a somewhat different approach. They defined “hot days” as those above a given percentile (without regard to the normal distribution), and instead of just counting hot days, they computed what fraction of the land area showed an excessive number of hot days during each year. They also included the entire globe for which data were available. Their overall result, however, is the same: that the number of hot days has increased more rapidly recently in spite of the global average temperature increasing more slowly (I’ve reproduced only part “b” of their figure 1):


They also found that the more extreme one defines “hot days,” the stronger is the most recent increase.

All of which goes to show that even if one defines climate only by the surface air temperature (which I don’t), and even if one accepts that the slowdown in global average temperature is significant (which I don’t), we still have to face the fact that we’ve witnessed climate change over the last decade and a half. It also emphasizes the point that it’s not just the mean that matters, it’s also the extremes. Some would say that the change in extremes is the greater threat to the habitability of our planet.

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28 responses to “Hot Days

  1. It would be interesting to see, for comparison, what a “number of cold days” analysis for the same data would look like.

    Is the increase in variation sufficient to increase the lows too? Or is the warming trend too large and/or is the data skewed?

    • SkS already put up for the USA the imbalance of record hot vs record cold and the record cold days are dropping in frequency.

      However, since the “record” becomes harder and harder to beat for each extra year you include in the record, you would expect it to drop a bit anyway. Whether it drops more because the variation doesn’t vary DOWN as much as it does UP in temperature would be shown, but the basic point is already well established.

  2. Words can’t express how much I value this Blog. Thanks for these posts, they are so interesting to read.

  3. I’d support Ernst K’s questions.
    Reading this, and supposing that the spread of the daily temps is “hard”, the question of a mechanism for this bigger spread arises.

  4. johnrussell40

    Very useful post. Thanks.

    Given the huge influence of the Atlantic ocean over the UK’s temperatures I’d love to see what difference climate change is making to the UK’s weather when compared to continental climates. I’ve certainly noticed that winters seem to be considerably warmer and summers wetter than they were in my youth; but maybe that’s an illusion.

  5. Martin Smith

    Is Skeptical Science under attack again? I haven’t been able to connect there for several hours.

  6. This is very interesting. Is one explanation is this phenomenon in which weather patterns get stuck more often, so you can get really cold periods and some really hot ones? I guess the way to check would be to look for changes to the lengths of strings of unusually hot and cold days, and see if the strings correlate to the increase in really hot and cold days?

    Note, I have assumed that cold days are increasing somewhat (after the trend is removed) in order to keep the mean from increasing as quickly. I think this is the best explanation because it’s in keeping with the idea that diurnal variation is shrinking (nights warming faster than days). Correct?

    Maybe I should just read the paper! If I can get my hands on it.

  7. Seems logical that in order for atmospheric warming to have been less pronounced, and for the number of hot days to have gone up, there must be a counter effect of increased low temperature days too, though less low than high, thus the upward trend. Increased noise. That jump around 2010 seems pretty big and sudden, to my eye. Any thoughts on the specific driver, or if it’s just a visual illusion because of the short term focus on the end of the graph?

  8. Just waiting for the accusation from one of the usual suspects that Kremsmuenster was cherry-picked.

    Something tells me Tamino has a reply already locked and loaded.

  9. Is there indication whether there were more “hot” days in summer or winter? Has the mix changed?

  10. Thank you for another clear and cogent walk-thru, Tamino.



  11. See the study From the press release
    To get their results, the researchers examined hot days starting from 1979. Temperatures of every day throughout the year were compared against temperatures on that exact same calendar day from 1979-2012. The hottest 10% of all days over that period were classified as hot temperature extremes.

    Globally, on average regions normally expect around 36.5 extremely hot days in a year. The observations showed that during the period from 1997-2012, regions that experienced 10, 30 or 50 extremely hot days above this average saw the greatest upward trends in extreme hot days over time and the area they impacted.

    The consistently upward trend persisted right through the “hiatus” period from 1998-2012.

  12. When getting the count of hot days are you counting 2.236 standard deviations above the red line in your first figure (i.e. hot compared to the mean at the time), or are you counting 2.236 standard deviations above the mean for the whole period?

    [Response: Above the mean for the whole period. That better emulates the procedure in Seneviratne et al. 2014.]

  13. This summer we had several very hot days in Melbourne, Australia. I did a post here which included a gadget showing all the summer day maxima back to 1855. Seven of our twenty hottest days in that time happened since start 2009.

  14. Michael Sweet

    (a href=””>Hansen has published a similar article using the three summer months: June, July and August. He shows that extreme hot summers are about 100 times more likely now when compared to the average heat from 1950-1980. This pattern continues in the 2010’s. Here is a recent update. The update shows three sigma deviations in summer about 14% of the time in the 2010’s compared to 0.1% in the base period. The winter time deviations are lower because there is more variation in temperature during the winter so the standard deviation is greater. Never the less, cold temperatures are rarer, the suggestions about cold above do not show up in the data. Hansen also finds no slowing down of increasing heat. He attributes this to the “hiatus” being primarily a cooling of the ocean and not land temperatures from La Nina. Hansen needs to find a new color or two for his graphs, 4 and 5 sigma is now common enough to be a significant part of the data. Eyeballing his graph on page 6 of the update, 4+ sigma is 4-5% and 5+ sigma (which runs off the graph) is 1%.

    If three sigma is “extremely hot” what do you call 4 and 5 sigma??

    At the time some scientists argues that Hansen’s data did not show that days were warmer, only seasons. This article follows up and shows days are also hotter. I noticed that Hansen uses a fixed baseline from the past. while this article uses a floating baseline including the period of warming. Didn’t tamino show that using a floating baseline baseline including the current time limits the standard deviations that the points can be off? Perhaps the use of the 10% of hottest days

  15. Thanks, I think this effect of increasing extremes could also be extracted from the arctic ice values, but this would of course not surprising.

    The ‘cold days’ analysis could also be interesting, probably you’d have to use another, lower cutoff value. There could be a notable seasonal difference in here, especially if you’d look the latitude bands separately. This would likely be similar to Prof. Francis’ analyses, and quite a huge bit of work, possibly enough to get a publication out of it, so a blog post would not be needed.

  16. Pete Dunkelberg

    This a powerful example of what you can do if you are Tamino and have all the statistical routines set up and know which are right to use and devote quite a bit of time to it.

    Meanwhile hot days are associated with drought and this is a hot topic at Eli’s.

  17. Much of the behavior shown in the second and third figures appears to arise directly from the the shape of the (near normal) distribution of the temperature anomalies about the trend, as that distribution is pushed higher and higher along trend line. That would strictly be a change in climatic mean, not variation. It’d be interesting to see the same figures drawn for detrended data, or for some synthetic data with similar trend and spread but no change in spread.

    You go on to show that there may be some stretching of the anomaly distribution at this site post-2010, which I found far more interesting. I hope you’ll take that further. Publishable?

    BTW, none of that is meant to deny the importance of the point that it is raw high temperatures which kill things, not detrended ones.

  18. Thomas Huld

    I’m not a statistician, so this may be wildly wrong, but here goes…

    I would think that if you have something like a normal distribution, then if you increase the mean but keep the variance constant, your number of extreme high events will grow in exponential fashion, if you define extreme events as being a certain number of standard deviations from the original mean. This would be true if you are far enough out in the tail.

    So the increase in number of hot days is in good agreement with the fact that the mean has gone up (as far as I know land temperatures have continued to increase at a pretty constant rate).

    [Response: But the mean hasn't increased at a constant rate. In fact one of the points of Seneviratne et al. is to show that the increase in hot days has accelerated even as the increase in global mean temperature has slowed.]

    However, I would think that this result is not in itself evidence for increased variance. To do that, wouldn’t it be better to calculate the variance for shorter time intervals?

    [Response: Yes, and that's what I did in the last three graphs of data from Kremsmuenster (Seneviratne et al. only computed hot days). But my results lack global coverage and proper area-weighting.]

    • I was thinking the same. The trend alone could account for a dramatic increase in hot days.

      But of course, if there is no trend in recent times (you could call it a pause), then there would not necessarily be an increase in hot days. But there clearly is a dramatic increase in hot days.

      My interpretation is that (a) there isn’t any pause, and (b) the variance is increasing.

      And that increase in variance is interesting.

      • Of course, the “if” is a conditional, but not considered conditional.

        What if it isn’t true?

        A question deniers and their lackeys never consider: it’s always the “if it’s not that bad”, never “if it’s worse than that”, which they label ALARMISM! whilst peddling the “If we do that, WE WILL BE LIVING IN CAveS!!!!!”. Which isn’t alarmism, it’s “stating the problem with CAGW”.

        How, nobody knows. Not even the deniers doing it know why.

      • Wow,
        Precisely, Aunt Judy never considers that uncertainty is a double-edged blade, and it cuts much deeper on the high side.

  19. The Australian Climate Council, which was defunded by Tony Abbott and now relies on public support, recently issued a report entitled “Heatwaves Hotter Longer More Often.”
    Below are a couple of the key findings:
    1. Climate change is already increasing the intensity and frequency of heatwaves in Australia. Heatwaves are becoming hotter, lasting longer and occurring more often.
    ›  Over the period 1971–2008, both the duration and frequency of heatwaves increased, and the hottest days during heatwaves became even hotter.
    ›  Hot days have doubled in Australia in the last 50 years. In the last decade, hot weather records have occurred three times more often than cold weather records.
    ›  The trend toward more frequent and more severe heatwaves in Australia is part of a larger global trend. Very severe heatwaves have occurred elsewhere, including Europe in 2003, Russia in 2010 and several regions in the south and central US in 2011 and 2012.
    [The report mentions that the number of extreme heat days for several of Australia's capital cities already has reached the number projected for 2030.]
    3. The climate system has shifted, and is continuing to shift, increasing the likelihood of more extreme hot weather.

    The report makes what I find to be a useful distinction between heatwaves and warm spells. It defines a heatwave as a period of at least three excessively hot days in a row. Warm spells “include excessively warm events (relative to time of year) that occur outside of summer.”

  20. Very hot again this Summer in Perth, Western Australia. Started mild, the turned nasty… and since Dec 1 we have had exactly 2mm of rain… Very hot and very dry… Nasty weather. Perth must be the climate change capital of the world.

  21. lewiscleverdon

    Tamino, I’ve been puzzling over a question that I hope may be of interest to you, arising from the PNAS paper:
    Observational determination of albedo decrease caused by vanishing Arctic sea ice
    Kristina Pistone, Ian Eisenman1, and V. Ramanathan

    From the Abstract:
    The analysis reveals a striking relationship between planetary albedo and sea ice cover, quantities inferred from two independent satellite instruments. We find that the Arctic planetary albedo has decreased from 0.52 to 0.48 between 1979 and 2011, corresponding to an additional 6.4 ± 0.9 W/m2 of solar energy input into the Arctic Ocean region since 1979. Averaged over the globe, this albedo decrease corresponds to a forcing that is 25% as large as that due to the change in CO2 during this period, considerably larger than expectations from models and other less direct recent estimates. Changes in cloudiness appear to play a negligible role in observed Arctic darkening, thus reducing the possibility of Arctic cloud albedo feedbacks mitigating future Arctic warming.

    A while back you kindly tested and affirmed the finding of the 2011 GRL paper that cryosphere decline was imposing a warming equal to that from “about 30%” of anthro-CO2 stock, which the PNAS paper’s focus solely on sea-ice may again confirm. Yet like the GRL paper this again leaves unstated the critical factor for mitigation strategy of the CO2eq of the current annual loss of Albedo.

    Using a range of simple exponents (4% to 10%) to find the starting CO2-equivalents in 1955 (when the first marginal Albedo decline appears to be indicated) that yield 123.1GtCO2eq for the ’79 to ’13 period (that amount being 25% of the gain in anthro-CO2 for the period) it was possible to show a range of notional CO2eq ‘outputs’ for current Albedo Loss. Expressed as percentages of the natural carbon sink effect (taken as a norm of 43% of annual emissions) these ranged from ~42% of sink capacity for a 4% exponent giving a 2013 CO2eq ‘output’ of 6.43GtCO2, up to ~75% for a 10% exponent giving an ‘output’ of 11.65GtCO2eq.

    However, the curve of Albedo Loss is plainly not a simple exponent, and this method was only useful as a general indicator in the absence of access to the actual data. Given the critical importance of the sinks not being offset by feedbacks for the effectiveness of mitigation by emissions control, I wonder whether you might care to assess the present ‘output’s’ scale and rate of growth from the data ? With your highly skilled calculation these could I think be particularly effective bits of information in terms of raising public & politicians’ awareness of the predicament, not simply among honest ‘flukers’ (assuming the deniers are a waste of time) but especially in energizing the majority who already acknowledge the science.