Tuesday, December 29, 2009

What would 1.5 °C of warming mean in Prague?

If you extrapolate the UAH climate record from the last 30 years to the following century, you will get something like 1.3 °C of warming. Let's generously call it 1.5 °C. Also, some people in the "fight against climate change" want to prevent Nature from warming by 1.5 °C.

Funny, I actually attended this concert at the Prague Castle and it was very good.

It's unlikely that the warming trend that was persistent in the last 30 years - but not in the last 60 years or the last 15 years - will continue. But let's assume it will. What would those 1.5 °C mean e.g. for Prague? Be sure that most of you live at places where the qualitative conclusions would be very similar.

If you open the Wikipedia page about Prague, you will learn that the average monthly temperatures range from -1.1 °C in January to 19.4 °C in July. Let's approximate this by
temp(x) = 9 - 10 Cos [(x-1/2) * Pi/6]
I have shifted "x" by 1/2 so that the minimum is at "x=1/2" which will be considered "mid January", and multiplied the argument by "Pi/6" so that the period of "temp(x)" is twelve. Now, let's consider the function "temp(x)+1.5", too. How do they differ?

Well, they differ by a shift in the y-direction. However, in the increasing (spring) and decreasing (fall) parts of the graph, you may also relate the two functions by a shift in the x-direction. Here is the result of the following plot:
Plot[{temp[x], temp[(x - 0.3)] + 1.5}, {x, 0, 12}, Filling -> {1 -> {2}}]

You see that the difference of 1.5 °C has the same effect in Spring as the early arrival of Spring by 0.3 months or 10 days. In the graphs, shifts in both directions were applied and they almost exactly canceled in the Spring.

This is true for approximately four months of the extended Spring, namely March, April, May, and June. Similarly, in the extended Fall, namely in September, October, November, and December, the effect of 1.5 °C of warming is equivalent to the delay by 10 days.

As far as the temperatures would go, the life in Prague in 2109 would be indistinguishable from the life in 2009 during these eight months: it would be simply sped up or delayed by 10 days. This is a statistical proposition: in individual years, you will clearly be unable to distinguish 4th of April from 14th of April in any reliable way.

After all, Prague has already seen a warming by more than 1 °C. The local meteorological noise is such that it allowed the Czech alarmists to argue that the Czech warming in the recent decade was by 1 °C. But did they succeed in spreading the fear? Well, they did not. Needless to say, 87% of this figure is weather - noise - because the global warming can only account for 0.13 °C per decade, as we have said at the beginning.

It means that the "catastrophic" change we have seen in the recent decade - namely no change worth talking about, as most Czechs realize very well - is exactly the same change that we can expect as a result of "global warming" in a whole century.

I have mentioned that approximately eight months would see no change of the life dynamics: Spring would only be sped up by 10 days and the Fall would be delayed by 10 days. This is not true for the remaining 2+2 months.

In the two hot summer months - July and August - we could see temperatures that were previous unusual. The July average would go from 19.5 °C to 21 °C or so. What a shock. I hope it is not necessary to explain that this would also be a non-issue - and this conclusion is true not only in Prague.

Estimating the changes to snow

Similarly, in the two cold winter months, January and February, some previously known regimes of cold weather would disappear (if you naively think about the average figures) or, more realistically, taking the noise into account, they would become less frequent. The most obvious change is the frequency of snowy days. Can we estimate how would the frequency of snowy days change if Prague got 1.5 °C warmer? Note that the current temperature average for January is -1 °C or so - very close to the freezing point - so the changes around 1 °C may matter.

I am going to solve this problem with a model. Our goal is to find out how many days would be "snowy". I will assume that it is a fixed fraction of the days whose average temperature is below 0 °C. It's not an exact rule but the numbers will be close. Consequently, we want to determine how the number of days whose average temperature is below 0 °C will change if the average temperature increases by 1.5 °C.

Obviously, the most important quantity to be determined is the natural day-to-day variability of the temperatures. And believe me or not, these variations are huge, too. I have said that the July-minus-January temperature difference in Prague is 20 °C. That's the difference between the average expected temperatures for the given months. And the typical temperature anomalies (the differences between the observed temperature and the expected temperature for the given place at the same time of the year) are actually comparable.

To have a good idea, I took all daily temperature anomalies in all January days from 1973 to 2009: 37 * 31 numbers. It's sensible to consider days (the timescale matters when you are determining the magnitude of the oscillations of the temperature averaged over the periods, e.g. days!) because the snow reacts to the temperature change within a day, so days are "marginally independent" but the average over a day is necessary to know whether the snow will survive. Amusingly, January 10th, 1999 was missing in Mathematica. So I interpolated it between January 9th and 11th. The resulting histogram looks like this:

By the way, you can see that the distribution is detectably asymmetric. The most likely January anomaly is around +1 °C. However, the negative anomalies can extend to much more extreme values than the positive ones so that the average is zero. The reason? When it gets really cold and the Earth is covered by snow, it's easy for the Earth to cool very quickly. The standard deviation of the distribution above is 5.25 °C.

Now, we can study the proportion of days whose average temperature is negative. Let's choose a random day in the year, calculate the sinusoidal "standard" temperature for the day (-1 + 10 times Cos), and add a random number from the normal distribution with the 5.25 °C standard deviation. Let's neglect the asymmetry of the distribution.

The random temperature, including the anomaly, at a random moment of the year is distributed like this:

Note that between 0 °C and 20 °C, the distribution is nearly uniform. It's because the underlying sinusoid actually likes to spend a lot of time near the extreme temperatures 0 °C and 20 °C. At any rate, it's simple to see that 17 percent of the days have average temperatures below 0 °C. It's even simpler to modify the Mathematica code by adding 1.5 °C to all the temperatures.

The result? 17 percent of below-freezing days will get reduced to 12.5 percent of below-freezing days. Here, 12.5 is about 1/4 lower than 17. We have said that we would assume the number of snowy days to be proportional to the number of below-freezing days. In other words, 1.5 °C of warming is expected to lead to a 1/4 decrease of the snowy days.

Even when you look at a quantity that is unusually sensitive to the temperature change - namely the frequency of snow in a place that barely visits below-freezing temperatures - the "catastrophic" change will only lead to a 25% change if the "huge" warming by 1.5 °C is expected.

Most people wouldn't be able to reliably determine the predicted difference in the snow cover of Prague. And most of them won't even remember how life looked like 100 years ago because a significant percentage of the population gets either dead or senile above the age of 100 if not earlier. ;-)

Proportionally speaking, the decrease of "cold events" will get more substantial if these events are rare to start with. So if a place had a tiny chance to ever see snow, warming would probably mean that the chance would get supertiny. However, important things in life usually don't depend on events that are hugely infrequent, anyway. On the other hand, the proportional decrease of snowy days would be much smaller at colder places - just a few percent.

People usually tend to think that a "Celsius degree" is a lot. After all, a degree may be a Bc, MSc, or Dr, and it makes a lot of difference. However, the Celsius or Kelvin degree as a unit of temperature difference is defined as one percent of the temperature difference between the freezing point and the boiling point of water. The difference between these two points is qualitative, of order one for life, if you wish, but one Celsius degree is just one percent of it, a negligible amount. Correspondingly, the impact of a 1.5 °C warming on virtually all things relevant for life is negligible, too.

A simple Mathematica notebook that was used to generate the results above: NB file, PDF preview.