Heat of the Moment

Ever since I finished my course requirements, people have been asking what I have been up to. I hope that posts like these will be something I can point others to as a response. These posts will follow the spirit of the XKCD what if? blogs and focus on ideas I have sunk some time into.


For the last few weeks, I have been slowly making my way through Wealth, Poverty and Politics by Thomas Sowell. (As a short aside, I recommend the book. It contains an immense amount of information as it maintains a quick pace for over four hundred pages. Of course, one has to keep in mind that Sowell is not an unbiased author.)

A passage that piqued my interest was in the section about climate where Sowell states

The highest temperature ever…. [has been] recorded outside the tropics….the sun shines more per day during the summer in the temperate zones than in the tropics, allowing a longer daily buildup of heat and fewer hours in which to cool down overnight

His initial statement is correct. The hottest undisputed (the world of meteorological records is filled with drama) temperature of 134 °F (56.7 °C) occurred on July 10, 1913 (a century of global warming has not made a dent on this record) at Death Valley, CA — located squarely outside the tropics at 36.5° N. However, I found it hard to believe that something as simple as the length of day could explain the position of the hottest place on Earth. The climate is exceptionally complicated, incorporating currents, precipitation, clouds, etc. Supercomputers are required to make accurate weather predictions. Like any reasonable person with no responsibilities, I decided to spend my entire day seeing how far this length of day argument could take me.

Theory

(This section might be a little technical, but the technical portions are not particularly relevant to the intuition)

I want to be able to predict the latitude of the hottest place on Earth on a given date.

I decided to model some abstract quantity I called H (for heat), which is monotonically increasing with temperature, but in some complicated way. So if some location has a higher H than another, the first location is predicted to be hotter by some unknown amount.

The most general model we can have is

A differential equation for the dynamics of our “heat” with some elaboration on the parameters

Clearly, if I want to solve this problem with my measly 4 watt, fan-less processor, we are going to have to make some assumptions. We can get rid of all the positional parameters other than latitude by focusing on an isotropic (uniform) Earth, so that moisture, biome, cloud cover, etc can be ignored — this gets rid of f. This is not as drastic an assumption as it seems. On Earth, a latitude spans a huge variety of regions. Thus, if we find that some latitude is conducive to high temperatures, then that latitude (or some close by latitude) will contain a region ideal for high temperatures. This region would correspond to the “hottest place on Earth”. As long as latitudes sample a variety of regions, ignoring other positional specific factors will not invalidate our “hottest” latitude estimate.

Next, we model the intraday sun intensity as a constant. For most of the day, the sun intensity is within half of the peak intensity. Also, the general shape of the intraday sun intensity does not vary much for latitudes ranging from 45° S to 45° N (outside this region, the angle of incidence is too shallow for sufficient sun intensity). This means we can just model the sun as a light bulb of fixed power that is on from sunrise to sunset with the power varying with the angle of incidence. We know that the amplitude of the sunlight is equal to the sine of the angle of incidence, so the intensity is the square of this sine.

Lastly, we can model the cooling as a fixed quantity. We know that the temperature in our region of interest varies by ~30° C, which is only about ten percent (~300K to ~330K). The mechanisms of cooling range from linear with temperature for conduction to a power law to the fourth for radiation. At most, this introduces a factor of less than two, which according to many of the fine physics professors at Caltech is pretty much one. Thus, in our model, every place on Earth is cooling at the same rate. Since we are interested in differences in H, terms that are uniform for all latitudes can be ignored. This means we can toss out the cooling term.

These assumptions have allowed us to simplify our dynamics of H to as follows

Simplified Dynamics

The θ corresponds to the angle of incidence at some latitude and time, L and t. The indicator is one during the day and zero during the night. Since the sun intensity value is constant we can choose to set it to one.

If we assume that everyday is independent, which is a weak assumption compared to our previous ones, we can solve this differential equation.

Solution to above differential equation. We can lose the proportionality since we do not care for constant factors. Subscripts are assumed

Where the t’s correspond to the time of sunrise and sunset, so the difference is the length of day.

To compute H(L,t), we need to find the angle of incidence and length of day throughout the year for all latitudes.

Computation

With this model in hand, coding it up is almost mechanical. stackoverflow had a nice class in Python for computing sunrise/sunset times. This class is fairly accurate. For today, Feb. 15, 2017, it said that the day length in Pasadena, CA would be 10.97 hours and Google said that its 11 hours and 1 minute.

Now we have to compute the angle of incidence. We know that without Earth’s tilt, the angle of incidence at the equator (latitude 0) would always be 90 degrees. This means the sine of the angle of incidence is the cosine of the latitude. A tilt will introduce a sinusoidal variation with amplitude equal to the magnitude of the tilt, which is 23.5 degrees for Earth. The period will be the length the year. The peak and trough correspond to the summer and winter solstice, respectively. This is all we need to parametrize the angle of incidence/”corrected” latitude.

How we generate the “corrected” latitude. The cosine of this is equal to the sine of the angle of incidence

So we have all we need to compute H(L,t). Here are the heatmaps from my code (please check it out!)

The two components that we need to convolve. The white section in length of day correspond to regions with 0 or 24 hour days (polar circle).
This is the end result of our model. We see hot patches at the solstices

Looking at the heat map, we see that the highest heat occurs at the summer solstices. A function that transforms these heat values to temperatures would incorporate some lag between the two to account for the heating effect (the Earth takes time to get hot). This is why the Death Valley record is a couple of weeks after the solstice. I suspect that our model is not too good for the spring/fall, because our length of day factor is useless. As a result, the other factors that we ignored begin to become more relevant.

We can attempt to fit the location of our global max heat to a sinusoidal function of the following form.

The parameter A would give us the latitude corresponding to the max heat. P will give us the period, which we want to be ~365 days and we want d0 to be near the winter solstice (10).

We get A = -32.8, P = 367, and d0 = 11.5. So P and d0 are about where we want them. All of this discussion was just trying to get A. First off, the negative sign is reassuring, the hottest place on New Years better be in the Southern Hemisphere. Good news for Sowell, 32.8 is ten degrees away from the tropics and only 3.2 degrees away from Death Valley (36° N). Our fit seems to work just as well for the Southern Hemisphere. The hottest temperature on Jan 6th, 2016 (I’m not cherry picking here, this is the closest waybackmachine gets to a recent southern summer solstice) was at Augrabies Falls (28.6° S). Plugging in d = 6, we get a max latitude of -31.3.

It is incredible that a model using only length of day and angle of incidence is able to get us so close to reality in a field as complicated as meteorology.

Here is a final plot to conclude

Hand waving can work!