Sunday, 7 January 2018

Celsenheit-Equivalent Wind Chill

During the recent and ongoing North American cold snap, I was reading posts from my friends in both Canada and the United States and realized that the temperatures were low enough that I couldn't tell whether they were Fahrenheit temperatures with windchill or just Celsius temperatures. This lead me to consider the conditions under which the wind is blowing strong enough to turn Fahrenheit into Celsius. This all assumes that temperatures are above -40 so that the Fahrenheit number is higher than the Celsius number, if they aren't then you have bigger problems than unit conversion.

Essentially the question is: at what wind speed can we say "It's -10!" and be correct for the ambient temperature in Celsius and the wind chill temperature in Fahrenheit.

The conversion between Fahrenheit and Celsius is well known and straightfoward, it is simply F=9/5 C+32 and C=5/9(F-32). The formula for wind chill is based on a fit to calculations of heat transfer from a person's skin while they're walking into the wind.  I looked up the formula on Wikipedia, there is one for Fahrenheit and miles per hour, and one for Celsius and kilometers per hour. They have four terms, one constant, one linear in temperature, one proportional to the 0.16 power of wind speed, and one proportional to both temperature and v to the 0.16. The formula for Celsius and kph is:

So we have a formula to transform the ambient temperature into a lower one given a wind speed, and we want to equate it to a formula that converts a Fahrenheit temperature into a numerically lower Celsius temperature. We can do this and solve for Vfc, and we get an ugly formula without much insight:

This is in kilometers per hour, you can divide by the natural logarithm of 5 to convert to miles per hour*. We can't learn to much from this formula except by looking at a graph or probing specific values. To answer the question above, if it's -10 C and there's a wind speed of 70 kph, it feels like -10 F, so you could say "It's -10!" and be right either way.

In the cold extreme, it doesn't actually go to zero at -40 as we expect, it just asymptotically approaches it. This is basically because the formula for windchill is just a fit to some modelling outputs and I even if it was calibrated that low, it's probably at the edge of where the fitting parameters are appropriate. Towards the warmer end, wind speeds get unrealistically high, like superhurricane fast. This is basically telling us that 0 C will never feel like 0 F no matter how fast the wind is blowing (or, it's blowing so fast that wind chill isn't your main problem.

We can also in principle do the same thing with the Humidex or Heat Factor, which are used to mark how much hotter it feels in the summer due to humidity, sort of the opposite of wind chill.  However, a day with 100% humidity will maybe make it feel 10-15 degrees hotter, but when it's warm out the Fahrenheit temperature is at least 50 degrees above the Celsius temperature, so this wouldn't really work ever.

*the number of kilometers in a mile is eerily close to the natural log of 5.