# What does "seasonal" mean?

So in followup to the ideas about "seasonal" variations, there's a fundamental philosophical question here. What is a seasonal variation?

Clearly, length of day in Bangkok is determined by the geometry of the earth and its spin about its axis and tilt relative to the orbit around the sun. Purely by being at a particular point on the earth and at a particular point in the orbit there will be a particular period of the day when light falls on Bangkok.

But in general, there are relatively regular but less strictly regular variations in things, such as for example the amount of cheese produced each month:

It rises and falls throughout the year but it's not periodic, more like quasi-periodic with a clear trend and a clear change in the pattern that occurs slowly in time, over a timescale of 5 to 10 years.

In general, you can break down every function into a part that changes "slowly" and a part that oscillates "quickly". There's a theorem that is related to this idea by Cartier and Perrin published in "Nonstandard Analysis In Practice" (Diener and Diener 1995). For the gist, consider the following idea.

Let be a standard, bounded periodic function of with period such that . Suppose that it's bounded by for all x. It is, basically a Fourier type series of and terms with no constant term.

Now consider the nonstandard function for a nonstandard integer.

Consider the integral over any standard length

Where and

On the right, inside the parens, the first term is zero due to the construction of as a zero mean oscillating periodic function, and the second term is bounded by . What is the conversion factor ? A small distance in the domain of the function represents a distance in the domain of , so the conversion factor is suggesting that the little bit at the end of the integral is bounded by which is infinitesimal.

So this nonstandard function is a function which when integrated over any appreciable domain takes on an infinitesimal value. In other words, although it is pointwise very different from zero by as much as which could be any appreciable number like 100 for example, over any observable distance its average is "essentially zero".

This is the essence of homogenization theory, and we don't need the function to be periodic necessarily, it just simplifies the construction here.

The point is, it's often natural to talk about two different timescales, a "slow" process and a "fast" process such that a model for what goes on at the scale of will average out to zero over times that are appreciable in the scale of . In other words, "daily" fluctuations in temperature (scale ) are irrelevant to modeling "annual" changes in seasonal temperature averages (scale ). When these scales are sufficiently far apart we can consider one scale the "seasonal" scale and another the "deviations".

The problem comes when "deviations" last for appreciable times on the "seasonal" scale, and now it's hard to separate the timescales meaningfully.