What does "seasonal" mean?

2016 May 2
by Daniel Lakeland

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 g(x) be a standard, bounded periodic function of x with period p such that \int_0^p g(x) = 0. Suppose that it's bounded by |g(x)| < B for all x. It is, basically a Fourier type series of \sin and \cos terms with no constant term.

Now consider the nonstandard function \hat g(x) = g(Npx) for N a nonstandard integer.

Consider the integral over any standard length L

\int_0^L \hat g(x) dx = \left ( \int_0^{Kp} g(s) ds + \int_{Kp}^{Kp+\epsilon} g(s)ds \right ) dx/ds

Where s = Npx and K= \lfloor (NpL/p)\rfloor

On the right, inside the parens, the first term is zero due to the construction of g(x) as a zero mean oscillating periodic function, and the second term is bounded by \epsilon B. What is the conversion factor dx/ds ? A small distance ds in the domain of the g function represents a distance ds/Np in the x domain of \hat g(x), so the conversion factor is 1/Np suggesting that the little bit at the end of the integral is bounded by B\epsilon/Np 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 B 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 t1 and a "fast" process t2 such that a model for what goes on at the scale of dt2 will average out to zero over times that are appreciable in the scale of t1. In other words, "daily" fluctuations in temperature (scale t2) are irrelevant to modeling "annual" changes in seasonal temperature averages (scale t1). 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.

 

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