Averages

I was discussing my thoughts on averages versus RMS the other day during happy hour.  I mentioned how the climate system was a lot like music.  I also threw in the Selvam Golden ratio implications.   I think that both went over his head, so I am making this quick post.  The chart above is just pure sine waves with the average of the full wave through the fifth or 5 times base frequency.  The average of all of the shapes is one.  The RMS value of all the pure waves is 0.707.  The RMS value of the average of all the pure sine waves, the blue curve, is 0.31569 +/- a touch since I only used 360 points in the spreadsheet for the sine waves.  With an RMS value of 0.707 and average of 1.0, there is a considerable range of variation possible depending on the length of time considered.  Adding this one section of time to that value, 0.707+0.31569= ~1, but it should be pretty obvious that during this one period of time, the apparent impact would seem much larger.

If I add a 45 degree lag to the base frequency, you can see the shift that would produce a step appearing function to the average wave shape.  The average value of all the pure sine wave curves is still one, the RMS values of all the pure sine waves is still 0.707, but the RMS and average value of the "average" blue curve changed, dramatically across the bifurcation point at zero.

I stopped at the fifth in these examples to show that the odd lower harmonics still have a noticeable impact while the even harmonics tend to cancel.


 Since energy has to move in any system, there will be delays in transfer and efficiency losses in the transfer and conversion of the energy.  For Earth there is a common 3 month lag.  3 months out of 12 months is a 1/4 lag pretty close to the 45 degree lag impact in the second graph.  Without knowing what is a true full cycle that would produce the "average", it would be better to consider RMS and Peak to Peak values as realistic limits.

This may seem counter intuitive since RMS is lower than average, but the summation of the harmonics and delays would produce the equivalent of a "constant" minimum value for any time frame.    In the second chart the "constant" is approximately 30% less than the "average" in the first half of the cycle.  This is without any capacity values, just internal lags.

So what to this have to do with the price of tea in China?  Variation is energy loss.  Compare an Earth year to a Venus year.  On Earth, there are 365.25 daily rotations per orbit with an eccentricity of 3.4% from peak to valley.  Venus has 0.92 daily rotations per orbit and an eccentricity of less than 1 percent.    With a Venus day nearly equal to a year and nearly no annual change in orbital forcing, there would be few harmonics and lags to consider at the surface, it would be nearly isothermal.  The "average" solar insolation calculation for both planets is still considered TSI/4 despite the huge orbital differences at their true "surfaces".  Earth's "surface" can likely never be isothermal, so it would always be cooler than "average" at its true "surface".