What is the Gain?

There are tons of scientists, real and not, looking at the huge mountains of data on climate change and a surprisingly large number think they have found, THE, A or SOME solution, including me.  I am in the A solution group.  A reasonable solution from one specific base period with a large but realistic error margin for one specific change.  If we increase atmospheric resistance to radiant cooling the "true" surface, the oceans, will increase in temperature by 0.8 +/- 0.2 C degrees.  Since the full impact of that increase will take 300 to 500 years roughly to be realized and the "solution" is only related to an increase in atmospheric resistance to radiant cooling, the range of possible change that could be experienced is roughly -3 to +3 at the "surface".  That could be larger or smaller, but without a better estimate of what the "true" absolute "surface" temperature is, who knows.

Since I am confident (or over confident if you like) in my "sensitivity", 0.8 +/-0.2, I am curious about the gain.  Since there are other impacts that can be amplified by atmospheric "signal", mainly regionally, which I don't consider major "feedbacks" to the longer term "equilibrium" condition, those might have an impact on "global" "sensitivity"  which would prove me wrong.

I completely expect there to be over shoots as the system hunts for some form of a new equilibrium.  Just looking at the paleo data standard deviation of 4.2 for the atmospheric surface and 0.82 for the ocean surface, there is a gain of 4.2/0.82~= 5 causing a fluctuation of +/-2.5 C degrees.  If anything, increased atmospheric resistance should damp that fluctuation somewhat.  In other words, atmospheric resistance changes the gain of the system.

This chart is a trial balloon.  The Golden Ratio is interesting in terms of a controls perspective. By using the square root of the Golden Ratio as the exponential value, you can get the blue curve and the orange is the ratio of the blue curve to the input value or x axis.  Between 1 and almost -1, would be a control range.  Between -2 and about 4 is the "slop" range where the system could be spiked and still return to "set point".  Below -3 there is zero gain.  Above ~4 there is a slow drift to complete loss of control.  Since an amplifier is limited by its applied voltage or available energy, it should take one hell of an impulse to drive it to complete loss of control.  Because of the two transitions at ~-1 and 1, there would be two semi-stable "set points".  So if the Golden ratio is the cat's ass of non-ergodic system control, the gain is {(1+5^0.5)/2}^0.5 or 1.2720196... which CO2 could possibly change, dunno.

That would mean that my +/-0.2 error is a little light, more like +/- 0.217, but I can live with that.

So we have ~240 Watts applied, with the gain, 240*1.2720..=~309 is the ocean amplification which is amplified by atmosphere yielding 309*1.2720..=~388Wm-2 if we lived on a perfect world.  388:240 with the proper decimal points, is the Golden Ratio.  In our imperfect world, the ocean half receives more energy, ~248 yielding 316 and 402Wm-2.  Since the oceans are the main energy storage, we get our more realistic range.  The ~306 and ~316 could be solid limits imposed by the freezing points of water and salt water, which agrees with what I had guessed before.  If CO2 increases the gain, I am wrong, if CO2 shifts the set points by reducing the gain, I am right. With ice as reference the shift would be limited by the 316Wm-2 upper and roughly 309Wm-2 lower ranges( 306 to 316 based on the GR).

All this is pretty much just numerology at this point.  I don't have any way to prove squat, but I can fine tune predictions and see what happens.  There may be some Golden Ratio fans out there that may, so go for it, but IMHO the GR does make a great controller gain.  The problem of how to relate the GR to the needed energy flux and distance in the phase space remains.

The spread sheet above shows one of my crazier ideas, using the GR powers and roots for distance.  309 is the rough mean of the heat of fusion range set by salt water.  The "distance" from 309 to the maximum sea surface effective energy (~500) is phi.  Half way between 500 and 309, 1/2 phi, is 393, a possible stable "average" based on water limited by the heat of fusion.  2 phi from 500 is 190, a possible radiant limit for air with 309 as an "average".  So if the Golden Ratio does produce semi-stable operation, factors of phi could be substituted for distance, at least, in a simple model.

Note though that since climate appears to be bi-stable, the second semi-stable operation region based on 316, fresh water heat of fusion, could produce the second strange attractor.  The two would by no means ensure a fixed range of operation, but could improve the estimate on the probability of operation within the ranges set by the 309 and 316 strange attractors.

As I have mentioned before, this could be a complete waste of time.  It is amazing to me though that the GR does produce a good "fit" with the original static model, but in a chaotic system you can find just about any relationship your little heart desires.

There may be more on this later, in any case, the dimensions in the chart could be used for a kick butt tile pattern :)

UPDATE:

I forgot to add the gains.  Unlike simple harmonics, the Golden Ratio would produce Golden Harmonics.  The chart above shows the values for each of the lines in the previous chart with the product from 1/16 to 2 phi.  A two phi world would have larger scale events.

In a one phi world, there are still large events that occur with synchronization and more noise, the smaller product.  A noisy world is a more stable world if the Golden Ratio is valid.

Since I have it handy, this is some more Golden Noise using the "frequencies" phi to 1/16 phi with the gain set to the "frequency".  The first signal is 1.618... times the cosine t*1/1.618... etc. for the other signals.  It appears to be random noise, but there is a signal.  My FFT module is prone to crash, but you can dig out a 41ka signal if you set t to be years.  It would be fun to see how many frequencies a diligent signal processing jukie could isolate :)