The Hoff takes a novel approach to explaining the way energy moves through the Earth’s climate system and raises surface temperature above the theoretical limit of Holder’s Inequality without the need for any radiative ‘greenhouse effect'; illustrated by a simple electrical circuit.
While the commonly accepted “average” insolation for the earth is 240 w/m2 (which in turn yields a black body temperature via SB Law of 255K), these figures have very little practical value. If you go to the links, I have two additional comments that illustrated this. The first one shows a very simple “step” function insolation curve that averages 240 w/m2, but, when broken down into hourly increments, yields a BB temperature on the order of 100 degrees below 255K. In a subsequent comment, I did pretty much the same modeling, but this time assumed some level od [of?] heat capacity. Again, the average watts/m2 comes out to 240, but when broken down into hourly increments, the temperature is much much higher. Two completely different temperatures from the exact same average insolation. That’s Holder’s Inequality in action, and the reason that averaging insolation or temperature across the earth in either time or space yields meaningless figures.
That said, Holder’s Inequality does in fact act as the mathematical concept of a “limit”. The more uniform the earth’s temperature is, and the more uniform insolation is, the closer to 255K the earth will be on “average”. Even when one considers conduction and convection processes that move energy from the tropics to the much colder high latitudes, the trading in of (for example) a one degree reduction in temperature in the tropics of several degrees increase in the high latitudes, one cannot come up with a radiatively balanced model that gets to more than 255K.
But the earth IS on “average” warmer that 255K, about 288K in fact (though that number results from averaging, which I’ve just explained is a misleading approach, but close enough in this case). The extra 33 degrees has always been attributed to GHG’s. But should it be?
In all the traditional analysis of SB Law and the earth’s equilibrium temperature, the manner in which the earth absorbs radiance, and emits radiance, are assumed to be equivalent in terms of how easily energy moves through the system. This doesn’t seem like a good assumption to me. In coming radiance is Short Wave, and hence slices right through the atmosphere for the most part, penetrating the oceans to a depth of as much as 100 meters. It gets there at the speed of light. How does it get back out?
Not at the speed of light, at least not most of it. It has to make it’s way up from the ocean depths via conduction and convection. At earth surface, some can escape via radiance, and a bunch of that gets intercepted by GHG’s and sent sideways, downward, backward, what have you. Most transfers to the atmosphere via conduction though, which in turn starts convective processes. In addition to all this transfer of energy upward and downward, both ocean and air currents move energy from tropics to higher latitudes, forcing energy that may have started upward from the ocean depths to travel horizontally as well as vertically before escaping to space.
In other words, SW doesn’t meet with much resistance on the way in, but LW meets with plenty of resistance on the way out.
Given that the paths in and out are completely different, they cannot be modeled by a single equation. While reading Doug Proctor’s article on this blog I noticed his point about the manner in which heating occurs on a completely different curve than cooling (see his figure 5, top graphic)
I realized that this is highly analogous to an electric circuit like the one below. AC voltage for example, is subject to Holder’s Inequality, just like insolation and temperature are. In fact, the only difference is that voltage is subject to a variable squared relationship while insolation is subject to a variable to the power of 4 relationship. Other than that, the exact same issues apply. For a 120 volt AC input, peak voltage is actually 170 volts.
In the circuit above, Power (Volts AC) is fed into the circuit and can only flow through the first diode. How fast it can charge the capacitor is subject to how high that resistance is. The higher the resistance, the longer it takes. Similarly, the bigger the capacitor, the longer it takes. Capacitance in this case operates exactly like heat capacity.
The second diode represents the discharge path, which has a completely different resistance and which represents Long Wave trying to exit the system. Again, Holder’s Inequality comes into play.
If the system was balanced, the voltage measured across the capacitor would approach a limit set by the “effective” average voltage, which would be 120 volts. Discharge would happen about the same as charge, and so the effective average would be the result. If, on the other hand, we increase the discharge resistance and decrease the charge resistance, a curious thing happens. The limit is no longer the effective average, the limit is the peak voltage. If the charge resistance is close to zero, and the discharge resistance close to infinity, the capacitor would, in fact, charge beyond the 120 V effective all the way to 170 Volts.
In other words, given that the paths for incoming and outgoing energy flux are completely different, and the resistance to the movement of energy completely different, we have to calculate equilibrium temperature in the context of resistance to energy movement that is different, plus heat capacity, plus Holder’s Inequality. But the sum total of doing so arrives at a temperature that is higher than the effective average of 255K