A Simpe Solution

In Zen and the Second Law of Thermodynamics I discussed claims of the non-existence of the so-called "greenhouse effect". Some of those claims use analogies or examples to "prove" the point. Most of these are misconceived at best and erroneous at worst. Most scientists and informed non-scientists accept the basic physics. I won't say "99%" or similar because I'm aware that 87.37% of statistics are made up on the spot, and in any case I don't know what the figure might be. This is not an "appeal to consensus" - the basic physics involved is well understood and founded on fundamental physical laws.

Here's such an analogy designed to "prove" that the infrared radiation ("back radiation") from the atmosphere to the Earth's surface either doesn't exist or cannot have the claimed effect. Not surprisingly, it's from one of the many authors of "Slaying the Sky Dragon" who thinks he's illustrated "perpetual feedback":

Say you have a blackbody plate (think of an electric heater) radiating 1000 W/m² toward another plate which, because of distance, absorbs half of that intensity, i.e., 500 W/m². At equilibrium, the receiving plate thus radiates 250 W/m² toward the 1000 W/m² plate. Question: Does the 1000 W/m² plate thereby rise to 1250 W/m²? If so, then, by raising the radiator's temperature without adding more energy, you've disproved the first law of thermodynamics. Effectively, you've made the radiator heat itself. Moreover, now at 1250 W/m², the radiator will heat the other plate still more, absorb another dose of back-radiated energy, and will reach 1562 W/m². And so on, ad infinitum.

This seems to prove a feedback causing a runaway heating in the system. However, the author (Alan Siddons) is confusing an infinite geometric series (with a finite sum) to a never-ending sequence of feedback. Let's look closer at the diagram. Something is missing (par for the course in these analogies). In this case it's the radiation lost from the system, and examining that will directly give us the equilibrium conditions.

At equilibrium, the two-plate system is receiving the equivalent of 1000 W/m² via the heated plate, so the system must lose that energy to the surroundings. If the heated plate is now radiating x W/m², then x/2 W/m² is being lost to the surroundings. The receiver is absorbing x/2 W/m², so must be radiating the same amount, with half of that radiation, x/4 W/m² lost to the surroundings. We have a simple expression to solve:

x/2 + x/4 = 1000   so 3x/4=1000  and x = 1,333.3

The heater radiates 1,333.3/2 = 666.7 (actualy 666.66 recurring) to the receiver, which radiates half, or 333.3 back:

Balance is restored and no runaway heating. But where does the "extra" radiation come from, which as claimed above "violates the First Law"? It comes from the heat energy stored in the system during the equilibrium process, and the amount depends on the specific heat of heater and receiver. There is no violation of the First Law, and no violation of the Second Law either. Net flow between the two plates is from hotter to cooler, with both plates heating up until overall equilibrium is restored. Al the heat energy in the system came from the heated plate, which has some of that returned to it from the receiver. The returned heat reduces the net heat lost from the heater, so because the input remains the same, it heats up until the heat lost from the system equals the heat input.

Does the cooler receiver "heat" the hotter heated plate? No, it simply replaces some of the energy radiated (and therefore lost) from the heater, so it's the continual 1000 W/m² energy input which does the heating. Mr. Siddons has himself said "a cooler body cannot heat a hotter body, it just slows the rate of cooing". This is a perfect example - initially the heater radiates 500 W/m² to the receiver, reducing to a net 666.7 - 333.3 or 333.3 W/m² at equilibrium, and the result is a hotter radiator, radiating a total of 1333.3 W/m².

The misconception arises from only considering the instantaneous input to the system and ignoring the heat stored during the equilibrium process, a common thread in such discussions. If you intend waving a big stick, make sure you've got hold of the right end of it first.

"Slaying the Sky Dragon" - a critique

I had intended to review the book "Slaying the Sky Dragon" (by a number of authors), but I've decided not to buy a copy. Three of its chapters are now available on the 'net in pdf format. I've read the two chapters authored by Claes Johnson, and I have to say that I'm staggered by their assertions. It doesn't take a climate physicist to see the erroneous assumptions that the flawed arguments are based on.

The chapter "Climate Thermodynamics - 4 Lapse Rate and Global Warming/Cooling" starts off badly:

The effective blackbody temperature of the Earth with atmosphere is -18C, which can be allocated to a TOA at an altitude of 5 km at a lapse rate of 6.5C/km connecting TOA to an Earth surface at 15C with a total warming of 5 × 6.5 = 33C. The lapse rate determines the surface temperature since the TOA temperature is determined to balance a basically constant insolation. What is then the main factor determining the lapse rate? Is it radiation or thermodynamics, or both?

Mr Johnson may be a mathematician, but he's not a climatologist nor a physicist. I'm none of those, but I can spot the equivalent of a "schoolboy howler" when I see one. Let's analyse the first sentence; TOA (Top of Atmosphere) is considered to be around 23 km altitude. The "-18C" has to have been calculated using the Stefan-Boltzmann equation, using the average radiation-to-space value of 240 W/m² (watts per sq. metre). This does indeed give a result of -18°C, and the average temperature at a height of 5 km is indeed -18°C, but there's a snag. The radiation to space comes from more than one source, and its profile is not a complete Planck curve, as required if the Stefan-Boltzmann equation is to be used.

Zen and the Second Law of Thermodynamics

The Clausius expression of the Second Law of Thermodynamics:

No process is possible whose sole result is the transfer of heat from a body of lower temperature to a body of higher temperature.

This is unequivocal, so how does the "Greenhouse Effect" in the atmosphere raise the temperature of the Earth's surface? The atmosphere is cooler than the surface, so the flow of heat must be from the surface to the atmosphere, and so it is. Yet a number of people claim that this is proof that the "Greenhouse Effect" does not exist; "A cooler body cannot heat a warmer body" they proclaim, while in fact no-one has even suggested that this is the case. Some even go so far as to claim the so-called "back radiation" from the atmosphere to the surface doesn't exist.

Such claims have no basis whatsoever. Atmospheric longwave infrared "back radiation" has been measured, and its spectrum plotted, routinely since the 1950s. A simple web search can reveal summaries of some of the hundreds of scientific papers on the topic. It is part of the fabric of atmospheric physics. To ignore or dispute the existence of the phenomenon can only be termed denialism. However, disputing the effect(s) of this radiation in detail is another matter entirely. Such arguments can be examined for scientific logic and therefore veracity.