Does the Greenhouse Effect Contradict the 2nd Law of Thermodynamics?

Occasionally I hear people claim that the greenhouse effect isn't real because it would contradict the 2nd law of thermodynamics. The idea behind this is that the atmosphere is cooler than the Earth's surface, and the 2nd law of thermodynamics says that heat always goes from hot to cold. Therefore, the Earth's atmosphere can't heat the Earth's surface. Therefore, the greenhouse effect can't exist in a universe in which the 2nd law of thermodynamics is operable.

A Simple Explanation

The silliness of this objection can become apparent with a simple analogy. If you walk outside with a thin t-shirt on a cold day (let's say 20°F), you'll quickly feel cold, and the surface of your skin will be significantly colder than normal. If you find a winter coat outside, its temperature will be 20°F. But if you put it on, you will begin to warm. It's not that the coat is generating heat and sending it to your body. It's simply slowing down the rate at which thermal energy leaves your body and allows your body to maintain a warmer temperature. With or without a blanket, the net flow of thermal radiation is from the your body (~98.6°F) to the outside (20°F).

In much the same way, the Sun radiates solar energy towards the surface of the Earth, and since the Sun is much hotter than the Earth, the net flow of thermal radiation is from hot to cold. Some of that solar energy is absorbed by the Earth and radiated back towards space as IR light. Since space is much colder than the Earth's surface, the flow of heat is from hot to cold. Greenhouse gases function as a jacket that slows down the rate of heat loss towards space, allowing the surface to maintain a warmer temperature than it otherwise would. That's the short answer to this objection. 

A Fuller Explanation

For those who want more than the above, a fuller explanation follows. I think some of the objections to my above explanation derive from some basic misunderstandings of thermodynamics, and possibly due to some perhaps oversimplified explanations of the greenhouse effect. I think there are two reasons why people object to the science on this: 

1. Simplistic ways of describing heat. It's true that heat flows from hot to cold. But every object with a temperature emits thermal radiation in all directions, including towards objects with a higher temperature. The net flow of thermal radiation is from hot to cold.
2. Simplistic explanations for the Greenhouse Effect. In short explanations of the greenhouse effect, we're told that greenhouse gases trap heat in the climate system to keep the earth about 33°C warmer than the Earth's effective temperature of -18°C. From this I suspect that people believe that the atmosphere is thought to be a generator of heat which it sends to the surface, rather than functioning like a jacket that slows down the rate of heat loss from the Earth's surface to space. 

Let's look at this a little more closely (I have more details here). The Earth's surface is constantly receiving energy from the Sun. If the Earth were just a rock hurling through space far away from any star, it's temperature would be ~2.7 K. But because we're orbiting a star and receiving heat from it, we're much warmer. The Sun is constantly shedding its heat towards us, and the Earth receives this at a rate of ~236 W/m^2. Let's call this Fin for the energy flux of incoming solar energy. However, like the Sun, the Earth's surface is also constantly shedding its heat towards space, and that rate is governed by Fs = εσT^4. If the Earth lacked an atmosphere and had an albedo of 0.3, we can solve for the Earth's temperature at ~255K, or ~252K warmer than a similar rock hurling through space. However, the Earth's temperature is actually ~288K, or ~33K warmer than its effective temperature. Since emissivity (ε) is near 1 at the surface (I used 0.98), Fs is largely determined by surface temperature. At 288K, Fs = 382 W/m^2. Notice that Fs > Fin by 146 W/m^2 (let's call this difference Fg). The Earth's surface is shedding heat, but something is causing Fs > 0 W/m^2. Since temperatures decrease with altitude (this is called lapse rate) the altitude where the Earth's radiating temperature (Fout) equals Fin is higher than the surface. In fact, we can approximate the altitude where Fout is observed. With a standard lapse rate of ~6.5K/km, that altitude is 33/6.5 = 5 km above the surface of the Earth. This is the phenomenon that is popularly called the Greenhouse Effect. We need an explanation for the fact that Fg = Fs - Fin = 146 W/m^2 and the altitude where Fout = 236 W/m^2 is 5 km above the surface.

Let's reorder our terms. Fs (energy flux at the Earth's surface) equals the Fin (incoming solar energy flux) plus Fg, or Fs = Fin + Fg. For those who reject the Greenhouse Effect, they need to explain why Fg > 0, and therefore they must imagine that Fg is generated out of nothing, since the Sun does not explain why Fg > 0. This violates the first law of thermodynamics (energy is neither created nor destroyed). So there must be something about the rate at which the Earth's surface sheds heat that explains Fg. This is the role of greenhouse gases (GHGs). GHGs in the atmosphere absorb a fraction of that outgoing longwave radiation (LWR) and radiate it in all directions. Some of that radiated LWR continues towards space and some is sent back towards the surface. This slows down the rate of surface heat loss, so more energy remains at the surface, and the radiating temperature of Earth is no longer at the surface; it's found higher in the atmosphere. As GHG concentrations increase, it becomes harder for LWR to escape to space; the atmosphere must be thinner for LWR to escape without being absorbed and emitted by GHG molecules at higher altitudes. That thinner atmosphere is found at higher altitudes where it's also colder, and colder air is a poorer emitter of thermal radiation than warmer air. So overall, more energy is trapped within the climate system, and the the altitude of the Earth's radiating temperature moves higher in the atmosphere. There is no energy or heat generated in this. All that is happening is that GHGs are making the Earth less efficient at shedding heat, so the Earth's surface can't cool to its effective temperature of ~255K. Greenhouse gases explain both why Fg = 146K and why the Earth's radiating temperature is found 5 km above the surface.

The net flow of thermal radiation here is always from hot to cold. The Sun is hotter than the Earth's surface, and the Earth's surface is warmer than space. Virtually all the energy that keeps the Earth's surface at ~288K is generated by the Sun (a tiny fraction comes from the Earth's interior); greenhouse gases just prevent some of that energy from escaping to space, so the surface can only cool to 288K instead of 255K. If we put this in terms of the effect of CO2, doubling CO2 concentrations reduces Fout by 3.7 W/m^2. That is, it causes an energy imbalance near the tropopause. The altitude at which the Earth's radiating temperature is observed moves higher in the atmosphere. The surface must warm in response; it does so according to the lapse rate, and Fs increases until Fin = Fout again; that is, the energy imbalance is reduced to 0 W/m^2.

Not all planets are as good at shedding heat as Earth. Venus has a much stronger greenhouse effect, so it doesn't shed heat very well at all, and it's a balmy ~750 K compared to our comparably comfortable ~288 K. The Earth's surface is colder than Venus and warmer than its effective temperature because 1) we have a greenhouse effect but 2) it's not nearly as strong on Earth as it is on Venus, and 3) the Earth Earth is a bit farther away from the Sun.

Let's summarize this without all the math. The Earth's actual surface temperature essentially depends on the rate at which it receives heat and sheds heat. If the Sun were to increase its output, the earth's surface would receive heat more rapidly than it loses it; the Earth's temperature must increase, and this increase in temperature causes the Earth's surface to shed heat at a higher rate, and a new equilibrium would be reached at a higher temperature. Likewise, if greenhouse gas concentrations increase, the Earth's surface will not shed heat as well, and the Earth's temperature must increase. As it increases, though, the Earth sheds heat more rapidly until the Earth the rate it sheds heat equals the rate it receives it again. At no point is there a net flow of thermal radiation from cold to hot; the laws of thermodynamics are observed.

The above chart shows incoming solar radiation at 342 W/m^2, of which 107 W/m^2 is reflected either in the atmosphere or from the surface, leaving Fin = 235 W/m^2 to be absorbed by the atmosphere and the surface. The Earth's surface at an equilibrium, stable temperature must therefore shed heat at a rate equal to the 235 W/m^2 it receives. Whenever an imbalance occurs, whether in the rate at which the earth receives or sheds energy, the Earth's temperature must change until the imbalance is closed. This is required by the laws of thermodynamics.

Conclusion

The greenhouse effect was essentially discovered by Fourier in 1824, even though he didn't know about greenhouse gases, because of thermodynamics. Fourier understood that the Earth's surface was much warmer than it should be given its distance from the Sun. Since energy can't be created out of nothing, he reasoned that the cause of this discrepancy had to do with some property of the atmosphere slowing down the rate at which the Earth's surface sheds heat. It wasn't until ~30 years later that experiments by John Tyndall discovered how gases like CO2 make the greenhouse effect work.

This is aspect of climate science about which there is essentially no disagreement among physicists and climate scientists. You can find essentially the same explanation, with different examples and emphases, from Roy Spencer, Stefan Rahmstorf at RealClimate and Skeptical Science. Denial of the greenhouse effect is now essentially a denial of observational evidence, since we now can observe the greenhouse effect with IR sensors and satellites. There is absolutely no merit to the claim that the greenhouse effect breaks the laws of thermodynamics; quite to the contrary, it depends on them.