# How can global warming occur in spite of Energy Balance?

The Greenhouse Gas Theory maintains that an increase in carbon dioxide will cause a corresponding increase in Earth's temperature. But this temperature increase will cause an increase in the flow rate of energy out to space. At the present temperature of 288 K, the power to space is 239 Wm^-2, and must be maintained at this value by the requirement of Energy Balance, whereby the output power is held equal to the power received from the Sun by the Earth's planetary system.
Therefore, no temperature change can occur. I should be pleased to receive help on this problem.
Eddie B.

• who says:" Energy Balance, whereby the output power is held equal to the power received from the Sun by the Earth's planetary system" no matter how the atmosphere is composed? Commented Jun 1, 2020 at 14:01
• If 100% of the sun's radiation falling one earth would be absorbed, then we would definitely not be at 288K...Why do you use shiny car window protectors in the summer? In your argument it would not matter if you put them or not, as your car will be at the same temperature no matter what... Commented Jun 1, 2020 at 14:07

The temperature increase does not cause an increase in energy flow away from the Earth, which must in equilibrium always equal the energy flux incident on the Earth's surface from the Sun (and from other negligible sources). Treating the Earth as a black body, it causes an increase in the amount of energy radiated out from the Earth, but part of this energy is blocked by the carbon dioxide molecules in the atmosphere which are opaque in the infrared.

To understand how this works in a toy model, suppose that the Earth were enclosed in an infinitesimally thin spherical shell fully transparent to visible light and opaque to infrared light. Suppose further that the energy flux incident on the Earth from the Sun may be decomposed as $$S = S_v + S_i$$ into its visible and infrared components, and do the same for the spherical shell (denoted by $$L$$) and the Earth (denoted by $$E$$) itself. Then, you have the energy balance relations

$$S = S_v + S_i = (L_i + L_v)/2 + E_v$$

$$L = L_v + L_i = S_i + E_i$$

$$E = E_v + E_i = S_v + (L_i + L_v)/2$$

We know that virtually all blackbody radiation emitted by both the spherical shell and the Earth itself will be in the infrared, i.e. approximately $$L_v = E_v = 0$$. This gives us

$$E = S_v + L_i/2 = S_v + (S_i + E)/2, \, \, \, E = 2 S_v + S_i$$

In other words, the spherical shell results in an effective increase of the energy flux hitting the Earth's surface by an amount $$S_v$$. This results in a corresponding increase in the Earth's temperature in thermal equilibrium.

The actual effect of carbon dioxide is much less pronounced than this - indeed, an increase of this order of magnitude in the energy flux incident on the Earth's surface would raise Earth's temperature by dozens of degrees celsius. This is because an atmosphere of carbon dioxide is not perfectly opaque and also because carbon dioxide itself is not perfectly opaque in the infrared, just in some concentrated regions of the spectrum corresponding to its spectral lines. Nevertheless, the mechanism that produces greenhouse heating is identical, and it's the same effect you see in actual greenhouses here on Earth (where it is achieved by glasses opaque in the infrared).

The missing consideration is the fact that greenhouse gasses work as a "selective blanket" that insulate the emission from Earth better than that from the sun.

The incoming solar radiation from the Sun is made up of certain wavelengths matching the Sun's surface temperature at around $$5600\,^\circ \mathrm C$$. The outgoing radiation emitted from Earth matches the Earth's surface temperature.

The key is now that the greenhouse molecules have a varying absorption spectrum. They more readily absorbs the radiation from Earth than that from the sun. Meaning, more solar energy is allowed through the atmospheric blanket than outwards going radiation from Earth. The yellow colours on the graph constitute Earth's emission spectrum - the red colours are what actually makes it through all the way through the atmosphere:

The graph is from this source: https://www.e-education.psu.edu/earth103/node/1006 where the Sun's spectrum can also be seen for comparison.

When the greenhouse molecules absorp radiation, they reemit it in all directions. A fraction of it is thus sent back to Earth - and this fraction is larger than the fraction of solar energy which is sent back into outer space.

It is thus harder for our planet to "get rid of" the solar energy than to receive it due to it's "selective blanket".

Thus, received solar energy accumulates here on Earth. This increases the temperature. Stefan-Boltzmann's law tells us that the emission from Earth increases with temperature, and so at some point the temperature will have risen to cause enough extra emission from Earth that the effect of the atmosphere is again balanced out. This balance will be found at a slightly higher temperature.

The assumption of energy balance is wrong. It's not a requirement, and it is currently violated by our observations.

For example, data from the Argo ocean floats informed a 2012 estimate that the Earth is absorbing roughly $$0.58\pm0.15$$ W/m$$^2$$ more energy than it emits.

• If one accepts this result, then my calculations show that it would produce a change of 0.16 K. Otherwise, the energy balance requirement should be maintained, and so no global warming should occur. Commented Jun 1, 2020 at 20:07
• @EddieBanner How are you getting just a constant temperature shift out of this? There are a ton of temperature-dependent forcing factors that affect the energy imbalance as a function of temperature. For example, higher temperature means less sea ice, which means the Earth reflects more light, which increases the energy imbalance over what it was at a lower temperature. Commented Jun 1, 2020 at 21:38
• Good point about my calculation. It would be interesting to know how long this small imbalance has exited, and how long we have enjoyed 288K. Please note, however, that Trenberth et al , NASA, and others do use the energy balance idea. Commented Jun 2, 2020 at 13:46
• @EddieBanner We know how long we've enjoyed 288K, because we can extract global temperature measurements from both the historical record and the archaeological/geological record. The answer is "not for very long at all"; a significant temperature rise, much faster than the gradual changes that have happened in the past, has occurred in the last century. Commented Jun 2, 2020 at 15:12
• @EddieBanner What do you mean, specifically, when you say that they "use the energy balance idea"? Commented Jun 2, 2020 at 15:12