85. The Greenhouse Effect

In the overall debate over global warming probably the most contentious area for many is the scientific validity of The Greenhouse Effect. Whether on Twitter or on climate sceptic sites like WUWT, there are many commenters who simply refuse to accept it, or fail to understand it. In fact many climate scientists (particularly non-physicists) also do not fully understand it, or misrepresent it. In this post I will outline some of the common misconceptions about it, and then explain how the greenhouse effect actually arises. 

 

Myth 1: Carbon dioxide causes an increased heating of the atmosphere

As the atmosphere can absorb heat that is radiated from the surface of the planet, the claim here is that the presence of greenhouse gases like carbon dioxide (CO₂) increases the amount of heat that the atmosphere can store. This is true(ish), but the amount of additional heat or thermal energy stored by CO₂ is so small relative to the total that it is irrelevant.

The key property here is the molar heat capacity of the gas. Every gas in the atmosphere has a different heat capacity, this being the additional amount of thermal energy stored in that gas per unit increase in temperature. However, they are generally very similar in magnitude (see here), although the molar heat capacity of CO₂ is about 25% greater those of nitrogen (N₂) and oxygen (O₂) due to its additional degrees of freedom in accordance with the equipartition theorem. But carbon dioxide only comprises about 0.042% of all the molecules in the atmosphere, so it can only increase the total heat capacity of the atmosphere by an insignificant 0.01%. But more importantly, even this small increase is irrelevant because it is the temperature of the atmosphere that determines its rate of thermal emission, not the quantity of heat that it stores. 

The amount of heat stored merely determines the rate at which the atmosphere will cool at night. This is why planets with thick atmospheres, like the Earth and Venus, cool less at night than planets with thin atmospheres like Mars. Their thick atmospheres mean that they store a lot more energy at a given temperature, but it is the temperature that determines the rate of energy loss. So two planets at the same temperature will cool at different rates if they have different densities of atmosphere, even though the initial rate of energy loss (as set by the temperature and the Stefan-Boltzmann law) will be the same for each. This is because the planet with the thinner atmosphere will run out of stored energy first.

Myth 2: The greenhouse effect is the result of a hot atmosphere heating the Earth's surface

This myth is related to, and dependent on, Myth 1. If the atmosphere is getting hotter because the CO₂ is trapping and storing heat emitted by the Earth's surface, then the temperature of the atmosphere will increase. Eventually the atmosphere will become hotter than the surface and so it will begin reheating the surface. So the surface temperature will also increase. Except this is not how greenhouse gases work. 

They don't trap heat for long periods, but instead just reflect it back to the surface. In essence they behave like a thermal mirror. The result is that the surface gets reheated by the atmosphere, but not because the atmosphere is hotter than the surface. The lower atmosphere, or troposphere, is never hotter than the surface. It is just that the photons of infra-red radiation emitted by the surface bounce off the CO₂ molecules, and some then get reflected back to the surface and reheat it.

Myth 3: Thermal radiation cannot move from a cold object to a hotter one

One reason many climate sceptics appear to reject the concept of the greenhouse effect is that they feel it violates basic principles of physics, not least the second law of thermodynamics. One of the many versions of this law states that net heat flow is always from a hot object to a cooler one, and not in the reverse direction. Many thus misinterpret this law because they fail to appreciate the importance of the term "net". It is not that heat or thermal energy cannot flow from a cold object to a hotter one: it does. In fact all objects emit (and absorb) thermal radiation irrespective of their temperature; the Stefan-Boltzmann law tells us that (see Post 12). The key point is that hotter objects emit more. In fact the Stefan-Boltzmann law dictates that the amount of radiation emitted is proportional to the fourth power of the thermodynamic temperature T measured in kelvins (see Eq. 12.6 in Post 12).

What the greenhouse effect does is increase the amount of energy that moves in the opposite direction by enabling the atmosphere to reflect back energy emitted by the surface. But the amount reflected back is always less than 100% of that emitted by the surface, so this still means that more energy is moving from the surface up into the atmosphere than is moving in the opposite direction. Thus, the net heat flow is still upwards into the atmosphere, moving from hot to cold. Consequently the second law of thermodynamics still holds.

Myth 4: There is too little carbon dioxide in the atmosphere to make a difference

Currently the concentration of carbon dioxide in the atmosphere is about 420 ppm, or 0.042% of all the gas molecules. This looks like a small number, but there are a lot of molecules in the atmosphere. In fact there are are 0.357 million moles of gas for every square metre of the Earth's surface (1 mole = 6.02 x 10²³ atoms or molecules). So 0.042% of that equates to 150 moles of carbon dioxide per square metre, or 9.04 x 10²⁵ molecules of carbon dioxide per square metre. 

As these molecules are typically 0.33 nm in diameter, this still means that every infra-red photon emitted from the surface of the Earth will expect to collide with at least ten million CO₂ molecules before it can escape into outer space. Sooner or later one of these molecules with absorb it and then re-emit it, and half of these re-emissions will be in a reverse direction towards the Earth's surface. That is why so few infra-red photons can escape. 

The exact number of collisions depends on the scattering cross-section of the molecule at the relevant wavelength of radiation. This cross-section represents the effective area of the molecule that the photon of radiation sees, or alternatively the actual area of the molecule multiplied by the probability of being absorbed at each potential collision. So while the actual cross-sectional area of the carbon dioxide molecule is about 10^-19 m², the effective area as measured by spectroscopy is much less, around 4x10^-24 m². This means that the number of collisions each photon can expect to make with a CO₂ molecule before it can escape into outer space is only about 360 (i.e. 9.04 x 10²⁵ x 4 x 10^-24). 

This, though, is still more than enough to block the emission path of virtually every photon with the necessary wavelength. In fact most will be blocked within 30 m of the surface (the result of dividing the effective thickness of the atmosphere of 10 km by 360). But as Fig. 85.1 below shows, only photons with wavelengths close to the CO₂ absoption bands at 2 µm, 2.7 µm, 4 µm and 15 µm can be absorbed by the carbon dioxide. For the rest the CO₂ molecules will be completely transparent.

Fig. 85.1: The absorption spectrum of different greenhouse gases in the visible and infra-red.

How the greenhouse effect really works

The starting point is electromagnetic radiation from the Sun which heats up the surface of the planet. As the surface warms it also gives off radiation, but because the temperature of the Earth's surface (288 K) is much less than that of the Sun (5778 K), the energy, or frequency, of the radiation emitted is much less than that of the incoming radiation. That means that its wavelength is longer - typically about twenty times greater. So while the incoming radiation from the Sun is mainly in the visible part of the electromagnetic spectrum (see the red curve in Fig. 85.1), the radiation emitted by the Earth's surface is generally in the infra-red (see the blue curve in Fig. 85.1).

The effect of greenhouse gases like carbon dioxide is to reflect back some of the outgoing infra-red radiation. This then gets re-absorbed by the surface and heats it further. This is the greenhouse effect. The key point is that the outgoing radiation is merely being reflected back by collisions with carbon dioxide (or water) molecules in the atmosphere. So these greenhouse gas molecules act like a mirror. In the next post I will outline the mechanism and mathematics of this in more detail. Any additional heating of the atmosphere only comes later as a result of the additional heating of the surface.

The result of this reflection of outgoing radiation is that the amount of radiation hitting the Earth's surface and being absorbed increases. For example, suppose the amount of radiation from the Sun that is being absorbed by the Earth's surface is I_o. As I showed in Post 13 (The Earth's energy budget) this equates to about 161 W/m². With no greenhouse effect in place the outgoing radiation will balance the incoming radiation, and so according to the Stefan-Boltzmann law (see Eq. 12.6 in Post 12) the surface temperature will be 231 K (or -42°C). 

However, if a fraction p of the initial outgoing radiation is reflected back (i.e. pI_o), then the total incoming radiation will be (1 + p)I_o . This will heat up the surface even more and result in a higher emission temperature, which in turn will increase the total intensity of the outgoing radiation I_T. If we then assume that the greenhouse gases reflect back a fraction f of the outgoing radiation, then the following equation must hold. 

(85.1)

This basically states that the total outgoing radiation must balance the incoming radiation plus the fraction of the outgoing radiation that is reflected back I_r = f I_T. Rearranging Eq. 85.1 gives the result

(85.2)

We know that the current mean surface temperature of the Earth is approximately 289 K (or 16°C), so this allows us to calculate I_T using the Stefan-Boltzmann law

I = σT⁴

(85.3)

where T is the temperature in kelvins and σ is the Stefan-Boltzmann constant. The result we obtain is that I_T = 396 W/m², and as I_o = 161W/m², this then implies that f = 0.59. In other words, the greenhouse gases reflect back about 59% of all outgoing infra-red radiation. 

Knowing the values of I_T and I_o also allows us to use Eq. 85.3 to determine the temperature of the Earth's surface both with and without the reflected radiation. These values will be 289 K and 231 K respectively. So the reflected radiation due to the Greenhouse Effect has increased the Earth's temperature by 58 K.