The earth’s atmosphere keeps the planet warmer than it would be otherwise. This is demonstrated by the much colder surface temperature of the moon at the same distance from the sun but without an atmosphere. This warming effect is called the greenhouse effect, but few people really understand what that actually means or how it works. There is so much contradictory information on the internet that I wanted to try and explain in words rather than equations how I see that the greenhouse effect works on earth.

Our atmosphere consists mainly of 78% nitrogen, 20% oxygen, 0.04% CO2 and roughly ~0.25% H2O (water vapour). H2O levels are concentrated in the lower atmosphere and vary on a daily basis, whereas the others components are nearly constant throughout the atmosphere. They are said to be “well mixed”. Recently CO2 has been increasing due to human activity mainly since the industrial revolution. So far CO2 levels have risen from about 300ppm to 400ppm since 1750, and may reach 600ppm by 2100.

**Density structure of the atmosphere**

The earth’s gravity keeps the atmosphere trapped to the planet. Despite this there is still a tiny loss of molecules to space in the upper reaches of the atmosphere. For this reason all the light hydrogen molecules originally in the atmosphere have already escaped to space. Gravity creates an exponential pressure gradient with the highest pressures at sea level at 15 lb/sq inch (1013 hPa). As you move upwards, for example when driving up a mountain, so the air pressure decreases and your ears begin to pop. This change in “hydrostatic pressure” and the exponential distribution of pressure with height is easy to derive.

Pressure changes with height as where rho is density and g is gravity which simply says that the change in pressure (force per unit area) moving up by a height DZ is equal to the change in “weight” of the air per unit area by moving to height Z+dz . This is . So applying the gas laws we get.

where T is the ‘temperature’ of the atmosphere

**Temperature structure of the atmosphere**

When a gas expands it cools. This is because temperature is just the average kinetic energy of all the molecules in one volume, so the gas loses internal energy by expanding into a larger volume. We know this intuitively because compressing air to pump up a bicycle tyre warms the inner tube. So as air rises in the atmosphere to lower pressure so it cools and likewise as it falls down to higher pressure so it warms up. Gravity maintains the pressure difference with height in the atmosphere but it is motion of air masses (convection) which sets up a temperature gradient so that it is warmer near the surface and cooler higher up. But as we shall see this does not happen by itself because you need the energy of the sun to drive that convection **and** you also need greenhouse gases. Simple thermodynamics shows that there is a perfect balance between convection and temperature gradient called the adiabatic lapse rate. When the atmosphere is stable the air is perfectly still and the temperature falls off exactly at the adiabatic lapse rate. If the sun rises in the morning and heats the surface so it induces a steeper temperature gradient thereby inducing convection. The sun’s energy drives a global heat engine moving heat from hot to cold according to the 2nd law of thermodynamics.

Remember that the atmosphere is stable at the adiabatic lapse rate. The dry adiabatic lapse rate is -g/Cp where Cp is the specific heat of air at constant pressure. Adiabatic means that that no external heat enters the gas so the work done in expanding a unit volume of air that rises upwards comes from the internal energy of the gas. Imagine a mass of atmosphere m that rises up a height DH against gravity then the work done is mgDH which is equal to the loss in internal energy(heat) or -mCpDT so DH/DT = -g/Cp. When the atmosphere is exactly at the lapse rate air can move up or down without external work. During the day the surface warms up fast increasing the lapse rate above this thereby inducing convection. If the environmental lapse rate is below the DALR then air will fall otherwise it rises until the DALR is restored. When evaporation is included we get weather such as storms, thunderstorms and precipitation.

**Effect of CO2**

The earth’s surface radiates heat as IR into the atmosphere just like a black body. CO2 absorbs photons of very specific wavelengths through quantum excitation of vibrational and rotational lines. These are concentrated in the 15micron band (see Fig 1).

This energy is quickly thermalised with surrounding air including other CO2 molecules. These quickly re-emit IR photons loosing vibrational energy according to the local thermalized temperature. As the temperature falls with increasing altitude so too does the relative emission of CO2 molecules. Multiple such re-emission steps “transfer” radiative energy in CO2 sensitive bands up through the atmosphere until the density falls sufficiently so that it escapes to space. This is because eventually there are so few CO2 molecules left higher up that the probability for absorption of a CO2 IR photon essentially falls to zero. During this process radiative energy is absorbed by the atmosphere tending to increase the temperature gradient because the density is highest near the surface. As this temperature gradient steepens away from the stable DALR so convection of air is induced to restore balance. IR radiation is the “energy source” that drives the convection heat engine that maintains the earth’s lapse rate.

The greenhouse effect works as follows.

- Solar energy during the day warms the surface.
- The surface radiates IR upwards into the atmosphere increasing the lapse rate by absorption of photons by CO2 ( and water) at different levels through radiative transfer.
- This absorption then drives convection and evaporation (latent heat) to restore the lapse rate back towards adiabatic stability.
- Most IR photons escape to space at higher and therefore colder altitudes.

The lapse rate and convection stops at the tropopause. Above that lies the stratosphere where temperatures first remain constant, and then increase because ozone absorbs sunlight to warm the upper atmosphere. What determines the height of the tropopause? The tropopause is the height where the net radiation loss to space exceeds the radiation absorbed from all lower levels. The greenhouse effect stops there and the atmosphere cools by radiation alone. Total radiation must balance incoming solar radiation so the average temperature is the “effective temperature” or about 225K. To look into more detail of how CO2 effects surface temperatures and how any increases will lead to global warming we need to study the wavelength dependence of the height where CO2 radiates to space. This then defines the effective emission height for CO2 in the atmosphere. The following is one way to calculate this height distribution.

The density of air with altitude is determined by barometric pressure. For a well-mixed gas the density of CO2 with height is determined by the overall concentration in ppm. Therefore at any altitude we know the number of CO2 molecules/m^3. The cross section for the absorption of IR photons of wavelength ? is given by the HITRAN database. Therefore we can calculate at which altitude a fraction = 0.5 of incident photons arriving from the TOA have been absorbed. Since absorption + transmission = 1 this height is exactly the same as the effective emission height for radiative transfer photons upwards of wavelength ? transmitted from all lower levels in the atmosphere and surface. The only criteria that determines the emission of IR photons of wavelength ? by CO2 molecules in all the optically thick layers below is the local thermalized temperature (Kirchoff’s law). This emission intensity is given by Boltzman’s distribution.

Now imagine a downward flux of IR photons originating from space. We assume a US standard atmosphere. Then for each wavelength using HITRAN we can calculate the height at which more than half of the incident photons are absorbed by CO2 molecules in the atmosphere. This is the same as the effective emission height for upwelling photons from lower atmospheric levels in local thermodynamic equilibrium. This is what you get

Then for each wavelength we can also calculate the emitted radiance and the result agrees almost perfectly with Nimbus spectra.

Furthermore by varying the concentration of CO2 you can calculate the change in OLR at the TOA for each wavelength. Integrating over wavelength gives you the net CO2 forcing as a function of CO2.Furthermore we can also allows “derive” the formula RF = 5.3 ln(C/C0) which had previously been a mystery to me (actually I got 6.0!). You can also calculate the net temperature effects of different CO2 concentrations on earth -ignoring all other feedbacks. The result is shown below.

**Summary**

The greenhouse effect depends on there being a lapse rate. This is stabilized at the (moist) adiabatic lapse rate through convection and evaporation. Convection is driven by solar heating of the surface and IR radiation absorption by the atmosphere. The lapse rate stops at the tropopause because this is where the radiative losses to space exceed the radiative absorption from below. The height of the tropopause effectively determines the surface temperature because radiative energy balance fixes the tropopause temperature at about 225K. CO2 just affects a narrow band of wavelengths centered on 15 microns. Increasing CO2 levels affect mostly the side lines leading to a weak logarithmic dependence of forcing because the central lines are already saturated way up to the stratosphere. This CO2 effect essentially increases the tropopause height somewhat leading to slightly larger surface temperatures at equilibrium. This is often called radiative forcing at the “top of the atmosphere” but really it is because the rise in the tropopause causes an energy imbalance until surface IR increases at the higher temperature to rebalance once again outgoing IR with incoming solar radiation.

12/6 update: corrected according to comment of Eli below.

13/6 update: corrected atmospheric pressure at sea level – see Ro below.

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The atmosphere