Radiation Balance and Greenhouse Effect

Greenhouse effect and CO2 forcing

The greenhouse effect is the warming of the surface and lower atmosphere that results because some gases absorb outgoing longwave (infrared) radiation and re-emit part of it downward. Carbon dioxide is the dominant long-lived anthropogenic greenhouse gas. Its influence is expressed as radiative forcing, the change in net downward radiative flux at the tropopause caused by a change in composition, measured in W m^{-2}. A positive forcing warms the climate.

Longwave heating of an atmospheric layer is governed by the convergence of the net longwave flux. With F_\uparrow and F_\downarrow the upward and downward longwave fluxes, the net upward flux is F_n = F_\uparrow - F_\downarrow. For a layer of density \rho and specific heat c_p, the energy budget gives \rho c_p \frac{dT}{dt} = -\frac{\partial F_n}{\partial z}. A layer that receives a net upward flux of 12 W m^{-2} at its base and loses 7 W m^{-2} at its top therefore absorbs 5 W m^{-2}. For a 500 m thick layer with \rho = 1 kg m^{-3} and c_p = 1000 J kg^{-1} K^{-1}, the heating rate is \frac{5}{1 \times 1000 \times 500} = 1\times10^{-5} K s^{-1}, or about 0.86 K per day.

The outgoing infrared flux to space originates near the level where the optical depth to space is about one. When a well-mixed greenhouse gas is added, the absorption coefficient of the layer rises and this emitting level moves upward. In the usual troposphere, where temperature falls with height, the emitting level becomes colder and the outgoing flux decreases. If temperature increases with height in the layer, the emitting level becomes warmer instead. Since emission follows \varepsilon\sigma T^4, the outgoing infrared flux then increases.

The global warming potential (GWP) compares the integrated warming effect of a unit mass of a gas with that of CO_2 over a fixed time horizon, usually 100 years, so CO_2 has a GWP of 1 by definition. Gases with long atmospheric lifetimes and high radiative efficiency per molecule have large GWPs. Nitrous oxide has the largest 100-year GWP among CO_2, CH_4 and N_2O, being a few hundred times that of CO_2. Water vapour is not assigned a standard GWP because it acts mainly as a fast feedback with a short residence time.

Emission inventories are built by multiplying activity by an emission factor. For road transport, annual fuel use equals distance travelled divided by fuel economy, and carbon release equals fuel volume multiplied by carbon content per litre. Since 1\ \text{Mt} = 10^{12}\ \text{g}, a fleet of 10 million vehicles, each travelling 10000 km per year at 20 km per litre, burns 5\times10^{9} L per year. At 2.5 g C per litre this releases 1.25\times10^{10} g, or 0.0125 Mt C per year.

Carbon budgets follow the same logic at the level of emissions. Only part of the annual anthropogenic CO_2 emission stays in the atmosphere, and the rest is taken up by the ocean and land. The concentration keeps rising as long as the net input is positive. If the airborne fraction is constant, the rise stops only when annual emissions themselves reach zero, not when they merely stop growing. For a steady reduction of emissions by a fixed amount each year, the year of zero emissions is found by solving E_0 - k\,\Delta E = 0 for the number of years k.

• Airborne fraction — the share of annual emissions that remains in the atmosphere. e.g. if 8 Gt of CO_2 are emitted and 4 Gt remain airborne, the airborne fraction is 0.5.
• Radiative forcing — the imbalance in net flux caused by a change in a forcing agent. e.g. a forcing of +2 W m^{-2} implies a net energy gain of 2 W m^{-2} at the tropopause before the climate adjusts.

Radiative equilibrium and Rayleigh scattering

A planet or small body is in radiative equilibrium when the absorbed shortwave energy equals the emitted longwave energy. For a sphere of radius R with albedo \alpha in a solar field of strength S, absorption is S(1-\alpha)\pi R^2 and blackbody emission is 4\pi R^2\sigma T^4. When the external source is removed, the body cools by releasing its stored heat, and the cooling rate depends on its heat capacity.

For a sphere of density \rho and specific heat c, the heat capacity is \rho c\,\frac{4}{3}\pi R^3 and the emitting area is 4\pi R^2. The energy budget therefore reduces to \rho c \frac{R}{3}\frac{dT}{dt} = -\sigma T^4. If the temperature change \Delta T is small compared with T, the emitted flux can be evaluated at the initial temperature, which gives a cooling time t = \frac{\rho c R\,\Delta T}{3\sigma T^4}. Note that the radius enters through the ratio of volume to emitting area, not through the size alone. For a body with R = 1 m, \rho c = 10^6 J m^{-3} K^{-1} at 200 K, cooling by 1 K takes about 3.7\times10^{3} s.

Rayleigh scattering applies when the scattering particles, such as air molecules, are much smaller than the wavelength of light. Its scattering efficiency is proportional to \lambda^{-4}, so the ratio of scattering efficiencies for two wavelengths is (\lambda_1/\lambda_2)^{-4}. A wavelength half as long is scattered 16 times more strongly. Because short wavelengths are scattered far more than long ones, the clear daytime sky appears blue, and the scattering of red light is comparatively weak.

• Radiative equilibrium — the state in which absorbed and emitted energy balance, so the temperature is constant in time.
• Rayleigh regime — particle size much smaller than the wavelength, where efficiency scales as \lambda^{-4}.