Idealized greenhouse model
The idealized model is a simple model of the greenhouse to determine the surface and the atmospheric temperature of the earth or of another planet.
The surface of the sun emits light and heat radiation. Radiation corresponding to a body of a temperature of about 5,500 ° C. The earth is much colder and radiates so at considerably longer wavelengths; This is primarily infrared radiation. The idealized greenhouse model based on the fact that certain gases in the atmosphere to absorb high-frequency electromagnetic rays of the sun (such as visible light) transparent for the radiation emitted from the surface low-frequency infrared radiation, however, are not very permeable. These gases include, for example, Carbon dioxide and water vapor. Heat can therefore easily penetrate into the atmosphere, but is there partially retained.
Kirchhoff's radiation law states that each gas of the atmosphere energy that it has absorbed, must emit again. Consequently, the atmosphere radiates in the long -wave infrared range in all directions, including towards the ground and towards all. Thermal equilibrium is when all the planets reach thermal radiation is re-emitted. In this idealized model, the greenhouse gases warm the planet's surface to a higher temperature than without it would be observable. This temperature offset leads to increased radiation, until ultimately the first retained part of the radiated heat to the top of the atmosphere is emitted.
The greenhouse effect can be illustrated with the aid of an idealized planet. It is a common textbook model.
The model Planet
The planet has a constant surface temperature Ts and an atmosphere of constant temperature Ta for better approximations to reality may, in addition, a temperature difference between the planet's surface and the atmosphere will be accepted. Alternatively, TS can be considered as the temperature of the earth surface and lower atmosphere and Ta as the temperature of the upper atmosphere. In order to justify that Ta and Ts are constant everywhere on the planet, strong ocean currents are assumed to lead to a strong mixing. In addition, it is assumed that there is no significant daily or seasonal variations of the temperature.
The model will find values for Ts and Ta such that the output from the top of the atmosphere is equal to the radiant power absorbed by the atmosphere solar energy. In view of the planet earth the power delivered by him is long-wave radiation and the incoming short-wave sunlight. Both radiation fluxes have their own, different emission and absorption characteristics. In the idealized model, we assume that the atmosphere is completely transparent to sunlight. The planetary albedo? P is the portion of incident solar radiation that is reflected back into space ( since it is assumed that the atmosphere of incident sunlight is completely transparent, it does not matter if the albedo or by reflection at the surface, at the top of the atmosphere is produced a mixture of both ). The flux density of incident solar radiation is specified by the solar constant S0. When planet earth is S0 = 1366 W m -2 and? P = 0:30. Since the surface of the earth is four times its cross-section ( of shadow ) is incident radiation S0 / 4
It is believed that the surface for long wavelength radiation has an emissivity of 1 ( which means that the earth in the infrared spectral region is a black body, which is a realistic assumption ).
Account
The surface emits radiation with a flux F, which is calculated from the Stefan- Boltzmann law as follows:
Where σ is the Stefan -Boltzmann constant is. To understand the operation of the greenhouse effect Kirchhoff's radiation law is elementary. At each wavelength, the absorption coefficient of the atmosphere is equal to its emissivity. Are emitted by the earth's surface radiation can, in comparison to the atmosphere a slightly different spectral composition. In the model, it is assumed that the mean emissivity ( = degree of absorption ) of the two beam currents is the same for the interaction with the atmosphere. Consequently, the symbol ε for emission and absorption coefficient of each infrared radiation flux of the atmosphere.
The radiation density of the top of the atmosphere exiting the infra-red radiation is:
(1) radiation at the top of the atmosphere ( a) up = radiation of the atmosphere ( a) upward not absorbed by the atmosphere of the radiation of the earth's surface ( s )
(1)
The last term is the fraction ε of upwardly directed and coming from the bottom of the radiation which is absorbed, so the degree of absorption of the atmosphere. In the first term on the right is ε the emissivity of the atmosphere, the adaptation of the Stefan- Boltzmann law to reflect the fact that the atmosphere is not optically dense. Consequently ε takes on the role of a clean mixture of the two streams in the calculation of the radiation outward flux density in the invoice.
So that the net radiation flux to the top of the atmosphere = 0 must be satisfied:
(2) - radiation from the sun emission of the atmosphere ( a) upward not absorbed by the atmosphere of the radiation of the earth's surface (s) = 0
(2)
Be given for a net radiation flux = 0 at the surface must:
(3) light from the sun emission of the atmosphere (A) downwards - radiating the surface (s) = 0
(3)
An energy balance of the atmosphere can be derived from either of the conditions described above or independently:
(4 ) Total emission of the atmosphere ( a) - radiation of the earth's surface (s) = 0
(4)
Note the important factor of 2, the erbibt from the fact that the atmosphere radiates both up as well as down. Consequently, the ratio of Ta to Ts of ε is independent. Equation (4) can be converted to:
(5) or
Ta can therefore be expressed as a function of Ts. With ( 5) used in (2) gives a solution for Ts as a function of input parameters:
Or changed:
(6)
A solution can also be expressed as a function of the effective temperature Te. This is the temperature that characterizes the radiance of the infrared radiation outgoing flow F, with the assumption that the spotlight would be a perfect radiator F = σTe4. In this model, it is simple to prepare. TE is also the solution to Ts for the case of ε = 0, that is, a lack of atmosphere. In this case, the right term vanishes in the square brackets of ( 6) and ( 6) to (7):
(7)
With the Te now defined in ( 7) results used in (6)
(8)
With a perfect greenhouse in which no radiation can escape from the surface and ε = 1:
With ( 8)
Using the one given above for the Planet Earth appropriate parameters yields:
With ( 8)
For ε = 1:
With ( 8)
For ε = 0.78:
With (8).
This value of Ts is coincidentally close to the frequently cited unbequellt indication of temperature of 288 K, which assertedly "global average surface temperature " should be. ε = 0.78 means that 22% of the radiation emitted from the surface escapes directly into space, which is in agreement with the assertion that escapes the greenhouse effect between 15% and 30 % radiation.
The difference resulting from a doubling of atmospheric carbon dioxide radiative forcing is with simple parameterization 3.71 W m -2 This is the value specified by the IPCC.
From the equation (1) follows:
With the values of Ts and Ta for ε = 0.78 is obtained for = -3.71 W m -2 with Δε = 0.019. Consequently, a change of ε of 0.78 to 0.80 in accordance with the radiation drive which arises from the doubling of the carbon dioxide. For ε = 0.80 is:
Consequently, this model predicts a global warming of Ts = 1.2 K for a doubling of carbon dioxide concentration. A prediction of a typical climate model results in a warming of the Earth by 3 K. This is primarily because climate models that take into account the positive feedback resulting from the water vapor feedback in the first place. With a simple tool, this effect can be taken into account. To this end, Δε is increased by 0.02 to a total of Δε = 0.04. Thus, the effect of a triggered by the warming increased water vapor concentration is approximately taken into account. This idealized model predicts a doubling of carbon dioxide then global warming of Ts = 2.4 K ahead, which is consistent with the IPCC about.
Extensions
This simple, single-layer atmosphere model can be directly converted into a multi-layer atmospheric model. For this, the equations for the temperature in a number of simultaneous equations to be transformed. This simple model always predicts a decreasing with increasing altitude the temperature and the temperature of all layers decreases with increasing concentration of greenhouse gases to. None of these assumptions is completely realistic: In the real atmosphere the temperatures rise above the tropopause and in increasing the concentration of greenhouse gases is expected ( and observed) that the temperatures drop there. The reason is that the real atmosphere does not have the same transmissivity for all light wavelength ranges.