MODTRAN is used to compute temperature changes due to greenhouse effect described in the page Converting Forcing to Temperature Change.
The greenhouse effect reduces the heat, as infrared light, emitted from earth to outer space. To maintain balance between incoming solar energy (less that reflected by albedo) and outgoing radiated energy, the earth’s surface temperature is forced to increase, and the energy to do so is called FORCING (watts/m²). Due to continuous variations of temperature, total pressure, and partial pressure of the various gases with altitude and their highly complex infrared absorption spectra (dependent on temperature and pressure), it is necessary to use a computer program to determine how much heat is lost to space. MODTRAN is free software (access at http://forecast.uchicago.edu/modtran.html.) The form at bottom shows the input to begin a study at present conditions. The “Model Output” box at the bottom contains the objective: “Upward IR Heat Flux”. The difference between heat flux value at the standard conditions and the heat flux value at the conditions you set is the FORCING (F). Note that the default physical parameters are 70 km altitude and “looking down.” The standard conditions (now 288.7 K ground temperature); “Locality” is either “no clouds or rain” for the maximum forcing or “stratus/strato CU begin 0.22 km top 6 km)” for more realistic forcing; and 400 ppm CO2. Other greenhouse gas concentrations listed can be varied as well.
To use 288.7 K as a standard earth surface temperature, change the “Temperature Offset” variable from 0 to 0.5. There is very little difference in FORCING calculations when using 288.2 K or 288.7 K as standard.
MODTRAN has been used for many years, and has been verified by satellite data. Its main application is to support airborne and spaceborne spectrometry and radiometry. NOTE: I have not been successful in computing ΔT by adjusting Temperature offset in the new conditions.
Converting Forcing to ΔT (Greenhouse Effect)
Step 1: Use Modtran to compute the Upward IR Heat Flux (Io)for initial and final conditions
Step 2: Subract Io for final conditions from Io for initial conditions, giving the Forcing (F) in watts/m²
Step 3: Compute ΔT using the formula: ΔT = F / (2.27E-07 * T³), where T is 288.2K (15C)
NOTE: You may want to repeat this calculation by changing T to T+ΔT/2 where ΔT is that computed from above. This gives a little more precision to the calculation. However these calculations are estimates.
Derivation of Formula:
The Stefan-Boltzman Law of Radiation (I) from a surface εat temperature T is I = εσT^4, where ε is the emittance (blackness) of the surface and σ is Stefan-Boltzman constant = 5.67E-08. The derivative of this equation is dI/dT = 4εσT³, rearranging and using finite differences (estimating) gives ΔT = ΔI / (4εσT³), assuming ε = 1 in the infrared region.
Example:
Estimate the increase in the earth’s temperature by doubling CO2 concentration in the atmosphere from 400 ppm to 800 ppm, using “No Clouds, No Rain” for “Locality” and leaving other variables at default. This gives for 400 ppm an upward IR Heat Flux (Io) of 267.622 watts/m². Then change only the CO2 ppm to 800 and repeat. This gives an Io of 264.859 w/m². The difference between the two, ΔIo, is 3.0 w/m². Dividing by 2.26E-07*288.2K³ yields ΔT =
The graph below demonstrates the accuracy of Modtran.
