How can we calculate the temperature of the atmosphere, including the greenhouse effect?

Question

I've been struggling to find equations that express how many degrees of warming greenhouse gases contribute, given the composition of an atmosphere (and solar insolation).

What I did find was the Stefan–Boltzmann Law, and that wiki article has good equations about black body radiation. It shows how to get a value of $279 K = 6 °C = 43 °F$ for the plain Earth with no atmosphere. Then to account for the albedo of Earth, the effective temperature is calculated at $255 K = −18 °C = -0.4 °F$.

Then the article basically states what the real values are without presenting any more equations or explaining how to calculate it.

However, long-wave radiation from the surface of the earth is partially absorbed and re-radiated back down by greenhouse gases, namely water vapor, carbon dioxide and methane. Since the emissivity with greenhouse effect (weighted more in the longer wavelengths where the Earth radiates) is reduced more than the absorptivity (weighted more in the shorter wavelengths of the Sun's radiation) is reduced, the equilibrium temperature is higher than the simple black-body calculation estimates. As a result, the Earth's actual average surface temperature is about 288 K (15 °C), which is higher than the 255 K effective temperature, and even higher than the 279 K temperature that a black body would have.

And there were only 2 cites that only cited the fact that H₂O, CO₂, and CH₄ are greenhouse gases.

I've been searching for how to calculate the warming due to GHGs, but no luck. I'm really hoping someone can shed some light on this. I'm looking for an equation that takes into account atmospheric composition and probably the rotation rate of Earth (or whatever body), and of course the solar insolation.

Please note I'm not really asking about climate change calculations. Climate change is more about how the atmospheric composition is changing (namely GHG increases). I'm simply looking for equations that dictate the temperature based on a given/constant composition of the atmosphere.

Answer 1

One can account for greenhouse gas effect for Earth's temperature in simple energy balance model in following manner.

Assume that fraction $f$ of longwave radiation emitted by Earth's surface is captured by green house gases in the atmosphere. Suppose Earth's surface temperature is $T_e$ and temperature of the atmosphere is $T_a$. The energy received from Sun should balance as follows $$\pi {r_e}^2 F_0 (1 - A) = 4 \pi r_e^2 (1-f) \sigma {T_e}^4 + 4 \pi r_e^2 \sigma {T_a}^4 $$ where $F_0$ is solar constant (Energy received from sun per unit area per unit time), A is Earth's albedo, $\sigma$ is StephenBotlzmann constant and $r_e$ is Earth's radius.

Since the black body radiation are isotropic, 50% of radiation emitted by atmosphere travels toward Earth's surface and remaining 50% of radiation travels toward the space. Also, assuming that atmosphere receives all of its energy from Earth's surface through longwave radiation, it gives following energy balance equation $$f \sigma {T_e}^4 = 2 \sigma {T_a}^4. $$ which leads to $$ T_a = T_e\left(\frac{f}{2}\right)^{\frac{1}{4}} $$

Nearly 80% of the radiation emitted from Earth's surface is absorbed by greenhouse gases (including water vapour) and clouds. Substituting value of $f$ 0.8 and solving it for $T_e$ will get you the answer you are looking for.

Reference

Chapter 7 The Greeenhouse Effect in Introduction to Atmospheric Chemistry by Daniel Jacob http://acmg.seas.harvard.edu/people/faculty/djj/book/bookchap7.html

Answer 2

The Milne-Eddington approximation lets you account for greenhouse gases, because they increase the longwave optical thickness $\tau$ of the atmosphere:

$$T_s = T_e \cdot \left(1 + \frac{3}{4} \tau\right)^{\frac{1}{4}}$$

where $T_s$ is surface temperature and $T_e$ is effective temperature (from sunlight and albedo alone).

$\tau$ can be found from curve fits to the partial pressures of the greenhouse gases involved. An example of a temperature model including greenhouse gases can be found here:

B.P. Levenson - Planet temperatures with surface cooling parameterized