# Is there a simple model for the interaction between greenhouse gases and infrared radiation?

I am thinking (in the simplest model) that the Earth emits $$N$$ photons per unit time, some proportion $$p$$ hit a greenhouse gas particle and will be re-emitted back towards the earth with a probability $$0.5$$. Therefore the more greenhouse gas particles there are, the larger $$p$$ will be, and the earth will heat up more.

For an improved model, I'm thinking the atmosphere acts more like a continuous media with the photons bouncing around between particles and heating them. In this case, is there a simple analogy or maybe a 1 dimensional differential equation model (like heat flow through a medium)? Does heat diffusing through a material behave in a similar way as radiation propagating through the atmosphere?

I am interested in simple and easy to understand and roughly accurate models for understanding greenhouse gases.

• Do you specifically want a particle based approach or does a simple model that quantifies the greenhouse effect suffice? If you insist on a particle based approach the general idea is that for short wave radiation you only consider scattering and absorption while for long wave radiation you neglect scattering but take emission into account. This can be done with Schwarzschilds equation that models the change of spectral radiance [$\text{W}\cdot \text{sr}^{-1}\cdot \text{m}^{-2}\cdot\mu\text{m}^{-1}$] and where you think of the atmosphere as being infinitely many slabs. May 28 at 7:58
• I would be interested in higher level abstractions if they are physically convincing and educational. Do you know any good/basic presentations of Schwarzschild's equation? May 28 at 17:04
• Yes there can certainly be a 1D equation. It will have some fudge factors to average out clouds, latitude, day/night variation etc. but it certainly can (and has been) written. See for example Wikipedia's Idealized greenhouse model
– uhoh
May 29 at 1:34

My answer focuses on a "simple and easy to understand and roughly accurate model for understanding greenhouse gases". This is the same model used in a comment to your question.

The most basic model for understanding the greenhouse effect is based on radiative equilibrium. Incoming solar radiation is balanced by outgoing (long wave) radiation. Stated in terms of energy: On large timescales outgoing energy must equal incoming energy.

A unit square facing the sun at the top of the atmosphere receives $$S_0 \approx 1360$$ $$\text{W}\text{m}^{-2}$$, with $$S_0$$ being the solar constant. The area hit by solar radiation is a disc with earths radius $$r$$. Thus, the energy/time earth receives is $$E_I = S_0\pi r^2$$. A fraction of the radiation is reflected (see albedo), yielding $$E_I = S_0(1-\alpha)\pi r^2$$, with $$\alpha$$ being the albedo. Earths albedo is around $$\alpha = 0.3$$.

If we think about earth being a black body with no atmosphere we can use the Stefan-Boltzmann law to estimate the outgoing radiative energy to be $$E_O = \sigma T_s^4 4 \pi r^2$$, with Stefan-Boltzmann constant $$\sigma = 5.67 \times10^{-8}$$ $$\text{W}\text{m}^{-2}\text{K}^{-4}$$ and earths surface Temperature $$T_s$$. The factor 4 occurs because for the outgoing radiation we need to consider earths whole surface (I gave an explanation on why this is a reasonable assumption here).

Since energy needs to be balanced we can equate $$E_I = E_O$$ and solve for $$T_s$$ which will yield us insight on how severe the greenhouse effect actually is.

$$\begin{equation} T_s = \left(\frac{S_0(1-\alpha)}{4\sigma} \right)^{\frac{1}{4}} \approx 255 \, \text{K} \approx -18°\text{C} \end{equation}$$

We observe that our estimated temperature is a lot colder than earths average Temperature, which is around $$15°\text{C}$$ or $$288.15 \, \text{K}$$. This corresponds to a greenhouse effect of approximately $$\sigma(288^4 \text{K}^4 - 255^4 \text{K}^4) \approx 150 \, \text{Wm}^{-2}$$. This is the amount of energy/time/unit square earth receives because of the atmosphere.

We could refine the model by adding atmospheric layers to the problem, however I think this captures the essence.

• This is a great quantifiable explanation of the effect of greenhouse gases. Is the long term storage of energy in organisms and the energy from the Earth's core insignificant in this equation? Jun 2 at 4:53
• Yes actually the ocean as a heat sink is a major factor and must be considered in more elaborate (toy) models. I've never read about anyone including earths core in their considerations (which doesn't mean a whole lot. My knowledge is limited from a few university courses), however energy fluxes to lithosphere are considered in climate models. I will try to adapt my answer over the course of the weekend when I have a little spare time. Jun 2 at 10:12