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

Question

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.

Answer 1

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.