Equations of idealized greenhouse model with $n$-layers?

Question

I wanna work with this climate model, the idealized greenhouse model. The Wikipedia page explains it very well with an example using one layer of atmosphere. Then it mentions that you can add more layers to model more powerful greenhouse effects. And there are some examples online with two and three layers. But what are the generalized equations to find the surface temperature $T_s$ with $n$-layers of atmosphere with this model?

Answer 1

Problem with the layers model

I cannot answer properly to your question, but let me tell you that this is a very interesting one.

There are no physical layers in atmosphere, optically speaking, because the optical index of gases varies continuously with altitude (air + other gas in troposphere, then less and less gases + ionized gas like ozone in the upper layers in very few quantity). For me, the layers model have not a clear physical sense.

Two layers problem

Take for example the 2-layers model: let's look a bit.

• The lower layer is clearly the earth surface, with its associated emissivity $\epsilon_s$ which is according to Kirchhoff's law equal to its absorption at thermal equilibrium, so in other words $\epsilon_s = 1 - t_s - r_s$, with $t_s$, $r_s$ being transmission and reflection coefficients of earth surface / air interface.
• The upper layer is undefined, but regarding to the temperature of the upper layer (around $-40$ °C), it seems that this is the upper troposphere altitude around $10$ km). Anyway, to this layer is associated an emissivity $\epsilon_a$, which does not make sense for a continuously varying optical index of gases ! There is no interface that means $\epsilon_a = 0$ (because the transmission between two very close layers of air is $1$), the emissivity is defined in everyday life by an interface between two media like a hot resistance and air, the earth surface and surrounding air, the wall of a oven and the surrounding air etc.

Absorption of a column of gases

As I discussed last time we can clearly compute the absorption $A$ by a column of length $h$ of gas (see this topic for example) with a Beer-Lambert like law to obtain something like (the exact expression exists also in terms of hypergeometric function): $$ A = 1 - e^{- k h \sum_i N_i \text{Im}(\alpha_i)} $$

with $k$ wave vector, $N_i$ number of particles per unit volume of species $i$ and $\alpha_i$ molecular polarizability of species $i$. We find $1$ roughly speaking just for the CO2, you see that $\epsilon_a$ and $A$ are completely different quantities since $A = 1$ just for CO2 and $\epsilon_a = 0.78$.

The "emissivity" of the upper layer $\epsilon_a$ is chosen to have $T_s = 288$ K

In this model, the emissivity $\epsilon_a$ is put equal to $0.78$ just to have the temperature surface $T_s = 288$ K. This constant has simply no physical sense, it is just to adjust the earth surface. The recent video of Sabine at 10:30 shows that we don't have to consider two or more surfaces, but a continuous set of surfaces that emit radiation (in computations of radiative transfer there are some contradictions and misunderstandings... but never mind).

Why doing multiple layers model when the two layers does not make sense ?

I don't know.

How to compute earth temperature? Heat equation and electromagnetic propagation

Let us focus on what is really solid: electromagnetism and heat equation. I am trying to do so at the moment, I will maybe publish the question to discuss with people from the forum... There are some issues to put in the solar flux inside the heat equation (as a boundary condition or inside the flux divergence?) and some others ... But clearly if you want to lose your time in days of nonsense computations, take a n-layers model and make a ton of assumptions... but this is not physics, this is called speculation.