What is the net radiative balance between two blackbodies with different temperature?

Question

Context

In trying to understand a crop productivity model, I want to figure out how to derive the equation for the net exchange of longwave radiation between two adjacent blackbodies (soil and atmosphere) over some discrete time step.

In order to calculate the net energy balance, longwave radiation ($L$) is calculated first (and can easily be calculated from first principles), so we can go from:

$$S_\downarrow + L_\downarrow = L_\uparrow+H+\lambda E+G$$

To here:

$$S_\downarrow + (L_\downarrow - L_\uparrow) = H+\lambda E+G$$

where $S$ is solar flux, $L$ is longwave radiation, $\lambda E$ is latent heat and $G$ is conduction.

Problem

This is the equation I am trying to derive:

$$L_{\text{soil}\rightarrow\text{air}}=4\sigma T^3\Delta T$$

Where

• $\sigma$ is the Stefan-Boltzman constant
• $L_{\text{soil}\rightarrow\text{air}}=L_\uparrow - L_\downarrow$ is the net heat flux over time step $\Delta t$
• $\Delta T=T_\text{air}-T_\text{soil}$

I am starting with the Stefan-Boltzman law:

$$L_\downarrow=4\sigma T^4$$

And I recognize the following:

• I am trying to calculate $L_\downarrow-L_\uparrow$ over some $\Delta t$ and
• $dL/dt = 4\sigma T^3$

However, I can't get back to the SB law. I have gotten as far as

$$L_\uparrow-L_\downarrow =\sigma T_\text{soil}^4-\sigma T_\text{air}^4$$

Answer 1

Firstly, I should point out that this is a mistake: $$dL/dt = 4\sigma T^3.$$ $L$ is the flux - it isn't changing in time. Or rather, it does (it's different during the day than during the night, for instance), but in doing this sort of calculation we normally assume it's constant, because the time scale over which it changes is long compared to the flux itself. If the flux was changing by such a large amount, it would mean that the ground was losing heat at a rapidly increasing rate, which would be odd.

All you need to do is note that the ground loses heat to the air at a rate $\sigma T_\text{soil}^4$ (note there is no factor of 4 here) whereas the air loses heat to the ground at a rate $\sigma T_\text{air}^4$. Thus this difference is $$ L_\downarrow = \sigma (T_\text{air}^4 - T_\text{soil}^4). $$

You could stop there, but it's inconvenient to work with those powers of 4, and it's not really necessary because $T_\text{air}$ and $T_\text{soil}$ are quite close in value (they're both around $300\:\mathrm{K}$). So let's let $T_\text{soil} = T$ and $T_\text{air} = T+\Delta T$. Then we have $$ L_\downarrow = \sigma (T+\Delta T)^4 - \sigma T^4 \\ = \sigma (T^4 + 4T^3\Delta T + 6T^2(\Delta T)^2 + 4T(\Delta T)^3 + (\Delta T)^4 \quad) -\sigma T^4. $$ The $T^4$ terms cancel out, and if we assume $\Delta T$ is small then we can ignore the $(\Delta T)^2$ term and all the other higher powers of $\Delta T$. This leaves us with $$ L_\downarrow = 4\sigma T^3\Delta T, $$ which is the expression you were trying to derive.

We could have let $T_\text{air} = T$ and $T_\text{soil} = T-\Delta T$ instead and it would have made no difference. The point is that we can do this trick whenever the temperatures are close enough that they can both be considered as small deviations from a single 'ambient' value.

Answer 2

But the air is the whole troposphere, you cannot choose a layer close to the ground that has the same temperature, and ignore the ice-cold mean temperature of -18C. The atmosphere`s contribution to soil is $σ(T^4air−T^4soil)=-150W/m^2$ as a whole, no matter convection or conduction. Since we are caclulating mean temperatures, we have to use radiated energy, we cannot include other ways of transfer since they are an effect of temperature and not a cause.

And the soil temperature includes everything down to the core, as it is balanced between the internal energy, the emitted energy and incoming solar energy, so just choosing a temperature of air that fits, is just a way to squeeze it into a faulty greenhouse-model. Start from the troposphere mean temperature, because that is what you do with soil. Then you will see that the atmosphere adds nothing, it costs energy.

I think it´s problematic to use flux inside a system, because the real "surface" is the tropopause. The troposphere and solid earth is more easily described as a state inside the system with $J/m^3$ for temperature. Then everything adds up and flux at the earth surface fits, as well as energy is allowed to transfer between layers in a stochastic way by convection etc. But the troposphere is always colder than the surface as a whole and can never be seen as adding energy to the surface.