The heat equation with energy source

We now continue to discuss the heat equation with an energy source. As an illustrative example we consider a rod with length L which is heated by an energy source E at position x = 0 and is maintained at a constant temperature T = 0 at position x = L by some external reservoir. The heat equation inside the rod reads

dU/dT = D*d^2U/dx^2

The reservoir is incorporated as the boundary condition:

U(L) = 0

In order to determine the appropriate boundary condition at position x = 0 we temporarily switch to a discretized mesh in the position coordinate with steplength 1:

Rate of change of energy per unit time = Heat gained by energy source per unit time - Heat lost to adjacent position per unit time

dU(0)/dt = E - D(U(0) - U(1))

Which in the continuum limit becomes:

dU(0)/dt = E + D*dU(0)/dx

As before we now ask for the stationary solution, that is the solution for which dU/dt = 0 everwhere. Inside the rod we have

D*d^2U/dx^2 = 0

with solution U(x) = Bx + C, which without the boundary conditions would  be independent of the diffusivity constant D. Taking into account the boundary condition we arrive at the solution

U(x) =  E/D*(L - x)

And in particular U(0) = E*L/D, which decreases as the diffusivity constant increases.

This could now maybe serve as the first babystep towards a heat equation for our atmosphere (which must also in the end incorprate convection etc etc) if the energy source is reinterpreted as the incoming sunlight. But there is a slight complication here. The incoming sunligt must pass through the upper atmosphere before it reaches the surface of the earth. Is this just "incidental", or would it perhaps be appropriate to include the incoming sunlight as simply a component of the total energy U(x). In that case the model system would be in equilibrium with a constant temperature. So which is the way to go? Well, I myself is not sure about this, but maybe you have some suggestion?...

The heat equation revisited

Let's recall the heat equation in one spatial dimension:

dU/dt = D*d^2U/dx^2

Here U is the heat-content or temperature at a given position and time, and D is the diffussivity constant. In order to completely solve the heat equation one needs to specify boundary conditions which might depend on various energy sources et.c., but we will not be so much concerned with that here. Often one is interested in finding a stationary solution that does not change in time, and for such a solution we have that dU/dt = 0 everywhere. The stationary one-dimensional heat equation is thus D*d^2U/dx^2 = 0.

Now let's try to construct a simple radiation model for an almost ideal gas. Many of the thermodynamic properties of an ideal gas, such as the energy content and pressure, are proportional to temperature so let's assume that we can interchangeably speak about the heat content and temperature. Suppose that the radiation is proportional to the temperature U and also proportional to a parameter A which is a measure of the emmisivity. Let's discretize the position variable with an index n. To start with, also suppose that every layer absorbs all incoming radiation (an unphysical assumption that we will later relax). If we now try to find a stationary solution, that is a solution that does not change in time, then there is no build-up or loss of energy and hence we must have balance between incoming and outgoing radiation. Now take the perspective of the layer at position n

In words:

Heat absorbed from the adjacent layers - Heat lost by radiation = 0

In numbers:

AU(n-1) + AU(n+1) - 2AU(n) = 0

Notice in particular the "back-radiation" term AU(n+1). However, what we have just written is simply the discrete form of the stationary heat equation:

A*d^2U/dx^2 = 0,

now with the diffusivity constant simply given by the emissivity parameter A. Relaxing the assumption that all incoming radiation is absorbed leads in its simplest form to the model I described in the post "A simple radiation model". One can of course take into account all sorts of other circumstances, for example a position dependent thickness of the gas and so on, but very quicly such more elaborate models become analytically intractable and one would probably need to use computers. 

However, the main message is that a model which incorporates "back-radiation" does not necessarily lead to a greenhouse effect.