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?...