Backradiation is Unphysical because it is Unstable

To understand that atmospheric "backradiation", the scientific basis of global warming, is an unphysical fictional phenomenon, it is useful to compare with heat conduction, or more generally diffusion.

The basic mathematical model of heat conduction/diffusion takes the following form with Q(x) rate of heat flow at point x (in one dimension for simplicity) and U(x) temperature

• Q(x) = - kappa (U(x+h)-U(x))/h (Fourier's Law)
• (Q(x) - Q(x-h))/h = f(x) (1st Law of Thermodynamics = energy balance)

where kappa is a positive coefficient, f(x) a heat source and h is a small (computational or physical) mesh size.

Setting kappa = 1 and f = 0 for simplicity (imposing some inhomogeneous boundary conditions), we obtain the following equation for U:

• (-U(x+h) + 2U(x) - U(x-h))/hh = 0. (1)

Suppose now that we forget the derivation of this formula and write it as:

• 2U(x)/hh = (U(x+h) + U(x-h))/hh (2)

with the following interpretation

• (U(x+h) + U(x-h))/hh = "incoming temperature" from x+h and x-h to x
• 2U(x)/hh = "outgoing temperature" from x to x+h and x-h.
• "incoming temperature" = "outgoing temperature".

Then "incoming temperature" would be like "backradiation". So far, so good.

But what is then the physical meaning of "incoming/outgoing temperature"? Is there really a physical "flow of temperature"? Is "flow of temperature" the same as "flow of heat"?

No! To understand this we rewrite the equation in computational form suitable for

explicit iteration from U(n,x) to U(n+1,x) with U(0,x) given:

• U(n+1,x) = U(n,x) - alpha (-U(n,x+h) + 2U(n,x) - U(n,x-h))/hh (3)

where alpha is a positive parameter to choose. This corresponds to time-stepping towards an equilibrium solution.

With alpha = 1 and U(n+1,x) = U(n,x) we then get back (1) along with the interpretation (2). But alpha =1 is too big for convergence of the iteration: Convergence requiers that that alpha ~ hh; more specifically the iteration convergence if alpha is smaller than hh/4, say alpha = hh/8. In this case (3) takes the form

• U(n+1,x) = U(n,x) - 1/8 (-U(n,x+h) + 2U(n,x) - U(n,x-h)) (4)

or

• U(n+1,x) = U(n,x) - h/8 (Q(n,x) - Q(n,x-h)). (5)

We can now interprete the equations (4) or (5) as expressing a form of physics effectively

establishing an equilibrium solution by iteration or time-stepping. In particular we can

interprete (5) as a (mild) relaxation decreasing U(n+1,x) as compared to U(n,x) if Q(n,x) is bigger than Q(n,x-1), that is if more flow is going out than in at x.

We next compare with (2):

• 2U(x)/hh =(U(x+h)+ U(x-h))/hh

with "out/inflow of temperature" 2U(x)/hh and (U(x+h)+ U(x-h))/hh. We now come to the punchline: There is no physics behind this massive "flow of temperature" because as h gets smaller and smaller, the "flow of temperature" increases without bound.

We can thus speak about flow of heat, but not "flow of temperature" with enormous amounts

of temperature flowing in and out (as h gets small), because this would corresponds to an unstable unobservable unphysical process.

In short: heat flows from hot to cold but there is no physics corresponding to a flow from cold to hot analagous to "backradiation". Heat can radiate from a hot to a cold body, but not from a cold to a hot. In other words:

• Backradiation corresponds to an unstable unobservable unphysical process.

You can make an analogy in economics with massive taxes (going to infinity) from the people and massive benefits (going to infinity) back to the people. Great idea, but unfortunately unstable!