There is no greenhouse effect: Part 2

Accounting for global temperature

Vincent J. Curtis

28 Sept 23

The following derivation is not original; it has been used by other workers for other purposes.  This derivation is intended to demonstrate the reasoning behind the claim that a global mean temperature can be accounted for entirely by insolation and albedo; no postulated greenhouse effect being necessary.

We begin with constructing a simple model familiar to those knowledgeable of the climate debate.  We will use the Stefan-Boltzmann law to derive a hemispherical mean temperature.  The hemisphere of radius R is assumed to be a blackbody at every point of its surface, and that it is illuminated by a constant light source with parallel rays.  The hemisphere forms a circular disk perpendicular to the incoming radiation, and hence the illumination that each point on the hemisphere receives depends upon the angle made by the vector normal to the surface of the hemisphere at that point with the income ray.

Thus the Stefan Boltzmann law, for any point on the hemisphere, can be written as follows:

                        T(Ɵ) = [(1-α)S_o cos Ɵ/σ]^1/4

where T(Ɵ) is the temperature along the circle on the surface of the hemisphere formed at angle Ɵ from the perpendicular incoming rays; α is the albedo of the hemisphere (assumed to be uniform across the hemisphere); σ is the Stefan-Boltzmann constant, and S_o is the insolation in Wm^-2. No assumption is made about an atmosphere; the whole idea of an atmosphere is neglected.

To obtain a mean temperature of the hemisphere, we need to integrate T(Ɵ) over the entire hemisphere.  An area element of the hemisphere can be expressed in terms of Ɵ as follows:

                        dA = 2πrdS

where r is the radius of the circle at angle Ɵ, and dS is the width of the circular band between r and r + dr.

Expressing in terms of Ɵ:

                        dS = R dƟ  ;  and r = R sin Ɵ

where R is the radius of the hemisphere.

Hence, dA can be expressed in terms of Ɵ as follows:

                        dA = 2πr dS = 2πR² sin Ɵ dƟ

Our Stephan-Boltzmann law is for a circular band around the hemisphere is:

                        σT⁴ = ƐS₀ cos Ɵ,  where Ɛ = (1-α)

Solving for T::

                        T(Ɵ) = (ƐS₀/σ)^1/4 cos^1/4Ɵ

Integrating:

(I’ve had to use a giant S in place of the integration symbol as the blog doesn’t support it.)

Substituting:    T_m = (2πR²)^-1  S (ƐS₀/σ)^1/4 cos^1/4Ɵ (2πR² sin Ɵ) dƟ

Cancelling, and integrating over Ɵ from 0 to π/2, we arrive at the following:

                        T_m = 4/5 * (ƐS₀/σ)^1/4

Hence the mean temperature of the hemisphere depends solely on S₀ and Ɛ, with σ being the Stefan-Boltzmann constant.

We assign the following values:

            Ɛ = 0.7

            S₀ = 1362 Wm^-2

            σ = 5.67 x 10^-8 Wm^-2K^-4

Hence:             T_m = 288K or 15℃.

The computed mean temperature for the simple hemisphere using the accepted figures for insolation and albedo lies within experimental error of the accepted IPCC figure of 16℃ for global average temperature, while the satellite measurement says it’s more like 14.6℃ in the lower troposphere.

Thus the accepted global average temperature can be arrived at using a very simple model of the problem, without reference to an atmosphere or greenhouse effect.  Hence, it is unnecessary to posit a greenhouse effect in order to achieve a global mean temperature of 15℃.  The IPCC model seems to require that the global mean temperature be 255K with a greenhouse effect contributing an addition 33 degrees.  There seems to be something wrong with the IPCC model, or with their computational data.

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It is worth observing that the IPCC global mean energy budget has 340 Wm^-2 as average insolation (1362/4), with 100 Wm^-2 lost due to reflection from clouds, and of that remaining, only 161 Wm^-2 actually is absorbed by the ground.  Yet, somehow, the ground radiates 398 Wm^-2 , 342 Wm^-2 hovers mysteriously in the atmosphere and radiates toward the ground, this being the alleged greenhouse effect; and 239 Wm^-2 radiates to outer space.  Needless to say, the budget doesn’t balance.  The IPCC global model is of a static, flat earth under constant illumination and with all heat transfer mechanisms averaged into a steady state.

Since the 342 Wm^-2 is allegedly in the atmosphere radiating, a striking question arises: why does radiation, which is supposed to be emitted in a random direction, only radiate downward, as indicated in the budget?  Why isn’t half downward, and half up to outer space? A figure of 239 Wm^-2 is given as that which escapes to space.