A seminal paper from Raval and Ramanathan (1989)^{1}The original paper was published in *Nature* in 1989, but is behind a paywall. Fortunately, it was reprinted in a conference proceeding soon after; that’s the link I’ve offered. defines the normalized greenhouse effect, , and indicates that, in the absence of lapse rate or water vapor feedbacks, wouldn’t be expected to change as the surface temperature increases. In this analysis, I investigate their mathematical analysis to try to understand what if any assumptions are involved.

## Their key equation

R&R present the following radiative transfer equation (TRE):

(1)

where is the pressure divided by surface pressure; the blackbody emission is ; and the effective absorptivity is the “integral of the monochromatic absorptivity weighted with , and normalized by .”^{2}For this, R&R cite Manabe & Wetherald (1967). However, I don’t see the equation in M&W (1967). “Thus is the absorptivity between the TOA () and the pressure level .”

Let’s try transforming the prior integral using integration by parts:

(2)

(3)

## Schwarzchild’s equation of radiative transfer

Let’s try to see how the above relates to Scharzchild’s equation of radiative transfer (which governs propagation of thermal radiation in the absence of scattering):

(4)

where is distance in a particular direction and is the wavelength-dependent absorption coefficient. This equation has the solution

(5)

where is the transmittance between and ,

(6)

If we define , then

(7)

Thus integration by parts leads to:

(8)

Applying the definition of leads to:

(9)

If we assume the surface air temperature matches the surface temperature and the surface is a blackbody, this becomes:

(10)

Now let’s change this to a more useful coordinate, by setting . This leads to:

(11)

(12)

where

(13)

Now let’s change variables again, to use where is pressure. This leads to:

(14)

where and

(15)

(16)

To get from intensity to flux, we need to integrate over all directions (within the upward hemisphere of solid angles). This involves an integration of the form where . This leads to:

(17)

where

(18)

If we define , , and compute , and this becomes

(19)

We can integrate this over wavelength, , and obtain R&R’s version of the RTE, but only if we assume that the absorption coefficient, is independent of wavelength, i.e., we have a “gray” atmosphere.

## Conclusions

Thus, the result that will not change as varies depends on these assumptions:

- Negligible scattering (as is implicit in Schwarzchild equation)
- Fixed lapse rate (as R&R name)
- Fixed atmospheric composition (including absolute humidity)
- Temperature varies linearly with altitude and pressure varies exponentially with altitude
- Surface is a blackbody with a temperature that matches the surface atmospheric temperature
- The atmospheric absorption/emission is “gray”, i.e., independent of wavelength