A seminal paper from Raval and Ramanathan (1989)1The 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.
Table of Contents
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
.”2For 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)
Initial Attempt
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)
Correspondence #1
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.
However, R&R indicated that is a “weighted average,” so perhaps we haven’t yet found the correspondence that they intended.
Correspondence #2
TBD
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
Second Attempt
I have found a source which derives an equation similar to the one offered by Raval and Ramanathan: Liou, K. N.. An Introduction to Atmospheric Radiation (International Geophysics, Volume 84) (Section 4.2). Elsevier Science. Kindle Edition. In particular, equation (4.2.10a) in that book is (with adjusted notation):
(20)
where is frequency (or wavenumber). However, there seem to be typos in this equation. I suspect what was meant was something like the following:
(21)
But, I’ll need to work through their derivation carefully to be certain…
The next step would be a change of variables, .
- 1The 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.
- 2For this, R&R cite Manabe & Wetherald (1967). However, I don’t see the equation in M&W (1967).