Schwarzschild's Equation
In this chapter we point out the physical basis of the direct surface-TOA flux relationships in the clear-sky in long-known university textbook radiative transfer equations and prove the validity of those equations in CERES datasets.
Two clear-sky equations
After the pioneering works of Schuster (1905) and Eddington (1905), Schwarzschild's
"On the Equilibrium of the Sun's Atmosphere" (1906) gave the
two-stream (plane-parallel), gray approximation for the transfer of radiation
under local thermodynamic equilibrium (LTE) conditions.
Eq. (11) consists of three terms, A for upward beam, B for downward beam, E for
emission of a layer, A0 for the emerging (outgoing) flux at the
upper boundary, and the tau for the optical depth:
The tau = 1 case gives E = A0, the effective emission (OLR) of the
system with A = 3A0/2 and B = A0/2.
If tau = 2 at the lower boundary (in planetary application: at the surface), A
= 2A0, E = 3A0/2 and B = A0.
Emden realized in 1913 that there is a "jump" (Temperatursprung,
discontinuity) at the surface in radiative equilibrium, since A – E = A0/2;
but in the same sentence he noted that this jump is greatly diminished by
convection and evaporation of water (this way, discovered radiative-convective
equilibrium). [Emden was a member of the Schwarzschild family, by marrying
Klara, sister of Karl.]
This solution is an approximation of the general integral equation, given in
Schwarzschild (1914).
[It is of historical interest that this paper was presented at a meeting
of the Prussian Academy of Sciences in Berlin in 1914 by Einstein in the
absence of the author who served as a soldier in World War I. The meeting was
chaired by Max Planck.]
An elegant deduction of the two-stream Eq. (11) from the integral equation is given by E.A. Milne (1930) On the thermodynamics of the Sun (in Handbook of astrophysics). He notes the equations may be derived from first principles. This solution is called the Eddington approximation (Goody 1964).
The size of discontinuity (that is, the net radiation at the surface) is equal to A0/2.
In other words, the non-radiative (convective) fluxes, the sum of sensible heat and latent heat, are unequivocally connected to half of the effective emission of the system (outgoing longwave radiation, OLR, at the top-of-atmosphere, TOA). This fact of the model is reproduced in standard textbooks like Houghton: The Physics of Atmospheres (1977, Eq. 2.13:
or Chamberlain: Theory of Planetary Atmospheres (1978)
Regarding the greenhouse effect, Houghton continues the derivation:
Here Eq. (2.15), with the optical depth of two (chosen above), χo* = 2, results Bg = 2ϕ/π, that is, A = 2A0 in Schwarzschild's notation.
The question is, are these equations:
Eq. (1) ΔA = A – E = A0/2
Eq. (2) A = 2A0
valid on Radiative Equilibrium (RE) models; and, mainly, are they
valid for the Earth's real Radiative-Convective Equilibrium (RCE)
atmosphere, in the annual global mean?