<em>39HYDRO</em> A Constraint on Convection

Zagoni, Miklos; Zagoni, Miklos

We compare four historic papers related to convection: Schwarzschild's (1906, Nachrichten), Emden's (1913, Sitzungsberichte), its English translation by Bateman (1916, Monthly Weather Review, MWR), and Manabe-Möller's (1961, MWR), from one point of view: how do they represent the net radiation at the surface (R_N) in radiative equilibrium, and the corresponding convection in radiative-convective equilibrium. We prove that while Emden’s work is based, declaratively, on Schwarzschild’s paper, the form he presents the basic equations differs slightly, leading to a further shift of emphasis in its English version, which finally culminates in an omission of a fundamental constraint relationship from the Manabe-Möller (1961) representation. We show that while the relationship in question is explicitly given in standard university textbooks on atmospheric physics and radiation, none of the follow-ups of Manabe’s radiative-convective approach mentions it in journal articles. This difference finally leads to a schism in climate physics: in theory, the relationship is unavoidable, but its application is missing from the practice of computations of convection. We justify the validity of the relationship, both in clear-sky and all-sky, on every available observation-based global mean energy flow datasets such as the two decades of CERES satellite mission, the IPCC global energy budget distributions and the GEWEX energy and water cycle assessment based on 30+ years of research.

In detail: Schwarzschild’s (1906) two-stream approximation gives the following terms for A (upward beam), B (downward beam), E (black-body emission of a layer) and A₀ (representing both absorbed solar and emitted longwave radiation at the top-of-atmosphere):

E = A₀(1+m)/2; A = A₀(2+m)/2; B = A₀m/2, (Schwarzschild 1906, Eq. 11)

where m is ‘Optische masse’, the optical depth. In Emden's treatment, m is the physical mass, k the absorption coefficient:

B = –kγm+B₀; A = –kγm+A₀; E = –kγm+E₀. (Emden 1913, Eq. 31)

with

γ = (B₀ – A₀)/2, E₀ = (B₀ + A₀)/2. (Emden 1913, Eq. 32),

stating that this approach and solution are given by Schwarzschild (1906). But while in Schwarzschild it is evident for the first sight that A – E = A₀/2, it is less obvious in Emden’s formulation — and even less clear in the English translation.

In the latter, the case is complicated further by the Translator’s footnotes (6) and (7), comparing Emden’s method to that of Gold (1909, Proc. Royal Soc.), emphasizing the differences, and thereby suggesting that Emden’s work is based on Gold’s; while in reality Emden’s method (and the two features mentioned) is the same as in Schwarzschild’s. This way, the attention of the Reader is further distracted away from looking at the original Schwarzschild paper. As a final outcome, the Manabe and Möller (1961, MWR) paper does not refer to Schwarzschild, but only to Gold and Emden, and does not include the explicit formula for the net radiation at the surface (connecting it to half of OLR)—and neither of its follow-ups (Manabe and Strickler 1964, Manabe and Wetherald 1967). For example, the latter two compute correctly the size (magnitude) of the temperature jump at the surface (Fig. 4, and Fig. 5., respectively), but the constraint on it – that it equals half of the effective emission – is never mentioned.

But it is there, explicitly, in handbooks like Goody (1964, Eq. 2.115); Houghton (1977, Eq. 2.13); Chamberlain (1978, Eq. 1.2.29 and Fig. 1.4); Goody and Yung (1989, Eq. 2.146); Salby (1996, Eq. 8.67); Visconti (2001, Eq. 3.49); Pierrehumbert (2008, Eq. 4.45); Vardavas and Taylor (2007); Andrews (2010); Ambaum (2021, Eq. 10.56), etc.

The question is, is it true in observations? When we first encountered this question, CERES EBAF Edition 2.8 was available, and we took the data from a presentation of F. Rose (2017) on a science team meeting at Langley; global means (March 2000 - February 2016). Let’s have

R_N = Net radiation at the surface = SFC (SW net + LW net) = SFC (SW down – SW up + LW down – LW up) = TOA LW up/2 , true or not?

Data from Rose et al. (2017), clear-sky:

R_N = 244.06 – 29.74 + 316.27 – 398.40 = 132.19 Wm^-2.

TOA LW up = 265.59 Wm^-2; TOA LW up /2 = 132.795 Wm^-2.

Yes, R_N = TOA LW up/2 – 0.60 Wm^-2. Notice, Earth’ heat uptake was 0.58±0.38 Wm^-2 at that time (Loeb et al. 2012).

The equivalence of R_N = A – E = A₀/2 triggers another question: how much are A and E, separately? As the TOA is defined at the optical depth m = 0, and OLR is released at m = 1, let’s try the surface with m = 2. Substituting it to Schwarzschild (1906, Eq. 11), we get A = 2A₀ and E = 3A₀/2. Using the same dataset, the total absorbed radiation at the surface

R_T = A = SFC (SW net + LW down) = 244.06 – 29.74 + 316.27 = 530.59 Wm^-2; 2A₀ = 2 × 265.59 = 531.18 Wm^-2; the difference is – 0.59 Wm^-2.

Further,

R_T – R_N = E = 398.40; 3A₀/2 = 398.385 Wm^-2, the difference is 0.015 Wm^-2.

These were for clear-sky. To create their all-sky versions is easy, by separating atmospheric LW radiation transfer from the LW effect of clouds. Hence, with all-sky data at both sides,

R_N (all-sky) = SFC (SW net + LW net) (all-sky) = (TOA LW up – LWCRE)/2, and

R_T (all-sky) = SFC (SW net + LW down) (all-sky) = 2 x TOA LW up (all-sky) + LWCRE.

Using the most recent, completed and archived CERES EBAF Edition 4.1 data product covering 264 monthly global means from April 2000 through March 2022, the differences of the individual equations are: -2.33; -2.82; 2.71; and 2.44 Wm^-2, respectively, and the mean bias of the four equations, using four decimal places from the netCDF file is 0.0007 Wm^-2, see Table 1. (Data were obtained from the NASA LaRC CERES ordering tool at https://ceres.larc.nasa.gov/data)

V40 39HYDRO A Constraint on Convection