Integer
solution
Here we show that the four equations have an evident geometric representation, defining their integer ratio solution.
There is no "groundtruth " for the radiation budget
(Barkstrom et al .1989)
Since
the equations are exact, we expect that their solution will also be exact.
The integer ratios for the clear-sky fluxes were given by this geometry:
We now multiply the unit by ten (since it was not specified yet):
and take also into consideration the all-sky energy flow system, which requires
a preparation also in the clear-sky fluxes:
Here L = 1
in red bold typeface represents LWCRE = UNIT ONE.
If OLR is 10
in the clear-sky, then it must be 9 in the all-sky;
and if DLR is 12 in the clear-sky, it must be 13 in the all-sky.
Clear-sky equations (1) and (2), and all-sky equation (3) and (3) are
satisfied.
The whole system as integer multiples of unit one is given below:
What we have to do is to calibrate the unit to observations.
Using CERES EBAF Edition 4.2 data, 22 full running years, April 2000 -
March 2022,
The best fit of the unit is 1 = 26.68 ± 0.01 Wm-2.
Table 2-1 and Table 4-1, similar to those in CERES EBAFEd4.1 Data Quality
Summary, give the clear-sky and all-sky fluxes, theory and observation:
EBAF Edition 4.1 (264 monthly means, April 2000 – March 2022), completed, archived, in MS Excel .xlsx format can be downloaded here.
EBAF Edition 4.2 (276 monthly means, July 2000 — June 2023) with integer positions, in MS Excel .xlsx format can be downloaded here.
Using the data again from the reference of AMS Meteorological Monograph (2019, Chapter 4), "81 ± 4 Wm-2 of latent heat transfer from the evaporation of water, 25 ± 4 Wm-2 of sensible heat transfer, and 399 ± 5 Wm-2 of cooling by thermal emission from the surface (L’Ecuyeret al. 2015)",.
Latent heat = 81 Wm-2 = 3 × 26.68 Wm-2 = 3 units (0.96 Wm-2 difference)
Sensible heat = 25 Wm-2 = 1 unit (-1.68 Wm-2 difference)
Surface thermal emission = 399 Wm-2 = 15 × 26.68 Wm-2 = 15 units (-1.04 Wm-2 difference),
each fits far within the stated range of uncertainty.
In the next page we offer a geometric deduction of this system from the
simplest greenhouse model.