All-sky versions and the four equations

Creating the all-sky versions

 In this chapter we create the all-sky versions of our clear-sky surface-TOA constraint relations and control them on CERES data.

“… establishing direct relationships between TOA radiative fluxes and surface fluxes and validating these
parametrizations using actual surface radiative flux and TOA flux measurements” (Wielicki et al. BAMS, 1996)

Eq. (1)     SFC Net = ΔA – E = A0/2     (clear-sky, net)

Eq. (2)     SFC Tot = A = 2A0   (clear-sky, total)

Separate atmospheric radiation transfer from the longwave cloud effect (LWCRE), use all-sky data in the left-hand side, all-sky data in A0, and include LWCRE:

Eq. (3)    SFC Net  = A – E = (A0 – L)/2  (all-sky, net, incl LWCRE)

Eq. (4)    SFC Tot  = A = 2A0 + L  (all-sky, total incl LWCRE)

As a first check, let us use the data from the AMS Meteorological Monograph (2019, Chapter 4) (with its reference) to control the all-sky equations (3) and (4).

 

"Collocated AMSR-E data reveal that these sources of surface heating are offset by 81 ± 4 Wm-2 of latent heat transfer from the evaporation of water (primarily from the oceans), 25 ± 4 Wm-2 of sensible heat transfer, and 399 ±  5Wm-2 of cooling by thermal emission from the surface (L’Ecuyer et al. 2015)."  Taking Outgoing LW = 240 Wm-2 and LWCRE = 28 Wm-2 from the same study, we get

Eq. (3)    SFC Net = 81 + 25  = (240  – 28)/2    (±0)

Eq. (4)    SFC Tot = 81 + 25 + 399 = 2 × 240 + 28  (-3)

Equation (3) is exact (zero difference), equation  (4) has -3 Wm-2 difference on that dataset.

This accuracy is insipring enought to continue their examination.

On CERES dataset, using their notation:


Verifying the four equations on CERES EBAF Ed4.1 data:


The bias of the individual equations is less than ±3 Wm-2.
The mean bias of the four equations together is indescernible from zero.

This extreme accuracy rises the question: Don't these equations represent an arithmetic identity? Or are they 'built-in' the CERES data production process?

The answer to both questions is: No.

There are large inter-annual fluctuations  [-0.8; + 1.4] Wm-2:

The mean bias of the four equations approaches zero and reaches it only after taking 17 years into the averaging and it remains there:

The mean deviation of the four equations from zero has no bias: it is a stochastic process.

EBAF Edition 4.1 is a completed, archived data product. 

For the record, it should be stated in the CERES Mission Final Report:

CERES EBAF Ed4.1 verified the four equations on its 22 years of operation (April 2000 — March 2022) with a mean difference of 0.0007 Wm-2.

The data in MS Excel .xlsx file format can be downloaded here. 

Earth's energy flow system follows the four equations on multi-decadal time scales exactly.

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