Does the Trenberth et al “Earth’s Energy Budget Diagram” Contain a Paradox?

DRAFT 5,  5/Oct/2011

Here we go; extracted from their 2009 paper, it being an update of that in the IPCC report of 2007 (& also 2001):

The unusual aspect of this diagram is that instead of directly showing radiative Heat Transfer  from the surface, it gives their depiction of the greenhouse effect in terms of radiation flux or Electro-Magnetic Radiation, (EMR [1]).  EMR is a form of energy that is sometimes confused with Thermal Energy or HEAT [1].  It will be explained later, that the 396 W/m2 surface radiation depicted above has very different behaviour to HEAT.  Furthermore, temperature change in matter can only take place when there is a HEAT transfer, regardless of how much EMR is whizzing around in the atmosphere.

A more popular schematic from various divisions around NASA and Wikipedia etc, is next, and it avoids the difficulty above: 

Figure 2                                                     NASA

Returning to the Trenberth et al paper, (link is in line 1 above), they give that the 396 W/m2 of EMR emitted from the surface in Fig.1 is calculated primarily by using the Stefan–Boltzmann law, and global year average conditions.  Putting aside a few lesser but rather significant issues therein, it is useful to know that:

1) The Stefan-Boltzmann law (S-B) describes the total emission from a flat surface that is equally radiated in all directions hemispherically [2].  (Stefan found this via experimental measurement, and later his student Boltzmann derived it mathematically). 

2) The validity of equally distributed hemispherical EMR is demonstrated quite well by observing the Sun. (with eye protection).  It appears to be a flat disc of uniform brightness, but of course it is a sphere, and at its outer circumference, (AKA limb), the radiation towards Earth is tangential from its apparent surface, not vertical.  It is not a perfect demo’ because of a phenomenon called limb darkening, due to the Sun not having a definable surface, but plasma with opacity effects.  However, it is generally not apparent to the eye and is adequate for the purpose here.

3) Whilst reportedly the original Stefan lab test was for a small flat body radiating into a hemisphere, its conclusions can be extended to larger areas by simple addition of many small flat bodies of collectively flat configuration, because of the ability of EMR waves to pass through each other without interference.   This can be demonstrated by night driving, when approaching car headlights do not change in brightness as a consequence of your own headlights opposing them.  (not to be confused with any dazzling effects and fringe illumination)

4) My sketch below demonstrates how radiation is at its most intense in the lateral directions.  It applies to both the initial S-B hemispherical surface radiation and to subsequent spherical radiation from the atmosphere itself. 

 5) Expanding on the text in Figure 3:  Air temperature decreases with altitude, (with lapse rate), but if we take any thin layer of air over a small region, and time interval, and with little turbulance, the temperature in the layer can be treated as constant.  Yet, the most intense radiation within the layer is horizontal in all directions, with net heat transfer of zero.  Where the radiation is not perfectly horizontal, adjacent layers will provide interception of it.  The horizontal vectors mostly exceed the vertical.  Another visualization in crude 2D is in Figure 4: 

Figure 4                   Sideways radiation

6) Figure 1 gives that there is 63 HEAT loss (W/m2) from the surface via radiation, 17 from thermals, and 80 from evapotranspiration.  What is not elaborated is that as a consequence of the latter two HEAT transfers, (65% of the total), additional infrared radiation takes place in the air column by virtue of warming it.  This initially starts as spherical emission and absorption, but as the air progressively thins upwards, absorption slows and ceases, and radiation then escapes directly to space.  Thus, the infrared radiation observable from space has complex sources, but has no labels to say where it came from, making some of the attributions “difficult”.

DISCUSSION;  So what to make of this?

The initial global average S-B surface emission, (Trenberth’s global 396 W/m2), would largely be absorbed by the greenhouse gases instantaneously near the surface. (ignoring some escaping directly to space).  However, a large proportion of the S-B 396 would be continuous and lateral, at the Trenberth imposed uniform conditions, without any heat transfer, and it CANNOT be part of the alleged 396 vertical flux. 

After that, the S-B law, which applied initially to the surface, no longer applies to the air above. (although some  clouds are sometimes considered to be not far off from a black body).  Most of the air’s initial absorption/emission is close to the surface, but the vertical distribution range is large, because of considerable variation in the photon free path lengths.  These vary with many factors, a big one being the regional and more powerful GHG water vapour level range which varies globally between around ~0 to ~ 4%.  (compared with CO2 at a somewhat comparatively constant ~0.04%).  The total complexities in attempting to model/calculate what may be happening are huge and beyond the scope of this here, but the point is that every layer of air at ascending altitudes continually possesses intense lateral radiation that is part of the S-B hemispherical 396, and therefore NOT part of the vertical 396 claimed in Figure 1.


The vertical or normal radiative flux of 396 W/m2 ascending from the surface to a high cloud level portrayed by Trenberth et al is not supported by first principle considerations of thermodynamics and elementary quantum theory.

Perhaps there is an alternative hypothesis that heat loss from surface radiation is converted from hemispherical to vertical transport via convective/evaporative heat loss processes, as in point 6) above, and which might amount to a significant outcome, but this is NOT the S-B radiative process described by Trenberth.


[1] There are conflicts in terminology between physics and engineering, (quantum theory and thermodynamics), but since the latter field is closer to common understandings, I lean towards that terminology.  For instance if we heat a pot on a stove, we say that it gets hotter, but a physicist may prefer to argue no; it has obtained more thermal energy, and HEAT is a transient condition.    

Surface radiation is also known as infrared light, or EMR, or Electro-Magnetic Radiation, or upwelling radiation, or infrared radiation, or long wave, and more, and is sometimes confused as thermal energy or heat radiation.  Some climatologists refer to thermal radiation as being restricted to the infrared, whereas it is a proven fact that solar visible light is also thermal, and more strongly so.

[2] The mathematical derivation of the S-B law is from integration of emission levels over the Planck frequency distribution together with integration of the solid angles.  {See link in item 1)}.  This does not result in a change of directionality from hemispherical to vertical as suggested by one commenter on another WUWT thread touching on this.  (start from here, in over 900 comments if you have stamina).   Interestingly, the S-B law that was initially determined for a small emitting flat surface can be extended to larger areas of flat configuration by simple addition of the areas.  This is because of the ability of light waves to pass through each other without interference.  (Or, in other words, a hemispherical element of radiation in the centre of a large non-concave area is not hindered by those towards the periphery). 

Another commenter (a physicist) suggested that Gauss’ law enabled the horizontal vectors of EMR to be integrated and changed in direction to vertical at the surface, and that in a transparent atmosphere, putting aside the planet’s surface curvature; it would remain constant and vertical to all heights.  Well, in a word: No.  Anyway, the surface considerations in the S-B law do not apply to the emissions from air close to the surface and above, which instantaneously annihilates a lot of the initial S-B surface emission.


About Bob Fernley-Jones

I'm a retired mechanical engineer, and I guess that because in my science, any bad assumptions can get people killed, I have an abhorrence of many things that are perpetrated by academics in some areas of science. In the case of so-called climate science, the culture and bias in some media is also repugnant to me. I'm hoping that the ABC will improve its self regulating policies and culture to eliminate bias, and this website is under development towards that end. (if necessary).

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