Stefan-Boltzmann, Kevin Trenberth, and Other Paradoxes

DRAFT 6, 7/Apr/12.    by Bob Fernley-Jones AKA Bob_FJ 

Here is the Trenberth et al energy budget diagram as 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, (AKA; EMR and a number of other descriptions of conflict between applied scientists and some academic physicists).  EMR is a form of energy that is sometimes confused with HEAT.  It will be explained later, that the 396 W/m^2 surface radiation depicted above has very different behaviour to HEAT.  Furthermore, temperature change in matter can only substantially take place when there is a HEAT transfer, or work-done, 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 issue 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 small flat surface that is equally radiated in all directions, (is isotropic/ hemispherical).  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 edge, the radiation towards Earth is tangential from its apparent surface, not vertical.  It is not a perfect demonstration because of a phenomenon called limb darkening, due to the Sun not having a definable surface, but actually plasma with opacity effects.  However, it is generally not apparent to the eye and the normally observed (shielded) eyeball observation is arguably adequate for demonstration here.

3) Whilst reportedly the original Stefan lab test was for a small flat body radiating into a concave 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.   This can be demonstrated by car driving at night, when approaching 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 greatest concentration  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 elemental (thin) layer of air over a small region, and time interval, and with little turbulence, the temperature in the layer can be treated as constant.  Yet, the most concentrated radiation within the layer is horizontal in all directions, but with a net heat transfer of zero.  Where the radiation is not perfectly horizontal, adjacent layers will provide neutralizing interception of it.

A more concise way of looking at it is with vectors, which put simply is a mathematical method for analysing parameters that possess directional information.  Figure 4, takes a random ray of EMR (C) at a modestly shallow angle, and analyses its vertical and horizontal vector components.  The length of each vector is proportional to the power of the ray, in that direction, such that A + B = C.  Of course this figure is only in 2D, and there are countless multi-directional rays in 3D, with the majority approaching the horizontal, through 360 planar degrees, where the vertical components also approach zero.

6) Trenberth’s figure 1 gives that 65% of the HEAT loss from the surface is via thermals and evapo-transpiration.  What is not elaborated by Trenberth is that as a consequence of this upward HEAT transfer, additional infrared radiation takes place in the air column by virtue of it being warmed.  This progresses through spherical emission and absorption, but as the air progressively thins upwards, absorption slows, and that radiation ultimately escapes directly to space.  Thus, the infrared radiation observable from space has complex sources from various altitudes, but has no labels to say where it came from, making some of the attributions arguably “difficult”.

DISCUSSION;  So what to make of this?

The initial isotropic S-B surface emission, (Trenberth’s global 396 W/m2), would in part be absorbed by the greenhouse gases instantaneously near the surface. (ignoring some escaping directly to space through the so-called “atmospheric window”).  However, a large proportion of the initial S-B 396 surface emission would be continuously lateral, at the Trenberth imposed constant conditions, without any heat transfer, and its horizontal vectors CANNOT be part of the alleged 396 vertical flux, because they are outside of the vertical field of view.  (and neither can they be seen from space laterally, because of absorption, so a spherical integration cannot be applied, whilst yes, it would otherwise be valid with a totally transparent atmosphere! )

After the initial atmospheric absorptions, the S-B law, which applied initially to the surface, no longer applies to the air above, which has no definable surface. (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 generally varies globally between around ~0 to ~4%.  (compared with CO2 at a somewhat 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 continuously possesses a great deal of lateral radiation that is partly driven by the S-B hemispherical 396, but cannot therefore be part of the vertical 396 claimed in Figure 1.


The vertical radiative flux portrayed by Trenberth et al of 396 W/m^2 ascending from the surface to a high cloud level is not supported by first principle considerations. (including the GHE)   The initial S-B 396 W/m^2 is by definition hemispherically isotropic as also is its loosely ascending spherical progeny also isotropic, with always prevailing horizontal vector components that are not in the field of view of the vertical.  The remaining vertical components of EMR from that source are thus less than 396 W/m^2.

It is apparent that according to Trenberth, HEAT loss from the surface via convective/evaporative processes is more substantial than from surface radiation losses.  It may be that there is a total resultant of similar order to 396 W/m^2, but that is NOT the S-B radiative process described by Trenberth.

This assumes that the consensus theory on the GHE is correct, unless that too contains a paradox.



I Seek your advice

In figure 5 below, note that the NIMBUS 4 satellite data on the left must be for ALL sources of radiation as seen from space, in this case, at some point over the tropical Pacific.  The total emissions, amount to the integrated area under the curve, which unfortunately is not given.  However, for comparison purposes, a MODTRAN calculator, looking down from 100 Km gives some interesting information for the figure, which is further elaborated in the tables below.  Unfortunately the calculator does not give global data or average cloud/sky conditions, so we have apples and pears to compare, not only with Nimbus, but also with Trenberth.  However, they all seem to be of somewhat similar order, and see the additional tabulations.

Compare MODTRAN & “Trenberth”, looking down from 2 altitudes, plus Surface Temperature
Location Kelvin 10 metres 100 Km. (Centigrade)
Tropical Atmosphere 300K 419 W/m^2 288 W/m^2 (27C)
Mid-latitude Summer 294K 391 W/m^2 280 W/m^2 (21C)
Mid-latitude Winter 272K 291 W/m^2 228 W/m^2 (-1C)
Sub-Arctic Winter 257K 235 W/m^2 196 W/m^2 (-16C)
Trenberth Global 288K ? 396  W/m^2 239 W/m^2 (15C ?)
Compare MODTRAN & “Trenberth”, looking UP from 4 altitudes:  W/m^2
Location From 10 m From 2 Km From 4Km From 6Km
Tropical Atmosphere 348 252 181 125
Mid-latitude Summer 310 232 168 118
Mid-latitude Winter 206 161 115 75
Sub-Arctic Winter 162 132 94 58
Trenberth Global 333     Shown as coming from  high cloud area  (= BS according to MODTRAN)
Be the first to like this post.

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).

3 Responses to “Stefan-Boltzmann, Kevin Trenberth, and Other Paradoxes”

  1. Bob, This is not my field of expertise ( nor many others it would seem) but from a discussion over at Tallbloke’s site some time ago I gathered that BB radiation from a flat surface was indeed in all hemispherical directions but the intensity fell of with the cosine of the angle to the normal. If so, the intensity would be 100% normal to the plane and 0% parallel to he plane. What this would do to your argument I know not and I cannot be sure I have got it right. But this was in connection with a paradox set up by Hans Jelbring which was apparently solved by a blogger using this argument. If you wish you can email me a response to cosserat at

    David Cosserat

  2. Hi David,
    Thanks for your interest.
    I think the seeming confusion is from Lambert’s Cosine Law which is to do with reflection brightness which depends on the incoming direction of the light. At normal, (90 degrees), it is at its maximum and when purely tangentially is at nil. (a different situation from a surface emitting EMR)

    Regards, Bob_FJ

    BTW, how did you find my website……it’s not intended as a blog but for drafting articles in WordPress format for publication elsewhere. Thanks anyway.

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

%d bloggers like this: