Figure 3 Revised earth energy budget diagram 2009
Tuesday, February 21, 2012
Earth energy budget is of fundamental importance for understanding what factors determine the climate and climate trends of the earth-atmosphere system. Figure 1 shows a diagram in the IPCC fourth report, AR4, for estimate of the Earth’s annual and global mean energy balance . This diagram is sometimes known as the K & T diagram.
Figure 1 Earth energy budget diagram of IPCC report AR4 2007
The diagram basically shows two giant energy trees: one is on the left hand side describing the incoming solar radiation energy and its components; and the other on the right hand side portraying the outgoing radiation and its components with respect to the earth-atmosphere system. In the middle are two lovely bunches of grass showing energy transfers due to evaporation and other thermal events.
This diagram has been widely discussed by, in particular, sceptical scientists. This article, however, will point out that there is a fundamental flaw in the diagram due to misuse of the Stefan-Boltzmann equation.
Examine first how 390 W/m2 surface radiation was obtained. The original paper of the diagram  states “for the outgoing fluxes, the surface infrared radiation of 390 W/m2 corresponds to a blackbody emission at 15°C.” Assuming the surface radiation being p, the statement can be best translated in a mathematical form:
(1) p = σT4 = 5.67 x 10-8 * (273.15 + 15)4 = 390.89 @ 390 (W/m2)
Where, σ is the Stefan-Boltzmann constant equal to 5.67 x 10-8 W/m2K4.
However, the earth surface is a gray rather than a black-body. For an object with any surface, the Stefan-Boltzmann equation should be multiplied by a coefficient called emissivity, ε:
(2) p = ε σT4 = ε * 5.67 x 10-8 * (273.15 + 15)4 @ ε * 390 (W/m2)
This is to say if ε is not 1, but 0.99, 0.95, 0.9 or 0.8, there will be an error for the surface radiation of 3.9, 19.5, 39 or 78 W/m2 respectively. Obviously, as long as the emissivity of the ground surface of the Earth, ε, is not accurately determined, the figure for the surface radiation is intrinsically uncertain; it is not a minor issue at all.
The error does not stop here. There is a further error that easily escapes from scrutiny, i.e. what temperature T stands for in the Stefan-Boltzmann equation - is 15°C correct?
It is true that the Earth’s mean near-surface air temperature, as measured by global weather stations using thermometers, is around 15°C. As such it is largely of the temperature of nitrogen and oxygen gases that consist of 99% dry air. Thus this temperature is not what one should use in the Stefan-Boltzmann equation. This is because nitrogen and oxygen are non-radiative (literally ε = 0 for transparent and white bodies); 0 multiplying anything leads to 0.
The T represents the temperature of the ground surface of the Earth, which is measured by infrared spectroscopy. Figure 2 shows a ModTran spectrum of the Earth outgoing radiation detected at 20 km high altitude with detector facing down under standard
air conditions. The spectrum was fitted with curves of the Planck’s law. The top envelop 285.04 K (11.89°C) is the temperature of the earth’s ground surface. One obtains: US
(3) p = ε σT4 = ε * 5.67 x 10-8 * 285.044 = ε * 374.31 (W/m2)
The above shows what is the proper methodology to apply the Stefan-Boltzmann law to quantify the surface radiation, though how to determine the emissivity and the mean ground surface temperature may need further investigations.
Figure 2 ModTran earth outgoing radiation spectrum for detector facing down at 20 km altitude under standard
air conditions. US
A number of values shown in the IPCC AR4 Earth Energy Budget have been revised in a paper published in Bulletin of the American Meteorological Society in 2009 [3, 4]. With respect to the 390 W/m2 surface radiation, effect of temperature variation from mean value was taken into consideration. This leads to a replacement of the 390 W/m2 with a new value of 396 W/m2. However, setting ε to 1 remains unchanged.
This revision may be an improvement, but is minor as compared with the uncertainty of emissivity and temperature of the ground surface of the Earth. As this figure is uncertain, many of the other figures on the outgoing radiation tree would be in doubt as well.
In summary, there are two technical errors in calculation of the surface radiation: 1) false assumption that the ground surface of the Earth is a black-body surface; 2) confusion of the near surface air temperature with the ground surface temperature in application of the Stefan-Boltzmann law. As long as ε can not be determined, the outgoing radiation from the ground surface of the Earth is not certain, casting doubts over the value of the diagram. (By J. Cao)
Figure 3 Revised earth energy budget diagram 2009
 Le Treut, H. et al. 2007: Historical Overview of Climate Change. In: Climate Change 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change [Solomon, S., D. Qin, M. Manning, Z. Chen, M. Marquis, K.B. Averyt, M. Tignor and H.L. Miller (eds.)].
Cambridge University Press, Cambridge, United Kingdom and . New York, NY, USA
 J.T. Kiehl and K.E. Trenberth; “Earth’s Annual Global Mean Energy Budget,” Bulletin of the American Meteorological Society, 78(2) pp197-208 (February 1997)
 K. E. Trenberth, J. T. Fasullo, and J. Kiehl, 2009: “Earth’s Global Energy Budget,” Bulletin of the American Meteorological Society, pp311-324 (March 2009)
 K. E. Trenberth and J. T. Fasullo; “Tracking Earth’s Energy: From El Nin˜o to Global Warming,” Surv. Geophys., DOI 10.1007/s10712-011-9150-2 (October 2011)