*by Kyoji Kimoto *(e-mail kyoji@mirane.co.jp)

**1. Climate sensitivity from the energy budget of the earth**

Figure1 is a diagram of global energy budget of the earth based on the latest study [Trenberth et al., 2009]. From the figure, natural greenhouse energy is calculated as follows to produce natural greenhouse effect of 34K.

Natural greenhouse energy:

Eb-Es = 333W/m2-78W/m2=255W/m2 (1)

Then, climate sensitivity factor is obtained as follows including climate feedbacks.

Climate sensitivity factor:

34K/255W/m2 = 0.13K/(W/m2) (2)

Since climate sensitivity is a perturbation of natural greenhouse effect due to CO2 increase, it is calculated as follows utilizing the radiative forcing of 3.7W/m2 for CO2 doubling cited in IPCC TAR.

Climate sensitivity : 0.13K/(W/m2)x3.7W/m2 = 0.5K (3)

Based on the data analysis of the Pinatubo eruption, Douglass et al. found that climate sensitivity factor is 0.22K/(W/m2) [Douglass et al.,2006]. Then, climate sensitivity is calculated as follows with the radiative forcing of 3.7W/m2 for CO2 doubling.

Climate sensitivity from Pinatubo eruption = 0.22K/(W/m2) x 3.7W/m2 = 0.8K (4)

This observational result is coincident with the climate sensitivity of 0.5K obtained from the energy budget of the earth by Eq.(3).

In contrast, IPCC claims that climate sensitivity is 3K with the range of 2~4.5K based on the GCMs simulation, which is 6 times larger than that obtained from the global energy budget of the earth. Theoretical analysis will be needed to elucidate a big difference in these climate sensitivities between model study and experimental data.

**2. Climate sensitivity based on IPCC’s theory**

Climate sensitivity is calculated with Eq.(5) utilizing the concept of Plank response and feedback parameters [Bony et al., 2006, Soden et al., 2006]. It includes all feedbacks, whereas Planck response includes surface temperature feedback but not lapse rate, water vapor, albedo and cloud feedbacks.

Climate sensitivity = Planck response *0/(0+lr+wv+a+c) (5)

Where,

Planck response = -3.7(W/m2)/0

0 = Planck feedback parameter = -4OLR/Ts

lr = lapse rate feedback parameter

wv = water vapor feedback parameter

Ts = surface temperature

OLR = Outgoing Long wave Radiation

In the literature, three groups of Planck feedback parameter 0 are found as shown in Table 1 together with Planck response factor [Kimoto, 2009].

Table 2 shows the comparison of Group A, Group B, and Group C in terms of Planck response and climate sensitivity calculated with averaged feedback parameters for IPCC AR4 obtained from 14GCMs simulation [Soden et al., 2006]. Table 2 also shows the following test results as to Ts and OLR of each group.

Test 1: Is the combination of Ts and OLR accordance with Stefan-Boltzmann law?

Test2: Is Ts surface temperature?

**Figure 1. Global energy budget of the earth (adapted from Trenberth et al. 2009)**

Er：long wave radiation

Eb：long wave back radiation

Ee：evaporation

Es：short wave absorbed by atmosphere

Et：thermal conduction

OLR: Outgoing Long wave Radiation

**Table 1. Three groups of L0 and Planck response factor [Kimoto, 2009].**

**Table 2. Planck response, climate sensitivity and two Test results.**

Climate sensitivity of Group A including IPCC is 3K, which is 6 times larger than climate sensitivity from the global energy budget of the earth. Furthermore, it is not in accordance with Stefan-Boltzmann law as to the combination of Ts and OLR.

Group B gives climate sensitivity of 2K. It is 4 times larger than the basis of 0.5K , and Ts is not the surface temperature.

Group C’s climate sensitivity is 0.75K, which is close to 0.5K from the global energy budget. Group C satisfies Stefan-Boltzmann law with the surface temperature Ts and the surface upward energy flow as OLR [Kimoto, 2009].

Based on the above arguments, Group C is the best choice of Ts and OLR, whereas Group A and Group B are theoretically failed in their combination of Ts and OLR.

Cess began the mathematical procedure of Group A , which has been followed by many researchers including IPCC AR4 as shown in Table1 [Kimoto 2009]. Based on the experimental data for OLR as a function of surface temperature and surface albedo measured by satellites, Cess expressed OLR with the modified Stefan-Boltzmann equation as follows [Cess, 1976]

OLR=Eeff * Sigma * Ts^4 (6)

Where,

Eeff = the effective emissivity of the surface-atmosphere system

Sigma = Stefan-Boltzmann constant

Cess obtained Planck feedback parameter 0 by differentiating Eq.(2) with the assumption that Eeff is a constant, as follows [Cess, 1976].

0 =-dOLR/dTs=-4EeffTs**3=-4OLR/Ts (7)

In Eq.(7), Cess took Ts=288K and Eeff=0.6 to obtain OLR=233W/m2, which gives the following L0 and Planck response factor [Cess, 1976].

0 =-4×233/288(W/m2)/K=-3.3(W/m2)/K

Planck response factor -1/L0 = 0.3K/(W/m2)

However, Cess’s procedure is a mathematical error since Eeff is not a constant. This can be shown with the following arguments [Kimoto, 2009].

Based on the energy balance of the atmosphere (see Figure 1), OLR can be expressed as follows.

OLR=Er+Ee+Et+Es-Eb (8)

From Eq.(6) and Eq.(8), the following equations are obtained [Kimoto, 2009].

Eeff * Sigma * Ts^4=EeffxEr=Er+Ee+Et+Es-Eb (9)

Eeff=1+(Ee+Et)/Er+(Es-Eb)/Er (10)

Therefore, Eeff is not a constant but a complicated function of Ts and the internal variables of the climate system [Kimoto, 2009]. Cess’s mathematical error causes the failure of Group A for the choice of Ts and OLR, which brings about 6 times larger climate sensitivity than 0.5K obtained from the energy budget of the earth..

**References**

Bony,S., Colman,R., Kattsov,V.M., Allan,R.P., Bretherton,C.S., Dufresne,J.L., Hall,A., Hallegatte,S., Holland,M.M., Ingram,W., Randall,D.A., Soden,B.J., Tselioudis,G. and Webb,M.J. 2006: Review Article How Well Do We Understand and Evaluate Climate Change Feedback Processes? J.Climate,Vol.19,3445-3482.

Cess,R.D., 1976: Climate Change: An Appraisal of Atmospheric Feedback Mechanisms Employing Zonal Climatology. J.Atmospheric Sciences, Vol.33,1831-1843.

Cess,R.D., Potter,G.L., Blanchet,J.P., Boer,G.J., DelGenio,A.D., Deque,M., Dymnikov,V., Galin,V., Gates,W.L., Ghan,S.J., Kiehl,J.T., Lacis,A.A., LeTreut,H., Li,Z.X., Liang, X.Z., McAvaney,B.J., Meleshko,V.P., Mitchell,J.F.B., Morcrette,J.J., Randall,D.A., Rikus,L., Roeckner,E., Royer,J.F., Schlese,U., Sheinin,D.A., Slingo,A., Sokolov,A.P., Taylor,K.E., Washington,W.M. and Wetherald,R.T., 1990: Intercomparison and Interpretation of Climate Feedback Processes in 19 Atmospheric General Circulation Models, J. Geophysical Research, vol.95, 16, 16,601-16,615.

Douglass,D.H.,Knox,R.S.,Pearson,B.D. and Clark,Jr.A.,2006:Thermocline flux exchange during the Pinatubo event. Geophysical Research Letters. Vol.33,L19711, doi:10.1029/2006GL026355.

Kimoto,K., 2009: On the confusion of Planck feedback parameters. Energy & Environment Vol.20, 1057-1066 (http://www.mirane.co.jp).

Ramanathan,V., 1981:The Role of Ocean-Atmosphere Interactions in the CO2 Climate Problem. J. Atmospheric Sciences, Vol.38,918-930.

Sclesinger,M.E., 1986: Equilibrium and transient climatic warming induced by increased atmospheric CO2. Climate Dynamics, Vol.1,35-51.

Soden,B.J. and Held I.M.,2006:An Assessment of Climate Feedbacks in Coupled Ocean-Atmosphere Models. J.Climate, Vol.19,3354-3360.

Trenberth,K.E., Fasullo,J.T. and Kiehl,J. 2009: Earth’s global energy budget. American Meteorological Society, March 2009 BAMS 311-323.

Tsushima,Y., Abe-Ouchi,A. and Manabe,S., 2005: Radiative damping of annual variation in global mean temperature: comparison between observed and simulated feedbacks. Climate Dynamics, Vol.24,591-597.

Wetherald,R.T. and Manabe,S., 1988: Cloud Feedback Processes in a General Circulation Model. J. Atmospheric Sciences Vol.45,1397-1415.

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For general information, see also Kimoto, 1983: Water Absorption and Donnan Equilibria of Perfluoro Ionomer Membranes for the Chlor-Alkali Process.