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Climate sensitivity 

The climate sensitivity (CS) according to the Earth's energy balance


The radiative forcing (RF change) at TOA has a linear relationship to the global mean surface temperature change dTs if two equilibrium climate states are assumed (L=lambda=climate sensitivity parameter:


dTs = L*RF                                                (1)


IPCC states (2007a) that L is a climate sensitivity parameter, which is nearly invariant parameter having a typical value about 0.5 K/(Wm-2). This value is based on rather old calculations (Ramanathan et al., 1985) before 1985, at which time narrow-band models were applied and not the accurate line-by-line methods of today. IPCC no longer keeps the climate sensitivity parameter as a nearly invariant parameter like in AR4. In AR5 its value varies in broad limits. The value of the climate sensitivity parameter is 0.811 K/Wm-2 for the CO2 forcing of 3.7 Wm.2 and the warming of 3.0 °C.


The author has used three different methods in calculating the CS and l values. The simplest analysis of CS and l is based on the total energy balance of the Earth by equalizing the absorbed and emitted radiation fluxes


SC(1-α) * (¶r2) = sT4 * (4¶r2),                  (2)


Where SC is solar constant (1368 W/m2), α is the total albedo of the Earth, s is Stefan-Bolzmann constant (5.6704*10-8), and T is the temperature (K). The temperature value of T can be solved:


T = (SC * (1 – α) (4s))^0.25                         (3)


Where T is the temperature corresponding the emitted longwave (LW) flux in the atmosphere. The average albedo (Ollila, 2013b; Ollila, 2014) is (104.2 Wm-2)/(342 Wm-2) = 0.30468. Using this albedo value, the temperature T would be -18.7 °C (=254.5 K) observed in the atmosphere in the altitude of about 9.5 km. According to the Planck’s equation, this temperature corresponds to LW radiation flux 237.8 Wm-2, which is the actual average emitted LW radiation flux of the Earth. The most common reported global mean surface temperature is 15°C, which means that the greenhouse effect would be 33.7 K. The surface temperature Ts can be calculated by adding 33.7 K into T:


Ts = T +33.7                                                (4)


The term SC(1-α)/4 is the same as the net radiative forcing (RF) and therefore Eq. (2) can be written in the form RF = sT4.  When this equation is derived, it will be d(RF)/dT = 4sT3 = 4(RF)/T.  The ratio d(RF)/dT can be inverted  transforming it into L:


dT/(d(RF)) = L = T/(4RF)= T/(SC(1-α))        (5)


In the all-sky conditions the total albedo flux 104.2 Wm-2 is the sum of the cloud reflected flux of 67.8 Wm-2, the surface reflected flux of 22.7 Wm-2 and the air reflected flux of 13.7 Wm-2. These values as well as the following three pairs of cloudiness and albedo values for clear, all-sky and cloudy sky conditions are based on energy balance analysis of global radiative fluxes (Ollila, 2013b; Ollila, 2014; Zhang et al., 2004; Bodas-Salcedo et al., 2008; Loeb et al., 2009): (0%, 53/342=0.155), (66%, 104.2/342=0.305), and (100%, 120/342=0.351). The second-order polynomial can be fitted through these points and the result is


α = 0.15497 + 0.0028623 * CL – 0.000009 * CL^2     (6)


where α is albedo and CL is cloudiness-%.


The differences between sky conditions are due to the degrees of cloudiness in different skies. This effect is generally called cloud forcing (CF). Normally the CF has been calculated at TOA as the difference between clear sky and all-sky conditions. Using the values of Ollila (2013b), the albedo flux change 53 - 104.2 = -51.2 Wm-2. The outgoing LW radiation decrease is the difference between OLR fluxes, which is 259 - 237.8 = 21.2 Wm-2. According to the most common definition, the CF is the sum of these two fluxes, which in this case is -30.0 W/m2, a cooling effect. This value is close to the values used in other studies (Ohring and Clapp, 1980; Harrison et al., 1990; Ardanuy et al., 1991; Zhang et al., 2004; Raschke et al., 2005; Loeb et al., 2009;  Stephens et al., 2012), which vary between -17.0 and -28 W/m2 average being -23.4 W/m2.


Spencer and Braswell (2011) have created a more complicated calculation method for cloud forcing by separating the effects and feedback of the clouds. Their final conclusion is that clouds have a negative impact on the surface temperature. Dressler (2010) has analysed the TOA radiation budget in response to short-term climate variations from the years 2000 to 2010, and his results showed positive feedback of the clouds. So the issue of cloud forcing still remains unclear without common acceptance and understanding but the big majority of CF studies show the cooling effect of cloudiness increase.


The specification of the common CF can be criticized, because it is based on the instant radiation flux changes after a cloudiness change and it does not recognize the dynamic delays of the climate system. Ollila (2014) has concluded that the real CF is based on the SW radiation changes only, because the Earth has yet to reach the radiation flux balance according to the 1st law of thermodynamics, which means that the OLR flux must be the same as the net solar input flux. This approach would increase the CF values by about 46 % (Ollila, 2014b).


The equation (6) does not mean that only the total cloudiness changes can cause albedo changes. The changes of other reflected fluxes (by surface and air and by different cloud types) have their effects on the total albedo but the numerical effects are not known. The equation (6) is well established because it is based on the measured fluxes in the global scale.


When the changes in radiative forcing are known, the equations (2), (3), and (4) can be used in calculating T, ECS (= Equilibrium CS) and l values for the variations of RF and α. The climate sensitivity parameter calculated according equation (5) is 0.268 K/(Wm-2) i.e 0.27  K/(Wm-2) 


The surface temperature is very sensitive for the cloudiness and albedo changes of the Earth, as one can see in Fig. 1. In the cloudiness range from 60 % to 70 %, the cloud forcing effect about 0.15 °C / cloudiness-%. This issues will be analyzed more closely in the section "Cloud forcing".


Figure 1

Dr. Ollila graph

The climate sensitivity (CS) according absorption and longwave radiation changes


The author has also calculated the CS and lambda (L) values applying two simulation tools available in the network, namely Modtran (Berk et al., 2013) and the Spectral Calculator (Gats, 2014). The results are collected in Table 1. The all-sky conditions have been calculated by combining the clear and cloudy sky values (Bellouin et al., 2003; Ollila, 2013b):


(1-CL/100) * Fclear + (CL/100) * Fcloudy = Fall-sky         (7)


Where F is a radiation flux of a sky in question and CL is a cloudiness-%. Also temperatures of different skies are combined according to this equation.


The  average global atmosphere’s (AGA) surface temperature is 15 °C, and the concentrations of the anthropogenic GH gases measured in 2005 (AGA 2005) or in 2012 (AGA 2012) have been used. The GH gas concentrations (2005/2012) are: CO2 (379/393 ppm), CH4 (1.774/1.866 ppm), and N2O (0.319/0.324 ppm), as reported by IPCC (2007c, 2013). The graphs in Fig. 1 are based on the AGA 2005 gas concentrations are based on the AGA 2012 conditions. The parameters and choices applied in Modtran simulations, are depicted in Table 2.




























The CS and L calculations are carried out to an altitude of 70 km. In these calculations, a few iterations are needed in both calculation tools in order to find the surface temperature, which compensates the increased absorption caused by a CO2 increase to 560 ppm, bringing the OLR flux exactly to the same the OLR flux caused by a CO2 concentration of 280 ppm. Because both the OLR change and the temperature change are calculated at the same time, the L value can be easily calculated. The cloudy sky values are calculated using the Modtran simulations, which show about 30 % lower OLR change than the clear sky simulations. This relationship has been used in estimating the cloudy sky values of Spectral Calculator simulations. IPCC’s report AR5 (2013) summarizes that according to several studies, the overall reduction of RF values in cloudy sky conditions is in average 25 % lower than the clear sky values.  The results of the simulations carried out by Modtran and Spectral Calculator are summarized in Table 2.



























The change of CO2 concentration from 280 ppm to 560 ppm would increase the total absorption of shortwave (SW) radiation by 0.40 Wm-2 according to the 1D model simulations. This change alone would mean an essential warming impact, but the situation is not straightforward, because this absorption directly decreases the SW radiation reaching the surface.


Myhre et al. (1998) have concluded that the absorption of solar radiation in the troposphere yields a positive RF at the tropopause and a negative RF in the stratosphere contributing to a net cooling effect of CO2 absorption of -0.06 Wm-2 for the concentration change from 280 ppm to 381 ppm. On these bases the author has not included the solar radiation absorption changes of CO2 into his calculations. The net effect of solar radiation absorption would slightly decrease the RF values of CO2 according to the analyses of Myhre et al. (1998).


The clear sky OLR change 2.69 Wm-2 calculated by Spectral Calculator at the TOA is the sum of transmittance flux change 1.12 Wm-2 and the radiance flux change 1.57 Wm-2. The OLR changes and the warming values of different CO2 concentrations are summarized in Table 4.  The global warming caused by the CO2 concentration increase from 280 ppm to 393 ppm calculated through OLR change is 0.24 °C without water feedback.


The logarithmic fitting gives the following equation between RF values and CO2 concentrations:


RF = 3.12 * ln(C/280),                        (8)


Where RF is the radiative forcing in Wm-2, C is the CO2 concentration in ppm.

table 4. the radiative forcing and warming values of different co2 concentrations (reference level 280 ppm). the clear sky values are calculated by spectral calculator and cloudy skies by Modtran.


Analyses of different CS results


Using Spectral Calculator simulation, a CO2 concentration of 393 ppm gives the  L value 0.230 and 1,370 ppm gives the L value 0.269. According to several studies (Zhang et al., 2004; Bodas-Salcedo et al., 2008; Loeb et al., 2009), the OLR flux varies between 233-240 Wm2 and using Eq. (3) shows that RF value 233 Wm-2 gives L value 0.270, and RF value 240 Wm-2 gives L value 0.265. The variation of l is relatively small but l is not invariant. The L values vary in totality from 0.230 to 0.319 in simulations. If Eq. (3) is applied for OLR changes calculated by the dRF 2.16 Wm-2 of Spectral Calculator, the ECS is 0.576 °C and L is 0.267. The same values using the dRF=1.834 Wm-2 of Modtran, the ECS is 0.49 °C and L is 0.267. The Modtran calculations’ results are not as accurate and reliable as the Spectral Calculator results, because the atmospheric conditions of Modtran cannot be specified with the same accuracy as in Spectral Calculator.


The author has also calculated the ECS value utilizing the IR absorption in the clear atmosphere; this value is 0.46 °C. Some other researchers (Miskolczi and Mlynczak, 2004) have calculated almost the same value, namely 0.48 °C. The most reliable results and best estimates are the values calculated by energy balance equations: ECS = 0.576 °C and
L = 0.268 K/(Wm-2) with the uncertainty ranges of 0.46–0.6 °C and 0.23–0.32 K/(Wm-2).


Some researchers have paid attention to the fact that the temperatures simulated by General Circulation Models (GCM) have departed from the real temperatures since 1998. There are several new research studies, which show lower ECS values than those of IPCC. According to these results, the best estimates and minimum values for ECS are: (Aldrin, 2012) 2.0 °C / 1.1°C; (Bengtson & Schwartz, 2012) 2.0 °C / 1.15 °C; (Otto et al., 2013) 2.0 °C / 1.2 °C and (Lewis, 2012) 1.6 °C / 1.2 °C.  Common features of these studies are mathematical methods like Bayes’s theorem to analyze the impact of CO2 based on the measured global data of radiative forcing factors, temperatures and ocean heat content.


These studies’ minimum values of ECS are practically same in the range 1.1-1.2 °C. Bengtson & Schwartz (2012) draw a conclusion that this value is the same as the no-feedback Planck sensitivity. An interesting point is that the ECS value of this study without any feedback mechanisms (including the Planck sensitivity calculation which is the same as equation (3)) is in the range 0.559…0.584 °C, and with water feedback the ECS according to the Plank’s equation is 1.1 °C. Is this a coincidence? There could be a very simple explanation. All the referred studies use the radiative forcing value of 3.7 Wm-2 for CO2 and they do not mention, whether or not water feedback has been used in their analyses.


The author’s conclusion is that the researchers of these studies have applied the RF value of 3.7 Wm-2 as in the study of Bengtson & Schwartz (2012). If this RF value has been calculated in the atmosphere, where is constant relative humidity, it would mean that it includes the positive water feedback duplicating the warming values. 



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