-1=e^ipi

Let's suppose that for the sake of argument I agree with you that the pentagon does not count as a civilian target. If a civilian plane is hijacked and used, then i counts as terrorism. Even more so if that plane contains civilians.

Okay, so I have tried to compile a 'better' data set to estimate this model with because I think the data prior to 1959 is not of good enough quality. I can get monthly Mauna Loa CO2 data here: ftp://aftp.cmdl.noaa.gov/products/trends/co2/co2_mm_gl.txt I'll use the seasonally detrended monthly data because I don't want the seasonal effects. For temperature, I can use the monthly HadCRUT4 data here: http://www.cru.uea.ac.uk/cru/data/temperature/HadCRUT4-gl.dat The problem is that I don't think this data set is seasonally detrended. To seasonally detrend the temperature data, I can simply perform the regression: temperature = β1*January + β2*February + ... + β12*December + β13*time + error I add the time variable because I need to take into account the fact that later months in the year occur later in my data set (since I am interested in January 1959 to December 2008). The month variables are just dummy variables. I can then seasonally detrend the data set if I take each temperature, subtract the dummy variable of that month, and then add the mean of the dummy variables. For CO2 emission data, I am still stuck with: http://cdiac.ornl.gov/ftp/ndp030/global.1751_2008.ems I can't seem to find freely available monthly CO2 emission data. To get estimate monthly CO2 emission data, I have to interpolate. However, the data given above is the emissions for the entire year (as opposed to the rate of emissions at a given point in time). Therefore, I have to turn this emission data (from 1959 to 2008) into 51 points on the integral from the start of 1959 to a later point in time. Once I do this, I can perform a cubic spline interpolation to get the estimates of this integral at the end of each month from January 1959 to December 2008. Then I can simply take the derivative to get approximations of the emissions for each month. I realize that doing this will remove season affects on CO2 emissions (for example, January probably has higher emissions than May because people in the Northern Hemisphere are heating their homes). However, I want to remove seasonal affects, so this isn't an issue for me. The other thing is that with monthly data, the months have an unequal number of days. So any regression I do will have to be weighted. I'm not sure how to do a Gauss-Newton non-linear regression with weights, but I can probably figure it out.

So I tried to perform additional restrictions. I tried fixing D = -0.016 and Ts(1876) = 13.5C (roughly pre-industrial temperatures). One thing I noticed is that with all the additional restrictions I am performing, I can rewrite the regression equation as: dCO2/dt = D*(CO2(0) - C*Ts(0) + C*T(t) - CO2(t)) + DH*(C*IT(t) + (G-1)*ICO2(t) + (– C*Ts(0) – G*CO2s(0) + CO2(0))*t) + H*(CO2(0) - G*CO2s(0) + (G-1)*CO2(t)) + E*Human(t) + E(D+H)*IHuman(t) + EDH*IIHuman(t) This has 6 explanatory factors for the 3 unknowns. So it is much easier to perform the regression (which means matlab has less rounding error when doing matrix inversions). Performing the Gauss-Newton estimation gives me the following 95% confidence intervals: E: (0.00025 +/- 0.0005) million tons of carbon per ppm D: (-0.001 +/- 0.012) per year H: (-0.007 +/- 0.012) per year E suggests that ~3.9 billion tons of carbon are needed to increase atmospheric CO2 by 1 ppm. The two decay times are not statistically significant. This isn't a very useful result, so I thought I would try to restrict E = 1/2180. With the new restriction I can rewrite the regression equation as: dCO2/dt – E*Human(t) = D*(CO2(0) - C*Ts(0) + C*T(t) - CO2(t) + E*IHuman(t)) + DH*(C*IT(t) + (G-1)*ICO2(t) + (– C*Ts(0) – G*CO2s(0) + CO2(0))*t + E*IIHuman(t)) + H*(CO2(0) - G*CO2s(0) + (G-1)*CO2(t) + E*IHuman(t)) This has 3 explanatory variables for 2 unknowns, so is even easier to estimate. But if I estimate this I get: D: (-0.02 +/- 0.02) per year H: (0.00 +/- 0.02) per year Again, I can't get statistically significant estimates of the two decay values I am interested in. Looking at the residual, I think the issue is data quality, especially before 1959 where I am using ice-core data. I tried to vary the restrictions and do other things, but get similar results. Maybe I should try to use only data after 1959 and use monthly data rather than annual data.

Isolating the Saudi's doesn't imply that the Saudi's will lose power. They have the 4th post powerful military in the world. Not that I think you can get much worse than the status quo. The Saudi's funding wahabbism with oil money are one of the main reasons why we have Al Queda, Al Shabab, Boko Haram, ISIS, and all these terrorists attacks in the western world.

And the pentagon contains civilians...

As far as I am concerned, something needs to satisfy 3 properties to count as terrorism: - must involve violent acts (such as murder, blowing up buildings, kidnapping, etc.) - must have an ideological motive (religious or political) - the civilian population has to be targeted These two criminal acts were not terrorist acts because the two criminals went out of their way to NOT target civilians. The recent attempt to perform a mass shooting in Halifax does not count as terrorism because it lacks an ideological motive. The recent murder of 3 muslims in North Carolina does not count as terrorism because it lacks an ideological motive. The mass shootings of 72 people by Anders Brevik counts as terrorism. The 911 hijackings count as terrorism. The kidnappings by the FARC rebels count as terrorism. The bombings that the IRA did several decades ago count as terrorism. Does that make sense?

Maybe, but I think that estimate is very reliant on the 2.6 billion tons value, which has a fair amount of uncertainty. Example: If you used 2.9 billion tons instead, then you would get 191 years instead of 300 years. There is also a fair amount of rounding error, so I don't think you can conclude much more than the order of magnitude is ~ a century.

@ Bonam - Yes I agree about your comment regarding 2.18 billion tons. However, I am trying to take into account the effect of the carbon sink absorbing some of that CO2 in my regression model. So the value of the rate of increase of atmospheric CO2 due to human emissions should be 1 ppm per 2.18 billion tons after taking into account the effect of the oceans (and maybe plant life) absorbing excess CO2. Edit: Yes with respect to your second comment. I am trying to measure that timescale. So far it appears like the decay time for this is ~70 years.

Also, if the ocean contains ~ 38000 billion tons of carbon (actually, I'll use a better estimate of 37,400 billion tons http://www.skepticalscience.com/human-co2-smaller-than-natural-emissions.htm) and at pre-industrial times there was ~600 billion tons (and assuming that the change in carbon in the ocean since pre-industrial times is relatively small compared to the total carbon in the oceans, which is reasonable given how much more carbon is in the ocean relative to the atmosphere and also the time difference of only 1.5 centuries), then the ratio of carbon in the atmosphere to carbon in the ocean in equilibrium is about 0.016. Therefore, by conservation of mass, the value of G in my recent regressions should be ~-0.016. Edit: Sorry, it G should be the inverse of this (-62.5).

@ Bonam - I am doing linear approximations as well. Though I am trying to get a rough idea of the time lags of global warming (particularly how long it takes for the oceans to catch up to surface temperatures). That is why your model (not really sure what else to call it, give me a different name if you prefer) is a bit insufficient for me (though a nice comparison). The reason for this is my assumption of constant decay did not appear to hold, which means that I couldn't properly estimate equilibrium climate sensitivity. As for graphing the model vs actual temperature / CO2, I haven't bothered to do that yet because the models I have presented so far to explain both of them are insufficient.

I'll try to do a rough estimation of the amount of CO2 (in terms of metric tons of carbon) that was in the atmosphere during pre-industrial times. As I explained in post #163, there is approximately 10329 kg of air above a square meter of Earth at sea level. The radius of the Earth is 6371 km. This means that there is approximately 4*pi*(6371000m)^2*10329kg = 5.268 x10^18 kg of air on Earth. If we assume an atmosphere of 78% N2, 21% O2 and 1% Argon, then the molar mass of the atmosphere is 28.96 g/mol. Thus there are approximately 1.819 x 10^20 mols of gas in the Earth's atmosphere. If pre-industrial times had 275 ppm of CO2, then this means that there were approximately ~ 5.00 x 10^16 mols of CO2 in the atmosphere during pre-industrial times. Since the molar mass of carbon is 12 g/mol, this means that the atmosphere had ~ 600 billion tons of carbon in the atmosphere during pre-industrial times. Dividing this by 275 gives 2.18 billion tons, which is the amount of carbon that needs to be burned in order to increase atmospheric CO2 by 1 ppm. Though most of the estimates I have done previously gave values higher than 2.18 billion tons. This could suggest that I am underestimating the effect of oceans to absorb CO2 (so it appears like more carbon is needed to raise atmospheric CO2).

Let's assume that the ~87 ppm CO2 feedback effect for a doubling of CO2 as the IPCC's central estimate is correct (posts 169-170 give justification for this). Furthermore, let's suppose that equilibrium climate sensitivity is 3 C and the carbon sink decays by 1.26% if its temperature is increased by 1 C. Then 87 = C*Integral(T=0 to 3; exp(-0.0126*T)dT) = -C/0.0126*(exp(-0.0126*3)-1) => C = 29.55, where C is the constant I have referred to in the last few posts that represents the increase in atmospheric CO2 due to increasing the heat sink by 1 degree. I'll assume that C = 29.55, because it is clear that my model in post #171 cannot distinguish between change in CO2 levels due to changes in the temperature of the heat sink and changes in CO2 levels due to changes in the carbon sink (since temperature and CO2 trends are very correlated). If I perform the regression of post #171, but fix C = 29.55, and CO2s(1876) = 275 (roughly pre-industrial levels) then my 95% confidence intervals are: E: 0.00034 +/- 0.00019 D: -0.61 +/- 0.72 G: -0.002 +/- 0.015 H: 0.013 +/- 0.005 Ts(1876): 12.7 +/- 8.2 Not bad, but I might need to use more a priori information for additional restrictions to get more reasonable results (especially since the effect of warming the heat sink on CO2 concentrations is so small). This does suggest that it takes ~ 2.9 billion tons of carbon to increase atmospheric CO2 by 1 ppm and that the decay time of the carbon sink towards equilibrium is ~75 years (comparable to the estimate in Bonam's model). I also checked the residual. I still see no evidence of variation in change CO2 concentrations due to volcanism.

So I re-estimated Bonam's model using 1876-2009 data (sorry about the temperature mistake) and I get the following 95% confidence intervals: K1 = (0.00039 +/- 0.00013) ppm per celcius K2 = (-0.058 +/- 0.428) ppm per million tons of carbon K3 = (-0.014 +/- 0.014) per year K4 = (4.3 +/- 4.2) ppm per year Make of that what you wish. If I were to assume that this model's assumptions are valid then this suggests that 2.6 billion tons of carbon is needed to increase atmospheric CO2 by 1 ppm and that atmospheric CO2 has a decay time of approximately 70 years.

Found a much more serious error in my code. I was accidentally using aerosol data instead of temperature data (sorry I mixed up a 6 with a 7). All the regressions since post #150 are nonsense. I tried the regression of post #168 but with the code correction. The 95% intervals are: E: 0.0028 +/- 0.0334 C: -61 +/- 549 D: 1.7 +/- 5.0 G: -0.0061 +/- 0.0421 H: 0.068 +/- 3.101 Ts(1876): 14.07 +/- 3.96 CO2s(1876): 272 +/- 115 Nothing is very significant; probably because the model is too complex for the data. I'll have to impose some restrictions (using a priori information of course).

I guess I need to consider permafrost as well, otherwise the 16.5 value in my last post is an underestimate. There are 1400-1700 billion tons of carbon in permafrost (http://en.wikipedia.org/wiki/Permafrost); I'll use the central estimate of 1550 billion tons. Due to lack additional a priori knowledge, I'll assume that the permafrost has a similar behaviour as the oceans (so 1 degree celcius increase releases ~1.26% of the carbon). Also, permafrost is located primarily in polar regions, so I should take into account polar amplification. The global polar amplification factor is ~2.0 (note that in the jetstream thread, I eventually used a polar amplification of 2.5; the reason was because I was considering the northern hemisphere, which has a higher polar amplification factor than the southern hemisphere). Putting these factors together (plus the assumption of 2.9 billion tons of carbon corresponding to 1 ppm) suggests that I should increase the 16.5 by ~13.5 to ~30. Note that if the equilibrium climate sensitivity is approximately 3 celcius then this suggests that doubling CO2 concentrations should release an additional 3*30 = 90 ppm of CO2 in the long run (due to ocean warming and permafrost melting). Note that this corresponds very well with the approximately 87 ppm value that is the central estimate of the ranges in the IPCC's AR4 (chapter 7). So the value of C in the previous regressions I did should be ~ 30.

Chemistry is not my strongest area, but as I understand it, Henry's constant determines the ratio between the solubility of gas in a liquid and the partial pressure of the gas above that liquid. http://en.wikipedia.org/wiki/Henry%27s_law#Temperature_dependence_of_the_Henry_constant From the wiki, one gets that the solubility of CO2 in water is proportional to exp(C*(1/T + 1/(298K))), where T is the temperature of the water in Kelvin and C = 2400K for CO2. Taking the natural logarithm of this gives C*(1/T + 1/(298K)). Plotting this from 0 C to 30 C gives a line of best fit of -0.0126 +/- 7.3095 and gives a very good fit (R^2 is 0.9993). Thus the amount of CO2 in water should decrease by about 1.26% if the temperature increases by 1 Celcius. Now the oceans contain around 38000 billion tons of carbon (http://worldoceanreview.com/en/wor-1/ocean-chemistry/co2-reservoir/). So if the temperature of the ocean were to increase uniformly by 1 Celcius, then this would release ~47.88 billion tons of carbon. If the value that 2.9 billion tons of carbon is needed to increase atmospheric CO2 by 1 ppm is correct (see post #168), then increasing the ocean temperature uniformly by 1 degree celcius should increase atmospheric CO2 by ~16.5 ppm. So the value of C in the last 2 regressions should be ~16.5. This is not a very large effect compared to Human emissions and to CO2 being absorbed by the ocean due to imbalance between CO2 concentrations in the Ocean vs CO2 concentrations in the atmosphere.

I've attempted a Gauss-Newton method estimate for the model described in post #164. Recall that the model was essentially the following 3 equations. dCO2/dt = E*Human(t) + C*dTs/dt + G*dCO2s/dt dTs/dt = D*(T(t) - Ts(t)) dCO2s/dt = H*(CO2(t) - CO2s(t)) The 95% confidence intervals for the estimates are: E: 0.00034 +/- 0.00018 C: 51 +/- 97 D: 0.35 +/- 0.63 G: -0.0027 +/- 0.0047 H: 0.068 +/- 0.090 Ts(1876): 13.97 +/- 0.06 CO2s(1876): 271 +/- 54 E is in ppm per million metric tones. It suggests that about 2.9 billion metric tonnes of carbon is needed to increase atmospheric CO2 by 1 ppm. Ts and CO2s represent the temperature of the characteristic heat sink and the CO2 concentration of the characteristic carbon sink in Celcius and ppm respectively. They are reasonable values. C, D, G, F are all unreasonable and not significant. As expected, the increase in CO2 concentrations due to the warming of the oceans is very correlated with the decrease in CO2 concentrations due to the atmosphere having excess CO2 relative to the ocean. Either I should try to get better data (either use all the data back to 1850 or use monthly data post 1950), or I should try to impose some reasonable conditions on the values of C and D.

I just noticed that I made an error in the code that I used to get the results for post #154. I miscalculated an integral. Below is the corrected 95% confidence intervals. C: -26 +/- 62 D: 0.043 +/- 0.026 E: 0.000257 +/- 0.000009 Ts(1876): 14.34 +/- 0.79 C is the CO2 emitted in ppm due to increasing the characteristic heat sink by 1 celcius. It's still negative and not statistically significant. D is in (year)^-1, and suggests that the decay time of the characteristic heat sink is 23 years. I guess getting it in the 60-70 range a few posts ago was a coincidence. In any case, this model is not accurate for reasons I explained in post #159. E is in ppm per million metric tones. It suggests that about 3.9 billion metric tonnes of carbon is needed to increase atmospheric CO2 by 1 ppm. Ts(1876) corresponds to the characteristic heat sink temperature in celcius for 1876.

Sorry, I made a mistake in my last post. It should be corrected now.

I thought I would try to solve this system of equations, to get something that I can estimate via a regression. For simplicity of notation, let Human be the CO2 emissions due to humans, IHuman be the integral of Human, IIHuman be the integral of IHuman, IT be the integral of T, and ICO2 be the integral of atmospheric CO2 (otherwise notation is the same as I previously had). Start with the equations: dCO2/dt = E*Human(t) + C*dTs/dt + G*dCO2s/dt (1) dTs/dt = D*(T(t) - Ts(t)) (2) dCO2s/dt = H*(CO2(t) - CO2s(t)) (3) Substituting (3) into (1) gives: dCO2/dt = E*Human(t) + C*dTs/dt + GH*(CO2(t) - CO2s(t)) (4) Integrating (1) with respect to time gives: CO2(t) – CO2(0) = E*IHuman(t) + C*(Ts(t) – Ts(0)) + G*CO2s(t) – G*CO2s(0) = E*IHuman(t) - C*(T(t) – Ts(t) + Ts(0) – T(t)) + G*CO2s(t) – G*CO2s(0) (5) Substituting (1) into (5) gives: CO2(t) – CO2(0) = E*IHuman(t) – C/D*dTs/dt + C*(T(t) – Ts(0)) + G*CO2s(t) – G*CO2s(0) (6) H*(6) + (4): dCO2/dt + H*CO2(t) – H*CO2(0) = E*Human(t) + C(1-H/D)*dTs/dt + GH*CO2(t) + EH*IHuman(t) – GH*CO2s(0) + CH*(T(t) – Ts(0)) => dCO2/dt = E*Human(t) + C(1-H/D)*dTs/dt + H(G-1)*CO2(t) + EH*IHuman(t) + CH*T(t) – CH*Ts(0) – GH*CO2s(0) + H*CO2(0) (7) Substituting (2) into (7) gives: dCO2/dt = E*Human(t) + C(D-H)*(T(t) - Ts(t)) + H(G-1)*CO2(t) + EH*IHuman(t) + CH*T(t) – CH*Ts(0) – GH*CO2s(0) + H*CO2(0) (8) Integrating (7) gives: CO2(t) – CO2(0) = E*IHuman(t) + C(1-H/D)*(Ts(t) – Ts(0)) + H(G-1)*ICO2(t) + EH*IIHuman(t) + CH*IT(t) + (– CH*Ts(0) – GH*CO2s(0) + H*CO2(0))*t (9) (8) + D*(9): dCO2/dt + D*CO2(t) – D*CO2(0) = E*Human(t) + C(D-H)*(T(t) - Ts(t)) + H(G-1)*CO2(t) + EH*IHuman(t) + CH*T(t) – CH*Ts(0) – GH*CO2s(0) + H*CO2(0) + ED*IHuman(t) + C(D-H)*(Ts(t) – Ts(0)) + HD(G-1)*ICO2(t) + EDH*IIHuman(t) + CDH*IT(t) + DH(– C*Ts(0) – G*CO2s(0) + CO2(0))*t => dCO2/dt = [(D+H)*CO2(0) – GH*CO2s(0) – CD*Ts(0)] + E*Human(t) + E(H+D)*IHuman(t) + EDH*IIHuman(t) +CD*T(t) + CDH*IT(t) + (H(G-1)-D)*CO2(t) + HD(G-1)*ICO2(t) + DH(– C*Ts(0) – G*CO2s(0) + CO2(0))*t; Explanatory Variables are: constant, Human, IHuman, IIHuman, T, IT, CO2, ICO2, t = 9 explanatory variables. Parameters are: E, C, G, D, H, Ts(0), CO2s(0) = 7 parameters So it is possible to estimate this.

What has happened in Eastern Ukraine was sad and easily avoidable. Unfortunately, the vast majority of western politicians have a pseudo-cold-war mentality and look at everything through the lens of moral absolutism. There was a peaceful way to deal with this, but the west chose the non-peaceful way in order to maintain moral absolutism. What is needed is for all sides (West, Russia, Western Ukrainians, Eastern Ukrainians) to accept the principle of right to self determination. Give each oblast in Ukraine the right to determine its own future and have referendums in each oblast giving people the choice between belonging to Ukraine, Russia, or being an independent state.

The government could try to completely isolate the Saudis, and point to the hypocracy of the US/EU and the oil money that leads to funding of Wahhabism that leads to terrorism. The main reason the US/EU can maintain their absurd positions is by ignorance of the population. If a strong ally such as Canada suddenly started strongly disagreeing with the US/EU and pointed out the insanity of current western foreign policy, it would be difficult for the media & foreign politicians to maintain the charade for very long.

You mean like sense of purpose? Wait, I thought terrorists had to target the civilian population. Stephen Harper can hardly be considered a civilian, nor a soldier at the war memorial, nor the police officers that were run over. The last two 'terrorist' attackers in Canada went specifically out of their way to NOT target civilians. So I don't think it is accurate to call the terrorists.

You have an optimistic view of the future. If religious people continue to outbreed non-religious people, if laws that prevent freedom of expression, press & religion continue, and if political correctness + cultural relativism continue then I am unsure this will occur on the timescale that you suggest it will.

I thought I would try to get a rough idea of what the timescale of the fast response (due to direct heating of the earth’s surface from increased CO2, as well as perhaps the water vapour feedback) is in order to get an idea of how this compares to the 60-70 year ‘ocean’ response time. Consider a representative square meter of the Earth’s surface at the global average surface temperature of 15 C and with a relative humidity of 75% (note that if you look at relative humidity across the Earth’s surface, it does not have much temperature dependence). First, let’s try to estimate the ‘fast’ heat capacity of this representative square meter. To do this I’ll use all of the air above the surface and the first 10 metres of ground/water below the surface. I justify the choice of 10 meters on the basis that this seems to be roughly the characteristic depth at which temperature changes rapidly to surface temperature (although I might be being generous; anyway, this is a rough calculation). The standard atmospheric pressure at sea level is 101325 Pascals. Since the acceleration due to gravity at the Earth’s surface is roughly 9.81 m/s2, this means there is roughly 10329 kg of air above this square metre of surface. Dry ‘air’ (78% N2, 21% O2 + other stuff) has a specific heat capacity of roughly 1006 J/kg/C at atmospheric pressure. However, the air isn’t dry. If we assume that all of this air is at 15 C and has a relative humidity of 75%, then by the clasius-clapeyron relation (specifically the August-Roche-Magnus formula, see http://en.wikipedia.org/wiki/Clausius%E2%80%93Clapeyron_relation for more details), such air would be 1.26% water vapour. The specific heat capacity of water vapour is roughly 1864 J/kg/C. The specific heat capacity of the 75% relative humid air is the weighted average, so is 1017 J/kg/C. However, if one were to maintain the 75% relative humidity and increase temperature, then one needs to take into account the fact that water vapour needs to be created by vaporizing water. The enthalpy of vaporization of water is roughly 2260 kJ/kg. If one takes the first derivative of the clasius-clapeyron relation then one obtains that at 15 C and atmospheric pressure that the increase in the percentage of air that is water that is required to maintain 75% relative humidity if one increases temperature by 1 C is 0.081%. This means that about 1831 J/kg/C are needed to maintain the relative humidity. This means that the effective specific heat capacity of the air is ~ 2848 J/kg/C. So that 10329 kg of humid air has an effective heat capacity of approximately 29417 kJ/C. Now let’s consider that 10 m of soil/water below the surface. The density of fresh water is 1000 kg/m3, the density of sea water is 1027 kg/m3, and the density of soil varies between 1000 kg/m3 and 1600 kg/m3. Furthermore, the specific heat capacity of soil is roughly 1480-2090 J/kg/C and the specific heat capacity of fresh water is 4182 J/kg/C. The soil has a higher density but lower heat capacity (and a lower heat capacity times density). To be on the safe side, I’ll round up slightly and assume that we have 10 m of fresh water below the surface. This corresponds to 10000 kg of water directly below the surface, which has a heat capacity of 41820 kJ/C. So the combined heat capacity that corresponds to this square meter of the Earth’s surface is ~71237 kJ/C. Assume (for simplicity) that the 10m of water underneath the surface and the air above the surface has a uniform temperature of 15C and radiates as a black body. Then the power of the radiation emitted by this square metre is m2*σT4, where σ is the Stefan-Boltzmann constant (5.6704 x 10-8Wm-2K-4). To get the power of radiation emitted to space, divide this by two (since half of the radiation will go upwards, and half will go downwards). In equilibrium, the blackbody radiation emitted must equal incoming radiation (from the sun, due to greenhouse gases, etc.), which we shall call F, the forcing. We get F = m2*σT4/2. Now consider small changes in the forcing such that we can use a taylor approximation to get ΔF = 2m2*σ T3ΔT, where ΔF and ΔT is the change in forcing and equilibrium temperature respectively. If we start in equilibrium and increase by a small amount ΔF, the temperature will slowly decay (exponentially) towards the new equilibrium from the old equilibrium. Therefore we can write the temperature as a function of time as T(t) = T(0) + ΔT*(1 - exp(-βt)), where t=0 corresponds to the time where there is the sudden change in forcing, and β is a parameter that determines the rate of decay. Taking the first derivative of this with respect to time gives dT(t)/dt = βΔT*exp(-βt). If we set time equals zero, we get dT(0)/dt = βΔT. Now we know that at t=0, the difference between outgoing and incoming radiation is ΔF, since right before t=0 things were at equilibrium. Given that the heat capacity is 41820 kJ/C, one can get the change in heat per unit time at t=0 as dT(0)/dt = ΔF/(41820 kJ/C). However, dT(0)/dt = βΔT and ΔF = 2m2*σ T3ΔT. Thus we get 2m2*σ T3ΔT/(41820 kJ/C) = βΔT. => β = 2m2*σ T3/(41820 kJ/C). Using T = (273.15 + 15) K, one obtains: β = 6.488 x 10^-8 s^-1 Inverting this gives the decay time, which is 15412904 seconds = 178.4 days = 0.488 years. Therefore, the fast decay time is roughly half a year, which is far less than the ocean decay time of 60-70 years. Of course, with water vapour the fast decay time might be slightly longer (since it would take time for water vapour to get into the air, then increase forcing, which causes more water vapour), though I may have been generous in my estimation of heat capacity. In any case, if we go with a CO2 + water vapour equilibrium climate response of 1.74 C (this is roughly the fast response that neglects various other feedback effects that would occur on longer time scales). Then after doubling CO2, within a year temperatures should increase to within 1/e^2 of this value (by about 1.5 C).