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We the people of Canada DEMAND change in Government. We the people of Canada demand that the government that belongs to us and to whom we lend our democratic power to lead this country and to sit in opposition, change the laws that govern the behaviour of those politicians that take the oath of office. We are tired of the way that parties spend our hard-earned tax dollars on things that do not benefit this country. We are tired of the lies by all parties and we are tired of the personal agendas that you all bring into the house of commons instead of working toward the common goal of bettering this great nation. We are tired of the ethics violations, the misappropriation of money that does not belong to you and the way you spend on other countries while here in Canada our own go without. We are tired of the way Canadians are left out in the cold as soon as you take office and the opposition all of a sudden becomes sanctimonious in the way they say they want to save us. We are tired of the Pandering politicians on all sides. We are tired of the insider trading. We are tired of the wealthy being the ones that you listen to and not us average Canadians because in fact most of you have never worked a solid day in your life. We are Tired of the way that the people we elect as our Members of Parliament stop speaking for us the moment they are told to tow the party line. We are tired of the way the country gets ignored because as soon as you take power and the house sits you then worry about the next election rather than worrying about us the people. We demand change and we have some changes we want: 1. We the people demand, not ask, that recall be brought into law. 2. We the people demand that restricted parliamentary privilege be restricted to times that it is a benefit to the country not the politicians. 3. We demand that under conditions that regular Canadians would lose their jobs that the same apply to politicians in the same fast manor. 4. We demand that if fired there is an end of your pension no if ands or buts. 5. We demand that any financial penalties that politicians receive be proportional to the wages they earn so as to sting them as we the citizens get stung. 6.We the people demand that when a politician ends his term that he not use political influence to enrich himself or family, if they do they are to be prosecuted. 7. We demand that all monies earned by the writing of books or speaking engagements by any serving politician or politician that has served (because they were only in that position to serve the country) should be paid to charity. 8. We demand that any politician caught deliberately lying to the citizens of this country be removed from office immediately. 9. We demand that any politician working against the people of this country be removed from office and if deemed serious enough be charged in a court of law. These demands are put before you by citizens of all political parties, All genders and all races. We the citizens of Canada have had enough of governments dividing the peoples and spending recklessly so as to force our taxes to a level that makes it hard to support our families.

I had some time this week, so I thought I would start a thread to discuss the question of what is the correct value of climate sensitivity for Earth with respect to the effects of CO2. For those that do not know, the definition of equilibrium climate sensitivity is the change in global average temperature in the long run due to a doubling of atmospheric CO2. This is partially a response to link to a scientific paper that estimated climate sensitivity that was linked in a post written by TimG back in July. http://www.mapleleafweb.com/forums/topic/23797-agwcc-deniers-fake-skeptics-their-mindset/?p=982902 http://www.sciencedirect.com/science/article/pii/S0304380014000404 The paper estimated transient climate response to be 1.093 °C and equilibrium climate sensitivity to be 1.99 °C. I mentioned to Tim that I thought that this was too low and did not agree with most other estimates of climate sensitivity that use other methods (though I did agree that many estimates, especially by the IPCC, were too high). I mentioned I would look at the details of the paper. If you haven't read the paper I recommend it (though you may need to pay or have access via some institution to read it). It is relatively short and uses a very simple approach to estimate climate sensitivity that relies primarily on empirical evidence and avoids making too many assumptions about the Earth's climate, which most models tend to do. Anyway, I looked into the paper and I saw a number of flaws in the methodology that could lead to an underestimation of climate sensitivity according to the empirical data. I'll discuss them and calculate a first-order correction to the estimates given in the paper. I was going to post this months ago but things got in the way. Anyway, what I find is that after first order corrections, the transient climate response is approximately 1.98 °C and the equilibrium climate sensitivity is 3.60 °C. Edit: Btw, if you can't access the paper, the methodology in the paper is basically: Look at global temperature record since 1850, detrend the data to take into account 60 year and 20 year climate cycles that correspond to events such as the Pacific Decadal Oscillation and the Atlantic Multidecadal Oscillation, assume that the anthropogenic warming after WW2 has been roughly linear since CO2 concentrations have been increasing roughly exponentially while solar forcing is logarithmic, assume that the vast majority of the warming prior to 1950 was natural due to the ending of the little ice age, obtain the anthropogenic warming since 1950 by subtracting the rate of natural warming, use the CO2 data since 1959 (I think Mauna Loa data is used) to give an estimate of transient climate sensitivity, and finally use the ratio of transient climate sensitivity to equilibrium climate sensitivity determined by various IPCC models to obtain an estimate of equilibrium climate sensitivity. WARNING: the rest of this post contains math and science. If you are afraid of math and science I suggest you run away and cry in a corner. A. Logarithm of Exponential So my biggest issue with the paper by Craig Loehle is the following statement: “The log of an exponentially rising function (as CO2 is) is of course a straight line” So I'll start by establishing the validity of the premises used for the benefit of the readers. Solar forcing is an approximately logarithmic function of CO2 concentrations; as a result, global average temperature is as well. That is, Temperature = A + B*ln(CO2 concentration), where A and B are some constants. There are many different justifications for this and the idea of this logarithmic relationship can be traced back all the way to 1896 by a guy called Svante Arrhenius. I may give a theoretical justification for this approximation in a later post (for the sake of the readers), but basically the logarithmic relationship is a very good approximation that is used by the IPCC and others. The second premise is that atmospheric CO2 has been exponentially rising since WW2. This is sort of true, although the statement can be misleading. If you take annual CO2 data from the Mauna Loa observatory (available at http://cdiac.ornl.gov/ftp/trends/co2/maunaloa.co2), then while the rate of change is approximately exponential a simple exponential fit with no constant added to the data set does not work very well. In order to get a good exponential fit to the data set you have to add a constant to the exponential. For example CO 2_ppm = 270 + 38.131*exp(0.0193(year-1950)) gives a very good fit to the data set with an R-squared value of 0.9984. Interestingly, 270 ppm corresponds to the 'pre-industrial' levels of atmospheric CO2. As for the claim that the log of an exponential function is a straight line, this is only true if there is no constant term added to the exponential. That is, ln(exp(x)) = x. But ln(a + exp(x)) != x if a!= 0. And in the case of an exponential fit to the CO2 record since WW2, we have a constant, the 270 ppm. This means that the logarithm of atmospheric CO2 concentrations is not a straight line. In an extreme case where the constant is much larger than the exponential term, the logarithm of the exponential is approximately an exponential. However, in the case of the logarithm of atmospheric CO2 concentrations since WW2, we have an intermediate situation where the magnitude of the constant is comparable with the magnitude of the exponential term. Since the time period used to estimate the anthropogenic warming is 1950-2010, for the sake of the discussion I'll take a Taylor approximation around the midyear (1980) as the logarithm of atmospheric CO2 concentrations: ln(270 + 38.131*exp(0.0193(year-1950))) ≈ ln(270 + 68.04*exp(0.0193(year-1980))) ≈ ln(338.04) + 68.04*0.0193/(338.04)*(year-1980) + 0.01932*270/68.04/(1+270/68.04)2/2*(year-1980)2 + higher order terms ≈ 58.2316423 + 0.00388466(year-1980) + 0.0000299417(year-1980)2 + higher order terms Note that the magnitude of (year-1980) is at most 30 since we are talking about 1950-2010. As a result, the magnitude of the linear term is at most 0.1165398 where as the magnitude of the quadratic term is at most 0.02694753. That is, the quadratic term is at most 23% the magnitude of the linear term. I would argue that the quadratic term being up to 23% as significant as the linear term suggests that the quadratic term should not be neglected. If this is the case then the logarithm of atmospheric CO2 concentrations should at least be considered a quadratic function of time, not a linear function of time. The higher order terms diminish in magnitude rapidly and are not very significant. For the sake of simplicity and to perform a first order correction to the Craig Loehle paper, let's suppose that the natural logarithm of CO2 is an approximately quadratic function of time, ln(CO2_ppm) ≈ 58.2316423 + 0.00388466(year-1980) + 0.0000299417(year-1980)2. Note that since Craig Loehle assumes that the change in the logarithm of CO2 concentrations have been approximately linear rather than quadratic, and since the coefficient in front of the quadratic term is positive, this means that Craig Loehle's assumption should bias his estimate of transient climate response downward and as a result, the estimate of equilibrium climate sensitivity is an underestimation. B. Definition of Transient Climate Response My second issue with the Craig Loehle paper is that the paper assumes that the warming since WW2 is the result of the transient climate response and then goes on to assume that what is calculated in the paper comparable with the transient climate responses calculated by others and given in the IPCC report. I understand why this was done, but ultimately the definition of transient climate response used by the IPCC differs from what is calculated by Craig Loehle. Let's look at the IPCC's definition of transient climate response: “A measure requiring shorter integrations is the transient climate response (TCR) which is defined as the average temperature response over a twenty-year period centered at CO2 doubling in a transient simulation with CO2 increasing at 1% per year.” http://www.ipcc.ch/ipccreports/tar/wg1/345.htm That is, the transient climate response according to the IPCC is approximately the magnitude of temperature change (assuming initially being in long run equilibrium) after the natural logarithm of CO2 concentrations have been increasing about ln(1.01) ≈ 0.00995 per year for ln(2)/ln(1.01) ≈ 69.66 years. However, the global situation since WW2 is that the natural logarithm of CO2 concentrations have been increasing quadratically over time at a rate of 0.002986 per year in 1950 to a rate of 0.004783 in 2010. That is, the rate of increase of the natural logarithm of CO2 concentrations since WW2 is much less than the scenario in the definition of transient climate response used by the IPCC and time period (1950 to 2010 = 60 years) of CO2 concentration increase is less than the 69.66 years used in the definition of transient climate response used by the IPCC. To be fair, Loehle tries to correct for this by taking into account that the increase in ln(CO2_ppm) was only about 0.326 of a doubling over 54 years. However, the overall effect of the difference in definition is that Loehle underestimates the transient climate response and I will explain why this is below. In order to perform a first order correction to the transient climate response value calculated by Craig Loehle and take into account the different definition of transient climate response used by the IPCC, I'll make the following assumption: the rate of change in global temperature in a given year is proportional to the difference between the current global temperature and the long run equilibrium global temperature at the current atmospheric CO2 levels. That is, global temperature should exponentially decay towards the long run equilibrium global temperature for a given level of atmospheric CO2. This is a reasonable assumption without further a priori information and I'll appeal to the scientific principle of Occam's Razor to justify it. Suppose that global temperature is an approximately logarithmic function of atmospheric CO2 concentrations (again this is a common assumption used by the IPCC and others). Furthermore, suppose that the effect of various positive and negative feedbacks acts as a multiplier to the initial change in temperature due to changing the atmospheric CO2 concentration (again a common assumption used by the IPCC and others). Then the long run equilibrium temperature for Earth at a given atmospheric CO2 concentration is a logarithmic function of atmospheric CO2. That is, long run temperature = A + S/ln(2)*ln(CO2_ppm), where A is some constant and S is the equilibrium climate sensitivity. Now if we use the assumption that the rate of change in global temperature in a given year is proportional to the difference between the current global temperature and the long run equilibrium global temperature at the current atmospheric CO2 levels then the change in global temperature over time is: dT/dt = k*(A + S/ln(2)*ln(CO2_ppm) - T), where k is some unknown positive constant and T is the global average temperature. C. IPCC Transient Climate Response Scenario Under the IPCC scenario for transient climate response, the natural logarithm of CO2 is a linear function of time, that is ln(CO2_ppm) = B + ln(1.01)*t. If we put this into the equation from part B, we get the differential equation dT/dt = kA + kS/ln(2)*B + kS/ln(2)*ln(1.01)*t - kT. Note that since the climate is in equilibrium at t = 0 under the transient climate response scenario, 0 = kA + kS/ln(2)*B - kT0 => T0 = A + S/ln(2)*B, where T0 is the global average temperature at t = 0. Thus the differential equation can be rewritten as dT/dt = k(T0 + S/ln(2)*ln(1.01)*t - T) . To solve for the above differential equation, make the substitution V = T - T0 - S/ln(2)*ln(1.01)*t. Then dV/dt = dT/dt - S/ln(2)*ln(1.01) and the above differential equation becomes dV/dt + S/ln(2)*ln(1.01) = -kV => dV/dt = -k(V + S/k/ln(2)*ln(1.01)). Now do the substitution U = V + S/k/ln(2)*ln(1.01). Then dU/dt = dV/dt and the differential equation becomes dU/dt = -kU. The general solution to this differential equation is U = C*exp(-kt), where C is some constant. Thus V = C*exp(-kt) - S/k/ln(2)*ln(1.01) and we get: T = C*exp(-kt) - S/k/ln(2)*ln(1.01) + T0 + S/ln(2)*ln(1.01)*t. Using the fact that the Temperature is T0 at t = 0 gives: T0 = C*exp(-k0) - S/k/ln(2)*ln(1.01) + T0 + S/ln(2)*ln(1.01)*0 => 0 = C - S/k/ln(2)*ln(1.01) => C = S/k/ln(2)*ln(1.01). Thus the differential equation can be rewritten as: T = S/k/ln(2)*ln(1.01)*(exp(-kt)-1) + T0 + S/ln(2)*ln(1.01)*t. Using a quadratic Taylor approximation of exp(-kt) gives: T = S/k/ln(2)*ln(1.01)*(1-kt+k2t2/2-1) + T0 + S/ln(2)*ln(1.01)*t => T = T0 + S/2/ln(2)*ln(1.01)*k*t2. Thus under the IPCC transient climate response scenario, global temperatures should increase approximately quadratically for the first 69.66 years. In order to get the temperature change at 69.66 years, plug in t = 69.66 = ln(2)/ln(1.01). => T = T0 + S/2/ln(2)*ln(1.01)*k*(ln(2)/ln(1.01))2 => T - T0 = S/2*k*ln(2)/ln(1.01). Note however that T - T0 is approximately the transient climate response. Thus the ratio of the equilibrium climate sensitivity to the transient climate response is S/(T - T0) = 2/k*ln(1.01)/ln(2). In the paper, Craig Loehle uses the IPCC ratio between equilibrium climate sensitivity and transient climate response of 1.81761. Using this value, one can calculate k. k ≈ 2*ln(1.01)/ln(2)/1.81761 ≈ 0.01579579. The inverse of k is 63.308, which means that the difference between the current global temperature and the equilibrium global temperature at current atmospheric CO2 decays by a factor of e in approximately 63.308 years. D. 1950-2010 Scenario Under the 1950-2010 scenario, the natural logarithm of CO2 is a quadratic function of time, that is ln(CO2_ppm) = 58.2316423 + 0.00388466(t-1980) + 0.0000299417(t-1980)2. If we put this into the earlier equation, we get the differential equation dT/dt = k(A + S/ln(2)*(58.2316423 + 0.00388466(t-1980) + 0.0000299417(t-1980)2) - T). Under the assumption of Craig Loehle, the climate is in equilibrium at t = 1950, so 0 = k(A + S/ln(2)*(58.2316423 + 0.00388466(-30) + 0.0000299417(-30)2) - T1950) => A = T1950 - S/ln(2)*58.14205003, where T1950 is the global average temperature in 1950. Thus the differential equation can be rewritten as dT/dt = k(T1950 + S/ln(2)*(0.08959227 + 0.00388466(t-1980) + 0.0000299417(t-1980)2) - T) . To solve for the above differential equation, make the substitution V = -T1950 - S/ln(2)*(0.08959227 + 0.00388466(t-1980) + 0.0000299417(t-1980)2) + T. Then dV/dt = dT/dt - S/ln(2)*(0.00388466 + 0.0000598834(t-1980)) and the above differential equation becomes dV/dt + S/ln(2)*(0.00388466 + 0.0000598834(t-1980)) = -kV => dV/dt = -k(V + S/k/ln(2)*(0.00388466 + 0.0000598834(t-1980))). Now do the substitution U = V + S/k/ln(2)*(0.00388466 + 0.0000598834(t-1980)). Then dU/dt = dV/dt + S/k/ln(2)*0.0000598834 and the differential equation becomes dU/dt - S/k/ln(2)*0.0000598834 = -kU => dU/dt = -k(U - S/k2/ln(2)*0.0000598834). Now do the substitution W = U - S/ k2/ln(2)*0.0000598834. Then dW/dt = dU/dt and the differential equation becomes dW/dt = -kW. The general solution to this differential equation is W = C*exp(-k(t-1980)), where C is some constant. Thus U = C*exp(-k(t-1980)) + S/k2/ln(2)*0.0000598834, V = C*exp(-k(t-1980)) + S/k2/ln(2)*0.0000598834 - S/k/ln(2)*(0.00388466 + 0.0000598834(t-1980)) and we get: T = C*exp(-k(t-1980)) + S/k2/ln(2)*0.0000598834 - S/k/ln(2)*(0.00388466 + 0.0000598834(t-1980)) + T1950 + S/ln(2)*(0.08959227 + 0.00388466(t-1980) + 0.0000299417(t-1980)2) Using the fact that T = T1950 when t = 1950 gives: 0 = C*exp(30k) + S/k2/ln(2)*0.0000598834 - S/k/ln(2)*(0.00388466 + 0.0000598834(-30)) + S/ln(2)*(0.08959227 + 0.00388466(-30) + 0.0000299417(-30)2) = C*exp(30k) + S/k2/ln(2)*0.0000598834 - S/k/ln(2)*(0.002088158) => C = exp(-30k)*S/k/ln(2)*(0.002088158 - 0.0000598834/k) Thus the differential equation can be rewritten as: T = exp(-k(t-1950))*S/k/ln(2)*(0.002088158 - 0.0000598834/k) + S/k2/ln(2)*0.0000598834 - S/k/ln(2)*(0.00388466 + 0.0000598834(t-1980)) + T1950 + S/ln(2)*(0.08959227 + 0.00388466(t-1980) + 0.0000299417(t-1980)2) For the sake of consistency with what I did with the IPCC scenario, I will take the quadratic Taylor approximation of exp(-k(t-1950)) around t = 1950: => T ≈ (1 - k(t-1950) + k2(t-1950)2/2)*S/k/ln(2)*(0.002088158 - 0.0000598834/k) + S/k2/ln(2)*0.0000598834 - S/k/ln(2)*(0.00388466 + 0.0000598834(t-1980)) + T1950 + S/ln(2)*(0.08959227 + 0.00388466(t-1980) + 0.0000299417(t-1980)2) => T - T1950 ≈ S/ln(2)*( (1 - k(t-1950) + k2(t-1950)2/2)/k*(0.002088158 - 0.0000598834/k) + 1/k2*0.0000598834 - 1/k*(0.00388466 + 0.0000598834(t-1980)) + 0.08959227 + 0.00388466(t-1980) + 0.0000299417(t-1980)2 ) = S/ln(2)*( 0.002088158/k - 0.0000598834/k2 - 0.002088158(t-1950) + 0.0000598834/k*(t-1950) + 0.001044079*k*(t-1950)2 - 0.0000299417*(t-1950)2 + 0.0000598834/k2 - 0.00388466/k - 0.0000598834/k*(t-1980) + 0.08959227 + 0.00388466(t-1980) + 0.0000299417(t-1980)2 ) Converting the (t-1980) terms to (t-1950) terms gives: T - T1950 ≈ S/ln(2)*( 0.002088158/k - 0.0000598834/k2 - 0.002088158(t-1950) + 0.0000598834/k*(t-1950) + 0.001044079*k*(t-1950)2 - 0.0000299417*(t-1950)2 + 0.0000598834/k2 - 0.00388466/k - 0.0000598834/k*(t-1950) + 0.001796502/k + 0.08959227 + 0.00388466(t-1950) – 0.1165398 + 0.0000299417(t-1950)2 – 0.00176502*(t-1950) + 0.02694753 ) = S/ln(2)*0.001044079*k*(t-1950)2 That is, the temperature response after 1950 due the 1950-2010 scenario should be approximately quadratic and the temperature change by 2010 should be approximately S/ln(2)*3.7586844*k. Note however that Craig Loehle assumed that the change in temperature response due to increased CO2 levels from 1950 to 2010 was linear. If the actual temperature is quadratic with a positive quadratic term then Craig Loehle underestimated the amount of warming from 1950-2010 when using a linear line of best fit. Using the value k ≈ 0.01579579 that was calculated earlier using the IPCC values, one obtains T - T1950 ≈ S/ln(2)*0.0000164920526*(t-1950)2. Plotting 0.0000164920526*(t-1950)2 from 1950 to 2010 and performing a least squares line of best fit gives a line equation of -1.9393 + 0.00098952t. This means that if one accepts the premises used by Craig Loehle (that the climate was basically in equilibrium prior to WW2 and the signal decomposition approach used to obtain the anthropogenic effect is correct) then the expected anthropogenic warming from 1950-2010 should be S/ln(2)*0.098952 per century. Craig Loehle calculates ‘transient climate response’ by taking this post WW2 warming, multiplies it by 0.54 and divides it by 0.326 to take into account the fact that according to CO2 records, only a 32.6% of a doubling in the CO2 levels occurred in the 54 year period from 1959 to 2013. This means that what Craig Loehle calculates as ‘transient climate response’ is actually S/ln(2)*0.098952*0.54/0.326 ≈ 0.23647*S. This means that the ratio between the true equilibrium climate sensitivity and what Craig Loehle calculates as ‘transient climate response’ is approximately 1/0.23647 ≈ 4.22887. But Craig Loehle assumes that what is measured is the true ‘transient climate response’ and uses the IPCC ratio of 1.81761 between equilibrium climate sensitivity and transient climate response. This means that Craig Loehle is underestimating both transient climate response and equilibrium climate sensitivity by a factor of 4.22887/1.81761 ≈ 2.3266. E. Earth was not in Equilibrium in 1950 My final issue with the Craig Loehle paper is that it assumes that the Earth was in climate equilibrium in 1950. This despite the fact that atmospheric CO2 concentrations were rising prior to 1950 and by 1950 were about 308 ppm, which is significantly higher than pre-industrial levels of 270 ppm. As a result, the assumption that the Earth was basically in climate equilibrium in 1950 is questionable. Furthermore, if the Earth was in climate equilibrium in 1950 and all of the anthropogenic warming prior to 1950 is due to CO2 emissions after 1950 then as shown in part D, the temperature response should be approximately quadratic and should be approximately 0 at 1950. However, what is actually observed is that the Earth is warming at a non-zero rate even at 1950. This suggests that some of the anthropogenic warming observed from 1950 to 2010 must be due to CO2 emissions that occurred before 1950. In order to separate warming due to emissions prior to 1950 to warming due to emissions after 1950, I used the information provided on Figure 1b of the Craig Loehle paper. Theoretically, I could try to reconstruct this data set and do the entire signal decomposition, but that is very time consuming. Instead, I tried to take the data points from the jpeg image of the graph. Of course, due to human error and pixilation, my values are slightly different from the true values. In any case, a linear line of best fit to the 1950-2010 data set I tried to obtain from Figure 1b is 0.040 + 0.0064*(year-1950). This corresponds to the actual line of best fit given of Figure 1b, which is 0.054 + 0.0066*(year-1950). The differences between the two values are most likely due to error when I tried to extrapolate the values of temperature anomaly from the graph. To try to account for this, I’ll multiply my values by 1.02, which gives a line of best fit of 0.041 + 0.0066*(year-1950). This gives the same slope, though a different intercept. However, the value of the intercept won’t be relevant for the calculations that I am about to do. I’ll assume that my modified values are ‘good enough’ and move on. So the reason I needed to infer the temperature anomaly data set used by Craig Loehle for the 1950-2010 period is because his simple linear fit doesn’t give me enough degrees of freedom to separate warming due to CO2 emissions prior to 1950 with warming due to CO2 emissions after 1950. Furthermore, it was argued in part E that the temperature anomaly over this time period should be approximately quadratic not linear. From figure 1b, it does appear that the rate of anthropogenic warming is increasing over the 1950 to 2010 time period. In any case, a quadratic least squares fit to my values gives an equation of temperature anomaly = 0.078918 + 0.0027006*(year-1950) + 0.0000644*(year-1950)2. Taking the derivative of the temperature anomaly with respect to the year gives the rate of temperature increase due CO2 emissions to after removing noise. This is 0.0027006 + 0.0001288*(year-1950). Note that when the year is 1950, the rate of temperature increase is nonzero. As explained earlier, this non-zero increase must be due to CO2 emissions that occurred prior to 1950. Furthermore, in part C it was shown that the climate decays by a factor of e towards its long run equilibrium in approximately 63.308 years. The temperature response after 1950 due to CO2 emissions that occurred prior to 1950 should be approximately E*(1 – exp(-k(year-1950))), where E is some unknown constant. The derivative of this with respect to the year is E*k*exp(-k(year-1950)). Setting the year to 1950 means that the rate of increase of temperature response per year is E*k. Setting this equal to the 0.0027006 that was obtained by the quadratic fit to the temperature response data suggests that E = 0.0027006/k. For the sake of consistency, I’ll again perform a quadratic taylor approximation of the exponential to get an approximation of the temperature response due to CO2 emissions prior to 1950, which gives: temperature response = 0.0027006/k*(k(year-1950)– k2(year-1950)2/2) ≈ 0.0027006*(year-1950) – 0.0013503*k*(year-1950)2. Subtracting the temperature response due to CO2 emissions prior to 1950 from the observed temperature response gives the temperature response due to CO2 emissions after 1950. This gives: temperature response due to CO2 emissions after to 1950 = 0.078918 + 0.0027006*(year-1950) + 0.0000644*(year-1950)2 - 0.0027006*(year-1950) + 0.0013503*k*(year-1950)2 ≈ 0.078918 + 0.000085729*(year-1950)2 if we use k ≈ 0.01579579, which was calculated in part C. Thus the difference between the temperature anomaly in a given year to the temperature anomaly in 1950 that is due to emissions that occurred after 1950 is approximately 0.000085729*(year-1950)2. However, in part D, it was shown that difference between the temperature anomaly in a given year to the temperature anomaly in 1950 that is due to emissions that occurred after 1950 should be roughly S/ln(2)*0.001044079*k*(year-1950)2. Equating this with the estimate from the last paragraph gives 0.000085729 = S/ln(2)*0.001044079*k. Isolating for the equilibrium climate sensitivity gives S = 0.000085729*ln(2)/ 0.001044079/k ≈ 3.603 °C. Dividing by the 1.81761 ratio suggests that the transient climate response is approximately 1.98 °C. F. Conclusion The Craig Loehle paper suggests that the transient climate response of Earth and the equilibrium climate sensitivity of Earth are approximately 1.09 °C and 1.99 °C respectively. While the Craig Loehle paper has a number of strengths such making few assumptions about the climate of the Earth, having a relatively small confidence interval and appealing directly to empirical data, the paper makes 3 assumptions significantly skew the estimation of these parameters. The 3 assumptions are that the logarithm of atmospheric CO2 concentrations is a linear function of time since WW2, Craig Loehle’s definition of ‘transient climate response’ is the same as what is used by the IPCC, and that the Earth roughly in climate equilibrium with respect to CO2 concentrations in 1950. In this post, I try to relax the three assumptions and perform a first order correction to the calculations of Craig Loehle. Relaxing the first two of these assumptions suggests that Craig Loehle underestimates both the transient climate response and equilibrium climate sensitivity by a factor of 2.33, though this gets offset slightly after relaxing the 3rd assumption. Overall, I find that after first order corrections, the transient climate response is 1.98 °C and the equilibrium climate sensitivity is 3.60 °C. These values are 81% higher than the values calculated by Craig Loehle. The 95% confidence intervals for these values are approximately the confidence intervals calculated by Craig Loehle multiplied by 1.81. Edit: My numbers are slightly off for a reason I explain in post #18. The corrected values are 1.62°C and 2.95°C for transient climate response and equilibrium climate sensitivity respectively. Edit 2: Later on, I make the claim that 2.95 C might be an overestimate of the ECS due to the assumption of constant decay towards equilibrium. This would only hold if I took the equilibrium to mean the Earth System Sensitivity rather than the ECS. So there is no reason to expect that 2.95 C is an overestimate of the ECS. Edit 3: Actually, since the rate of change in solar irradiance from 1850-1950 is larger than from 1950-2010, the assumption of constant 'natural warming' by Loehle results in an underestimate of climate sensitivity since natural warming should have been slower during the second period. As a result, 2.95 C is an underestimate of the best estimate of equilibrium climate sensitivity (due to the uncertainty, the result should not be significantly different from 3C). Edit 4: Actually, if you take into account the fact that only about 76% of changes in greenhouse gas forcing is due to CO2 (with CH4, N2O and other gases contributing the rest) then this suggests a climate sensitivity of approximately 2.25 C.

As requested by Michael Hardner, this is a thread to discuss emission scenarios and economic impacts of climate change. I’ll try to keep the original post relatively short, but I want to spend a paragraph or two on relevant aspects of this topic. Climate Sensitivity: Equilibrium climate sensitivity (ECS) is how much global temperatures will increase in the long run after a doubling of atmospheric CO2 due to all but the very slow feedbacks (primarily albedo feedbacks that have decay times on the order of centuries or millennia). ECS is the most common measure of climate sensitivity and the IPCC’s ECS range is from 1.5 C to 4.5 C. However, more recent studies strongly suggest that ECS is in the lower half of this range (1.5 C to 3.0 C) and I would argue that empirically evidence all but excludes the possibility of an ECS greater than 3 C. It takes a bit over a century to reach go from equilibrium and reach the new ECS after a doubling of atmospheric CO2. For shorter time scales, the transient climate response (TCR) is usually a better measure of how much warming you should expect due to CO2 doubling; the TCR is probably somewhere between 1.2 C and 2 C. The Earth System Sensitivity (ESS) is how much global temperatures change in the very long run due to a doubling of atmospheric CO2; ESS is probably between 2 C and 4C. Btw, radiative forcing due to CO2 concentrations is a roughly logarithmic function of CO2, which means the first doubling of CO2 concentrations causes roughly the same temperature increase as the second doubling of CO2 concentrations. You can see a more in depth discussion about climate sensitivity here: http://www.mapleleafweb.com/forums/topic/24202-what-is-the-correct-value-of-climate-sensitivity/ Uptake of CO2: After CO2 has entered the atmosphere, all of it doesn’t stay there forever. Some of it gets absorbed by plants, some of it gets absorbed by rocks, and some of it gets absorbed by oceans. By far oceans are the dominant source of up taking additional CO2. The behaviour of oceans roughly follows Henry’s law (https://en.wikipedia.org/wiki/Henry%27s_law), which basically means that oceans will try to be in equilibrium with the atmosphere with respect to how much dissolved CO2 the oceans hold. In the long run, Oceans absorb roughly 85% of emitted CO2, but in the short run the effect of oceans is only half. If one combines the effect of temperature increase due to emitted CO2 and the rate at which CO2 gets absorbed, maximal warming due to emitting CO2 occurs after roughly 1 decade (http://iopscience.iop.org/1748-9326/9/12/124002/article). Below is a diagram of the BERN model, which gives you an idea of what percentage of emitted CO2 remains in the atmosphere after a certain amount of time: Temperature-CO2 Feedback: As temperatures warm, oceans are less able to hold CO2 and permafrost melts, releasing CO2. As a result, warmer temperatures release CO2, which causes more warming. This is a positive feedback. Contrary to what the media may tell you, the strength of this feedback isn’t that large. Based on empirical data (be it Pleistocene ice core data or papers such as this http://www.nature.com/nature/journal/v463/n7280/full/nature08769.html)the strength of this feedback is very likely below 30 ppm/C. Polar Amplification: Polar regions are expected to warm faster according to both empirical data and climate models. The polar amplification factor is roughly 2.5. Which means that if global temperatures increase by 1C, then the temperature of polar regions increases by roughly 2.5 C; by contrast equatorial regions increase in temperature by approximately 2/3 the increase in global temperature. Below gives you a rough idea of which regions of Earth warm faster: Hurricanes and Extreme Weather Events: There is a lot of misleading information out there about the relationship between extreme weather events and climate change. Climate change doesn’t necessarily cause an increase in extreme weather events. In fact, since the temperature difference between poles and equator is expected to decrease due to global warming, and weather events on Earth try to act as heat engines that transport heat from equator to poles, many extreme weather events will decrease in magnitude or frequency. For example, Tornadoes in Tornado alley are expected to decrease. Most of the increase in hurricane frequency from 1980-2005 is actually due to the Atlantic Multidecadal Oscillation (AMO), not due to increases in greenhouse gases. Since studies discussing various extreme weather will likely be brought up in this thread, I would encourage skepticism about a study’s definition of extreme weather. Many studies will define extreme weather as any large deviation in observed weather/climate from the historic mean. These types of definitions are problematic because by definition any deviation (be it cooling or warming) from the historic mean is going to result in more extreme weather so these definitions don’t even satisfy the transitive property. These types of definitions would also suggest that a winter in Ottawa of -20C is less extreme than a winter of -5C, even though most people would consider -20C to be more extreme than -5C. Jet streams: Jet streams are expected to weaken due to climate change since climate change lessens the Earth’s temperature gradient. This could lead to more static weather conditions primarily in the Northern Mid-latitudes. For example, an increase in global temperatures by 3C may cause an increase in the frequency of prolonged droughts in Northern midlatitudes by 28% due to this effect. A thread on this topic can be found here: http://www.mapleleafweb.com/forums/topic/23461-effectsimplications-of-climate-change-on-jetstreams/. Precipitation Patterns: Contrary to what the media may suggest, climate change doesn’t mean that everything will become desert like. In fact it is arguably the opposite. As temperatures warm, air can hold more water (this follows the Clausius-Clapeyron relation which basically suggests that carrying capacity increases by 7% per 1 celcius). As a result, there is greater moisture transfer between oceans and continents due to climate change, so the continents get on average wetter even after taking evaporation into account as the Earth warms. This effect is clearly evident if one compares Earth’s vegetation today with Earth’s vegetation during the Last Glacial Maximum when global temperatures were 4 C cooler. However, things aren’t as straightforward as everywhere gets wetter. Due to how air circulates around the world, many deserts occur around 30 degrees North or South of the equator. Climate change may move this region poleward, which may cause places like California to become drier but Subsaharan Africa to get wetter. Also, depending on a place’s location relative to mountain ranges and land masses, places may become wetter or drier in response to changing wind patterns. Sea Level Rise: Sea rise this century will be half a meter with a confidence interval of plus or minus a quarter meter. This result is from the IPCC’s fifth assessment report and is fairly robust across a large range of emission scenarios. In the very long run (like after a two millenia) sea level rise will be more (approximately 3 m per degree of warming according to sea level differences compared to the Eemian (the last interglacial)). Ocean Acidification: As oceans absorb excess atmospheric CO2, they will become more acidic. There is a bit of a trade-off in that more ocean acidification means that there is less CO2 in the atmosphere so there is less global warming. Since the 18th century, Ocean pH has dropped by 0.11. It is expected to drop by about 0.25 by 2100. Even after the acidification, oceans will still be very alkaline in 2100 (pH of over 7.8). You can read more here: https://en.wikipedia.org/wiki/Ocean_acidification CO2 Fertilization Effect: As one increases atmospheric CO2, one increases the CO2 available for plants to perform photosynthesis. As a result, increased atmospheric CO2 results in more plant growth. Optimal atmospheric CO2 for plant growth is 1000-1500 ppm (which corresponds roughly to the higher atmosphere CO2 levels that were seen for most of the past 600 million years; our low atmospheric CO2 levels are relatively recent geologically speaking). Some types of plants are more affected by CO2 fertilization than other plants. C4 plants like corn evolved to adapt to our very low CO2 environment so aren’t as affected by the CO2 fertilization effect as C3 plants like Rice. Discount Rates: In order to compare present costs/benefits with future costs/benefits, one needs a discount rate. A discount rate basically accounts for the fact that a dollar today is more valuable than a dollar tomorrow even after one accounts for inflation. The reason for this is that you could take a dollar today and invest it and get a return. For developed countries the social discount rate is generally considered to be between 3% and 7%; it is even higher for developing countries. The Canadian Treasury Board and the US Office of Management and Budget typically suggest a discount rate of 7% for cost benefit analyses. However, many cost-benefit analyses for climate change will use absurdly low discount rates (usually 2.5% to 3.5%), although they don’t really give much basis for this. Using very low discount rates for climate change studies yet much higher discount rates of 7% for everything else seems really inconsistent to me. I suspect that one of the reasons climate changes studies use very low discount rates is due to people trying to dogmatically obtain a conclusion that favours mitigation policy. If you see a result in this thread that uses a very low or very high discount rate, please be skeptical. Based on studies I have seen in the past that try to obtain the social discount rate based on the saving-consumption behaviour of people in the US (which gives a discount rate between 5% and 6%) I think that a discount rate of 5% is reasonable. 2 C Target: Inevitably, this thread will likely bring up the 2 C target. Many people have this belief that we are safe if warming is below 2C relative to pre-industrial levels and then once we pass 2C all hell breaks loose and the world is doomed. This is not the case, the impact of climate change is not a step function, it is continuous; 2.1 C of warming will have slightly more impact than 1.9 C of warming, 2.3 C of warming will have slightly more impact than 2.1 C of warming, etc. There is no scientific basis or even economic basis for the 2C target, the 2C target is an arbitrary political target that was created 20 years ago because a German scientist (Hans Joachim Schellnhuber) felt that politicians were too stupid to understand all the nuance and complexity of climate change. RCP 8.5: The representative concentration pathways (RCPs), which are emission scenarios created by the IPCC, will inevitably get mentioned in this thread. I want to clarify something about RCP 8.5. RCP 8.5 is NOT a REPRESENTATIVE business as usual scenario, despite what articles or even scientific papers may say. It technically a business as usual scenario, but it was creating using extreme assumption after extreme assumption so it is not representative. I’ll list 3 reasons why it is not representative. Firstly, the RCPs are based on CMIP5 climate models. CMIP5 climate models have numerous problems (such as 30 W/m^2 zonal oscillations), but the biggest issue is that they are over sensitive. They have a median ECS of 3.2 C, which is inconsistent with empirical evidence, and the predictions made by CMIP5 models are very close to being falsified by recent data (as I explained in this thread http://www.mapleleafweb.com/forums/topic/24584-arcticantarctic-sea-ice-what-to-make-of-it/?p=1065868). Secondly, RCP 8.5 clearly overestimates future CO2 emissions based on empirical data and barely shows any impact of the population slowdown/decline that is expected to occur midcentury. For example, if I expect global population to follow a roughly logistic trend (this probably overestimates future population since the UN predicts global population will peak midcentury) and emissions per capita follow a roughly exponential trend, I get that global CO2 emissions per year will be 15 Gt of carbon in 2100. However, the RCP 8.5 has global annual CO2 emissions in 2100 of 27 Gt of Carbon. RCP 6.0 more closely resembles what is expected to happen under a no mitigation scenario. Lastly, RCP 8.5’s predictions of CH4 and N2O emissions are grossly inconsistent with what one should expect based on empirical evidence. RCP 8.5 predicts very large increases in CH4 and N2O after a rapid acceleration in emission rates, yet emissions per capita of both CH4 and N2O are decreasing. Rate of increase of atmospheric N2O and CH4 have both plateaued. Given that these are the next two most important greenhouse gases, RCP 8.5 is grossly overstates future warming. Realistically, the climate in 2100 should be approximately 2C warmer than current temperatures under a no-mitigation scenario.

Hi, My guest post on Judith Curry's blog has been posted here: http://judithcurry.com/2015/11/29/decision-making-under-uncertainty-maximize-expected-social-welfare/ This is a shorter version of a longer version here: https://curryja.files.wordpress.com/2015/11/expected-social-welfare-maximization-2.pdf The longer version has a lot more nuance, but I had to keep the word limit to around 4000 for the shorter version. If you want to make any comments on it, please post here. Expected social welfare maximization is a way to make decision making under uncertainty. This is especially applicable to climate change, where uncertainty is fairly high. Though it can be applied to pretty much any policy question, from immigration, to education, to healthcare.

http://arstechnica.com/science/2013/12/billion-dollar-climate-denial-network-exposed/ Strange how conservative posters on MLW have also been so blindly in support of bad science funded by corporate dollars. On one hand, we have climate change researched by scientists. On the other hand we have climate change deniers being pushed by billions of dollars donated from corporations and ultra wealthy conservatives into conservative think tanks. ... And for some reason... conservative partisans choose to support the people with the most to lose from changing business practices...

According to a new study by Stanford researcher Mark Jacobson, existing alternative energy technology can replace fossil fuels at a roughly comparable cost. This corrects the assertions made by some at MLW that we need fossil fuels for the forseeable future.

A new report commissioned by the World Bank paints a picture of a world convulsed by rising temperatures. We're doomed if we do nothing about climate change say experts. The thing is, even those who are making money from the status quo will be effected. So will they look at the impact this wil have on their grand children's world, and act? Will people show any urgency and stop pretending that it's not happening? Humans must immediately implement a series of radical measures to halt carbon emissions or prepare for the collapse of entire ecosystems and the displacement, suffering and death of hundreds of millions of the globe’s inhabitants, according to a report commissioned by the World Bank. The continued failure to respond aggressively to climate change, the report warns, will mean that the planet will inevitably warm by at least 4 degrees Celsius (7.2 degrees Fahrenheit) by the end of the century, ushering in an apocalypse. Link