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What is the correct value of Climate Sensitivity?

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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.

Edited by -1=e^ipi

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The correct answer is 17

17 what? 17 Celcius? What is your basis for this? 17Celcius is way outside even the IPCC's most alarmist claims and I would be surprised to see even 1 peer reviewed Journal that had 17 Celcius in its 95% confidence interval for equilibrium climate sensitivity.

Edited by -1=e^ipi

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17 what? 17 Celcius? What is your basis for this? 17Celcius is way outside even the IPCC's most alarmist claims and I would be surprised to see even 1 peer reviewed Journal that had 17 Celcius in its 95% confidence interval for equilibrium climate sensitivity.

He meant to say 1.7.

Nah - we're both pulling your leg.

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Sorry - correction. Re evaluated the formulas and ran it through my greatest resource - Deep Thought ;

THE ANSWER IS ........... 42!

Is trolling a thread allowed according to the forum rules?

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Can we look forward to the IPCC announcing a complete reassessment of it's conclusions on the basis of this pivotal paper?

I haven't seen anything in the news yet...I just don't understand it. Why is that?

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Here is a justification for the logarithmic relationship between CO2 and temperature for those who were wondering about the origin of the relationship.

G. Simple justification for logarithmic relationship between CO2 concentrations and temperature:

Originally, I was going to derive a constant lapse rate (rate at which temperature decreases with altitude) and derive that air density decreases approximately exponentially with altitude in order to show that increasing CO2 by a factor α increases the altitude at which you have a certain CO2 concentration by ln(α) and as a result the black body radiation that escapes into outer space decreases by a factor proportional to ln(α) (using the Stefan-Boltzman law and the fact that temperature changes are small relative to the absolute temperature). However, that derivation fails once you take into account the fact that the amount black body radiation at a certain altitude depends on the density of air at that altitude and not just temperature. It seems like there are a lot of ‘derivations’ out there that give the logarithmic relationship, but many of them aren’t valid.

Probably the most compelling justification of the logarithmic nature is as follows: Consider the absorption spectra of CO2. Because of CO2, the Earth’s atmosphere is basically opaque to radiation of around 15 μm, and mostly transparent at other wavelengths. When atmospheric CO2 increases, the opaque wavelengths will remain opaque and the transparent wavelengths will remain transparent. Only a small set of transparent wavelengths will become opaque.

By the same physics that justify the Beer-Lambert law, the transmission coefficient of radiation of wavelength λ through the atmosphere (assuming only the 15 μm absorption line for CO2 is relevant) is exp(-k*CO2_ppm*α(λ)), where k is some positive constant and α(λ) is a function of the wavelength that depends on the spectral line of CO2 in the Earth’s atmosphere. I.e. more CO2 means that more radiation is absorbed so a lower transmission coefficient.

If the spectral line shape of CO2 absorption around 15 μm is approximately Gaussian, then α(λ) ≈ c*exp(-(λ-a)^2/b), where a, b and c are positive constants. This means that the transmission coefficient is approximately exp(-k*CO2_ppm*c*exp(-(λ-a)^2/b)). Suppose that the edge of the opaque wavelengths λ0 corresponds to a transmission coefficient of τ. Then τ = exp(-k*CO2_ppm*c*exp(-(λ0-a)^2/b)) => ln(τ) = -k*CO2_ppm*c*exp(-(λ0-a)^2/b) => ln(τ-1)/k/c/CO2_ppm = exp(-(λ0-a)^2/b) => ln(ln(τ-1)/k/c/CO2_ppm) = -(λ0-a)^2/b => b*ln(k*c*CO2_ppm/ln(τ-1)) = (λ0-a)^2 => (λ0-a)^2 = u + v*ln(CO2_ppm), where u and v are constants. => |λ0-a| = sqrt(u + v*ln(CO2_ppm)). This means that the bandwidth of the opaque wavelengths is 2|λ0-a| = 2*sqrt(u + v*ln(CO2_ppm)).

Since the bandwidth is relatively small, the intensity of black body radiation from the Earth over the opaque wavelengths is approximately constant. As a result, the radiative forcing due to CO2 is proportional to sqrt(u + v*ln(CO2_ppm)).

However, the assumption of a Gaussian spectral line shape isn’t that realistic. Phenomena that cause Lorentzian broadening plus multiple CO2 spectral lines in the 15 μm region that are caused by all the different modes of vibration of the CO2 molecule broaden the tails of the spectral line shape beyond Gaussian. Figure 1 in http://www.princeton.edu/~lam/documents/LamAug07bs.pdf shows the natural logarithm absorption factor of CO 2 in the Earth’s atmosphere. From the figure it appears that the natural logarithm of the absorption factor of CO2 decreases approximately linearly rather than quadratically in the tails.

Since the edge of the opaque wavelengths should be in the tails, this suggests that if one performs similar calculations as I did with the Gaussian spectral line shape, one does not end up with the square root (the Princeton pdf shows this using a simple Taylor approximation). So radiative forcing due to CO2 is proportional to x + y*ln(CO2_ppm), where x and y are unknown constants.

If the Earth is in radiative equilibrium then 0 = h*T4 – J – (x + y*ln(CO2_ppm)), where h is some positive constant, the first term is the black body radiation per unit area that the Earth emits into outer space due to the Stefan-Boltzman Law, J is the amount of solar radiation per unit area heating the Earth plus other constant factors, and the 3rd term is the radiative forcing. Isolating for temperature gives T4 = q + r*ln(CO2_ppm), where q and r are constants.

If we let T = T0 + ΔT, where ΔT is the increase in global temperature due to CO2 then since T0 >> ΔT, one can perform a Taylor approximation to get (T0 + ΔT)4 ≈ T04 + 4ΔT*T03 ≈ q + r*ln(CO2_ppm) => T0 + 4ΔT ≈ q*T0-3 + r* T0-3*ln(CO2_ppm) => 4T0 + 4ΔT ≈ 3T0 + q*T0-3 + r* T0-3*ln(CO2_ppm) => T0 + ΔT ≈ (3T0 + q*T0-3 + r* T0-3*ln(CO2_ppm))/4 => T = T0 + ΔT ≈ m + n*ln(CO2_ppm), where m and n are constants.

Also, here is an oxford paper that gives a better justification: https://dl.dropboxusercontent.com/u/101453219/radiative-transfer.pdf

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Have you tried getting that published, -1=e^ipi?

No I haven't. I have other priorities right now. Also, I would have to polish things up a lot if I wanted it published.

Can we look forward to the IPCC announcing a complete reassessment of it's conclusions on the basis of this pivotal paper?

I haven't seen anything in the news yet...I just don't understand it. Why is that?

How is anything I wrote contradicting the IPCC? Rather I demonstrated that a very low estimate of equilibrium climate sensitivity (1.99 Celcius, which TimG referred to in the past) is an underestimate and performed a correction to show that the actually climate sensitivity of the data set is closer to 3.60 Celcius, which is in line with the claims of the IPCC.

Edited by -1=e^ipi

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Also, another reason why I'm really skeptical of these very low (less than 2°C) estimates of climate sensitivity is because they would suggest a huge divergence between what is expected by basic physics and what is observed.

http://www.globalwarmingequation.info/global%20warming%20eqn.pdf

The warming effect from CO2 alone suggests and equilibrium climate sensitivity of 1.15 °C.

The add water vapour feedback effect increases this by about 0.59°C to 1.74 °C.

Edit: this is a mistake. the water vapour feedback would be an additional 1.14 °C, not 0.59°C, which brings sensitivity to 2.29 °C.

If the less than 2°C estimates of climate sensitivity are correct then it basically suggests that the effect of feedback effects other than water vapour is basically zero. I understand there are negative feedback effects such as cloud formation, but I still find it hard to believe that the effect would be so strong to get a climate sensitivity estimate that low.

Edit: woops, I meant basically zero not less than zero.

Edited by -1=e^ipi

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No I haven't. I have other priorities right now. Also, I would have to polish things up a lot if I wanted it published.

Seems like it might be worth considering for you. Seems like you have a reasonable body of original research/calculations there, and MLW isn't exactly the best audience for it.

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No I haven't. I have other priorities right now. Also, I would have to polish things up a lot if I wanted it published.

How is anything I wrote contradicting the IPCC? Rather I demonstrated that a very low estimate of equilibrium climate sensitivity (1.99 Celcius, which TimG referred to in the past) is an underestimate and performed a correction to show that the actually climate sensitivity of the data set is closer to 3.60 Celcius, which is in line with the claims of the IPCC.

I thought you might be trying to address the alarmism which would be a pretty noble and important goal. Has it occurred to you that saving the human economy trillions might be worth a little of your time? You'd be a hero, set for life, women would throw themselves at your feet and schools would be named after you. Just sayin'.

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Okay, I see 1 flaw with what I did in part D. When the quadratic Taylor approximation after solving for the differential equation doesn’t properly take into account that the rate of increase ln(CO2_ppm) is increasing from 1950 – 2010. So I might be overstating how much Craig Loehle is understating his estimate of climate sensitivity. Instead I’ll do a simple numeric approximation of what should occur from 1950 – 2010 and take a linear regression of that.

dT/dt = k*(A + S/ln(2)*ln(CO2_ppm) - T), where k ≈ 0.01579579
ln(CO2_ppm) ≈ 58.2316423 + 0.00388466(t-1980) + 0.0000299417(t-1980)2
≈ 58.14205003 + 0.002088158*(t-1950) + 0.0000299417*(t-1950)2

Setting dT/dt = 0 when T = 1950 gives:
0 = k*(A + S/ln(2)*58.14205003 – T1950)
=> A = T1950 - S/ln(2)*58.14205003
=> dT/dt = k*(T1950 + S/ln(2)*(0.002088158*(t-1950) + 0.0000299417*(t-1950)2) - T)

Let ΔT = T - T1950 be the temperature anomaly. Then
dΔT/dt = 0.01579579*(S/ln(2)*0.002088158*(t-1950) + 0.0000299417*(t-1950)2 - ΔT).

I’ll perform a simple numerical approximation of this with the initial condition ΔT1950 = 0 and recursion relation ΔTt – ΔTt-1 = 0.01579579*(0.002088158*(t-1950) + 0.0000299417*(t-1950)2 - ΔTt-1).

Note that I have omitted S/ln(2) here to make it comparable to what I did in part D.

A least squares line of best fit to this numerical approximation gives temperature anomaly = -2.36655 + 0.0012073*t, where as I used 1.9393 + 0.00098952t in part D. This means that I underestimating the ratio of warming from 1950 to 2010 and the equilibrium climate sensitivity by about 28.04%. This means that my values of 1.98°C and 3.60°C for transient climate response and equilibrium climate sensitivity are overestimates. After this correction, transient climate response and equilibrium climate sensitivity should be 1.62°C and 2.95°C respectively. This is more in line with the ~3°C values which appear to be the best estimates of climate sensitivity in the scientific literature.

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Seems like it might be worth considering for you. Seems like you have a reasonable body of original research/calculations there, and MLW isn't exactly the best audience for it.

I'm considering it. I would like to publish, but the issue is just finding time in my busy schedule. As for audience, the calculations are more motivated by self-interest than desire for an audience.

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This is more in line with the ~3°C values which appear to be the best estimates of climate sensitivity in the scientific literature.

Why should sensitivity be a constant?

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Why should sensitivity be a constant?

It isn't constant, but constant climate sensitivity is a simplifying assumption many make to perform empirical estimations. Though I would argue that constant climate sensitivity is a reasonable approximation when discussing the expected anthropogenic climate change.

I would expect that climate sensitivity decreases with temperature. Water vapour is an approximately exponential function of temperature according to the clausius-clapeyron relation. The radiative forcing effect is a logarithmic function of concentrations, so this means that the water vapour forcing effect is roughly a linear function of temperature. However, if cloud volume is roughly proportional to water vapour, and the surface area of clouds is roughly proportional to volume to the power of 2/3, then cloud area should be roughly an exponential function of temperature. Since cloud cover is a negative feedback mechanism by increasing albedo, where as water vapour radiative forcing is a positive effect, the overall effect should be that climate sensitivity decreases with temperature.

In addition, as the polar ice caps melt, eventually you run out of ice to melt and the remaining ice is closer to the poles where it gets less direct sunlight so the albedo effect of melting glaciers becomes less relevant. This is one of the reasons why I think that many of the paleoclimate climate sensitivity estimations by people such as James Hansen greatly overstate climate sensitivity (when they claim climate sensitivity is 4-6°C) since the feedback effect of melting glaciers is much higher when you have Glaciation all the way to 40°N in an ice age than what we have today. I was going to create another thread on the absurdity of the 2°C / 450 ppm target, but that effort got derailed a few months back because I became busy with other things.

Edited by -1=e^ipi

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I'll also point out that the simplifying assumption of constant decay rate towards equilibrium that I made in my calculations doesn't necessarily hold. In reality, some feedback mechanisms have a much lower timescale than other feedback mechanisms. The net effect is that initial decay rate towards equilibrium should decrease over time in response to an initial displacement. As a result, I may have slightly underestimated the percentage of equilibrium warming that should have occured since 1950. If this is the case then 2.95°C might be slightly too high. Though probably not by much since the definition of transient climate response occurs over roughly the same time scale (69.66 years vs 60 years) as the 1950-2010 data set.

I could try to relax the assumption of constant decay rate, but then in order to perform calculations such as accounting for warming from 1950-2010 that is due to CO2 emissions prior to 1950) I may need additional degrees of freedom to work with. But if I try to get more degrees of freedom from the temperature anomaly data set, I will start to reduce statistical significance and the cubic or higher order terms may not be statistically significant.

The overall message I want to convey to other readers is to be skeptical of all scientific papers and not just blindly accept what is in the abstract or conclusion as absolute truth (as I have seen some posters such as Waldo perform many times). I have seen good papers with mediocre abstracts and terrible papers with misleading abstracts; as the expression goes 'the devil is in the details'. There are often preverse incentives in the scientific community (such as the flawed peer review process, presence of confirmation bias, the mechanism by which researchers get funding, and the mechanism by which professors get tenure) that result in scientists using unscientific methodology, misrepresenting data or sometimes comitting fraud. This occurs even in fields that are not politicized such as global warming (I have seen it first hand).

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I was planning to get around to reading that magnum opus OP, but still haven't gotten around to attempting to wade through it yet. But, there is something that jumped out at me early in:

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.

So, first I'd like to know what is or how long is "the long run?" Because one uncomfortable fact about the effects of warming I've come across lately is that it takes oceans literally centuries to completely circulate heat that's been diffused down to lower layers back up to the surface and back into the atmosphere. And so far, the reports that are being produced for the IPCC and most other climate studies only ask what for a prediction of warming 10 to 50 years into the future/ not 2, 3 or more centuries! So, the takeaway for me is that the full effects of the CO2 and other GHG's we've dumped into the atmosphere so far, are not going to be completely realized until long after we are dead and gone...but, somebody in the future will have to deal with them.

In the meantime, if you make a claim that you...and apparently you alone, have the correct value for climate sensitivity, how far into the future are those residual heat effects being calculated?

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If he and he alone has the correct values for climate sensitivity, then he should be sending them to an academic journal for publication.

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Given the economic and geo-political stakes surrounding the issue of action vs inaction on AGW this could be a case of eminent domain. I can understand how fame, recognition and being thrust into the limelight might give him/her pause to consider but I really think the prudent and responsible thing to do is to alert authorities about -1=e^ipi's research.

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If the less than 2°C estimates of climate sensitivity are correct then it basically suggests that the effect of feedback effects other than water vapour is less than zero. I understand there are negative feedback effects such as cloud formation, but I still find it hard to believe that the effect would be so strong to get a climate sensitivity estimate that low.

Why? Clouds are affected by the concentration of water vapour. It seems fairly logical that an increase in water vapour will lead to an increase in clouds which, in turn, provide a negative feedback effect. The logic is only complicated because a lot depends on the type of clouds that form (i.e. some increase warming and some decrease it). Also, sensitivity is likely non-linear (i.e. higher at the end of ice ages, slower as ice retreats and plants grow) and you could see different types of clouds form as temperatures change. Edited by TimG

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