Guest post by Pat Frank
Among other things, the paper showed that the air temperature projections of advanced GCMs are just linear extrapolations of fractional greenhouse gas (GHG) forcing.
The paper presented a GCM emulation equation expressing this linear relationship, along with extensive demonstrations of its unvarying success.
In the paper, GCMs are treated as a black box. GHG forcing goes in, air temperature projections come out. These observables are the points at issue. What happens inside the black box is irrelevant.
In the emulation equation of the paper, GHG forcing goes in and successfully emulated GCM air temperature projections come out. Just as they do in GCMs. In every case, GCM and emulation, air temperature is a linear extrapolation of GHG forcing.
Nick Stokes’ recent post proposed that, “Given a solution f(t) of a GCM, you can actually emulate it perfectly with a huge variety of DEs [differential equations].” This, he supposed, is a criticism of the linear emulation equation in the paper.
However, in every single one of those DEs, GHG forcing would have to go in, and a linear extrapolation of fractional GHG forcing would have to come out. If the DE did not behave linearly the air temperature emulation would be unsuccessful.
It would not matter what differential loop-de-loops occurred in Nick’s DEs between the inputs and the outputs. The DE outputs must necessarily be a linear extrapolation of the inputs. Were they not, the emulations would fail.
That necessary linearity means that Nick Stokes’ entire huge variety of DEs would merely be a set of unnecessarily complex examples validating the linear emulation equation in my paper.
Nick’s DEs would just be linear emulators with extraneous differential gargoyles; inessential decorations stuck on for artistic, or in his case polemical, reasons.
Nick Stokes’ DEs are just more complicated ways of demonstrating the same insight as is in the paper: that GCM air temperature projections are merely linear extrapolations of fractional GHG forcing.
His DEs add nothing to our understanding. Nor would they disprove the power of the original linear emulation equation.
The emulator equation takes the same physical variables as GCMs, engages them in the same physically relevant way, and produces the same expectation values. Its behavior duplicates all the important observable qualities of any given GCM.
The emulation equation displays the same sensitivity to forcing inputs as the GCMS. It therefore displays the same sensitivity to the physical uncertainty associated with those very same forcings.
Emulator and GCM identity of sensitivity to inputs means that the emulator will necessarily reveal the reliability of GCM outputs, when using the emulator to propagate input uncertainty.
In short, the successful emulator can be used to predict how the GCM behaves; something directly indicated by the identity of sensitivity to inputs. They are both, emulator and GCM, linear extrapolation machines.
Again, the emulation equation outputs display the same sensitivity to forcing inputs as the GCMs. It therefore has the same sensitivity as the GCMs to the uncertainty associated with those very same forcings.
Propagation of Non-normal Systematic Error
I posted a long extract from relevant literature on the meaning and method of error propagation, here. Most of the papers are from engineering journals.
This is not unexpected given the extremely critical attention engineers must pay to accuracy. Their work products have to perform effectively under the constraints of safety and economic survival.
However, special notice is given to the paper of Vasquez and Whiting, who examine error analysis for complex non-linear models.
An extended quote is worthwhile:
“… systematic errors are associated with calibration bias in [methods] and equipment… Experimentalists have paid significant attention to the effect of random errors on uncertainty propagation in chemical and physical property estimation. However, even though the concept of systematic error is clear, there is a surprising paucity of methodologies to deal with the propagation analysis of systematic errors. The effect of the latter can be more significant than usually expected.
“Usually, it is assumed that the scientist has reduced the systematic error to a minimum, but there are always irreducible residual systematic errors. On the other hand, there is a psychological perception that reporting estimates of systematic errors decreases the quality and credibility of the experimental measurements, which explains why bias error estimates are hardly ever found in literature data sources.”
“Of particular interest are the effects of possible calibration errors in experimental measurements. The results are analyzed through the use of cumulative probability distributions (cdf) for the output variables of the model.
“As noted by Vasquez and Whiting (1998) in the analysis of thermodynamic data, the systematic errors detected are not constant and tend to be a function of the magnitude of the variables measured.
“When several sources of systematic errors are identified, [uncertainty due to systematic error] beta is suggested to be calculated as a mean of bias limits or additive correction factors as follows:
“beta = sqrt[sum over(theta_S_i)^2],
“where “i” defines the sources of bias errors and theta_S is the bias range within the error source i. (my bold)”
That is, in non-linear models the uncertainty due to systematic error is propagated as the root-sum-square.
This is the correct calculation of total uncertainty in a final result, and is the approach taken in my paper.
The meaning of ±4 W/m² Long Wave Cloud Forcing Error
This illustration might clarify the meaning of ±4 W/m^2 of uncertainty in annual average LWCF.
The question to be addressed is what accuracy is necessary in simulated cloud fraction to resolve the annual impact of CO2 forcing?
We know from Lauer and Hamilton, 2013 that the annual average ±12.1% error in CMIP5 simulated cloud fraction (CF) produces an annual average ±4 W/m^2 error in long wave cloud forcing (LWCF).
We also know that the annual average increase in CO₂ forcing is about 0.035 W/m^2.
Assuming a linear relationship between cloud fraction error and LWCF error, the GCM annual ±12.1% CF error is proportionately responsible for ±4 W/m^2 annual average LWCF error.
Then one can estimate the level of GCM resolution necessary to reveal the annual average cloud fraction response to CO₂ forcing as,
(0.035 W/m^2/±4 W/m^2)*±12.1% cloud fraction = 0.11%
That is, a GCM must be able to resolve a 0.11% change in cloud fraction to be able to detect the cloud response to the annual average 0.035 W/m^2 increase in CO₂ forcing.
A climate model must accurately simulate cloud response to 0.11% in CF to resolve the annual impact of CO₂ emissions on the climate.
The cloud feedback to a 0.035 W/m^2 annual CO2 forcing needs to be known, and needs to be able to be simulated to a resolution of 0.11% in CF in order to know how clouds respond to annual CO2 forcing.
Here’s an alternative approach. We know the total tropospheric cloud feedback effect of the global 67% in cloud cover is about -25 W/m^2.
The annual tropospheric CO₂ forcing is, again, about 0.035 W/m^2. The CF equivalent that produces this feedback energy flux is again linearly estimated as,
(0.035 W/m^2/|25 W/m^2|)*67% = 0.094%.
That is, the second result is that cloud fraction must be simulated to a resolution of 0.094%, to reveal the feedback response of clouds to the CO₂ annual 0.035 W/m^2 forcing.
Assuming the linear estimates are reasonable, both methods indicate that about 0.1% in CF model resolution is needed to accurately simulate the annual cloud feedback response of the climate to an annual 0.035 W/m^2 of CO₂ forcing.
This is why the uncertainty in projected air temperature is so great. The needed resolution is 100 times better than the available resolution.
To achieve the needed level of resolution, the model must accurately simulate cloud type, cloud distribution and cloud height, as well as precipitation and tropical thunderstorms, all to 0.1% accuracy. This requirement is an impossibility.
The CMIP5 GCM annual average 12.1% error in simulated CF is the resolution lower limit. This lower limit is 121 times larger than the 0.1% resolution limit needed to model the cloud feedback due to the annual 0.035 W/m^2 of CO₂ forcing.
This analysis illustrates the meaning of the ±4 W/m^2 LWCF error in the tropospheric feedback effect of cloud cover.
The calibration uncertainty in LWCF reflects the inability of climate models to simulate CF, and in so doing indicates the overall level of ignorance concerning cloud response and feedback.
The CF ignorance means that tropospheric thermal energy flux is never known to better than ±4 W/m^2, whether forcing from CO₂ emissions is present or not.
When forcing from CO₂ emissions is present, its effects cannot be detected in a simulation that cannot model cloud feedback response to better than ±4 W/m^2.
GCMs cannot simulate cloud response to 0.1% accuracy. They cannot simulate cloud response to 1% accuracy. Or to 10% accuracy.
Does cloud cover increase with CO₂ forcing? Does it decrease? Do cloud types change? Do they remain the same?
What happens to tropical thunderstorms? Do they become more intense, less intense, or what? Does precipitation increase, or decrease?
None of this can be simulated. None of it can presently be known. The effect of CO₂ emissions on the climate is invisible to current GCMs.
The answer to any and all these questions is very far below the resolution limits of every single advanced GCM in the world today.
The answers are not even empirically available because satellite observations are not better than about ±10% in CF.
Present advanced GCMs cannot simulate how clouds will respond to CO₂ forcing. Given the tiny perturbation annual CO₂ forcing represents, it seems unlikely that GCMs will be able to simulate a cloud response in the lifetime of most people alive today.
The GCM CF error stems from deficient physical theory. It is therefore not possible for any GCM to resolve or simulate the effect of CO₂ emissions, if any, on air temperature.
Theory-error enters into every step of a simulation. Theory-error means that an equilibrated base-state climate is an erroneous representation of the correct climate energy-state.
Subsequent climate states in a step-wise simulation are further distorted by application of a deficient theory.
Simulations start out wrong, and get worse.
As a GCM steps through a climate simulation in an air temperature projection, knowledge of the global CF consequent to the increase in CO₂ diminishes to zero pretty much in the first simulation step.
GCMs cannot simulate the global cloud response to CO₂ forcing, and thus cloud feedback, at all for any step.
This remains true in every step of a simulation. And the step-wise uncertainty means that the air temperature projection uncertainty compounds, as Vasquez and Whiting note.
In a futures projection, neither the sign nor the magnitude of the true error can be known, because there are no observables. For this reason, an uncertainty is calculated instead, using model calibration error.
Total ignorance concerning the simulated air temperature is a necessary consequence of a cloud response ±120-fold below the GCM resolution limit needed to simulate the cloud response to annual CO₂ forcing.
On an annual average basis, the uncertainty in CF feedback into LWCF is ±114 times larger than the perturbation to be resolved.
The CF response is so poorly known that even the first simulation step enters terra incognita.
The uncertainty in projected air temperature increases so dramatically because the model is step-by-step walking away from an initial knowledge of air temperature at projection time t = 0, further and further into deep ignorance.
The GCM step-by-step journey into deeper ignorance provides the physical rationale for the step-by-step root-sum-square propagation of LWCF error.
The propagation of the GCM LWCF calibration error statistic and the large resultant uncertainty in projected air temperature is a direct manifestation of this total ignorance.
Current GCM air temperature projections have no physical meaning.