Today, once again, I will show that Dr. Pielke Sr. is correct:

Spatial variations in the Global Mean Surface Temperature do contribute to leading order to the anomalous radiative balance of the earth.

Why am I writing my third post on this?

Because, in response to my second post, a vistor in comments suggested that mine analysis had been refuted by Horatio Algernon’s attempted to estimate the size of the effect. Horatio concluded the effect is minuscule: this is incorrect. The incorrect answer was then disseminated in comments at a variety of places, and I think I need to illustrate the correct result here. (Horatio found his error and posted while I was writing this.)

For those who are fascinated by the entire “Series Expansion Kerfuffle”, this third analysis will have a side benefit: I will show that the terms I identified in the previous blog post on this subject, give a rather decent estimate of the effect of spatial variations on the anomalous forcing for the earth’s climate’s anomalous energy balance.

In today’s post, before embarking on “the boring proof”, I will discuss the the precise way in which this issue is important, so as to put it in context.

In what sense is this important? Empirical Estimates of Climate Sensitivity.

Recall the issue is dispute is a very precise one. The question is whether it is possible to obtain accurate estimates of climate sensitivity λ using this approximate equation and data for T’ alone.

(1)dH/dt = f -T’/λ

where H is the heat content of the earth’s climate system, f is the anomalous radiation forcing the climate and is defined relative to a reference baseline, and T’ is the global means surface temperature anomaly, also defined relative to a reerence baseline.

How is the anomalous energy budget used?

This equation is referred to as th anomalous radiation budget from the earth, which was labeled equation (1) in both Dr. Pielke’s article, and Eli Rabett’s article. In this equation

This equation has been used by a variety of scientist to estimate climate sensitivity, λ, from empirical data, including for example James Hansen, (2003) who, using the quasi-steady assumption, estimated λ= 3/4 ± 1/4 °C per W/m². Dr. Hansen, of NASA GISS, considered this empirical estimate sufficiently important to highlight with the attractive graphic shown to the left; he also took the time to compare the accuracy of estimates based on climate models to empirically based methods using these words:

Although climate models yield a similar climate sensitivity, the empirical result is more precise and reliable because it includes all the processes operating in the real world, even those we have not yet been smart enough to include in the models.

Others who have published estimates of λ based on empirical data for T’ include: Gregory et al. (2002) and Schwartz (2007). An alternative way equation (1) can be used is shown in, Raper et al. (2002) where a form of equation (1), with the approximation discussed here, was used to estimate the heat uptake of the ocean.

Note that, in the case of Raper et al. (2002), the estimate of climate sensitivity λ comes from GCM’s which do not use (1) in their radiation balance model. However, the approximate equation (1) was subsequently used outside the context of a GCM to estimate the current magnitude of dH/dt. When (1) is used to post-process output of GCM data in this way, treating it as “data”, then uncertainty is introduced in the subsequent estimate.

Why?

Because, even if λ is perfectly estimated by GCMs, the approximation Q_rad~T’/λ remains somewhat inaccurate in equation (1). Consequently, if (1) is used to estimate any other quantity during post processing, uncertainty is introduced during the post processing. In the case of Raper et al. (2002) the uncertainty would be introduced into the estimate of the uptake of heat in the ocean.

In that particular case, the uncertainty in equation (1) would in turn, affect climate scientists ability to estimate how much heat is “in the pipeline”.

So, we can see that equation (1) not only appears in the IPCC TAR, it is also used in various ways by climate scientists when they are trying to gain understanding of the earth’s climate response, and when they try to make projections of future behavior.

On to the boring equations!

The boring equation part will explore this question only: How much error is introduced in the anomalous radiation emitted from the earth’s surface by using the approximation Q_rad= T’/λ where T’ is the anomaly in the global mean surface temperature (GMST) and λ is the climate sensitivity? Those familiar with the physics of radiation know that for a surface with a spatially invariant emissivity, the full expression for the radiative anomaly Q_rad is

(2)Q_rad~ ε σ ∫∫T⁴– T_o⁴dA

where σ is the Stefan-Boltzman constant, ε is the emissivity of the earth’s surface and A is the surface area of the earth, T is the spatially varying temperature of the earth and and T_o is the spatially varying temperature in a baseline case .

Some will recall that in my previous post, I suggested that, mathematically, this approximation holds to leading order in temperature differences, (indicated with primes),

(3a)Q_rad~ (T’ /λ_o) ( 1 + 3 <δT_o²>/<T_o>²)+ (3 / λ_o)<δT_oδT’>/<T_o>

I immediately suggested that <δT_o²>/<T_o>²<<1. So, as a practical matter, the following approximation was suitable for the earth's climate: (3b)Q_rad~ T’ /λ+ 3 (<T_o> / λ)(<δT_oδT’>/<T_o>²)

where the terms in bold are additional terms that I believe should be included in the anomalous budget for the earth’s climate system when it is used to obtain estimates of items of interest to climate scientists, for example, climate sensitivity, λ.

I then performed a back of the envelop calculation to illustrate the magnitude of the terms in bold is roughly 15%-25% that of the T’ /λ term. When (1) is used to estimate λ, the error introduced by the approximation introduced an uncertainty of ±0.7K in the estimate for the climate sensitivity to doubled CO₂; this is one half the full uncertainty in the estimate for climate sensitivity to doubled CO₂, which the IPCC report estimates to fall between 1.5K and 4.5K.

However, my argument, based on a series expansion, and a back of the envelop estimate, did not convince all. In particular, Horatio Algernon, suggested that my math could be dis-proven using a simple thought experiment.

Horatio’s Thought Experiment

Horatio Algernon proposed that we could test whether or not spatial variations of the earth’s temperature contribute significantly to anomalous forcing by comparing three cases to the following baseline case:

Baseline case:
Let us assume the earth’s surface temperature temperature varies only with latitude and obeys this functional relationship:

(4)T = T_r – Δ_T cos(2φ)

where T_r is a reference temperature, required to specify the problem, Δ_T is the half the temperature difference from pole to equator and φ is the polar angle. Horatio suggested that we use T_r=275K, and Δ_T = 25K as the base-case. (I will henceforth add subscripts ‘o’ to all quantities corresponding to the basecase.

For this baseline, we can find the average temperature by integrating over area. Because temperature varies only with latitude, the integral formulation for the average is:

(5) <T> = (1/2) ∫ {T_r – Δ_T cos(2φ) }sin(φ) dφ

From now on, I will often express the area integral using <>, which is more compact. This symbol is commonly used as an averaging operator. So, by definition:

(6) <Y> =(1/2) ∫ Y sin(φ) dφ
for any quantity ‘Y’.

To obtain the average temperature, (5) can be integrated however you wish. Because additional more complicated integrals are coming up, I did so numerically. Using the reference values T_ro= 275K, and Δ_To = 25K, the area average of the surface temperature is <T_o>= 288.3 K.

Next, using the previously described based case, Horatio suggested we compare the anomalous radiation balance for three cases; which I will modify slightly to match values I consider more applicable to the earth’s surface temperature:

• Case 1: The temperature profile of the earth’s surface temperature shifts such that the average temperature remains <T>= 288.3 K but the temperature of the pole warms 1 K relative to the mean and the temperature of the equator decreases 1K relative to the mean. For this case T’=0K, and Δ_T = 23K. (Note, from a computation point of view, when Δ_T changes, the this case the reference temperature of T_r must be adjusted to maintain <T>= 288.3 K.)
• Case 2: The temperature of the earth’s surface increases +0.6 K everywhere. That is, T’=+0.6K; Δ_T = 25K. ¹
• Case 3: The earth’s GMST temperature rises +1K, and the poles warm 1K relative to the equator. So, T’=1K; Δ_T = 23K

Horatio suggested that we compare the anomalous forcing estimated in Case 2 is approximately equation to Case 3. If it is, then the extra terms I propose in equation (2) don’t matter. However, the two don’t match, then the extra terms, which correspond to the temperature shift in Case 1, do matter.

Let’s just do it with no approximation!

Calculate the total radiation loss and anomalous forcing.

Returning to explicit integral formulation for averaging, for a surface with constant emissivity of 1, the total energy emitted per unit area is:

(7) F = (σ/2) ∫ { (T_r – Δ_T cos(2φ))⁴ }sin(φ) dφ

This can be integrated a number of ways. I did so numerically and obtained F_o= 3.7E+02 W/m^2 ; this is included in table 1 below and represents the base case. The results for cases 1-3, using appropriate values for T_r and Δ_T are also included.

Note when examining total forcing, F_o, the differences between all cases are less 1%.

To obtain the anomalous forcing of interest in equation (1) and to this whole discussion, we subtract the total forcing in the base case from that found in any other case. The values for f=F-F_o are shown in table 1.

Case	<T> K	ΔT K	F  Watt/m2	T’ =<T>-<To> K	f = F-Fo Watt/m2	%
Base Case	283.33	25	3.71E+02	0	0	—
Case 1	283.33	23	3.72E+02	0	0.5	13%
Case 2	283.93	25	3.74E+02	0.6	3.1	87%
Case 3	283.93	23	3.75E+02	0.6	3.6	100%

Now, let us compare the error introduced by neglecting spatial variation in the earth’s surface temperature when estimating the anomalous forcing. This can be done by comparing the anomalous forcing for case 2 to case 3. Examining the table, we see that case 2 captures only 87% of the total forcing in case 3.

This means that if we fail to account for the spatial variations in temperature in this case, we will introduce an error of 13% in any estimate of the climate sensitivity obtained using equation (1).

How does this compare to the estimate based on Lucia’s Series Expansion?

I know anyone who plowed through this far now wishes to discover how the today’s 13% error compares to the estimate based on my series expansion. It’s easy enough to compute for any particular profile.

Recall, in my earlier post, I said the ratio of the effect of the temperature shift alone to that of a uniform increase in the earth’s surface temperature could be estimated using the equation:

(8)3 <δT_oδT’>/(<T_o>T’)

So, for this profile, I computed the covariance <δT_oδT’>= 8.9K^2numerically for case 3 Inserting, I find f1/f2 ~ 16.2%; this corresponds rather well with f1/f2 =15.3% found using the exact computation. ( The difference, which results in an error of roughly 1% when computed the desired value f2, is easily attributed to the <δT_o²> terms whose contribution I advised neglecting as small.)

So, this exercise confirms that

1. The spatial variations in temperature do matter when one uses IPCC equation (1) to study the anomaly in the earth’s energy balance. (That is: Dr. Pielke Sr. is correct.)
2. The terms I suggested result in a reasonable approximation of this effect.

So, why did Horatio get the wrong answer.

On the one hand, I’d like to simply say “Mistakes were made.” and leave it at that. (While i was typing this, Horatio himself appeared and admitted he had made a mistake.)

However, I think it’s worthwhile to explain the error, as it appears that several bloggers want desperately to do prove things using series expansions of functions inside averaging operators. So, it is useful to explain the main error I think Horatio made. (I may be mistaken though. He is explaining what he considers to be his main mistake in comments, and I may be misunderstanding what he did in his original post. .

I think that in Horatio’s original post, he assumed he could estimate a term that looked like this

(9a) ∫ {T_local,o³ T’_local }sin(φ) dφ
where the subscript ‘local’ indicates a spatially varying quantity

using

(9b) { ∫ T_local,o³ sin(φ) dφ }{ ∫ T’_local sin(φ) dφ
}
Unfortunately, this doesn’t work because the quantities he had defined, T_local,o and T’_local vary with polar angle φ.

In fact most statisticians and engineers familiar with statistical notation will recognize my <> notation as an averaging operator. Translating the integrals to the more compact notation, and substituting more abstract variables X and Y, I think Horatio did this: <X³ Y>=<X³><Y> where both X and Y vary locally, and can be thought of as a randomly varying quantity. (I suspect the difficulty was not conceptual, but was the result of failing to use explicit notation to distinguish already averaged quantities and locally varying quantities.)

Interestingly, it is possible to relate the “error” Horatio due to neglecting spatial gradients reported in first post was proportional to the 3 <δT_o²>/<T_o>² terms in (3a), which are, indeed, quite small.

Conclusions

Just to end this post on something other than math, the conclusions are:

1. Dr. Pielke Sr. correctly identified that effect of spatial variations in earth’s surface temperature should be considered when using the IPCC equation for the anomalous energy balance of the earth’s climate.
2. The terms I derived to estimate this effect give good results when compared to a full solution of a simplifed analytical temperature profile suggested by Horatio Algernon.
3. Horatio Algernon found his own error before I posted. He communicated the error to me immediately. (I once discovered to my horror that a result posted on a blog was incorrect becaused I’d used log() instead of ln(). So, I can hardly say I don’t make similar errors!)
4. I hope to heaven I don’t have to reprove this again!

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End Notes:

1. I modified case 2 to reflect the earth’s current temperature anomalie which is approximately 0.6K. temperature shift from pole to equator has been 1K or greater, based on monthly average values. The full analysis requires instantaneous values, which suggests even greater shifts. So, my numbers bring Algernon’s value into better alignment with the changes in the earth’s climate system, while still tending to underestimate the effect of spatial variations.

Update

Update: Feb. 21, 3:30 pm CST: I updated some text to reflect that Horatio did not write his post in response to mine. He was commenting on Dr. Pielke’s post. The post came to my attention when a visitor alerted me to it– and I am under the impression the visitors intention was to tell me my analysis had been refuted.