In my previous post, I discussed a Global Climate Change Blog Kerfuffle over the IPCC’s equation describing the radiative balance. The Kerfuffle involves one of the conclusions Dr. Roger Pielke Sr. reported in a recent peer reviewed article and two blog posts. .

Dr. Pielke’s fuller point is rather arcane and relates to the ability of scientists to accurately estimate the magnitude of the climate sensitivity, λ, based on Global Mean Surface Temperature anomalies, T’, as measured and reported using a very specific equation in an IPCC document. Dr. Pielke Sr. made several points regarding the uncertainties associated with determining the λ, all related to issues associated with the anomaly, T’.

The following paragraph in particular bothered Eli Rabett

“Indeed, it is easy to show that weighting by (T+T’)**4 significantly emphasizes the lower latitudes, since the relationship is to the 4th power of temperature. I look forward to your analysis as we have recommended.”

Eli’s response was:

[This] is just wrong, as anyone who learned about series expansions of functions in Cal I would know.

Turns out Eli is wrong; Dr. Roger Pielke Sr. is exactly right.

The correctness of Dr. Roger Sr.’s claim can be shown either using series expansions of functions or the simpler arithmetical method Dr. Roger Sr. used. Both give the same results for the magnitude of the effect Dr. Roger Sr.’s describes. (as they should.)

Now, it is my opinion, that when possible, it is much wiser to rely on simple arithmetic to illustrate the importance of a phenomena: One is less likely to go horribly wrong. However, as Eli claims to have dis-proven Dr. Roger Sr.’s arithmetic through use of more complicated series expansions of non-linear functions, I think it’s necessary to illustrate how to do this correctly. The terms describing the phenomenon Dr. Roger Sr. will be shown to be leading order effects.

For those who don’t want to read the math, the main result, in term of phenomenology follow. With regard to the question “Do neglecting spatial variations in surface temperature introduce uncertainty when estimating climate sensitivity using the IPCC equation in question?”

1. Spatial variations in temperature do matter. If I’m not mistaken in my math, we could fix up up the IPCC equation to include their effect, resulting in:

   (1)dH/dt = f -( T’/λ + (3/λ) <δT_oδT’> /<T_o> )

   where H is the heat content of the land-ocean-atmosphere system, f is the radiative forcing (i.e. the radiative imbalance), λ is called the “climate feedback” parameter, <T_o> is the absolute value of the global means surface temperature in the reference case, δT_o is the difference between a local surface temperature and the GMST in the reference case (denoted with ‘o’ subscripts) , and δT’ is the difference between the local surface temperature and the GMST at the current time (t).

   The final term on the right hand side of (1), shown in bold, describes the effect of variations in the spatial distribution of temperature that exist, and may change as the planet warms (or cools.) Mathematically, the term is leading order in temperature differences, noted with primes ‘.

2. A back of the envelope estimate also indicates the magnitude of the effect of spatial variation is sufficiently large to retain in (1). Using current measured values of the anomlaie T’ and the spatial variations in temperature suggest the new term is roughly 15%-25% the size of the original linear term. Neglecting this physical effect could account for a roughly 1/3 to 1/2 the uncertainty in the estimate for the climate sensitivity to doubled CO₂ in the IPCC estimate of climate range. (Note to those who read yesterday’s post: This uncertainty is identical to that obtained using Atmos’s estimate of the error in T’)
3. If someone wishes to obtain an empirical estimate of λ_2xCO2 with uncertainties less than ±0.7K, it is important to account for the effect of spatial surface temperature variations in some way. Since the current uncertainty range is thought to be ±1.5K, this means any sensible researcher should consider these variations.

The main results, with regard to the blog kerfuffle, are:

1. Roger Sr.’s arithmetic was correct.
2. Eli’s series expansion was inadequate.

The Boring Proof.

The blog kerfuffle is related to this IPCC equation, which is supposed to be an approximation for the energy balance of the earth, expressed in terms of the “Global Mean Surface Temperature Anomalie”, T’, which I will define more precisely later. The equation in question is:

(2)dH/dt = f -T’/λ
where H is the heat content of the land-ocean-atmosphere system, f is the radiative forcing (i.e. the radiative imbalance), and λ is called the “climate feedback” parameter.

The “Series Expansion Kerfuffle” relates only to one term, which I will call Q_rad. It’s supposed to describe the excess radiation losses from the earth that occur as its surface temperature responds to forcing due to greenhouse gases (or anything for that matter.) In (2) this term is approximated as: Q_rad ~ T’/λ.

That approximation for Q_rad accounts for extra heat lost by radiation when the average surface temperature of the planet warms. In so far as it describes that effect, the term is linearized. As far as I can tell, no one is worried about that linearization.

So, what’s the problem? Roger Sr. is concerned that we can’t use historical records for T’ to estimate λ for a number of reasons. The one important to this kerfuffle is this: Equation (2) is missing terms that arise as a result of the spatial variations in the earth’s surface, and Dr. Roger Pielke Sr. believes these terms matter.

Fuller Representation of Radiative Losses

How does Dr. Roger Pielke Sr. describe the issue?

As an example, assume a region of the Earth with a base temperature of 270K and another region with a base temperature of 300K. The difference in the outgoing long wave radiation (assuming blackbody behavior where the emission is proportional to T**4) results in a 34% greater emission from the warmer location. Adding a temperature increase of 1K to each location results in a 38% greater change when this increase is applied to the warmer temperature (i.e. comparing the difference between the incremental change in outgoing long wave radiation at the cold and warm locations).

Dr. Roger Sr. is i correct. Full expression for the radiative flux should include a T⁴ dependence where T is an absolute temperature, not the temperature anomaly and it should also account for the spatial variations.

So, lets’ fix up equation (2) to account for both features.

Since temperature varies over the surface of the earth, a more detailed representation of the total radiative heat losses should be replaced by a surface integral of the form;

~ σ ∫∫ εT⁴ dA
where σ is the Stefan-Boltzmann constant, ε and T are respectively the spatially varying emissivity and integration is performed over the surface area of the earth, A.

However, equation (2) was obtained by taking a difference with a reference case which means that f is a forcing relative to some absolute value of forcing, F_o tht causes the temperature on the surface of the earth to achieve some reference value T_o. So, any instantaneous local anomaly T’ is defined relative to this temperature. That is T’= T-T_o.

Recognizing this, the more finicky blogger might wish to replace the simple linear T’/λ term with the more complicated integral:

(3)dH/dt = f – ε σ ∫∫T⁴– T_o⁴dA
where A is surface area.

This results in an equation sufficiently horrifying to repel the average blog reader. Luckily, we aren’t interested in the full equation, but only the portion that describes what might be called the “radiative anomaly”, Q_rad, which I’ll define as:

(4)Q_rad~ ε σ A < (T⁴– T_o⁴)>

Here, the angle brackets are used <Y> to indicate the surface area average of any quantity “Y” that may vary over the earth’s surface; which can be written more formally as <Y> =A ^-1∫∫ Y dA

We now have an equation that not only accounts for the full T⁴ dependence for radiation but also permits us to consider the effect of spatial variations in surface temperature.

However, everyone would prefer an simpler, approximate equation, that does not contain a surface integral. We seek something more like (2) but we also wish to do the analysis in sufficient detail to see if the extra terms in (1) appear.

Expand local temperature, T, into an average (GMST) and spatially varying portions.

Let’s begin by defining the temperature anomaly in (1) T’ in (1) in terms of global mean surface temperature. Recall that T’ in (1) & (2) itself is a global average anomaly; it happens to be the quantity

(5)T’ = <T>-<T_o>

where the angle brackets describe surfae area averaging as above and T_o describes the temperature that existed at a point of the surface of the earth under reference conditions.

We know the poles and equator differ in temperature (even under reference conditions). To explore the effect described by Dr. Roger Sr., we’ll define a local temperature deviation, δT, as the difference between the instantaneous local absolute temperature at any point T, and the instantaneous global average temperature <T> as δT = T – <T>. Applying the definition to the reference case, we get δT₀ = T₀ – <T₀>)

Let us further define a dimensionless temperature distribution function function Θ = δT / Δ_To where Δ_To is the standard deviation in the earth’s surface temperature in the reference case. This definition is convenient because Θ is sort of a “shape” function, it’s magnitude is order 1. Later on, we’ll also be able to magnitude of terms containing the powers in Δ_To relative to those containing <T_o> noting that Δ_To is much smaller than <T_o> .

At any time, the Global Mean Surface Temperature is defined as:

(6)<T>= A-1 ∫ ∫ ( <T> +Δ_To Θ) dA

But since the definition of any surface average temperature <T> requires <T>= A^-1∫ ∫ <T> dA we know that:

(7)0 =<Θ>=< em>A^{-1∫ ∫ Θ dA}

Equation (7) result is unremarkable, but it gets used later, so I’m numbering it should anyone later have questions.

Insert expansion into the integral (4)

Our next step is to insert the expansion into the integral describing the anomaly in radiative forcings (4), retain only just enough of the leading order terms to ultimately capture the leading order effect desired for (2) and simplify.

Using the definition of Θ (the dimensionless local instantaneous temperature deviation), we can either use the binomial theorem or MacLaurin Series to further expand Q_rad. At this point in the analysis, we’ll immediately neglect terms that are in higher order than Δ_To², or contain temperature difference to similar order as small compared to contributions that scale as the the reference temperature, <T_o>³. We will also assume variations in emissivity are unimportant to this particular calculations. The result is:

(8)Q_rad~ ε σ < (T⁴– T_o⁴)> ~
ε σ < (<T>⁴ + 4 <T>³Î”_ToΘ +6<T>²(ΘΔ_To)²..)   –   (<T_o>⁴ + 4<T_o>³ Δ_ToΘ_o +6<T>_o²(Θ_oΔ_To)² +..) >

Next recall that, for any Y, taking the average of the average returns the average, so <<Y>> = <Y>, and <Y – (Y- <Y>)> = 0. With a small amount of manipulation, the leading order terms for Q_rad are found:

(9)Q_rad~ ε σ <( T⁴– T_o⁴)>
~ 1/ λ_o {T’ + ( Δ_To/<T_o> )² [3T’ <Θ_o²> + 1.5 <T_o>( 2 <ΘΘ’> ) ]}

where the constant is defined as λ_o = <T_o>³ / (4 ε σ )

Now for the final equation!

Equation (9) contains the leading order and first correction terms in “T’/T” required to determine whether the IPCC approximation can be applied to estimate the radiative heat loss. It’s useful substitute dimensional variables (δT), rather than Θ:

(10a)Q_rad~ (T’ /λ_o) ( 1 + 3 <δT_o²>/<T_o>²)+ (3 / λ_o)<δT_oδT’>/<T_o>

Or if we prefer this to resemble the IPCC equation, we can define λ= λ_o/ ( 1 + 3 <δT_o²>/<T_o>²) and write

Given the range of temperatures on earth, it turns out we can simplify this equation further, and for all practical purposes, the appropriate approximation is:
(10b)Q_rad~ T’ /λ+ 3 (<T_o> / λ)(<δT_oδT’>/<T_o>²)

Leading order terms describing the effect of spatial variations on the radiant heat loss from the earth’s surface are highlighted in bold. These terms describe the effect Dr. Roger Sr. was discussing, and which are missing from the IPCC equation (2).

The new terms appear when one

1. Writes down the functional form for radiation losses from the earth’s surface, (this is the horrible integral) including the effect of spatial variations
2. Expands temperature into mean and deviations in a series and
3. Simplifies to leading order in temperature anomalies.

The terms describing the effect of spatial variations did not appear in Eli’s expansion because he skipped the first step: He did not write down the integral equation. Instead, he wrote down the functional form for radiation losses for an isothermal body, with no spatial variations in temperature. He did include the T⁴ dependence, expanded that already simplified equation, and then wrote done the leading order term.

So, how much do spatial variations matter?

Whether or not these spatial variations “matter”, depends, of course, on what phenomena one is investigating.

In Dr. Roger Sr.’s case, he was trying to use GMST data to estimate climate sensitivity to doubling of CO₂, λ_2xCO2. Current estimates suggest λ_2xCO2 falls between 1.5K and 4.5K. The relevant question then, is, “Relative to a range of 3K, how much difference does neglecting spatial variations make?”

To truly answer that question precisely requires downloading data from a whole bunch of GCM’s and or obtaining possibly non-existent data describing the earth’s surface temperature and calculating things like the standard deviation in the earth’s instantaneous surface temperature, <δT_o²> or more detailed things.

I’m not going to do that on a blog. (I’ll probably never do it. But if someone else would like to, I’d find the results interesting.)

Still it is worth doing a back of the envelope estimate using accessible data; my goal is to determine if the terms contribute less than 1%, 10%, 100% or so on. So, let’s do that.

Suppose:

1. The baseline temperature for the earth’s surface <T_o>~ 280K (60
2. the global temperature anomaly is T’ ~ + 0.6K which is approximately equal to the current anomaly.
3. To estimate the value of <δT_oδT’>, let’s first assume that, we can divide the earth’s surface into three regions: 1/3 of the area (that near the poles)is ‘cold’ , with a base temperature deviation of δT_o=-20K , 1/3 (near the equator) is ‘hot’ with a baseline deviation of δT_o=+20 K and 1/3 is average δT_o=0 K

   (Note: This temperature distribution returns a standard deviation of 16K, which appears roughly consistent, compared to the temperature ranges of 90K in the adjacent figure showing annual average temperature from Nasa GSFC. Estimates of the temperature variations, δT, based on annual averaged values will tend to underestimate the instantaneous variation in temperatures about the instantaneous global mean, which is what is actually required to evaluate δT_o. Using a low value will tend to underestimate the magnitude of effect of the variance in actual temperatures. So, we will be obtaining a lower bound. Those wishing to estimate an upper bound might use δT_o=±30K as might be more appropriate for instantaneous variations.)

4. 

   Assume the temperature in the “cold” part of the earth 1K compared to the ‘average’ anomaly and that at the equator dropped 1K compared to average anomaly. This is somewhat consistent with this map of the Surface Temperature Anomaly for 2005. (Source: NASA, )

   This results in <δT_oδT’> = ( 20K (-1K)+ 0K (-0)K + (-20K)(+1K) )/3 = -20K²(2/3).

Now, comparing the relative magnitude of the neglected terms in 10b, we find the error associated with neglecting this phemonmenon is of the order:

error ~ -3(2/3) (-20K² ) /(300K 0.6K ) ~ 17%

If we use a spatial variation for temperatures of 30K, as might be a more typical value on an individual day, we estimate the error due to neglecting the spatial terms is roughly 25% compared to the term actually retained in the IPCC equation.

Of course, this is only a back of the envelop estimate of the importance of the term. But generally speaking, errors of 15%-25% arising from missing terms in an equation are thought to be important. These sorts of uncertainties tend to introduce bias, and cannot be eliminated by taking lots of data. The only fix is to correct the functional relationship to include the physical phenomenon that has been neglected.

But does an error of 15%-25% matter in the case Dr. Roger Sr. was considering?

The error of 15%-25% is quite important if one is to use the IPCC equation (1) to estimate the climate sensitivity, λ. Typically, those doing this computation try to obtain climate data at near steady state. In this case, neglecting the additional terms, the error will propagate into the computation of climate sensitivity, λ.

The range for the λ_2xCO2 is thought to fall in the range of 1.5K to 4.5K. So, the current uncertainty range is ±1.5K. In contrast, if we were to use the IPCC equation, and empirical data for T’ to estimate the λ_2xCO2, our uncertainty due to neglecting this physical process alone could be as large as ±0.75 K: That’s 1/2 the current uncertainty interval. Worse, measurement uncertainty, or other errors would widen the uncertainty interval in any estimate.

Clearly, if we wish to reduce the current uncertainty in λ_2xCO2, below the current value of ±1.5K, a careful scientist or engineer one would wish to account for the existence of spatial temperature gradients, particularly the large ones between the poles and the equator!

Dr. Roger Pielke Sr. was right; and proved himself so using simple arithmetic. It’s a bit sad to see series expansions dragged into the whole mess. Naturally, when done correctly, the results of series expansions agree with arithmetic.

Advice to bunnies young and old: When a result can be proven with simple arithmetic, don’t expect series expansions to make the result go away.

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Update: 2/15/2007: I found a better image for anomalies.