The Cold Equations

Guest Post by Willis Eschenbach

I’ve tried writing this piece several times already. I’ll give it another shot, I haven’t been happy with my previous efforts. It is an important subject that I want to get right. The title comes from a 1954 science fiction story that I read when I was maybe ten or eleven years old. The story goes something like this:

A girl stows away on an emergency space pod taking anti-plague medicine to some planetary colonists. She is discovered after the mother ship has left. Unfortunately, the cold equations show that the pod doesn’t have enough fuel to land with her weight on board, and if they dump the medicine to lighten the ship the whole colony will perish … so she has to be jettisoned through the air lock to die in space.

I was hugely impressed by the story. I liked math in any case, and this was the first time that I saw how equations can provide us with undeniable and unalterable results. And I saw that the equations about available fuel and weight weren’t affected by human emotions, they either were or weren’t true, regardless of how I or anyone might feel about it.

Lately I’ve been looking at the equations used by the AGW scientists and by their models. Figure 1 shows the most fundamental climate equation, which is almost tautologically true:

Figure 1. The most basic climate equation says that energy in equals energy out plus energy going into the ocean. Q is the sum of the energy entering the system over some time period. dH/dt is the change in ocean heat storage from the beginning to the end of the time period. E + dH/dt is the sum of the outgoing energy over the same time period. Units in all cases are zettajoules (ZJ, or 10^21 joules) / year.

This is the same relationship that we see in economics, where what I make in one year (Q in our example) equals what I spend in that year (E) plus the year-over-year change in my savings (dH/dt).

However, from there we set sail on uncharted waters …

I will take my text from HEAT CAPACITY, TIME CONSTANT, AND SENSITIVITY OF EARTH’S CLIMATE SYSTEM, Stephen E. Schwartz, June 2007 (hereinafter (S2007). The study is widely accepted, being cited 193 times. Here’s what the study says, inter alia (emphasis mine).

Earth’s climate system consists of a very close radiative balance between absorbed shortwave (solar) radiation Q and longwave (thermal infrared) radiation emitted at the top of the atmosphere E.

Q ≈ E                                                                       (1)

The global and annual mean absorbed shortwave irradiance Q  = γ J, where γ [gamma] is the mean planetary coalbedo (complement of albedo) and J is the mean solar irradiance at the top of the atmosphere (1/4 the Solar constant) ≈ 343 W m-2. Satellite measurements yield Q ≈ 237 W m-2 [Ramanathan 1987; Kiehl and Trenberth, 1997], corresponding to γ ≈ 0.69. The global and annual mean emitted longwave irradiance may be related to the global and annual mean surface temperature GMST Ts as E = ε σ Ts^4 where ε (epsilon) is the effective planetary longwave emissivity, defined as the ratio of global mean longwave flux emitted at the top of the atmosphere to that calculated by the Stefan-Boltzmann equation at the global mean surface temperature; σ (sigma) is the Stefan-Boltzmann constant.

Within this single-compartment energy balance model [e.g., North et al., 1981; Dickinson, 1982; Hansen et al., 1985; Harvey, 2000; Andreae et al., 2005, Boer et al., 2007] an energy imbalance Q − E arising from a secular perturbation in Q or E results in a rate of change of the global heat content given by

dH/dt = Q – E                                                               (2)

where dH/dt is the change in heat content of the climate system.

Hmmm … I always get nervous when someone tries to slip an un-numbered equation into a paper … but I digress. Their Equation (2) is the same as my Figure 1 above, which was encouraging since I’d drawn Figure 1 before reading S2007. S2007 goes on to say (emphasis mine):

The Ansatz of the energy balance model is that dH/dt may be related to the change in GMST [global mean surface temperature] as

dH/dt = C dTs/dt                          (3)

where C is the pertinent heat capacity. Here it must be stressed that C is an effective heat capacity that reflects only that portion of the global heat capacity that is coupled to the perturbation on the time scale of the perturbation. In the present context of global climate change induced by changes in atmospheric composition on the decade to century time scale the pertinent heat capacity is that which is subject to change in heat content on such time scales. Measurements of ocean heat content over the past 50 years indicate that this heat capacity is dominated by the heat capacity of the upper layers of the world ocean [Levitus et al., 2005].

In other words (neglecting the co-albedo for our current purposes), they are proposing two substitutions in the equation shown in Figure 1. They are saying that

E = ε σ Ts^4

and that

dH/dt = C dTs/dt

which gives them

Q = ε σ Ts^4 + C dTs/dt                     (4)

Figure 2 shows these two substitutions:

Figure 2. A graphic view of the two underlying substitutions done in the “single-compartment energy balance model” theoretical climate explanation. Original equation before substitution is shown in light brown at the lower left, with the equation after substitution below it.

Why are these substitutions important? Note that in Equation (4), as shown in Figure 2, there are only two variables — radiation and surface temperature. If their substitutions are valid, this means that a radiation imbalance can only be rectified by increasing temperature. Or as Dr. Andrew Lacis of NASA GISS recently put it (emphasis mine):

As I have stated earlier, global warming is a cause and effect problem in physics that is firmly based on accurate measurement and well established physical processes. In particular, the climate of Earth is the result of energy balance between incoming solar radiation and outgoing thermal radiation, which, measured at the top of the atmosphere, is strictly a radiative energy balance problem. Since radiative transfer is a well established and well understood physics process, we have accurate knowledge of what is happening to the global energy balance of Earth. And as I noted earlier, conservation of energy leaves no other choice for the global equilibrium temperature of the Earth but to increase in response to the increase in atmospheric CO2.

Dr. Lacis’ comments are an English language exposition of the S2007 Equation (4) above. His statements rest on Equation (4). If Equation (4) is not true, then his claim is not true. And Dr. Lacis’ claim, that increasing GHG forcing can only be balanced by a temperature rise, is central to mainstream AGW climate science.

In addition, there’s a second reason that their substitutions are important. In the original equation, there are three variables — Q, E, and H. But since there are only two variables (Ts and Q) in the S2007 version of the equation, you can solve for one in terms of the other. This allows them to calculate the evolution of the surface temperature, given estimates of the future forcing … or in other words, to model the future climate.

So, being a naturally suspicious fellow, I was very curious about these two substitutions. I was particularly curious because if either substitution is wrong, then their whole house of cards collapses. Their claim, that a radiation imbalance can only be rectified by increasing temperature, can’t stand unless both substitutions are valid.

SUBSTITUTION 1

Let me start with the substitution described in Equation (3):

dH/dt = C dTs/dt                          (3)

The first thing that stood out for me was their description of Equation (3) as “the Ansatz of the energy balance model”.

“And what”, sez I, “is an ‘Ansatz’ when it’s at home?” I’m a self-educated reformed cowboy, it’s true, but a very well-read reformed cowboy, and I never heard of the Ansatz.

So I go to Wolfram’s Mathworld, the internet’s very best math resource, where I find:

Ansatz

An ansatz is an assumed form for a mathematical statement that is not based on any underlying theory or principle.

Now, that’s got to give you a warm, secure feeling. This critical equation, this substitution of the temperature change as a proxy for the ocean heat content change, upon which rests the entire multi-billion-dollar claim that increased GHGs will inevitably and inexorably increase the temperature, is described by an enthusiastic AGW adherent as “not based on any underlying theory or principle”. Remember that if either substitution goes down, the whole “if GHG forcings change, temperature must follow” claim goes down … and for this one they don’t even offer a justification or a citation, it’s merely an Ansatz.

That’s a good thing to know, and should likely receive wider publication …

It put me in mind of the old joke about “How many legs does a cow have if you call a tail a leg?”

…

“Four, because calling a tail a leg doesn’t make it a leg.”

In the same way, saying that the change oceanic heat content (dH/dt) is some linear transformation of the change in surface temperature (C dTs/dt) doesn’t make it so.

In fact, on an annual level the correlation between annual dH/dt and dTs/dt is not statistically significant (r^2=0.04, p=0.13). In addition, the distributions of dH/dt and dTs/dt are quite different, both at a quarterly and an annual level. See Appendix 1 and 4 for details. So no, we don’t have any observational evidence that their substitution is valid. Quite the opposite, there is little correlation between dH/dt and dTs/dt.

There is a third and more subtle problem with comparing dH/dt and dTs/dt. This is that H (ocean heat content) is a different kind of animal from the other three variables Q (incoming radiation), E (outgoing radiation), and Ts (global mean surface air temperature). The difference is that H is a quantity and Q, E, and Ts are flows.

Since Ts is a flow, it can be converted from the units of Kelvins (or degrees C) to the units of watts/square metre (W/m2) using the blackbody relationship σ Ts^4.

And since the time derivative of the quantity H is a flow, dH/dt, we can (for example) compare E + dH/dt to Q, as shown in Figure 1. We can do this because we are comparing flows to flows. But they want to substitute a change in a flow (dT/dt) for a flow (dH/dt). While that is possible, it requires special circumstances.

Now, the change in heat content can be related to the change in temperature in one particular situation. This is where something is being warmed or cooled through a temperature difference between the object and the surrounding atmosphere. For example, when you put something in a refrigerator, it cools based on the difference between the temperature of the object and the temperature of the air in the refrigerator. Eventually, the object in the refrigerator takes up the temperature of the refrigerator air. And as a result, the change in temperature of the object is a function of the difference in temperature between the object and the surrounding air. So if the refrigerator air temperature were changing, you could make a case that dH/dt would be related to dT/dt.

But is that happening in this situation? Let’s have a show of hands of those who believe that as in a refrigerator, the temperature of the air over the ocean is what is driving the changes in ocean heat content … because I sure don’t believe that. I think that’s 100% backwards. However, Schwartz seems to believe that, as he says in discussing the time constant:

… where C’ is the heat capacity of the deep ocean, dH’/dt is the rate of increase of the heat content in this reservoir, and ∆T is the temperature increase driving that heat transfer.

In addition to the improbability of changes in air temperature driving the changes in ocean heat content, the size of the changes in ocean heat content also argues against it. From 1955 to 2005, the ocean heat content changed by about 90 zettajoules. It also changed by about 90 zettajoules from one quarter to the next in 1983 … so the idea that the temperature changes (dT/dt) could be driving (and thus limiting) the changes in ocean heat content seems very unlikely.

Summary of Issues with Substitution 1:   dH/dt = C dT/dt

1. The people who believe in the theory offer no theoretical or practical basis for the substitution.

2. The annual correlation of dH/dt and dT/dt is very small and not statistically significant.

3. Since H is a quantity and T is a flow, there is no a priori reason to assume a linear relationship between the two.

4. The difference in the distributions of the two datasets dH/dt and dT/dt (see Appendix 1 and 4) shows that neither ocean warming nor ocean cooling are related to dT/dt.

5. The substitution implies that air temperature is “driving that heat transfer”, in Schwartz’s words. It seems improbable that the wisp of atmospheric mass is driving the massive oceanic heat transfer changes.

6. The large size of the quarterly heat content changes indicates that the heat content changes are not limited by the corresponding temperature changes.

My conclusion from that summary? The substitution of C dT/dt for dH/dt is not justified by either observations or theory. While it is exceedingly tempting to use it because it allows the solution of the equation for the temperature, you can’t make a substitution just because you really need it in order to solve the equation.

SUBSTITUTION 2: E = ε σ Ts^4

This is the sub rosa substitution, the one without a number. Regarding this one, Schwartz says:

The global and annual mean emitted longwave irradiance may be related to the global and annual mean surface temperature GMST Ts as

E = ε σ Ts^4

where ε (epsilon) is the effective planetary longwave emissivity, defined as the ratio of global mean longwave flux emitted at the top of the atmosphere [TOA] to that calculated by the Stefan-Boltzmann equation at the global mean surface temperature; σ (sigma) is the Stefan-Boltzmann constant.

Let’s unpick this one a little and see what they have done here. It is an alluring idea, in part because it looks like the standard Stefan-Boltzmann equation … except that they have re-defined epsilon ε as “effective planetary emissivity”. Let’s follow their logic.

First, in their equation, E is the top of atmosphere longwave flux, which I will indicate as Etoa to distinguish it from surface flux Esurf. Next, they say that epsilon ε is the long-term average top-of-atmosphere (TOA) longwave flux [ which I’ll call Avg(Etoa) ] divided by the long-term average surface blackbody longwave flux [ Avg(Esurf) ]. In other words:

ε = Avg(Etoa) /Avg(Esurf)

Finally, the surface blackbody longwave flux Esurf is given by Stefan-Boltzmann as

Esurf = σ Ts^4.

Substituting these into their un-numbered Equation (?) gives us

Etoa = Avg(Etoa) / Avg(Esurf) * Esurf

But this leads us to

Etoa / Esurf = Avg(Etoa) / Avg(Esurf)

which clearly is not true in general for any given year, and which is only true for long-term averages. But for long-term averages, this reduces to the meaningless identity Avg(x) / Avg(anything) = Avg(x) / Avg(anything).

Summary of Substitution 2: E = ε σ Ts^4

This substitution is, quite demonstrably, either mathematically wrong or meaninglessly true as an identity. The cold equations don’t allow that kind of substitution, even to save the girl from being jettisoned. Top of atmosphere emissions are not related to surface temperatures in the manner they claim.

My conclusions, in no particular order:

• Obviously, I think I have shown that neither substitution can be justified, either by theory, by mathematics, or by observations.

• Falsifying either one of their two substitutions in the original equation has far-reaching implications.

• At a minimum, falsifying either substitution means that in addition to Q and Ts, there is at least one other variable in the equation. This means that the equation cannot be directly solved for Ts. And this, of course, means that the future evolution of the planetary temperature cannot be calculated using just the forcing.

• In response to my posting about the linearity of the GISS model, Paul_K pointed out the Schwartz S2007 paper. He also showed that the GISS climate model slavishly follows the simple equations in the S2007 paper. Falsifying the substitutions thus means that the GISS climate model (and the S2007 equations) are seen to be exercises in parameter fitting. Yes, they can can give an approximation of reality … but that is from the optimized fitting of parameters, not from a proper theoretical foundation.

• Falsifying either substitution means that restoring radiation balance is not a simple function of surface temperature Ts. This means that there are more ways to restore the radiation balance in heaven and earth than are dreamt of in your philosophy, Dr. Lacis …

As always, I put this up here in front of Mordor’s unblinking Eye of the Internet to encourage people to point out my errors. That’s science. Please point them out with gentility and decorum towards myself and others, and avoid speculating on my or anyone’s motives or honesty. That’s science as well.

w.

Appendix 1: Distributions of dH/dt and dT/dt

There are several ways we can see if their substitution of C dT/dt for dH/dt makes sense and is valid. I usually start by comparing distributions. This is because a linear relationship, such as is proposed in their substitution, cannot change the shape of a distribution. (I use violinplots of this kind of data because they show the structure of the dataset. See Appendix 2 below for violinplots of common distributions.)

A linear transformation can make the violinplot of the distribution taller or shorter, and it can move the distribution vertically. (A negative relationship can also invert the distribution about a horizontal axis, but they are asserting a positive relationship).But there is no linear transformation (of the type y = m x + b) that can change the shape of the distribution. The “m x” term changes the height of the violinplot, and the “b” term moves it vertically. But a linear transformation can’t change one shape into a different shape.

First, a bit of simplification. The “∆” operator indicates “change since time X”. We only have data back to 1955 for ocean heat content. Since the choice of “X” is arbitrary, for this analysis we can say that e.g. ∆T is shorthand for T(t) – T(1955). But for the differentiation operation, this makes no difference, because the T(1955) figure is a constant that drops out of the differentiation. So we are actually comparing dH/dt(annual change in ocean heat content) with C dT/dt (annual change in temperature)

Figure 2 compares the distributions of dH/dt and dT/dt. Figure A1 shows the yearly change in the heat content H (dH/dt) and the yearly change in the temperature T (dT/dt).

Figure A1 Violinplot comparison of the annual changes in ocean heat content dH/dt and annual changes in global surface temperature dT/dt. Width of the violinplot is proportional to the number of observations at that value (density plot). The central black box is a boxplot, which covers the interquartile range (half of the data are within that range). The white dot shows the median value.

In addition to letting us compare the shapes, looking at the distribution lets us side-step all problems with the exact alignment of the data. Alignment can present difficulties, especially when we are comparing a quantity (heat content) and a flow (temperature or forcing). Comparing the distributions avoids all these alignment issues.

With that in mind, what we see in Figure A1 doesn’t look good at all. We are looking for a positive linear correlation between the two datasets, but the shapes are all wrong. For a linear correlation to work, the two distributions have to be of the same shape. But these are of very different shapes.

What do the shapes of these violinplots show?

For the ocean heat content changes, the peak density at ~ – 6 ZJ shows that overall the most common year-to-year change is a slight cooling. When warming occurs, however, it tends to be larger than the cooling. The broad top of the violinplot means that there are an excess of big upwards jumps in ocean heat content.

For the temperature changes, the reverse is true. The most common change is a slight warming of about 0.07°C. There are few examples of large warmings, whereas large coolings are more common. So there will be great difficulties equating a linear transform of the datasets.

The dimensions of the problem become more apparent when we look at the distributions of the increases (in heat content or temperature) versus the distributions of the decreases in the corresponding variables. Figure A2 compares those distributions:

Figure A2. Comparison of the distribution of the increases (upper two panels) and the decreases (lower two panels) in annual heat content and temperature. “Equal-area” violinplots are used.

Here the differences between the two datasets are seen to be even more pronounced. The most visible difference is between the increases. Many of the annual increases of the ocean heat content are large, with a quarter of them more than 20 ZJ/yr and a broad interquartile range (black box, which shows the range of the central half of the data). On the other hand, there are few large increases of the temperature, mostly outliers beyond the upper “whisker” of the boxplot.

The reverse is also true, with most of the heat content decreases being small compared to the corresponding temperature decreases. Remember that a linear transformation such as they propose, of the form (y = m x + b), has to work for both the increases and the decreases … which in this case is looking extremely doubtful.

My interpretation of Figure A2 is as follows. The warming and cooling of the atmosphere is governed by a number of processes that take place throughout the body of the atmosphere (e.g. longwave radiation absorption and emission, shortwave absorption, vertical convection, condensation, polewards advection). The average of these in the warming and cooling directions are not too dissimilar.

The ocean, on the other hand, can only cool by releasing heat from the upper surface. This is a process that has some kind of average value around -8 ZJ/year. The short box of the boxplot (encompassing the central half of data points) shows that the decreases in ocean heat content are clustered around that value.

Unlike the slow ocean cooling, the ocean can warm quickly through the deep penetration of sunlight into the mixed layer. This allows the ocean to warm much more rapidly than it is able to cool. This is why there are an excess of large increases in ocean heat content.

And this difference in the rates of ocean warming and cooling is the fatal flaw in their claim. The different distributions for ocean warming and ocean cooling indicate to me that they are driven by different mechanisms. The Equation (3) substitution seen in S2007 would mean that the ocean warming and cooling can be represented solely by the proxy of changes in surface temperature.

But the data indicates the ocean is warming and cooling without much regard to the change in temperature. The most likely source of this is from sunlight deeply heating the mixed layer. Notice the large number of ocean heat increases greater than 20 ZJ/year, as compared to the scarcity of similarly sized heat losses. The observations show that this (presumably) direct deep solar warming both a) is not a function of the surface temperature, and b) does not affect the surface temperature much. The distributions show that the heat is going into the ocean quickly in chunks, and coming out more slowly and regularly over time.

In summary, the large differences between the distributions of dH/dt and dT/dt, combined with the small statistical correlation between the two, argue strongly against the validity of the substitution.

Appendix 2: Violinplots

I use violinplots extensively because they reveal a lot about the distribution of a dataset. They are a combination of a density plot and a box plot. Figure A3 shows the violin plots and the corresponding simple boxplots for several common distributions.

Figure A3. Violin plots and boxplots. Each plot shows the distribution of 20,000 random numbers generated using the stated distribution. “Normal>0” is a set comprised all of the positive datapoints in the adjacent “Normal” dataset.

Because the violin plot is a density function it “rounds the corners” on the Uniform distribution, as well as the bottoms of the Normal>0 and the Zipf distributions. Note that the distinct shape of the Zipf distribution makes it easy to distinguish from the others.

Appendix 3: The Zipf Distribution

Figure A3. Violinplot of the Zipf distribution for N= 70, s = 0.3. Y-axis labels are nominal values.

The distinguishing characteristics of the Zipf distribution, from the top of Figure A3 down, are:

• An excess of extreme data points, shown in the widened upper tip of the violinplot.

• A “necked down” or at least parallel area below that, where there is little or no data.

• A widely flared low base which has maximum flare not far from the bottom.

• A short lower “whisker” on the boxplot (the black line extending below the blue interquartile box) that extends to the base of the violinplot

• An upper whisker on the boxplot which terminates below the necked down area.

Appendix 4: Quarterly Data

The issue is, can the change in temperature be used as a proxy for the change in ocean heat content? We can look at this question in greater detail, because we have quarterly data from Levitus. We can compare that quarterly heat content data to quarterly GISSTEMP data. Remember that the annual data shown in Figures A1 and A2 are merely annual averages of the quarterly data shown below in Figures A4 and A5. Figure A4 shows the distributions of those two quarterly datasets, and lets us investigate the effects of averaging on distributions:

Figure A4. Comparison of the distribution of the changes in the respective quarterly datasets.

The shape of the distribution of the heat content is interesting. I’m always glad to see that funny kind of shape, what I call a “manta ray” shape, it tells me I’m looking at real data. What you see there is what can be described as a “double Zipf distribution”.

The Zipf distribution is a very common distribution in nature. It is characterized by having a few really, really large excursions from the mean. It is the Zipf distribution that gives rise to the term “Noah Effect”, where the largest in a series of natural events (say floods) is often much, much larger than the rest, and much larger than a normal distribution would allow. Violinplots clearly display this difference in distribution shape, as can be seen in the bottom part of the heat content violinplot (blue) in Figure A4. Appendix 3 shows an example of an actual Zipf distribution with a discussion of the distinguishing features (also shown in Appendix 2):

The “double” nature of the Zipf distribution I commented on above can be seen when we examine the quarterly increases in heat and temperature versus the decreases in heat and temperature, as shown in Figure A5:

Figure A5. Comparison of the distribution of the increases (upper two panels) and the decreases (lower two panels) in quarterly heat content (blue) and quarterly temperature (green)

The heat content data (blue) for both the increases and decreases shows the typical characteristics of a Zipf distribution, including the widened peak, the “necking” below the peak, and the flared base. The lower left panel shows a classic Zipf distribution (in an inverted form).

What do the distributions of the upward and downward movements of the variables in Figure A5 show us? Here again we see the problem we saw in the annual distributions. The distributions for heat content changes are Zipf distributions, and are quite different in shape from the distributions of the temperature changes. Among other differences, the inter-quartile boxes of the boxplots show that the ocean heat content change data is much more centralized than the temperature change data.

In addition, the up- and down- distributions for the temperature changes are at least similar in shape, whereas the shapes of the up- and down- heat content change distributions are quite dissimilar. This difference in the upper and lower distributions is what creates the “manta-ray” shape shown in Figure A4. And the correlation is even worse than with the annual data, that is to say none.

So, as with the annual data, the underlying quarterly data leads us to the same conclusion: there’s no way that we can use dT/dt as a proxy for dH/dt.

Appendix 5: Units

We have a choice in discussing these matters. We can use watts per square metre (W m-2). The forcings (per IPCC) have a change since 1955 of around +1.75 W/m2.

We can also use megaJoules per square metre per year (MJ m-2 y-1). The conversion is:

1 watt per square metre (W m-2) = 1 joule/second per square metre (J sec-1 m-2) times 31.6E6 seconds / year = 31.6 MJ per square metre per year (MJ m-2 yr-1). Changes in forcing since 1955 are about +54 MJ per square metre per year.

Finally, we can use zettaJoules (ZJ, 10^21 joules) per year for the entire globe. The conversion there is

1 W/m2 = 1 joule/second per square metre (J sec-1 m-2) times 31.6E6 seconds / year times 5.11E14 square metres/globe = 16.13 ZJ per year (ZJ yr-1). Changes in forcing since 1955 are about +27 ZJ per year. I have used zettaJoules per year in this analysis, but any unit will do.