# 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

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

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.

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Roger Carr
January 28, 2011 2:58 am

I am going to enjoy your autobiography (of a cowboy) more, when it is published, Willis, because I believe I will actually understand it… but I am trying, here.

kim
January 28, 2011 3:04 am

They seem to think that extra energy can only turn into heat. Anybody ask the energy about such slavery?
==============

Ceri Reid
January 28, 2011 3:32 am

This is interesting. You should submit it to a peer reviewed publication. If you’ve gone wrong somewhere, I’m sure a reviewer will point it out. If you haven’t made a mistake, then it should be made known in the conventional academic way.

cal
January 28, 2011 3:38 am

I have not finished reading all the post yet but I had to stop at this point to make a comment.
“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.”
I think this is a mistake. Temperature is not a flow. It is not “converted” to a power by multiplying by sigma. The output of the equation is a power density because sigma has the dimensions of Wm^2K^4.
Most of your arguments still stand but it is a pity that you make this sort of error since it is easy for others to dismiss some very good points because of it. I can imagine some other physicists just stopping at this point, which would be a shame.

Ken Hall
January 28, 2011 3:50 am

Outstanding work there Willis as usual. I am most grateful to you for transposing an almost meaningless gibberish of mathematical slights of hand and fraudulent equations into a clearly understandable format and by so doing, highlighting the weakness at the heart of the alarmist’s mathematics. Math itself does not lie, people sometimes use math to lie, and when they need an Ansantz to make their sums work, then you know that they are in trouble. This amounts to fraud on a massive global scale.

January 28, 2011 3:50 am

Thank you! I read that story about 50 years ago and it’s come to mind every since, but I could never recall the name of it. I remember fighting with the conclusion for days after, trying to come up with some alternative. I figured if they’d jettisoned all their clothes and all the packaging for their supplies, not to mention various personal items that they should have compensated for her weight. Still, it was a sober look at the inexorability of numbers.
Oh — and an excellent article. I think the Ansatz Club would be an excellent name for a global warming rock band.

January 28, 2011 3:51 am

The Cold Equations plot sounded familiar to me. There is an 1989 episode of The Twilight Zone, Season 3, Episode 16 that uses that story, and has that title.

David Baigent
January 28, 2011 3:54 am

I am impressed..
“Apples and oranges”
db..

Hector M.
January 28, 2011 4:00 am

Willis,
your comparison with income and savings at the beginning of your piece is somewhat imprecise. At the individual level, as you said, your income equals the sum of your expenditure and the the net amount saved during a period. But at the level of the economy, the amount saved by everybody should equal the total amount invested in new infrastructure or equipment; therefore in the end all goods and services produced are sold, and product equals income. To achieve that equality, prices may have to go up or down (it is an ex post identity). The whole Keynesian theory is based on the idea that planned investment may be lower than available savings, thus producing unintended unemployment, and a product that is lower than potentially could be.
As the analogy is not perfect, I suggest you drop it entirely.

LazyTeenager
January 28, 2011 4:08 am

The difference is that H is a quantity and Q, E, and Ts are flows
-/——-
You were doing OK until you got to this bit. T is not a flow. It has the wrong units to be a flow.
At a more fundamental level it is a form of energy and more particularly it is that portion of the energy in a substance that is due to random kinetic energy of the molecules that make it up,

Baa Humbug
January 28, 2011 4:24 am

Thanks once again Willis, I look forward to the “Joe Sixpack” summary.
I guess the AGW theory would work out fine if 70% of the surface of the Earth wasn’t covered with deep oceans.
Also, if the atmosphere could heat the oceans, then the mean T of oceans would be 14-15DegC, (same as the atmosphere, but alas the mean T of oceans is about 3DegC) especially since it’s been about 12,000yrs since the height of the last ice age. I guess atmosphere needs more time to keep heating the oceans.
By the way, Ansatz = Arrhenius

LazyTeenager
January 28, 2011 4:26 am

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:
———-
It’s like coming across a kitten tangled in a ball of wool. How do I disentagle this mess. Do I try or just cut it apart with a pair if scissors?
Let’s be patient.
The assumption that the ocean heat content changes are driven by transfer of heat from the atmosphere is probably a misinterpretation of what Schwartz says. I suspect that it really means transfer of heat from surface water to deeper water.

cal
January 28, 2011 4:30 am

Further to my last post I think the your sentence “So if the refrigerator air temperature were changing, you could make a case that dH/dt would be related to dT/dt.” is also misleading.
dH/dt is always proportional to the temperature difference and not the rate of change of temperature. This is just not a good analogy. In a refrigerator the contents will cool asymptotically to the temperature of the chamber. If the chamber is itself cooling then the equation is much more complex but is very unlikely to be proportional to the rate of cooling of the refrigerator. The situation you describe is only close to proportional if the contents of the refrigerator were already at the same temperature as the refrigerator. Then fluctuations in the regrigerator temperature would cause proportional flows of heat into and out of the contents.
The same argument can be used against the equations used by Scharwtz. This could possibly be true if the temperature of the sea were always in dynamic equilibrium with the surrounding air. However we know from experience (a shallow pond for example) that the absorption of energy is mainly dependent on how sunny it is. You only have to see the speed that frost vanishes on a cold sunny day compared to how it hangs around on a slightly warmer but cloudier day to see that conduction cannot compete with direct radiation.
I am not trying to argue against the point you are making. I think you are saying that the sea heats the air and not the other way round and I think this has got to be 99% true. I just don’t think that it is particularly well argued.

January 28, 2011 4:31 am

Actually the substitutions are fine — eps sig Ts^4 is just the blackbody law, and C dTs/dt is definitional. The problem — the reason it’s ONLY an Ansatz — is that they assume (a) that epsilon, the coalbedo, is a constant (as opposed to being a function of temperature and/or something else), and (b) they assume they can average temperature and heat capacity into a point radiator and heatsink.
Since neither of these assumptions is true, it’s totally illegitimate to talk about an effect that’s 0.0027 of the main flow as if the physics demanded it. But there’s nothing wrong with the math, per se.

Baa Humbug
January 28, 2011 4:37 am

cal says:
January 28, 2011 at 3:38 am

I think this is a mistake. Temperature is not a flow. It is not “converted” to a power by multiplying by sigma.

I’m sure Willis will be along in a minute, but until then…
From http://www.chem1.com

Heat is the quantity of thermal energy that enters or leaves a body.
Temperature measures the average translational kinetic energy of the molecules in a body.
Energy is measured in joules, and temperature in degrees. This difference reflects the important distinction between energy and temperature:
* We can say that the 10 g of water we heated on the stove now contains 100 J more energy than it did before. And because energy is an extensive quantity, we know that a 5-g portion of this warmer water contains 50 J more energy than it did originally.
* Temperature, by contrast, is not a measure of quantity; being an intensive property, it is more of a “quality” that describes the “intensity” with which thermal energy manifests itself.

I hope physicists keep reading to the end and provide feedback.

January 28, 2011 4:37 am

“And what”, sez I, “is an ‘Ansatz’
Coincidently, on January 8, I came upon that word while reading a post by the physicist Luboš Motl at:
http://motls.blogspot.com/2011/01/climate-sensitivity-from-linear-fit.html
I even emailed the mathworld web page link I found after doing a search to myself so I could review it again, and again later (on my smartphone).

RalphieGM
January 28, 2011 5:00 am

As to energy, why is sun/ earth gravity neglected in radiation balances?
I thought differential gravity was the mechanism that roils the core forcing huge temperatures a few miles below our feet ( not quite the million degrees Mr. Gore has spoken about ), but additional heat nevertheless.

BargHumer
January 28, 2011 5:00 am

I am an engineer, not a scientist, so I understand some of this but not all. The comment from Cal is interesting and seems to call in to question the statement “The difference is that H is a quantity and Q, E, and Ts are flows.” Perhaps this needs some clarification by the author.
I can measure the temperature of an air flow in degrees without defining the flow rate, and I can measure the temperature (degrees) of a heating element in order to control it without considering it to be a flow.
Anyway, this was a good read, and I wish I could understand it all.

P. Solar
January 28, 2011 5:08 am

Willis, there is something fundamentally wrong with the basic physics of this treatise:
dH/dt = Q – E (2)
where dH/dt is the change in heat content of the climate system.
There is a principal called dimensional analysis that states that both sides of an equation must be of the same dimensional nature. Dimension here does not mean “size” dimension but the nature of the quantities treated. ie if one side is energy the other side is energy. If one side a force the other side must be able to resolved to similar units. The base “dimensions” are mass length and time, but the principal can be applied in derived quantities as well, like energy.
So , the crux: you can’t equate a rate of change of energy dH/dt to and energy , Q-E.
>>
where dH/dt is the change in heat content of the climate system.
>>
This is clearly wrong. The d/dt nomenclature depicts a rate of change not a change.
Equation 2 could have a *change* in heat content and would make sense this would be dH or delta H NOT dH/dt. That could be just a carless error in nomenclature, but for the fact that it is used as a *rate of change* in what follows:
Equation 3 is correct dimensionally.
dH/dt = C dTs/dt (3)
but when this is substitued to give equation 4 you’re back in trouble.
Q = ε σ Ts^4 + C dTs/dt (4)
Now you’re back to adding incompatible quantities. The RHS is adding an energy to a rate of change of energy.
This is just nonsense. It’s like adding a distance to a speed.
eg.
distance to end position = starting postition + 30 mph !?
It’s gobbledy-gook.
I appreciate you seem to have cribbed this analysis from the quoted papers and I don’t have time to download them and check over the papers, so either you have misunderstood and misquoted what was in the papers or the authors that you cribbed don’t understand basic physics they are using either.
That latter possibility would not totally surprise me in climate science.
I also do not have time now to go through the rest of this lengthy post so I’ll just inform you of the error and let you dig through the implications. Since this is the very foundation it likely means the rest is false.
If you’ve had difficulty putting this together , maybe this is why…
Hope this info helps.

P. Solar
January 28, 2011 5:27 am

OK , I’ve seen that Q seems to change nature depending where you use it. You say it is in ZJ but summed over a year. It appears it is mean to be rate of change of energy elsewhere it’s and intensity:
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
caption to figure 1 :
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.
So is Q in ZJ/year or W/m2 ??
ZJ/year is demensionally equivalent to W not to W/m2
This may just be inconsistency rather the the error I first thought above.

Fernando
January 28, 2011 5:29 am

Insensate = Ansatz
makes sense in Portuguese.
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..
If this statement is true
Q= mc∆T
c= ci – co …[a positive number (to avoid further confusion)]
ci (input) = heating
co (ouput) = cooling
ci and co are not linear.
Maybe that’s how the planet warms or cools

David Falkner
January 28, 2011 5:32 am

Why doesn’t conservation of energy apply to the ocean?

Dave
January 28, 2011 5:33 am

“the meaningless identity Avg(x) / Avg(anything) = Avg(x) / Avg(anything).”
In computer programming, it’s not terribly unusual to see a line of code something like:
If [value A] = [value A] Then [do stuff]
It’s usually a relic left as the result of changes or some such, and is obviously completely pointless (although sometimes not worth digging out and redoing the code). It’s commonly known as an ‘alternate-reality check’.
[is there much more of this?]

P. Solar
January 28, 2011 5:36 am

E = ε σ Ts^4
again this is a power , not an energy so E is confusing . You need to be clearer when talking about a power term.
“Q is the sum of the energy entering the system over some time period.”
No, the way you are using it is not *energy” it is power. You are not using the sum of the energy over that period as you state, you are using the average rate of change of energy over that time. That is power or rate of change of energy not energy. Units in watts not joules.
Sorry , really have to drop this now, but you need to get a clearer idea of what all these terms are.
regards.

Brian H
January 28, 2011 5:36 am

Fatal. You politely say things like, “Issues”, “argue strongly against”.
But what this analysis does is explode a massive “petard” under the foundations. You have earned the “Sapper’s Medal of Honor”.
Another “issue” you might want to address at some point is the problem of averaging 4th power products, vs taking (assuming) 4th power products of averages.
Take the numbers 2,3,4. The average of 2 & 4 is 3.
But 2^4 is 16, 4^4 is 256, total 272, average 136.
3^4 is 81.
Any distribution of values with wider spread will have a significantly higher average of fourth power products than a centrally compressed one.

Basil
Editor
January 28, 2011 5:41 am

w.
Could you clarify this:
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.
Isn’t it the case that most “heat transfer” — from the tropics, where it is mostly received, to the poles where it mostly escapes — is driven mostly by atmospheric currents (anywhere from 60 to 75%) than by ocean currents?

Editor
January 28, 2011 5:42 am

I read The Cold Equations, a long time ago, I think before I understood the some of the math involved, but at least understood the concepts.
Changed my life, it did. I decided right then and there to never stowaway on a spaceship, and never have. Sadly, I’ve never had the opportunity to test my resolve.

Vince Causey
January 28, 2011 5:46 am

I think a misunderstanding has been made of what C dTs actually means. My own reading of the Levitus paragraph is that he is really talking about some hybrid thing which consists mostly of ocean and a smidgen of atmosphere. Levitus actually says “this heat capacity is dominated by the heat capacity of the upper layers of the world ocean .” Now, if the heat capacity is dominated by the ocean, then which temperature do you use for dTs in that equation? Common sense would say the ocean, if that is the dominant heat sink. If you take an average of ocean and atmosphere then you get into all sorts of difficulties trying to weight each dT according to heat capacities.
So, in practical terms, equation 3 is saying that the heat stored in the worlds oceans in a year is equal to its heat capacity multplied by its temperature increase in a year. But as ocean temps haven’t increased much, if at all, in the last 7 years then doesn’t that make dH/dt practically zero?

Joe Born
January 28, 2011 5:56 am

I agree with the comments to the effect that calling Ts a flow is a problem, at least from the standpoint of effective exposition. I understand what you mean, but calling average translational kinetic energy a flow places barriers in the road to understanding/acceptance. Lose that sentence, is my vote.
Commenter P. Solar is all wrong, of course, but I, too, was inclined to interpret Q and E as heat before I read the context. Maybe that choice of variable names confuses other readers, too.
On the other hand, I’m happy with your income = expenditure + savings analogy; it’s just that your use of the term economics enticed one commenter off onto a macroeconomics tangent. Maybe you could drop that term?
Finally, a drive-by suggestion from one who only rare grasps what you write completely: you may want to consider adding a nod to one of your previous themes, i.e., the “governor.” My understanding of that (alleged-by-you and plausible-to-me) phenomenon is that a small change in surface temperature results in a great change in transport of latent heat to the upper atmosphere, where it changes to sensible heat and greater radiation into space. Maybe this is very roughly like an emissivity that is much more temperature-dependent than the Schwartz equations assume? In the (admittedly unlikely) event that what I’ve just said makes any sense, referring to it may give the reader an addition handhold by which to grasp your concepts.

Tregonsee
January 28, 2011 5:56 am

Reminds me of the famous “Then a miracle occurs!” cartoon.

Charles Higley
January 28, 2011 5:59 am

I have a quick, maybe simple, question. The discussion mentions a connection between the upper level temperature (emissivity) and the surface temperatures. However, since heat which goes into evaporation is temperature neutral (not being observed as a temperature change in the lower atmosphere) and water vapor transported upwards by convection condenses at altitude, releasing this heat, how can they be related without realizing or recognizing this effectively hidden transport of energy. The energy disappears into the water phase change and reappears at altitude.
How is this accounted for in these equations? I have not read the whole thing, but I have not detected it here.

January 28, 2011 6:01 am

Wow. I am actually learning something here.
A long-forgotten sensation.

Ian W
January 28, 2011 6:10 am

To pick up what cal says: January 28, 2011 at 3:38 am Temperature is not Heat content.
But where everyone has gone wrong so far is that in the atmosphere, unlike the oceans, the enthalpy can vary greatly dependent on the amount of water vapor.
Therefore, in the atmosphere Temperature is NOT EQUAL TO nor is it PROPORTIONAL TO (at a fixed rate) Heat Content.
Any simplistic formula that assumes a constant relationship between atmospheric temperature and heat content is FALSE due to the varying humidity (look at GOES water vapor imagery).
Then there is the misapplication of Stefan Boltzmann’s power law to radiation from the Earth’s atmosphere. Take the example of the winds blowing over the Himalayas the orographic uplift caused by the mountains (not convection) causes the air to rise and as the air cools water changes state from vapor to liquid and from liquid to ice at each stage releasing the latent heat of state change. Consider the floods in Pakistan all caused by this uplift in the monsoons: all that water was vapor but then released its latent heat to become ice/liquid then rain – a LOT of energy. This heat release is not governed by radiation laws based on surface temperatures. Similarly convective uplift can take humid air and liquid water to more than 30,000ft in the tropics and the ITCZ where the water at height rapidly changes state to ice; again releasing large amounts of latent heat in the upper atmosphere. The heat released is not governed by Stefan Boltzmann’s radiative laws either.
Simple back of the envelope models based on flawed assumptions seem to underly much of AGW theory.

P. Solar
January 28, 2011 6:11 am

E = ε σ Ts^4
I agree this equation , with the definition of ε is a meaningless identity , effectively saying 1=1.
What it really represents is more “watch the pea” tricks. Since ε in the S-B equation is a constant there is a subliminal messages that this epsilon is also constant. That is not said explicitly nor justified.
All the climate physics is contained in what this “effective” emissivity is , how it varies with time and temperature etc. How does it change from day to day with cloud cover …?
This one pseudo constant that quietly gets brushed under that carpet is in fact a multi-dimensional matrix quantity that represents most of what happens in climate from the ocean surface to the TOS !!
I think you have put your finger on something here.

Juice
January 28, 2011 6:12 am

Temperature isn’t a “flow.” In the simplest sense, it’s a function. It relates the heat content of an object to its heat capacity.
C = ΔQ/ΔT
ΔT = ΔQ/C
An object has some quantity of heat. It’s heat capacity is a ratio of the quantity of heat units to the number of temperature units (Kelvin”s” if you will). So you could call temperature a quantity too if you like.

In Burrito
January 28, 2011 6:37 am

Apologies in advance for sounding like a pretentious snob, but novel thermodynamic analysis is a dangerous game if you haven’t had it at the graduate level. This is far too lengthy for a detailed critique, but two points of confusion that stand out are the radiative dissipation and heat capacity equations.
First, the equation dH/dt = Cp dT/dt is absolutely valid, but you need to be very precise about your defined control volume. When folks talk about ocean heat content, they’re talking about the top 10s to 100s of feet of the ocean surface. The assumption (and a decent one) for using this control volume is that its thermal mass far exceeds that of the atmosphere, thereby rendering the atmosphere insignificant. Roy Spencer explains it in his global warming blunder book.
Ocean heat content is fairly simple to “average” over the surface of the earth since it’s linear with Ts.
The radiation term is missing the surface area of the Earth. Qualitatively speaking, it should be Qrad = Area(epsilon)(sigma)Ts^4. However, epsilon and Ts are going to vary over the surface of the earth. Therefore, to get total long wavelength radiative loss you really need the integral of (eps)(sigma)Ts^4 dAs. Because it goes as T^4, it’s much more dangerous to assume an “average” surface temperature. That’s because a 1 degree increase at the poles will yields a relatively greater change in long wavelength radiation that a 1 degree increase at the equator, keeping in mind that the loss from warmer areas of the earth is much much greater than colder areas. The point being, it’s complicated, non-intuitive, and needs to be approached quantitatively, not qualitatively.

stagioni
January 28, 2011 6:52 am

Willis Eschenbach has his mental GPS tracking nicely. Well done.
1) He has not been hoodwinked by the comment in the article about the conservation of energy;
2) He has pointed out one of the core weaknesses in the entire edifice of AGW
without having to drill one borehole anwhere or tax anyone! Cost = zero trillion dollars.

D. Patterson
January 28, 2011 7:01 am

Willis, I have the DVD in my collection. See:
The Cold Equations, Alliance Atlantis Presents, Bill Campbell, Poppy Montgomery, 93 min. Color, DVD, Echo Bridge Home Entertainment, c2003.

tmtisfree
January 28, 2011 7:17 am

The paper by Stephen E. Schwartz has been commented in 2008:
Knutti et al : http://www.iac.ethz.ch/people/knuttir/papers/knutti08jgr.pdf
Nicola Scafetta : http://www.fel.duke.edu/~scafetta/pdf/2007JD009586.pdf
and the reply by Stephen E. Schwartz is:
http://www.ecd.bnl.gov/pubs/BNL-80226-2008-JA.pdf

tmtisfree
January 28, 2011 7:20 am
cal
January 28, 2011 7:21 am

P. Solar says:
January 28, 2011 at 5:08 am
Willis, there is something fundamentally wrong with the basic physics of this treatise:
dH/dt = Q – E (2)
where dH/dt is the change in heat content of the climate system…..So , the crux: you can’t equate a rate of change of energy dH/dt to and energy , Q-
I think the explanation for this can be found in the next paragraph….
“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. ”
Thus it is clear that Q and E are power densities measured in watts per metre squared or joules per second per metre squared. Thus the dimensions are right as far as I am concerned as long as H is also per square metre.
I am not sure who is to blame for the rather sloppy way the equations are presented since I have also not read the original.

Vince Causey
January 28, 2011 7:22 am

P Solar,
“So , the crux: you can’t equate a rate of change of energy dH/dt to and energy , Q-E.
>>
where dH/dt is the change in heat content of the climate system.
>>”
I thought that as well, but then I realised it doesn’t mean that. dH/dt is not meant to be the rate of change of heat with respect to time, but the amount of heat that accumulates in 1 year. That is what the dt part is meant to be. This was done to make dH comparable with Q and E, which are both quantities of energy input and output in 1 year. So the equation is simply saying that the heat accumulated in 1 year is equal to the heat coming in in 1 year minus the heat going out.

kwik
January 28, 2011 7:30 am

Willis, I think you might be interested in reading this;
http://www.csc.kth.se/~cgjoh/blackbodyslayer.pdf

David L. Hagen
January 28, 2011 7:36 am

Great explorations Willis
Encourage you to explore relationships between the distributions in ocean heating with the distributions in insolation – primarily (1-cloudcover.)
There should be more marked differences by hemispherical winter vs summer. The differences N/S in land mass vs ocean should relate to differences in albedo – primarily from clouds, snow, and biomass.

oeman50
January 28, 2011 7:46 am

When I refreshed myself on Stefan-Boltzmann, my reference (Wikipedia – I know: http://en.wikipedia.org/wiki/Stefan%E2%80%93Boltzmann_law) indicated it is a description of the energy flux density. Cal says:
“I think this is a mistake. Temperature is not a flow. It is not “converted” to a power by multiplying by sigma. The output of the equation is a power density because sigma has the dimensions of Wm^2K^4.”
The equation does not attempt to make temperature a flow, it is simply a mathematical relationship describing how the energy flux density changes with temperature. And by definition, a “flux” decribes a flow, there is no flux without a flow of energy.

Vince Causey
January 28, 2011 7:47 am

“The global and annual mean absorbed shortwave irradiance Q = γ J, corresponding to γ ≈ 0.69. ”
From that they conclude “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.”
The thing I noticed is that in they have treated γ as a constant. It follows in that case that a ‘forcing’ due to say co2 increasing, would perturb Q upwards so that it becomes greater than E and leading to the heat accumulation dH. But there is no rational basis for making γ constant. It has been argued by others, including Willis and Dr Spencer, that γ is not constant, due to changes in cloud cover. In that case, Q may not even be perturbed by co2, in which case the whole edifice is based on a flawed premise.

Roger Longstaff
January 28, 2011 7:51 am

I think that the article demonstrates that simple assumptions can not be used to model climate – I assume that this was the point of the author. Planetary rotation is a key, missing ingredient, and also………….
I’ve looked at clouds from both sides now,
From up and down, and still somehow
It’s cloud illusions I recall.
I really don’t know clouds at all.

D. Patterson
January 28, 2011 8:30 am

cal says:
January 28, 2011 at 7:21 am
P. Solar says:
January 28, 2011 at 5:08 am

How many joules of energy are transferred between Earth’s planetary environment and the interplanetary environment? What is the net balance of these energy transfers? What portion of these energy transfers measured in joules and by percentage of the whole occur by means other than radiation?
What are the proper elements to be considered in determining the energy state in joules of a planetary body at a given moment in time and space and a given time period in time and space?
Do the current modeling equations include all significant elements of a planetary body’s energy state or not?

Leonard Weinstein
January 28, 2011 8:32 am

Willis,

Nolo Contendere
January 28, 2011 8:33 am

Just a side note to say that I really do enjoy the intellectual give and take on this site and quite often learn something. Thanks everybody. Now back to digesting Willis’ article….

Coldish
January 28, 2011 8:40 am

Ansatz: Thanks, Gary (4.37am) for the link to Lubos Motl who has used the term as follows:
“First, let me offer you an excellent Ansatz for the concentration of CO2 in the year “year” during the industrial era. We won’t usually need it but it’s simply:
conc(year) = 280 ppm + 22.3 ppm * exp[(year-1920) / 57]”.
If you want a single word equivalent in English for ‘Ansatz’ you could use ‘formulation’. However the German word ‘Ansatz’ seems an apt term, especially with the helpful definition which Willis has found in Wolfram’s Mathworld.

banjo
January 28, 2011 8:44 am

[snip. Completely off topic. ~dbs, mod.]

RobM
January 28, 2011 8:47 am

Willis, Avery good post in terms of the basic concepts explained. However the lack of attention to definitions has led to a lot of confusion to your readers. I think is arises from starting with Q and E as derivatives of energy when such symbols would usually refer to energy.
Because you ‘normalized’ to 1 yr time units and suppressed the dt (1) ? Then, later, you use flow to indicate time derivative ? All the equations will be dimensionally ok if you pay attention to this. Unfortunately many will attack you for this problem, which does not invalidate any of your basic points. It does indicate lack of respect if you believe that the authors you criticize did not use dimensionally valid equations.

Laurie Bowen
January 28, 2011 9:02 am

David Baigent says:
January 28, 2011 at 3:54 am
I am impressed..
“Apples and oranges”
Whenever I see this argument . . . I must assert . . . that may be true sir, but it is all still fruit to me . . . (for me) It’s a rational reference to a Venn diagram.

Vanguard
January 28, 2011 9:07 am

Willis,
You have a problem in your first figure. You have the incident energy labeled as Q and the outgoing energy labeled as E+dH/dt. Now, Q does indeed equal E+dH/dt, but the outgoing energy is NOT E+dH/dt. No, it is only E. You have the amount of outgoing energy mislabeled in the figure. When dH/dt is positive it means that energy is being transferred into the oceans at the rate of dH/dt, so that over a total period of Delta_T the amount stored is dH/dt times Delta_T.
When dH/dt is negative, it means that energy is being transferred from the oceans into the atmosphere, such that E = Q-dH/dt > Q and more energy is leaving than is entering.

mkelly
January 28, 2011 9:20 am

“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.”
“… 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.”
They have two “heat capacity” terms. C(air) and C(docean). For C(air) to change via composition then Cp of air must be increasing. I have not read of this. Also given the chemical reaction of oxidation for example methane CH4 + O2 -> CO2 + 2H2O. Given CO2 Cp of .8(something) and H2O of 2.(something) a change in heat capacity is being caused by water not CO2. 44.1 * .8 not equal to (2*18)*2.
Setting up a Cp*m*T comparison to find out how hot air would have to be or how much air would be needed to warm the deep ocean should be possible.

mcates
January 28, 2011 9:27 am

P. Solar,
I read this site every day and appreciate all opinions, but I think one can communicate disagreement without being quite so rude.
Especially, after someone obviously put alot of effort into the post.
“This is just nonsense.

It’s gobbledy-gook.”
There is so much information to process and these kinds of comments are just a distraction.

KR
January 28, 2011 9:28 am

Two comments on the two substitutions.
Substitution 1: Ocean heat content, due to the long time constant involved in heating/cooling the oceans will have very poor correlation to yearly temperatures, let alone quarterly. You will need to analyze correlation to decadal or multi-decadal temperatures and ocean heat content for any meaningful result. Yearly/quarterly correlations are far too short a period to be meaningful.
Substitution 2: Effective TOA IR emissivity and it’s relationship to surface temperatures come from satellite observations and line-by-line computations of upwards and downwards IR – the relationship is quite strong, unless you have successfully disproven the last 100 years of radiative and thermodynamic physics. Furthermore, you have dropped the surface emissivity in your presentation – the surface emissivity is not a blackbody by any means, although most ground cover emissivity is between 0.9-0.95 in IR.
Neither issue is, as presented, a valid objection to the basic energy equations.

dp
January 28, 2011 9:31 am

The refrigerator analogy is an interesting proposition. I don’t know that it matters to your presentation, but presume we have two identical refrigerated boxes chilled to 0º C, and two cans of room temperature beer. We put one can in each box, and we evacuate one box leaving it essentially airless.
After a period of time both beers will have reached the temperature of the chamber. One through pure radiation, the other through radiation and convection. It’s a minor difference but probably non-trivial. Radiated energy was not mentioned in your ‘fridge example or in describing how ocean heat transfers to/from the atmosphere.

Tom Jones
January 28, 2011 9:34 am

I just cannot get past dH/dt = C dTs/dt. That is just flat not true unless there is a linear relationship between Ts and H, which there is no reason to believe. Maps of sea surface temperature say that it is not true. One needs to integrate C T over all space to get H. I don’t see how that substitution can be made.

KR
January 28, 2011 9:40 am

In regards to my post #59:
Effective TOA emissivity is also directly calculable from TOA spectra and surface IR spectra. Surface spectra show the differences between a black-body (known curve from temperature) and actual emissions, while TOA satellite spectra compared to a black-body of surface temps gives the emissivity at TOA.
We know the surface temps (measurement), we know the black-body curves for that (basic physics), we know the TOA IR spectra (decades of satellite measurements). Hence we know the effective emissivity to space of the Earth relative to surface temperature. There’s no wiggle room there – your objection to TOA emissivity is invalid.

kwinterkorn
January 28, 2011 9:46 am

The atmosphere cannot transfer heat to the oceans (except in negligible amounts). But it must be true that as the the temperature of the atmosphere increases, the net rate of heat transfer from the oceans to the atmosphere will slow (for radiative and conductive heat transfer). Rising atmosphere humidity will slow the evaporative cooling of the oceans. Hence a warm, wet atmosphere is a more effective “thermal blanket” for the oceans, than a cool, dry atmosphere.
If a rising CO2 level in the air slows the exit of heat from the atmosphere to space, then rising CO2 will lead to rising air temps and consequently higher ocean heat content.
This is a decent, simple theory, easy to understand. The problem for the CAGW proponents is that the measured data is not cooperating with this theory. The air is not heating enough and the ocean heat content is not rising enough. There must be other processes countering the CO2 effect. Candidates for such negative feedback include: saturation effects of CO2 absorbtivity, cloud cover increases due to rising humidity, increasing air turbulence and storms lifting heat more rapidly into upper atmosphere (bypassing much CO2), and no doubt others.
The history of the Earth’s climate demonstrates a system dominated by complex negative feedbacks. The climate has been amazingly stable over a wide range of “forcings”. This is why the CAGW theory defies common sense, regardless of any set of equations or models used to predict an apocalypse.

FrankK
January 28, 2011 9:54 am

Willis I have not gone through your entire post yet but let me make the following point.
Temperature is not flow it is equivalent hydraulic head in groundwater hydrology and Voltage in electrical theory.
Heat flow in a solid medium for example is Q=K dT/dl where Q is the flow; K is the thermal conductivity; T the temperature and l is the distance (i.e. Fouriers Law) is it not? The same as Darcy’s law in groundwater hydrology and Ohms law in electricity.

January 28, 2011 9:55 am

Will take about a week to grasp all of this ,but how much “sunlight “is converted into biomass ,is it relevant?photosynthasis must use a lot of energy.

January 28, 2011 10:11 am

One doesn’t have to scroll down your piece to find that you have discovered some of the basic mistakes that modeling climate scientists have made. Most come from the idea that there is some “dynamic climate equilibrium factor” that can be measured as a long-term global average. Missdefining the e term as they have done is a good example of forgetting basic science. The e term can be considered admittance thru the atmosphere. The reciprocal of e is resistance. That resistance varies considerably globally. So does the surface temperature. Properly used, e can tell us the relative contributions of water in the air and CO2 in slowing down energy loss to space. A global average e is about as useful as tits on a bull. http://www.kidswincom.net/CO2OLR.pdf.

In Burrito
January 28, 2011 10:14 am

anuary 28, 2011 at 9:49 am
In Burrito says:
January 28, 2011 at 6:37 am
Apologies in advance for sounding like a pretentious snob, but novel thermodynamic analysis is a dangerous game if you haven’t had it at the graduate level. This is far too lengthy for a detailed critique, but two points of confusion that stand out are the radiative dissipation and heat capacity equations.
First, the equation dH/dt = Cp dT/dt is absolutely valid, but you need to be very precise about your defined control volume.
The problem isn’t that you sound like a pretentious snob. It is that you haven’t said anything. It is useless to simply say “dH/dt = Cp dT/dt is absolutely valid” without a single theory, fact, or observation to back up your claim. I have provided a host of different reasons, both theoretical and observational, to show that it is not valid. Those observations and theoretical considerations are on the table, and against that, your claim that their work is “absolutely valid” without citations or observations is meaningless handwaving from an anonymous burrito … which on my planet, at least, does not comprise falsification of anyone’s claims.
w.
100+ years of thermodynamic analysis says that dH/dT = Cp dT/dt is valid, and I *did* explicitly state my assumption and limit my application to temperature changes in the ocean surface (no atmosphere, no phase changes, no expansion, etc.). This is the same assumption Roy Spencer makes in his model. I’m not sure if we’re talking past each other with you saying it’s mathematically invalid or not a valid assumption for the macroscopic energy balance.

Ron Cram
January 28, 2011 10:29 am

Willis,
Thank you for your efforts. This is another very interesting post. I do not know if you are correct or not, but I am glad you are putting it out there to be discussed. I love the way you show your work.
I wish it were possible to entice Stephen Schwartz to comment here, but unfortunately he is convinced his writing time should be devoted to journal articles. I am a fan of Schwartz. We exchange emails on occasion. Your goal is probably to get some feedback before you submit to a journal. If you were to publish your critique in a journal, Schwartz may reply. And I think he is able to modify his position if convinced a mistake has been made.
My question is what difference does this make? For example, Schwartz was using these equations to estimate a climate sensitivity. Are you saying if the substitutions were not made, the estimate would be different? Or are you saying if the substitutions were not made, an estimate would not be possible? I’m sorry for being a little dense here. I’m rushed for time and thought it might be easier to just ask.

George Tetley
January 28, 2011 10:32 am

Willis,
if you told me that time, and our universe is in the hand of some great personage lighting a match, I would expect you to be the one to accurately calculate the amount of breath to blow it out !
I am on my knees, keep up the phenomenal posting, you are the greatest.
George

Robbo
January 28, 2011 10:35 am

wrt to the Sub-rosa equation:
Ts is a mean. The mean of a fourth power is decidedly not the fourth power of the mean. ( two readings, 1,3 mean = 2 fourth power of mean=16; 4th powers 1 and 81; mean of fourth powers = 41) This expression is not valid.

cal
January 28, 2011 10:36 am

D. Patterson says:
January 28, 2011 at 8:30 am
cal says:
January 28, 2011 at 7:21 am
P. Solar says:
January 28, 2011 at 5:08 am
How many joules of energy are transferred between Earth’s planetary environment and the interplanetary environment? What is the net balance of these energy transfers? What portion of these energy transfers measured in joules and by percentage of the whole occur by means other than radiation?
What are the proper elements to be considered in determining the energy state in joules of a planetary body at a given moment in time and space and a given time period in time and space?
Do the current modeling equations include all significant elements of a planetary body’s energy state or not?
This is my pencil sketch of how it works. Previous posts go into more detail but they may contain more detail than you need.
The only way earth loses heat is by radiation. If the world is in thermal equilibrium the energy received from the sun (and any radioactive decay in its core) must equal the energy lost to space. About 70% of the energy incident at the top of the atmosphere is actully absorbed, the rest is reflected into space. This is all short wave radiation. In practice the earth oscillates around a short term equilibrium point due to the very powerful negative feedback of the Stefan Boltzman law which says that the outgoing radiation is proportional to the fourth power of the absolute temperature. This means that a small change in temperature results in a massive increase in energy flow acting in the opposite direction to the temperature change. The equilibrium point is mainly dependent on the earths albedo (reflectivity) and the effect of the atmosphere which increases the height and therefore lowers the temperature at which radiation into space takes place.
As to whether the models reflect the science the jury is out. I am sure they include the basic science as described above but the world is much more complex than this simple model. Climate models have to predict (amongst other things)the way the equilibrium temperature will change given changes in land use and atmospheric composition. But heat is lost to the upper atmosphere though convection and evaporation and previous posts have gone into the implications of this in great detail. Not only do these transport energy but the generation of clouds have both negative and positive impacts on the equilibrium temperature. Add to this the fact that the world is not in long term equillibrium due to the absorption and release of energy from the oceans and the ice caps and you have a complex problem to solve. My own opinion is that they are far from solving it.
Hope this helps

MikeW
January 28, 2011 10:43 am

The cold Internet bestows immortality to all things it touches, even to little girl who meets a fate she didn’t deserve.
http://www.spacewesterns.com/articles/105/
I’d not read that story before, thanks. A powerful story with a very nostalgic flavor of the post WWII era Sci-Fi style.

Dishman
January 28, 2011 10:48 am

I believe Substitution 1 is demonstrably incorrect, from the physics.
A more correct substitution would be the temperature of Tropopause. The first problem with working there is that Tropopause is not a fixed surface.
Makarieva et al [2010] kinda starts down the road to deriving the correct substitution, but, to my read, stops well short.
If nothing else, I’ll write it up myself when I run out of bigger fish to fry.

cal
January 28, 2011 10:56 am

oeman50 says:
January 28, 2011 at 7:46 am
When I refreshed myself on Stefan-Boltzmann, my reference (Wikipedia – I know: http://en.wikipedia.org/wiki/Stefan%E2%80%93Boltzmann_law) indicated it is a description of the energy flux density. Cal says:
“I think this is a mistake. Temperature is not a flow. It is not “converted” to a power by multiplying by sigma. The output of the equation is a power density because sigma has the dimensions of Wm^2K^4.”
The equation does not attempt to make temperature a flow, it is simply a mathematical relationship describing how the energy flux density changes with temperature. And by definition, a “flux” decribes a flow, there is no flux without a flow of energy.
I think you missed my point. Willis was saying that dT/dt was a rate of change of flow. It is not.
My guess is that the original writers were saying that the energy in the sea is proportional to temperature. The proportionality is the thermal capacity. It is not an equation that is designed to explain why or how the sea warms it is simply a statement of how thermal capacity is defined.

FrankK
January 28, 2011 10:57 am

Willis,
Reading a little further regarding dH/dt = C dT/dt
In groundwater hydrology (equivalent to Heat Theory)
dV/dt = S dh/dt This is an valid equation where V is the change in volume (also eq Heat); S is the Storativity (eq heat capacity) h is the hydraulic head (eq Temperature) and t is time.
I think what you are objecting to is not the validity of that equation per se but the substitution of the surface temperature Ts for the ocean temperature which you indicate are not related. Is that correct ?

dwb
January 28, 2011 11:12 am

well, but what about heat exchange with the earth’s crust too, not just the oceans? is there some reason to discount the effects of steam vents, volcanoes, and the like?

P. Solar
January 28, 2011 11:21 am

>>
wrt to the Sub-rosa equation:
Ts is a mean. The mean of a fourth power is decidedly not the fourth power of the mean. ( two readings, 1,3 mean = 2 fourth power of mean=16; 4th powers 1 and 81; mean of fourth powers = 41) This expression is not valid.
>>
but the global temperature is not going to triple (unless Hansen turns out to be correct 😉 ) so what it the error in say comparing 300K to 310K (0C=273K)
the difference of the forth powers is about 14% whereas the diff in temp is around 3%.
So it is not negligible but not an order of magnitude off.
Maybe a important error is the lack of account for heat coming from the interior of the Earth that was mentioned above.
It seems however that Schwanz 2007 was only a first attempt at a ball park empirical figure. It is not the corner stone of AGW.
At least someone is trying to use data rather than computer models.
It all goes to underline how grossly over-simplified all this work is and just how much climate science is in its infancy. Nothing you’d want to trust to redesign life on Earth of start screwing around with geo-engineering.

stephen richards
January 28, 2011 11:23 am

random kinetic energy of
Lazy teenager has discovered a new form of energy. Even Newton missed that one.

Gunnar Strandell
January 28, 2011 11:27 am

Thank you Willis,
The term “Ansatz” is spelled Ansats in Swedish and hav some interesting explanations
in the dictionary of the Swedish Academy, that selects Nobel Prize winners in litterature:
http://g3.spraakdata.gu.se/saob/
freely translated by me:
2) attack on the enemy
7) start of something
8) impulse, initiative, idea
10) the lips position before you blow the horn
Nevertheless your article has given light to my sense of AGW being built on a vicious circle.

Graham Green
January 28, 2011 11:27 am

You must have been raised eating magic beans ol’ cowboy.
This post represents a triumph in clarity of thought.
Ansatz is defined by Google as ‘an educated guess that will be verified later’. Ansatz science from ersatz ‘scientists’.

January 28, 2011 11:34 am

Willis, what “cal” says at January 28, 2011 at 3:38 am, holds for me. Temperature is not a flow.
I think you are on to something here, but I cannot determine exactly what because of several problems.
First, it is standard procedure to identify each and every term in every equation, with that term’s appropriate physical units. For example, it is not clear what “Ts” is. Usually, T with a subscript is a temperature of some sort, with the subscript designating what location has that temperature. Of course, temperature has units of F, or R, or C, or K (Fahrenheit, Rankine, Celsius, or Kelvin, respectively). I had assumed Ts is the surface temperature of the Earth, until I read that Ts is a flow.
Second, the appropriate equation for heat flow, Q, related to heat capacity is
Q = m x Cp x (T2 – T1); where
Q is heat flow in Btu/hour (or similar heat per time units),
m is the mass, in pounds
Cp is the heat capacity constant, expressed in Btu/hr/pound/degree F,
T1 is temperature of the mass at its initial condition, in degrees F,
T2 is the temperature of the mass at its final condition, in degrees F.
It should be noted that Cp is not truly a constant, but has a slight dependence upon temperature. However, for temperatures that vary over a small range, Cp can be considered a constant.
Only by clarifying the exact units of each term may the reader follow the unit conversions and thus determine the validity of the exercise and thus the conclusions.

FrankK
January 28, 2011 11:59 am

Its a little early here so a correction I posted before is required:
I wrote:
“Willis,
Reading a little further regarding dH/dt = C dT/dt
In groundwater hydrology (equivalent to Heat Theory)
dV/dt = S dh/dt This is an valid equation where V is the change in volume (also eq Heat); S is the Storativity (eq heat capacity) h is the hydraulic head (eq Temperature) and t is time.
I think what you are objecting to is not the validity of that equation per se but the substitution of the surface temperature Ts for the ocean temperature which you indicate are not related. Is that correct ?”
===================================================
The above should read: where dV/dt is the change in volume with respect to time.
While I’m here. In groundwater hydrology (as in heat theory)
Inflow volume-Outflow volume = Change in Volume (1)
(i.e. theMass Balance equation). Dividing each side by dt
Inflow Rate – Outflow rate = dV/dt = S dh/dt (2)
This is the basis for S2007
From (2) above in heat theory therefore :
Q (rate) -E (rate) = C dT/dt (3)
This is a basic physically correct equation. The point is how this equation is interpreted is the critical issue and as you suggest what substitutions are used.
Thought provoking Willis and its getting me more interested in digging deeper
into whole basis of AGW “theory”.

FrankK
January 28, 2011 12:04 pm

Oh incidentally
Only when S dh/dt is zero in GW hydrology is the system at steady state or the heat flow equation Q=E

jae
January 28, 2011 12:08 pm

Great post, as usual!
One very minor point. You say:
“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):”
Don’t you mean “…can only be rectified by increasing OR DECREASING temperature.” ? (apologies if someone else mentioned this; I don’t have time to read all the comments).

Juice
January 28, 2011 12:09 pm

I don’t know what the fuss is about. I can’t see anything wrong with the general relationship of:
dH/dt = C dT/dt
The rate of change of temperature must be linear with the rate of change in heat, proportional to the heat capacity. For real systems, the relationship might not be entirely linear, but pretty close. You might have some boundary issues depending on the thermal conductivity of the system, but generally speaking I can’t find a problem with the above relation, although it is oversimplified.
This may be interesting:
http://en.wikiversity.org/wiki/Nonlinear_finite_elements/Linear_heat_equation

Fernando
January 28, 2011 12:13 pm

dV/dt = S dh/dt
Closed cycle,
dV/dt = 0,0000000
S # 0,00000
…dh/dt = 0,000000
delta (MSL mm) = 50 mm or 40 mm

Laurie Bowen
January 28, 2011 12:36 pm

Mr. Eschenbach you say . .
. . . “whole colony will perish … so she has to be jettisoned through the air lock to die in space.”
I say,
I guess I am “witarted” as I don’t understand why this is relevant . . . as it is a partial information story . . .
Mr. Eschenbach you say . .
most basic climate equation says that energy in equals energy out plus energy going into the ocean.
I say,
This I understand . . . because you said “basic”
Mr. Eschenbach you say . .
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).
I say,
true . . . . unless you are going in debt . . .
which leads me to that old accounting trick called. . . . 9, 8, 7, 6, + 5 = 11
as a math guy you should know it. . . if not, to understand the error . . . it must be demonstrated.
I am at the point that maybe I should just “jettison” myself from these discussions. The entire concept of understanding climate change is fairly simple to me . . . the big arguments are quantifications. . . but I need to understand the relevence of arguing about say 1000 when your dealing with 1,000,000,0000,000,000,000 . . . . with very few exceptions. I guess it takes more that a few trees for a forest to be made. . . . and of course that leads to talking about angles and pins. But, for me the quantifications are misleading until the qualifications are correct. . .

January 28, 2011 12:38 pm

Hmm. One thing that might help the reader (it sure helps me) is to carry along the units in every equation, term by term. This lets me verify that all terms have consistent units. For instance:
Example I: Force = mass times acceleration
F [in Newtons (i.e., kg.m/sec²)] = m [in kil’ograms] * a [in meter/sec²]
Example II: Net Force = Applied Force – Friction
Net Force [in Newtons] = Applied Force [in Newtons ] – (Normal Force [in Newtons] * Coefficient of Friction [unitless])
I also recall seeing somewhere that the temperature exponent of 4.0 in the Stefan-Boltzmann Law was only valid for perfect black bodies. The Earth is not a perfect black body.

Tim Folkerts
January 28, 2011 12:45 pm

The mere fact that so many people are commenting about the units indicates that there is a problem — whether the problem is with nomenclature or the science is mostly a matter of semantics.
* traditionally, “E” and “Q” would be used for energy (e.g. J or ZJ)
* in this discussion, “E” and “Q” are generally used for power (energy/time; e.g. ZJ/year)
* in E = εσ(Ts)^4, “E” would be power/area (e.g. W/m^2). To make this work, either ε would have to include the surface area of the earth somehow, or the equation would have to change to εσA(Ts)^4.
My suggestion would be to more precise in usage and more precise is stating exactly what each quantity is being used for.
On a different front, I started to wonder about different energy contributions …
Human energy use from burning fuels is about 0.5 ZJ per year; we could call this “F”.
Geothermal energy flow up from below is about 1 ZJ per year; we could call this “G”
This would make the equation: E + dH/dt = Q + F + G. Of course, both of these are pretty small, but they would add to about 10% as much energy as is entering/leaving the oceans. Are there other significant sources/sinks of energy?

E.M.Smith
Editor
January 28, 2011 12:45 pm

Very well done. Very well done indead.
Yeah, I perked up on Ansatz as well. A reasonable translation would be “guess”…
We’re basing all this on a GUESS.
Also: 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.
Caught my eye. This is another of the Great Leap Forward moments… 😉
They assume they have a clue what the heat capacity is. This is just so full of errors I don’t know where to start. It assumes there will be no long term change in the formation or destruction of ice, for example. It assumes there will be no long term change in the overturning of the oceans. (They even dance around saying that further down with the bit about heat stored in the surface layers; yet changes in the curcumpolar currents can change the rate of overturning of the world ocean and cause significant changes in surface temperatures and thus assumed or implied heating / cooling. Just look at ENSO variations). And so much more.
They (warmer “climate scientists”) have built a fantasy world on guesses, assumptions, and broken physics (averaging an intensive variable, temperature, from different objects / places) and think it has meaning. All it has is such confusion that it’s hard to show how broken it is. Like the person speaking complete bafflegab, it’s so broken you can’t begin to get a handle on straightening it out.
What you have done is found that handle to grab…
Oh, I’d also note in passing that they assume all heat must be radiated as IR from the TOP of the atomosphere. This implies that there is NO IR transmissive window at all from the surface. Yet surfaces will drop to very cold temperatures very quicking (with variance by object… that old fractal surface problem…) as they see an apparant very low IR night time sky temperature. Point an IR thermometer at the sky some cold dark night. Surprised at how “cold” it is? If that “top of the sky” is doing all the radiating, it’s gotten darned cold up there…
And what happens to all their assumptions if the CO2 is NOT CO2 at altitude?
This Just In: As of 11 Jan 2011 we’ve found out that CO2 and H2O can combine to form stable carbonic acid gas at temperatures as high as -30 C.
So what happens to all that CO2 “forcing” if the CO2 isn’t there? …
http://chiefio.wordpress.com/2011/01/27/fizzy-sky-ir-spectrum-is/
No, it’s not “settle science”, but that’s the whole point. We just don’t know what’s going on. So maybe those “way below -30 C” night sky IR temps are simply the result of the CO2 and water making a substance that does not block IR very well at night. Now that whole “top of the atmosphere” assumption exits and we get something rather like what we really see. My windshield frosts over from direct IR radiation even at air temperatures that do not support frost. Direct surface to space IR transport of heat?
I’m beginning to think more scientists need to be sent off to Cowboy Camp for a year or three to get them oriented to what a night sky feels like…

KR
January 28, 2011 1:04 pm

Willis Eschenbach @ 80
The upper 90 m of the ocean (the “well mixed” zone where wind driven circulation moves energy around relatively quickly) would alone add about a 6 year time factor for an instantaneous change in heating. Considering the deep layers with thermohaline circulation adds another 10-100 years (would be ~230 without circulation).
Hence my statement that yearly or worse yet quarterly air temperature correlations are worse than useless – you’re essentially looking at the correlation with noise. You really need decadal or multi-decadal trends to see what’s going on in climate, as opposed to weather. Inherent system variability (El Nino, for example) will slop energy around enough on shorter timescales that you simply cannot see the trends.
As to the second “substitution” objection: As I posted in @64, we know the surface temperatures (repeated, multiple direct measurements). We know the expected black-body radiation in IR based upon that temperature. And we have decades of satellite observations of the actual space-going IR to show us how much energy is actually emitted.
Put those three data together and you directly get the effective emissivity of the Earth to space.
E = ε σ Ts^4 is the Stefan-Boltzmann equation; the classic thermal radiation relationship. I suspect it was not numbered in the original because it’s accepted science, not a newly introduced equation to be justified. My reaction about “disproving” was in relation to your apparent issues with the Stefan-Boltzmann relationship.
We know the temperature of the surface, we know what is emitted to space, and via the S-B equation we get the effective emissivity. There’s simply no issue there.

Ron Cram
January 28, 2011 1:04 pm

Willis,
I was just reading a paper by Chylek and wondered what you would think of the equations in that paper. I’m not giving you homework. I’m just saying I would be interested in your analysis.
http://www.knmi.nl/~laagland/KIK/Documenten_2008/2007JD008740.pdf

Tim Folkerts
January 28, 2011 1:06 pm

Heat capacity, usually denoted with a capital “C”, has dimensions of [Energy/Temperature], for example J/K or BTU/˚F
Specific heat capacity, usually denoted with a lowercase “c”, has dimensions of [Energy/Temperature/mass], for example J/(kg*K) or BTU/(lb˚F)
I have never seen heat capacity include time, nor Q include time.
Roger Sowell says: January 28, 2011 at 11:34 am
Q is heat flow in Btu/hour (or similar heat per time units),
Cp is the heat capacity constant, expressed in Btu/hr/pound/degree F

Perhaps that is the convention in some areas of engineering, but again, that highlights the need to be very explicit in defining how the symbols are used, since different people my be interpreting the symbols differently

KR
January 28, 2011 1:07 pm

Willis – “You have the lovely habit of being specific about your objections and your thoughts.” – Thank you, I make an attempt to do so, and appreciate the feedback.

Brian H
January 28, 2011 1:40 pm

Charles Higley says:
January 28, 2011 at 5:59 am
I have a quick, maybe simple, question. The discussion mentions a connection between the upper level temperature (emissivity) and the surface temperatures. However, since heat which goes into evaporation is temperature neutral (not being observed as a temperature change in the lower atmosphere)

Say what? Evaporation causes powerful cooling of any surface, and nearby air. Else what’s perspiration for?

January 28, 2011 1:42 pm

What part does heat emanating from the Earth’s Core play in all this?
ntesdorf

Malcolm Miller
January 28, 2011 2:08 pm

“The Cold Equations” was a story in Analog Science Fiction magazine. I can’t remember the author, but if I had the time to examine my complete collection of ‘Astounding/Analog Science Fiction’ which has never been indexed I’m sure I could find it. I remember both the title and the story from about 50 or so years ago.

Tim Folkerts
January 28, 2011 2:11 pm

Willis,
I disagree a little with you here … “It is useless to simply say “dH/dt = Cp dT/dt is absolutely valid” without a single theory, fact, or observation to back up your claim. ”
Pretty much the definition of heat capacity (at constant pressure) is [heat that flowed into an object]/[change in temperature]. Traditionally this is written as
Cp = Q/ΔT
but with the nomenclature used here this becomes
Cp = – dH/d(Ts) [dH is the energy LEAVING, not entering the ocean].
Rearranging and finding the energy per year gives
– dH/dt = Cp d(Ts)/dt
Other than the minus sign, the biggest problem to me is that they use d(Ts) (the surface temperature change) as a proxy for the change in temperature of the some nebulous amount of upper ocean. Clearly the two ought to be related (if the top meter is warming, then some warming throughout the relevant layers should also be occurring, although the correlation may not be great.)

John Phillips
January 28, 2011 2:14 pm

In Burrito said “That’s because a 1 degree increase at the poles will yields a relatively greater change in long wavelength radiation that a 1 degree increase at the equator, ”
Isn’t the opposite true? Since energy is a function of T**4? and temps are higher near the equator?

Alan S. Blue
January 28, 2011 2:20 pm

The Cold Equations available as an ebook where there’s an explanation and history in the preface. The actual short story is not among the available ‘sample chapters.’

Laurie Bowen
January 28, 2011 2:22 pm

For Malcolm Miller
The Cold Equations – Wikipedia, the free encyclopedia
“The Cold Equations” is a science fiction short story by Tom Godwin, first published in Astounding Magazine in 1954. In 1970, the Science Fiction Writers of …
Summary – Reactions – Allegations of borrowing – Adaptations
en.wikipedia.org/wiki/The_Cold_Equations – Cached – Similar

George E. Smith
January 28, 2011 2:25 pm

“”””” 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 49 times in three short years. 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. “””””
And here’s where I would disagrre with you Willis:- “”””” and longwave (thermal infrared) radiation emitted at the top of the atmosphere E. “””””
I suggest that only a small portion of the LWIR emission to space, actually comes from the top of the atmosphere; where the molecular and atomic density is evanescent, and hardly capable of emitting anything like 390 W/m^2 upwards.
A good bit of the outgoing LWIR originates from the very surface of the earth, corresponding to the various ranges of surface Temperature that can be found on earth all at the same time.
The heat capacity at the top of the atmosphere has to be so low, that any emission at all, would cause the temperature to plummet, if it wasn’t being constantly replenished from below.
Do we actually have something like a measured complete spectroscopic plot of the outgoing radiation from earth say from 0.1 to 100 microns, at a resolution capable of showing all the important components ? I have some “simulated” ones for a much smaller part of the external view, that clearly don’t have all the expected components present and accounted for.

Editor
January 28, 2011 2:33 pm

Malcolm Miller says:
January 28, 2011 at 2:08 pm
> “The Cold Equations” was a story in Analog Science Fiction magazine. I can’t remember the author,

“The Cold Equations” is a science fiction short story by Tom Godwin, first published in Astounding Magazine in 1954. In 1970, the Science Fiction Writers of America selected it as one of the best science fiction short stories published before 1965, and it was therefore included in The Science Fiction Hall of Fame, Volume One, 1929-1964.

Some sources, including Kurt Busiek, have alleged that Godwin essentially took the story from a story published in EC Comics’ Weird Science #13, May–June 1952, called “A Weighty Decision,” scripted by Al Feldstein. In that story there are three astronauts who are intended to be on the flight, not one, and the additional passenger, a girl that one of the astronauts has fallen in love with, is trapped aboard by a mistake rather than stowing away. As in “The Cold Equations”, various measures are proposed but the only one which will not lead to worse disaster is for the unwitting passenger to be jettisoned. Algis Budrys said that “The Cold Equations’ was the best short story that Godwin ever wrote and he didn’t write it.”
Other sources note that the theme of Feldstein’s story is itself strikingly similarly to the story “Precedent”, published by E.C. Tubb in 1949; in that story, as in the others, a stowaway must be ejected from a spaceship because the fuel aboard is only enough for the planned passengers. These sources argue that neither Feldstein nor Godwin intentionally “swiped” from the stories that came before, but merely produced similar variations on an ancient theme, that of an individual being sacrificed so that the rest may survive.

James Barker
January 28, 2011 3:08 pm

Great post, Willis. The Cold Equations was written by Tom Godwin, and published by Analog,

Jim D
January 28, 2011 3:25 pm

It seems to me that these equations for C and epsilon are definitions in terms of long-term trends. If ocean heat content increases, so will Ts, and C is the proportionality constant. If surface temperature increases, so will outgoing radiation and epsilon is the proportionality constant. It is very clear to me that these were not intended to describe time-scales as short as inter-annual variability, but more the climate time-scales. Their Ansatz is therefore to define these proportionality constants for a simplified model of climate-scale variations.

otter17
January 28, 2011 3:28 pm

If the assertions in this post are true, these seem like pretty egregious thermodynamics errors. Why not publish these results in a real journal instead of a blog? Be sure to post again if the paper is approved. Maybe this would overturn parts of AGW.
I hope the articles below aren’t true, otherwise credentials would be questionable:
[snip]
[You can link to opinions that Willis Eschenbach is “lying” on many different blogs. But not here. ~dbs, mod.]

January 28, 2011 3:44 pm

Willis, thanks for the math showing the fallacy of regarding the earth’s radiative balance as a relatively simple function of its average surface temperature (whatever that is; note that Dr. Hansen’s GISStimate of the “average” temp for a given year — e.g. 1940 — changes as time passes; who knew Ts varies both with and without time?).
With regard to the story, like others here I was impressed by it when I read it as a kid. The Wiki article references an interesting critique of the story’s premise here:
http://www.writing.com/main/view_item/item_id/1204039-BS-Flag-on-The-Cold-Equations
I also recall reading a probably derivative novel “High Vacuum” in which a lovestruck female stows away on the first moon mission, causing it to crash for lack of fuel:
http://en.wikipedia.org/wiki/High_Vacuum
To this day I suspect the problems on Apollo 13 were really caused by a stowaway. 🙂

Geoff Sherrington
January 28, 2011 4:14 pm

One of my first posts on CA years ago was the impossibility of inferring too much from T measurements that were half daily max + min. This early act assumes a distribution that is artificial and it affects all downstream discussions about distributions. Specifically I objected to this measure being used as a stand-in for H.
This rather limits the discussion on distributions to the period where daily T is measured many times a day, say from 1990 onwards or even later.
Willis, your post questions the T to H equivalence and rattles the cage of orthodoxy again. These early “balance” authors were trying to get away with plots as bad as a movie script – like “The Heat of the Night” which was popular for black body radiation of a magic property named charisma (units unknown).

wayne
January 28, 2011 4:35 pm

Willis points out:
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
Summary of Substitution 2: E = ε σ Ts^4
This substitution is, quite demonstrably, either mathematically wrong or meaninglessly true as an identity.

Willis, this seems half-science, I don’t agree with the top and the bottom question depends on the limits and exact definition. I have commented hear long enough that you know I usually have a hard time placing in words what is in my thoughts, but I’ll try here on one isolated aspect and this is not strictly addressed toward you for I’m sure you already know everything I am about to say, maybe.
What does Ts, the temperature at the surface, tell us at the surface? Well, it is a temperature so it merely tells us at that very point what the kinetic energy is by the mean velocity of the molecules. But says absolutely nothing of where that energy goes during the next dt (increment of time), does it?
I can immediately see at least four aspects that occur and let’s start with only the oceans being 71% of the surface. Some will always go into the water below dictated by specific heat and conductivity and the temperature below. Some will go into (or be added coming out) via latent heat for there is always evaporation/condensation occurring at the ocean interface. This is largely dictated by the wind velocity and turbulence to remove the saturated air from the surface but it is always there. Some of that energy will go into the creation of winds itself, and ocean currents though tiny.
After subtracting all of these energy fluxes (uses) from Ts then you can compute (maybe only by numeric integration for I can’t imagine a closed form set of equation here) what energy is left that can then be transferred by radiation, but, the Ts value is no longer the same as before, that is, the Ts that can radiate by that equation above. If you just use the equation above then ALL of the energy from Ts is going to radiation, nowhere else, and that clearly, and logically, is not the case. The same happens at the soil interface but with different parameter values.
We all know in the tropics it is evaporation rules. All of the energy, some 2350 J/cm3 of evaporated water, must be subtracted from the energy at the surface before radiative equations can come into play. It’s a figment. They are way overstating the radiation in these cases.
Yes, as they did, you can smear this over such a long period of time that most, if not all, of these effects cancel but you have then smeared out all of the resolution to be able to say if anything is actually different for one month or year to the next. Sounds a little like the uncertainty principle doesn’t it? That curious, though not really the same.
One more big point. If energy is spread evenly over a surface, the radiation emitted at T will ALWAYS be less that if the exactly same quantity of energy is lumped in certain areas with other areas cooler. That is what the fourth power causes instantly and that is why you really can’t average the energy evenly over the Earth in the first place and hope to get true science out. Even though radiation rules all aspects as viewed far away as from space, input equals output, it is the non-radiative energy fluxes that rule the movement of energy upward so radiation to space can easily occur (yes, I know of the window and optical thickness, altitude dependence).
IPCC and certain climatologists seem to toss all of that reality out of the window and into the wind and this warped view is carried along (sometimes by many here) without realizing all of the subtle mistakes that just occurred in their statements, intentional or not. When real physics get into this arena, I hope soon, we will finally start to get some correct answers.
I’ll stop rambling. I have one more comment but will stop here at a good breakpoint.

crosspatch
January 28, 2011 5:05 pm

And Dr. Lacis’ claim, that increasing GHG forcing can only be balanced by a temperature rise, is central to mainstream AGW climate science.

The problem with that is it assumes all the heat radiated transits all the CO2. It doesn’t. If you have vapor condensing into ice at very high altitude, a large amount of heat is lost well above the CO2. Same thing even with water vapor condensing at anything >20,000 feet. It is well above most of the CO2. If all the heat were radiated from the surface, the argument would have some validity but a large amount of heat is moved via evaporation and condensation that transits a large portion of the CO2 before it is released as longwave IR.
For example, a large amount of the heat generated by the human body is transported thousands of feet aloft by the wind and then released when our sweat condenses at high altitude.
If you can quantify how much heat is released at which altitude, you can then begin to get a realistic grip on the impact of atmospheric CO2. Absorption of heat by CO2 isn’t really a matter of the relative quantity of the total atmosphere (ppm of the atmosphere made up of CO2), it is a function of the absolute density of CO2 between where the heat is released and the top of the atmosphere.
If you release longwave IR in an area where there are few molecules of CO2 to absorb it, it doesn’t matter what the percentage of CO2. Some small percentage of almost nothing is less than almost nothing.

January 28, 2011 6:02 pm

Interesting thought.
I agree:
The T,E and Q are flows statement is off.
The fridge analogy needs a little work.
If you define the units earlier you should see that the dH/dt = C dT/dt equation is right.
The plots of OHC and temperature change are interesting. Rapid change in OHC of the magnitude you show goes against what I would expect. It would be interesting to see what areas had the greatest short term changes in OHC.
In Burrito’s comment about how OLR changes from the poles to the tropics brings up a good point about emissivity and the uncertainty in determining Sensitivity. Dry air 2XCO2 Sensitivity is approximately 1 to 1.6 C. So at the poles, CO2 would be the dominate GHS. But the estimated ~3.3 C sensitivity (Gavin) is moist air. So I expect that the average emissivity is over estimated. How much I don’t know.

Steve Reynolds
January 28, 2011 6:20 pm

Willis: “My body is at about 37°C. Like anything at that temperature, there is a constant flow of radiative energy emanating from my body, 525 W/m2. How, then, is a temperature of 37°C not equivalent to a flow of 525 W/m2?”
Because the flow is not a linear function of temperature. In fact in your example, if your temperature increases much more, you will start sweating, and total energy flow will continue to increase while temperature will not (unless you are dehydrated).

Steve Fitzpatrick
January 28, 2011 6:58 pm

Willis,
I do not think Schwartz believes that the ocean is warmed by the air. I believe he understands that it is mostly the other way around. After all, the majority of total solar energy reaching the earth’s surface is absorbed by the ocean surface layer. Most of that energy finds it’s way to space through the atmosphere.

Hoser
January 28, 2011 7:19 pm

I guess my problem with this is it implicitly assumes all physics and no chemistry or biology. That might be a mistake. How do the numbers add up?
In the oceans, algae absorb light and perform chemical reactions that convert light energy to biomolecules that require X kcal/mole to synthesize. Not all of that stored energy is returned to space. Some of it sinks to the bottom of the ocean. What percentage of the incoming radiation is captured and doesn’t get converted to heat?
On land, soil is formed through a decay process involving vegetable matter accumulated on land and grasslands. Coal is an example of stored solar energy from millions of years ago.
Changes in the amount of energy stored through chemical synthesis might be comparable to the effects of other factors in the energy balance. After all, we do have a lot of photosynthetic organsims on the planet, and quite a bit of biomass. It doesn’t all burn or decay back to CO2, H2O, NO3-, HSO4-, H2PO4- and so on. Living things are composed of cells, and the bulk of these cells need to divide regularly. Therefore, they need to take up nutrients. The majority of the biosphere depends on photosynthesis ultimately. But how many kg of biomass are left behind each year?
A very rough estimate of Net Primary Production is 100 Gt/yr of C (http://en.wikipedia.org/wiki/Biomass_(ecology)). If we take ‘C’ as referring to the typical photosynthetic reaction:
H2O + CO2 + light -> CH2O + O2, (http://www.solarnavigator.net/photosynthesis.htm)
each ‘C’ corresponds to 112 kcal/mole of stored energy. Working out the math, I get about 9 ZJ/yr of stored solar energy. Some of it goes back into the system as heat during decay.
Perhaps 9 ZJ out of 5500 ZJ is small, but I am not yet convinced it is negligible. How much of that 5500 just reflects off of clouds or the ocean surface? And biological effects can have other impacts. Transpiration certainly is a significant contributor to humidity, and thus cloud formation.
Obviously, the real world is not so simple as the energy equation we are talking about in the main post. But certainly I enjoyed the discussion!

Udar
January 28, 2011 7:30 pm

Steve Reynolds says:
January 28, 2011 at 6:20 pm
Willis: “My body is at about 37°C. Like anything at that temperature, there is a constant flow of radiative energy emanating from my body, 525 W/m2. How, then, is a temperature of 37°C not equivalent to a flow of 525 W/m2?”
Because the flow is not a linear function of temperature. In fact in your example, if your temperature increases much more, you will start sweating, and total energy flow will continue to increase while temperature will not (unless you are dehydrated).

Energy flow is not linear function of temperature, but it is a KNOWN function of temperature, at least for radiative heat transfer, which is the only one that is being discussed here. Evaporative cooling, convection and conduction are not being used in radiative balance equations.

January 28, 2011 7:52 pm

@ Tim Folkerts at January 28, 2011 at 1:06 pm
Tim, my mistake, and you are correct, that the Cp should not have included units of time. It should be Btu/hr/degree F. The time unit should properly be with the mass, in pounds per hour. Thus, we have Q in Btu/hr = mass in pounds/hr x Cp in Btu/lb/degree F x (T2 – T1) in degrees F.
The continuous process industries use Q as Btu/hr and mass in pounds/hr. These industries include things such as chemical plants, petrochemical plants, natural gas processing plants, oil refineries, and many others. Every industrial plant with a fired heater or fired furnace uses these calculations.
The point remains that every element of each equation must be labeled and its units provided.

Alan McIntire
January 28, 2011 8:00 pm

See Trenbeth’s figures at
http://stephenschneider.stanford.edu/Climate/Climate_Science/EarthsEnergyBalance.html
He has a total flux of 492 watts, with 102 in latent heat of vaporization of water vapor and convection.
E = ε σ Ts^4 is wrong! It should be something like
E = ε σ Ts^4 + Latent heat, and their argument doesn’t give any indication of how this should be broken down. I suspect that as E increases, the sensible heat should increase at a lower rate than latent heat.
As to
dH/dt = C dTs/dt
“The total mean mass of the atmosphere is 5.1480×1018 kg with an
annual range due to water vapor of 1.2 or 1.5×1015 kg depending on
whether surface pressure or water vapor data are used; somewhat
smaller than the previous estimate. The mean mass of water vapor is
estimated as 1.27×1016 kg and the dry air mass as 5.1352
A 4C rise or higher this century would see the world warm almost as
much in 100 years as it did during the 15,000 years since the end of
the last ice age.”
During last ice age, there were 71.3 million k3 ice *0.917 vol ice/vol
water=
65.3821 million cubic kilometers of water.
1 cubic meter= 1000 kg.
1 cubic km = 10^12 kg
65.3821 million cubic km= 65.3821*10^18 kg
Total heat to melt glaciers =65.3821 *10^18 *1000*334 kj=2.18*10^25
joules
Cp air= 1.012 joules/gram K
1012 Joules/kg K * 5.148^10^18 =5.209776 *10^21 joules
4degree increase=2.0839 *10^22 joules
So about 1000 times as much heat went into melting the glaciers at
the end of the Pleistocene as went into heating the atmosphere.
C in the above equation is on the order of 0.001
too small to be measured over any reasonable interval- random fluctuations will
overwhelm any attempted measurement of correlation, and as you stated, it would be the ocean heat driving air temperature rather than vice versa.

Tim Folkerts
January 28, 2011 8:19 pm

Willis,
As I understand it, a “flow” is a transfer of something from one place to another. [Eg “there was a flow of 2 gallons of water into my sink.”]
A “flow rate” would be a measure of the rate at which the flow occurred. [Eg “there was a 0.5 gal/min flow rate of water into my sink.”]
Q is a flow rate of energy (at least it is in this article; more typically “Q” is used for a flow). E is a flow rate of energy. dH/dt is a flow rate of energy. [ie some number of joules of energy are transferred from one place to another place in a given amount of time]
That would mean that C dTs/dt is a flow rate. C dTs would be a flow. dTs by itself is neither a flow nor a flow rate (although it is related to both). Ts is also not a flow or a flow rate, but it is related to both via Stefan-Boltzmann.
If you have a different definition of “flow” or “flow rate” that you would like to propose, I will listen.

Jim
January 28, 2011 8:20 pm

I think they meant that equation was the starting point.
***
Ansatz (pronounced [ˈanzats], English: “onset”; today, “approach, setup, starting point”; plural: Ansätze) is a German noun with several meanings in the English language. [1] It is widely encountered in physics and mathematics literature. Since ansatz is a noun, in German texts the initial a of this word is always capitalised.
***
http://en.wikipedia.org/wiki/Ansatz

crosspatch
January 28, 2011 9:29 pm

Given a constant amount of CO2, if weather conditions change so that there are more rainstorms, more of the heat will bypass the atmospheric CO2. Everyone says the equations take it into account, but how can it when it can vary by such a large amount from year to year?
How much heat was released at high altitude by the rainstorms in Australia? I don’t think the amount of heat released by weather at high altitude is consistent from one year to the next.

wayne
January 28, 2011 9:32 pm

Willis, from Appendix 1 downward, very interesting as usual. What a great way to visualize the fallacy in the use of such simple equations without all of the other factors never addressed.
Probably you have already asked yourself, can the dissimilarity of the shapes of warming to cooling now be brought closer together by adding or subtracting a proper factor of the cloud cover data over this period or maybe some of the other major factor?
It’s so logical, and real, that the deep absorption of short wave radiation would be the primary cause, and the thermal conductivity and diffusivity, being quite low, greatly slows the movement of that energy back out of the ocean. I’m surprised those violin graphs don’t look even further apart in shape.

Crispin in Waterloo
January 28, 2011 9:36 pm

Leonard Weinstein says:
…2) The air temperature does have an effect on ocean heat retention. It is a small effect, and not dominate (thus the low correlation), but don’t say there is no effect at all. The mechanism is that the ocean loses the absorbed solar radiation three ways….
++++
I agree Leonard that the radiation of heat from the water surface is often misunderstood or overlooked. Water is a powerful radiator of heat, about the same as liquid black oil!
++++
kwinterkorn says:
…But it must be true that as the the temperature of the atmosphere increases, the net rate of heat transfer from the oceans to the atmosphere will slow (for radiative and conductive heat transfer). Rising atmosphere humidity will slow the evaporative cooling of the oceans.
++++
I believe this is a mistake, confusing specific humidity with relative humidity. It is a common error and many CO2-global warming arguments contain it. Yes it is true that the warmer ocearn (however heated) will give off more water vapour into a warmer atmosphere, but the warmer air will remove more moisture (total mass) to achieve the same relative humidity.
If the whole system, air and water, increases 0.1 degerees in temperature, the rate of evaporation is the same. Evaporation is not blocked by air that is now ‘full’. By ‘full’ I mean, stabilising at the same relative humidity as before the increase. As the air is warmer, it can hold more water.
I recently mentioned elsewhere a related error which is to think that if it contains more moisture (in g/m^3) is will rain more. If the land over which the cloud passes is also 0.1 deg warmer, then the cooling effect will be down to a temperature that is also 0.1 degrees higher than before, so the amount of rain is the same, the dried air moving away at 0.1 degrees higher, retaining its little increase of moisture as to goes because the old rained-out _relative_ humidity has been reached. This misunderstanding drives claims about New York snow being caused by ‘warmer air holding more water in a heated world’.
Leonard’s point is that there is an insulative effect from the higher absolute humidity. This is correct. The resulting warmer air can contain more moisture (etc) and is part of a positive feedback. All things considered, however, the resulting _relative_ humidity in the end is the same, for the temperature changes with which we are concerned.

FrankK
January 28, 2011 10:14 pm

Willis Eschenbach says:
January 28, 2011 at 5:14 pm
Hi, Frank. No, that’s not what I’m saying.
In a situation where temperature differential is driving the temperature change, or equivalently where hydraulic head is driving the volume change, that relationship holds. But in this situation, atmospheric temperature is not driving ocean heat content changes, it simply doesn’t have enough thermal mass.
======================================================
Willis, Well I don’t disagree with you, in a sense that is what I was saying that you disagree that Ts is not a suitable proxy.
I have not read S2007 whether that’s what author(s) are saying but my interpretation would be that the sun is heating the ocean (that is part of Q going into the ocean) and cooling is part of E heat rate being dissipated. So wouldn’t the surface temperature also be a reflection of what the sun is also adding to the ocean over the long term.
The fact that they don’t agree (surface temps and ocean) in the short term surely is not the measure since there are time lags involved etc. What bothers me more is the next point:
The Stefan- Boltzmann equation to me with the epsilon parameter is that this parameter has a value somewhere less than 1 and would seem to take into account the fact that the earth is not a truly black body and well as other factors. But where does CO2 and water vapour fit into all of this? It seems all to be lumped into epsilon. One parameter that has no definite value!! In other words this is a fitting parameter and they are trying to extract the contribution that CO2 is making (apart from a more important water vapour) to limit the radiation to space.!! Its open slather – take your pick about the episilon value, but if you are CO2 driven s there’ a clear path available to “validate” your model.
If I have misinterpreted what the S2007 authors have done or intended my apologies.

Austin
January 28, 2011 10:44 pm

If it is windier, you will get a lot more water vapor off the oceans.
This is like leaving the lid off the pot and it takes longer to boil.
The colder periods on the Earth were much more windier according to the Greenland ice cores.
Maybe the key is to find the speed control?

Tim Folkerts
January 29, 2011 12:16 am

Willis,
I’m not really sure what your latest objections are.
“Tim, I’ve said it before, but I’ll say it again. That formula is true and valid … but only in the case where the gain/loss of heat is governed by ∆T.”
It takes energy to warm an object. The amount of energy is C∆T. The heat gained (called dH here) is governed by the temperature change of the object; the temperature change is governed by the heat gained. dH = C∆T. Unless there is also a change of phase, then this equation will hold. (Or there would be potential problems if the mass of the object changed, but I don;t think the overall change in the mass of the ocean due to evaporation will be enough to be a serious concern, since an almost exactly equal amount is returned by precipitation.)
“they’re not even using ∆T (difference between air and ocean)”
They SHOULDN’T be sung the difference between air and ocean. ∆T would be the amount that the ocean warms on average during the year (or cools if ∆T is negative). It is not the temperature difference between air and water. Are you thinking about heat conduction, where the amount of energy that flows is proportional to the temperature difference across some insulating layer? Then the difference between ocean and air would play a role, but that is not the ∆T (or “dTs” as it is called here) that is being used in this paper.
“as you point out, they’re using the change in T with time (dT/dt).”
That is what they should be using. If the temperature of the ocean starts the year at T1 and ends at T2, then the the total energy change is dH = C(T2-T1) = C∆T = C d(Ts). Since we want dH per year, we divide both sides by (t2-t1) = ∆t = dt = 1 year.
The problem that I was trying to point out is that the “object” being heated is not clearly delimited. Just how much of the ocean should be included? A quick look at the actual suggest they try to address that issue, so I am not going to go into more detail here now.

January 29, 2011 3:09 am

About the units – just a question about dH/dt = Q-E. H is the ‘heat content’ of something. It’s dimension is joules (energy). Rate of change of H with time dH/dt has dimension joules/second. That is watts. OK but Q (irradiance) is not watts but watts/meter^2. So is H heat per square meter? What is the physical meaning of this – a two dimensional surface of no thickness cannot have heat can it?

peter_ga
January 29, 2011 3:22 am

I wonder if the nub of the argument here is that “global warming theory” says that everything should be warming, the atmosphere, the surface, and the oceans.
However if the atmosphere and surface is warming, but the oceans are cooling, then it is surely simply a transient natural phenomena, to do with the phase of the ocean flow oscillations. The oceans are dumping heat into the atmosphere, and cooling as a result.

January 29, 2011 3:34 am

Willis,
I think at least four things have been left out of the equation, which make the theory indeterminate.
Firstly there are layers in both the oceans and the atmosphere, which are in turbulent interaction.
Second there is an interchange of heat into and out of all living matter (animal, bacteria & vegetable) as it grows and decays. Individually this is imaterial, but there sure is a lot of it in total.
Third there is the latent heat of melting and evapotation going in both directions. These store and release energy in an indefinite timetable.
Clouds, thunderstorms, volcanos add to the chaotic mix.
Fourth, there are the short term chaotic atmospheric and oceanic perturbances, the Le Nino / La Nina events that happen every few years and then the major 60 year cycles, changing each 30 years as we are seeing at the moment. These also store and release heat.
So it’s easy to say that the physics is settled – case closed, as we hear all the time, which just means that:
The energy coming in, LESS the energy going OUT,
EQUALS ( and must by all the laws of physics always equal) plus or minus the CHANGE in what’s temporarily left WITHIN the system in a particular year.
Now that’s ALL that physics can tell you.
The actual equation, let alone the numerical values is written in the stars.
Now what did Hamlet say to Horatio – something about “more than you can know or understand” I think.
Bother – my memory of English literature is even worse than my physics.
And I was a physics scholar once upon a time when the world was young.
No matter.
My principle is right – I am afeared that you are on a wild goose chase, meboy.
Warmists often talk reeeeele egucated like, but its all spin and make believe.
Balancing global heat budgets in detail is for the birds.

cal
January 29, 2011 5:03 am

Willis Eschenbach says:
January 28, 2011 at 5:08 pm
A number of people have said that temperature is not a flow. However, temperature can be converted to the equivalent blackbody radiation flow using the familiar Stefan-Boltzmann formula.
Think about it this way. My body is at about 37°C. Like anything at that temperature, there is a constant flow of radiative energy emanating from my body, 525 W/m2. How, then, is a temperature of 37°C not equivalent to a flow of 525 W/m2?
On this logic a voltage is a current! A hill is an acceleration!
I explained in a previous post that the “flow” is encapsulated in the stefan boltzman constant which has the dimensions of watts/m^2T^4. If you do a dimensional analysis of the Stephan Boltzman equation you will clearly see that the dimensions on the left hand side are watts/m^2 and these dimensions are reflected in the Stephan Boltzman constant itself not in T.
To repeat, it is the Stefan boltzman constant which governs the radiative flux generated by the Temperature just as the resistance of a wire governs the flow of current caused by a voltage. Temperature is not a flow.
If you still don’t believe me do the calculation of doubling T. Will the flow double? I don’t think so!

izen
January 29, 2011 5:09 am

Willis Eschenbach says:
January 28, 2011 at 5:08 pm
“Think about it this way. My body is at about 37°C. Like anything at that temperature, there is a constant flow of radiative energy emanating from my body, 525 W/m2. How, then, is a temperature of 37°C not equivalent to a flow of 525 W/m2?”
Is this making the assumption that the skin surface is a ‘black-body’ radiator? -grin-
If this was the ONLY energy flow I think you would need to consume several thousand calories and HOUR to maintain body temperature.
However net flow means that most of us require far less energy input to maintain our surface temperature.
1)-The ambient temperature determines how much energy is flowing BACK to the body surface, so while the 37degC determines the outgoing flow of 525W/m2 the temperature difference is proportional to the net flow.
2)-We usually modify this further by the use of clothes that absorb the outgoing energy radiated from the body, warm in response and radiate a portion of that energy back to us. The clothes also ensure that the outer radiating surface of the person is at a lower temperature than the skin surface so that energy loss between the system of body+clothes radiates at less than 525W/m2
The parrallel with surface and tropopause temperatures is obvious I hope.

Alan McIntire
January 29, 2011 6:30 am

In reply to Izen. The calories humans consume are actually Kilocalories. Given that,
depending on your weight and lifestyle , you may consume about 2000 calories per day, that 2000 actual kilocalories is actually 2,000,000 calories, so yes you DO have to consume several thousand calories per hour to stay alive.
I’m reminded of the joke-when you drink a liter, about 1.09 quarts, of ice water, you
body heats heats it up to 37C. That works out to 37,000 calories. So why can’t you lose weight by sitting around drinking ice water. That 37,000 calories works out to only
37 kilocalories, about 37/2000 or 1.85% of a reasonable daily intake.

Peter
January 29, 2011 7:09 am

Wills:

Think about it this way. My body is at about 37°C. Like anything at that temperature, there is a constant flow of radiative energy emanating from my body, 525 W/m2. How, then, is a temperature of 37°C not equivalent to a flow of 525 W/m2?

Except, if there isn’t an equivalent flow of energy into your body, it wouldn’t stay at 37°C for long.
Perhaps a living body is a bad example, as it regulates its temperature, but the principle still holds – for a constant temperature, Ein = Eout.
I discovered the basic principles of this when I was around eight years old. Armed with a clinical thermometer, I filled a bathtub with water at a temperature of 37°C, and was then very surprised that I couldn’t immerse my body in the tub for more than a few seconds before starting to feel unbearably hot.

izen
January 29, 2011 7:29 am

@- Alan McIntire says:
“In reply to Izen. The calories humans consume are actually Kilocalories. ”
I am well aware of the conversion factors between food calories and kilocalories as used in human nutrion.
But I must admit to a big mistake in my calculation. To expend 525W/m2 ‘only requires around 900 ‘human’ calories (kilocalories) an hour.
Or something over 20,000 calories a day, around ten times the usual human consumption.
In very cold climates increasing the calorie intake is required because the temperature difference increases the NET loss of energy.
But when the temperature difference between your body and the environment is around 15degC then you don’t need to eat 2 big Mac’s an hour to maintain body temperature.

Peter
January 29, 2011 8:31 am

Cal,

On this logic a voltage is a current

As power is voltage x current, W/m2 can be regarded as a measure of flow (current) , provided that either the voltage or resistance is known.

Laurie Bowen
January 29, 2011 8:42 am

ausiedan said
January 29, 2011 at 3:34 am
“I think at least four things have been left out of the equation, which make the theory indeterminate.” . . . . . “Balancing global heat budgets in detail is for the birds.”
My question is, why is it SO important a thing to do . . . get it exactly right?
Is it because the ability to predict accurately gives the illusion of the power to control? “likened unto a god”
or
Is it because the ability to predict accurately gives the illusion of the power to influence behavior and say SEE it worked?
or
Is it just the ability to instill fear?
I’d like to think it is for forwarning . . . but, thus far this kind of knowledge has historically been used for the former three, (at least from my perspective).

Robbo
January 29, 2011 8:59 am

@ P. Solar says:
“.. the global temperature is not going to triple (unless Hansen turns out to be correct 😉 ) so what it the error in say comparing 300K to 310K (0C=273K)
the difference of the forth powers is about 14% whereas the diff in temp is around 3%.
So it is not negligible but not an order of magnitude off.”
– Spot on, I only used 1 and 3 to clearly illustrate the faulty maths. To use Stefan-Boltzman to calculate radiative heat output you have to integrate by time and space to account for the fact that different locations have different temperatures at any specific time, and that each location’s temperature varies with time on diurnal and annual cycles. Also, I wonder if there is a difference between land and sea. IIRC land heats up and cools down faster than the sea, so a priori is likely to to emit more radiative heat than sea at the same average temperature.
With regard to the magnitude of the error, I could understand if this was offered as an estimate, with the caveat “we know it’s not correct, but it’s pretty close for practical purposes”, but its not put that way. Further, even a 3% error can be highly significant (try underpaying your taxes by 3% if you don’t believe me). Lastly, my biggest objection is that this is such a crass mistake, in the one little bit of the picture that I understand, that all the stuff I don’t understand, let’s say I am not taking it at face value.
“It all goes to underline how grossly over-simplified all this work is and just how much climate science is in its infancy. Nothing you’d want to trust to redesign life on Earth of start screwing around with geo-engineering.”
– Very nicely put, thank you

Laurie Bowen
January 29, 2011 9:53 am

Robbo said January 29, 2011 at 8:59 am
“It all goes to underline how grossly over-simplified all this work is and just how much climate science is in its infancy.”
I am sorry, but I completely disagree with this statement . . . In my opinion, Climate Science is fairly old and understood. Wars have been planned and fought based on Climate Science for one . . . To me, what is going on is there has been a “break” in the accurate predictions, and those that plan, based on climate science, are trying to figure out where ‘they’ went wrong. Sad, but true. Climate Science, has been “proprietary” information for many, many, many generations. It’s just like the “secrets” of banking, only different.
Now, how do you prove a secret is a ‘secret’!?

Peter
January 29, 2011 10:43 am

Laurie Bowen:

Climate Science is fairly old and understood

How many climate scientists were there 30 years ago?
What makes you think that we know any more about how the climate works than we know about the far side of the Moon?

John in NZ
January 29, 2011 11:13 am

Very interesting Willis. I think you are onto something. It will take me some time to get my head around all of it.
However, my first problem. The assumption that there is a “close radiative balance”.
“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”
Over longer periods (i.e. centuries) I accept that absorbed shortwave radiation Q will approximately equal longwave (thermal infrared) radiation emitted at the top of the atmosphere E.
In other words
Q ≈ E
But over shorter periods (i.e.years), is it reasonable to assume that albedo approximately constant? Some years there will be more cloud cover making Q lower and vice versa.
So on an annual basis it is unreasonable to assume that Q ≈ E
Even if E is approximately constant, which I doubt, the annual absorbed shortwave radiation Q is quite variable.
Which means everything that follows is questionable as well.

Laurie Bowen
January 29, 2011 11:26 am

Peter, The best I can tell you (at this point) is . .
Many books have been written, for many years. Before books, it was handed down from generation to generation. From farmers, from herders, seafarers, travalers, and the scientists. “Merlin” was one of the first chemists/physicist, back then they called them magicians, today he is a mere legend.
To demonstrate the way I see it. Gun powder was first written about 3,000 years ago, by the Chinese.
Knowledge and civilizations CAN run in cycles. It is something that even Ayn Rand alludes to. It does not have to be that way as some cycles are behavioral (wars) and some cycles are beyond our control (climate).
Ask yourself, why the Libraries of Alexandria were destroyed? I will say it is not the first time in our long human hystory that knowledge has been lost.
Without a camera, how do you prove a bird just flew by?

John in NZ
January 29, 2011 11:37 am

Is the difference between Q and E is merely dH/dt?
Q is defined in terms of an annual irradience and mean albedo. but year to year the albedo may be sufficiently variable to upset the balance.
In other words, E varies dependant upon dH/dt but Q varies dependant upon γ .

Peter
January 29, 2011 11:42 am

Laurie,
It all depends on what you mean by, “climate science”

John in NZ
January 29, 2011 11:45 am

Vince Causey says:
January 28, 2011 at 7:47 am
What I was trying to say but he said it better.

3x2
January 29, 2011 12:08 pm

Willis,
I always look forward to your posts – keep them coming. (feeling guilty about not commenting more often)
The main comment I do have is not directly related to the post itself but something that seems to crop up monotonously in comments here lately – “warm air holds…”. Guys (‘n’ galls) – ‘air’ ‘holds’ NOTHING. Energised Water Molecule leaves (the surface), de-energised Molecule returns (to the surface) defying gravity as it goes – please read Dalton and his inconvenient gas laws. If you are still having trouble with “warm air holds…” then head off here for an edu-muk-ashon
So, Willis I have spent some time looking at your post and (bottom line) I think that you long ago recognised the flaw in the “all that’s happening between the surface and TOA is theoretical ‘black body’ radiative physics” POV. You had it right with your “Thermostat Hypothesis”. It is a 1.4 Billion cubic kilometre of liquid water bypass of anything else going on in “climate”. The equations don’t match reality because they are ‘Ansatz’ and wrong.
As an aside – do you really believe that, before ARGO, we really had any clue as to what OHC might or might not have been?

KR
January 29, 2011 12:11 pm

Willis:
I’ve just read through the Schwartz paper – very interesting, with some counter-intuitive results, such as a very low calculated heat capacity for the oceans and a very low climate sensitivity result, essentially a zero-feedback value on top of CO2 doubling. He states that given his results the climate is essentially in lock-step with forcings, with a 5 year time constant for response.
My impression is that he is looking at the transient response only, and is failing to take into consideration deep ocean heating. There are a couple of papers indicating distinct heating in benthic flows out of the Antarctic, for example. That’s the “in the pipeline” energy, which we haven’t seen the full results of yet.
That said, having read the paper, the effective emissivity of the Earth has very little to do with his results – he’s primarily speaking of ocean temperatures, rather than radiative balances; I would have to consider the top of atmosphere IR really irrelevant to the methods he’s using.
So:
I followed up on (not being a German speaker) the various definitions of “Ansatz” – the primary one used in physics and math I found was “an educated guess that is verified later by its results”. Hardly the horror you describe. I think we would have to ask Schwartz which definition he was using before judging that term.
You are correct, the surface temps do not seem to have a direct relationship with TOA radiation over the last 15 years – I’m going to do some digging on that. I am guessing that there may be a correlation to increases in CO2 (decreasing effective emissivity) over the last 15 years, but I’m going to have to run some numbers if I can. That’s a very reasonable point. One of the things I like about your posts is that you have sufficient content to have things worth discussing!
As to the energy “in the pipeline” – I get an odd feeling that when this comes up in discussion people think that the energy being discussed is supposed to be hidden under a rock somewhere, and that the absence of this hidden cache is somehow the invalidation of AGW. The thing is, the energy “in the pipeline” is energy that hasn’t arrived yet! It’s energy that is accumulating due to an imbalance between incoming and outgoing energy, energy that is slowly warming the oceans as it arrives, but it’s not here yet. If it was here, it wouldn’t be in the pipeline. So what we’re seeing is a _slow_ change in temperature as the imbalance warms the ocean, an imbalance that will exist until surface temps are high enough to radiate energy to space equal to that coming in. Trenberth’s issues with OHC are that we don’t seem to see the energy accumulation expected _so far_, which may be an issue of imprecise measurements, more benthic circulation than expected, whatever, but points to limits on either our measurements or our models of heating.
Keep in mind that we also have internal variability, like the ENSO, Arctic Oscillation, etc., moving energy around in the ocean and climate system. These provide cyclic changes in heat distribution several times larger than the yearly trend in heat content, and hence make it important to look at multi-decade data to ascertain imbalance trends – my original point on short correlations, and an issue concerning me with Schwartz’s paper as well.

Jim Masterson
January 29, 2011 12:43 pm

>>
peter2108 says:
January 29, 2011 at 3:09 am
. . . two dimensional surface of no thickness cannot have heat can it?
<<
This is not true. In classical thermodynamics, heat is defined as the transfer of energy across a system boundary due to a temperature difference (from hotter to cooler). System boundaries are generally two dimensional surfaces. An object/system cannot contain heat as heat is transient phenomenon.
Jim

Richard Sharpe
January 29, 2011 12:53 pm

Laurie Bowen says on January 29, 2011 at 11:26 am

To demonstrate the way I see it. Gun powder was first written about 3,000 years ago, by the Chinese.

Hmmm, can you point me to a reference, because this is some 2000 years earlier than other claims for Chinese priority on gunpowder. Were they corning their powder? What were their cannon casting techniques like?

Laurie Bowen
January 29, 2011 2:28 pm

Peter says:
January 29, 2011 at 11:42 am
It all depends on what you mean by, “climate science”
Gee, that is correct . . . and I will tell you is that I did not involve myself in these arguments in my community until Al Gore showed up with his (climate science) and bright idea of blaming climate change on human activity and proposing to tax the common citizenry (without apportionment) into oblivion. providing a very very “comfortable” ‘income’ to people like himself . . . .
I am not a scientist, although I would have like to have been, but I am a student of Cycles of all kinds . . . and cycles are everywhere . . . . and I learned a long time ago, that if you are any good at predictions, people will try to ‘steal’ your model before they will ever pay you for it.
I learned along time ago, a sheep skin does not guarantee competence, and the lack of one therein, does not make one “witarted”.
and Richard Sharpe you say . . . Hmmm, can you point me to a reference, because this is some 2000 years earlier than other claims for Chinese priority on gunpowder. Were they corning their powder? What were their cannon casting techniques like?
I am not your Tom Sawyer . . . I read the statement a book “Timetables of History” (I believe that is the name) and it was around 1000 BC if I recall correctly. The point of the mention was that knowledge, if found can be hidden, known, lost, forgotten or just plain monopolized.
The problem I am trying to articulate . . . It’s like people asking, how many times can we re-invent the wheel?. . . . OR would it be better just to acknowledge the wheel as invented and stop “acting” like it’s new or even improved.
Oh, and the Chinese have the oldest preserved writings . . . also, . . . . quite a feat for a bunch of “communists”
I apologize to the rest of the group for interjecting a qualitative argument into what is meant to be a quantitative discussion . . . . . but, for me math is nothing but quantitative reasoning, which is qualitative first. For example, I have never understood why anyone would want to solve for PI . . . when it equals 22/7.

Jim Masterson
January 29, 2011 3:34 pm

>>
Laurie Bowen says:
January 29, 2011 at 2:28 pm
For example, I have never understood why anyone would want to solve for PI . . . when it equals 22/7.
<<
At the risk of feeding the trolls, I have to correct your basic misconception. Pi is an irrational number; therefore it can’t equal a rational number like 22/7. It’s also transcendental, but that is really going OT.
Jim

Alan McIntire
January 29, 2011 3:49 pm

In reply to Izen : My grand uncle used to say
” Five foot eight and eight score weight” is a good build for a man.
Using that figure, plus the formula for surface area
http://www.halls.md/body-surface-area/refs.htm
BSA (m²) = ( [Height(in) x Weight(lbs) ]/ 3131 )½
68*160 /3131 = 3.4749 and the square root of that is 1.86 square meters.
The ratio in watts between 37 C and 15 C would be about
(310/288)^4 = 1.342,
A naked 160 lb 5′ 8″ man standing around in 15 C weather would lose
390.7 * 0.342* 86,400*1.86 = 21,473,172 joules /4.186 calories per joule=5,129,759
or 5130 Kcalories per day, a reasonable figure considering most people don’t stand around naked in 15 C weather.

Joel Shore
January 29, 2011 4:57 pm

Willis Eschenbach says:

A number of people have said that temperature is not a flow. However, temperature can be converted to the equivalent blackbody radiation flow using the familiar Stefan-Boltzmann formula.
Think about it this way. My body is at about 37°C. Like anything at that temperature, there is a constant flow of radiative energy emanating from my body, 525 W/m2. How, then, is a temperature of 37°C not equivalent to a flow of 525 W/m2?

Sorry, Willis, but this simply is not right. The Stefan-Boltzmann Equation just says that an object emits radiation by virtue of its having a nonzero temperature. That doesn’t make temperature a flow … It is more like the cause of the flow.
What Lazy Teenager said (and stephan richards for some reason mocked) is much closer to the truth, namely that temperature is “that portion of the energy in a substance that is due to random kinetic energy of the molecules that make it up”. I would quibble with this statement a little bit…In particular, you would want to say that temperature is a measure of that energy (as it doesn’t have units of energy itself) and equating it in some simple linearly proportional way to the kinetic energy is only true for an ideal gas, but the basic gist of it is correct.
You can also think of temperature as being that quantity that two substances that are allowed to exchange energy through heat with each other will try to equalize. (Hence, it is really differences in temperature, and not temperature itself, that drives the exchange of heat. Remember that the Stefan Boltzmann Equation only describes the emission from an object…In general, an object can absorb radiative energy from its surroundings too.)
This site is not a bad introduction to temperature: http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/temper2.html And, the wikipedia entry is a bit more ponderous but useful: http://en.wikipedia.org/wiki/Temperature

otter17
January 29, 2011 5:23 pm

“You can link to opinions that Willis Eschenbach is “lying” on many different blogs. But not here.”
Got it. The validity of such claims in those articles may not have been checked in a rigorous journalistic environment.
But this is ok?
http://wattsupwiththat.com/2011/01/28/the-met-office-and-the-bbc-caught-cold/
“The Met Office has been caught ‘cold’ lying about its winter forecast in a disgraceful attempt to salvage its reputation.”

P. Solar
January 29, 2011 6:32 pm

KR says:
“I followed up on (not being a German speaker) the various definitions of “Ansatz” – the primary one used in physics and math I found was “an educated guess that is verified later by its results”. Hardly the horror you describe. I think we would have to ask Schwartz which definition he was using before judging that term.”
The problem is , it is not verfied by results, it just remains “an educated guess “. For you other comments you really should check out the published comments and responce posted by someone near the top. It covers a lot of what you raise.
However, the big problem as always is that any and all warming is attributed to CO2. If it’s bigger than CO2 should be , then it’s hand wavy cloud or something “feedback”.
After all the fancy formulae the bottom line seems to be : it has to be CO2, what else could it be?
Hardly a scientific proof, although it probably qualifies as one of those unverified ansatz things.

Brian H
January 29, 2011 8:33 pm

Robbo;
About the land and sea emissions, two words: “specific heat”.
Water is much harder to cool or warm. At a specific temperature, land will cool faster because it’s easier to cool, which will then drop its emissions. The sea is harder (takes more emissions) to cool. So it will emit more.

cba
January 29, 2011 9:12 pm

Willis,
two possible balance mechanisms
1. cloud cover fraction
2.cloud reflectivity as a function of particulate sizes
both of these change the incoming absorbed amount and #1 also changes the outgoing LWR. The cloud cover fraction can determine or create a new balance point.

FrankK
January 29, 2011 9:13 pm

Willis Eschenbach says:
January 28, 2011 at 5:08 pm
Think about it this way. My body is at about 37°C. Like anything at that temperature, there is a constant flow of radiative energy emanating from my body, 525 W/m2. How, then, is a temperature of 37°C not equivalent to a flow of 525 W/m2?
=========================================================
This is really like saying in an electrical circuit the voltage at the positive end is say 100 volts.. Now put a resistance in the circuit. Ok now we get a current I = V/R lets say 1 amp. Now your equivalently saying:
“ How then is a voltage of 100 volts not equivalent to a flow of 1 amp”? .
No voltage is the potential that causes the flow to occur its not the “flow” itself-that’s the current or the energy transfer in the Boltzmann equation not the temperature.
The temperature is the absolute temperature in Kelvins relative to the absolute minimum possible. There in lies the potential difference operating to produce energy flow.
Well that’s how I remember it. Cheers.

Father Guido
January 29, 2011 9:49 pm

So if CO2 isn’t causing CAGW, which I agree with, what is?
I keep hearing it’s natural and cyclic in nature, but until we can show what causes it we’re no better than the Al Gore crowd. It’s happening now, so what has changed in the last 150 years? Do all ice ages behave the same way, if not why not. What has happened to the sun over the last 10,000 years etc. etc.

Brian H
January 29, 2011 11:26 pm

Father G;
Not so. It is not necessary to know all to show that the CAGW crowd knows little or nothing. It is only necessary to show that the independent variables (solar influx, CO2, etc.) and the dependent variables (temperature, storms, etc.) are within historical normal bounds to keep the “Null Hypothesis” rooted where it should be: nothing unusual or drastic is happening; mankind is not Destroying the Climate; de-industrializing the planet is unwarranted (and mass-murderously dangerous). It is up to the CAGW crowd to disprove all of those things, and it has not done so, or come close.

FrankK
January 29, 2011 11:34 pm

Father Guido says:
January 29, 2011 at 9:49 pm
So if CO2 isn’t causing CAGW, which I agree with, what is?
I keep hearing it’s natural and cyclic in nature, but until we can show what causes it we’re no better than the Al Gore crowd. It’s happening now, so what has changed in the last 150 years? Do all ice ages behave the same way, if not why not. What has happened to the sun over the last 10,000 years etc. etc.
====================================================
What we know is that the Sun’s minimum activity created the most recent Little Ice Ages (Maunder and Dalton).
We have been recovering from that so-called Maunder (and Dalton) period in temperature since, except just recently.
SEE:
http://en.wikipedia.org/wiki/Maunder_Minimum
Temperature rises before those periods in the Medieval Period had nothing to do with CO2 levels so why should they now.

Oliver Ramsay
January 30, 2011 12:00 am

Clearly, the internal temperature of homeostasis is not the same as that of the skin surface, so the analogy is not quite apt, but if metabolic processes and environmental conditions did conspire to maintain a constant skin temperature, then the radiative flow would also be constant.
Real human caloric requirements might look quite different from the putative thousands suggested, if one used an actual surface temperature, rather than 37 C, especially for the naked man who has hypothermia before lunchtime.
And, is there a scenario where a body at 37C doesn’t emit 525W m^2?

cba
January 30, 2011 4:21 am

one correction to concepts,
stefan’s law is the integration over all wavelengths (or frequencies) and over a hemisphere (half sphere) angle wise of planck’s law. The more detailed version is P/A = epsilon sigma (T^4 – To ^4) which gives the net energy flow out and T is the black body object and To is the temperature of the other objects surrounding the black body object. For Earth, we have the cold of space surrounding us except for the sun that is about 1/2 degree across.
Father Guido,
the daily temperature record from the CRU has been well described by Lean & Kopp using a simple simple model that uses ENSO, volcanic activity, TSI, and a straight line slope increase attributed to humans. It is extremely accurate with an r=0.92 correlation for the time of the model 1980-2010.
there are multiple considerations concerning that line. During that time urban heat island effects have increased, manipulation of data has been done by most parties involved in their collection that adds temperature to recent records and takes it away from earlier records. that line is also an error catch all rather than being based on any specific human effects.

Joel Shore
January 30, 2011 7:53 am

Oliver Ramsey says:

Real human caloric requirements might look quite different from the putative thousands suggested, if one used an actual surface temperature, rather than 37 C, especially for the naked man who has hypothermia before lunchtime.
And, is there a scenario where a body at 37C doesn’t emit 525W m^2?

The physics textbook that we teach from used 33 C as a typical skin temperature. Also, as cba notes, you have to account for not only the energy emitted but that absorbed. It works out that the net rate of heat transfer via radiation in a room that is at 24 C (~75 F) is about 100 W, which the book claims is close the typical human resting metabolism and indeed 100 W works out to just over 2000 kilocalories per day, which is a reasonable number.

P. Solar
January 30, 2011 8:20 am

Trenberth et al : http://www.cgd.ucar.edu/cas/papers/2000JD000298.pdf
A major part of the ocean heat loss to the atmosphere is through evaporation and thus is realized in the atmosphere as latent heating in precipitation, which drives teleconnections.
Evaporation and then condensation in the atmosphere are determined by a number of factors: temperature (not T^4) , wind speed , humidity, etc. The whole idea of outward heat transfer being represented by a T^4 relationship seems totally flawed.
All of Schwantz’s analysis is based on that “Ansatz” and implicitly, if you look at his “solutions” of the equations, *assumes* the magic epsilon is a temperature independent scalar. He seems to think he can ignore evaporation and then only add it in later as a feedback whereby it adopts a T^4 dependence. This is never even discussed nor justified and is clearly flawed.
I find it hard even to say that that is a gross simplification. I’m more inclined to say it is a gross Umsatz.

Jeff
January 30, 2011 9:45 am

this assumption that the Earths core generates no energy is interesting … false but interesting ….

izen
January 30, 2011 10:22 am

@-Alan McIntire says:
January 29, 2011 at 3:49 pm
“The ratio in watts between 37 C and 15 C would be about
(310/288)^4 = 1.342,
A naked 160 lb 5′ 8″ man standing around in 15 C weather would lose
390.7 * 0.342* 86,400*1.86 = 21,473,172 joules /4.186 calories per joule=5,129,759
or 5130 Kcalories per day, a reasonable figure considering most people don’t stand around naked in 15 C weather.”
That you for doing an accurate calculation, I hoped someone would, but didn’t expect it.
In part because it confirms the objection to Willis Eschenbach’s claim that body temperature can be regarded as an energy flow…
As you show here it requires ANOTHER temperature to give a temperature difference before the (net) energy flow can be calculated.
Because we are not standing exposed in deep space with an ambient temperature near 0 deg K we receive back around 80% of the energy we emit. All of the energy we emit from the body surface eventually makes it to space. But the back radiation from the surroundings reduces the net flow of energy to a magnitude were we have some chance of compensating by the metabolic generation of heat from food.
Earlier in the thread someone scorned the idea that drinking cold water would use up calories and prevent obesity. Drinking cold water certainly IS a good method of dropping the core temperature, but recent research may indicate that a warmer enviroment does result in more of the food we eat going into storage than maintaining body temperature. :-
http://www.redonline.co.uk/news-views/in-the-news/central-heating-causes-obesity
Perhaps the next media hyperbole’ would be ‘AGW causes obesity!’…. -grin-

January 30, 2011 11:43 am

From Laurie Bowen on January 29, 2011 at 11:26 am:

Without a camera, how do you prove a bird just flew by?

The feathers and meaty chunks falling down near the base of the wind turbine constitute definitive proof.
So does the easily-recognizable whitish mostly-liquid blob that just splat on my car. The bird must have been flying as there are no trees or power lines nearby for it to have been roosting on.
The hole in the old window screen can be considered proof that a bird flew into the house, there were feathers nearby. Currently I can’t find anything else that could have done it. Can’t find the bird either, guess the cats are transforming it into energy and mostly other substances.
And thus go the major forms of particle detection in nuclear physics.

January 30, 2011 12:54 pm

Jeff said on January 30, 2011 at 9:45 am:

this assumption that the Earths core generates no energy is interesting … false but interesting ….

Who said that? The issue is that as far as Earth’s climate is concerned, the heat from the core is negligible. There are vast reserves built up, and more heat is being generated, but at a very slow rate. And this layer of solidified rocky scum, the crust, is a good bulk insulator on the scale involved, with the rate of heat transfer being incredibly low.
Just look at how the winter temperatures go. Here on the surface they can be running around 20 to -40°F here in the mid-latitudes, yet you can dig down into the ground just 10 or so feet and have temperatures around 50°F, all year long. This is true even in the Arctic, might have to go down just a little bit further. Does that indicate we’re getting enough heat to the surface from the core to matter?

izen
January 30, 2011 1:15 pm

Willis Eschenbach says:
January 30, 2011 at 12:26 pm
“OK, I seem to be caught in a semantic question here. Let me rephrase and say that
σ T^4 is a flow, and that Q is a flow, and that E is a flow, but that Q is not a flow. Can you speak to that issue?”
Not sure the second part of the sentence makes sense…
But σ T^4 is NOT a flow. A flow only happens across a boundary with two different temperatures. Energy does not flow from one part of your body to another if they are both at 37degC.
T^4 is just one term required to define a flow, another temperature is required to calculate the magnitude of that flow across the boundary where the temperature difference exists. All this omits any contribution from conduction to energy flows, it is concerned only with how the random kinetic energy of the molecules/atoms of a physical object relate to its emission of photon energy because of quantum effects.
Only in the exceptional case of the other temperature at the boundary being absolute zero – 0degK is (σ T^4) sufficient to describe the flow of energy.

KR
January 30, 2011 1:49 pm

Willis – In my last posting I believe I went a bit off topic, digging into the Schwartz paper you mentioned. Your actual objection, I believe, is to the use of the E = ε σ Ts^4 equation?
From basic conservation of energy, if E[in] != E[out], there will be a change in internal energy, energy in the climate, which will manifest as a change in temperature. There’s a huge limiting feedback on that temperature change, the T^4 temperature relationship with energy emitted (IR to space), so it doesn’t take a lot of temperature change to make a fairly significant change in energy emitted.
Now, the Stefan-Boltzmann relationship you seem to have issues with, E = ε σ Ts^4, or more properly E = ε σ (Ts^4 – Tspace^4), expressed as a per square meter value, is about as basic and established as thermodynamics gets. The factor ε discussed by Schwartz as a relative value is simply the simplified relationship between surface temperatures and what we see emitted to space from satellite measurements.
Now, that’s not, perhaps, the most accurate value – the realistic and more accurate results come from line-by-line computation across the depth of the atmosphere (with checks coming from real measurements, radiosondes, TOA values, etc.), starting from basic physics. This, not incidentally, is equivalent to performing a Runge-Kutta numeric integration (something I’ve done repeatedly) of a function that doesn’t have an analytic (symbolic) solution – work it through line by line, add up the incremental results, and see what you get. And what you get for a doubling of CO2 (which decreases total emissivity of the Earth through band widening and raising the effective stratospheric altitude of final emission to space, meaning it comes from colder CO2) is a radiative imbalance of ~3.7W/m^2. Which would lead (without feedbacks) to a temperature rise of 1.1C.
That’s starting from basic physics, checked against measurements of surface and TOA values. There’s your theoretic support – if you find issues with the SB equations and the last 100 years of physics, then by all means publish it!
As to the “heating in the pipeline” (I’ve always hated that expression; I would prefer “heating we expect to occur”) heating – that’s the heat we expect to see (but haven’t yet) due to the changes in the climate system, that will have to accumulate before the long term temps stop changing. That energy isn’t all here yet. Some is, some doesn’t seem to be (the Trenberth “missing heat”), and as I said earlier that indicates either an error in our math, or limits of our measurements.
But if you want real-world measurements, take a look at http://www-argo.ucsd.edu/levitus_2009_figure.jpg and http://www.argo.ucsd.edu/global_change_analysis.html#temp – sure looks like a somewhat noisy measure of continuing warming to me. And the thermal expansion reflects that as well – see http://academics.eckerd.edu/instructor/hastindw/MS1410-001_FA08/handouts/2008SLRSustain.pdf in particular Figure 3a.
Personally, I find basing billion dollar decisions on hiding from observations and the well understood science to be a very sad thing.
Science ain’t perfect, it never has been – certainty is for religion. I don’t know that a rock dropped from my hand will fall the next time. But would it be reasonable to base my decisions on hoping it won’t?

RobM
January 30, 2011 4:57 pm

Willis,
Sorry my last post was a bit too quick and not well thought out. I have now read the paper. I think that what threw me and many others off is this. There are really two different temperatures involved here. One is Ts the effective radiating temperature of the surface. The definition of Ts is not given in the paper. But to be the radiative temperature it must be the temperature within a mm or less of the surface. Another temperature (shall we call it Tv) used but not even explicitly named is the average temperature over the ocean averaged over the world to some poorly determined depth (order ~ 100M stated in the paper). Simply saying that average heat energy is proportional to average T gives H=CTv ( definition of C if you like). from this it trivially follows that dH/dt=CdTv/dt The ‘ansatz’ is actually that one may say dTv/dt=dTs/dt and then use the same symbol for both Ts and Tv Wow!! what a ‘trick’. This equality seems very unlikely specifically because of the large difference in time variation to be expected between the two. The measurements cited essentially show that this assumption is very inaccurate.

RobM
January 30, 2011 7:08 pm

KR says, That’s starting from basic physics.
Yes That is basic physics, but the basic physics issues are not the problem. The basic radiation / absorption effects were worked out a century ago and more accurate calculations are not improving things much. The problem is the fact that these (accurate) radiation calculations are done with a way oversimplified model of the world. To improve on the understanding the ‘physics’ is not basic. To become more accurate the model now needs to account for all the effects of global convection (vertical and horizontal) the ‘one compartment’ model discussed on this for sure won’t cut it.
Fixation on ‘global average’ anything is probably the wrong approach. With all the dynamics going on, delta T’ in tens of k day to night region to region, Atmospheric water content varying drastically in time and space. You cant just average all this nonlinear stuff and expect to get tenth degree accurate models.
It is just as likely that the ‘local’ effects circulations etc have more effect on global mean temp than global mean temp has on them. These effects are driven by differences in pressure, temperature etc. Which, in turn are driven by things like non uniform insolation. Since Co2 tends to be more uniformly distributed it could reduce these pole to equator differences. (reduced heat loss from the low h2o polar regions) Will this cool or warm the planet? Who knows but we won’t find out with radiation heat transfer solutions alone.

January 30, 2011 8:39 pm

Willis Eschenbach says:
January 28, 2011 at 5:08 pm
A number of people have said that temperature is not a flow. However, temperature can be converted to the equivalent blackbody radiation flow using the familiar Stefan-Boltzmann formula.
Think about it this way. My body is at about 37°C. Like anything at that temperature, there is a constant flow of radiative energy emanating from my body, 525 W/m2. How, then, is a temperature of 37°C not equivalent to a flow of 525 W/m2?

Simply because it isn’t. Let’s consider an equivalent argument:
Voltage is equivalent to current.
How so?
Well voltage can be converted to current by the relationship V = IR.
I hear you object, hang on, what about R? Different Rs give different Is for the same V, so the analogy is not correct.
But, I reply, neither does your equation give the same flow for a given temperature, because no real body is a black body. Your body will have some a different radiation because of the emissivity, which, although it is a constant for your body, is a material property of the body, not a universal constant, and in that sense is exactly like R, which is also a property of the body through which current passes.
So if you insist that T is a flow, one can equally insist that voltage is a current. It is a question of word usage, but for me it is a highly confusing usage that could not be successfully sustained throughout all of physics. Can you imagine the confusion if people habitually referred to voltage measurements as current measurements? The same applies here, as the many comments from confused readers shows. Temperature is a potential, like voltage, and the flow in each case (whether radiation or current respectively), while it depends upon the potential, is not a potential in either case. The potential determines the flow but it is not equivalent to the flow.

HR
January 31, 2011 2:29 am

willis,
The violin plots look interesting but I’m struggling with their interpretation. It seems quite subjective!
Things are “quite dissimilar” etc. How similar would things have to be for you to describe them as “quite similar”? Maybe better is there anyway of objectively quantify the similarity? What might help me is if you showed something that does correlate such as detrended ENSO and temperature. What would that comparsion look like? Nobody would expect the correlation between two climate data sets to be perfect so subject descriptions seems problematic.

RobM
January 31, 2011 9:03 am

Willis,
This discussion about flows T etc prompted me to go back to a good thermodynamics text. Zemansky, ‘Heat and Thermodynamics’ fourth Edition McGraw-Hill, 1957. Sorry that I am reluctant to scan and post the relevant page. But on page 72 there is a figure, which, I maintain is the proper diagram for Schwartz’ ‘Single compartment’ model. Make the resistive heater into Solar input, Cooler surroundings into ‘Radiation’ into space and the make Zemansky’s ‘system’ box the top layer (~100 M) of the ocean. The T^4 stuff just gives some (poor) tie to reality. Schwartz may have done some perturbation type analysis to relate an expansion of T^4 around some T0 to get a physical approximation, but he does not say so. In the end the model is just assumes a linear relationship between ‘radiated’ energy and temperature.
The equation below figure 4.6 in Zemansky is essentially the one used in Schwartz
Zemansky’s equation is dQ=dQ1-dQ2=CpdT

Laurie Bowen
January 31, 2011 9:43 am

Jim Masterson said:
Pi is an irrational number; therefore it can’t equal a rational number like 22/7. It’s also transcendental, but that is really going OT.
Jim, If you work on base 7 it is always fine. . . . That is why, in my itty bitty brain a complete circle is 360 degrees even it is an oval. For example, a year is 365.25 days but it is still 360 degrees . . . Irrational in this case, is an illusion, or relative. . . .
PS. . . I thought this observation was to be forth coming . . . . at least by someone.

KR
January 31, 2011 10:08 am

Willis – The “effective planetary emissivity” is a useful parametrization of how much energy leaves the Earth for space. It’s measurable, computable, and demonstrable.
We also know how increasing greenhouse gases will change the effective emissivity, at least in terms of direct forcings/imbalances in radiation to/from the planet. (Feedbacks, of course, are a different matter entirely; a great deal of uncertainty there.) Looking at TOA satellite spectra from the Earth shows the distinctive notches from greenhouse gases reducing IR emissions to space – and with some of the spectra suppressed in this manner, the entire spectra must be higher (from a warmer surface) to radiate the same amount of energy as without the GHG’s.
There’s nothing wrong with a trivial identity – especially if it’s a useful one. If that useful number changes, we can expect long term changes in the temperature of the climate.
I’m sorry you don’t like the effective planetary emissivity – but it’s a really useful relation.

George E. Smith
January 31, 2011 11:12 am

“”””” KR says:
January 31, 2011 at 10:08 am
Willis – The “effective planetary emissivity” is a useful parametrization of how much energy leaves the Earth for space. It’s measurable, computable, and demonstrable. “””””
Well I have no earthly idea what the word “parametrization” means; well actually that’s a statement that the word has no meaning to me. I know what “parametric equations” are; for example the set:-
x = Cos (theta) ; y = Cos (n.theta) is a parametric equation form for the Tchebychev Polynomials; y = Tn(x) in terms of the parameter (theta). This form is valid only for the range -1 to +1 for x and y , or theta = 0 to 2 pi (radians).
The non parametric forms would be:- T0(x) = 1 , T1(x) = x , T2(x) = 2.x^2 -1 etc.
Or in general one could have:- F(xyp) = 0 where p is a parameter that is set for any specific xy function; and of course one could have higher order sets with multiple parameters ( p)i .
But “Effective Planetary Emissivity” would not in any case be a measure of “how much energy leaves earth for space.” ; because that would not include any albedo effect, which is solar spectrum energy that is incident on planet earth; but is scattered or reflected out into space; but is in no way emitted by earth.
Emissivity is a specifically defined measure that relates the emittance of an actual physical body, to that of a black body at the same Temperature; both emitting thermal radiation, that is entirely a consequence of the Temperature of that body.

George E. Smith
January 31, 2011 11:36 am

“”””” Willis Eschenbach says:
January 30, 2011 at 8:31 pm
KR says:
January 30, 2011 at 1:49 pm
Willis – In my last posting I believe I went a bit off topic, digging into the Schwartz paper you mentioned. Your actual objection, I believe, is to the use of the E = ε σ Ts^4 equation?
From basic conservation of energy, if E[in] != E[out], there will be a change in internal energy, energy in the climate, which will manifest as a change in temperature. There’s a huge limiting feedback on that temperature change, the T^4 temperature relationship with energy emitted (IR to space), so it doesn’t take a lot of temperature change to make a fairly significant change in energy emitted.
Now, the Stefan-Boltzmann relationship you seem to have issues with, E = ε σ Ts^4, or more properly E = ε σ (Ts^4 – Tspace^4), “””””
“”””” or more properly E = ε σ (Ts^4 – Tspace^4), “””””
Now I have a problem with that. The “Stefan-Boltzmann” relationship gives the total emittance of a black body as a function of its Temperature and nothing else. The total emittance of a black body is in no way affected by the Temperature of “space” or anything else; only the Temperature of the black body.
If you want to say that the net energy loss from a black body radiating into space is:- “”””” E = ε σ (Ts^4 – Tspace^4), “””, then you should say that; not that it is the effective emittance given by the S-B relationship.
For a start, the Temperature of space is somewhat variable, depending on where you look. presumably it is somewhere in that 3K range they talk about. A 3 K “black body” would be emitting a completely different spectrum from a BB at say 288 or 255 or some other imagined earth Temperature. If this sytem was totally in equilibrium, the the earth would have to also be emitting that same 3K spectrum back into space. Well of course it isn’t and it can’t; and it isn’t in thermal equilibrium anyway; which is why it is emitting far more energy than it receives from space (not counting that sun that is out there in space).
The emittance of a BB does not depend on what radiation is hitting it; it depends only on the Temperature of the body.

KR
January 31, 2011 11:51 am

George E. Smith – IR emissions to space don’t include albedo reflections – the solar spectrum has almost no IR at all, and it’s actually quite easy to account for albedo. That’s energy not absorbed in the first place.
The “effective planetary emissivity” describes the amount of energy emitted as thermal IR to space by Earth at a particular temperature, and has been measured at about 0.62 relative to a black-body. Increasing GHG’s lower that emissivity.
Any measure of emissivity needs to account for the temperature of the emitting body and the integral of energy across the (occasionally complex) emission spectra relative to a black-body. The top of atmosphere number for that, given a surface temperature, summarizes that spectra and is about 0.62.

January 31, 2011 12:01 pm

“Perhaps you could give us a useful, non-trivial identity … and while you are at it, please give us an example of an identity which is subject to “long term changes” …”
Well for the first: the morning star = the evening star or tully = cicero

Steve
January 31, 2011 1:35 pm

KR says:
January 31, 2011 at 11:51 am
“The “effective planetary emissivity” describes the amount of energy emitted as thermal IR to space by Earth at a particular temperature, and has been measured at about 0.62 relative to a black-body. Increasing GHG’s lower that emissivity.”
I think you’ve got a typo or something there. Increasing GHGs should increase energy emitted as thermal IR to space, right? (after heat capacity is reached) Otherwise the atmosphere hasn’t increased in temperature.

In Burrito
January 31, 2011 3:29 pm

Willis Eschenbach says:
January 30, 2011 at 1:16 pm
KR says:
January 29, 2011 at 12:11 pm

I followed up on (not being a German speaker) the various definitions of “Ansatz” – the primary one used in physics and math I found was “an educated guess that is verified later by its results”. Hardly the horror you describe. I think we would have to ask Schwartz which definition he was using before judging that term.
If that were the case, you’d expect them to provide the verification that dH/dt actually is equal to C dT/dt. But they don’t. Nor do you. So we’re back to just an “educated guess”, which is basically what the definition I used says.
Cp = dH/dT *by definition*. Therefore, dH/dt = Cp dH/dt. There is absolutely no arguing the mathematical validity of this. To be techically correct, Cp *is* a function of temperature; however, over the few degrees that the temperature is varying, Cp can be reasonably taken as a constant.
Discussion of this requires absolute precision. We’re discussing the energy balance of the earth which is basically Power In – Power Out = Rate of Energy Accumulation. Power In is incoming light unreflected (by clouds) sunlight. Power Out is infrared radiation (driven by T^4). Rate of Energy Accumulated is m Cp dT/dt of the upper layer of the oceans where m is the mass of this layer. I’m using “Power” instead of “Energy” because we’re talking about *rates* of energy transfer, and power is energy per unit time.
The reason that climatologists use “x” meters of the ocean surface for “m” is that its thermal mass is much greater than the atmosphere. The dT in the accumulation term is the change in water temperature as there is a net accumulation (or loss) of energy stored in the oceans (NOT the difference in air and water temperature). I’m guessing that heating of land surfaces is also neglected because heat only penetrates to a very shallow depth. Geothermal energy, etc. are neglected as well (I assume) because they are small compared to sunlight and IR.
Now if you want to debate the validity of the choice of control volume (i.e. the ocean surface layer), that’s fine. It’s absolutely fair game to challenge simplifying assumptions that go into a model. But there is *no basis* for questioning the mathematical validity of dH/dT = Cp dT/dt.

Tim Folkerts
January 31, 2011 5:05 pm

Steve says: January 31, 2011 at 1:35 pm
I think you’ve got a typo or something there. Increasing GHGs should increase energy emitted as thermal IR to space, right? (after heat capacity is reached) Otherwise the atmosphere hasn’t increased in temperature.

No, this is right. The extra GHG will block some energy until a new equilibrium is reached. At equilibrium, the earth will still radiate the same energy, but the surface temperature will be higher, Same energy from higher temperature = lower emissivity.

In Burrito
January 31, 2011 5:43 pm

Now if you want to debate the validity of the choice of control volume (i.e. the ocean surface layer), that’s fine. It’s absolutely fair game to challenge simplifying assumptions that go into a model. But there is *no basis* for questioning the mathematical validity of dH/dT = Cp dT/dt.
And of course I screwed this up, should be:
dH/dt = Cp dT/dt

Tim Folkerts
January 31, 2011 6:08 pm

To me, the biggest single challenge in this discussion is that the use of symbols from the original paper is rather befuddled.
* “E” is traditionally used for energy [eg joules or zettajoules]
* “E” in the diagram at the top is the rate that energy leaves the earth = power [eg joules/second = Watts or ZJ/year]. If they want to redefine “E” that is cool, but it can lead to confusion.
* “E” in E = ε σ Ts^4 is irradiance = power divided by area [eg Watts/m^2]
To match the use of “E” as in “E = 5500 ZJ/yr”, then we should really have E = ε σ A Ts^4 where A is the surface area of he earth. Then both would be power.
The same problem shows up for Q
* “Q” is traditionally a symbol for heat, with the same units as energy
* “Q” in the first figure is power
* “Q” in Q = γ J is irradiance = power/area
Then there is the sign error in equation 3: dH/dt = C dTs/dt
If dH/dt extra energy leaves the earth, then dH/dt extra energy leaves the ocean, which means the ocean must be COOLING => dTs is negative => C dTs/dt is negative. To fix this, we need dH/dt = – C dTs/dt.
I’m surprised this made it thru peer review when such Freshman-level mistakes in dimensional analysis and signs are glossed over.
That said, I pretty much agree with KR & In Burrito in their analysis of the situation.

wayne
January 31, 2011 6:18 pm

In Burrito, no biggie. The vast majority commenting here are very intelligent and knew even without an expilicit correction that the T was t.

January 31, 2011 7:06 pm

The very first equation stopped me cold.
Q=E+dH/dt
Q is the energy entering the earth atmosphere system from the sun.
E is the long wave energy emitted from the top of the atmosphere.
dH/dt is the annual change in ocean heat content. This is a strange notation for this variable. H implies units of energy, and the dt implies a change of temperature. The actual units of this variable are units of energy, rather than energy per unit time.
The notation of this equation is off putting for anyone who has studied science given the definitions provided by Eschenbach.
The next section written by Eschenbach relates to the Stephen Schwartz paper on calculation of climate sensitivity from heat capacity of the ocean.
In Schwartz’s paper, the units of Q and E are not energy, but rather Watts/M^2/second, ie energy flux which is different from energy.
Then lower down Eschenbach admits confusion about the units of his equations.
http://www.ecd.bnl.gov/steve/pubs/HeatCapacity.pdf
Then lower down Eschenbach admits confusion about the units of his equations.
He claims the paper is valid because of the number of citations. He neglects to mention that most of those who cite this paper do so in order to refute its conclusions, and that even Schwartz admitted that its conclusions were not correct.
http://www.ecd.bnl.gov/steve/pubs/HeatCapCommentResponse.pdf

“Reanalysis of the autocorrelation of global mean surface temperature prompted by the several
10 Comments, taking into account a subannual autocorrelation of about 0.4 year and bias in the
11 autocorrelation resulting from the short duration of the time series has resulted in an upward revision of
12 the climate system time constant determined in Schwartz [2007] by roughly 70%, to 8.5 ± 2.5 years (all
13 uncertainties are 1-sigma estimates). This results in a like upward revision of the climate sensitivity
14 determined in that paper, to 0.51 ± 0.26 K/(W m-2), corresponding to an equilibrium temperature
15 increase for doubled CO2 of 1.9 ± 1.0 K, somewhat lower than the central estimate of the sensitivity
16 given in the 2007 assessment report of the Intergovernmental Panel on Climate Change, but consistent
17 within the uncertainties of both estimates.”

If you google “schwartz HEAT CAPACITY, TIME CONSTANT, AND SENSITIVITY OF EARTH’S CLIMATE SYSTEM criticism” you get a lot of peer reviewed papers,many of them behind a paywall.
The central objection is summarized in the following link.
http://www.skepticalscience.com/Stephen-Schwartz-on-climate-sensitivity.html
Schwartz calculates sensitivity as the quotient of the climate “time constant” and global heat capacity. The “time constant”, or time for the climate system to return to equilibrium after a perturbation, is a key aspect of the paper and Schwartz estimates around 5 years.
However, as Schwartz points out in his study, climate recovers at different rates depending on the nature of the forcing causing the perturbation. Short term changes such as a volcanic eruption result in a short time constant of a few years. A long term increase in CO2 levels results in a recovery spanning decades. Schwartz rightly points out “as the duration of volcanic forcing is short, the response time may not be reflective of that which would characterize a sustained forcing such as that from increased greenhouse gases because of lack of penetration of the thermal signal into the deep ocean.”
In spite of that, Schwartz filters out long term changes by detrending the time series data which has the effect of biasing the result towards a shorter time constant. The time constant for non-detrended data yields a time constant of 15 to 17 years. Consequently, the estimated time constant of 5 years is questionable – a value the final result hinges on.

It seems that Eschenbach is totally ignorant of the science underlying the subject of his post and can’t even get the units straight. In addition he is not familiar with the flaws in the paper that he cites.
It seems that he tries to make up for this ignorance by peppering his article with wise cracks.
Eschenbach would never get his work published in any peer reviewed journal, because he hasn’t done his homework.
[Reply: Willis Eschenbach is a peer reviewed author. ~dbs, mod.]

February 1, 2011 12:55 am

From eadler on January 31, 2011 at 7:06 pm:

The very first equation stopped me cold.
Q=E+dH/dt
Q is the energy entering the earth atmosphere system from the sun.
E is the long wave energy emitted from the top of the atmosphere.
dH/dt is the annual change in ocean heat content. This is a strange notation for this variable. H implies units of energy, and the dt implies a change of temperature. The actual units of this variable are units of energy, rather than energy per unit time.
The notation of this equation is off putting for anyone who has studied science given the definitions provided by Eschenbach.

Why? First off, if you’ve done enough of even just general physics involving differential calculus, you’d automatically assume that dX/dt is a time derivative unless it is explicitly stated otherwise, and if it is then you’d gripe they should have used something else and reserved “t” for time as is standard.
Then we look at Q and E. The units for Q and E are clearly stated in the paper as “W m-2” or Watts per meter squared. Note that a Watt is a Joule per second, thus Q and E are in Joules/(m^2*sec), thus time-based rates.
Then we assemble the energy balance equation, with Q and E being simple rates.
Qt = Et + H(t) where H is an unstated equation with time as the variable.
Take the first derivative with respect to time:
Q = E + dH/dt
or if you insist:
Q = E + dH(t)/dt
See, just simple easy-to-understand math.
You might be getting hung up on the caption to Figure 1: “Units in all cases are zettajoules (ZJ, or 10^21 joules) / year.” Fine. Take the equation above from the Schwartz paper, multiply both sides by the number of seconds in a year to get J/(m^2*year). Then since we’re discussing a global value, multiply both sides by the area of the virtual globe that represents the referenced top of the atmosphere. Voila, the units match.
Don’t get hung up on Willis’ Q, E, and H(t) in Fig 1 (in ZJ/yr) being different from the Q, E, and H(t) of the Schwartz paper (in W per m^2) as shown in Fig 2, since the difference is basically just multiplying both sides of the Schwartz equation with two constants. The critique went from the Schwartz paper anyway.

Bruckner8
February 1, 2011 6:21 am

These are my favorite articles on WUWT, where people (Willis in this case) put out clear reasoning, encourage others to poke holes in it, and come back with counter-arguments. It feels like real science.
But I can’t always follow the cases where the original author decides to “let this or that go,” and does NOT respond to some arguments.
My takeaway from this article is: Either the Warmists are using assumptions found to be false, or Willis is dead wrong on something. Many counter-posters are trying to show how Willis is dead wrong, and Willis has done a fine job pushing them back, but he has not responded to all of them.
The fact that this isn’t even clear is why AGW will continue to live. It’s hard to argue when the other side’s core premise is “let’s assume 1+1 =3, and see where it takes us.”
It just boils down to logical exercise, reality be damned, no matter how expensive.

February 1, 2011 6:52 am

I have to admit some errors in my previous post. I was in too much of a hurry and was didn’t check my work adequately.
I meant dH/dt implies a rate of change with respect to time, not temperature, which is what I wrote.
Also, the energy flux should have been Watts/M^2. I put in an extra factor of “/” which would have been correct if I had written Joules instead of Watts which contains the factor by definition. Looking at what I wrote, I am aghast at the errors.
My errors do not alter the fact that Eschenbach’s definitions were wrong.
Eschenbach makes the following argument:
“…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.”
What is correct physics is not decided by a show of hands or a vote by people who do not understand the subject. This kind of rhetoric is uncalled for in a scientific paper.
In fact the air above the ocean, which contains GHG’s is not transparent to the long wave radiation emitted from the surface of the ocean. It emits long wave energy itself, and the flux that it emits is determined by the temperature of air above the ocean. These energy fluxes have been measured, and their magnitude is not determined by a “show of hands”.
http://content.imamu.edu.sa/Scholars/it/net/trenbert.pdf
Looking at google scholar, there are no peer reviewed articles by Eschenbach. He wrote something for Energy and Environment, which is not regarded scientific community as peer reviewed. There is a 1 paragraph comment that was published in nature and that is it.
He has written articles, but not in the peer reviewed literature.

Tim Folkerts
February 1, 2011 9:40 am

Willis Eschenbach says: January 31, 2011 at 10:32 pm
Tim, you are falling into the same trap as S2007, which is the assumption that the only way to re-establish the balance is to change the temperature.

I don’t assume that this is the only way, but it is certainly a way. If it is indeed the way that energy gets balanced, then the “effective emissivity” would have to fall, not rise, as I was explaining in the original post. Even if GHGs are only part of the way that the energy gets balanced, then the effective emissivity would still fall, just not as much.

February 1, 2011 11:22 am

From eadler on February 1, 2011 at 6:52 am:

Looking at google scholar, there are no peer reviewed articles by Eschenbach. He wrote something for Energy and Environment, which is not regarded scientific community as peer reviewed. There is a 1 paragraph comment that was published in nature and that is it.
He has written articles, but not in the peer reviewed literature.

Truly, either you are being deliberately disingenuous, or you are so appallingly ignorant you must be working at avoiding knowledge.
Seven listings total. Confirmation from Willis that the training manual is also his would be definitive, however the “Introduction to Training” section certainly does seem to follow his writing style.
Four listings for Energy & Environment, which is more than “something.”
The 2004 Nature piece:
http://www.geo.arizona.edu/web/Cohen/pdf/63%20OReilly%20et%20al%202004%09Nature.pdf
Far more than one paragraph.
Then you (proudly?) display the high-handed conceit and deceit that has poisoned climate science for far too long:
1. Energy & Environment is a peer-reviewed journal that has peer-reviewed and published Willis’ work.
2. The scientific community (as in The Climate Consensus?) does not regard E&E as peer-reviewed.
Therefore Willis has not published in peer-reviewed literature.
EBSCO is a long-established service for researchers. From their About Us page:

The providers of EBSCOhost, the most-used premium research service in libraries and other institutions worldwide, EBSCO offers a suite of more than 300 full-text and secondary research databases covering all subject areas, levels of research, and user communities – from schools, public libraries and universities, to hospitals, corporations and government agencies.

Check out their Environment Index™, specifically the Coverage List. E&E is, according to EBSCO, a peer-reviewed academic journal. And EBSCO better know what they’re talking about, as their business depends on it. Therefore Willis has published in peer-reviewed literature.
As Willis said above, calling a tail a leg doesn’t make it a leg. What you’re trying to serve us may have come from a cow, but it sure ain’t beef. But please, have yourself yet another helping, we can tell from your grin you must really like the taste.

In Burrito
February 1, 2011 2:58 pm

In Burrito says:
January 31, 2011 at 3:29 pm

Cp = dH/dT *by definition*. Therefore, dH/dt = Cp dH/dt. There is absolutely no arguing the mathematical validity of this.
Two things. First, I don’t follow your math at all. I think you made a typo, and that you meant “Therefore, dH/dt = Cp dT/dt”. Is that correct?
Second, In Burrito, it took a while for me to figure out the problem. The difficulty is, they are not using that equation. In your equation above, the correct definition, T and dT/dt refer to the temperature and the temperature change of the object whose heat content is changing. In our case, that would be the temperature of the ocean, since we are measuring the heat content of the ocean.
But they are not doing that, they are not using T to refer to ocean temperature. They are using T to refer to the global mean surface air temperature, which is a totally different animal.
And you can’t simply substitute one “T” for another and claim that the equation is still valid.
Willis – correct on catching my typo. I think we’re debating what the control volume of Cp dT/dt is…. A quick perusal of the Schwartz paper looks like Cp dT/dt refers to *everything*, so that the temperature of everything increases/decreases by deltaT, even though the effective thermal mass is dominated by the ocean surface. So even though the temperature lag behind the forcing is determined by the mass of the ocean surface, the model assumes that the atmosphere tracks with deltaT of the ocean surface. I don’t think this is necessarily wrong…but I need to review Schwartz in more detail.