Where Did I Put That Energy?

Guest Post by Willis Eschenbach

[UPDATE 2 AM Christmas morning, and of course Murphy is still alive and his Law is still in operation. I find a decimal point error in my calculations … grrr, I hates that, ocean energy flows shows at 1/10th size. Public exposure of error, the bane of any scientific endeavor.

And Murphy being who he is, the correction doesn’t solve the puzzle at all. It only makes it more complex. I have updated Figure 2 and some of the text, and added a third figure. The only good news is, it doesn’t affect my conclusions, there’s still something very wrong in the canonical climate equations.

Merry Christmas to all, it can only get better from here.]

In the Climategate emails, Kevin Trenberth wrote:

How come you do not agree with a statement that says we are no where close to knowing where energy is going or whether clouds are changing to make the planet brighter.  We are not   close to balancing the energy budget.  The fact that we can not account for what is happening in the climate system makes any consideration of geoengineering quite hopeless as we will never be able to tell if it is successful or not!  It is a travesty!

Although I sympathized with him, I was unclear about exactly where the hole was in the energy budget. However, my research into the climate sensitivity of the GISS model has given me some new insights into the question. Intrigued by the findings I reported in “Model Charged With Excessive Use of Forcing“, I wanted to look closer at the results from the NASA GISS climate model. As you may recall, I was trying to understand the low sensitivity I had calculated for the GISSE model. I went to the CMIP archive to see if I could get the top-of-atmosphere (TOA) forcing for the GISS model month by month, but the GISS folks didn’t archive that data. Rats.

Figure 1 (may take a moment to load). Anomalies in the heat content of the top 700 metres of the ocean from 1955 to 2003. Units are zettaJoules (10^21 Joules).

Someone pointed out on that previous thread that I was neglecting the ocean in my calculations … guilty as charged. The basic energy equation for the planet is that energy added to the climate system equals energy leaving the system plus energy going into the ocean. Energy can’t just disappear, it has to go somewhere. It either leaves the system, or it goes into the ocean. So I went off to see what the change in the heat content of the ocean has looked like over the period of record. The National Oceanographic Data Center (NODC) has the data. Figure 1 shows a movie of what I found. Not much of a movie, but it’s the first one that I’ve made in R, so I was happy about that. The legend says “∆H” where it should say “H”, but it’s 3AM and I’m not going to fix it. So how can this ocean heat content information be related to the question of climate sensitivity?

As you can see in Fig. 1, nature is a puzzle. Things happen in blobs and patches, without immediately obvious reasons. However, we can see that the heat content of the top layer of the ocean has increased since 1955 by a total of 154 ZJ.

First, a bit of math. Not much math, and not complex math. We’re looking at one of the fundamental equations of the current climate paradigm. The statement above was that:

Energy added to the climate system equals energy leaving the system plus energy going into the ocean.

Mathematically this can be restated as ∆Q (change in energy added, Joules/year) = ∆U (change in energy lost, Joules/year) + ∆Ocean (change in energy in/out of ocean, Joules/year), or

∆Q = ∆U + ∆Ocean (Joules/year)                          (Equation 1)

Note that this is different from a statement about a general equilibrium, which may or may not be satisfied in any given year. This is an absolute requirement, because energy cannot be created or destroyed. If we add extra energy to the system, it has to either leave the system via increased radiation or get stored in the ocean. There is no “lag” or “in the pipeline” possible with Equation 1. The atmosphere has far too small a thermal mass to store a significant amount of energy. The earth warms too slowly to serve as a reservoir for annual changes. Global ice amounts are fairly stable (although they might make a very small change over the long-term, global annual variations are small). So any large annual change of incoming energy has to either change the ocean storage or leave the system.

Now, the current climate paradigm holds that “U”, the energy leaving the system, is equal to the surface temperature “T” divided by the climate sensitivity “S” (∆U=∆T/S). This is another way of stating the idea that the surface temperature is linearly related to changes in the top-of-atmosphere radiation. [See e.g. Kiehl (PDF). Be aware that Kiehl uses lambda (λ) as sensitivity, which in my terminology would be 1/Sensitivity].

The current paradigm also holds (wrongly, in my opinion) that the sensitivity “S” is a constant. The IPCC says that the central value for the climate sensitivity constant “S” is about 0.8 °C per W/m2 (or 3°C per doubling of CO2). So according to the current paradigm, we can replace ∆U (change in energy leaving the system) with ∆T/0.8. This gives us:

∆Q = ∆T / 0.8  + ∆Ocean  (Joules/year)                 (Equation 2)

It struck me when I was looking into this that we actually have the means to test this claim of mainstream climate science. We have the historical forcings, from the GISS tables. We have the historical GISS temperatures. And we have the historical heat content of the ocean. (The conversion from Watts/m2 to joules/year is covered in the Appendix.)

Figure 2 shows annual changes in incoming energy (∆Q, red), outgoing energy (∆T/S, light blue), and energy moving into and out of the ocean ∆Ocean (dark blue). We can express them either in joules per year or in W/m2. I have chosen joules per year, to emphasize that this is the movement of actual energy that cannot be created or destroyed. It has to go somewhere, and there’s not many choices.

Figure 2. The missing energy puzzle. Every year, the amount of energy entering the system (red) should equal the energy leaving the system (light blue) plus the energy going into/out of the ocean (dark blue). It doesn’t.

Figure 3. Annual Energy Budget Error, ∆T/S + ∆H – ∆Q. Positive errors indicate excess heat in the ocean. Some folks have commented that they don’t like having photos in the background. This Figure’s for you.

As you can see, something is really, really off the rails in this. The total forcing Q is known through observation to take large drops after volcanic eruptions (from the volcanic aerosols reflecting away the sunlight), with similarly large and fast recoveries. But this is not reflected in the sum of the outgoing energy (∆T/S) plus the ocean changes. In other words, the forcing drops because of the volcanoes, but there is no corresponding drop in temperature or ocean heat storage as you would expect. The forcing springs back when the stratosphere clears after the eruption, but there is no corresponding rise in either temperature or ocean storage.

The real surprise is the absolute size of the missing energy. It is often more than 20 ZJ. This means that something very fundamental is wrong here.

Some of the possibilities for unraveling this koan are:

• Foolish math or logic error on my part. I don’t think so, as I have checked and rechecked my figures, units, and logic. But I’ve made plenty of mistakes in my life. Please check my numbers and everything else. [UPDATE – well, I sure called that one …]

• Bad data in one or more of the datasets. Always possible. However, the huge size of the discrepancy argues against that. Even though there are errors in all datasets, these would have to be very large errors. Even the forcings dataset is mostly based on observations (CO2 and volcanic aerosol changes). So bad data seems doubtful, it would have to be really, really bad.

• One of the datasets is off by one year, so the timing is wrong. That doesn’t work, though, correlation doesn’t improve with a lag or a lead.

• IPCC climate sensitivity is too large. If it were smaller, ∆T/S would be larger to help balance out the ∆Q. The problem is, the temperature changes are not well correlated with the forcing changes. In addition, the regression of (∆Q – ∆Ocean) on ∆T has an R^2 of 0.01. This means that the climate sensitivity has no explanatory power in respect to the error, regardless of its value.

• The change in energy at the top of the atmosphere (∆U) is not represented by ∆T/S. I would say that this is the most likely explanation. I think that the current paradigm, in which the temperature is linearly related to the forcing, is highly unlikely. Simple consideration of the complexity of the system discourages assumptions of linearity.

• The change in energy at the top of the atmosphere ∆U is correctly represented by ∆T/S, but S in turn is not a constant but a function of T “f(T)”. Thus the substitution in Eqn. 1 should actually be

∆U = ∆T/f(T)

This is a refinement of the previous possibility. I put this forward because of the obvious daily change in climate sensitivity in the tropics, with the sensitivity dropping as the day progresses and the temperature increases. Since that variation in the climate sensitivity occurs daily over about a third of the planet, the part of the planet where the energy enters the system, it is not unreasonable to think that the global climate sensitivity should be a function of temperature. (Note that even here the sensitivity is unlikely to be a linear function of temperature, as the natural situation contains clear thresholds at which the climate sensitivity changes rapidly.)

• Something else that I haven’t thought of yet.

I make no hard claims about any of this, as I don’t know where the missing energy really is. I don’t even know if this is the missing energy that Trenberth was talking about. My theory is that the energy is not missing, but that Equation 2 is wrong. My hypothesis is that the earth responds to volcanoes and other forcing losses by cutting back on clouds and thunderstorms. This lets in lots of energy, and as a result neither the air temperature nor the ocean heat storage change very much. I have detailed that hypothesis here.

About the only solid thing I can say out of this analysis is that if my numbers and logic are correct, then one of the fundamental equations of the current climate paradigm is falsified …

We’ll see how it plays out. All comments and explanations gladly accepted.

w.

[UPDATE: This discussion continues at Some of the Missing Energy]

APPENDIX: Converting Joules/year to W/m2 involves the fundamental relationship:

1 Joule is the application of 1 Watt for 1 second

So … one Watt/m2 applied for one year gives us 1 *  31.6E6 Joules/m2 per year. (Watts/m2 times seconds in 1 year.)

To get total Joules for the planet, we need to multiply that answer by 5.1E14 square metres, to include the total surface area. So one Watt/m2 of forcing, acting on the planet for 1 year, delivers 16.3E21 Joules/year (16.3 zettaJoules). This allows us to convert easily between Joules/year and W/m2