Spencer on Pinatubo and climate sensitivity

Revisiting the Pinatubo Eruption as a Test of Climate Sensitivity

By Roy W. Spencer, PhD.

The eruption of Mt. Pinatubo in the Philippines on June 15, 1991 provided a natural test of the climate system to radiative forcing by producing substantial cooling of global average temperatures over a period of 1 to 2 years. There have been many papers which have studied the event in an attempt to determine the sensitivity of the climate system, so that we might reduce the (currently large) uncertainty in the future magnitude of anthropogenic global warming.

In perusing some of these papers, I find that the issue has been made unnecessarily complicated and obscure. I think part of the problem is that too many investigators have tried to approach the problem from the paradigm most of us have been misled by: believing that sensitivity can be estimated from the difference between two equilibrium climate states, say before the Pinatubo eruption, and then as the climate system responds to the Pinatubo aerosols. The trouble is that this is not possible unless the forcing remains constant, which clearly is not the case since most of the Pinatubo aerosols are gone after about 2 years.

Here I will briefly address the pertinent issues, and show what I believe to be the simplest explanation of what can — and cannot — be gleaned from the post-eruption response of the climate system. And, in the process, we will find that the climate system’s response to Pinatubo might not support the relatively high climate sensitivity that many investigators claim.

Radiative Forcing Versus Feedback

I will once again return to the simple model of the climate system’s average change in temperature from an equilibrium state. Some call it the “heat balance equation”, and it is concise, elegant, and powerful. To my knowledge, no one has shown why such a simple model can not capture the essence of the climate system’s response to an event like the Pinatubo eruption. Increased complexity does not necessarily ensure increased accuracy.

The simple model can be expressed in words as:

[system heat capacity] x[temperature change with time] = [Radiative Forcing] – [Radiative Feedback],

or with mathematical symbols as:

Cp*[dT/dt] = F – lambda*T .

Basically, this equation says that the temperature change with time [dT/dt] of a climate system with a certain heat capacity [Cp, dominated by the ocean depth over which heat is mixed] is equal to the radiative forcing [F] imposed upon the system minus any radiative feedback [lambda*T] upon the resulting temperature change. (The left side is also equivalent to the change in the heat content of the system with time.)

The feedback parameter (lambda, always a positive number if the above equation is expressed with a negative sign) is what we are interested in determining, because its reciprocal is the climate sensitivity. The net radiative feedback is what “tries” to restore the system temperature back to an equilibrium state.

Lambda represents the combined effect of all feedbacks PLUS the dominating, direct infrared (Planck) response to increasing temperature. This Planck response is estimated to be 3.3 Watts per sq. meter per degree C for the average effective radiating temperature of the Earth, 255K. Clouds, water vapor, and other feedbacks either reduce the total “restoring force” to below 3.3 (positive feedbacks dominate), or increase it above 3.3 (negative feedbacks dominate).

Note that even though the Planck effect behaves like a strong negative feedback, and is even included in the net feedback parameter, for some reason it is not included in the list of climate feedbacks. This is probably just to further confuse us.

If positive feedbacks were strong enough to cause the net feedback parameter to go negative, the climate system would potentially be unstable to temperature changes forced upon it. For reference, all 21 IPCC climate models exhibit modest positive feedbacks, with lambda ranging from 0.8 to 1.8 Watts per sq. meter per degree C, so none of them are inherently unstable.

This simple model captures the two most important processes in global-average temperature variability: (1) through energy conservation, it translates a global, top-of-atmosphere radiative energy imbalance into a temperature change of a uniformly mixed layer of water; and (2) a radiative feedback restoring forcing in response to that temperature change, the value of which depends upon the sum of all feedbacks in the climate system.

Modeling the Post-Pinatubo Temperature Response

So how do we use the above equation together with measurements of the climate system to estimate the feedback parameter, lambda? Well, let’s start with 2 important global measurements we have from satellites during that period:

1) ERBE (Earth Radiation Budget Experiment) measurements of the variations in the Earth’s radiative energy balance, and

2) the change in global average temperature with time [dT/dt] of the lower troposphere from the satellite MSU (Microwave Sounding Unit) instruments.

Importantly — and contrary to common beliefs – the ERBE measurements of radiative imbalance do NOT represent radiative forcing. They instead represent the entire right hand side of the above equation: a sum of radiative forcing AND radiative feedback, in unknown proportions.

In fact, this net radiative imbalance (forcing + feedback) is all we need to know to estimate one of the unknowns: the system net heat capacity, Cp. The following two plots show for the pre- and post-Pinatubo period (a) the ERBE radiative balance variations; and (b) the MSU tropospheric temperature variations, along with 3 model simulations using the above equation. [The ERBE radiative flux measurements are necessarily 72-day averages to match the satellite’s orbit precession rate, so I have also computed 72-day temperature averages from the MSU, and run the model with a 72-day time step].

As can be seen in panel b, the MSU-observed temperature variations are consistent with a heat capacity equivalent to an ocean mixed layer depth of about 40 meters.

So, What is the Climate Model’s Sensitivity, Roy?

I think this is where confusion usually enters the picture. In running the above model, note that it was not necessary to assume a value for lambda, the net feedback parameter. In other words, the above model simulation did not depend upon climate sensitivity at all!

Again, I will emphasize: Modeling the observed temperature response of the climate system based only upon ERBE-measured radiative imbalances does not require any assumption regarding climate sensitivity. All we need to know was how much extra radiant energy the Earth was losing [or gaining], which is what the ERBE measurements represent.

Conceptually, the global-average ERBE-measured radiative imbalances measured after the Pinatubo eruption are some combination of (1) radiative forcing from the Pinatubo aerosols, and (2) net radiative feedback upon the resulting temperature changes opposing the temperature changes resulting from that forcing– but we do not know how much of each. There are an infinite number of combinations of forcing and feedback that would be able to explain the satellite observations.

Nevertheless, we do know ONE difference in how forcing and feedback are expressed over time: Temperature changes lag the radiative forcing, but radiative feedback is simultaneous with temperature change.

What we need to separate the two is another source of information to sort out how much forcing versus feedback is involved, for instance something related to the time history of the radiative forcing from the volcanic aerosols. Otherwise, we can not use satellite measurements to determine net feedback in response to radiative forcing.

Fortunately, there is a totally independent satellite estimate of the radiative forcing from Pinatubo.

SAGE Estimates of the Pinatubo Aerosols

For anyone paying attention back then, the 1991 eruption of Pinatubo produced over one year of milky skies just before sunrise and just after sunset, as the sun lit up the stratospheric aerosols, composed mainly of sulfuric acid. The following photo was taken from the Space Shuttle during this time:

There are monthly stratospheric aerosol optical depth (tau) estimates archived at GISS, which during the Pinatubo period of time come from the SAGE (Stratospheric Aerosol and Gas Experiment). The following plot shows these monthly optical depth estimates for the same period of time we have been examining.

Note in the upper panel that the aerosols dissipated to about 50% of their peak concentration by the end of 1992, which is 18 months after the eruption. But look at the ERBE radiative imbalances in the bottom panel – the radiative imbalances at the end of 1992 are close to zero.

But how could the radiative imbalance of the Earth be close to zero at the end of 1992, when the aerosol optical depth is still at 50% of its peak?

The answer is that net radiative feedback is approximately canceling out the radiative forcing by the end of 1992. Persistent forcing of the climate system leads to a lagged – and growing – temperature response. Then, the larger the temperature response, the greater the radiative feedback which is opposing the radiative forcing as the system tries to restore equilibrium. (The climate system never actually reaches equilibrium, because it is always being perturbed by internal and external forcings…but, through feedback, it is always trying).

A Simple and Direct Feedback Estimate

Previous workers (e.g. Hansen et al., 2002) have calculated that the radiative forcing from the Pinatubo aerosols can be estimated directly from the aerosol optical depths measured by SAGE: the forcing in Watts per sq. meter is simply 21 times the optical depth.

Now we have sufficient information to estimate the net feedback. We simply subtract the SAGE-based estimates of Pinatubo radiative forcings from the ERBE net radiation variations (which are a sum of forcing and feedback), which should then yield radiative feedback estimates. We then compare those to the MSU lower tropospheric temperature variations to get an estimate of the feedback parameter, lambda. The data (after I have converted the SAGE monthly data to 72 day averages), looks like this:

The slope of 3.66 Watts per sq. meter per degree corresponds to weakly negative net feedback. If this corresponded to the feedback operating in response to increasing carbon dioxide concentrations, then doubling of atmosphere CO2 (2XCO2) would cause only 1 deg. C of warming. This is below the 1.5 deg. C lower limit the IPCC is 90% sure the climate sensitivity will not be below.

The Time History of Forcing and Feedback from Pinatubo

It is useful to see what two different estimates of the Pinatubo forcing looks like: (1) the direct estimate from SAGE, and (2) an indirect estimate from ERBE minus the MSU-estimated feedbacks, using our estimate of lambda = 3.66 Watts per sq. meter per deg. C. This is shown in the next plot:

Note that at the end of 1992, the Pinatubo aerosol forcing, which has decreased to about 50% of its peak value, almost exactly offsets the feedback, which has grown in proportion to the temperature anomaly. This is why the ERBE-measured radiative imbalance is close to zero…radiative feedback is canceling out the radiative forcing.

The reason why the ‘indirect’ forcing estimate looks different from the more direct SAGE-deduced forcing in the above figure is because there are other, internally-generated radiative “forcings” in the climate system measured by ERBE, probably due to natural cloud variations. In contrast, SAGE is a limb occultation instrument, which measures the aerosol loading of the cloud-free stratosphere when the instrument looks at the sun just above the Earth’s limb.

Discussion

I have shown that Earth radiation budget measurements together with global average temperatures can not be used to infer climate sensitivity (net feedback) in response to radiative forcing of the climate system. The only exception would be from the difference between two equilibrium climate states involving radiative forcing that is instantaneously imposed, and then remains constant over time. Only in this instance is all of the radiative variability due to feedback, not forcing.

Unfortunately, even though this hypothetical case has formed the basis for many investigations of climate sensitivity, this exception never happens in the real climate system

In the real world, some additional information is required regarding the time history of the forcing — preferably the forcing history itself. Otherwise, there are an infinite number of combinations of forcing and feedback which can explain a given set of satellite measurements of radiative flux variations and global temperature variations.

I currently believe the above methodology, or something similar, is the most direct way to estimate net feedback from satellite measurements of the climate system as it responds to a radiative forcing event like the Pinatubo eruption. The method is not new, as it is basically the same one used by Forster and Taylor (2006 J. of Climate) to estimate feedbacks in the IPCC AR4 climate models. Forster and Taylor took the global radiative imbalances the models produced over time (analogous to our ERBE measurements of the Earth), subtracted the radiative forcings that were imposed upon the models (usually increasing CO2), and then compared the resulting radiative feedback estimates to the corresponding temperature variations, just as I did in the scatter diagram above.

All I have done is apply the same methodology to the Pinatubo event. In fact, Forster and Gregory (also 2006 J. Climate) performed a similar analysis of the Pinatubo period, but for some reason got a feedback estimate closer to the IPCC climate models. I am using tropospheric temperatures, rather than surface temperatures as they did, but the 30+ year satellite record shows that year-to-year variations in tropospheric temperatures are larger than the surface temperatures variations. This means the feedback parameter estimated here (3.66) would be even larger if scaled to surface temperature. So, other than the fact that the ERBE data have relatively recently been recalibrated, I do not know why their results should differ so much from my results.

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72 thoughts on “Spencer on Pinatubo and climate sensitivity

  1. Interesting. Thank you, Dr. Spencer.
    It will be interesting to learn why your results are different to those by Forster and Gregory.
    Is there any indication of the size of the error in your lambda estimation? I did notice that that scatter diagram has a lot of variation.

  2. Dr. Spencer: I am having some difficulty interpreting the following statement:

    As can be seen in panel b, the MSU-observed temperature variations are consistent with a heat capacity equivalent to an ocean mixed layer depth of about 40 meters.

    If you have time, I would appreciate some expansion on the meaning of this.
    /dr.bill

  3. Any areosols added would do for cooler temperatures, never for warmer temperatures, as the atmosphere, that mix of gases called AIR, cannot “hold” heat as water (The Oceans) does: Nobody can change this fact. Air heat volumetric capacity=0.00192 joules, water=4.186 (3227 times than that of AIR).PERIOD.
    Sorry if I repeat this fact too many times but it makes all models simply stupid…unless you prefer to warm your feet with a bottle filled with hot air, instead of water!!

  4. The net reduction in solar radiation as a result of Pinatuba is on the order of 2.9 W/m2 (from GISS) to as high as 5 W/m2. 18 months later, 50% of the reduction was still in effect. Temperatures fell by about 0.4C. The Ocean Heat Content numbers don’t show any particular large decline.
    So there is no way to take -3.0 W/m2 (from ERBE) (reduced by 50% 18 months later) which then translated into a reduction of 0.4C and say that is consistent with global warming’s predicted long-term lambda value of 1.25 W/m2/K (or 3.7 W/m2 results in 3.0C of warming). Just taking the short-term impact into consideration, Pinatuba gives very low sensitivity values.
    The reason the papers find the Pinatuba numbers to be consistent with global warming’s propositions is because they are very bad at basic math (or because they didn’t want to end up on some “List”).

  5. “Note that even though the Planck effect behaves like a strong negative feedback, and is even included in the net feedback parameter, for some reason it is not included in the list of climate feedbacks. This is probably just to further confuse us.”
    =============
    Who needs confusion, when you have chaos.
    DNA constantly changes, I.E. is in chaos/random mutation.
    Why would climate be different, DNA is just trying to adapt to a chaotic system.
    A 4 billion year model run, and we end up with Al Gore 😉

  6. So, if the weak Solar Activity was to have a negative forcing upon the climate, it would be greeted by a positive feedback of heat energy from 40m depth of the oceans. What we would eventually see is a depleted ocean heat content, as the ocean heat is expended in holding off the loss.
    Conversely, high Solar Activity would be compensated for by the oceans absorbing the extra heat (negative feedback.
    None of which has anything to do with C02, which is merely along for the ride in this model.

  7. Where is the proof that climate sensitivity is constant like gravity. At least, in this study, the time frame is short. Often you hear people say the climate sensitivity must be high to account for the end of the last ice age. I would like if someone could do a year by year best estimate of sensitivity in the last 500 000 years. I think we would see sensitivity gets lower when the temperature gets over a certain high(compatible with the thermostat hypothesis).

  8. @Enneagram
    Some time before, I was intrigued that temperature in the Philippines didn’t seem to have gone down in Summer 1991 compared to other years. Now after this pesentation, it becomes more clear that temperatures during the rainy season in the Philippines depend much more on ocean temperature. It took time for the ocean to cool after the Pinatubo eruption. This seems to be the feedback with the huge time lag that affects the whole world. So to me it doesn’t matter if you take CO2 as a warming forcing or aerosol as a cooling forcing. Aerosols are very effective to cool the oceans but with the opposite effect of an instanteneous positive forcing such a warming event that would decrease cloud optical depth. So I don’t understand your point.

  9. Marc77,
    The climate’s sensitivity [temperature response] to CO2 operates on a log scale. So the more CO2 that is added to the atmosphere, the less effect each additional molecule has. That’s why the current climate is indifferent to more CO2. It doesn’t really matter much, beyond ≈100 ppmv.

  10. David says:
    June 27, 2010 at 4:33 pm
    [SNIP]
    —————–
    David do you like getting snipped? Argue the science or make a point. There are plenty of commenters who believe in AGW and they argue their case quite well in most cases. Try it!!! Oh, please read WUWT Policy as well.
    http://wattsupwiththat.com/policy/

  11. The equation presented seems to leave out one aspect I rarely read about in discussions or even many papers on climate change. That is simply that in the end all – every picojoule – of energy the Earth receives ends up being radiated out into space. All of the fuss, fury and math about the “greenhouse effect” seems to omit that basic fact. I must be missing something, since I don’t have six post-graduate degrees in some variation on “climate science” after my name. Over a relatively short period of time, any increase in carbon dioxide or any other gas which absorbs in-coming or out-going energy originally received from the Sun re-radiates it all back into space. Whatever “warming” is attributable to these gases seems necessarily to be a short-term phenomenom. I ignore the alleged increase in water vapor as having a further warming, simply because the claimed effect cannot exist, else we would have all “fried” – to indulge in a bit of hyperbole, myself – long since. Not to mention we would already have seen any such effect every year, given the temperature changes from winter to spring to summer of at least two orders of magnitude larger than the claimed temperature increases from carbon dioxide. No such “run-away” effect appears in nature.
    So what, in my ignorance and skepticism, am I not understanding? This all seems to me to be “much ado about nothing”, were it not that the politicians and their beneficiaries – which apparently includes many of the core of climate scientist who remain intent on alarming us with their “projections, not predictions; but really we’re all doing to die!!” tales – have seized upon this as a means of increasing their political power. By the way, we’re actually reaching a “tipping point” in that area of human activity, but that’s another comment for another blog.

  12. Marc77 says:
    June 27, 2010 at 6:46 pm
    “Where is the proof that climate sensitivity is constant like gravity.” […]
    Good point. It fits well with something anna v wrote several months ago to the effect of, “You can never cross the same river twice.”
    As far as I can know from my readings, the earth’s climate has never really been the same twice in geological time scales. From that perspective, the recent spate of NH glaciations and interstadials is just weather.
    So, might I add to your post; what must the sensitivity be (oh, and sensitive to what?) to plunge the earth into another ice age or get us out of one?

  13. There always seems to be a misunderstanding that energy systems are the sum of multiple components, which include input (solar, cosmic, etc.) , storage (oceans, land mass, atmosphere, etc.) and dissipation (clouds, water vapor, etc).
    Systems like these are really a hybrid of multiple impulse functions with varying length tails based on the composition and structure of the materials involved.
    Looking at these as simple linear systems will always render a sub-optimal understanding because these systems are neither simple or linear, especially when the storage elements are subject to varying principles of fluid dynamics which are dominated by the topology of the environment.

  14. Ike – Planets each have differing capacities to radiate incoming energy back to space. Consider the Moon vs Earth. Approximately the same distance from the Sun, similar composition, very differing abilities to retain radiative energy from the Sun. The Earth then is warmer. Mars less so, Venus more so. It’s all about air and clouds.
    Now consider this concern – suppose there is a condition on Earth that, when heated, creates a greater ability of our planet to absorb more energy than it radiates in a given unit of time – say a human lifetime. Say, for example, that component is CO2. Let’s say that some perturbation causes, briefly, the Earth to absorb more energy than it radiates back to space, and that increase in heat causes the release of additional perturbing components that is additive. Oh, say, permafrost over peat bogs. The added heat warms the peat which releases more CO2 which causes more absorption until, finally, the oceans heat up.
    Let’s say that this causes seafloor methane mats to become unstable and to begin releasing methane and other GHG’s, and this, added to the earlier warming influences, cause additional warming. Warming to the point that life becomes stressed, the ice caps melt, and Gavin Schmidt utters on his last breath “I was right!”.
    You are correct that all these picojoules will ultimately be released to space, but the rate at which this is done can be sufficiently slow as to end life as we know it by allowing heat to build up for a time. Let’s say this goes on for a thousand years.
    It’s important to me this not happen before I retire, but after I die I really won’t care and I don’t like shoveling snow that much anyway, but to return to the story…
    Perhaps all is not lost meaning it may in fact be true, for example, that adding heat to the system through perturbation will create more clouds which will have a moderating effect on temperature rise and this dooms day scenario is unachievable. Or it may be true that what we see as a perturbation is, in fact, not. The possibility exists for us to be misreading the root cause of warming because we’re not intelligent enough to see the big picture. CO2 and clouds, for example, affect the temperature of the atmosphere. The temperature is changing. CO2 is changing. Is CO2 the cause, or a result? What if it’s actually cosmic rays causing the problem?
    Here’s the real problem – clouds and cosmic rays don’t have a political solution. CO2 does. Therefore, CO2 is the problem of choice – no math needed. CO2 is an empowering influence that clouds will never be. There is no shortage of people ready to exploit that. Politicians are empowered by it and they have budget because they create it as needed. Government budgets create markets where markets would not otherwise exist, so there is a growing “green” economy sector. That includes public education as you might guess, so it can perpetuate. The “green” economy sector feeds back by way of lobbying to create greater need for action – and I assure you this far exceeds anything nature can do until the current interglacial period ends – and the loop is closed.
    Eisenhower was closer to the truth than Dr. Jones.

  15. re Ike: June 27, 2010 at 7:26 pm
    Hi Ike. I do have a bunch of letters after my name, but they’re all in Physics, not ‘climate science’. Given the situation in that area these days, however, that doesn’t make me unhappy. 🙂
    Regarding your question, I think the critical issue that sometimes gets forgotten in the ‘input = output’ statement is that this applies only when the system is in equilibrium. If there is a change in input or output, the system will respond by changing something so as to try and direct things either back to the old equilibrium or to a new one, depending on what changes, and in what way.
    On another thread last week, Willis Eschenbach used an example to illustrate this notion. The ‘system’ (as I set it up afterwards) consists of an empty cylinder into which a substantial stream of water is directed at the top. There is also a hole at the bottom, through which some of the water escapes. As it happens, this setup can be solved exactly to find things like the height of the water at any time, the amount of water currently in the cylinder, the amount currently leaving it, and the cumulative amount that has passed through the system from start to finish. You can also examine what happens if anything changes.
    You can find, for example, the ultimate stable height to which the water will rise, and when it reaches this level, the inflow and outflow rates will be exactly the same. Before reaching that state, however, the cylinder is still filling, and the outflow is less than the inflow. Ultimately there is one cylinder-full of water that is retained in the system, and it will stay there as long as the system stays the same, even though it is never the same water molecules for very long.
    Now, if you change the inlow rate, what happens? Well, that creates a new stable level, and the water level will increase or decrease by a certain amount until it reaches that new stable height, after which the ‘input = output’ statement is again true. Same thing happens if the outflow rate is changed (bigger or smaller hole).
    Another thing you can do is to give the system a ‘one time shot of extra water’, perhaps by dumping in a bucketful at the top all at once. This makes the level rise right away, but the ‘regular’ inflow hasn’t changed, so the stable level will be the same, and what happens is that the outflow speeds up a little, and after a while the water level is back down to normal, and things are as they were before. If you had scooped out a bucket of water, the outflow would have slowed down a bit until the water level was back up to normal, and you can calculate the amount of time needed for these things to happen.
    This cylinder of water isn’t, of course, a ‘climate model’, but it illustrates the behaviour of a large number of systems in Nature, including biological and thermodynamical ones, and many, many others. If you wanted to, you could think of the water height as analogous to temperature, the inflow and outflow as radiation, and the water currently in the cylinder as the energy stored in various ways on the planet, including the bit that makes the atmosphere livable.
    One notable difference is that it is quite possible, at least in principle, to maintain this cylinder in a state of equilibrium (they do it all the time in mixing vats in factories), whereas the Earth is never in such a state, if for no other reason than the fact that we spin on our axis every 24 hours. On the other hand, it is quite possible for the Earth to ‘hunt’ up and down towards a stable level as the inputs and outputs change back and forth. As far as anyone can tell, it has been doing this for vast periods of time, and in a manner that has given an acceptable place for biological organisms to exist and evolve.
    In any case, I have no reason to think that this will change because we inject a bit of CO2 into an already vast system. The CO2 system is just about max’ed out anyway, and while anything we, or the whales, or the ants, or the volcanoes might do, will have some kind of temporary effect, it is likely to be as significant as painting the water cylinder a different colour.
    /dr.bill

  16. Rapid cooling following certain volcanic eruptions create wide spread suffering. The Year without a Summer in 1816 is an example. Can any case be made that higher CO2 levels may act as a buffer against the cooling effect of SO2 and other volcanic aerosols?

  17. Bravo:
    “[…] but we do not know how much of each. There are an infinite number of combinations of forcing and feedback that would be able to explain the satellite observations.”

    Credibility is eroded when confidence in allocation is projected authoritatively. Climate scientists throw linear decompositions around far too haphazardly.

  18. “As can be seen in panel b, the MSU-observed temperature variations are consistent with a heat capacity equivalent to an ocean mixed layer depth of about 40 meters.”
    But this author, using the same model, talks about 700 m:
    R. W. S., “The Great Global Warming Blunder”, p. 115-116
    Did I miss something, or is this the current state of the science?

  19. It seems to me that the Cp*[dT/dt] would be better estimated using the ocean temperature anomaly rather than the troposphere temperature anomaly or better yet using the buoy system temperature vs depth profiles to estimate the change in energy. We will have better data for the next major eruption in the tropics.

  20. rbateman says:
    June 27, 2010 at 6:35 pm (Edit)
    So, if the weak Solar Activity was to have a negative forcing upon the climate, it would be greeted by a positive feedback of heat energy from 40m depth of the oceans. What we would eventually see is a depleted ocean heat content, as the ocean heat is expended in holding off the loss.
    Conversely, high Solar Activity would be compensated for by the oceans absorbing the extra heat (negative feedback.
    None of which has anything to do with C02, which is merely along for the ride in this model.

    Someone else gets it. Nice one Rob.

  21. Bill Illis says:
    June 27, 2010 at 5:55 pm
    The net reduction in solar radiation as a result of Pinatuba is on the order of 2.9 W/m2 (from GISS) to as high as 5 W/m2. 18 months later, 50% of the reduction was still in effect. Temperatures fell by about 0.4C. The Ocean Heat Content numbers don’t show any particular large decline.

    I doubt Pinatubo affected the level of solar radiation at all, the sun being 93,000,000 miles away from the erruption. It would have had a big effect on insolation though, as you enumerate. We need to avoid conflating the two as some people round here exploit the confusion in small changes of total solar irradiation and changes in the insolation, energy recieved at the surface of Earth, and specifically the ocean.
    I have calculated that the ocan was getting around 4W/m^2 nore than the long term average from the sun (and lowered cloud cover) in the 93-2003 decade. This would be around enough to offset the cooling from Pinutubo. There was a big El nino around ’89. I think the ocean would have been heading into a negative anomaly around Pinutubo’s erruption anyway, given the timing of it’s general ocscillations.
    The “compensation” of the additional heat going into the ocean either side of the erruption explains why Ocean Heat Content didn’t drop much as a result of Pinatubo.

  22. I would suggest you americans to tell your EPA to order to shut down all those inconvenient volcanoes!!
    Who knows if they will end, someday, ejecting noxious MILK!

  23. Bookmarked.
    Am I correctly reading your conclusion: to predict that this lower value will even further reduce the impact of a doubling in CO2? What was their stream of logic (their calculation) that shows an a higher value for feedback?

  24. tallbloke says:{June 28, 2010 at 5:14 am)
    “I doubt Pinatubo affected the level of solar radiation at all, the sun being 93,000,000 miles away from the erruption. It would have had a big effect on insolation though, as you enumerate. We need to avoid conflating the two as some people round here exploit the confusion in small changes of total solar irradiation and changes in the insolation, energy recieved at the surface of Earth, and specifically the ocean”
    Thank you for once again bringing up this very important distinction. I think this is the most misunderstood concept of the “change in TSI affects climate, no it doesn’t” argument.

  25. I think there is a math error in the way you are graphically integrating the equation
    Cp*[dT/dt] = F – lambda*T
    when you graph the dT/dt part, dt is allways the interval of suceeding measurements rather than the whole time interval, so that you always get a lower value.
    thanks

  26. We are about to reach our tipping point of sensitivity regarding climate changers fools. This would be, by far, a much more dangerous “tipping point”, so our advice for them would be not to abuse in preaching non-sense. I guess you don´t want to see your pseudo-prophet naked and beautifully adorned with tar and feathers.☺

  27. Some call it the “heat balance equation”, and it is concise, elegant, and powerful. To my knowledge, no one has shown why such a simple model can not capture the essence of the climate system’s response to an event like the Pinatubo eruption.
    Cp*[dT/dt] = F – lambda*T

    Several people have shown it already .
    Basically this equation is just the first law of thermodynamics with major unsaid implicit assumptions .
    There is already a VERY bad inconsistency in units .
    The units of Cp are J/kg/°K yet the right hand side doesn’t contain mass .
    The establishment of this equation goes like that :
    1: dU = delta Q (first principle of thermodynamics assuming no work is involved . Of course if we deal with fluids like oceans and atmosphere , work is always involved and already this assumption is wrong !)
    2: rho.Cp.dT.dV = heat in – heat out where :
    rho is the volumic mass and dV is a small volume in neighbourhood of a generic point P(x,y,z) . The heat in and heat out are evaluated at the boundary of dV e.g on a surface dS .
    3) rho(x,y,z,t).Cp.dT(x,y,z,t).dV/dt = g(x,y,z,t,T).dS/dt where g is some function representing the net specific heat flow through the surface dS (units J/m²)
    We now Taylor develop g(x,y,z,t)/dt at first order in T what gives :
    g(x,y,z,t,T)/dt = F – lambda (x,y,z,t,T).T and lambda is just a partial derivative of g.
    4) So now we got :
    rho(x,y,z,t).Cp.dT(x,y,z,t).dV/dt = (F-lambda(x,y,z,t,T).T(x,y,z,t)).dS
    5) The following step is to integrate this equation over a sphere (f.ex Earth surface) for a very thin layer of thickness dz assuming rho constant . This gives :
    rho(z,t).Cp.dz.Integral[dT(x,y,z,t).dx.dy/dt] = F.S – Integral[lambda.T.dx.dy]
    This step is actually illegal because there is no way one can integrate a Taylor development at first order . It is like saying that a parabole is a straight line .
    6) But hey this is climate science so we will go farther . We will suppose that lambda is constant . Farther we will suppose that we can get the differentiation on the left hand side out of the integral . This is messy to say the least because the partial derivatives of T are not continuous . Anyway it gives then :
    rho(z,t).Cp.dz.d/dt[Integral[T(x,y,z,t).dx.dy] = F.S – lambda.Integral[T(x,y,z,t).dx.dy]
    But Integral[T(x,y,z,t).dx.dy] is per definition S.GMST (GMST global mean surface temperature and S Earth surface area) . Therefore :
    rho(z,t).Cp.dz.dGMST/dt = F – lambda.GMST
    7) Almost there . Now we need that famous step where a miracle happens . There is still that annoying density (rho) and the dependence on altitude z of all variables (F, rho , GMST) . Well we make them disappear . It gives .
    Cp.dGMST/dt = F – lambda.GMST
    Let’s resume what we did :
    – we dealt with fluids and supposed there is no work . But there is always work with fluids .
    – we integrated a Taylor expansion of a function which is only valid in a neighbourhood of a point . This is illegal .
    – we have got the differentiation out of an integral . This is invalid because the partial derivatives are not continuous on all interfaces (solid- liquid , solid – gas , liquid – gas) .
    – we have supposed that lambda is constant . Lambda being a partial derivative of a heat flow function , there is no reason it should be constant .
    – we have evacuated both the mass and the altitude z dependencies . There is no justification for that .
    – There is also no reason to assimilate the constant term F in the Taylor developpement of g (the net heat flow) to “radiative forcing” and its partial derivative lambda to “feedback”. Besides these terms are not only radiative but contain convection and conduction in reality . Here of course as the assumption is no work , convection and conduction doesn’t exist .
    I think that it is more than enough to consider that despite that this formula is concise , it is neither elegant nor powerful . Certainly such an accumulation of wrong assumptions is rare and it would be a miracle if all errors cancelled .

  28. I thought that Smokey had a pretty good point. I would not be surprised if CO2 had a log effect on temprature – in other words, to reach the next degree of warming, you would have to pump 10 times the quantity of CO2 into the atmosphere. (If I understand him correctly). But is there evidence for this? Has anyone a contrarian view? Sure, the vast majority of persons writing to this blog are skeptics (as am I) but I am always willing to look at evidence that shows I am wrong. That’s what skeptic means in contrast to the AGW ‘believers’.

  29. TomVonk says: June 28, 2010 at 9:42 am
    Please lookup thermal mass, bulk temperature and latent heat! Then remember that all physical laws are approximations only…

  30. Grumpy Old Man says:
    June 28, 2010 at 10:06 am
    I thought that Smokey had a pretty good point. I would not be surprised if CO2 had a log effect on temprature – in other words, to reach the next degree of warming, you would have to pump 10 times the quantity of CO2 into the atmosphere

    Then we would have a colder earth covered with coniferous forests.
    And if we could keep on increasing CO2, forests would surpassed CO2 increase. No alternatives left for bedwetters but to cry and pee a lot!

  31. Thanks to dr. bill and dp for your explanations of what goes on with received radiation from the Sun and re-radiation of that energy outward. I decline to enter the arguments about the mathematical validity of the equation as it has been … hmmm … more than 45 years since I studied calculus of any sort.
    Your explanations then lead me ask another ignorant question. I gather then that the temperature(s) in the air, oceans, etc of the Earth changes in direct relation to solar radiation in frequencies which are or can be absorbed by the Earth – by which I intend to include atmosphere, hydrosphere and lithosphere and whatever else there is named or discovered since my last science class – and is some relationship to the time required for that energy to be re-radiated. Judging from my readings of the published papers about “climate change”, there isn’t much being done to ascertain that time period; is there? Or is that one of the phenomena which are not directly measureable and therefore must be determined indirectly from others whose quantities are directly measureable? Efforts in that direction would be more fruitful, or so it seems to me, than most of what has been going on, particularly with the I.P.C.C. and their evident use of published propaganda from interested – even financially interested – NGO’s as with the error in the loss of glacial ice in the Himalayan chain and loss of forest in the Amazon, just to mention two recent exposures of that method of “proof”. Too much politics; too little science, it seems to me. Well, thank you again, and I’ll refrain from cluttering up WUWT with questions. Hopefully by reading I’ll get more understanding.

  32. Ike says:
    June 28, 2010 at 1:26 pm

    Common sense is by far more valuable than deceiving post normal science. How else could you explain, for example, that Democritus, 600 years BC could have found that water was icosahedrical and Pitagoras, with a humble monochord, found the laws governing nature. And, last but not least, calculus operations made by computers are made using arithmetics. So, don’t cheat yourself, pigs still cannot fly though some pretend to.

  33. “”” Invariant says:
    June 28, 2010 at 12:40 pm
    TomVonk says: June 28, 2010 at 9:42 am
    Please lookup thermal mass, bulk temperature and latent heat! Then remember that all physical laws are approximations only… “””
    Approximations to what ?
    My Physical Chemistry Text Book says that one Calorie is exactly 4.184 Joules. Now admittedly it doesn’t also specify just what a calorie is; as in raises one gram of water by once deg C (under such and such conditions).
    But no some Physical laws are exact; but they are exact descriptions of the behavior of some MODEL or other. The inexactitude; if there is such, is in the connection between the calculated behavior ofTHE MODEL and the actual behavior of the real universe.
    It simply wouldn’t do to have different practitioners get different answers when predicting the behavior of the same model. It’s ok that different experimental methodologies may yield different observed values of what is purportedly the same thing; for then we can seek to uncover the discrepancy in the meothods; but our models should always produce the same answers in the hands of different workers; or what good are they.
    George

  34. “”” Grumpy Old Man says:
    June 28, 2010 at 10:06 am
    I thought that Smokey had a pretty good point. I would not be surprised if CO2 had a log effect on temprature – in other words, to reach the next degree of warming, you would have to pump 10 times the quantity of CO2 into the atmosphere. (If I understand him correctly). But is there evidence for this? Has anyone a contrarian view? Sure, the vast majority of persons writing to this blog are skeptics (as am I) but I am always willing to look at evidence that shows I am wrong. That’s what skeptic means in contrast to the AGW ‘believers’. “””
    Well it may well be true that it takes more and more CO2 to get the next increment of effect; but when somebody says something is logarithmic; that implies a very specific and non negotiable mathematical formulation.
    Purportedly atmospheric CO2 has enjoyed about five doublings since the Pre-Cambrian era 600 million years ago; well to be more correct that would be about five halvings. Yet the corresponding Temperature proxy information gives not even the vauest hint of any logarithmic association with the CO2 proxies.
    And for the modern era, where we have observed less that 1/3 of one doubling with any kind of accurate measurment methodology; ther’s not a shred of evidence for a logarithmic linkage there either; the error bands in the data are so wide, that virtually any well behaved function can be made to fit equally well with either a logartithmic or straight line approximation.
    But when I look at the Current-Voltage relationship of well made semi-conductor diodes, and see a truly logarithmic relationship over 20 doublings of the current; then I take a jaundiced view of claims that CO2/T is logarithmic; besides what is the theoretical Physical basis for expecting to get a logarithmic relationshsip; I don’t see any such theoretical basis either.
    The driving force between any CO2 -Temperature causation has to start with the surface emittance of the earth surface which itself must generally follow a fourth power of Temperature Law of some kind; so it varies by more than an order of magnitude over the earth surface, and all at the same instant of time. So already We have some highly Temperature dependent variability of the very forcing mechanism that is supposed to initiate the cO2 atmospheric warming phenomenon; before we even get to returning some effect back to the surface.
    So I don’t place any faith whatsoever in claims of a logarithmic connection. I am prepared to believe that mayba a little CO2 does a lot and a lot does not do a lot more; but I don’t buy into a lot of the “Saturated CO2” arguments either; since re-absorption/re-radiation cascades; have to be a part of the Physical mechanisms.
    But bottom line, I think it is all completely irrelevent anyway since I believe that H2O regulates the whole thing via the cloud modulation process, and I don’t think CO2 is anything more than just another perturbator like solar TSI variations, or volcanic or other aerosol episodes, and the like. “IT’S THE WATER !!”

  35. “To my knowledge, no one has shown why such a simple model can not capture the essence of the climate system’s response to an event like the Pinatubo eruption.”
    Because the term ‘equilibrium’ is totally wrong; the terms “steady state” or “quasi-steady state” are appropriate. On cannot apply equilibrium thermodynamics to a steady state system, nor would one need to.
    Your use of the term ‘equilibrium’ to describe a ‘average temperature’ of a rotating Earth is not only wrong, it is nonsense.
    I do enjoy reading your posts Roy, but why use this howler?

  36. ” Ike says:
    That is simply that in the end all – every picojoule – of energy the Earth receives ends up being radiated out into space. All of the fuss, fury and math about the “greenhouse effect” seems to omit that basic fact. I must be missing something”
    Its a shell game. Take a small piece of plutonium aboard the international space station, then place a pair of small mirrored hemispheres around it, then another pair, and another,….and so on, like Russian dolls.
    Within a day the center will be hotter than the center of a super nova as each shell reflects half its incoming heat back inside. This must be true as Zeno was right and you can in fact avoid being hit by a bullet by flinching.

  37. TomVonk,
    Dr. Roy Spencer is doing a back-of the envelope calculation. This means that he takes a complex system and makes a number of zeroth order assumptions in order to develope a simple model to describe it. You have completely misunderstood the system and environment that Dr. Spencer is using.
    Firstly, Dr. Spencer assumes that:
    a) the atmosphere can represented by a verticle column of gas with a cross-section of
    1 metre^2,
    b) the volume and mass of the “model atmosphere” are fixed,
    c) the model atmospheric column of gas can be characterized by an average
    temperature and pressure (which determine its mean density because of the
    assumption of constant volume).
    d) once a perturbation has been established by the eruption of Mt. Pinatubo, the net
    forcing and net feedback paramters are a linear function of time.
    Correcting for assumption a) ONLY requires a mutiplicative constant to allow for the actual three-dimensional geometry of the atmosphere.
    Assumption b) allows him to make the approximation to the first Law of Thermodynamics that:
    Change in heat energy = Change in Internal energy
    delta Q = delta U
    dQ/dt = dU/dt
    m * Cv * dT/dt = dU/dt where m = the fixed mass of the column of gas
    Cv * dT/dt = (1/m) dU/dt
    = constant * dU/dt since delta W = 0
    as the workdone by the environment on the gas column in the small time interval of the experimental perturbation dt is effectively zero.
    Now, if the internal energy of the column of gas is being subject to a small perturbation in the net (internal) radiative forcing (measure in Watts/m^2) over a small period of time dt, that is being opposed by a net (internal) radiative feed back (also measured in Watts/m^2) over the same small period of time dt, then:
    dU/dt = [Radiative Forcing]/dt – [Radiative Feedback]/dt
    However, if you refer to the last figure in Dr. Spencer’s presentation, you will see that once the pertubation has been established, the net forcing and net feed back are both linearly decreasing with time of the short period of the purtubation. Thus,
    dU/dt = constant * ([Radiative Forcing] – [Radiative Feedback])
    = constant * (F – lambda * dT)
    Hence,
    Cp*[dT/dt] = const*(F – lambda*T) QED

  38. re Ike: June 28, 2010 at 1:26 pm
    I doubt that I can answer your questions in a way that is satisfactory, Ike, mostly because I don’t know how to work out the answers myself, but I can make a few more comments that might help.
    The warm surface of the Earth gets rid of energy in two principal ways. One is by direct conductive transfers to the air molecules in contact with it. This warms the air, and it then undergoes convection processes, mainly vertical (‘thermals’), but also horizontal (‘winds’), which move that energy to other places that are cooler. Some of the energy possessed by the convecting molecules, particularly those that rise high enough, will be radiated out into space. The rate of transfer of energy by conduction is very much affected by the specific heat and thermal conductivity of the surface, and this has different values all over the place.
    The other process is for the Earth to radiate to space directly. The rate of emission depends on the temperature at the surface. You may have read that this depends on the 4th power of the surface temperature, and that would be true if the Earth were truly a blackbody and radiated at all frequencies. In fact, though, neither of these things is applicable, and the actual temperature dependence is somewhere between T¹ (if you’re dealing with just low-frequency stuff) and T⁴ (if the whole spectrum is involved), or some temperature polynomial that depends on, among other things, the emissivity at every frequency for every part of the Earth, which also varies quite a lot from one time and place to another.
    So then, all I can offer is the preceding ‘hand-waving’ to describe what’s going on. Trying to deal with this quantitatively, however, is something of a nightmare.
    /dr.bill

  39. Niderthana
    I have misunderstood nothing and I have shown quite rigorously I believe what assumptions are necessary to obtain this equation from the 1st principle of thermodynamics .
    It indeed shows , beyond any reasonable doubt that the “zeroth order model” can’t capture any significant feature of the atmosphere-ocean dynamics .
    You just restated some of those wrong assumptions and omitted many .
    a) the atmosphere can represented by a verticle column of gas with a cross-section of 1 metre^2,
    No . The column must also contain water because most of the surface are oceans .
    b) the volume and mass of the “model atmosphere” are fixed,
    No . The volume can be fixed but the mass not . The density is variable depending whether the column contains solids (above land) or liquids (above oceans) .
    c) the model atmospheric column of gas can be characterized by an average
    temperature and pressure (which determine its mean density because of the
    assumption of constant volume).

    Same comment as a) and b) . Atmosphere is just a negligible part of the heat capacity of the system . Besides fluid dynamic processes are not governed by “means” anyway .
    d) once a perturbation has been established by the eruption of Mt. Pinatubo, the net forcing and net feedback paramters are a linear function of time.
    This is a joke ? And why just this arbitrary separation in “net forcing” and “feedback” ? What is phase change – forcing or feed back ? There is no linear function of time over the 3 years either .
    delta Q = delta U
    I already commented on it . Even the assumption of “constant mass” doesn’t imply that delta W = 0 and we have seen that the mass is not constant for every column anyway . The work of gravity is certainly not 0 . Neither the one of viscous forces for that matter .
    dU/dt = [Radiative Forcing]/dt – [Radiative Feedback]/dt
    However, if you refer to the last figure in Dr. Spencer’s presentation, you will see that once the pertubation has been established, the net forcing and net feed back are both linearly decreasing with time of the short period of the purtubation. Thus,
    dU/dt = constant * ([Radiative Forcing] – [Radiative Feedback])
    = constant * (F – lambda * dT)

    This is so confused that one really wonders if you know what you are talking about .
    Out of a dozen of comments I will mention only the most important and I have already mentionned them in a more rigorous way in the first post .
    – We don’t deal with any infinitesimal dt . The time scale in the figures is months !
    – the T in dU/dt is supposed to be an average temperature of the column (where is its top ? stratosphere ?) . The measures are surface temperatures , actually low troposphere anomalies .
    – What about heterogenous columns above oceans that contain water and air ? What is the T ?
    – Why is the “feedback only radiative” ? What about latent heat changes ?
    – How happens the miracle that transforms [Radiative Feedback]/dt in
    lambda*T . Where does the T come from ?
    – The “feedback” processes contain per definition everything what is not radiative forcing e.g latent heat , work , albedo etc) . The figures show that the sum of everything (ERBE imbalances) is anything but linear in time over the 3 years of observation . There is no reason for “forcing” and “feedback” to be each separately linear with time . There is no reason for lambda or for F for that matter to be independent of any other variable (space , density , temperature etc) either .
    In summary there are so many wrong assumptions in this formula that whatever it describes , it doesn’t belong to our Universe .

  40. The main factor in temperature change during 1992 has been overlooked here, that being changes in the solar signal, and I mean solar wind speed/density and not TSI.
    Planetary Ordered Solar Theory indicates a very cool January and October for 1992. The much colder January/February 1991, made that year colder than 1992 overall, just as an indicator of the extent of solar forced variation.
    @Tallbloke, ” There was a big El nino around ’89.” ? http://www.cpc.ncep.noaa.gov/products/analysis_monitoring/ensostuff/ensoyears.shtml

  41. I am more concerned with the effect of Pinatubo on global temperature than climate sensitivity. Much nonsense has been written about it, starting with Stephen Self et al. in the big Pinatubo book “Fire and Mud.” In their article “The Atmospheric Impact of the 1991 Mount Pinatubo Eruption” they claim an observed surface cooling in the Northern Hemisphere of up to 0.5 to 0.6 degrees Celsius and a cooling perhaps as large as -4 degrees over large parts of the earth in 1992-93. But when you look at where these numbers come from he shows you global temperature curves from 1991 to 1994 (his Figure 12A) for stratosphere, troposphere and surface temperatures. The troposphere and surface temperatures both show a peak exactly where the eruption is and temperature descends from there into a valley that bottoms out in 1992. The depth of the valley is about 0.6 degrees Celsius and this must be the source of his numbers. He goes on to pontificate that “The Pinatubo climate forcing was stronger than the opposite, warming effects of either the El Nino event or anthropogenic greenhouse gases in the period 1991-1993.” Unfortunately he is dead wrong both on temperature as well as on forcing. He does not understand that temperature peaks and valleys like the one he shows are a normal part of global temperature oscillations whose cause is the ENSO system in the Pacific. The satellite record of lower tropospheric temperatures shows five such El Nino peaks before 1998. The peaks correspond to the El Nino periods and the valleys in between are La Ninas. It so happens that Pinatubo erupted exactly when an El Nino peaked and the temperature was just beginning to descend into a La Nina valley. Obviously Pinatubo did nothing to suppress an El Nino but just got a free ride when a convenient La Nina was appropriated to give it cooling power. But Self also wonders about “…why surface cooling is is clearly documented after some eruptions (for example, Gunung Agung, Bali, in 1963) but not others – for example El Chichon, Mexico, in 1982.” Apparently what we have is pot luck: if a volcano erupts when the El Nino has peaked and temperature is going down you can report cooling. If it erupts when a La Nina has just bottomed out and temperature is going up there is no cooling to report. This is what happened to poor El Chichon: it erupted when a La Nina had just bottomed out and there was no chance for a free ride since an El Nino was building up. Unfortunately the misinformation about Pinatubo cooling has spread far and wide by now and the 1991-92 La Nina is still mismarked “Pinatubo cooling” on many temperature charts.

  42. I am more concerned with the effect of Pinatubo on global temperature than climate sensitivity. Much nonsense has been written about it, starting with Stephen Self et al. in the big Pinatubo book “Fire and Mud.” In their article “The Atmospheric Impact of the 1991 Mount Pinatubo Eruption” they claim an observed surface cooling in the Northern Hemisphere of up to 0.5 to 0.6 degrees Celsius and a cooling perhaps as large as -0.4 degrees over large parts of the earth in 1992-93. But when you look at where these numbers come from he shows you global temperature curves from 1991 to 1994 (his Figure 12A) for stratosphere, troposphere and surface temperatures. The troposphere and surface temperatures both show a peak exactly where the eruption is and temperature descends from there into a valley that bottoms out in 1992. The depth of the valley is about 0.6 degrees Celsius and this must be the source of his numbers. He goes on to pontificate that “The Pinatubo climate forcing was stronger than the opposite, warming effects of either the El Nino event or anthropogenic greenhouse gases in the period 1991-1993.” Unfortunately he is dead wrong both on temperature as well as on forcing. He does not understand that temperature peaks and valleys like the one he shows are a normal part of global temperature oscillations whose cause is the ENSO system in the Pacific. The satellite record of lower tropospheric temperatures shows five such El Nino peaks before 1998. The peaks correspond to the El Nino periods and the valleys in between are La Ninas. It so happens that Pinatubo erupted exactly when an El Nino peaked and the temperature was just beginning to descend into a La Nina valley. Obviously Pinatubo did nothing to suppress an El Nino but just got a free ride when a convenient La Nina was appropriated to give it cooling power. But Self also wonders about “…why surface cooling is is clearly documented after some eruptions (for example, Gunung Agung, Bali, in 1963) but not others – for example El Chichon, Mexico, in 1982.” Apparently what we have is pot luck: if a volcano erupts when the El Nino has peaked and temperature is going down you can report cooling. If it erupts when a La Nina has just bottomed out and temperature is going up there is no cooling to report. This is what happened to poor El Chichon: it erupted when a La Nina had just bottomed out and there was no chance for a free ride since an El Nino was building up. Unfortunately the misinformation about Pinatubo cooling has spread far and wide by now and the 1991-92 La Nina is still mismarked “Pinatubo cooling” on many temperature charts.

  43. “”” dr.bill says:
    June 28, 2010 at 8:25 pm
    re Ike: June 28, 2010 at 1:26 pm
    ……………………..
    The other process is for the Earth to radiate to space directly. The rate of emission depends on the temperature at the surface. You may have read that this depends on the 4th power of the surface temperature, and that would be true if the Earth were truly a blackbody and radiated at all frequencies. In fact, though, neither of these things is applicable, and the actual temperature dependence is somewhere between T¹ (if you’re dealing with just low-frequency stuff) and T⁴ (if the whole spectrum is involved), or some temperature polynomial that depends on, among other things, the emissivity at every frequency for every part of the Earth, which also varies quite a lot from one time and place to another. “””
    Well I’m not sure you are giving radiation the credit it is due dr. bill.
    First of all; I do NOT discount the energy transport effects due to conduction, convection and evaporation; those are all legitimate thermal energy transport processes that DO come into play in moving energy around the planet.
    But Radiation is ultimately the only way for it to exit to space; absent the exodus of large amounts of material.
    And the Earth may be much more “Black Body” like than you think. For one thing, aboiut 73% of the earth surface is oceans, and the radiant energy absorption by water is almost total. The optical reflectance is only 2% for normal incidence, and maybe averages 3% over all angles so about 97% of incident energy; certainly in the solar spectrum, does enter the water. It is either absorbed by the water, or propagates deeper, until something else absorbs it. Well shallow waters around beaches, will have some small bottom reflectance; which then must then return to the surface through the same absorbing water. And the reflected energy is diffuse; so a good fraction of it will find itself trapped in the water by Total Internal Reflection. I’ve never actually calculated the total TIR trapping by a water surface; but it’s an 8th grade optics calculation.
    So the deeper ocean areas are quite good as near total absorbers of solar energy.
    The Black Body; Stefan Boltzmann calculation of emitted radiation sets a maximum envelope to a surface emission. No surface can emit more than a Black Body, due to Thermal Radiation alone (as a result of its Temperature).
    The 4th power of T integral is not strongly dependent on the actual emission spectrum. The peak of the BB spectrum increases as the fifth power of the Temperature, and the actual spectrum of the particular surface gets applied to that as a spectral emissivity factor.
    But I think you will find, that for most real terrain surfaces, any elemental area will have a total emittance that does vary as the 4th power of the Temperature; modified only by a spectral emissivity. The oceans would look quite black if it wasn’t disguised by the blue light scattering in the atmosphere.
    I have looked into really deep and really clean Sea of Cortez water at sea level with overhead cover to remove local direct sunlight; and that water looks plain black to me.
    Total spectral coverage ius not necessary to get close to 4th power response.
    Anybody familiar with BB spectra knows that 98% of the Total energy is contained between 0.5 of the peak wavelength, and 8 times the peak wavelength; so for the incoming solar spectrum, that is about 250 nm out to 4.0 microns; with 1% straggle at each end. For the 15 deg C global mean thermal radiation, that range would be from about 5 microns min to 80 microns max, and over that spectral range, water is pretty much totally absorbing in just a few microns of thickness.
    Even with a quite narrow emission spectrum (can’t imagine what material) the emitted power is hardly likely to ever be linear with Temperature; I would think it is more likely to be higher than 4th power than lower, because of the spectral peak emittance being 5th power.
    People keep talking about how the higher colder atmospheric layers will radiate to space at lower emittances; and Dr Spencer even mentions an effective radiating Temperature of 255 K. I don’t disagree with that number based on how it is defined; but effective Temperature or not; far more energy is raidated from surfaces that are more like 330 K or higher, than 255 K.
    At Vostok station and similar Antarctic highlands locations; the radiating Temperature might be as low as 185K, which is a pitiful contribution to cooling the planet.
    Dr Roy’s 255 K is maybe the equilibrium earth orbit black body Temperature under one assumption; but that doesn’t mean that most of the earth is emitting such a spectrum; it’s not.
    The highest energy losses are from the hottest most arid tropical desert surfaces, with peak spectral emittances around 8-9 microns right in the water window (with little water to block anyway); and such spectra are less captured by CO2 which operates at 15 microns (maybe 13.5 to 16.5).
    The idea that surface energy is transported by various processes to the upper Troposphere, and then radiated to space from there at some low temperature BB spectrum rate, is quite wrong. Those processes do occur of course and are useful contributions; but direct surface radiation to space; is much more prevalent than is suggested by Trenberth’s isothermal planet cartoon energy budget.

  44. tallbloke says:
    June 28, 2010 at 5:14 am
    I have now watched Dr. Scafetta’s presentation here;
    http://yosemite.epa.gov/ee/epa/eed.nsf/vwpsw/360796B06E48EA0485257601005982A1#video
    And let me say this; It was very interesting. Scafetta only points out certain correlations, and does not in any way say that all can be explained, he just says these correlations are interesting, and shows us some interesting ideas.
    I can understand that the warmers (taking into account the objective paragraph of the IPCC) do not want ANYONE to hear about this.
    I am sure Dr. Scafetta is safely on the blacklist?
    In fact, we can use the blacklist to find interesting papers on the climate from now on?
    Because we can hope that the people on the blacklist is not on the HockeyTeam,
    and therefore looks at interesting science, not related to insignificant trace-gases?
    Yes?

  45. Tom Vonk,
    I think the readers of this blog have gotten the message that you are incapable of doing a simple heuristic calculation. You have left no stone unturned in your head long rush to hide your ignorance with (totally) uneccesary complexity.
    You do not seem to understand that zeroth or first order calculations require broad approximations. The trick is to know what approximations to make. The whole idea of making first order calculations is not give you a definitive answer that is chiseled in stone but to direct you towards a useful estimate of the quatity that you are seeking.
    I do not necessarily agree with Dr. Spencer’s confidence in his answer but I do applaud him for at least trying to get a hand on a possible estimate of the feadback parameter.
    I am not going make the effort of pointing out all the absudities in your post but I will try address a few. Now where to start…?
    Let start with these zingers…
    a) the atmosphere can represented by a verticle column of gas with a cross-section of 1 metre^2,
    No . The column must also contain water because most of the surface are oceans .

    In Thermodynamics you can define a system. How you define the system depends on the problem you are addressing. If you are talking about the Earth’s atmosphere, it is usual to define the system as being a vertical column of gas, that has a lower boundary in contact with either the Earth’s Ocean or Earth’s surface and an upper boundary in contact with space. It is quiet valid to regard the Earth’s Oceans and space as being external to the system you have chosen. If you do so then you can regard any energy or mass transport across the upper or lower boundaries as transfers from the environment. If on the other hand, you regard the system as a coupling between the atmosphere and the top 100 m of the oceans, then you would have to define your upper and lower boudaries accordingly.
    b) the volume and mass of the “model atmosphere” are fixed,
    No . The volume can be fixed but the mass not . The density is variable depending whether the column contains solids (above land) or liquids (above oceans) .

    Have you every heard about the approximation that 2/3 of all the Earth’s surface is water ? When you are doing first order calculations – you do rough and ready approximations like this.
    delta Q = delta U
    I already commented on it . Even the assumption of “constant mass” doesn’t imply that delta W = 0 and we have seen that the mass is not constant for every column anyway . The work of gravity is certainly not 0 . Neither the one of viscous forces for that matter .

    You do not seem to understand even the most basic thermodynamic principles. If you define a system, work done inside that system between particles in not part of the dW
    term in the First Law:
    dQ = dU + dW
    dQ referes to the heat energy gained or lost by the system (i.e. the atmsophere) and dU referes to the change in internal energy of this system.
    dW referes to work done on the system by the environment (dW is postive) or work done by the system on the environment (dW is negative). Mechanical work is achieved by moving a force applied to (or by) the system through a distance. It is a reasonable assumption to assume that on time scales of a few months or years that the net work done by (or to) the system (i.e. the atmosphere) on the environment (i.e. the oceans and space) is negligible. Under these circumstances it is entirely valid to claim that
    dQ = dU

    dU/dt = [Radiative Forcing]/dt – [Radiative Feedback]/dt
    However, if you refer to the last figure in Dr. Spencer’s presentation, you will see that once the pertubation has been established, the net forcing and net feed back are both linearly decreasing with time of the short period of the purtubation. Thus,
    dU/dt = constant * ([Radiative Forcing] – [Radiative Feedback])
    = constant * (F – lambda * T)

    From your rantings, I have to assume that your mathematical training is in Pure rather than Applied Mathematics.
    You are right in pointing out that I made a mistake in this part of my analysis. I should have said that the assumption that Dr. Spence had (unknowingly) made is that both the forcing and radiative feedback terms used need to exponentially decreasing functions of time [and not linear] in order for his formular to make any sense.
    First, if the radiative forcing (F) can be approximated by function a short sharp increase followed by an exponentially decreasing function of time, then it is a reasonable assumption to represent F as :
    F(t) = Fo * exp(-kt) where Fo=the initial forcing term and k=constant
    related to the e-folding time of the exponential decay,
    Hence, dF/dt = -k Fo * exp( -kt)
    dF/dt = -k F
    Of course, for my final statement to make sense, the e-folding decay rates for the radiative forcing and feed-back would have to be in the same ball-park i.e.
    dU/dt = [Radiative Forcing]/dt – [Radiative Feedback]/dt
    dU/dt = constant * ([Radiative Forcing] – [Radiative Feedback])
    and hence, Cp*[dT/dt] = const*(F – lambda*T)
    As for your comments on dt being infinetesimally small, all that required for this to be true is that the time interval you are considering for the perturbation (dt ~ 3 years) must be small compared climatic time scales (> 30 years).

  46. TomVonk says:
    June 29, 2010 at 2:42 am
    In summary there are so many wrong assumptions in this formula that whatever it describes , it doesn’t belong to our Universe .

    I think you are on thin ice here. Please go through the derivation of (4) in the below paper that is based on sound physics – and you will understand why…
    http://www.ecd.bnl.gov/steve/pubs/HeatCapacity.pdf
    I guess that you have a solid bacground in mathematics, but that you have little experience with common engineering physics that is always based on a number of approximations. Usually mathematicians cannot tell the difference between approximations and erroneous assumptions, so their conclusions are not always valid…

  47. @Arno Arrak says:
    June 29, 2010 at 9:12 am
    Well observed. There also are a number of earlier eruptions where no cooling can be seen.
    Read;
    Ulric Lyons says:
    June 29, 2010 at 8:37 am

  48. re George E. Smith: June 29, 2010 at 9:17 am
    I think that everyone with any interest in these matters has long since learned that the only way to permanently get energy off-planet (in the absence of giant rail-guns) is by means of radiation, but that wasn’t the point of my note to Ike. You will also note that I stated explicitly that there are many variables involved, that few of them ‘stand still’, and that quantitative calculations are difficult. Your own list of complications, conditionals, caveats, and uncertainties only serves to emphasize that conclusion. If this were easy to do, we’d all have long since done it, and this blog likely wouldn’t exist, at least with its current focus.
    Regarding the temperature dependence that you mention, it is an uncontested fact that the simple T⁴ behaviour is only true when the complete spectrum is involved. In the case of absorption and re-radiation by H2O, CO2, and the other gases in the atmosphere, the relevant ranges are quite specific and limited, but can be affected by things like pressure broadening, which varies with altitude, time of day, time of year, and many other quantities, and thus the re-radiation spectrum definitely isn’t complete.
    The T³ dependence you mentioned for the peak height of the irradiance formula (Planck’s Law) is not relevant. It is a spurious consequence of using wavelength as a variable to obtain Wein’s Displacement Law, which gives the location of the peak in the Planck’s Law expression. The other common forms of Planck’s Law use frequency or photon energy as variables, and give a different location for the peak ‘irradiance colour’. They also give a peak height that it proportional to T³. This is a well-known ‘problem’, but it doesn’t really matter.
    What does matter, is not the irradiance, but the integrated irradiance, which gives the same result for the emitted power, regardless of what variable is used, and that result is strictly limited to a maximum temperature dependence of T⁴. If there are gaps in the spectrum, the temperature dependence is either reduced to a lower order, or at the very least, lower orders are present in the result.
    /dr.bill

  49. Tom Vonk provides an excellent deconstruction of the derivation of Spencer’s
    elementary linear DE feedback model. I simply add the analytic observation
    that, under the best of circumstances, such a model is incapable of
    revealing anything meaningful about climate sensitivity, let alone about heat from
    ocean depths.
    Let’s assume, for sake of discussion, that GMST is an (unknown) analytic
    function T = u(f) of the insolation flux f. Then, by the Chain Rule, its
    time derivative is given exactly by
    dT/dt = (du/df) x (df/dt)
    where the first r.h. term represents the sensitivity and the second term is
    the time derivative of the observed insolation. Adding the feedback term
    (-lambda x u) destroys that exact multiplicative relationship. Furhermore,
    it tries to explain an effect due entirely to a transient decrease in
    insolation (excitation) by ascribing it to a self-induced dissipation of
    temperature (response). The feedback formulation of the model is
    fundamentally wrong.
    Many climate scientists seem to labor under the misapprehension that
    feedback is somehow necessary in order for the system response to input to
    be distributed over time. Nothing can be further from the truth. In
    systems with elements of storage (capacity or memory), the output can be
    time-distributed without any feedback whatsover. For any linear system,
    the output y(t) in response to any bounded input x(t) can always be
    represented exactly by a convolution time-integral (from minus infinity to
    the present time) of the impulse response function h(tau) and the input
    time-history x(t). The characteristic “time-constant” of the decay of
    impulse-response function, rather than feedback, is what controls the
    duration of effects from past values of input.
    BTW, it was IPCC’s fantastic claim that the entire climate system can be
    represented by a transfer function expressed as the product of the reciprocals (!) of the transfer functions of its subsystems is what convinced me that
    they lack basic analytic understanding of systems. It made me an instant
    sceptic after their first report.

  50. Just noting we can also estimate the climate sensitivity by how much temperatures have increased to date compared to the forcing increase to date.
    Temps are up 0.7C or so (give or take an artificial 0.2C or 0.3C added by the adjustments of Tom Karl) and the forcings have increased by 1.6 watts/m2 or up to 1.9 watts/ m2 by the time we include the newest numbers.
    1.6 / 0.7 =

  51. Accidently hit post comment before I was finished.
    1.6 / 0.7 = 2.3 (double to 50% more than expected depending on how much lag one wants to build in and maybe up to three times too high if you don’t accept the high Aerosols negative forcings numbers).
    Trenberth published about this recently and included a new term “Negative Radiative Feedback” to explain the missing energy or the missing temperature response.
    http://img638.imageshack.us/img638/8098/trenberthnetradiation.jpg
    What is the “Negative Radiative Feedback” – it could be clouds, it could be missing energy absorption in the deep oceans or it could be that the theory is just wrong to start with.

  52. sky says: The characteristic “time-constant” of the decay of impulse-response function, rather than feedback, is what controls the duration of effects from past values of input.
    Good point! An intuitive explanation of a thermal time constant is provided by MIT,
    “The time constant tau is in accord with our intuition, or experience; high density, large volume, or high specific heat all tend to increase the time constant, while high heat transfer coefficient and large area will tend to decrease the time constant.”
    http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node129.html (Equation (18.18) is the most important)
    Now, I wonder whether our spinning earth may have a large number of different thermal masses, all with different time constants, and that an analogy with a forced oscillator may be useful? Obviously the earth is a dissipative system, but still there may be natural oscillations that can be sustained with very little effort.
    Energy is always conserved but comes in so many forms that it is highly nontrivial to do a detailed energy budget. Still, it is possible to have an idea about the most important contributions; we know that the thermal mass of the oceans is the largest thermal mass, and we know that most of the energy dissipation is radiation to space. Then equation (4) in this paper follows,
    http://www.ecd.bnl.gov/steve/pubs/HeatCapacity.pdf
    Sure this is an approximation only, but it is a valid approximation and not an erroneous assumption. It is also a meaningful equation, it tells us that,
    1. Temperature is an accumulative property.
    2. It takes time to change the temperature due to ocean thermal mass.
    3. The temperature may or may not be close to the equilibrium temperature.
    4. Energy is dissipated mainly due to radiation back to space.
    I think one of Dr. Spencer’s main points is that the temperature here on our planet may be far from the equilibrium temperature and that it may take several centuries to oscillate between cold and warm periods on each side of the equilibrium.
    It seems to me that “Tom Vonk” and partly “sky” lack the required physical understanding and intuition to imagine a simplified heat balance equation for our planet. Then, I cannot see how you picture our planets climate variations in simplified mathematical terms and you should possibly go back to the Feynman Lectures on Physics once more and re-read the chapters about Energy.

  53. Invariant (3:56pm):
    Despite your self-projected proficiency in physics, you fail to notice a crucial difference between Spencer’s model formulation and Schwartz’s self-admitted Ansatz (premise without rigorous basis). The latter involves T^4 in a dimensionally-consistent algebraic restatement of the energy difference between outgoing and incoming energy. It tells us nothing about system operation. Spencer’s simple model involves T to the FIRST power as the explicitly-proclaimed “feedback” and is dimensionally inconsistent!
    I find your reference to Feynman ironically amusing. I’ll let you guess from whom I learned my physics.

  54. Invariant (3:56pm):
    Despite your professed proficiency in physics, you fail to notice a
    crucial difference between Spencer’s model formulation and Schwartz’s
    self-admitted Ansatz (premise without rigorous basis). The latter involves
    T^4 in a dimensionally-consistent algebraic restatement of the
    difference between outgoing and incoming energy fluxes. It tells us nothing
    about system operation. Spencer’s simple model involves T to the FIRST
    power as the explicitly-proclaimed “feedback” and is dimensionally
    inconsistent!
    I find your reference to Feynman ironically amusing. I’ll let you guess
    from whom I learned my physics.

  55. sky: June 30, 2010 at 5:44 pm
    If you look again at the Schwartz paper, you will note that he seemed pleased with his Ansatz (eq’n 3), not ashamed of it in the sense of something to be ‘admitted‘. It is also not a bad approximation anyway, and has been used by many others.
    This leads to the T⁴ term in his eq’n 4, but he notes that the solution (eq’n 6) is not his, but was obtained by others who had also used this approximation.
    If you check the Math, you will see that eq’n 6 is not a solution to eq’n 4, but to a linearized version of eq’n 4, in which the T⁴ term has been replaced by the linear term of an expansion around the mean temperature.
    Finally, to get to Dr. Spencer’s equation, you just need one more simplifying approximation to eq’n 6, i.e. to take the time interval involved to be small in comparison to the time constant τ.
    All in all, it’s just a regular back-of-the-envelope calculation of a type that is carried out by physicists and engineers every day in order to get a ballpark estimate of the main effects, as has been pointed out by several other readers.
    /dr.bill

  56. dr. bill:
    These sort of back-of the-envelope approximations may satisfy academics, but when they result in dimensionally inconsistent model equations they wind up misleading everyone about how real-world physics operates. Furthermore, the assumption that the time-constant of the interacting uppermost ocean layer (which should not be confused with the lag-variable tau in the convolution integral) is small relative to the duration of the insolation disturbance due to Mt. Pinatubo is particularily onerous. Few realize that it is dependent on the strength of winds and is typically on the order of weeks to months in the upper mixed layer. That, rather than mathematical justifications of academic handwaving, is what physically matters.

  57. sky: July 1, 2010 at 3:11 pm
    Actually, it’s pretty clear that Dr. Spencer is a ‘real-world’ person using real-world data, and working on real-world problems. Whether he works at a university (say UAH) or at a private corporation (say RSS) doesn’t change that at all. You might also want to check out the underpinnings of your vaunted sense of omniscience and your erroneous sense of what other people do or don’t know.
    /dr.bill

  58. dr. bill:
    I’m not impugning Spencer’s work with real-world data, which I much admire. The analytic mis-formulation of his model is the issue at hand. And decades of experience in analyzing and modeling real-world processes is what I bring to the discussion. Enough said.

  59. sky: July 1, 2010 at 5:21 pm
    dr. bill:
    I’m not impugning Spencer’s work with real-world data, which I much admire. The analytic mis-formulation of his model is the issue at hand. And decades of experience in analyzing and modeling real-world processes is what I bring to the discussion. Enough said.

    Seems like a good idea. Here’s an article I read yesterday.
    Perhaps there’s something in there for both of us. ☺
    /dr.bill

  60. but when they result in dimensionally inconsistent model equations they wind up misleading everyone about how real-world physics operates.
    Challenges to “sky”
    1. Ensure WUWT readers that Dr. Spencer has a unit/dimension of lambda that leads to a dimensionally inconsistent model equation.
    2. Write down your favourite heat balance equation for our planet, use as few letters/symbols as possible.

  61. That “Pinatubo cooling” is abject nonsense. Pinatubo eruption happened to coincide with peak warmth of the 1991 El Nino that was immediately followed by a temperature drop of half a degree to the bottom of the 1991/92 La Nina. The eruption was perfectly timed to make it look like it caused that temperature drop and Self et al. fell for the illusion. But that La Nina was a perfectly normal La Nina and had nothing to distinguish it from the previous two as Figure 7 in my book demonstrates. That figure is an analysis of satellite temperatures and brings out the five El Nino peaks and their accompanying La Ninas in the eighties and nineties. They can be clearly identified and all are part of the ENSO oscillation in the Pacific. ENSO has a global temperature influence and shows up in all accurate global temperature records. But Self et al. [in “Fire and Mud” edited by Newhall & Punongbayan (University of Washington Press, 1996), pp. 1089-1115], however, had no idea that the temperature oscillations in the satellite record belonged to ENSO and simply appropriated that particular La Nina cooling for their volcano. To justify it they show an out of context segment of the satellite record in their Figure 12A and if you don’t know what you are looking at it is easy to see why they thought the volcano had done it. And since they are the big experts everyone just copied them. Their data also showed that the volcanic aerosols that were blasted into the stratosphere first warmed it and that stratospheric cooling did not start until 1993. That’s two years after the tropospheric cooling they claimed. Also, an observation they report should have alerted them. They start to wonder why surface cooling is “…clearly documented after some eruptions (for example, Gunung Agung, Bali, in 1953) but not others – for example, El Chichon, Mexico, in 1982)…” The answer is pot luck. If the eruption coincides with the start of a La Nina cooling it looks like the volcano did it. But when it takes place when a La Nina period just ended and an El Nino is building up there is no observable cooling. It’s all in timing. Pinatubo erupted precisely when a La Nina was just starting to form. But for El Chichon the timing was inopportune: it erupted just when an El Nino was beginning to build up. No one could find its cooling simply because it did not exist, and neither did Pinatubo cooling exist. But Self et al. think they have found it and pontificate: “Pinatubo climate forcing was stronger than the opposite warming effects of either the El Nino event or anthropogenic greenhouse gases in the period 1991-1993.” Complete bullshit but it passes for science among the global warming gang.

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