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.
climatepatrol says:
I mean CO2 AND aerosols won´t make for any warming.
It´s the Sun, though TSI were “conveniently adjusted”:
IPCC “Consensus” on Solar Influence was Only One Solar Physicist who Agreed with http://climaterealists.com/?id=5910
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!
tallbloke says:
June 28, 2010 at 5:14 am
Something really happened about 1989. See Dr.N.Scafetta conference at EPA
http://yosemite.epa.gov/ee/epa/eed.nsf/vwpsw/360796B06E48EA0485257601005982A1#video
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?
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.
Dr.Bill @ur momisugly 9:24PM.
That’s elegantly stated. Thanks.
================
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
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.☺
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 .
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’.
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…
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!
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.
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.
How accurate are the estimations of volcanic output?
“”” 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
“”” 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 !!”
“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?
” 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.
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
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
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 .
Tom V
“within the limits” “at the limit”
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
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.