Stacked Volcanoes Falsify Models

Guest Post by Willis Eschenbach

Well, this has been a circuitous journey. I started out to research volcanoes. First I got distracted by the question of model sensitivity, as I described in Model Climate Sensitivity Calculated Directly From Model Results. Then I was diverted by the question of smoothing of the Otto data, as I reported on in Volcanoes: Active, Inactive, and Retroactive. It’s like Mae West said, “I started out as Snow White … but then I drifted.” The good news is that in the process, I gained the understanding needed to direct my volcano research. Read the first of the links if you haven’t, it’s a prelude to this post.

Unlike the situation with say greenhouse gases, we actually can measure how much sunlight is lost when a volcano erupts. The volcano puts reflective sulfur dioxide into the air, reducing the sunlight hitting the ground. We’ve measured that reduction from a variety of volcanoes. So we have a reasonably good idea of the actual change in forcing. We can calculate the global reduction in sunlight from the actual observations … but unfortunately, despite the huge reductions in global forcing that volcanoes cause, the global temperature has steadfastly refused to cooperate. The temperature hasn’t changed much even with the largest of modern volcanoes.

Otto et al. used the HadCRUT4 dataset in their study, the latest incarnation from the Hadley Centre and the Climate Research Unit (CRU). So I’ll use the same data to demonstrate how the volcanoes falsify the climate models.

All VolcanoesFigure 1. Monthly HadCRUT4 global surface air temperatures. The six largest modern volcanoes are indicated by the red dots.

This post will be in four parts: theory, investigation, conclusions, and a testable prediction.


Volcanoes are often touted as a validation of the climate models. However, in my opinion they are quite the opposite—the response of the climate to volcanoes clearly demonstrates that the models are on the wrong path. As you may know, I’m neither a skeptic nor a global warming supporter. I am a climate heretic. The current climate paradigm says that the surface air temperature is a linear function of the “forcing”, which is the change in downwelling radiant energy at the top of the atmosphere . In other words, the current belief is that the climate can be modeled as a simple system, whose outputs (global average air temperatures) are a linear function of the SUM of all the various forcings from greenhouse gas changes, volcanoes, solar changes, aerosol changes, and the like. According to the theory, you simply take the total of all of the forcings, apply the magic formula, and your model predicts the future. Their canonical equation is:

Change in Temperature (∆T) = Change in Forcing (∆F) times Climate Sensitivity

In lieu of a more colorful term, let me say that’s highly unlikely. In my experience, complex natural systems are rarely that simply coupled from input to output. I say that after an eruption, the climate system actively responds to reductions in the incoming sunlight by altering various parts of the climate system to increase the amount of heat absorbed by other means. This rapidly brings the system back into equilibrium.

The climate modelers are right that volcanic eruptions form excellent natural experiments in how the climate system responds to the reduction in incoming sunlight. The current paradigm says that after a volcano, the temperature should vary proportionally to the forcing. I say that the temperature is regulated, not by the forcing, but by a host of overlapping natural emergent temperature control mechanisms, e.g. thunderstorms, the El Nino, the Pacific Decadal Oscillation, the timing of the onset of tropical clouds, and others. Changes in these and other natural regulatory phenomena quickly oppose any unusual rise or fall in temperature, and they work together to maintain the temperature very stably regardless of the differences in forcing.

So with the volcanoes, we can actually measure the changes in temperature. That will allow us to see which claim is correct—does the temperature really follow the forcings, or are there natural governing mechanisms that quickly act to bring temperatures back to normal after disturbances?


In order to see the effects of the volcanoes, we can “stack” them. This means aligning the records of the time around the volcano so the eruptions occur at the same time in the stack. Then you express the variations as the anomaly around the temperature of the month of the eruption. It’s easier to see than describe, so Figure 2 shows the results.

stacked temperatures six major volcanic eruptionsFigure 2. Stacked records of the six major volcanoes. Individual records show from three years before to five years after each eruption. The anomalies are expressed as variations around the temperature of the month of the eruption. The black heavy line shows the average of the data. Black vertical lines show the standard error of the average.

The black line is the average of the stacked records, month by month. Is there a signal there? Well, there is a temperature drop starting about six months after the eruptions, with a maximum of a tenth of a degree. However, El Chichon is clearly an outlier in this regard. Without El Chichon, the signal gets about 50% stronger.

stacked temperatures five major volcanic eruptionsFigure 3. As in Figure 3, omitting the record for El Chichon.

Since I’m looking for the common response, and digging to find the signal, I will leave out El Chichón as an outlier.

But note the size of the temperature response. Even leaving out El Chichon, this is so small that it is not at all clear if the effect shown is even real. I do think it is real, just small, but in either case it’s a very wimpy response.

To properly judge the response, however, we need to compare it to the expected response under various scenarios. Figure 3 shows the same records, with the addition of the results from the average models from the Forster study, the results that the models were calculated to have on average, and the results if we assume a climate sensitivity of 3.0 W/m2 per doubling of CO2. Note that in all cases I’m referring the equilibrium climate sensitivity, not the transient climate response, which is smaller. I have used the lagged linear equation developed in my study of the Forster data (first cite above) to show the theoretical picture, as well as the model results.

stacked modeled observed theoretical temperaturesFigure 4. Black line shows the average of the monthly Hadcrut temperatures. Blue line shows the average of the modeled annual temperatures from the 15 climate models in the Forster paper, as discussed here. The red line shows what the models would have shown if their sensitivity were 2.4°C per doubling of CO2, the value calculated from the Forster model results. Finally, the orange line shows the theoretical results for a sensitivity of 3°C per doubling. In the case of the red and orange lines, the time constant of the Forster models (2.9 years) was used with the specified sensitivity. Tau ( τ ) is the time constant. The sensitivity is the equilibrium climate sensitivity of the model, calculated at 1.3 times the transient climate response.

The theoretical responses are the result of running the lagged linear equation on just the volcanic forcings alone. This shows what the temperature change from those volcanic forcings will be for climate models using those values for the sensitivity (lambda) and the time constant (tau).

Now here, we see some very interesting things. First we have the model results in blue, which are the average of the fifteen Forster models’ output. The models get the first year about right. But after that, in the model and theoretical output, the temperature decreases until it bottoms out between two and three years after the eruption. Back in the real world, by contrast, the average observations bottom out by about one year, and have returned to above pre-eruption values within a year and a half. This is a very important finding. Notice that the models do well for the first year regardless of sensitivity. But after that, the natural restorative mechanisms take over and rapidly return the temperature to the pre-eruption values. The models are incapable of making that quick a turn, so their modeled temperatures continue falling.

Not only do the actual temperatures return to the pre-eruption value, but they rise above it before finally returning to the that temperature. This is the expected response from a governed, lagged system. In order to keep a lagged system in balance, if the system goes below the target value for a while, it need to go above that value for a while to restore the lost energy and get the system back where it started. I’ll return to this topic later in the post. This is an essential distinction between governors and feedbacks. Notice that once disturbed, the models will never return to the starting temperature. The best they can do is approach it asymptotically. The natural system, because it is governed, swings back shortly after the eruption and shoots above the starting temperature. See my post Overshoot and Undershoot for an earlier analysis and discussion of governors and how they work, and the expected shape of the signal.

The problem is that if you want to represent the volcanoes accurately, you need a tiny time constant and an equally tiny sensitivity. As you can see, the actual temperature response was both much smaller and much quicker than the model results.

This, of course, is the dilemma that the modelers have been trying to work around for years. If they set the sensitivity of their models high enough to show the (artificially augmented) CO2 signal, the post-eruption cooling comes out way, way too big. If they cut the sensitivity way, way down to 0.8° per doubling of CO2 … then the CO2 signal is trivially small.

Now, Figure 4 doesn’t look like it shows a whole lot of difference, particularly between the model results (blue line) and the observations. After all, they come back close to the observations after five years or so.

What can’t be seen in this type of analysis is the effect that the different results have on the total system energy. As I mentioned above, getting back to the same temperature isn’t enough. You need to restore the lost energy to the system as well. Here’s an example. Some varieties of plants need a certain amount of total heat over the growing season in order to mature. If you have ten days of cool weather, your garden doesn’t recover just because the temperature is now back to what it was before. The garden is still behind in the total heat it needs, the total energy added to the garden this season is lower than it would have been otherwise.

So after ten days of extra cool weather, your garden needs ten days of warm weather to catch up. Or perhaps five days of much warmer weather. The point is that it’s not enough to return the temperature to its previous value. We also need to return the total system energy to its previous value.

To measure this variation, we use “degree-days”. A degree-day is a day which is one degree above from some reference temperature. Ten degree-days could be five days that are two degrees warmer than usual, or two days that are five degrees warmer than usual. As in the example with the garden, degree-days accumulate over time, with warmer (positive) degree days offsetting cooler (negative) degree days. For the climate, the corresponding unit is a degree-month or a degree year. To convert monthly temperature into degree-months, you simply add each months temperature difference from the reference to the previous total. The record of degree-months, in other word, is simply the cumulative sum of the temperature differences from the date of the eruption.

What does such a chart measure? It measures how far the system is out of energetic balance. Obviously, after a volcano the system loses heat. The interesting thing is what happens after that, how far out of balance the system goes, and how quickly it returns. I’ve left the individual volcanoes off of this graph, and only shown the stack averages.

stacked cumulative modeled observed and theoretical temperaturesFigure 5. Cumulative record of degree-months of energy loss and recovery after the eruptions. Circles show the net energy loss in degree-months four years after the eruption. 

Remember that I mentioned above that in a governed system, the overshoot above the original temperature is necessary to return the system to its previous condition. This overshoot is shown in Figure 3, where after the eruptions the temperatures rise above their original values. The observations show that the earth returned to its original temperature after 18 months. The results in Figure 5 show that it took a mere 48 months to regain the lost energy entirely. Figure 5 shows that the actual system quickly returned to the original energy condition, no harm no foul.

By contrast, the models take much larger swings in energy. After four years, the imbalance in the system is still increasing.

Now folks, look at the difference between what the actual system does (black line) and what happens when we model it with the IPCC sensitivity of 3° per doubling, or even the model results … I’m sorry, but the idea that you can model volcanic eruptions using the current paradigm simply doesn’t work. In a sane world, Figure 5 should sink the models without a trace, they are so very far from the reality.

We can calculate the average monthly energy shortage in the swing away from and back to the zero line by dividing the area under the curve by the time interval. Nature doesn’t like big swings, this kind of response that minimizes the disturbance is common in nature.  Here are those results, the average energy deficit the system was running over the first four years.

average energy deficit 4 yearsFigure 6. Average energy deficit over the first four years after the eruption.

In this case, the models are showing an average energy deficit that is ten times that of the observations … and remember, at four years the actual climate is back to pre-eruption conditions, but the models’ deficit is still increasing, and will do so for several more years before starting back towards the line.


So what can we conclude from these surprising results?

The first and most important conclusion is that the climate doesn’t work the way that the climate paradigm states— it is clearly not a linear response to forcing. If it were linear, the results would look like the models. But the models are totally unable to replicate the rapid response to the volcanic forcings, which return to pre-existing temperatures in 18 months and restore the energy balance in 48 months. The models are not even close. Even with ridiculously small time constant and sensitivity, you can’t do it. The shape of the response is wrong.

I hold that this is because the models do not contain the natural emergent temperature-controlling phenomena that act in concert to return the system to the pre-catastrophic condition as soon as possible.

The second conclusion is that the observations clearly show the governed nature of the system. The swing of temperatures after the eruptions and the quick return of both temperature and energy levels to pre-eruption conditions shows the classic damped oscillations of a governed system. None of the models were even close to being able to do what the natural system does—shake off disturbances and return to pre-existing conditions in a very short time.

Third conclusion is that the existing paradigm, that the surface air temperature is a linear function of the forcing, is untenable. The volcanoes show that quite clearly.

There’s probably more, but that will do for the present.


Now, we know that the drops in forcing from volcanoes are real, we’ve measured them. And we know that the changes in global temperature after eruptions are way tiny, a tenth of a degree or so. I say this is a result of the action of climate phenomena that oppose the cooling.

A corollary of this hypothesis is that although the signal may not be very detectable in the global temperature itself, for that very reason it should be detectable in the action of whatever phenomena act to oppose the volcanic cooling.

So that was my prediction, that if my theory were correct, we should see a volcanic signal in some other part of the climate system involved in governing the temperature. My first thought in this regard, of course, was the El Nino/La Nina pump that moves warm Pacific water from the tropics to the poles.

The snag with that one, of course, is that the usual indicator for El Nino is the temperature of a patch of tropical Pacific ocean called the Nino3.4 area. And unfortunately, good records of those temperatures go back to about the 1950s, which doesn’t cover three of the volcanoes.

A second option, then, was the SOI index, the Southern Oscillation Index. This is a very long-term index that measures the difference in the barometric pressures of Tahiti, and Darwin, Australia. It turns out that it is a passable proxy for the El Nino, but it’s a much broader index of Pacific-wide cycles. However, it has one huge advantage. Because it is based on pressure, it is not subject to the vagaries of thermometers. A barometer doesn’t care if you are indoors or out, or if the measurement location moves 50 feet. In addition, the instrumentation is very stable and accurate, and the records have been well maintained for a long time. So unlike temperature-based indices, the 1880 data is as accurate and valid as today’s data. This is a huge advantage … but it doesn’t capture the El Ninos all that well, which is why we use the Nino3.4 Index.

Fortunately, there’s a middle ground. This is the BEST index, which stands for the Bivariate ENSO Timeseries. It uses an average of the SOI and the Nino 3.4 data. Since the SOI has excellent data from start to finish, it kind of keeps the Nino3.4 data in line. This is important because the early Nino3.4 numbers are from reanalysis models in varying degrees at various times, so the SOI minimizes that inaccuracy and drift. Not the best, but the best we’ve got, I guess.

Once again, I wanted to look at the cumulative degree-months after the eruptions. If my theory were correct, I should see an increase in the heat contained in the Pacific Ocean after the eruptions. Figure 6, almost the last figure in this long odyssey, shows those results.

stacked cumulative BEST el nino indexFigure 6. Cumulative index-months of the BEST index. Positive values indicate warmer conditions. Krakatoa is an obvious outlier, likely because it is way back at the start of the BEST data where the reconstruction contains drifts.

Although we only find a very small signal in the global temperatures, looking where the countervailing phenomena are reacting to neutralize the volcanic cooling shows a clearer signal of the volcanic forcing … in the form of the response that keeps the temperature from changing very much. When the reduction in sunlight occurs following an eruption, the Pacific starts storing up more energy.

And how does it do that? One major way is by changing the onset time of the tropical clouds. In the morning the tropics is clear, with clouds forming just before noon. But when it is cool, the clouds don’t form until later. This allows more heat to penetrate the ocean, increasing the heat content. A shift of an hour in the onset time of the tropical clouds can mean a difference of 500 watt-hours/m2, which averages over 24 hours to be about 20 W/m2 continuous … and that’s a lot of energy.

One crazy thing is that the system is almost invisible. I mean, who’s going to notice if on average the clouds are forming up a half hour earlier? Yet that can make a change of 10 W/m2 on a 24-hour basis in the energy reaching the surface, adds up to a lot of watt-hours …

So that’s it, that’s the whole story. Let me highlight the main points.

• Volcanic eruptions cause a large, measurable drop in the amount of solar energy entering the planet.

• Under the current climate paradigm that temperature is a slave to forcing with a climate sensitivity of 3 degrees per doubling of CO2, these should cause large, lingering swings in the planet’s temperature.

• Despite the significant size of these drops in forcing, we see only a tiny resulting signal in the global temperature.

• This gives us two stark choices.

A. Either the climate sensitivity is around half a degree per doubling of CO2, and the time constant is under a year, or

B. The current paradigm of climate sensitivity is wrong and forcings don’t determine surface temperature.

Based on the actual observations, I hold for the latter.

• The form (a damped oscillation) and speed of the climate’s response to eruptive forcing shows the action of a powerful natural governing system which regulates planetary temperatures.

• This system restores both the temperature and the energy content of the system to pre-existing conditions in a remarkably short time.

Now, as I said, I started out to do this volcano research and have been diverted into two other posts. I can’t tell you the hours I’ve spent thinking about and exploring and working over this analysis, or how overjoyed I am that it’s done. I don’t have a local church door to nail this thesis to, so I’ll nail it up on WUWT typos and all and go to bed. I think it is the most compelling evidence I’ve found to date that the basic climate paradigm of temperatures slavishly following the forcings is a huge misunderstanding at the core of current climate science … but I’m biased in the matter.

As always, with best wishes,




Climate sensitivity is measured in one of two units. One is the increase in temperature per watt/m2 of additional forcing.

The other is the increase in temperature from a doubling of CO2. The doubling of CO2 is said to increase the forcing by 3.7 watts. So a sensitivity of say 2°C per doubling of CO2 converts to 2/3.7 = 0.54 °C per W/m2. Using the “per doubling” units doesn’t mean that the CO2 is going to double … it’s just a unit.


Let’s see, what did I use … OK, I just collated the Otto and Forster net radiative forcings, the Forster 15 model average temperature outputs, the GISS forcing data, and the dates of the eruptions into a single small spreadsheet, under a hundred k of data, it’s here.


The method depends on the fact that I can closely emulate the output of either individual climate models, or the average output of the unruly mobs of models called “ensembles” using a simple lagged linear equation. The equation has two adjustable parameters, the time constant “tau” and the climate sensitivity lambda. Note that this is the transient sensitivity and not the equilibrium sensitivity. As you might imagine, because the earth takes time to warm, the short-term change in temperature is smaller than the final equilibrium change. The ratio between the two is fairly stable over time, at about 1.3 or so. I’ve used 1.3 in this paper, the exact value is not critical.

Using this lagged linear equation, then, I simply put in the list of forcings over time, and out comes the temperature predictions of the models. Here’s an example of this method used on the GISS volcanic forcing data:

lagged conversion of forcing to temperature

Lambda (a measure of sensitivity) controls the amplitude, while tau controls how much the data gets “smeared” to the right on the graph. And sad to say, you can emulate any climate model, or the average of a bunch of models, with just that … see my previous posts referenced above for details about the method.


Here are the most recent six eruptions, eruptions that caused large reductions in the amount of sunlight reaching the earth, with the date of the eruptions shown in red.

NovaruptaMt Agungel chichonSanta MariaKrakatoa


Even Krakatoa, which was supposed to be the cause of the “Year Without A Summer”, didn’t raise a ripple on the global scale.


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Huub Bakker

That’s an amazing analysis Willis. As you say, the clearest evidence yet that the models not only don’t get the numbers right but don’t even get the form of response right. You should look to get this published.

Great post Willis. I think you’ve hit the nail squarely on the head. One possible point why Krakatoa creates a different effect (Fig 6) could be dust. Didn’t it throw up enough dust to create red sunsets in London for months after (a la Turner paintings)? Now i don’t know about the others in that respect and I guess it depends on the VEI.
Would dust and sulphur have different but synergistic effects? Does it depend on the height of the ejecta? Does it depend on the latitude of the volcano? Does this in turn affect how long the dust/sulphur stays in the system and how far and how fast it is spread around the globe?
These are just some of the questions forming in my head after reading this. I have seen discussions of such elsewhere and I’m sure you will have considered them. I love things that make me think. Thank you.
Like you say –

One crazy thing is that the system is almost invisible. I mean, who’s going to notice if on average the clouds are forming up a half hour earlier? Yet that can make a change of 10 W/m2 on a 24-hour basis in the energy reaching the surface, adds up to a lot of watt-hours…

Often answers are right in front of us – for those who take the time to notice.


Willis’ analysis is very interesting. Thanks Willis.
It seems to show the impact of volcanoes is not at all large.
There aren’t a huge number of large volcanoes going off around
the world at the same time. They tend to be solitary phenomena.
Why should a volcano be considered to have much effect on the
global climate?
It’s a point source.
Some of its output may meander around the globe but not most
of it. It falls to the surface within a few hundreds of kilometers from
its source.
Does that which does stay airborne, mix evenly and permeate every
where or does it track according to prevailing winds/air-streams?
If the latter, then its impact can’t be anything but relatively minor.

Baa Humbug

We live in a gas medium.
We measure “surface” temperature at around 2 metres above the ground.
Any gas will rise when its temperature is increased.
No wonder there is hardly any change in measured temperatures after eruptions.
I’m about to light a big bon fire in the front paddock. I’ll be sitting around that fire with some friends eating and drinking. My side that’s facing the fire will be toasty warm, whilst the side away from the fire will be cool.
ALL the air that’s warmed by the fire will quickly rise and move away from me, to be replaced by cool air. A small, very local breeze will be created.
Even though my bon fire is about 2 metres in diametre (as much as my local council will allow) and not quite a metre high, it will generate some very high temperatures. However only a couple of metres away from the fire, there’ll be no difference in the air temperature at all. Any gas molecules warmed by the fire will quickly shoot up up and away.
We can not expect to measure an increase in the temperature of an unconstrained gas. It doesn’t “stand still” long enough for us to measure it.

David Archibald

“Either the climate sensitivity is around half a degree per doubling of CO2” This agrees with what Modtran derives. So I am going with A.

Peter Pond

Love it! Willis, you have a talent for explaining your reasoning in clear, concise terms that just about anyone can understand. I don’t know if you are correct, but at least you have provided your argument and its underlying reasons simply.

Ian H

Brilliant. Very convincing and clear. Well done. I’m convinced. Until a better explanation comes along I’m going to go with your one.
The next step then is to ask what CAN cause the climate to change if it is governed as you describe. Because as we all know climate can and does change. For example what could cause the LIA or the late 20th century warming if the system is indeed governed. A governed system is likely to be quite insensitive to changes in the input energy. To get it to change you would need something that “tweaks” the settings on the governor.
You suggest that the governor is the timing of daily cloud formation in the tropics. So the question becomes what could tweak the timing and speed at which clouds form each day in the tropics? Which brings us back to the solar wind and galactic cosmic rays. The difference under your theory of a governed system is that we understand now why the climate seems to be so sensitive to this change. Unlike other changes to forcing which have little effect this change turns the knob on the thermostat.

“Their canonical equation is:
Change in Temperature (∆T) = Change in Forcing (∆F) times Climate Sensitivity
In lieu of a more colorful term, let me say that’s highly unlikely.”

Whose canonical equation? Could you give a reference? Certainly no GSM works that way.


Superficially, this seems pretty convincing. Thanks for the work that’s gone into this.

Outstanding analysis, Mr. Eschenbach, thank you very much.

Ian H

Change in Temperature (∆T) = Change in Forcing (∆F) times Climate Sensitivity

Nick Stokes asks:
Whose canonical equation? Could you give a reference? Certainly no GSM works that way.

Certainly Nick. Glad to help you with your obviously sincere hunt for enlightenment. I suggest you simply search for a definition of climate sensitivity. For example one can be found at however you can find the same equation in plenty of places. Indeed anyone who uses climate sensitivity must define it and in doing so will write down an equation pretty much exactly like the one above. Oh it might be written in the form
Sensitivity = (∆T)/(∆F)
but I’m sure a clever guy like you can see that these are the same.

J Martin

I was going to ask if you could work it backwards and derive sensitivity from it, but I guess you did and got a half degree.
A. Either the climate sensitivity is around half a degree per doubling of CO2, and the time constant is under a year,
B. The current paradigm of climate sensitivity is wrong and forcings don’t determine surface temperature.

I do think that the or needn’t be so exclusive and that or / and is valid. Myself, I go for and.

Lance Wallace

Very clearly explained as always. However, removing a data point (El Chichon) when all you have is 6 such data points, is a very serious decision. It’s not clear to me that El Chichon is that much of an outlier in Figure 2. It’s pretty much in the middle of the pack most of the time. Pinatubo is more consistently at the bottom, and Novarupta is low before the eruption and very high after. You say you’re dropping it because you’re “looking for a signal” but that’s the same reason Briffa uses for dropping out Khadyta River from Yamal.
So what happens if you leave El Chichon in? It weakens the “signal” to the point where one wonders if there is any signal at all. Then when you get to the stacked El Nino index (Figure 6), you drop out yet another volcano (Krakatoa). You’ve now dropped 33% of your data! Suppose you added El Chichon to this Figure 6–what would it look like? And then suppose you draw your black line using all the data–what would that look like? Not asking you to change your conclusions, but just provide all the data for your readers.


Reality can be a real swine when your pet theory is shown to be false.
Good post Willis.


Willis, I agree with you, the effect on temperature from these volcanos have been rather small. But the explanation via the BEST el Nino index looks strange to me (figure 6). The index starts increasing 18 months before the eruption. Is the index smoothed (3 years)?
Another detail, the “year without summer” occurred after the eruption of Tambora in 1815.

“Ian H says: May 25, 2013 at 2:36 am”
Your Wiki reference says:
“For a coupled atmosphere-ocean global climate model the climate sensitivity is an emergent property: it is not a model parameter, but rather a result of a combination of model physics and parameters. By contrast, simpler energy-balance models may have climate sensitivity as an explicit parameter.
Δ T = λ RF

A simpler model. That doesn’t make it a canonical equation.
In fact, if you look up the definitions of the various sensitivities, they relate to particular situations. Equilibrium response to a move from one fixed forcing to another. Or TCR, which mentions a period of seventy years. None entails a claim that Δ T is proportional to Δ F with constant factor.
Gregory and Forster 2008 do say something like that:
“Observations and AOGCM simulations of twentieth century climate change, and AOGCM experiments with steadily increasing radiative forcing F, indicate a linear relationship F = ρΔT, where ΔT is the global mean surface air temperature change and ρ a constant ‘‘climate resistance’’.”. But that’s an observation, not a presumption. They show the results, with scatter.
And when it comes to prediction, they say (6.2):
“If F = ρΔT holds, we can use ρ to make projections of ΔT given F.”
Doesn’t sound so canonical. They go on to describe some situations where ρ might be constant.

Grey Lensman

Great post, clever theory, but one small error.
for Krakatau read Tambora, much bigger bang and source of year without summer

Rob L

Probably useful to subtract the 10-20 temperature year trend spanning each volcanic eruption, to better isolate the impact of the volcano. Perhaps also think about a second variable for size of volcano (obviously not all equal)
That said bravo. This is a very simple and powerful reproof for status quo climate modelling (at least of volcanos and aerosols), and will be hard to argue against. Worthy of an academic paper.

Chris Wright

Another superb piece of work by Willis. When fully developed and refined, this simply screams out to be a peer-reviewed research paper in a science journal (if any real science journals are left).
I would tend to go for Willis’ first option. But in a sense it doesn’t matter too much: whether one option or the other is true, or both, the final result is the same: the warming caused by CO2 is very small and possibly negligible. This is also consistent with the lack of warming for almost two decades, the fact that almost half of the 20th century warming occurred when there wasn’t enough CO2 (1900 to 1945), and the evidence from the ice cores (CO2 follows the temperature and not the other way around).
One thought did occur, though. The whole thrust of these investigations is into the very largest eruptions, for obvious reasons. But volcanoes are erupting all the time and presumably more eruptions occur in some years compared to others, so a graph of eruption intensity (a bit like the ACE measurement for hurricanes) might have some structure over the years, as opposed to random noise. If so, could there be any correlation between mean volcanic activity and global climate. If so, this could provide more evidence on this question. It would also remove problems with small numbers of data points and the need to remove data points that look like outliers.
By the way, if anyone can give a link to any AVE data (Accumulated Volcanic Energy) I’d be very grateful.
Anyway, many thanks to Willis for an excellent, beautifully written and thought-provoking piece of research.
It does rather look like yet another dagger in the heart of AGW!

David Longinotti

I think this is a strong challenge to the orthodoxy regarding climate sensitivity, but the posited correction mechanism doesn’t appear to cohere with the data shown. The claim is that “When the reduction in sunlight occurs following an eruption, the Pacific starts storing up more energy.” But the timing seems to challenge this assertion – in Figure 6 the change in the slope of the cumulative Best Index occurs about 20 months BEFORE the eruptions, and there is no change in slope around the time of the eruption. Is the implication that the Pacific starts storing energy in anticipation of the eruption, or have I misunderstood the proposed correction phenomenon (or its measurement)?

“First we have the model results in blue, which are the average of the fifteen Forster models’ output.”
I’ve looked through your post, appendices, the spreadsheet, and Forster’s paper, and I still can’t work out what this means. Are they actually outputs from the models? Or are they outputs from your model of the models?


Nick Stokes says:
May 25, 2013 at 2:18 am
“Their canonical equation is:
Change in Temperature (∆T) = Change in Forcing (∆F) times Climate Sensitivity
In lieu of a more colorful term, let me say that’s highly unlikely.”
Whose canonical equation? Could you give a reference? Certainly no GSM works that way.
It doesn’t matter “how they work”, Nick, if that’s the result they give. Which it is. I believe there was a post here over the past week or so doing the sums on that so I won’t bother repeating them.
One of the first concepts taught in computer modelling is the “black box”. Essentially, any model can be replaced by any other model provided they both give the same output for the same inputs. If they do then they are “functionally equivalent” and it doesn’t matter in the least how they transform the input to the output.
It’s the principle used in many of the old “think of a number” tricks – which pre-date computers by a very long time. You get someone to choose a number (the input) and then ask them to do complicated maths with it (effectively running it through a model) over several stages. You can then tell them what the input was from their answer (the output). They work because all those complicated steps in your “model” can be replaced by a much simpler, but functionally equivalent, equation that you can do instantly in your head.
However the GCMs “work” internally, their (smoothed) output for any initial input forcing can be obtained using Change in Temperature (∆T) = Change in Forcing (∆F) times Climate Sensitivity, iterated over time.
Since we’re told not to worry about all those little wiggles of variablilty, because they’re just weather and it’s the long-term (smoothed) change that matters, the models are functionally equivalent to that cannonical equation.

Greg Goodman

Excellent article Willis. The response to volcanoes is what I’ve been saying for well over a year. Thanks for taking the effort to put all this into a coherent whole.
Full credit to you for your “governor” hypothesis and the mechanism, I think that is the key as you say. The system is governed by NON-linear feedbacks not the simplistic linear feedback that is behind all the models.

Delighted to see this challenge to the device at the heart of GCMs. I refer to that gift to the programmers involved of the use of ‘external forcing’ as a wheeze to step around the difficulties of actually modelling CO2 or aerosols or any of the other things on the list of ‘forcings’. Willis is focusing on the crude assumption also deployed – what he refers to as their canonical equation above. Even with it, the variability of the outputs, even from apparently pampered models, is very large indeed. Take it away, and suddenly the well from which so many grandiose prognostications about this that and the other would dry up, and many a would-be prophet would be deprived of fuel for their lucrative scaremongering.

Nick Stokes says: May 25, 2013 at 4:24 am
“Are they actually outputs from the models?”

I see now that they are the ones you digitized from the Otto et al paper; I couldn’t find the data in the Forster paper. Incidentally, I found a version of the Forster paper as published here.

Margaret Hardman

In the middle of marking university entrance individual studies/investigations. I would not give this a good mark. There are lots of confounding factors not taken into account, for example: how long do the particles erupted into the atmosphere remain there? What criteria are used for selecting the eruptions (there were twelve VEI 5 or 6 eruptions in the 20th century)? 14 eruptions would have given a better sample. Are the results statistically significant?
What effect did the eruption of Mount Hudson have in such close temporal proximity to Mt Pinatubo (VEI 5+ and 6 respectively)? Can you unpick the effects of the two eruptions? How much material was erupted into the atmosphere at these eruptions and the other eruptions? Can we see local effects and how far do those effects extend?
The conclusion needs to have strong valid evidence to support it which appears to be missing. Because unsubstantiated assumptions are made in this post, the assertions remain speculative at best and most probably wrong. There will be those that pick that last sentence out for criticism so I shall answer it now. Firstly, taking a simple equation that models part of the behaviour of the climate system and showing it might be wrong does not invalidate all climate models. I believe there are 19 models referenced by the IPCC. Even if this equation is fundamental to them all, the evidence here does not invalidate it.
I know nothing of the author but I would suspect they have not used their full scientific training in the production of this short article. Had they done so, they would have spent a considerable time looking in detail at the possible confounding factors surrounding the global temperatures for each of these eruptions and those that were left out.


I might point out that different eruptions, and different types of volcanos, produce significantly different volumes of SO2. Pinatubo was in fact quite rich in the amount of SO2 spewed into the atmosphere for its relative size. There is also the issue of tropical versus arctic eruptions, eruptions in the arctic produce less effect on sunlight because the emitted S02 travels far less around the globe. Not sure if these effects your analyses.
There is a book ‘Eruptions that shook the world’ which gives a good summary of volcanism, and of course looks at many historical eruptions. There is some good stuff on investigations into eruptions which show up in the ice record, but where it is still not known which volcanoes caused them. Krakatoa appears to have a long history, as does the Pompeii area and a few in the Kurils.
Also, the ‘year without a summer’ was 1816 with Tambora in Sumbawa, not Krakatoa.

Greg Goodman

Having applauded the effort, now to point out some inaccuracies and short comings that I noted.
Excluding El Chicnon is a bit dubious. It is worth pointing out that it is the only one that does seem to show a possible cooling signal. It would be more instructive to look at why that is and whether it is simple confounded with some other independent effect that happened at about the same time.
Looking at LOD we see there is a change around El Chicnon. Is that a confirmation of a real volcanic cooling or witness to something else causing an “apparent” volcanic cooling?
As I pointed out in your “spot the volcano” post last year, once clear effect of volcanoes is warmer winters. In fact if we split CRUTem4 into tropical and ex-tropical we can see this clearly for both Chichon and Pinatubo.
There is a very strong warming due El Chicnon so don’t be too quick to regard it as an outlier.

Nick Stokes

Joe says: May 25, 2013 at 4:36 am
“However the GCMs “work” internally, their (smoothed) output for any initial input forcing can be obtained using Change in Temperature (ΔT) = Change in Forcing (ΔF) times Climate Sensitivity, iterated over time.”

That’s actually not true, and Willis has shown that it isn’t. In his version, ΔT is quite well approximated by an exponentially smoothed ΔF, with time constant of about three years. Considerably lagged.
Here’s what Otto et al say – it’s their eq 2:
“Both equations (1) and (2) assume constant linear feedbacks and (2) further assumes that the ratio of ΔQ to ΔT for the observed period is the same as that at year 70 of a simulation in which atmospheric CO2 levels increase at 1% per year, which is approximately the case over the past few decades if we exclude periods strongly affected by volcanic eruptions (see Supplementary Fig. S2).”
(my bold) – lot of caveats.

Greg Goodman

Now looking at SST, we see no notable effect in terms of warming / cooling but do see a reduced annual variation after Pinatubo causes a negative “anomaly” : warmer winters cooler summers for about three years. There is a slight dip but that process was already underway at least a year beforehand.
This failure to account for existing trends is often the cause of false attribution of cooling to volcanoes. Some way is needed to remove this.
Your stacking idea is one crude way to hope that different periods will cancel out to reveal volcanic signal. This is good simple way to get a first look but you can’t start removing “outliers” otherwise you are open to selection bias.

Greg Goodman

Why are you using ECS to calculate a transient response ?!

Bill Illis

The climate responds to volcanoes as if the feedbacks are actually negative versus the positive that is assumed in the global warming movement.
The climate responds as if the total of all feedbacks is negative 70% versus the 55% positive that global warming theory is based on.


@Nick Stokes
The Forster’s paper gives a 15-models mean for time from 1850 so I think Willis just did the with this output as for real temperatures when Figure 2. As for the red and yellow curves, he shows that the output of Foster models are quite the same as the one of a simple linear model in this article, so he surely did exactly that

Greg Goodman

“Degree days: What does such a chart measure? It measures how far the system is out of energetic balance. ”
No, that’s wrong. Temperature is a measure of heat (energy) . Degree.days is an integral of energy in some units like joule.second , it is not a measure of energy.
If a system comes back to the same temperature is has recovered to the same energy content.
The plot is still useful to look at differences in behaviour.
“Remember that I mentioned above that in a governed system, the overshoot above the original temperature is necessary to return the system to its previous condition. ”
Again you are misunderstanding the physics a bit here. This is the response of an “under-damped” system. An over-damped system will also return without overshoot but will return. Both are “governor” type systems with non-linear feedback.
It is NOT necessary to have overshoot to have a governor, but a bit of over shoot will recover quicker. Too much and it will show multiple oscillations before settling.
A linear negative feedback will NEVER come back it will settle to a new equilibrium with an exponential decay. GCMs responses only comes back up at all because of a) GHG and b) other +ve forcings which they have that produce a positive effect.
That does not alter your fundamental point but you need to better understand all this in order to avoid being dismissed.


Nick Stokes says:
May 25, 2013 at 4:56 am
Joe says: May 25, 2013 at 4:36 am
“However the GCMs “work” internally, their (smoothed) output for any initial input forcing can be obtained using Change in Temperature (ΔT) = Change in Forcing (ΔF) times Climate Sensitivity, iterated over time.”
That’s actually not true, and Willis has shown that it isn’t. In his version, ΔT is quite well approximated by an exponentially smoothed ΔF, with time constant of about three years. Considerably lagged.
No, Willis showed that it’s quite well approximated using sensitivity and lag. The lag has no bearing on the magnitude of the response, only on its timing. So the fundamental formula is still as above, with the lag (as its name implies) simply delaying things slightly.
I’m sure you understand that really.

Ian H

You point out that sensitivity is computed in some models and an explicit parameter in some simpler models and claim that this somehow renders the equation non-canonical. A canonical equation is simply an equation that is so by definition. This equation is the defining equation for sensitivity so it is canonical. Are we going to end up arguing about the meaning of the word canonical? That seems to be a fairly pointless argument to have.
Any function can be locally approximated by a linear one (\cite{Newton}). If temperature is expressed as a function of forcing (of any kind) then it can be locally approximated by the linear equation Willis used. The real issue is whether or not climate models predict temperature as a function of forcing. I would be astonished to see you try to defend the position that climate models do not in general take the point of view that temperature is a function of forcing. In fact you have just told me that ” … AOGCM simulations of twentieth century climate change, and AOGCM experiments with steadily increasing radiative forcing F, indicate a linear relationship F = ρΔT, where ΔT is the global mean surface air temperature change and ρ a constant ‘‘climate resistance”. So clearly most do predict that temperature is a function of forcing.
If you want to argue the point I will concede that a GCM need not be limited to expressing temperature as a function of forcing. Indeed a GCM could even be tweaked to act according to the mechanism that Willis describes with the onset of cloud formation in the tropics having a strong temperature dependence so that it acts as a thermostat. Clouds are an admitted weakness of most GCMs. Perhaps Willis has the fix! Ponder that fact before putting too much faith in GCMs. There are far too many choices to be made in building such a thing and we can very easily make the elephant wiggle his trunk. Such models mostly tell us which way the builders of the model thought the trunk should go.

Willis: I see you’ve caught the volcano-ENSO bug. I’ve looked at that before, using NINO3.4 SST anomalies and the Extended Multivariate ENSO Index (MEI), but never published a post. I had also not looked at the old (GISST-based) version of the BEST ENSO index, which you used in this post, but I will say that the newer (HADSST2-based) version of the BEST ENSO index…
…is too noisy to be of value to you.
Of the others, the extended MEI…
…appears to show the best relationship. You may consider trying it in your analysis. The following are a series of graphs that compare the Extended MEI data to arbitrarily scaled (scaling factor = 10) GISS Aerosol Optical Depth data as our volcano proxy.
Mount Pinatubo preceded the El Niño, but El Niño conditions had been reached earlier in the year:
El Chichon coincided with the El Niño:
Mount Agung coincided with the El Niño:
Novarupta doesn’t work (with all ENSO indices, BTW), but that eruption occurred at high latitudes:
The El Niño conditions preceded Santa Maria (with all ENSO indices), but the early start of that El Niño looks awkward:
There were minor El Niño conditions lagging Krakatoa:
I seem to recall reading papers that suggested that a volcanic eruption could initiate the downwelling Kelvin wave that starts an El Niño, and that seems to make sense.
In your post, you put the early portion of the SOI in a good light. However, the Tahiti portion of the SOI data is also questionable before 1935.
Before Trenberth’s alarmists days and before he was off looking for missing heat, he was well known for his ENSO research. In his 1997 “The Definition of El Niño”…
..Trenberth writes:
“Various versions of the SOI exist although, in recent years, most deal only with atmospheric pressures and usually only those at Darwin and Tahiti. In using the SOI based on just two stations, it must be recognized that there are many small scale and high frequency phenomena in the atmosphere, such as the Madden-Julian Oscillation, that can influence the pressures at stations involved in forming the SOI, but which do not reflect the Southern Oscillation itself. Accordingly, the SOI should only be used when monthly means are appropriately smoothed (Trenberth 1984, Trenberth and Hoar 1996a). For many years, Tahiti data were available only after 1935. Ropelewski and Jones (1987) outline an extension of the SOI prior to then using newly discovered Tahiti data, and they also discuss different ways of standardizing the data for use in the SOI. However, there are questions about the integrity of the Tahiti data prior to 1935 (Trenberth and Hoar 1996a), as the Tahiti-Darwin correlation is much lower in spite of strong evidence that the SO was present from other stations, and the noise level and variance in the early Tahiti data is higher than in the more recent period.”
Also, Willis, I was confused by something in your post. You wrote, “When the reduction in sunlight occurs following an eruption, the Pacific starts storing up more energy.”
Because the El Ninos are occurring at the same times as the volcanos, are you saying that an El Niño is causing the Pacific to store heat? The opposite is actually occurring.

Ray C

“And how does it do that? One major way is by changing the onset time of the tropical clouds. In the morning the tropics is clear, with clouds forming just before noon. But when it is cool, the clouds don’t form until later. ”
Vegetation produces green leaf volatiles, GLVs, adding volatile organic compounds, VOCs, which form secondary, gas to particle, aerosols and so c.c.n. , cloud condensation nuclei.
The rate of this production is, I suspect, linked to the degree of ‘stress’ the biosphere is under at any given time. More heat, greater soil moisture deficit, more VOCs produces and equates to more cooling cloud formation. Conversely Cool conditions, less stress, less VOCs, less clouds formed.
“A recent estimate of global SOA , secondary organic aerosol, production based on VOC fluxes indicated that there may be significant missing SOA precursors that are currently unknown (Goldstein and Galbally, 2007). The results obtained in this study indicate that GLVs may be an important part of this unidentified global source of SOA, which have been overlooked as a consequence of their volatile first generation oxidation products.”
Is this an example of the climate system actively responding to reductions in the incoming sunlight by altering cloud amounts via an overlooked mechanism of aerosol production and cloud formation by plants.

I agree with you that B. is more likely. Try using a negative CO2 sensitivity and see what the models “predict”.

Greg Goodman

Now to reiterate a point I made on one of your other threads, where does this energy go once the governor has evacuated it from the tropical SST?
Firstly it goes to troposphere where is creates a slight warm spot ( but not the predicited hot-spot since they amplified the GHG with water vapour, but there is some small warmth in TMT ).
Then a large amount will end up radiated to space and the rest goes polewards. That atmospheric transport is fairly classic meteorology.
Now what happens at the pole?
So as not to deviate discussion from Willis’ important article, I’ll leave that for another day.
To be continued ….

Ian H says: May 25, 2013 at 5:23 am
“This equation is the defining equation for sensitivity so it is canonical.”

But it isn’t. If you look up the definitions for ECS and TCR, they are specific to notional experiments. ECS is the eventual equilibrium response to a prescribed step change in F, uniformly sustained. TCR postulates a 1% rise over 70 years (doubling). Neither of these implies this “canonical” equation. Or even linearity.
But the real point is, the sensitivities are after the fact summary statements descriptive of the system. No-one claims that their supposed constancy (if it is supposed) can be used as a general purpose predictor of temperature. That was the “canonical” equation cited. If it’s used that way, surely someone could cite someone doing it.

The Year without a summer was 1816 after Mt Tambora erupted – nothing to do with Krakatoa.


When you work in the real world with controlled processes and the PID controllers which keep the processes in control, you see strong evidence of bounce, lag and settling time which you refer to. Once the process is initiated, and the controller brought online to guide the process, there is always a deviation away from the control point in the opposite direction of the deviation which triggers a response from the controller. Dozens of companies make these PID controllers, there are even tunable control blocks programmed into commonly used PLC machine controls which take the place of the stand alone controllers. provides a very good overview on tuning PID controllers, and the discussion points out the effects of changes in tuning parameters on rise time (time from induced perturbation to hit 90% of steady state operating point) Overshoot (how much the tracked parameter deviates beyond the steady state operating point before returning to that point) and Settling Time, which is how long it takes for the process to regain stability after being perturbed.
One thing that practice in tuning these controllers teaches you is that the larger the quantity of the controlled substance is, the slower the responses to perturbation will be and the more stable the system itself will be. IE, a 10000 gallon aquarium at the local zoo will vary in temperature much less and a substantial change in its’ temperature will occur much more slowly than that of the 10 gallon aquarium sitting on it’s stand in your home.
It’s amazing to me how much stability there is in the climate of the earth, and how well the mechanisms work to perform, without human interference, the job that these PID controllers do under the direction of those tasked with keeping industrial processes (or even the temp in your home) under control, and how the climate does not oscillate totally out of control, producing either iceberg earth or barbecue earth.

Greg Goodman

Another thing your stacker plots show is that , on average, major eruptions happen at times when temperature have already been cooling for several years. This means that a part of the post eruption cooling would probably have happened anyway.
Those seeing cooling usually forget to account for that and incorrectly attribute it to the volacano.
This particularly true for Mt Pinatubo which in fact created no net cooling.

Greg Goodman

Forrest Mims III falls into this trap in his comments on your earlier thread.
His carefully monitored datasets could be useful here but he has not published yet so all we get are a few pics and his comments absent of even his own analysis.


Far be it from me to nit-pick over an article that Willis has put so much thought and time and intelligence into. I do have a few questions and observations, and maybe they will turn out to be irrelevant.
First, it would make sense to me that the chemical composition of the materials ejected from a volcano and thrown into the atmosphere would affect how the eruption affects the atmosphere and its ability to capture solar radiation. How much does that vary from volcano to volcano? Clearly some stuff that goes up will come down quickly, and some not so quickly, and chemical composition will be a relevant factor in identifying the difference.
A companion question is that of how high the ejected materials go. For major volcanic eruptions, this may be relatively constant, but I don’t know.
A second companion question is how much the latitude of a volcano matters in the apparent global distribution of its ejected gases and ash.
All that said, I find Willis’s explanation to be compelling, and the answers to these questions would only serve to refine the hypothesis, and maybe explain some of the differences between the selected volcanic eruptions.

David L. Hagen

Thanks for thought provoking comparisons. On first impressions, it clearly shows major physics missing in current global climate models. You provide a hypothesis. Now to see how well it holds up or if others improve on or complement it to further the science.
Re “but I’m biased in the matter.” No need to berate yourself. Put your best argument forward.

Greg Goodman

“Since I’m looking for the common response, and digging to find the signal, I will leave out El Chichón as an outlier.”
I think you need to remove that idea. It is not necessary, does not change the lack of cooling and is a clear case of selection bias.
Even with the full set of data there is a downward trend before the eruption and at the end : no downward trend and no net cooling. That sounds to me like volcanoes cause at net positive effect if anything.
In reality I think that net volcanic effect is indistinguishable from zero , the rest is natural variations that , on average, happen to be tending downwards when major eruptions happen.
Whether that is simple coincidence, in view of the very small sample of events , or indicates some correlation between cooling phases and the cause that triggers eruptions will likely remain beyond our understanding for a long time yet.
Anyway, I think if you correct some theoretic inaccuracies you have made a good summary of the lack of cooling that is a fundamental inaccuracy of current modelled behaviour.
Like I’ve been saying for a long time the incorrect model response to volcanism is pillar on which exaggerated CO2 forcing stands. Long over due that someone kicks it away. Congratulations on putting this together.

Greg Goodman

Here’s another one showing that cooling often precedes these events rather than being caused by them. Note, I’ve flipped stratosphere temps which works the other way.

Not only do the actual temperatures return to the pre-eruption value, but they rise above it before finally returning to the that temperature. This is the expected response from a governed, lagged system.
We see this for example in the autopilot in boats. When the gain is turned up to quickly return to course, there is always an overshoot before the original course is resumed.
The climate models are behaving like an autopilot with the gain turned right down, so that if a wave (forcing) pushes the boat off course it will take a very long time to return to course. If a second wave hits before the boat is back on course, the boat goes even further off course. Thus, the climate models predict catastrophic warming – the boat will go dangerously off course in response to forcings.
However, in a boat where the gain is turned up, as we are seeing in the volcano response, then the boat quickly returns to course before a second wave can hit, and maintains a relatively stable course even in very large waves. There is no catastrophe if the gain on the course correction is high.
Anyone that has sailed small boats offshore in waves will be familiar with how this works. Overshoot is a result of the gain in the course correction mechanism. What Willis has shown is that there is evidence for a high level of gain in the temperature course correction in climate. The temperature response to volcanoes shows that the climate models are underestimating this gain.
I think Willis has hit the nail on the head. The volcano data shows that climate (temperature) does not act like a linear combination of the forcings. Rather it is more like an autopilot or pendulum. When a forcing pushes the temperature in one direction, the climate system works to oppose the forcing.