Guest Post by Willis Eschenbach
I see that Susan Solomon and her climate police have rounded up the usual suspects, which in this case are volcanic eruptions, in their desperation to explain the so-called “pause” in global warming that’s stretching towards two decades now. Their problem is that for a long while the climate alarmists have been shouting about about TWO DEGREES! PREPARE FOR TWO DEGREES OF DOOM BY 2100!! But to warm two degrees by 2100, you have to warm at 0.2°C per decade, or around 0.4°C during “the pause” … so they are now left trying to explain a missing warming that’s two-thirds of the 20th century warming of 0.6°C. One hates to confess to schadenfreude, but I’m sworn to honesty in these pages …
In any case, I got to thinking about their explanation that it wuz the volcanoes what done it, guv’nor, honest it wuz, and I did something I’d never thought to do. I calculated how much actual loss of solar energy occurs when there is a volcanic eruption. I did this by using the Mauna Loa atmospheric transmission data. These observations record what percentage of the solar energy is being absorbed by the atmosphere above the observatory. I multiplied this absorption percentage by the 24/7 average amount of solar energy (after albedo) which strikes Mauna Loa, which turns out to be 287 W/m2. (As you’d expect from their tropical location, this is larger than the global average of 240 W/m2 of sunlight after albedo). Figure 1 shows that result, which was a surprise to me:
Figure 1. Amount of solar energy absorbed by the atmosphere above Mauna Loa, Hawaii. Data Source
Now, before I discuss the surprising aspects of this graph, let me note that the Mauna Loa data very sensitively measures the effect of volcanic eruptions. Even small volcanoes show up in the record, and the big volcanoes are clearly visible. Given that … is there anyone out there foolish enough to buy the Susan Solomon explanation that the cause of the pause can be found in the volcanoes? I guess there must be people like that, the claim has been uncritically accepted in far too many circles, but really … who ya gonna trust? Susan Solomon, or your own lying eyes?
I’ll return to the question of the pause, but first let me talk of surprises. The thing that was surprising to me in this was the size of the loss of solar energy. The El Chichón and Pinatubo eruptions reduced the downwelling solar energy by maxima of forty and thirty watts per square metre at Mauna Loa. This is a huge reduction, much more than I would have guessed.
One measure of how much energy is lost is the total loss until such time as the absorption returns to its pre-eruption value. It turns out that in the case of both El Chichon and Pinatubo, the net loss of solar energy was about 450 watt-months per square metre. The loss was spread more widely (5 years) in the case of El Chichon than in the case of Pinatubo (3 years) before it returned to normal.
This means that for the period 1982-1987, Mauna Loa was running at 450 W-months/m2 divided by 60 months equals an average deficit of no less than 7.5 W/m2 of incoming energy over the five-year period … and it’s worse for Pinatubo, since that involved the same total energy but only lasted for three years. So for the three years from 1991-1994, Mauna Loa was running at a whacking great average solar energy deficit of 14 W/m2 …
Now, how much difference did this surprisingly large lack of incoming energy make? According to the IPCC, climate sensitivity is 3° per doubling of CO2, and a doubling of CO2 is a forcing increase of 3.7 W/m2 … and Mauna Loa was running at 14W/m2 shy of normal, that’s almost four doublings of CO2. So according to the IPCC, that kind of a decrease in forcing should have lead to a temperature drop of 11°C … so what actually happened?
Well, we’re in fantastic luck, because the temperature records at Mauna Loa are very good. Here’s what they say (study here):
Figure 2. Mauna Loa temperatures. Vertical red lines show the dates of the El Chichon (March 1982) and Pinatubo (June 1991) Graph from B. D. Malamud et al.: Temperature trends at the Mauna Loa observatory, Hawaii.
As you can see, despite the large decrease in incoming sunshine, there is absolutely no visible change in either the noon or the midnight temperatures … go figure. What happened from the volcano is nothing at all. No effect.
Now, y’all may recall that I have argued over and over against the concept of climate sensitivity. This is the widely-accepted hypothesis that the changes in temperature are determined by the changes in forcing. I’m a climate heretic—I don’t think climate works that way at all.
In particular, despite widespread skepticism, I have persisted in saying that volcanoes basically don’t do jack in the way of affecting the global temperature. I can finally demonstrate that unequivocally because I’ve stumbled across a very well-documented and precisely measured natural experiment.
At Mauna Loa we have a clear example of a measured decrease of 7 W/m2 in the average incoming solar energy for five years (1982-1987), and a decrease of 14 W/m2 for 3 years (1991-1994) … and there is absolutely no sign of either forcing decrease in the temperature record of the very place where the solar decrease was measured.
As I’ve said over and over, the emergent phenomena of the climate system respond instantly (hours or days, not months or years) to any change in the temperature. If it cools, we rapidly get a drop in albedo, which allows in more sun, and the balance is restored. If it warms, very soon thereafter albedo increases, we get less sun, and again the balance is restored. So while I was surprised by the size of the drop in downwelling solar energy, I was not surprised that we can’t find the signal of the solar drop in the temperature records.
Setting that question aside, let me return to the “pause”. Solomon et al. used the Vernier aerosol optical depth (AOD) dataset, which is available here. It is a calculated global dataset based on various observations. The explanation of the calculations is here. If anything, there is less recent variation in that dataset than in the Mauna Loa dataset. Figure 3 compares the two over the period of the satellite temperature observations.
Figure 3. Compares the negative of the aerosol optical depth with the Mauna Loa transmissivity data. Mauna Loa data rescaled to match AOD data for comparison purposes only.
So it doesn’t much matter which one we use to compare to the temperature data. Let me use the Mauna Loa transmissivity data, since the native units are in the same range as the temperature anomaly. Figure 4 shows the comparison of the Mauna Loa transmission data with the UAH MSU satellite-based lower troposphere temperature data:
Figure 4. Satellite lower tropospheric temperatures (blue) and Mauna Loa solar transmission (black line). Note that while Pinatubo happened at the start of a temperature drop, El Chichon happened at the start of a temperature rise. In addition, in neither case are the rise or the drop notable—the drop 1988-1989 or 2007-2009 is indistinguishable from the post-Pinatubo drop.
Finally, lest some folks claim that because Mauna Loa is in the northern hemisphere we can’t compare it to the global temperature changes, Figure 5 shows the comparison of the Mauna Loa with the northern hemisphere temperatures:
Like I said … I know there must be folks out there that can be convinced that the changes in the black line, the known effects of the volcanoes, are the reason that there is a “pause” in the global temperatures … I’m not one of them.
CONCLUSIONS:
• I may never find better evidence of the lack of connection between changes in forcing and changes in temperature than the measured large drop in solar forcing and the total lack of corresponding temperature change at Mauna Loa. It is a superb natural experiment, and has been very precisely measured for over half a century. It provides strong evidence in favor of my hypothesis that the temperature is controlled by emergent phenomena, and has very little to do with forcing.
• The change in forcing from the 21st century volcanoes is trivially small in both the Vernier AOD dataset and the Mauna Loa dataset. It is far too small to have the effect that they are claiming. I don’t care what the climate models told Solomon et al., the post-2000 changes in volcanic forcing are meaningless.
• My oft-repeated claims about the lack of effect of volcanoes on the global temperature are completely borne out by these results.
My regards to all,
w.
AS ALWAYS: If you disagree with me or anyone, please quote the words you disagree with. That way we can all know exactly what it is you have a problem with. Vague handwaving claims go nowhere.
MAUNA LOA TRANSMISSION DATA: From their website
The “apparent” transmission, or transmission ratio (Ellis & Pueschel, Science, 1971), is derived from broadband (0.3 to 2.8um) direct solar irradiance observations at the Mauna Loa Observatory (19.533 ° N, 155.578 ° W, elev. 3.4 km) in Hawaii. Data are for clear-sky mornings between solar elevations of 11.3 and 30 degrees.
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I enjoy Willis’s ability to take existing data( no matter how skewed it might be) and still be able to make a point. He has a patience with the human race that I don’t. I hope his heart condition is better now and he is getting enough clips over the head from his family to keep him mellow.
No, no, no. Willis you are giving the planet 24 hour sunshine. This is NOT reality.
Reality has a night time with heat loss and zero heat gain. You TOA radiation can ONLY be spread over a HEMISPHERE giving an average of one half the total. This is where the IPCC got it wrong, and introduced the GHG theory to make up the enormous difference and makes the GHG effect HOTTER than the sun.
As I explained, but you obviously did not read, your 240W/m2 does not give enough heat to run the water cycle because you need liquid water for this.
My quoted figures for surface energy is VERIFIABLE by measurement. Or do you prefer to massage figures to fit your crap theory.
Again GET REAL.
Willis Eschenbach says, February 25, 2014 at 4:16 pm:
“But that doesn’t mean that the El Nino pump somehow explains the variations of the planetary temperature. The El Nino pump RESPONDS to high temperatures, it doesn’t cause them.”
I’m talking about the ENSO process. The ENSO process involves the entire tropical/subtropical Pacific basin.
What drives the ENSO process in either direction (towards cooling or towards warming) over multidecadal periods is inconsequential for this discussion.
There is no question that global temperatures are held on a tight leash by the ENSO process, the administrator of our planet’s largest energy reservoir by far. Since 1970, global temperatures have only parted permanently with the NINO3.4 SSTa on three (3) occasions:
http://i1172.photobucket.com/albums/r565/Keyell/GWexplained_zps566ab681.png
All of these sudden global upward shifts (first pointed out, described and explained by Bob Tisdale, as you most likely know) occurred at specific points where the pan-Pacific climate regime underwent abrupt and fundamental changes. The first one happened in the direct aftermath of the Great Pacific Climate Shift of 1976/77. It came about when the SO all of a sudden decided to plunge to a new mean level and stay there for the next 30 odd years. This saw a relative warming (OHC and SST) of the East Pacific and a relative cooling (OHC) of the West Pacific. (The SST of the West Pacific stayed flat.)
The next two shifts were clearly of West Pacific origin. They are not seen in the East Pacific. Bob Tisdale has thoroughly and meticulously explained how exactly these shifts came about.
It is also fascinating to see how the AMO, or rather the SSTa of the North Atlantic, is directly affected by the ENSO process through so-called atmospheric bridges – it varies with NINO3.4 (East Pacific) during the major El Niños, but is affected by the massive surface heat accumulation in the West Pacific directly on the heels of these specific El Niños, so that the following Atlantic La Niña response is significantly reduced.
http://i1172.photobucket.com/albums/r565/Keyell/PacificvsAMO_zpsa3a19420.png
To accompany my last posting:
http://i1172.photobucket.com/albums/r565/Keyell/WarmhorseshoeB_zps08d09467.png
http://i1172.photobucket.com/albums/r565/Keyell/N-AtlvsNINO_zpsbb990151.png
http://i1172.photobucket.com/albums/r565/Keyell/Trinn_zps234f2e4e.png
I would think that the tropics would stay relatively warm, the mid latitude temps would drop most. This would increase easterlies and therefore drive toward la nina conditions. Winds in the region would probably also mute the cloud/temp resonse in the region some, since they increase evaporation and the aerolos will prevent the vapor from reaching the upper atmosphere. This would be a neutral/positive feedback for the region (cooling the ocean/surface despite the low heat input), and a negative feedback for the world, preventing transfer of heat to the poles to be radiated away.
Rolf, the greenhouse effect dimishes as concentration rise, so each doubling give the same 3.7W/m^2.
Could I impose upon you to explain the meaning of “linear” in this context and what the implications of that feature are?
I mean specifically that R and F only occur as first powers in the set of equations. The cross terms provide for an effectively nonlinear coupling, but the individual equations are (holding e.g. F constant and looking at the DE for R) 1st order linear equations.
Without the product in the cross term, related coupled first order linear ODEs can be reduced to second order linear ODEs — for example the pair of linear 1st order equations that reduce to the linear 2nd order harmonic oscillator equation. With such a cross term, one cannot really reduce the pair to a single second order equation at all, but if one did it would probably be nonlinear, I agree. In any event, the linearity or nonlinearity per se are at issue only in the sense that linear systems — if I recall correctly — typically do not have chaotic solutions, and the Lotke-Volterra equation has true Poincare cycles (indeed, trivial ones) rather than chaotic trajectories with divergences in phase space for all positive values of the parameters. This is characteristic of a wide class of mechanical systems to which Liouville’s theorem can be applied, but there are always ODE systems that are formally very “close” to linear systems with (eventually) closed trajectories that will exhibit chaos — there are a number of examples of oscillators that in a limit are classical closed systems, and for many values of the parameters are just such a system, but for certain values of the parameters or initial conditions become chaotic. My students often play with some of the systems — “Bender Bouncers”, double penduli, harmonically driven damped mass-on-a-rod penduli — numerically. It is great fun.
Mind you, I am far from expert in nonlinear ODEs and chaos. I know a bunch of stuff about it, and have played with at least some such equations extensively, but this is really difficult stuff and mathematicians spend careers working on it now — it isn’t like the linear stuff that was largely worked out in the 19th and early 20th centuries.
rgb
Rolf says:
February 25, 2014 at 11:29 pm
Grab a cup of coffee. I was discussing four doublings of CO2, not four doublings of the forcing. CO2 acts logarithmically. It increases the forcing by 3.7 watts per doubling of CO2 … but the 3.7 W/m2 doesn’t double, the amount of CO2 (the atmospheric concentration) is what doubles.
w.
johnmarshall says:
February 26, 2014 at 3:13 am
First, let me invite you stuff your GET REAL up your fundamental orifice.
Second, since you have not provided a quotation of my words I don’t have a clue what you are getting all nasty and snarky and babbling on about. 24 hour sunshine? Say what?
Get a grip, grab a handful of politeness, quote my words and try again. But lose the attitude, it’s unbecoming in a person such as yourself, one so clueless as to try to attack me without quoting me …
w.
@Steve Fitzpatrick-I think most of the discrepancy is more due to there being a big difference between the local scale figures, and the global scale ones.
Which actually implies that there is a significant spatial gradient to the volcanic forcing. It would be a bad mistake to view such a case as analogous to the effect of a uniform forcing like CO2…
Good sensible stuff again. Your look at the unexpectedly minor temperature response to a huge reduction in surface insolation associated with volcanic activity over several years is a “QED” of the thermostat hypothesis. However I think it more compelling if it is possible to show from Ceres data the evidence of corresponding albedo reduction in the ITCZ caused by reduction (probably disappearance) of clouds at those precise times. It would also make you the best forecaster on the planet for the weather in the tropical zone looking forward 3yrs after a major volcanic eruption! What about those weather records during those years?
Gary Pearse says:
February 26, 2014 at 10:49 am
Sadly, CERES data only goes back to 2000 … however, I showed the changes in NH albedo following Pinatubo above, viz:
w.
Hi Willis,
Great post, as usual. And, I totally agree with your emergent phenomenon theory. It’s one of those “I thought everyone already knew this” kind of a thing… especially in climate study.
I spent much of my formative years in the Denver area, and, much like the tropics (but to a lesser degree), we experienced the exact same kind of temperature regulating phenomena as you describe in the tropics. The 2pm thunderstorm is a regular occurrence on the front range (everyone that lives there knows what I’m talking about)… but only if it’s hot enough (and humid enough, but that’s a separate thing).
That said, I would like a clarification on what “johnmarshall” is asking about… albeit I will try to ask much less rudely.
I think what he’s not getting, and frankly neither am I, is why in your calculation you divide the incoming radiation (minus albedo, etc.) by the entire surface area of the planet, rather than just the surface area of the exposed half of the sphere.
I will grant you, that it probably wouldn’t matter in terms of what you were looking for in this post. But, in general, shouldn’t we be looking at an average of 480 W/m^2 instead of 240 W/m^2 for the irradiated half of the planet? So, I would have thought that it should be…
1360 pi r2 / (4 pi r2) / 2 = 2720 / 4 ≈ 680 W/m2 average solar radiation.
I’m sure there’s some simple reason why it’s not… but I will be darned if I can see what it is? Am I missing something here?
Thanks. And thanks again for your diligence, and your patience.
rgb,
if your rabbits get out of kilter and reproduce to the level that they over exploit their food source, to the level where it cannot recover in the presence of rabbits, then you may lose both rabbits and foxes. Both gone and even then the food source may not recover as it might have been replaced by hardier and less succulent plants.
Not sure what this has got to do with climate except in relation to unstable equilibrium
If you divide the insolation by 4 you are spreading it over the entire planet’s surface. This is NOT reality. Reality has a 12 hour period that receives zero energy but still radiates. The model has to relate to reality or you get the wrong answer, as you have.
That 240W/m2 does not come from the surface. Using the adiabatic environmental lapse rate you can calculate that this radiation is from 5-6 Km ABOVE the surface, or the cloud tops which would be sensible given all the heat locked up in those clouds thanks to latent heat.
In the sun’s zenith position radiation, as measured, is around 960W/m2 which gives, using S/B formulae a possible temperature of 88C Temperatures have been measured near this on the surface in some deserts. Convection is the Primary Heat Remover from the surface not radiation. Mixed with latent heat explains why rainforests are cooler than deserts, the very opposite of that predicted by the GHG effect.
timetochooseagain says:
February 26, 2014 at 9:24 am
“@Steve Fitzpatrick-I think most of the discrepancy is more due to there being a big difference between the local scale figures, and the global scale ones.”
There is some influence of local versus global, of course, since average solar intensity varies with latitude and season. But that does not explain almost an order of magnitude overestimate of the net volcanic forcing made here by Willis. The issue is mainly that AOD (Tau) is not a measure of light lost to space via scattering by aerosol particles, it is a measure of the total fraction of incident light scattered by those aerosol particles, in all directions. Most of the scattered light is scattered in forward directions and ends up being absorbed by the Earth as diffuse sunlight rather than direct sunlight, and is not lost to space. Only backward scattered light (which is a small fraction of the total scattered) is what is lost to space.
johnmarshall says:
February 27, 2014 at 3:02 am
John, you are correct that an average is not reality. Reality is reality. Averages are averages.
That said, averages can be useful. For example the 24/7 global TOA average sunlight, which is 340 W/m2. It’s both useful and accurate despite not being reality.
In particular, it allows us to do things like global energy budgets, or calculate how much energy is going into the ocean, that kind of thing.
It’s useful in part because it is accurate. By that I mean if I use say the 340 W/m2 global average insolation, I can multiply that by the time involved (say a month), and that will give an accurate answer in watt-months/m2. By accurate, I mean that if you measure the radiation for the month, that’s the number you’ll actually get.
So I fear that your argument, that it is NOT reality doesn’t matter—it’s both useful and accurate, meaning it matches observations.
w.
Steve Fitzpatrick says:
February 27, 2014 at 10:04 am
Thanks, Steve. Since I didn’t use AOD (aerosol optical depth) except for comparison purposes, but instead I used the Mauna Loa transmission data (percentage of sunlight making it through clear air), I’m not sure what your point is.
w.
rgbatduke: “[T]he Lotke-Volterra equation has true Poincare cycles (indeed, trivial ones) rather than chaotic trajectories with divergences in phase space for all positive values of the parameters.”
A belated thank you for your response. (I was traveling, and your response escaped my attention.)
I’m afraid I’ll go to my grave knowing nothing further about non-linear differential equations, so for the excerpt above was enlightening.
Willis,
“I’m not sure what your point is.”
My point is that your calculation of energy loss is wrong.
If you look at the NASA AOD historical curve (http://data.giss.nasa.gov/modelforce/strataer/) and compare it to the Mauna Loa direct transmission data (http://www.esrl.noaa.gov/gmd/grad/mloapt.htm) you can see that the two histories (where they overlap) are essentially the same data: the Mauna Loa transmission is close to (1 – AOD). Whether you consider AOD or consider direct solar transmission intensity, the calculation of energy lost to space is MUCH less than you calculated. The problem with your calculation is that you assumed a reduction in direct solar intensity means that reduction is 100% lost to space. It is not. Most of the intensity lost from the sun’s direct transmission is scattered in forward directions. That forward scattered light still reaches the Earth as diffuse light; it is not lost to space as your post suggests. Your error is not small, it is near an order of magnitude, because the vast majority of light scattering by volcanic aerosols is forward, not backward.
Willis,
That second address should have been: http://www.esrl.noaa.gov/gmd/grad/mloapt.html.
Dropped the last letter in the above comment, sorry.
Steve Fitzpatrick says:
February 28, 2014 at 2:50 pm
Thanks, Steve. Although that makes your claim clearer, it doesn’t provide any actual support for your claim.
We know for a fact that some of the energy is reflected, some of it is absorbed, some of it is scattered, and some is transmitted. I have assumed that the majority of what is not transmitted is either reflected or absorbed.
And indeed, the usual explanation of the effect of sulfates in the stratosphere is that they are white in color, and they cut down the sunlight.
There is another, more fundamental difference between the AOD and the MLO data. The AOD data is solely for the stratosphere. The MLO data includes part of the troposphere as well.
Now, I agree that I’ve not accounted for scattered light. You are 100% correct about that. So let’s assume that it’s 1/2 scattered forwards, 1/4 absorbed, and 1/4 reflected. That’s likely not too far from reality. In that case, instead of reductions of 7.5 watts per square metre on average (El Chichon) or 14 W/m2 (Pinatubo), it would be about half of that. That calculates to 3.7 W/m2 for El Chichon and 7 W/m2 for Pinatubo.
My point remains. The IPCC and the prevailing climate theory says that “climate sensitivity” is 1.5-4.5 W/m2 for a forcing increase of 3.7 W/m2 from a doubling of CO2.
To be conservative I’ll use 2° per doubling. Now that’s equilibrium sensitivity, and according to the models transient sensitivity is about three-quarters of that. So for a doubling of CO2 (3.7 W/m2) they say we should see cooling of 1.5°. This means for the El Chichon revised reduction in incoming solar (3.7 W/m2) we should see about 1.5° of cooling … and for the 7 W/m2 reduction from Pinatubo we should see about twice that, a 3°C drop … but we see nothing.
Despite whatever numbers you might put on it, or what interpretation you might give it, observations show that at Mauna Loa there was no visible effect on the temperature from the volcanic eruptions.
Best regards,
w.
Willis,
Thank you for your reply. You say “So let’s assume that it’s 1/2 scattered forwards, 1/4 absorbed, and 1/4 reflected.” Unfortunately, it is nothing like that. The longer lived volcanic aerosols are mainly sulfuric acid (liquid) droplets, and are mainly in the stratosphere. Tropospheric aerosols disappear pretty quickly. In the size range of ~0.5 to ~1 micron diameter, the total forward scattering by droplets is on the order of 100 times greater than backward scattering (Borhen and Huffman, ‘Absorbtion and Scattering of Light by Small Particles’, Wiley, 1998, page 115). Now, there are some solid volcanic particulates which absorb some light and cause stratospheric warming. These can be non-spherical in shape, and those same solid particles may reflect a little more light than can a liquid droplet, but the overwhelming majority of the measured reduction in solar intensity following a volcanic eruption is in fact forward scattered light. The NASA GISS estimate for net reduction in solar intensity (23 times the AOD, or about 3.7 watts/M^2 for a brief peak reduction after Pinatubo) seems perfectly reasonable. Your estimate of ~30 watts/M^2 reduction at Mauna Loa after Pinatubo is not at all reasonable.
We disagree about the influence of Pinatubo on surface temperatures, because, IIRC, you discount the influence of the El Nino warming which followed very shortly after Pinatubo. If that warming is taken into account (see my post: http://rankexploits.com/musings/wp-content/uploads/2013/06/Figure5.png, where ENSO is accounted for) the cooling influence of Pinatubo is clear.
Willis,
Your computed forcing from Pinatubo of 14 W/m^2 is not global. Granting this is caused by aerosols circulating in the stratosphere. The aerosols are carried by the jet stream passing over Hawaii and the Pacific ocean. The stream is about 800 km wide or covers 6% of earth’s area. Convert your 14 W/m^2 to global forcing = 0.878 W/m^2.
How much change in global temperature will that cause? Using the SB equation, a cooling of 0.17 C. How much global cooling was observed in 1991-92? Lo and behold exactly 0.17 C! Your computed forcing is correct, if you convert it to global.
Dr. Strangelove says:
March 1, 2014 at 6:19 am
Your computed forcing from Pinatubo of 14 W/m^2 is not global. Granting this is caused by aerosols circulating in the stratosphere. The aerosols are carried by the jet stream passing over Hawaii and the Pacific ocean. The stream is about 800 km wide or covers 6% of earth’s area. Convert your 14 W/m^2 to global forcing = 0.878 W/m^2.
Nice theory … but it runs aground on the reef of ugly facts. From the link I gave in the head post we have the distribution of the aerosols for the volcanoes:
Note that the distribution of the aerosols of recent volcanoes, particularly of Pinatubo, is global, extending all the way to the poles …
Doc, I put those links in my head post for a reason … can you guess what it might be?
Lo and behold, your calculation is nonsense, the forcing was global. Do your homework first, it avoids this kind of embarrassment.
w.