Guest Post by Willis Eschenbach
After I published my previous post, “An Observational Estimate of Climate Sensitivity“, a number of people objected that I was just looking at the average annual cycle. On a time scale of decades, they said, things are very different, and the climate sensitivity is much larger. So I decided to repeat my analysis without using the annual averages that I used in my last post. Figure 1 shows that result for the Northern Hemisphere (NH) and the Southern Hemisphere (SH):
Figure 1. Temperatures calculated using solely the variations in solar input (net solar energy after albedo reflections). The observations are so well matched by the calculations that you cannot see the lines showing the observations, because they are hidden by the lines showing the calculations. The two hemispheres have different time constants (tau) and climate sensitivities (lambda). For the NH, the time constant is 1.9 months, and the climate sensitivity is 0.30°C for a doubling of CO2. The corresponding figures for the SH are 2.4 months and 0.14°C for a doubling of CO2.
I did this using the same lagged model as in my previous post, but applied to the actual data rather than the averages. Please see that post and the associated spreadsheet for the calculation details. Now, there are a number of interesting things about this graph.
First, despite the nay-sayers, the climate sensitivities I used in my previous post do an excellent job of calculating the temperature changes over a decade and a half. Over the period of record the NH temperature rose by 0.4°C, and the model calculated that quite exactly. In the SH, there was almost no rise at all, and the model calculated that very accurately as well.
Second, the sun plus the albedo were all that were necessary to make these calculations. I did not use aerosols, volcanic forcing, methane, CO2, black carbon, aerosol indirect effect, land use, snow and ice albedo, or any of the other things that the modelers claim to rule the temperature. Sunlight and albedo seem to be necessary and sufficient variables to explain the temperature changes over that time period.
Third, the greenhouse gases are generally considered to be “well-mixed”, so a variety of explanations have been put forward to explain the differences in hemispherical temperature trends … when in fact, the albedo and the sun explain the different trends very well.
Fourth, there is no statistically significant trend in the residuals (calculated minus observations) for either the NH or the SH.
Fifth, I have been saying for many years now that the climate responds to disturbances and changes in the forcing by counteracting them. For example, I have held that the effect of volcanoes on the climate is wildly overestimated in the climate models, because the albedo changes to balance things back out.
We are fortunate in that this dataset encompasses one of the largest volcanic eruptions in modern times, that of Pinatubo … can you pick it out in the record shown in Figure 1? I can’t, and I say that the reason is that the clouds respond immediately to such a disturbance in a thermostatic fashion.
Sixth, if there were actually a longer time constant (tau), or a larger climate sensitivity (lambda) over decade-long periods, then it would show up in the NH residuals but not the SH residuals. This is because there is a trend in the NH and basically no trend in the SH. But the calculations using the given time constants and sensitivities were able to capture both hemispheres very accurately. The RMS error of the residuals is only a couple tenths of a degree.
OK, folks, there it is, tear it apart … but please remember that this is science, and that the game is to attack the science, not the person doing the science.
Also, note that it is meaningless to say my results are a “joke” or are “nonsense”. The results fit the observations extremely well. If you don’t like that, well, you need to find, identify, and point out the errors in my data, my logic, or my mathematics.
All the best,
w.
PS—I’ve been told many times, as though it settled the argument, that nobody has ever produced a model that explains the temperature rise without including anthropogenic contributions from CO2 and the like … well, the model above explains a 0.5°C/decade rise in the ’80s and ’90s, the very rise people are worried about, without any anthropogenic contribution at all.
[UPDATE: My thanks to Stephen Rasey who alertly noted below that my calculation of the trend was being thrown off slightly by end-point effects. I have corrected the graphic and related references to the trend. It makes no difference to the calculations or my conclusions. -w.]
[UPDATE: My thanks to Paul_K, who pointed out that my formula was slightly wrong. I was using
∆T(k) = λ ∆F(k)/τ + ∆T(k-1) * exp(-1 / τ)
∆T(k) = λ ∆F(k)(1 – exp(-1/ τ)) + ∆T(k-1) * exp(-1 / τ)
The result of the error is that I have underestimated the sensitivity slightly, while everything else remains the same. Instead of the sensitivities for the SH and the NH being 0.04°C per W/m2 and 0.08°C per W/m2 respectively, the correct sensitivities should have been 0.05°C per W/m2 and 0.10°C per W/m2.
-w.]
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The question will come up so I think it needs to be addressed. If the temperature is only influenced by albedo and TSI then does that mean there is no greenhouse effect? And, if there is, then why doesn’t its effect show up here.
My own view, given earlier, is that GHGs work to provide a thermostatic effect. Hence, once the temperature gets close to the “setting” then adding more GHGs will have no effect. As long as sufficient GHGs exist in the atmosphere the thermostat works fine and adding more GHGs won’t make any difference. This means that N&Z claims are not sufficient and that theory can be ignored. However, it does mean the planetary correlations they found may be meaningful since they are essentially based on the same factors as Willis has determined (since the gravity and mass of the atmosphere are more or less constant for the Earth at this time).
The bottom line is GHGs are required to raise the temperature of the Earth above 255K, the GHE is real but only half the story and, finally, adding more GHGs will have almost no impact on climate.
richardscourtney says:
June 1, 2012 at 2:51 am
—————————–
Apologies for missing your earlier post.
Eschenbach has produced a method that is incapable of finding the correct climate sensitivity. He and you appear to be incapable of realising that. He is not the first to fall into this trap: Schwartz (2007) created a method with similar failings. The comment by Foster et al (2007) on Schwartz (2007) (http://www.jamstec.go.jp/frsgc/research/d5/jdannan/comment_on_schwartz.pdf) is relevant to Eschenbach latest half-backed effort.
You seem allergic to the idea that climate is more complex than a one-box model. How long would I have to search before I could find a quote from you or your fellow lobbyists proclaiming that climate models are insufficiently complex? Do you not notice the contradiction here, or do you not care? To claim that a one -box model is sufficient, is to claim that either all components of the climate system (such as the subsurface ocean, and the atmosphere) warm at the same rate, or that the more slowly changing parts of the system cannot subsequently affect the other parts. Which of these two absurd positions do you hold?
Willis, I wrote a post on the proper way to calculate trends for seasonal (and anomalised data) here several years ago.
The trends calculated are unaffected by the starting and ending months of the data.
I found my calcs from 2005 that gave a climate sensitity to CO2 (doubling) of 0.3C.- as stated in my post on a related earlier thread.
But what if temperature primarily drives CO2, not the reverse, as I proposed in January 2008?
Conclusions:
CO2 drives temperature,
AND
Tail wags Dog.
News at 11.
JohanS says:
June 1, 2012 at 3:49 am
Another reference. I know I’ll meet a lot of resistance from laymen here including the author but there really is nothing new in this posting. Albedo is just another unit of measure for the earth’s equilibrium temperature and it should therefore come as no surprise that they should line up perfectly.
http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/grnhse.html
My emphasis:
An issue of major concern is the possible effect of the burning of fossil fuels and other contributers to the increase of carbon dioxide in the atmosphere. The action of carbon dioxide and other greenhouse gases in trapping infrared radiation is called the greenhouse effect. It may measurably increase the overall average temperature of the Earth, which could have disastrous consequences. Sometimes the effects of the greenhouse effect are stated in terms of the albedo of the Earth, the overall average reflection coefficient.
JohanS if you are able to point to observed metrics that the infra red radiation from the Earth is reduced by GHG you will be the first person to do so. However, there are observations that show that the outgoing long wave radiation appears not to be attenuated in this way.
The quotes you have provided are unsupported hypotheses unless they can provide observational evidence. . .
“If the temperature is only influenced by albedo and TSI”
I would say that temperature at the surface is set by surface atmospheric pressure and TSI but albedo changes the amount of solar energy (TSI) that gets into the oceans AND is involved in the rate at which energy flows from oceans to space.
Low albedo allows more energy into the oceans due to the widening of the equatorial air masses but also represents a faster hydrological cycle working to reduce or cancel out the effect of more energy into the oceans.
High albedo allows less energy into the oceans due to the narrowing of the equatorial air masses but also represents a slower hydrological cycle working to reduce or cancel the effect of less energy into the oceans.
Thus high sensitivity to albedo changes but low sensitivity overall because the albedo changes apply a negative system response to any other forcing effect.
Oceanic heat release from below and solar changes from above (other than raw TSI) both work to alter the width of the equatorial air masses in relation to the size of the polar air masses. Sometimes offsetting and sometimes supplementing one another in a constant dance around the equilibrium system energy content set by surface pressure and TSI.
When the polar air masses are ‘winning’ the troposphere is cooling and when the equatorial air masses are ‘winning’ the troposphere is warming.
Willis Eschenbach says:
I think you are misstating the objections to your original post. I don’t think I saw anyone complain that you were looking at the average annual cycle rather than the actual annual cycle over several years. Whether you look at the average annual cycle or the annual cycle over several years, you are still looking at a cycle of a particular frequency. I would in fact be shocked if your model fit the data for the average annual cycle but failed for the annual cycle when looked at over the whole data set…which is pretty much just a repetition of that cycle over and over again! I don’t see how that would be possible.
The one thing that is new here is the attempt to see how well you do with the linear trend in the data over time. However, there are things about that which are unclear to me: You claim to have fit the linear trend well but you don’t show in detail how well that is or how your fit to that trend changes as you change the time constant….Or, if you can use a longer time constant coupled with a greater sensitivity in order to do well with that trend. My point is that I don’t believe this trend over time is providing much of a constraint on your model. So, you really aren’t showing anything different from what you were showing us before.
As has been pointed out to you, there are ways you really could test your model, such as seeing if the model with the same time constant and lambda_0 can accurately represent the diurnal cycle…or, looking to see if your model with these fitting parameters can correctly diagnose the ACTUAL KNOWN sensitivity in a climate model.
Global warming theory depends on increased GHGs reducing Albedo (clouds and ice surfaces and vegetation).
Low cloud cover declines and high cloud cover increases which provides 1.0 W/m2/C of feedback (and results in 0.6C of the 3.0C per doubling expected). Ice and vegetation are lower and take longer to have an impact. Clouds might only take a day or two.
—–
Now the Earth’s Albedo certainly does change through time. How much does it increase in the ice ages for example. What about the Cretaceous when there is no ice and the ocean covers 35% of the continents. Snowball Earth conditions produce an Albedo of 50% versus 30% today.
Clouds are the big uncertainty. They are over 50% of the total Albedo at -53.0 W/m2. Doubled CO2 +3.7 W/m2. The cloud feedback signal can easily impact your doubled CO2 temperature impact by +/- 100%.
RomanM says:
June 1, 2012 at 5:45 am
Thanks for the info on the calculation method, not only was it informative in and of itself. The comments, particularly of Nick Stokes, show how hard some of the AGW tribe work to avoid correct analysis of the problem. Wow, the whole show in a microcosm.
Excellent paper, and thanks very much for it. One of Irving Copi’s elements of a successful hypothesis — the one too many climate scientists forget or ignore — is simplicity. All other things being equal (and that’s something you show pretty well), the simpler model, has to be the best. This approach looks like a bloody good bet to confusticate and bebother The Team. If albedo and insolation are good enough to explain observations on their own then, in logic, anything else should be considered superfluous … at least until somebody shows that some of the various forcings improve the model. Apparently, they don’t … at least in relatively recent times. I accept that the model’s not predictive, but surely insolation follows some fairly well-known trends. And I imagine future albedo can be predicted from trends as well … within limits: testable over the next decade or three, and subject to correction with real-time measurements as they come in. I suppose one could estimate albedo and insolation in past as well, though maybe without much precision. (Better than tree rings, anyhow.) This looks like a significant advance: something against which a lot of dodgy ‘research’ might be tested. One wonders why it hasn’t been done before (or, if it has, in which swamp the results were buried). Very well done, and I hope you’re able to whack a few thick skulls with it.
ice ages have usually been attributed to orbital mechanics, variations in the Earth’s tilt and orbital path. this would lead to a very precise cycle of glaciation and thawing. However, while cyclical, the timing is not that precise and it doesn’t align really well to the orbital mechanics. It would seem other items are actually the trigger for both glaciation and for melting. Albedo changes make for a tremendous opportunity to explain this as well as short term variations.
As stated above, bond albedo is the type of albedo we need to describe the amount of power being absorbed by Earth and its atmosphere, (1-a)*incoming solar. Bond albedo is all scattered/reflected energy coming from a body at all wavelengths of the em spectrum. Earth is around 0.3 venus is about 0.9, and the moon is about 0.123 (wikipedia bond albedo page has the references).
Venus is total cloud cover while the moon is essentially rock and dirt of the same materials as Earth. Earth is 70% liquid water on the surface and 62% cloud cover obscuring the surface leaving rock, ice/snow, dirt/sand, and vegetation to be a small fraction of the total area.
If Earth had 0.9 bond albedo like Venus but located on the surface as in a cloudless snow ball Earth scenario, Earth’s mean temperature would be down around 157K or 116 deg C below freezing. This is the Stefan Boltzman T for the emission of 34 W/m^2.
If Earth had an albedo of 0.123 like the Moon, its surface T average would have to be only about 269K, or -4 deg C were it not for atmospheric effects warming the surface. This is the Stefan Boltzman T for the emission of 299 W/m^2.
Earth’s actual albedo of 0.3 which, without an atmosphere, would require a surface radiating temperature of 255K, -18 deg C which is what is required to emit 239 W/m^2. Note that the average incoming power minus albedo reflected power is (1-0.3)*341 = 239 W/m^2.
Earth’s albedo is quite complex due to its unique make up. Ice (opaque) and fresh snow are quite high, possibly in the 0.6 to 0.8 range. Old snow is much lower and clear ice approaches the albedo of water which is under 0.04. Liquid water is under 0.04 for high angles of incidence for the incoming light energy. For low angles it increases dramatically but then there is not much power coming in per unit area at such low incidence angles. The remaining surface, under 30% is land and if it is rock, is likely to be similar to the lunar albedo, 0.12. Sand, especially wet sand, can be over 0.3 and vegetation also runs in the 0.1 to 0.2 range. However, at any one time around 62% of the Earth is covered in clouds and thick ones totally obscure surface albedo, substituting their own instead. Clouds range from around 0.4 to 0.9 albedo, depending on thickness and type of cloud. Note too that scattering in the atmosphere amounts to around a 0.04 albedo contribution as I recall.
This all adds up to our 0.3 albedo with the usual interglacial snow and ice contribution and land use changes being hardly worth mentioning as part of the whole. That 70% ocean coverage on the surface means that all land is a small part of the whole, especially considering that cloud cover of 60% amounts to the vast majority of our albedo. Snow and ice have little impact globally because it is located now at higher elevations where the incoming solar power per unit area is far less and where open water’s albedo starts to become significant. Getting rid of our clouds and atmospheric scattering (while keeping the ice and oceans) would result in about 0.09 albedo which is significantly less than the Moon’s 0.123 albedo.
All of this combined with Willis’ calculations points to a negative feedback or more aptly, a setpoint control system for Earth’s climate. Cloud cover (albedo) is the dominating factor and it adjusts by water cycle (evaporation, clouds, rain) for the incoming power and regulates the temperature rather well for changes in incoming power. However, during glaciation periods the feedback loop gets fouled up due to lots of extra fresh ice and snow which short circuits this cloud controlling feedback. Having lots of fresh ice and snow at lower lattitudes means that the albedo can stay very much the same with and without clouds present.
As the ice and snow ages and packs down the albedo drops. Also it gets dirty with soot from forest fires and volcanoes, and gets dirty from incoming cosmic debris and possibly windstorm born dust and dirt and sand until eventually one also starts to get melting pools which results in very low albedo of liquid water and the low albedo of clear ice, exacerbated by a lack of cloud cover due to the low temperatures. These are what eventually gets us out of the ice age glaciation period.
As for getting into one, there are always the possibility of a quiet sun, changes in extra solar cosmic rays that affect cloud cover and cloud albedo (due to cloud forming nuclei size variations affecting the reflectivity), orbital parameters, volcanic emissions, forest fires, and cosmic debris impacts and plain old serious weather pattern flukes that can combine into a perfect storm of albedo increase that drives us into starting a glaciation cycle.
richard telford:
Your post addressed to me at June 1, 2012 at 5:41 am does not address the points I have repeatedly put to you. Instead, it provides twaddle like this
NO!
I am allergic to unjustified stupidity.
Willis’ has demonstrated that a “one box” model works.
You claim a more complex model is needed.
It is your responsibility to justify your claim.
Your waffle, irrelevancies and insults suggest you cannot justify your claim.
Richard
JohanS:
re. your post addressed to me at June 1, 2012 at 5:20 am.
Either you have misunderstood Carl R. (Rod) Nave or he is wrong.
In either case I have no intention of contacting him.
Richard
joeldshore:
You really are good at ignoring everything which does not fit your prejudices. For example, at June 1, 2012 at 6:38 am you assert to Willis.
You know that is nonsense because it was repeatedly explained to you in the previous thread.
There is no “ACTUAL KNOWN sensitivity”.
Each climate model uses a different value of climate sensitivity.
Willis has assessed reality so a climate model can be tested against Willis result.
A climate model is a representation of an idea about reality so it tells NOTHING about the validity of Willis’ result.
Richard
The following is an attempt to find some common ground between Willis and Dr Telford, which might foster further discussion.
Dr Telford correctly notes that sceptical commentators have often criticised the climate models as being too simple to adequately model the extraordinarily complex system that is global climate. He seems disturbed that Willis has produced a model which is almost trivially simple when compared to the other climate models.
The root of the complexity is that a large number of factors contribute to our climate and their relationships frequently appear very complex.
If I understand Willis correctly, I believe he is taking a different approach. He postulates that a good approximation may be reached with a very small number of parameters; excluding many factors may as either
1. Sufficiently small to be ignored or
2. Bound to other (included) parameters, such that the included parameters might act as a proxy for the affect of the excluded parameters.
Because this is intended as a fresh approach, its inconsistency with earlier approaches does not inherently invalidate it.
If we wish to say that Willis’ simplification is invalid, we need to identify specific ways in which it fails, rather than just dismissing it because it is different.
If the above is characterisation is correct, may I encourage Dr Telford (and others who share his opinion) to identify specific flaws.
richardscourtney says:
June 1, 2012 at 8:19 am
————–
If Eschenbach had done his analysis on a week of hourly data from Epsom, he would have found a time constant of a few hours, with no evidence that longer time constants are required. Would you still claim that a one-box model was sufficient, even though the results of hourly, monthly and annual data all suggest different time constants?
I hope the advice you claim to give to MPs is of better quality than what you write here. Or does advising MPs amount to sending uninvited letters?
Willis Eschenbach says:
May 30, 2012 at 12:55 pm
[UPDATE—ERROR] I erroneously stated above that the climate sensitivity found in my analysis of the climate models was 0.3 for a doubling of CO2. In fact, that was the sensitivity in degrees per W/m2, which means that the sensitivity for a doubling of CO2 is 1.1°C. -w.
Willis, in the previous analysis you got 0.4 and 0.2 which averaged to 0.3 then subsequently posted the above correction are the results you post here actually ºC or ºC/W/m2, or is it just coincidence that the magnitudes are so similar? Thanks
Sunlight and albedo seem to be necessary and sufficient variables to explain the temperature changes over that time period.
And presumably, Svensmark can explain a good deal about the albedo.
BREAKING NEWS – CERN Experiment Confirms Cosmic Rays Influence Cloud Seeds
http://wattsupwiththat.com/2011/08/24/breaking-news-cern-experiment-confirms-cosmic-rays-influence-climate-change/
Climate sensitivity of 0.3° C/decade (Figure 1) for the NH. That’s 3°C/century for the NH. And 0.14°C/decade for the SH. That’s 1.4°C/century for the SH. Averaging it out at 1.4+3.0=4.4/2=2.2°C global temperature increase for a doubling of CO2. And that’s not catastrophic?
It means we will definitely skip the coming Ice Age and the so-called coming global cooling is out of the question if we continue to burn fossil fuels as usual.
Thanks for the info Mr. Eschenbach: I made a mistake in my comment in your previous post. I thought you meant a sensitivity of just 0.3°C for a complete CO2 doubling and not a trend per decade.
richard telford says:
June 1, 2012 at 2:18 am
Richard, I’m done with you. Your attitude is insufferable, as is your assumption that I am acting in bad faith. And referring to me as “Eschenbach”, as if I wasn’t here, is a calculated insult.
I have provided all of the data and calculations for you or anyone else to examine. If you want to do something with it, please do so, and report back. If you think that there are a range of time constants, that’s quite possible, I don’t deny that. So if that’s your claim, then get up off of your dead okole, buckle down, do the work, and demonstrate that to us. I have demonstrated that for the time periods in question (one month to fourteen years) a single time constant does a fine job of replicating both the larger trend in the NH and the smaller trend in the SH, as well as the monthly variations. Does this establish that my model is correct? No, it just shows that it does a good job.
Heck, I’m not claiming that my model is the be-all and end-all. For starters, I suspect that a “fat-tailed” distribution would do a better job than a standard exponential distribution. Or there may be some other model entirely that works better, perhaps your model, I don’t know. I just do what I can and put it out for comment.
But insulting me is not the way forwards.
Sadly,
w.
cba says:
June 1, 2012 at 7:20 am
ice ages have usually been attributed to orbital mechanics, variations in the Earth’s tilt and orbital path. this would lead to a very precise cycle of glaciation and thawing. However, while cyclical, the timing is not that precise and it doesn’t align really well to the orbital mechanics…..
___________________________________________________________
You missed the more recent developments that take care off that problem.
Luboš Motl has a good article with links to the 2006 paper In defense of Milankovitch by Gerard Roe
What you are referring to is shown in the graph of Ice Volume vs the June theoretically calculated insolation at 65 °N. This is the GRAPH. As Motl says “The graphs above are just unimpressive. A lag of 8,000 years has to be added by hand to make it at least remotely plausible. There’s no real agreement”
Gerard Roe realized a trivial mistake had been made.
So when the correct comparison is made, the rate of change, the derivative of the ice volume you get this GRAPH
So I am sorry but that dog don’t hunt no more.
JohanS says:
June 1, 2012 at 2:57 am
Cite?
While there is reflectivity or “albedo” in the longwave frequencies, it is typically quite small for solids and liquids, and even smaller for gases. Clouds and liquid water, for example, are essentially a blackbody at longwave frequencies, with very little being reflected. And for gaseous CO2 itself, the albedo is effective zero.
Several people have pointed out that the albedo contains the signals of e.g. the indirect aerosol effect [effect of aerosols on clouds], and the land-use change. They are correct.
But CO2 directly? Not happening.
Indirectly? Perhaps, through some theoretical CO2 –> temperature increase –> greater evaporation –> more clouds –> greater albedo … but there are a host of assumptions in that, and it doesn’t work on short time scales (e.g. decadal or shorter) because the CO2 changes are small in that length of time.
w.
cba says: “ice ages have usually been attributed to orbital mechanics, variations in the Earth’s tilt and orbital path. this would lead to a very precise cycle of glaciation and thawing. However, while cyclical, the timing is not that precise and it doesn’t align really well to the orbital mechanics.”
This is because the variations in summer insolation in the Northern Hemisphere correlate with the time derivative of the global ice volume, not the ice volume itself. It’s quite clear that the orbital theory is right:
http://earthweb.ess.washington.edu/roe/GerardWeb/Publications_files/Roe_Milankovitch_GRL06.pdf
JohanS says:
June 1, 2012 at 3:20 am
Thanks, Johan. I’m sorry, but that’s just the uncited, unsupported, and unsubstantiated opinion of one man, an associate professor at Georgia State. It is certainly not the definition used by most scientists.
For example, the following result on google says:
Notice that it says nothing about longwave radiation. This is the common definition of albedo, the one most people use.
More to the point, it is the definition of albedo used by the authors of the study from which I took the albedo measurements that I used. They are talking about albedo as shortwave reflection, not longwave. So no, I am not “rolling CO2 into my calculations” as you claim, because the figures I am using are only of shortwave albedo.
There’s an interesting discussion of albedo and long-wave radiation here. Generally, they are inversely related, but this is because of clouds affecting the longwave radiation, it’s not from CO2.
All the best,
w.
Also, note that it is meaningless to say my results are a “joke” or are “nonsense”. The results fit the observations extremely well. If you don’t like that, well, you need to find, identify, and point out the errors in my data, my logic, or my mathematics.
I agree. The test will be in the future, say perhaps 20 years from now. The record of F and T, hence their first differences, will be available. True “predicted” temperatures for the future years and months will be available, conditional on the observed changes in F, and the predicted temperatures will be tested against the observed temperatures.
What’s peculiar about the model is that the change in temperature for May (or any month) depends on the change in forcing for May, rather than on the mean or other integral of forcing for May or the preceding April. A change in forcing has about 90% of its total effect during the month of the change in forcing (the 90% approximation is from your spreadsheet, showing the decay of the effect in subsequent months.)
lambda and tau of your model are strictly monotonic functions of the parameters a and b of a linear vector autoregressive model (displayed in my posts on the earlier thread), and don’t have anything to do with lag or delay. Any pair of strictly monotonic functions would do as well.