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Comments 126 to 150 out of 230:

126. Falkenherz at 22:01 PM on 5 October, 2012
     Hi! Would you be so kind to include something on the following points: "To the extent that evaporation dominates over the surface-sensible heat flux, one can, in fact, argue that changes in the net radiation at the surface control the sensitivity of the global hydrologic cycle (the mean rate of precipitation or evaporation) rather than the sensitivity of surface temperatures." and "The level of temperature is dominated by factors in the stratosphere. We have very little water vapour in the stratosphere, so it in fact cannot be a dominant driver of global warming." Both are arguments from sceptics, but I did not find any hints here how to put them into context.
127. Bob Lacatena at 23:35 PM on 5 October, 2012
     Falkenherz, I'm having a hard time parsing either of your quotations. Can you direct me to an actual place where a skeptic makes either of these claims (more clearly)? Right now they both look (to me) like pure gobbledygook. It's hard to argue against complete nonsense. [Changes in the global hydrologic cycle? Temperatures is dominated by the stratosphere? Huh?]
128. Philippe Chantreau at 03:08 AM on 6 October, 2012
     Falkenherz, do not waste your time with arguments that rea not supported. This: "The level of temperature is dominated by factors in the stratosphere. We have very little water vapour in the stratosphere, so it in fact cannot be a dominant driver of global warming." Is nonsense. The first part is, and the second as well. There is enough water vapor in the stratosphere to make it a major factor in the stratospheric radiative balance, see Iacono and Clough (1995).
129. Falkenherz at 18:35 PM on 8 October, 2012
     I am sorry if that seems so much nonsense that it isn't even worth an explanation to you. I know that this article here starts with the assupmtion of water vapour being dominant, because a lot of sceptics say this. But I think you could discern and explain a bit more between the ratiative processes and the evaporation processes and the clouds caused by more water vapour, alll of which are very strong effects of water vapour. The sceptic I encountered refers to the radiative element, which is the only possible wv process for a positive forcing. My first quote is from Held 2000, p. 446 http://maths.ucd.ie/met/msc/ClimSyn/heldsode00.pdf who is trying to disprove this argument; sadly, I don't get it. The second quote is a translated argument I encountered from a sceptic who seems to know a lot about radiative physics, but then he suddenly comes with queer assumptions like this; and I found so far no source to directly disprove him with a scientific argument (Held 2000, p. 446, was a try, but I cannot quote things which I don't understand myself). As to temperature dominated by stratosphere, I believe this refers to the absorption length, which might not be long enough by just counting the troposphere. Thanks for the referral to Iacono and Clough, but it seem they do not adress this issue directly, but to follow up on it, as they are just "introducing an improved, LW radiative transfer model". I need an explanation why water vapour in the stratosphere is very much driving global temperature. With google I just found articles which report that a recent decrease of water vapour in the stratosphere was the cause of some cooling, but no explanation or experimental proof as to why.
130. Falkenherz at 18:37 PM on 8 October, 2012
     Ah, nevermind, I think I found a good source: http://www.sciencemag.org/content/327/5970/1219.full
131. AlanSE at 05:43 AM on 10 January, 2013
     Near the beginning of this article it gives "carbon dioxide contributes 32 W/m2" But that's not right, is it? I looked at the paper, and yes, the numbers are in there. But just a cursory look at what else is out there gives different information. http://scienceofdoom.com/2010/02/19/co2-an-insignificant-trace-gas-part-seven-the-boring-numbers/ This data is extremely straight forward and extremely well quantified, but they give 8 W/m^2, even at 1000 ppm of CO2. No where near 30 W/m^2. So what's going on? Are these two sources reporting figures that represent different things?
132. Bob Loblaw at 05:59 AM on 10 January, 2013
     AlanSE: Took a bit to find the 32 W/m^2 - it in the intermediate version of the rebuttal, which disappears when you go to the third page of comments. ...but to answer the final question you have: yes, they are different things. The 32 W/m^2 value refers to what is coming back down to the surface from the atmosphere. The other numbers (8 W/m^2 for 1000ppm CO2 - you'll also see other references that mention about 4 W/m^2 for doubled CO2, i.e. 600ppm) refer to the change in flux at the top of the atmosphere (i.e. the exchange with space). It is this latter value that creates the imbalance that leads to global warming - reduced losses to space require a warmer system to restore balance with absorbed solar. The failure to distinguish between surface fluxes and top-of-atmosphere fluxes is a common error.
133. KR at 05:59 AM on 10 January, 2013
     AlanSE - The 32 W/m^2 value is for the difference between no CO2 at all and current values. 8 W/m^2 is for two doublings of CO2, with a logarithmic effect, estimated at: ΔForcing = 5.35 W/m^2 (ln [CO2_new/CO2_previous]) or ~3.7 W/m^2 per doubling of CO2. So these are indeed two different numbers for two different situations. The ΔF of 0ppm -> 390ppm does not equal the ΔF of 250pp -> 1000ppm, in particular since at low concentrations of a greenhouse gas (near zero) it will have a linear rather than logarithmic effect.
     Response: [DB] Fixed link.
134. Tom Curtis at 07:13 AM on 10 January, 2013
     AlanSE @132, as noted by Bob and KR, the 32 W/m^2 is the total strength of the greehouse effect from CO2, whereas the 8 W/m^2 is the change in the total strength of the greenhouse effect as atmospheric concentration increases from 278 to 1000 ppmv. I notice in reading the graph from Science of Doom, you use the highest estimate of radiative forcing (IPCC 1990). Subsequent to the publication of the IPCC first and second reports, Myrhe showed that models where then overestimating the forcing from doubling of CO2. The correct value is 6.85 +/- 0.68 W/m^2 for an increase in CO2 from preindustrial levels to 1000 ppmv. Further, since the intermediate article above was last updated, Schmidt et al, (2010) (PDF) have come up with a more accurate estimate of the all sky CO2 contribution to the GHE of 29.5 W/m^2 for 1980 concentrations. That will have increased by 0.8 W/m^2 since then, to 30.3 W/m^2. A further increase of CO2 concentration to 1000 ppmv will increase the CO2 contribution by 5.0 W/m^2 relative to 2010 levels, taking the total CO2 contribution up to 35.3 W/m^2. It should be noted that most of the non-CO2 contribution to the total greenhouse effect comes from water and clouds, which would largely disappear from the atmosphere given a lack of CO2, and hence constitute a feedback rather than a forcing. I should also add that while the formula given by KR above is accurate for calculating change in forcing for levels of CO2 found in the atmosphere over the last 600 million years, but becomes inaccurate for very low levels of CO2 and so cannot be used to calculate the total contribution.
135. Tom Curtis at 07:29 AM on 10 January, 2013
     Bob Loblaw @132, the 32 W/m^2 is the CO2 contribution to the net difference between surface upward longwave radiation and top-of-atmosphere upward longwave radiation, and hence its contribution to the total greenhouse effect as measured at the TOA. As such, KR's account of the situation is accurate. I initially skimmed your post and did not pick up that your account differed from that of KR and myself. As previously noted, the estimate of Kiehl and Trenberth (1997)given in the intermediate article has since been superceded by the estimate given by Schmidt et al, (2010).
136. Bob Loblaw at 07:44 AM on 10 January, 2013
     Yes, KR's explanation is better. I was looking at the text in the rebuttal where it said "carbon dioxide contributes 32 W/m2 (Kiehl 1997). These proportions are confirmed by measurements of infrared radiation returning to the Earth's surface (Evans 2006).", and got messed up by the "earth's surface" part. Of course, with no atmosphere at all, there isn't any IR towards the earth's surface, but that still doesn't apply to an atmosphere with no CO2... which would have downward-direct IR at the surface. I was hasty, and messed up. The point that you need to be careful about surface vs. TOA fluxes is valid, but I'm not explaining the actual numbers that were asked about.
137. AlanSE at 05:50 AM on 19 January, 2013
     Thanks for your replies @KR, @Tom Curtis, @Bob Loblaw. I'm much further in my understanding. Here is my understanding: all the number we've talked about are TOA values. The graph I linked to (albeit outdated) gives the change in TOA from pre-industrial to future concentrations. This would be the change in the heat flux through the atmospheric window. We'll say that number is 7 W/m^2 for a change to 1000 ppm. Then the currently discussed 32 W/m^2 is the figure for radiative forcing from all CO2. Actually, it's a little more complicated since the absorption overlaps with other gases, and without the cross-correlation the paper gives 29 W/m^2. My understanding is that if you removed CO2 from the atmosphere entirely you would increase the flow through the infrared window by the 29 number. This now makes sense to me because the differential power of adding CO2 is much much greater at lower concentrations. The entire infrared window seems to be 40 W/m^2. So if industrial emissions decreases this by 7 W/m^2, we're left with 32 W/m^2. Now I have to revise my picture of the change in temperature to compensate. I previously considered an increase in surface temperature as compensating for the additional radiative forcing. Now it seems that most of this would come from a higher temperature of the atmosphere. If you look at this reference: http://www.sierrapotomac.org/W_Needham/TheGreenhouseEffect.htm You can see that longwave radiation from the atmosphere into space is 235 W/m^2, whereas the longwave directly from the surface is only that paltry 40. So as we increase CO2 the former number will increase much more. I'm still holding the assumption that the surface and atmosphere will increase in temperature by similar amounts from AGW. It's just that the additional radiation that compensate for it will mostly come from the atmosphere. Is there anything wrong with this picture? Of course, I haven't gotten into anything about feedback effects.
138. Tom Curtis at 11:31 AM on 19 January, 2013
     AlanSE @137: 1) Greenhouse effect: The explanation of the greenhouse effect at sierrapotomac.org is of poor quality, and will only confuse you if you are trying to understand it correctly. In particular, it describes the greenhouse effect as causing a greater absorption of heat. In fact, an increase in atmospheric CO2 would result in a greater increase in heat only while the Earth was not a radiative equilibrium. Once the Earth reached radiative equilibrium, there would be no further net gain in heat, but the greenhouse effect would still be enhanced relative to the condition with less CO2. It also says,"The heating of the earth due to the radiant heat of the sun is called the greenhouse effect", which is egregiously wrong. The radiant energy received from the Sun at the Earth's current albedo is 239 W/m^2. In the absence of a Greenhouse effect, the temperature of the Earth's surface would rise till it emitted 239 W/m^2, ie, approximately 255 degrees Kelvin (-18 C). A low IR emissivity would raise the temperature slightly; while a less than even temperature distribution across the surface would lower it. In practice, the second is the stronger effect so that the mean global surface temperature would be less than 255 K. As it happens, the Earth's global mean surface temperature is approximately 288 K (15 C). The higher temperature results in a much higher outward IR flux at the surface than the energy received from the Sun, and indeed, much higher than the outward flux at the top of the atmosphere. The difference between the outward flux at the surface and that at the top of the atmosphere is the atmospheric greenhouse effect. Trenberth et al (2010) give that difference as 156 W/m^2 (see diagram below), while Schmidt et al (2010) gives it as 155 W/m^2. It is very important that the IR radiation at the TOA is less than that at the surface only because greenhouse gases absorb outgoing IR radiation; and because those gases are cooler than the surface, so that when they emit IR radiation it has a reduced flux. It is also important to recognize that the TOA flux can only be smaller because energy absorbed by the atmosphere is also transferred to the surface. Without the energy transfer to the surface both the reduced flux at TOA, and the flux at the surface greater than solar radiation absorbed would violate conservation of energy. As it happens, the energy equations do balance (see diagram above). The energy transfer from atmosphere to surface is in the form of back radiation, but could be in another form and you would still have a greenhouse effect. 2) Atmospheric Window: You appear to be confused by the "atmospheric window". "Atmospheric windows" are frequencies within the electromagnetic spectrum in which there is almost no atmospheric absorption, so that radiation in the window can go directly from the surface to space (or vise versa). As can be seen in the modtran image below, there is an IR atmospheric window between wavenumbers 800 cm^-1 and 1000 cm^-1, and another smaller window around 1100 cm^-1. As it happens, about 40 W/m^2 escapes from the surface to space through these "atmospheric windows", but they should not be confused with the total radiation to space (236 W/m^2, most of which comes from the atmosphere) or with the atmospheric greenhouse effect. As noted in section one, the atmospheric greenhouse effect is the difference between the upward IR radiation from the surface and the upward IR radiation to space from the Top of the Atmosphere. Doubling CO2 concentration creates a radiative forcing of 3.7 W/m^2, ie, it reduces the IR radiation to space by 3.7 W/m^2. The reduced upward flux at the TOA creates an energy imbalance which warms the Earth until the imbalance ceases to exist. Ignoring all feed backs, that requires a warming of approx 1.1 C at the surface to accomplish; or in other words an increase in the upwards IR flux at the surface by about 6 W/m^2. Because equilibrium is reached (ignoring feedbacks), the upward IR flux at the TOA will have returned to 239 W/m^2, so a radiative forcing of 3.7 W/m^2 will have caused a total change in the net atmospheric greenhouse effect of 6 W/m^2. In practice, the net change will be approximately two to three times that amount, with much of the increase attributable to the water vapour feed back. This may seem confusing, but only if you mistake radiative forcing, ie, the net change in TOA radiative flux before temperature adjustments with the net atmospheric greenhouse effect, ie, the difference between surface and TOA upward IR flux. Unfortunately, that is a mistake I made in my previous post @134. I apologize for any confusion I have caused as a result. 3) Increased CO2: Increasing CO2 in the atmosphere widens the large valley in outgoing radiation between 600 and 700 cm^-1 (see modtran graph above). To maintain equilibrium, the total area of the graph, which represents the total upward flux, must remain constant. Because the area is reduced near 600 and 700 cm^-1, it must be increased elsewhere, including in the atmospheric windows. As the upward flux in the atmospheric windows comes from the surface, this means the surface temperature must increase. Consequently your assumption that most of the warming would occur in the atmosphere (ignoring feed backs) is mistaken.
139. Jose_X at 09:33 AM on 10 March, 2013

     Tom Curtis #138, I wanted to add (correct it if wrong) that the forcing value is a theoretical value. It's as if the entire change were done instantly. In reality, we experience CO2 increases, not from 1x (of some arbitrary value) to 2x overnight, but as small increments over time, so the imbalance is likely never to get very large at all. The 1x is a reference point and the 3.7 forcing value is a theoretical imbalance that would exist if the 2x happened right away ("with surface and tropospheric temperatures and state held fixed at the unperturbed values").

140. Tom Curtis at 10:23 AM on 10 March, 2013

     Jose_X @139, that is mostly correct. 

     As noted @138 (point 2) , the CO2 forcing is the total net change in TOA upward IR flux before (or excluding) temperature adjustments.  Clearly if we add the CO2 incrementally, there will be temperature adjustments so that the TOA energy imbalance will be significantly less than the CO2 forcing.  But, the forcing is the same at 2xCO2 regardless of whether it is added incrementally or as a single slug.

     That does not differ from your explanation except in terms of what is meant by "theoretical variable".  Roughly (as it is a while since I studied this), a theoretical variable in physics is a value which is an element of an empirical theory that is not directly measurable itself, but whose values have direct implications for measurable variables.  In this sense, forcing is "theoretical".  You appear to treat "theoretical" as a synonym for "hypothetical" which would be incorrect.   

141. Jose_X at 08:43 AM on 11 March, 2013

     Tom Curtis, thanks for the response, but I was thinking of "theoretical" just as you explained, meaning that if we could actually do 2x over night then we would observe the full forcing value.

142. Tom Curtis at 09:17 AM on 11 March, 2013

     Jose_X @141, fair enough.  In that case, however, you should have written "the 3.7 forcing value is a theoretical imbalance that would exist only be directly measurable if the 2x happened right away"  or possibly, "the TOA energy imbalance would only equal the forcing if the CO2 was doubled instantaneously".

     Even that is not perfectly accurate, in that short term fluctuations in temperature result in short term departures from radiative equilibrium, but they are small relative to the 3.7 W/m^2 forcing from doubling CO2.

143. Jose_X at 03:30 AM on 13 March, 2013

     Tom Curtis, thanks for the response, but I was thinking of "theoretical" just as you explained, meaning that if we could actually do 2x over night then we would observe the full forcing value.
     Tom Curtis #142. Rethinking the meaning of the words now after your 142 comment, I see that you were right and I did mean hypothetical. I didn't mean that the value couldn't exist under any circumstance, but rather that it could but would not unless the CO2 were eg done "over night". I concluded this from the statement by the IPCC (quoted on Wikipedia) that a reference value is used to define the forcing ("with surface and tropospheric temperatures and state held fixed at the unperturbed values") so that perhaps a perturbation "forcing" is not an entity (theoretical or measurable) that is manifested in reality if there isn't an imbalance at TOA of that magnitude at any given point in time.
     
     Let me ask something about the meaning of the word "forcing". Would "yes" be the answer to the question, "is the forcing at TOA from CO2 a value greater than 5 W/m^2?" [ie, is it clearly greater than the 2x forcing quoted by the IPCC because it would include contributions from all CO2 in the atmosphere today?]

144. KR at 03:55 AM on 13 March, 2013

     Jose_X - It's a bit difficult to understand what you are discussing here. Are you asking for the current sum value of CO2 forcing? According to NASA numbers CO2 represents ~20% of the current greenhouse effect, water vapor and clouds ~75%, and other gases (methane, CFCs, ozone, etc) another 5%. 

     33C of greenhouse effect, or 6.6C directly attributable to CO2 by those numbers, means about 22-23 or so W/m^2 of total CO2 forcing at this time. 

     On the other hand, "forcing" is generally referred to as the change from previous conditions, usually 1750AD, in which case as of 2005 CO2 accounts for ~1.66 (1.49-1.83) W/m² as a delta. Add in all the other factors +/-, and the total net anthropogenic forcing change since 1750 is about 1.6 (0.6-2.4) W/m².

     

     [Source]

     But again, I'm not entirely certain what you are asking about. 

145. Jose_X at 04:52 AM on 13 March, 2013

     I was not looking at "forcing" as something like a gravitational force value associated with a body on the earth (which tends to exist all the time) but more like (net) heat, a quantity defined as the summation of other (sign-dependent) quantities and which might frequently approach or be zero. I guess that was wrong.
     
     So, to confuse everyone still further, I guess forcing is somewhat analogous to pressure in that it tends to be nonzero and significant whether the system is approximately in equilibrium or not and under a context where "pressure" would refer to partial pressure of a species or, more specifically, to the additional contribution to partial pressure that would be provided as the species moves from one reference concentration to a different target point separated from the first by a stadardized multiple (of 2).

      

146. Jose_X at 08:56 AM on 13 March, 2013

     Are the following observations correct?
     
     And does someone have an answer for the two questions at point 3.
     
     1. The forcing value depends on the reference sun intensity. If the sun was providing approximately a stable 50 W/m^2 TOA, then the CO2 forcing calculations would yield different numbers.
     
     2. For the lambda (climate sensitivity proportionality coefficient) between forcing at TOA and (yearly global mean) temperature on the surface to be approximately constant, we should use a reference point for forcing such that the delta temperatures involved are at most a few K. [This criteria would include a "near" constant solar intensity requirement that has easily been met by sun+albedo across spans of a few centuries.] This would allow, for example, the relevant Stefan Boltzmann relationships to remain in an approximately linear region, eg, where each degree change in temp would correspond approximately to a constant change in W/m^2 (currently approaching 5.4 W/m^2 per C at ground level).

     Also, I was guessing that, although the lambda and forcing values might be determined from computer programs that may potentially take a long time to run in order to achieve a high level of accuracy, that using simple models (esp linear) afterward with those derived values allows researchers to ask and answer many questions involving lots of combinations of forcing agents and quantities without having to run involved simulations for every such combination of values. The simple models also help us understand the main factors in climate change and together with all models used help keep a sanity check on each other.
     
     3. Calculating the contributions from CO2 to the greenhouse effect (using data from KR's recent comment):
     
     287.5 - 255.5 = 33 (relevant temperature values to nearest 0.5 C)
     33 / 6.6 = 5 (aka, 20% contribution by CO2)
     387.38 - 241.63 = 145.75 (Stefan Boltzmann W/m^2 at surface corresponding to 33 C increase and e=1)
     145.75 / 5 = 29.15 (contribution by CO2 to total W/m^2 at surface)
     
     So there have been 29.15 W/m^2 of warming due to CO2 for its 6.6 C contribution to the total greenhouse effect of 33 C.
     
     Question 1: How do we calculate the total forcing by CO2 that led to this 29 W/mSq?
     
     Put differently, if we revert to an atmosphere with almost no ghg effect, meaning that the TOA and surface are roughly at 255K, how many times will we have to apply a forcing of 1 W/m^2 (TOA) from CO2 additions to get to our current atmosphere? Can someone roughly (including hand waving) show the steps in unit increments that achieve this and so that we can see how the ground temp or ground irradiance change at each step? [For the purposes of addressing this question, on may optionally assume H2O is liquid all throughout the 33 C range.]
     
     Question 2: Whatever approximation method is used to answer Question 1, will we get roughly the same answer if we go in 5 W/m^2 increments?
     
     My confusion is that forcings are defined at TOA yet TOA always reverts to about 240 W/m^2. And there isn't to me an obvious way to derive that total forcing value (eg, from CO2.. or even from all ghg) simply from knowing how much change was experienced on the ground (the greenhouse effect). I can see how that question might be answered possibly by calculating in delta/differential steps and taking a limit in order to try and get a unique result (although I don't know how to make that calculation). Alternatively, it might not be possible to get sequence/limit convergence to a unique value on a "delta" analysis. I don't know, and before I spend more time thinking about this problem, I'm curious if someone has access to an online reference where I can get the answer or if they can provide the answer themselves.
     
     Resolving these questions makes it easier for me to address more definitively some skeptical questions I have been getting related to "forcings" math (and in understanding this for its own sake).

147. KR at 11:00 AM on 13 March, 2013

     Jose_X - Some misconceptions here, if I might point them out (hopefully correctly).

     "How do we calculate the total forcing by CO2 that led to this 29 W/m^2?"

     If you start from zero CO2, the forcing per increase in CO2 concentration starts linear, and becomes logarithmic as various bands reach saturation (with increases coming from band widening, rather than peak increases), so the relationship is not consistent over concentrations. Methane (IIRC) is still in the linear region, CO2 is not. The accurate answer comes from line-by-line radiative codes such as MODTRAN using the HITRAN spectral database - essentially numeric integration. You have to do the math. There is no simple equation. 

     "My confusion is that forcings are defined at TOA yet TOA always reverts to about 240 W/m^2"

     Yes, it does, as 240 W/m^2 is what is incoming from the sun. When enough time elapses for GHG forcing changes to come to equilibrium, energy out = energy in at TOA, although at a different surface temperature depending on those changes. The only reason for that equilibrium number to change would be changes in incoming energy, perhaps from albedo/land use or significant cloud percentage/distribution. I do expect that the melt of the Arctic ice cap, for example, will raise that equilibrium number somewhat by decreasing summer albedo. 

148. Tom Curtis at 11:11 AM on 13 March, 2013

     Jose_X @146:

     1)  Because the CO2 modulates the outgoing LW radiation, it depends on the strength of that radiation, and hence on surface temperature.  That does mean it depends on the strength of the incoming solar radiation, but only indirectly.  In contrast, changes in aerosol opitical depth, which change albedo, are directly dependent on the strength of the incoming solar radiation for the strength of the forcing.

     2)  Across the range of temperatures and conditions experienced in the Phanerozoic (approx -6 to +8 C relative to present values) λ has been fairly constant, with changes in continental configurations having a much larger effect on changes in λ than variation withing the temperature range.  In broader terms, λ becomes significantly larger as ice sheets approach 30^o Latitude (North or South) due to the much enhanced albedo, and as temperatures rise significantly beyond 8 C above current values (due to enhanced water vapour feedback).

     Other than noting that your second equation in section (3) should be 33/5 = 6.6, I will return to it when I have a bit more time.

149. Jose_X at 11:26 AM on 13 March, 2013

     Tom Curtis, actually the equation should probably have been 6.6/33 = .2 (aka, 1/5) and then rather than divide the 5 into the 145.75 I would have multiplied the .2 and 145.75. Point is that 6.6 is 20% of the whole change as expressed in C so I wanted to find that same 20% of the whole change in W/msq.

150. Jose_X at 11:34 AM on 13 March, 2013

     KR, calculating the forcing for CO2 early on in the process (near 0% CO2) and continuing, eg, using at each step the integration process you mentioned, is what I imagined might be done. But I am curious if that procedure would lead to the same total forcing result no matter in how many steps we calculate the increments.
     
     As an extreme example, if we calculate the forcing of one additional molecule of CO2 at a time (assuming the standard model we use now applies, eg, ignoring quantum or other effects that might be present at very low concentrations or what not, and assuming that computation could finish some day), what total forcing would we get and would this value be the same if after an initial start we switch over to steps where we double the prior value.
     
     Why do I care? I would like to know if the forcing calculations adhere to linear superposition like linear operators do with vector (tensor?) quantities (ie, partitioning into component parts arbitrarily, operate on these separately, and then combined back additively into a unique whole). The definition of forcing makes me a little nervous about that.
     
     Does someone have software that can simulate several reasonable partitions for the current CO2 to test if each partitioning path leads to the same answer? [An example might be to calculate 1 W/msq of forcing at each step vs 2 W/msq at each step. In fact, there is no need to do this until the current atmosphere. We can just compare 1 W/sqm for 40 steps vs 2 W/sqm for 20 steps.]

     [In these replies I giving, m^2 is same as mSq is same as sqm ...etc. I get lazy with the keystrokes sometimes.]

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