The diminishing influence of increasing Carbon Dioxide on temperature

Guest essay by Ed Hoskins

Using data published by the IPCC on the diminishing effect of increasing CO2 concentrations and the latest proportional information on global Man-made CO2 emissions, these notes examine the potential for further warming by CO2 emissions up to 1000ppmv and the probable consequences of decarbonisation policies being pursued by Western governments.

The temperature increasing capacity of atmospheric CO2 is real enough, but its influence is known and widely accepted to diminish as its concentration increases. It has a logarithmic in its relationship to concentration. Global Warming advocates and Climate Change sceptics both agree on this.

IPCC Published reports, (TAR3), acknowledge that the effective temperature increase caused by growing concentrations of CO2 in the atmosphere radically diminishes with increasing concentrations. This information has been presented in the IPCC reports. It is well disguised for any lay reader, (Chapter 6. Radiative Forcing of Climate Change: section 6.3.4 Total Well-Mixed Greenhouse Gas Forcing Estimate) [1]. It is a crucial fact, but not acknowledged in the IPCC summary for Policy Makers[2].

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The rapid logarithmic diminution effect is an inconvenient fact for Global Warming advocates and alarmists, nonetheless it is well understood within the climate science community. It is certainly not much discussed. This diminution effect is probably the reason there was no runaway greenhouse warming caused by CO2 in earlier eons when CO2 levels were known to be at levels of several thousands ppmv. The following simplifying diagram shows the logarithmic diminution effect using tranches of 100ppmv up to 1000ppmv and the significance of differing CO2 concentrations on the biosphere:

§ Up to ~200 ppmv, the equivalent to about ~77% of the temperature increasing effectiveness of CO2. This is essential to sustain photosynthesis in plants and thus the viability of all life on earth.

§ A further ~100 ppmv was the level prior to any industrialisation, this atmospheric CO2 made the survival of the biosphere possible, giving a further 5.9% of the CO2 Greenhouse effect.

§ Following that a further 100ppmv, (certainly man-made in part), adding ~4.1% of the CO2 effectiveness brings the current level ~400 ppmv.

§ CO2 at 400pmmv is already committed and immutable. So CO2 has already reached about ~87+% of its potential warming effect in the atmosphere.

Both sceptics and the IPCC publish alternate views of the reducing effect on temperature of the importance of CO2 concentration. These alternates are equivalent proportionally but vary in the degree of warming attributable to CO2.

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The IPCC have published views of the total effect of CO2 as a greenhouse gas up to ~1200ppmv, they range in temperature from +6.3°C to +14.5°C, shown below:

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There are other views presented both by sceptical scientists and CDIAC, the Carbon Dioxide Information and Analysis Centre. What these different analysis show the is the amount of future warming that might be attributed to additional atmospheric CO2 in excess of the current level of ~400ppmv. Looking to the future in excess of 400ppmv, wide variation exists between the different warming estimates up to 1000ppmv, see below.

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A comparison between these estimates are set out below in the context of the ~33°C total Greenhouse Effect.

This graphic shows in orange the remaining temperature effect of CO2 up to 1000ppmv that could be affected by worldwide global decarbonisation policies according to each of these alternative analyses.

Some of the IPCC data sets shows very large proportions of the temperature effect attributable solely to extra CO2. The concomitant effect of those higher levels of warming from atmospheric CO2 is that the proportion of the total ~33°C then attributable the water vapour and clouds in the atmosphere is displaced so as to be unrealistically low at 72% or 54%.

It has to be questioned whether it is plausible that CO2, a minor trace gas in the atmosphere, currently at the level of ~400ppmv, 0.04% up to 0.10% achieves such radical control of Global temperature, when compared to the substantial and powerful Greenhouse Effect of water vapour and clouds in the atmosphere?

There are the clearly divergent views of the amount of warming that can result from additional CO2 in future, but even in a worst case scenario whatever change that may happen can only ever have a marginal future effect on global temperature.

Whatever political efforts are made to de-carbonize economies or to reduce man-made CO2 emissions, (and to be effective at temperature control those efforts would have to be universal and worldwide), those efforts can only now affect at most ~13% of the future warming potential of CO2 up to the currently unthinkably high level of 1000ppmv.

So increasing CO2 in the atmosphere can not now inevitably lead directly to much more warming and certainly not to a catastrophic and dangerous temperature increase.

Importantly as the future temperature effect of increasing CO2 emissions can only be so minor, there is no possibility of ever attaining the much vaunted political target of less than +2.0°C by the control of CO2 emissions[3].

Global Warming advocates always assert that all increases in the concentration of CO2 are solely man-made. This is not necessarily so, as the biosphere and slightly warming oceans will also outgas CO2. In any event at ~3% of the total[4] Man-made CO2 at its maximum is only a minor part of the CO2 transport within the atmosphere. The recent IPCC report now admits that currently increasing CO2 levels are probably only ~50% man-made.

On the other hand it is likely that any current global warming, if continuing and increased CO2 is:

§ largely a natural process

§ within normal limits

§ probably beneficial up to about a further 2.0°C+ [5].

It could be not be influenced by any remedial decarbonisation action, however drastic, taken by a minority of nations.

In a rational, non-political world, that prospect should be greeted with unmitigated joy.

If it is so:

· concern over CO2 as a man-made pollutant can be mostly discounted.

· it is not essential to disrupt the economy of the Western world to no purpose.

· the cost to the European economy alone is considered to be ~ £165 billion per annum till the end of the century, not including the diversion of employment and industries to elsewhere: this is deliberate economic self-harm that can be avoided: these vast resources could be spent for much more worthwhile endeavours.

· were warming happening, unless excessive, it provides a more benign climate for the biosphere and mankind.

· any extra CO2 has already increased the fertility of all plant life on the planet.

· if warming is occurring at all, a warmer climate within natural variation would provide a future of greater opportunity and prosperity for human development, especially so for the third world.

De-carbonisation outcomes

To quantify what might be achieved by any political action for de-carbonization by Western economies, the comparative table below shows the remaining effectiveness of each 100ppmv tranche up to 1000ppmv, with the total global warming in each of the five diminution assessments.

The table below shows the likely range of warming arising from these divergent (sceptical and IPCC) views, (without feedbacks, which are questionably either negative or positive: but probably not massively positive as assumed by CAGW alarmists), that would be averted with an increase of CO2 for the full increase from 400 ppmv to 1000 ppmv.

The results above for countries and country groups show a range for whichever scenario of only a matter of a few thousandths to a few hundredths of a degree Centigrade.

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However it is extremely unlikely that the developing world is going to succumb to non-development of their economies on the grounds of reducing CO2 emissions. So it is very likely that the developing world’s CO2 emissions are going to escalate whatever is done by developed nations.

These figures show that whatever the developed world does in terms of decreasing CO2 emissions the outcome is likely to be either immaterial or more likely even beneficial. The table below assumes that the amount of CO2 released by each of the world’s nations or nation is reduced universally by some 20%: this is a radical reduction level but just about conceivable.

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These extreme, economically destructive and immensely costly efforts by participating western nations to reduce temperature by de-carbonization should be seen in context:

§ the changing global temperature patterns, the current standstill and likely impending cooling.

§ the rapidly growing CO2 emissions from the bulk of the world’s most populous nations as they continue their development.

§ the diminishing impact of any extra CO2 emissions on any temperature increase.

§ normal daily temperature variations at any a single location range from 10°C to 20°C.

§ normal annual variations value can be as much as 40°C to 50°C.

§ that participating Europe as a whole only accounts for ~11% of world CO2 emissions.

§ that the UK itself is now only about ~1.5% of world CO2 emissions.

As the margin of error for temperature measurements is about 1.0°C, the miniscule temperature effects shown above arise from the extreme economic efforts of those participating nations attempting to control their CO2 emissions. Thus the outcomes in terms of controlling temperature can only ever be marginal, immeasurable and thus irrelevant.

The committed Nations by their actions alone, whatever the costs they incurred to themselves, might only ever effect virtually undetectable reductions of World temperature. So it is clear that all the minor but extremely expensive attempts by the few convinced Western nations at the limitation of their own CO2 emissions will be inconsequential and futile[6].

Professor Judith Curry’s Congressional testimony 14/1/2014[7]:

“Motivated by the precautionary principle to avoid dangerous anthropogenic climate change, attempts to modify the climate through reducing CO2 emissions may turn out to be futile. The stagnation in greenhouse warming observed over the past 15+ years demonstrates that CO2 is not a control knob on climate variability on decadal time scales.”

Professor Richard Lindzen UK parliament committee testimony 28/1/2014 on IPCC AR5[8]:

“Whatever the UK decides to do will have no impact on your climate, but will have a profound impact on your economy. (You are) Trying to solve a problem that may not be a problem by taking actions that you know will hurt your economy.”

and paraphrased “doing nothing for fifty years is a much better option than any active political measures to control climate.”

As global temperatures have already been showing stagnation or cooling[9] over the last seventeen years or more, the world should fear the real and detrimental effects of global cooling[10] rather than being hysterical about limited, beneficial or now non-existent warming[11].


References:

[1] http://www.grida.no/publications/other/ipcc%5Ftar/?src=/climate/ipcc_tar/wg1/222.htm

[2] http://www.powerlineblog.com/archives/2014/05/why-global-warming-alarmism-isnt-science-2.php

[3] http://www.copenhagenconsensus.com/sites/default/files/ccctolpaper.pdf

[4] http://www.geocraft.com/WVFossils/greenhouse_data.html

[5] http://www.spectator.co.uk/features/9057151/carry-on-warming/

[6] http://hockeyschtick.blogspot.fr/2013/11/lomborg-spain-wastes-hundreds-of.html

[7] http://www.epw.senate.gov/public/index.cfm?FuseAction=Files.View&FileStore_id=07472bb4-3eeb-42da-a49d-964165860275

[8] http://judithcurry.com/2014/01/28/uk-parliamentary-hearing-on-the-ipcc/

[9] http://www.spectator.co.uk/melaniephillips/3436241/the-inescapable-apocalypse-has-been-seriously-underestimated.thtml

[10] http://www.iceagenow.com/Triple_Crown_of_global_cooling.htm

[11] http://notrickszone.com/2010/12/28/global-cooling-consensus-is-heating-up-cooling-over-the-next-1-to-3-decades/

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mpainter
August 10, 2014 1:23 pm

Concerning the absorption of IR by the surface of water (top2-3 microns), there is another way to examine the matter: by rate of evaporation. A typical rate in hot climates is one cm per day; that works out to about 8 microns per minute. This figure is averaged over 24 hrs and actual daytime evaporation rates would be higher because of the much greater amount of heat involved. The point is this approach allows a calculation of the residence time of the heat in the upper two or three microns and behold, the microns are gone in a few seconds. No time for conduction to lower levels, period. A simple experiment will show that is impossible to heat water by IR.

acementhead
August 10, 2014 1:32 pm

Raymond says: August 10, 2014 at 9:29 am
higley7 says:
“…….. The computer models do not do night-time, …….”
This statement can not be true, if it is could someone explain why these models have any validity at all?

Raymond higley7’s post comes in the category of “not even wrong”(sorry I don’t know who coined the expression), Posts such as his allow warmists to categorise us realists as idiots. I’m opposed to censorship but in some things it is best e.g. the owner of this site properly forbids discussion of “*hem*rails” for obvious reasons.

mpainter
August 10, 2014 1:34 pm

Could be that I’ m wrong, but I simply cannot take decarbonization as a serious notion. For me it is not in the realm of possibilities, not even in this loony world of enviro-wackos.

Edward Richardson
August 10, 2014 1:39 pm

mpainter says:
August 10, 2014 at 1:23 pm
“A simple experiment will show that is impossible to heat water by IR. ”
http://www.tellusa.net/index.php/tellusa/article/view/15675

August 10, 2014 1:47 pm

Curt says:
August 10, 2014 at 12:54 pm
Genghis, you say with respect to the DWLWIR hitting the ocean:
“The only explanation that I can think of is that evaporation increases in direct response to the increased radiation.”
So you agree that the overall earth/ocean/atmosphere energy level is higher with this radiation than without it?
****************
No, the energy in the Ocean is not increased at all, because the DWLWIR never gets past the first couple of microns, before it gets evaporated away just like boiling water doesn’t get any hotter no matter how much extra heat gets applied to it. Also the ocean surface temperature acts as an upper limit on the atmospheric temperature.

August 10, 2014 1:54 pm

Donald L. Klipstein says:
If the top 3 microns warms up without the 10 micron level warming up, then heat conduction through the extremely thin temperature gradient between 3 and 10 microns down will cool the top 3 microns faster than increased evaporative cooling. Evaporative cooling won’t outpace heat conduction until the temperature gradient expands over a much greater range of depth, at which point mixing becomes significant at transferring heat to deeper levels.
************
You have it wrong. The top couple of microns don’t heat up. There is no temperature gradient change.
Again the best analogy is measuring the temperature of boiling water while increasing the amount of heat being applied, the liquid water will stay at exactly the same temperature, increased evaporation instantly compensates for the extra heat.

Greg Goodman
August 10, 2014 2:06 pm

TonyB: Your CET work is interesting, however: “Please note that the graphing package somewhat inflates the warmth in the decade around 1540”
Don’t blame the “graphing package” you are not centering then running average ( which is a crap filter to start with.). The reason 1540’s looks too high is because it’s too late ! It is obvious to the natked eye. Shift the “smoothed” version back 5y and both the 1540 and y2k (in fact the whole graph) will line up a lot better.
Unusually, there don’t seem to be the typical peak inversions that RM often creates, just luck of the draw in the periodiciteis present in the data.
One thing that does look wrong in the cira 2000 peak if far too pointy in realation to the unfiltered data.
Try triple running mean of half the length and it will be just as “smooth” and probably follow better.
https://climategrog.wordpress.com/wp-admin/post.php?post=989&action=edit

August 10, 2014 2:07 pm

Mosher says
Why are you so obsessed with arguing with warmest using their play book? They are a bunch of liers trying to control you and me. When you argue inside the constraints they set they can simply say you accept their argument but misunderstand its complexities and misinterpret their results.
You let them be the authority and they can dismiss you as a cub who has strayed from the fold.
One should argue what they believe and can back up, not what is accepted by the established.
They have an agenda and could careless where the truth lies.

Greg Goodman
August 10, 2014 2:09 pm

Genghis says:
August 10, 2014 at 1:54 pm
Donald L. Klipstein says:
If the top 3 microns warms up without the 10 micron level warming up, then heat conduction through the extremely thin temperature gradient between 3 and 10 microns down will cool the top 3 microns faster than increased evaporative cooling. Evaporative cooling won’t outpace heat conduction until the temperature gradient expands over a much greater range of depth, at which point mixing becomes significant at transferring heat to deeper levels.
************
You have it wrong. The top couple of microns don’t heat up. There is no temperature gradient change.
Again the best analogy is measuring the temperature of boiling water while increasing the amount of heat being applied, the liquid water will stay at exactly the same temperature, increased evaporation instantly compensates for the extra heat.
======
Everyone seems to love making assertions and contra assertions about this subject. Anyone got anything more concreate that back of envolop arguments and assertion?

Tonyb
August 10, 2014 2:19 pm

Greg
Thanks for your comments. I always read your material.
There was a sharp peak in Cet around 2000 as can be seen in the met office Hadley CET 1772 figures.
I met up with David Parker at the Met office last year, who created the record. They haveRecently changed the stations being used as they felt they were running too warm. My estimate would be that the upwards incline to 2000 was too sharp and the decline since too exaggerated. The temperatures have started rising again but are still nowhere near their peak.
Incidentally, whilst it was an artefact of the graphing package to some extent! further research has shown that the pre 1540 period showed a sharp temperature increase. There was a sharp decline from around 1450 to 1500
Tonyb

Curt
August 10, 2014 2:19 pm

Genghis:
I asked the question:
“So you agree that the overall earth/ocean/atmosphere energy level is higher with this radiation than without it?”
and you responded:
“No, the energy in the Ocean is not increased at all, because the DWLWIR never gets past the first couple of microns, before it gets evaporated away…”
which is not an answer to my question at all. I carefully asked about “the overall earth/ocean/atmosphere energy level”.Let’s grant, at least for the sake of argument, the the DWLWIR does not affect the temperature, and therefore the energy level, of the ocean itself. But by evaporating water from the surface, it does add the latent heat of evaporation to the overall system.
The reason I make a point of this is that there are many (and I don’t necessarily include you in this) who believe that because the liquid water body temperature does not apparently increase in the presence of increased DWLWIR, that the energy in this radiation is somehow lost. And that would be a blatant 1st Law violation.

August 10, 2014 2:22 pm

There is substantial research showing that significantly increased or decreased temperatures reduces crop yields. That same research showed that increasing atmospheric CO2 could more than off-set the yield reductions resulting from those temperature changes.
Unless it can be shown 1) that temperatures have risen at least in part due to Man’s CO2 contributions; 2) that that rise in temperature produced an unacceptable change in climate; and 3) reducing CO2 absolutely will reduce temperatures, then the precautionary principle dictates that we must NOT reduce atmospheric CO2.
In other words: if any temperature change up or down is caused primarily by natural variation, then we will only be reducing our global food supplies by reducing CO2.

August 10, 2014 2:38 pm

AGAIN with this absurd idea that DWLWIR can do thermodynamic work on the surface?! It’s not a heat flux, folks! If it warms the top 3 microns of the surface skin, it means it’s heat. If it provokes more evaporation, it means it’s heat. Because only ‘heat’ (and ‘work’) are real, thermodynamically working flows of energy. Actual, detectable transfers of energy. DWLWIR is not heat! Energy is transferred radiatively from the warm surface to the cool atmosphere only. As radiative heat. Period. Stop pretending DWLWIR is a separately working flux of energy, operating as if it were heat, distinct from the UWLWIR within the same, integrated radiation field. It’s not.
richard verney says, August 10, 2014 at 11:09 am:
“Genghis says:
August 10, 2014 at 10:11 am
Infrared (atmospheric radiation) is absorbed by the top few microns of the surface, which does in fact heat the ocean.
///////////////////////
Does the energy absorbed in those 3 microns heat the ocean? For it to do so, it needs to be dissipated (and hence diluted) to depth at a rate faster than the energy absorbed in the top 3 microns would power/drive evaporation from the surface layer of the ocean.”

No, Richard. It needs to be HEAT. And it’s not heat. DWLWIR is not heat and hence it cannot heat the top 3 microns of the skin layer and it cannot provoke more evaporation.

August 10, 2014 2:46 pm

Climatology is the art of looking for trouble, finding it everywhere, diagnosing it incorrectly and applying the wrong remedies. (apologies to Groucho Marx)

August 10, 2014 2:49 pm

Curts says
If DWLWIR is not being absorb by the oceans (which it is not) for it to raise the overall earth/ocean/atmospher energy level you would have to see a rise is atmospheric temperatures. Which is not happening. This can only mean there is an increase of energy transfer in to space.
There is no effect model of how AGW can transfer heat to the oceans and not have a corisponding rise in atmospheric temperatures at the same time.

John Slayton
August 10, 2014 2:52 pm

cnxtim says:
Is this anything more than a “face saving” exit plan for all the CAGW evangelists?
“Yo wagons ho!, thar be the real … (insert what you will here).

I think the word you want may be ‘elephant’.
http://www.wordorigins.org/index.php/site/comments/elephant_to_see_the/
: >)

August 10, 2014 2:59 pm

Curt,
The atmosphere resting on top of the solar-heated surface does of course make less energy go out from the surface per unit of time at equal temperature than if there were no atmosphere, only the vacuum of space. This is because the atmosphere has a mass. It thus has a ‘heat capacity’. It is able to warm. Space isn’t. It also, for the same reason, weighs down on the surface (space doesn’t), setting a limit to buoyant acceleration and evaporation rates at a certain temperature. This is what forces Earth’s surface to be much warmer than the Moon’s. It’s got nothing to do (it couldn’t have) with energy INPUT to the surface from a cooler place. It has to do with a smaller energy OUTPUT from the surface to this same cooler place. Because this cooler place is still warmer than space. The atmosphere still gains energy from the surface, by the simple fact of being the cooler of the two systems. It’s all a matter of how much it gains per unit of time. And this is set by the temp gradient and the weight of the atmosphere on the surface.
Come back when you can show us how the presence of radiatively active gases in our atmosphere – specifically through their radiative properties – actually reduce (or work towards reducing) the tropospheric temperature profile. If they don’t, then they don’t contribute to the warming of the surface. Simple as that.

Nick Stokes
August 10, 2014 3:06 pm

“The rapid logarithmic diminution effect is an inconvenient fact for Global Warming advocates and alarmists, nonetheless it is well understood within the climate science community. It is certainly not much discussed.”
It is much discussed. Whenever people refer to sensitivity as 2&dseg;C/doubling, or whatever, they are invoking the logarithmic behaviour.
“CO2 at 400pmmv is already committed and immutable. So CO2 has already reached about ~87+% of its potential warming effect in the atmosphere”
The basis for this percent arithmetic (here and earlier) is not stated, and it makes no sense. A logarithmic curve has diminishing slope, but no maximum or minimum. If sensitiivity is 2K/doubling, and if 400 ppmv corresponds to equilibrium 288 K, then 400 * 2^-144 ppmv corresponds to 0K. Or 400 * 2^144 ppmv to 576 K. Of course, what that shows is simply that logarithmic behaviour is just an approximation that works in a limited range.
“The recent IPCC report now admits that currently increasing CO2 levels are probably only ~50% man-made.”
Where?

Bill Illis
August 10, 2014 3:14 pm

We are now at 50% of the logarithmic calculated theoritical doubling of GHGs.
50% of doubling —> +0.6C temperature increase (once you factor in the Mosher-sanctioned fake temperature adjustments).
Half-way, +0.6C.
Half-way and plants are growing much better as in yields are way-up, even your grass is growing faster.
Now we can also say the warming in the pipeline held by the oceans is only 0.2 W/m2/year after the newest results or nothing to worry about.
Full-way to doubling –> +1.2C? plus a little more once the oceans catch up that little bit and plants are more productive.
The numbers make increased CO2 a positive development for the planet. Simple as that.

August 10, 2014 3:29 pm

Greg Goodman says:
August 10, 2014 at 2:09 pm
Everyone seems to love making assertions and contra assertions about this subject. Anyone got anything more concreate that back of envolop arguments and assertion?
+++++++++++
I am sitting in Manjack, in the Abacos this very moment with an IR gun reading the surface temperature, it is 31.4˚ C. with a clear sky, almost dead calm conditions. If conditions stay the same, when I take a reading sometime tonight, when I check the anchor, the temperature will be the same whether it is a clear sky or cloudy. In the morning it will most likely be cloudy and the same temperature.
If the wind picks up a little the surface temperature will go down. If the wind picks up a lot the temperature will stabilize at a new point, probably warmer.
I have been doing this for a couple of years now in various anchorages and I have seen the surface temp stay the same for days and even the occasional week or two. When I say the same temp I mean within a few tenths.
As a farm boy from the high deserts in Idaho, I have to admit that I was blown away by the temperature stability of the ocean surface.

August 10, 2014 3:29 pm

So Nick Stokes has gotten permission from Hansen to reappear?
H. Grouse, John Carter and John Finn will be around any minute now.

James McCown
August 10, 2014 3:30 pm

I discussed this issue of how much infrared energy is being absorbed by CO2 with Hu McCulloch a couple months ago. He had an interesting insight. Whatever function we use to compute the radiative forcing for CO2 has got to be a bounded function. Once all the infrared in some frequency band is absorbed, adding more CO2 to the atmosphere will not result in any additional radiative forcing. Therefore, instead of ln(X), the formula for radiative forcing should something more like 1 – e^(-x).

JohnWho
August 10, 2014 3:34 pm

Nick Stokes says:
August 10, 2014 at 3:06 pm
The basis for this percent arithmetic (here and earlier) is not stated, and it makes no sense. A logarithmic curve has diminishing slope, but no maximum or minimum.

Help me out with that, Nick.
Wouldn’t the minimum be either at 0% atmospheric CO2 or, at least, 1 part CO2 to the entire atmosphere; while the maximum would be at either 100% CO2 or, at most, a CO2 atmosphere with only 1 part non-CO2?
At 50% CO2 there is only one doubling left. Wouldn’t the maximum then be reached?
The bigger question would be: at what point do further doublings add only a barely measureable amount of warming?
If all of this isn’t in the “settled science” realm after all the discussions over the years, will it ever be?
Just wondering.

August 10, 2014 3:34 pm

This is all well and good as a purely radiative argument. However, the derivation of the thermal gradient within a gravitationally bound atmosphere pays no heed to inter atmospheric radiative exchange. The tropospheric lapse rate is dT/dh=-g/Cp gives a monumentally solid depiction of the bulk if the lower atmosphere up to the point where direct atmospheric heating (solar) drives the system out of the reversible adiabatic profile. This can be derived through equating kinetic and potential energy to a dQ=0 (adiabatic) condition or thermodynamically from the gas laws.
So, theoretically, back radiation ‘heats’ the lower atmosphere with ‘special molecules’ whilst these same molecules reduce the intensity of solar energy reaching the surface and increase the upper atmospheric emissivity. Even though we haven’t used these to calculate the thermal relationship between the upper and lower troposphere, well, no further than how by mass density the molecules affect Cp. Increasing upper atmospheric emissivity cools the upper, but the upper and lower are tied by the mechanical lapse which for a given heat capacity ‘fixes’ the gradient set by gravitational containment for long term stability.
Radiative exchange never produces a thermal gradient. The exchange, reduces the thermal gradient. Gravity is responsible for the enhancement of the surface temperature and the mass aloft, as all interacting matter radiates, accounts for the necessary decrease in surface emissivity. The transmission of the atmosphere to short wave and long wave radiation is not that dissimilar, the Sun producing most of its radiative emissions in the infra red (51% IR, 37% visible) which then covers ever spectral line of every GHG. More GHG’s, less heat reaching the surface, more upper atmospheric cooling, some ‘calculated back radiation’. Net effect, zero measurable.