And to get a new increase of this order of magnitude (3°-4), it will be necessary to alter the quantity of carbonic acid till it reaches a value nearly midway between 2 and 2.5.
That seems very much in line with current estimates of ~3°C/doubling.
Unsettled Scientistsays
John E Pearson,
Yes, I agree, it’s not a simple formula you plug things into with our modern understanding, we’re way past that. But Six-Legged specifically asked for the Arrhenius one, which is why I brought up that paper. Certainly I would not rely upon centuries old science for a modern value, but for someone looking to do the math by hand, that’s what Arrhenius did. If one is trying to calculate it themselves, it will have to be a very simple idealized model. If we want the best science has to offer, it’s going to involve highly complicated models run on computers. I almost included this in my previous replies. If you’re looking for a formula that has one variable (delta CO2 ppm) and spits out degrees celsius delta, you’re not understanding the science.
Honestly, I’m really enjoying this AIP link. It does cover this stuff. It gives the story of the study of CO2 as a GHG, with links describing the early simple models, radiation maths, and even more modern GCMs. The story really is enlightening. Arrhenius doing absurdly large calculations with significant uncertainties, basically to take his mind of his wife leaving him and taking the kid. He wasn’t trying to understand global warming or even necessarily predict it, people were trying to solve the problem of the ice ages. It was through studying that phenomenon that the prediction of man-made global warming came about, not some conspiracy. It’s like how one of the most fundamental concepts in physics, Entropy, came out of trying to make steam engines work better.
But yeah, I wasn’t trying to say our modern understanding was easy, just pointing Six Legged to what I think he is trying to get. Simple questions like that involve a lot of assumptions, so it’s hard to know exactly what to give people.
Unsettled Scientistsays
KR,
That 3-4 was a regional/continental value? Arrhenius got something much higher for arctic regions, like 8-9. I think the 5-6 was the average for the globe. I’d have to go reread to confirm this.
Patrick 027says
Re 347 John E. Pearson I’ve wondered if I ought not to be able to putz around and do something iterative to first get the 1.3C no feedback temperature, then include H2O vapor and get a correction to atmospheric temp, then correct the H2O feedback and get a new atmospheric temp, etc. but I’ve never tried.
This doesn’t capture the nonlinear effect of variable effective heat capacity (over longer times, greater amounts of heat capacity come into play), nor any slow feedback effects, but assuming or approximating as a linear response (PS I may not be using standard notation here; sorry):
B is the Planck response at TOA per unit change in global average surface temperature T (increase in OLR at TOA (assuming instant stratospheric adjustment?), and assuming constant tropospheric lapse rate, with no compositional or optical-property feedbacks or any other feedbacks)
F is the decrease in OLR per unit T due to those other (non-Planck) feedbacks (including (important or not) any change in horizontal temperature variation at any level, etc.)
where B and F are in such units as W/m2 per K
C is the heat capacity of the climate system per unit area
RF is the imposed radiative forcing (relative to the baseline where equilibrium ∆T = 0)
H is enthalpy (~~heat) per unit area
then
dH/dt = C*dT/dt = RF + (F-B)*∆T
or
dH = C*dT = RF*dt + (F-B)*∆T*dt
—
∆T = ∆Teq (the equilibrium ∆T) when dT/dt = 0, thus
RF + (F-B)*∆Teq = 0
∆Teq = RF/(B-F)
(whereas with iteration, you start with ∆T1 = RF/B, then add ∆T(n+1) = (F*∆Tn)/B repeatedly)
The way to estimate RF by hand – get a global average OLR graph (that’s the part you can’t do easily by hand), identify the CO2 valley, find a spectrum of log(cross section) CO2 (can’t do that easily by hand either), find the interval over which the optical cross section halves outward from ~15 microns (on each side if they differ; if the slope varies look around where optical path length for the atmosphere is a moderate value ~1, give or take). Multiply the depth of the CO2 valley by the interval; that’s TOA RF before stratospheric adjustment (subtract the widenning and increase in height of the little hill in the center of that value to get a more accurate result – this probably requires some more info about the temperature profile in the stratosphere). Then do the same as if the CO2 valley OLR = 0 – that’s approximately tropopause RF before stratospheric adjustment. Subtract TOA value from tropopause value – that’s the stratospheric cooling (some additional. Assume some fraction of that is realized as a reduction in downward radiation at the tropopause, and subtract that from tropopause RF – that’s your tropopause RF (and TOA RF) after stratospheric adjustment. (Perhpas you could try to estimate B by shifting the rest of an OLR graph (outside where it reaches the tropopause-level Planck function) up by 1 K and looking at the change in area).
Or use this http://forecast.uchicago.edu/Projects/modtran.html (PS some error due to assumption of perfect blackbody surface (based on last time I used it) – this shouldn’t be a large error for tropopause and TOA RF because it is masked by H2O, clouds, CO2, etc, already; more significant if you start with no GHGs. Advantage – you can do different latitudes and conditions seperately and then find a global average from that. PS I haven’t tried it but you might be able to estimate B from this website too. The GH(LW) portion of F as well if you make some assumptions – but you can look at F values from climate models (and observations) for hints.
Martin Vermeersays
KR #351:
And to get a new increase of this order of magnitude (3°-4), it will be necessary to alter the quantity of carbonic acid till it reaches a value nearly midway between 2 and 2.5.
That seems very much in line with current estimates of ~3°C/doubling.
Eh, no… it says new increase (and 3°.4, not 3°-4). He talks about multiplying by 1.5 a second time, giving 2×3.4 degrees for 2.25x the amount of CO2.
Have a look at Table VII on page 266, under the header “Carbonic acid = 2.0”. It really is between 5 and 6 degrees per doubling.
@357
The linked story sports a graphic of “Scenario of potential sea level rise within 100-200 years” showing Europe under 100m of water. So me droning on with my usual ‘there simply isn’t enough energy about to raise sea level over a couple of metres per century unless you fill the oceans with icebergs’ likely won’t go down to well here.
I think someone made a mistake when adding text to that image that accompanies Chris’s article at Joe Romm’s place. I did an image search and the picture appears several other places (French and Italian sites) — but never with that English text overlaying it.
One of those captions the same image with “If Antarctic were to melt completely, it should give us 61 meters of sea level rise worldwide….” which makes more sense.
I’m sure neither Chris nor Joe is predicting that. It’s got to be a typo on the image, it’s inconsistent with Chris’s text.
MARodger, the image is a bit sensational i guess. But there are studies done on several meters ( around 20) SLR within a few years or decades. Following this, 1-2 centuries may yield 100 meters. The main point to take home with this image is that we require new maps for country shorelines, in the coming decades. With BAU we are on a path to an ice free state.
JCHsays
MARodger – if I understand the basis of Hansen’s prediction of up to 5 meters by ~2100, it’s simply that it has happened in the past. Are you saying it could not have happened in the past, or that there was more energy available then to melt it, or that just a lot of land-based ice must have been quickly dumped into the oceans to have achieved that rate?
Dan H.says
MA,
Without a serious influx of icebergs, there simply is not enough energy to raise sea levels that much. Even with the recent loss of ice from Greenland, SLR has slowed in the last few years. Estimates much beyond the current 2-2.5 mm/yr seem unachievable, or sensational (as Chris pointed out).
[Response: More unsourced wishful thinking masquerading as informed commentary? You know the deal – provide justifications for scientific statements or don’t bother. Where for instance do you get a current ‘2-2.5 mm/yr’? You know as well as I do that all of the satellite data sets show more than 3mm/yr. – gavin]
KRsays
Martin Vermeer – Thanks for the reference – I clearly misread that section. However, looking at the latitude dependencies in the paper, it appears to be a global value of ~5.2C to 5.3C/doubling, given equatorial at 4.9-5C/doubling, polar amplification at ~6C/doubling.
I have to say, though I _still_ find that Arrhenius piece an excellent work in physics extrapolation.
JCH @363
What I’m saying is that when I calculate the net rate of energy entering the climate system (about 0.6 W/sq m yields 9.5 zJ pa) it would only be enough to melt 28,500 cu km of ice and raise sea levels 80mm pa. Now, assuming I haven’t put a decimal point upside down (as I do occasionally), the question this raises is how do the ice & energy get together to allow the melt to happen? I’m suggesting only a portion of the energy (say 25%) can do this (ie 20mm SLR pa or 2m per century) without iceberg-strewn oceans.
The implications of my suggestion here are still very green. Past discussion about it have become quite heated at times and not very constructive. However it is safe to say that my suggestion does sit very uncomfortably with Fig 7 of Hansen & Sato 2010 which appears to suggest SLR of 500mm pa by the end of the century.
Increased energy that is used up melting ice rather than warming the climate will increase the residual climate forcing levels as CO2 levels rise and thus allow the 80mm figure to increase (a maximum of +4mm pa if all warming stopped).
One consideration I have yet to explore (but raised by your questioning @363) is that the average SLR from the last ice age I measure to be 13mm pa yet the rate of warming was far slower.
“Abstract. Meltwater pulse 1A (mwp-1A) was a prominent feature of the last deglaciation, which led to a sea level rise of [around] 20 meters in less than 500 years….”
You can’t extend that to 300 meters in any time span, let alone 100-200 years, there isn’t enough ice above sea level.
Seriously, your text makes sense as cautionary discussion.
That one illustration — which it looks like someone borrowed from a different site and added misleading numbers to — bugs me.
The most recent (today’s) skeptical science article on the same subject refers to
“… rapid sea level rises of 9 m in 500 years during the Meltwater pulse 1a event 14,600 years ago and 2.5 m in the second event, 8,200 years ago.”
I will try to clarify, why i believe a SLR of 100 meters can be possible within the given time. Please post your input under that article… will comment back once i updated it.
PS for Chris — the above is nitpicking (the highest form of primate social bonding, and so meant).
The article is good reading and has and cites credible numbers from sources. If you agree the illustration with the added claim on it wasn’t supported by the science sources you relied on there’s no reason to be reluctant to clarify the piece.
Science is an endless process of corrections — not the defense of anything forever in its original form.
Showing people we do that (even us kibitzers and commenters) is part of the lesson about how science works.
About the Image: Scenario of potential sea level rise within 100-200 years. This is based on the current rate humans put Co2 into the atmosphere, which is 10,000 times faster than the natural processes.
For that matter and potential singular positive feedbacks the rate of Sea Level Rise (SLR) can be assumed to rise with a similar rate. And for that matter there is no equivalent in the earth recorded history. The main SLR rise is likely not to come from melted water, rather then thermal expansion, which is attributed to be 70-75% of current observed SLR.
If you feel that this theory is not justified i welcome your feedback over at my blog.
Patrick 027says
Is there a way to post an image directly into a comment (ie copy and paste from one’s own file, rather than first posting it elsewhere on the internet and then using that to get it into the comment)?
PS AGW: this is the point in the movie where that girl says “It’s he-ere”.
Re – past rapid sea level rise references, “Deglacial rapid sea level rises caused by ice-sheet saddle collapses”Lauren et al 2012 (abstract) gives the MWP-1A as “a sea level rise of 14–18 metres over 350 years” from a P Deschamps et al reference. Lauren et al suggest a mechanism for half this rise, their paper discussed at Skeptical Science.
I saw the “climateprogress” name, and the article’s link with Joe Romm’s name, and mistook your blog for Climate Progress.
That’s what I get reading climate blogs while traveling and rushed.
Memo to self: the website for “Climate Progress” is named thinkprogress.net.
My mistake.
I can’t follow all the blogs out there, I try to stick to the science sites and always, always read the cited sources.
I believe you’ll find thermal expansion is already figured into the published science, and you’re counting it twice, to come up with your estimate for depth. I can’t imagine how you’re getting your estimate for timespan.
I doubt you’ve discovered the scientists have missed something obvious. See what the actual scientists say; I noticed Martin Vermeer already responded to you. You can’t do much better; listen to him.
siddsays
Mr. M. A. Rodger writes on the 22nd of July, 2012, at 1:59 PM:
“… how do the ice & energy get together to allow the melt to happen?”
1) From the Gregoire paper, as surface melt reaches the low point of a saddle, the amount of area available for melting jumps, surface mass balance goes negative, and the saddle melts and lowers deeper into the ablation zone, melts faster and over longer fractions of a year, rinse and repeat; voila: MWP1A
2) Add creep and basal sliding, ice only moves downhill, deeper into ablation zone.
3)ELA (equilibrium line altitude) is rising all the time, with it the ablation area, shrinking the accumulation zone. Swiss camp at 1150 m is no longer in accumulation zone, i see that ELA is up above 1700m already.
4)Your calculation for absorbed solar energy is for net albedo change. Consider that there is plenty of available energy to melt ice in the air in the ablation zone, and worse yet, rainfall delivers energy from warming ocean to the ice.
5)I think from the ASCAT images, that melt has reached not only the saddle at 67N (as we knew already,) but is now (July 2012) at the summit.
Chris, what’s your source for this? Who made the picture? Who added the time span to the picture? Why do you trust your source to the point you’ve blogged the claim without a cite? There’s no basis in the science for that claim that I can find.
Seriously, it looks like a mistake. Martin gave you a source to do the numbers in his comments, on your blog.
sidd @378
Yes but no but…
Moving Greenland’s ice surface down towards sea level will get more of it melted, but at what cost to the local energy budget, the global energy budget even? The energy has got to come from somewhere.
Let’s be ridiculous & tow Greenland to the equator. That will get the melt going but where you tow it too has be cooler as a result. That cooling will suck energy from surrounding regions. This is the same process when saddles are lowered.
Let’s be more ridiculous & cut up Greenland & scatter it (ice and all) round the tropics. Now that will up the melt rate but also cool the entire tropics, the entire globe.
As a simple matter of energy budgets, if this tropical Greenland (or a dropping saddle) is raising sea level by more than 80mm pa, it will be sucking more heat from the rest of the world than the world is being heated up by global warming. That means that even with global warming’s forcing, the whole globe will actually be cooling!
And remember Greenland is big. At 80 mm per year, Greenland has 100 years of ice to melt, be it starting at 3,000m in the Arctic Circle or at sea level in the tropics. Antarctica is ten times bigger still.
Martin Vermeersays
KR 365:
I have to say, though I _still_ find that Arrhenius piece an excellent work in physics extrapolation.
KR count me in as a fan of old Svante! The way he used the Langley measurements was brilliant, using the greater path length through the atmosphere as the Moon stands low in the sky as a surrogate for greater GHG concentrations.
Why Arrhenius got so high a sensitivity value? Besides his way-too-simple vertical atmospheric structure, I have the following suspicion. He only considered two gases: carbon dioxide (“K”) and water vapour (“W”). In reality of course, there are many more greenhouse gases: methane, ozone, nitrous oxides; and then there are aerosols. None of these belong to the CO2 feedback system, and should thus not be included in any computation of CO2 doubling sensitivity. But all of them are present in the sky over Arizona, masquerading in A’s computation as “K”, i.e., carbon dioxide.
But in summary, A already found the following fundamentals:
1) the logarithmic nature of GHG forcing with concentration
2) the water vapour feedback effect, strongly positive — from the skies of Arizona through the cycle of the seasons, not from physics, although he undoubtedly knew Clausius-Clapeyron (there is a similar equation in chemistry bearing his name, ask Eli ;-) )
Chris @374 & @380
Sorry to add to the objections I’ve already made about the graphic. Then, I think it is actually the same objection but from a different direction of approach (or perhaps that’s ‘directions’.) as the problems I have stem from the timings indicated by the graphic.
Two things. First, why 10,000 times faster? If the last ice age ended over a period of 10,000 years via a 6 Wm^-2 forcing, our 3 Wm^-2 forcing over 100 years is only 50 times faster. (Note this does leave some questions even at 50 times.)
Second, I would suggest that if my protest over energy poverty for rapid SLR from melting land ice has merit, rapid SLR from thermal expansion is not the answer as it is energy-wise far more intense. A simple calc (ignoring the pressures of the deep oceans) gives expansion requiring 60 times the energy for the same SLR. This perhaps overstates the discrepancy by a few tens of percent. Indeed, measured OHC is rising by something like 7.5 zJ pa but only achieving SLR of 1.5 mm. If that energy were applied to melting (from 0 deg C), it would have managed 65mm SLR pa.
Dan H.says
Sorry gavin,
Usually people do not require references for that which is well known, but I will comply.
[Response: People require references from you because of your history of a) making things up, and b) misinterpreting data… PDSI anyone? – gavin]
[Response: See how easy was that? (Better figure here, updated paper Church&White2011) Now the reason why you think that the 2009 data is ‘true’, and the 2011 update (which shows 2.8 ± 0.8 mm year−1 over the satellite period) and the satellite data itself (which are more current) are ignorable is perhaps worth discussing… – gavin]
If one were to choose a period of precisely 20 years, then a SLR of ~3mm/yr can be deduced. Over the past 30 years, SLR is ~2.5 mm/yr, while even the satellite data shows a decline to ~2 mm/yr over the past decade. Clearly not a sign of rapid acceleration.
[Response:Really? You don’t even need to read German to know that is crap. But now your point is over the definition of ‘rapid’. Movement of sorts I suppose. – gavin]
Dan H.says
Gavin,
Wow! Talk about misinterpretations! I did not misinterpret PDSI, nor have I ever accused you of not knowing how to read. Perhaps, if you were a little more forthcoming and honest, people would have more respect for your opinions. Sadly, I do not feel this will change. So ist das Leben!
@MARodger
It seems you are not taking into account the counter-intuitive effect of a drop in surface temperatures: that would increase the amount of energy available for melting ice by increasing the global radiative disequilibrium. I suppose this effect would be moderated by the atmospheric H2O feedback (as well as by other feedbacks) but it’s not negligible.
A change in oceanic circulation or in the location of large ice masses could therefore raise the sea level, even without a CO2/solar/whatever forcing. Or it could trigger a glacial episode if the magnitude of the feedbacks temporarily overwhelmed the drop in energy radiated by the surface… this ain’t trivial (see Heinrich events)!
siddsays
Mr. M.A. Rodger writes on the 23rd of July, 2012 at 3:09 AM:
“…raising sea level by more than 80mm pa, it will be sucking more heat from the rest of the world than the world is being heated up by global warming”
So stipulated.
Then what might be the largest plausible rate for SLR today? MWP1A had 50mm/yr for 500 yr, is that possible today ?
Because the rate humans put Co2 into the atmosphere today is of that order, when you compare it to the natural process from volcanoes.
JCHsays
If land-based ice suddenly goes nonlinear and ends up as floating ice a decades ahead of time, it seems like you guys are saying SLR has to wait for melting. In the case of icebergs, isn’t SLR instantaneous?
JCH,
You are correct. Any land-based ice which becomes seaborne will contribute to SLR immediately.
siddsays
Mr. JCH says on the 23rd of July, 2012 at 12:48 PM:
“In the case of icebergs, isn’t SLR instantaneous?”
GRIS would find it difficult to calve icebergs thru constricted outflow channels, easier to melt in place, export meltwater. WAIS, on the other hand…
sidd
Patrick 027says
Re Chris Machens –
1000s of years to rise 100 ppm deglaciation (5000 from memory, is this right? – IPCC has a nice graph (Ch 6 in AR 4, WGI, I think))
~100 years or less (depends on when you measure the slope) to rise 100 ppm anthropogenic
from memory, 0.2 Gt C/yr geologic emissions – tends to result in no change because there’s silicate weathering going on (but when you get a Snowball Earth, different story. 0.2 Gt C/yr would add ~ 0.094 ppm CO2 to the atmosphere if it is trapped there (a bit over 1000 years to get 100 ppm). Typically, redistribution among all surface reservoirs occurs relatively fast relative to geologic emission and sequestration by silicate weathering + organic C burial. Proportions of course can vary with the total, but if proportions were held constant (except for vegetation and soil, and marine biomass/organic), it would take about 60 (*) times longer (~ 64,000 years (*)) to get to 100 ppm – (* – actually a bit longer than that, as this is based on a situation in disequilibrium – see below). Deglaciation C changes were significantly faster.
Ratios:
100 ppmv CO2 ~= 213 Gt C ~= 781 Gt CO2, but forget sig.fig.s becauses I only used first two digits of molar masses (29 g/mol average atmosphere, 12 g/mol C, 44 g/mol CO2); based on atmosphere of 5.148 E18 kg (see http://en.wikipedia.org/wiki/Atmosphere_of_Earth , under “Density and mass”)
from http://earthguide.ucsd.edu/virtualmuseum/climatechange1/05_2.shtml , diagram shows 750 Gt in atmosphere and 44953 Gt in atmosphere, ocean (inorganic), continental and marine biomass, and dissolved organic carbon. The deep ocean in particular isn’t in equilibrium with the atmosphere due to recent emissions (and this is old data already – 750 Gt atmospheric C was the good old days).
See also link provided by KR @ 356 above.
Patrick 027says
Re 388 JCH – well, it would be as fast as Tsunamis can travel around the world (not that you would necessarily get Tsunamis, that’s just an indication of the speed over which sea level can equilibrate). There isn’t a hard and fast rule that says you can’t have global average surfacde temperature rise halt or even reverse during the overall process – I think what some people might be arguing is that outside short-term fluctuations, it doesn’t seem likely (that the heat accumulation would be so concentrated on the land-based ice or that the land-based ice would flow and calve into the sea so rapidly) – it’s interesting to consider, though, that if this happened (or if meltwater lakes flowed out and, via being fresh water, spread out over the surface of the ocean, on top of warmer water – although inevitable mixing would occur), heat accumulation would increase more or not decrease as rapidly as radiative disequilibrium would not decay as fast/persist longer/grow.
Patrick 027says
me, two comments ago: “(except for vegetation and soil, and marine biomass/organic),” – I forgot to do that in calculating the number, but it wouldn’t make a huge difference.
Patrick 027says
“(or if meltwater lakes flowed out “… – of course, that implies heating already occured.
Chris Machens @387
The impact of volcanoes on climate as a source of CO2 is surely balanced by its extraction from the climate system by rock weathering. Which means its impact on sea level is thus zero.
And my memory is that volcanoes emit an average of about 70 MtC pa while anthropogenic emissions are currently 10,000 MtC pa. If this were a meaningful ratio, which I do not see, it would be 140 times, not 10,000 times.
In this video James Hansen explains the rate and how volcanoes only put 1/10000 of a part per million Co2 into the air. Which means 1 000 000 years for the natural process to reach 100 ppm.
John E. Pearson – “Arrhenius came up with something like a 6C warming for a doubling of CO2”
Where was this? Looking at the 1896 paper, “On the Influence of Carbonic Acid in the Air upon the Temperature of the Ground” (http://www.rsc.org/images/Arrhenius1896_tcm18-173546.pdf), he states on page 275 that:
That seems very much in line with current estimates of ~3°C/doubling.
John E Pearson,
Yes, I agree, it’s not a simple formula you plug things into with our modern understanding, we’re way past that. But Six-Legged specifically asked for the Arrhenius one, which is why I brought up that paper. Certainly I would not rely upon centuries old science for a modern value, but for someone looking to do the math by hand, that’s what Arrhenius did. If one is trying to calculate it themselves, it will have to be a very simple idealized model. If we want the best science has to offer, it’s going to involve highly complicated models run on computers. I almost included this in my previous replies. If you’re looking for a formula that has one variable (delta CO2 ppm) and spits out degrees celsius delta, you’re not understanding the science.
Honestly, I’m really enjoying this AIP link. It does cover this stuff. It gives the story of the study of CO2 as a GHG, with links describing the early simple models, radiation maths, and even more modern GCMs. The story really is enlightening. Arrhenius doing absurdly large calculations with significant uncertainties, basically to take his mind of his wife leaving him and taking the kid. He wasn’t trying to understand global warming or even necessarily predict it, people were trying to solve the problem of the ice ages. It was through studying that phenomenon that the prediction of man-made global warming came about, not some conspiracy. It’s like how one of the most fundamental concepts in physics, Entropy, came out of trying to make steam engines work better.
But yeah, I wasn’t trying to say our modern understanding was easy, just pointing Six Legged to what I think he is trying to get. Simple questions like that involve a lot of assumptions, so it’s hard to know exactly what to give people.
KR,
That 3-4 was a regional/continental value? Arrhenius got something much higher for arctic regions, like 8-9. I think the 5-6 was the average for the globe. I’d have to go reread to confirm this.
Re 347 John E. Pearson I’ve wondered if I ought not to be able to putz around and do something iterative to first get the 1.3C no feedback temperature, then include H2O vapor and get a correction to atmospheric temp, then correct the H2O feedback and get a new atmospheric temp, etc. but I’ve never tried.
This doesn’t capture the nonlinear effect of variable effective heat capacity (over longer times, greater amounts of heat capacity come into play), nor any slow feedback effects, but assuming or approximating as a linear response (PS I may not be using standard notation here; sorry):
B is the Planck response at TOA per unit change in global average surface temperature T (increase in OLR at TOA (assuming instant stratospheric adjustment?), and assuming constant tropospheric lapse rate, with no compositional or optical-property feedbacks or any other feedbacks)
F is the decrease in OLR per unit T due to those other (non-Planck) feedbacks (including (important or not) any change in horizontal temperature variation at any level, etc.)
where B and F are in such units as W/m2 per K
C is the heat capacity of the climate system per unit area
RF is the imposed radiative forcing (relative to the baseline where equilibrium ∆T = 0)
H is enthalpy (~~heat) per unit area
then
dH/dt = C*dT/dt = RF + (F-B)*∆T
or
dH = C*dT = RF*dt + (F-B)*∆T*dt
—
∆T = ∆Teq (the equilibrium ∆T) when dT/dt = 0, thus
RF + (F-B)*∆Teq = 0
∆Teq = RF/(B-F)
(whereas with iteration, you start with ∆T1 = RF/B, then add ∆T(n+1) = (F*∆Tn)/B repeatedly)
ECS (equilibrium climate sensitivity) = ∆Teq/RF = 1/(B-F)
—
if RF is constant, a solution is (noting dT = d(∆T))
H/C = ∆T = ∆Teq * (1 – exp[-t/τ])
so
exp[-t/τ] = 1-∆T/∆Teq
d(∆T)/dt = ∆Teq * (1/τ) * exp[-t/τ]
d(∆T)/dt = ∆Teq * (1/τ) * (1-∆T/∆Teq)
subs: C*dT/dt = RF + (F-B)*∆T
C*d(∆T)/dt
= RF + (F-B)*∆T = ∆Teq * (C/τ) * (1-∆T/∆Teq)
subs: RF/(B-F) = ∆Teq
RF + (F-B)*∆T = RF/(B-F) * (C/τ) * (1 – ∆T*(B-F)/RF)
div by RF:
(1 – (B-F)*∆T/RF) = 1/(B-F) * (C/τ) * (1 – ∆T*(B-F)/RF)
1 = 1/(B-F) * (C/τ)
τ = C/(B-F) = C*ECS
therefore:
H/C = ∆T
= ∆Teq * (1 – exp[-t/(C*ECS)])
= RF*ECS * (1 – exp[-t/(C*ECS)])
————
The way to estimate RF by hand – get a global average OLR graph (that’s the part you can’t do easily by hand), identify the CO2 valley, find a spectrum of log(cross section) CO2 (can’t do that easily by hand either), find the interval over which the optical cross section halves outward from ~15 microns (on each side if they differ; if the slope varies look around where optical path length for the atmosphere is a moderate value ~1, give or take). Multiply the depth of the CO2 valley by the interval; that’s TOA RF before stratospheric adjustment (subtract the widenning and increase in height of the little hill in the center of that value to get a more accurate result – this probably requires some more info about the temperature profile in the stratosphere). Then do the same as if the CO2 valley OLR = 0 – that’s approximately tropopause RF before stratospheric adjustment. Subtract TOA value from tropopause value – that’s the stratospheric cooling (some additional. Assume some fraction of that is realized as a reduction in downward radiation at the tropopause, and subtract that from tropopause RF – that’s your tropopause RF (and TOA RF) after stratospheric adjustment. (Perhpas you could try to estimate B by shifting the rest of an OLR graph (outside where it reaches the tropopause-level Planck function) up by 1 K and looking at the change in area).
Or use this http://forecast.uchicago.edu/Projects/modtran.html (PS some error due to assumption of perfect blackbody surface (based on last time I used it) – this shouldn’t be a large error for tropopause and TOA RF because it is masked by H2O, clouds, CO2, etc, already; more significant if you start with no GHGs. Advantage – you can do different latitudes and conditions seperately and then find a global average from that. PS I haven’t tried it but you might be able to estimate B from this website too. The GH(LW) portion of F as well if you make some assumptions – but you can look at F values from climate models (and observations) for hints.
KR #351:
Eh, no… it says new increase (and 3°.4, not 3°-4). He talks about multiplying by 1.5 a second time, giving 2×3.4 degrees for 2.25x the amount of CO2.
Have a look at Table VII on page 266, under the header “Carbonic acid = 2.0”. It really is between 5 and 6 degrees per doubling.
JMcDonald – Lifetime of CO2?
I would suggest Archer 2009 (http://ctserver.uchicago.edu/~archer/reprints/archer.2009.ann_rev_tail.pdf) for a good look at it.
Possible Sea Level Rise of 1-3 meters (or more) within the next 50 years http://climateprogress.net/item/possible-sea-level-rise-of-1-3-meters-or-more-within-the-next-50-years.html
@357
The linked story sports a graphic of “Scenario of potential sea level rise within 100-200 years” showing Europe under 100m of water. So me droning on with my usual ‘there simply isn’t enough energy about to raise sea level over a couple of metres per century unless you fill the oceans with icebergs’ likely won’t go down to well here.
Chris, can you look again at your article on Joe Romm’s page that you’ve linked to?
The text within the featured image says “Europe 100 meter sea level rise approx. 100-200 years.”
Your article refers “… 1-3 meters of sea level rise within the next 50 years.”
100 meters in one or two centuries? What’s that from?
I think someone made a mistake when adding text to that image that accompanies Chris’s article at Joe Romm’s place. I did an image search and the picture appears several other places (French and Italian sites) — but never with that English text overlaying it.
One of those captions the same image with “If Antarctic were to melt completely, it should give us 61 meters of sea level rise worldwide….” which makes more sense.
I’m sure neither Chris nor Joe is predicting that. It’s got to be a typo on the image, it’s inconsistent with Chris’s text.
This search will work for a few hours: http://www.tineye.com/search/695aaf5a0aa883b4a2a8f05141d3de3ac089e0dd/?pluginver=firefox-1.1
Chris, I tried repeatedly to comment under your article at Climateprogress but got session timeouts, invalid tokens, etc. each time I tried.
MARodger, the image is a bit sensational i guess. But there are studies done on several meters ( around 20) SLR within a few years or decades. Following this, 1-2 centuries may yield 100 meters. The main point to take home with this image is that we require new maps for country shorelines, in the coming decades. With BAU we are on a path to an ice free state.
MARodger – if I understand the basis of Hansen’s prediction of up to 5 meters by ~2100, it’s simply that it has happened in the past. Are you saying it could not have happened in the past, or that there was more energy available then to melt it, or that just a lot of land-based ice must have been quickly dumped into the oceans to have achieved that rate?
MA,
Without a serious influx of icebergs, there simply is not enough energy to raise sea levels that much. Even with the recent loss of ice from Greenland, SLR has slowed in the last few years. Estimates much beyond the current 2-2.5 mm/yr seem unachievable, or sensational (as Chris pointed out).
[Response: More unsourced wishful thinking masquerading as informed commentary? You know the deal – provide justifications for scientific statements or don’t bother. Where for instance do you get a current ‘2-2.5 mm/yr’? You know as well as I do that all of the satellite data sets show more than 3mm/yr. – gavin]
Martin Vermeer – Thanks for the reference – I clearly misread that section. However, looking at the latitude dependencies in the paper, it appears to be a global value of ~5.2C to 5.3C/doubling, given equatorial at 4.9-5C/doubling, polar amplification at ~6C/doubling.
I have to say, though I _still_ find that Arrhenius piece an excellent work in physics extrapolation.
> I’m sure neither Chris nor Joe is predicting that.
Well, Chris thinks it’s possible, so I’m at least half wrong.
I’m still sure Joe Romm wouldn’t agree with Chris on that.
Chris, did you run that by the sea level researchers whose work you are discussing there, and has been discussed here?
Let’s see if I’m wrong again.
… You can also have land-based fresh water surging into the ocean at times; not sure how voluminous that would be exactly.
JCH @363
What I’m saying is that when I calculate the net rate of energy entering the climate system (about 0.6 W/sq m yields 9.5 zJ pa) it would only be enough to melt 28,500 cu km of ice and raise sea levels 80mm pa. Now, assuming I haven’t put a decimal point upside down (as I do occasionally), the question this raises is how do the ice & energy get together to allow the melt to happen? I’m suggesting only a portion of the energy (say 25%) can do this (ie 20mm SLR pa or 2m per century) without iceberg-strewn oceans.
The implications of my suggestion here are still very green. Past discussion about it have become quite heated at times and not very constructive. However it is safe to say that my suggestion does sit very uncomfortably with Fig 7 of Hansen & Sato 2010 which appears to suggest SLR of 500mm pa by the end of the century.
Increased energy that is used up melting ice rather than warming the climate will increase the residual climate forcing levels as CO2 levels rise and thus allow the 80mm figure to increase (a maximum of +4mm pa if all warming stopped).
One consideration I have yet to explore (but raised by your questioning @363) is that the average SLR from the last ice age I measure to be 13mm pa yet the rate of warming was far slower.
Hansen & Sato 2010 link
http://www.columbia.edu/~jeh1/mailings/2011/20110118_MilankovicPaper.pdf
Chris, the closest I can come to support your rate of change isn’t close: Meltwater pulse 1A from Antarctica as a trigger of the Bølling-Allerød warm interval — AJ Weaver et al. (2003) Science Magazine.
“Cited by 263” says Scholar.
“Abstract. Meltwater pulse 1A (mwp-1A) was a prominent feature of the last deglaciation, which led to a sea level rise of [around] 20 meters in less than 500 years….”
You can’t extend that to 300 meters in any time span, let alone 100-200 years, there isn’t enough ice above sea level.
Seriously, your text makes sense as cautionary discussion.
That one illustration — which it looks like someone borrowed from a different site and added misleading numbers to — bugs me.
The most recent (today’s) skeptical science article on the same subject refers to
“… rapid sea level rises of 9 m in 500 years during the Meltwater pulse 1a event 14,600 years ago and 2.5 m in the second event, 8,200 years ago.”
Hank Roberts, the Captcha is working. (Tested it and others could reply) Please try again.
Hank Roberts, CPN climateprogress.net is not Joe Romm’s place, as you should know because you comment on Romm’s place too.
I will try to clarify, why i believe a SLR of 100 meters can be possible within the given time. Please post your input under that article… will comment back once i updated it.
PS for Chris — the above is nitpicking (the highest form of primate social bonding, and so meant).
The article is good reading and has and cites credible numbers from sources. If you agree the illustration with the added claim on it wasn’t supported by the science sources you relied on there’s no reason to be reluctant to clarify the piece.
Science is an endless process of corrections — not the defense of anything forever in its original form.
Showing people we do that (even us kibitzers and commenters) is part of the lesson about how science works.
Best writing I know on the process, by Peter Watts: Because As We All Know, The Green Party Runs the World.
I updated the article
About the Image: Scenario of potential sea level rise within 100-200 years. This is based on the current rate humans put Co2 into the atmosphere, which is 10,000 times faster than the natural processes.
For that matter and potential singular positive feedbacks the rate of Sea Level Rise (SLR) can be assumed to rise with a similar rate. And for that matter there is no equivalent in the earth recorded history. The main SLR rise is likely not to come from melted water, rather then thermal expansion, which is attributed to be 70-75% of current observed SLR.
If you feel that this theory is not justified i welcome your feedback over at my blog.
Is there a way to post an image directly into a comment (ie copy and paste from one’s own file, rather than first posting it elsewhere on the internet and then using that to get it into the comment)?
PS AGW: this is the point in the movie where that girl says “It’s he-ere”.
Re – past rapid sea level rise references, “Deglacial rapid sea level rises caused by ice-sheet saddle collapses” Lauren et al 2012 (abstract) gives the MWP-1A as “a sea level rise of 14–18 metres over 350 years” from a P Deschamps et al reference. Lauren et al suggest a mechanism for half this rise, their paper discussed at Skeptical Science.
> climateprogress.net … not Joe Romm’s
Fooled me! won’t happen again.
I saw the “climateprogress” name, and the article’s link with Joe Romm’s name, and mistook your blog for Climate Progress.
That’s what I get reading climate blogs while traveling and rushed.
Memo to self: the website for “Climate Progress” is named thinkprogress.net.
My mistake.
I can’t follow all the blogs out there, I try to stick to the science sites and always, always read the cited sources.
I believe you’ll find thermal expansion is already figured into the published science, and you’re counting it twice, to come up with your estimate for depth. I can’t imagine how you’re getting your estimate for timespan.
I doubt you’ve discovered the scientists have missed something obvious. See what the actual scientists say; I noticed Martin Vermeer already responded to you. You can’t do much better; listen to him.
Mr. M. A. Rodger writes on the 22nd of July, 2012, at 1:59 PM:
“… how do the ice & energy get together to allow the melt to happen?”
1) From the Gregoire paper, as surface melt reaches the low point of a saddle, the amount of area available for melting jumps, surface mass balance goes negative, and the saddle melts and lowers deeper into the ablation zone, melts faster and over longer fractions of a year, rinse and repeat; voila: MWP1A
2) Add creep and basal sliding, ice only moves downhill, deeper into ablation zone.
3)ELA (equilibrium line altitude) is rising all the time, with it the ablation area, shrinking the accumulation zone. Swiss camp at 1150 m is no longer in accumulation zone, i see that ELA is up above 1700m already.
4)Your calculation for absorbed solar energy is for net albedo change. Consider that there is plenty of available energy to melt ice in the air in the ablation zone, and worse yet, rainfall delivers energy from warming ocean to the ice.
5)I think from the ASCAT images, that melt has reached not only the saddle at 67N (as we knew already,) but is now (July 2012) at the summit.
sidd
Re image 100 meter SLR. The point about this image is not the timeframe, but that we will get there on current emission path.
Chris, what’s your source for this? Who made the picture? Who added the time span to the picture? Why do you trust your source to the point you’ve blogged the claim without a cite? There’s no basis in the science for that claim that I can find.
Seriously, it looks like a mistake. Martin gave you a source to do the numbers in his comments, on your blog.
I”m done.
sidd @378
Yes but no but…
Moving Greenland’s ice surface down towards sea level will get more of it melted, but at what cost to the local energy budget, the global energy budget even? The energy has got to come from somewhere.
Let’s be ridiculous & tow Greenland to the equator. That will get the melt going but where you tow it too has be cooler as a result. That cooling will suck energy from surrounding regions. This is the same process when saddles are lowered.
Let’s be more ridiculous & cut up Greenland & scatter it (ice and all) round the tropics. Now that will up the melt rate but also cool the entire tropics, the entire globe.
As a simple matter of energy budgets, if this tropical Greenland (or a dropping saddle) is raising sea level by more than 80mm pa, it will be sucking more heat from the rest of the world than the world is being heated up by global warming. That means that even with global warming’s forcing, the whole globe will actually be cooling!
And remember Greenland is big. At 80 mm per year, Greenland has 100 years of ice to melt, be it starting at 3,000m in the Arctic Circle or at sea level in the tropics. Antarctica is ten times bigger still.
KR 365:
KR count me in as a fan of old Svante! The way he used the Langley measurements was brilliant, using the greater path length through the atmosphere as the Moon stands low in the sky as a surrogate for greater GHG concentrations.
Why Arrhenius got so high a sensitivity value? Besides his way-too-simple vertical atmospheric structure, I have the following suspicion. He only considered two gases: carbon dioxide (“K”) and water vapour (“W”). In reality of course, there are many more greenhouse gases: methane, ozone, nitrous oxides; and then there are aerosols. None of these belong to the CO2 feedback system, and should thus not be included in any computation of CO2 doubling sensitivity. But all of them are present in the sky over Arizona, masquerading in A’s computation as “K”, i.e., carbon dioxide.
But in summary, A already found the following fundamentals:
1) the logarithmic nature of GHG forcing with concentration
2) the water vapour feedback effect, strongly positive — from the skies of Arizona through the cycle of the seasons, not from physics, although he undoubtedly knew Clausius-Clapeyron (there is a similar equation in chemistry bearing his name, ask Eli ;-) )
3) the polar / high-latitude amplification.
…and look Ma, no computers! Pretty damn cool.
Chris @374 & @380
Sorry to add to the objections I’ve already made about the graphic. Then, I think it is actually the same objection but from a different direction of approach (or perhaps that’s ‘directions’.) as the problems I have stem from the timings indicated by the graphic.
Two things. First, why 10,000 times faster? If the last ice age ended over a period of 10,000 years via a 6 Wm^-2 forcing, our 3 Wm^-2 forcing over 100 years is only 50 times faster. (Note this does leave some questions even at 50 times.)
Second, I would suggest that if my protest over energy poverty for rapid SLR from melting land ice has merit, rapid SLR from thermal expansion is not the answer as it is energy-wise far more intense. A simple calc (ignoring the pressures of the deep oceans) gives expansion requiring 60 times the energy for the same SLR. This perhaps overstates the discrepancy by a few tens of percent. Indeed, measured OHC is rising by something like 7.5 zJ pa but only achieving SLR of 1.5 mm. If that energy were applied to melting (from 0 deg C), it would have managed 65mm SLR pa.
Sorry gavin,
Usually people do not require references for that which is well known, but I will comply.
[Response: People require references from you because of your history of a) making things up, and b) misinterpreting data… PDSI anyone? – gavin]
http://notalotofpeopleknowthat.files.wordpress.com/2012/05/church-white-2009_htm_7f8ac800.jpg
[Response: See how easy was that? (Better figure here, updated paper Church&White2011) Now the reason why you think that the 2009 data is ‘true’, and the 2011 update (which shows 2.8 ± 0.8 mm year−1 over the satellite period) and the satellite data itself (which are more current) are ignorable is perhaps worth discussing… – gavin]
If one were to choose a period of precisely 20 years, then a SLR of ~3mm/yr can be deduced. Over the past 30 years, SLR is ~2.5 mm/yr, while even the satellite data shows a decline to ~2 mm/yr over the past decade. Clearly not a sign of rapid acceleration.
http://www.kaltesonne.de/?p=1742
[Response:Really? You don’t even need to read German to know that is crap. But now your point is over the definition of ‘rapid’. Movement of sorts I suppose. – gavin]
Gavin,
Wow! Talk about misinterpretations! I did not misinterpret PDSI, nor have I ever accused you of not knowing how to read. Perhaps, if you were a little more forthcoming and honest, people would have more respect for your opinions. Sadly, I do not feel this will change. So ist das Leben!
[Response: Oh, how soon we forget: – gavin]
@MARodger
It seems you are not taking into account the counter-intuitive effect of a drop in surface temperatures: that would increase the amount of energy available for melting ice by increasing the global radiative disequilibrium. I suppose this effect would be moderated by the atmospheric H2O feedback (as well as by other feedbacks) but it’s not negligible.
A change in oceanic circulation or in the location of large ice masses could therefore raise the sea level, even without a CO2/solar/whatever forcing. Or it could trigger a glacial episode if the magnitude of the feedbacks temporarily overwhelmed the drop in energy radiated by the surface… this ain’t trivial (see Heinrich events)!
Mr. M.A. Rodger writes on the 23rd of July, 2012 at 3:09 AM:
“…raising sea level by more than 80mm pa, it will be sucking more heat from the rest of the world than the world is being heated up by global warming”
So stipulated.
Then what might be the largest plausible rate for SLR today? MWP1A had 50mm/yr for 500 yr, is that possible today ?
sidd
MARodger “..why 10,000 times faster?”
Because the rate humans put Co2 into the atmosphere today is of that order, when you compare it to the natural process from volcanoes.
If land-based ice suddenly goes nonlinear and ends up as floating ice a decades ahead of time, it seems like you guys are saying SLR has to wait for melting. In the case of icebergs, isn’t SLR instantaneous?
Just popping in. Saw this earlier.
Giant Beijing Rainstorm Triggers Citizens’ Anger
http://www.businessweek.com/articles/2012-07-23/giant-beijing-rainstorm-triggers-citizens-anger
Then I saw the RC article on the Miocene. Thanks.
JCH,
You are correct. Any land-based ice which becomes seaborne will contribute to SLR immediately.
Mr. JCH says on the 23rd of July, 2012 at 12:48 PM:
“In the case of icebergs, isn’t SLR instantaneous?”
GRIS would find it difficult to calve icebergs thru constricted outflow channels, easier to melt in place, export meltwater. WAIS, on the other hand…
sidd
Re Chris Machens –
1000s of years to rise 100 ppm deglaciation (5000 from memory, is this right? – IPCC has a nice graph (Ch 6 in AR 4, WGI, I think))
~100 years or less (depends on when you measure the slope) to rise 100 ppm anthropogenic
from memory, 0.2 Gt C/yr geologic emissions – tends to result in no change because there’s silicate weathering going on (but when you get a Snowball Earth, different story. 0.2 Gt C/yr would add ~ 0.094 ppm CO2 to the atmosphere if it is trapped there (a bit over 1000 years to get 100 ppm). Typically, redistribution among all surface reservoirs occurs relatively fast relative to geologic emission and sequestration by silicate weathering + organic C burial. Proportions of course can vary with the total, but if proportions were held constant (except for vegetation and soil, and marine biomass/organic), it would take about 60 (*) times longer (~ 64,000 years (*)) to get to 100 ppm – (* – actually a bit longer than that, as this is based on a situation in disequilibrium – see below). Deglaciation C changes were significantly faster.
Ratios:
100 ppmv CO2 ~= 213 Gt C ~= 781 Gt CO2, but forget sig.fig.s becauses I only used first two digits of molar masses (29 g/mol average atmosphere, 12 g/mol C, 44 g/mol CO2); based on atmosphere of 5.148 E18 kg (see http://en.wikipedia.org/wiki/Atmosphere_of_Earth , under “Density and mass”)
from http://earthguide.ucsd.edu/virtualmuseum/climatechange1/05_2.shtml , diagram shows 750 Gt in atmosphere and 44953 Gt in atmosphere, ocean (inorganic), continental and marine biomass, and dissolved organic carbon. The deep ocean in particular isn’t in equilibrium with the atmosphere due to recent emissions (and this is old data already – 750 Gt atmospheric C was the good old days).
See also link provided by KR @ 356 above.
Re 388 JCH – well, it would be as fast as Tsunamis can travel around the world (not that you would necessarily get Tsunamis, that’s just an indication of the speed over which sea level can equilibrate). There isn’t a hard and fast rule that says you can’t have global average surfacde temperature rise halt or even reverse during the overall process – I think what some people might be arguing is that outside short-term fluctuations, it doesn’t seem likely (that the heat accumulation would be so concentrated on the land-based ice or that the land-based ice would flow and calve into the sea so rapidly) – it’s interesting to consider, though, that if this happened (or if meltwater lakes flowed out and, via being fresh water, spread out over the surface of the ocean, on top of warmer water – although inevitable mixing would occur), heat accumulation would increase more or not decrease as rapidly as radiative disequilibrium would not decay as fast/persist longer/grow.
me, two comments ago: “(except for vegetation and soil, and marine biomass/organic),” – I forgot to do that in calculating the number, but it wouldn’t make a huge difference.
“(or if meltwater lakes flowed out “… – of course, that implies heating already occured.
CO2 increase something like 5x as fast as during the PETM, according to secondary sources I found, which cite that to
Trends, Rhythms, and Aberrations in Global
Climate 65 Ma to Present
James Zachos, et al., Science 292, 686 (2001);
DOI: 10.1126/science.1059412
(search will find that full text if you look through the first few pages of results; it’s been cited upwards of 2,000 times)
10,000 is a big, round number.
5 x the PETM rate of change is, to me, terrifying.
What matters isn’t how much is added, what matters is how much stays in the atmosphere and oceans after natural cycling.
We’re doing something like 2x the natural cycling rate now, if my poor memory is correct. Traveling, busy, someone else may get a better number.
Educate people with these numbers, help them mean something.
Chris Machens @387
The impact of volcanoes on climate as a source of CO2 is surely balanced by its extraction from the climate system by rock weathering. Which means its impact on sea level is thus zero.
And my memory is that volcanoes emit an average of about 70 MtC pa while anthropogenic emissions are currently 10,000 MtC pa. If this were a meaningful ratio, which I do not see, it would be 140 times, not 10,000 times.
Re Patrick 027
In this video James Hansen explains the rate and how volcanoes only put 1/10000 of a part per million Co2 into the air. Which means 1 000 000 years for the natural process to reach 100 ppm.
http://youtu.be/TSexmRu3kbM?t=6m28s