## Lansner Reloaded

__Introduction__

Frank Lansner has presented an interesting derivation of climate response to a doubling of CO2. He uses current values for Greenhouse warming and MODTRAN radiative forcing calculations to derive the temperature response. While his approach is interesting, his presentation is confusing and poorly presented. Here I drop most of his discussion to draw out the basic math underlying his main insight – using current Greenhouse conditions to derive a temperature climate response to CO2 doubling.

__Derivation of Climate Reponse to CO2 Doubling__

The Greenhouse effect accounts for about 33C warming.

CO2 accounts for about 18% of the Greenhouse effect.

The warming due to CO2 is **dT_total_CO2 = 33C * 0.18 = 6C.**

Starting at 5ppm CO2, there are 6 doublings to 320ppm CO2.

**(5,10,20,40,80,160,320)**

Using MODTRAN, at 0ppm CO2, the total forcing is:

**RF0 = 318.396**

Using MODTRAN, at 5ppm CO2, the total forcing is:

**RF5 = 308.631**

Using MODTRAN, at 320ppm CO2, the total forcing is:

**RF320 = 288.597**

The change in forcing from 0ppm to 320ppm is:

**RF0 – RF320 = 29.799**

The change in forcing from 5ppm to 320ppm is:

**RF5 – RF320 = 20.034**

The change in forcing from 5ppm to 320 ppm is xx%:

**(RF5-RF320)/(RF0-RF320) = q = 0.6723044**

Assuming that the change in temperature is

directly proportion to the change in forcings,

then the change in temperature over the last 6 doublings is

**dT_6doubles_CO2 = dT_total_CO2 * q = 6C * 0.67 = 4C**

Assuming that the change in temp is roughly the same

between each doubling (and the MODTRAN model supports this)

then the change in temperature due to a single doubling

is simply the change in temp over 6 doublings divided by 6

**dT_6doubles_CO2/6 = dT_1double_CO2 = 4C / 6 = .67C**

This is the final answer:

each doubling of CO2 results in a change of temperature of .67C

__Discussion__

Lansner’s presentation is confusing. The 9.25 factor thingee doesn’t seem to make much sense. It appears to simply be the inverse of the fraction of CO2 greenhouse warming over a single doubling within a range which includes an intial warming + 6 doublings. But we could easily take an additional doubling or drop one (starting with 2.5ppmCO2 or 10ppmCO2) and this factor will change by a large degree. It is an arbitrary factor, a magic number based on starting the doubling counts at 5ppm.

But what about this version of the derivation? Is it similiarly arbritrary? Let’s test it.

First, we check just the previous 5 doublings.

RF10 = 305.522

**(RF10-RF320)/(RF0-RF320) = q = .57**

**dT_5doubles_CO2 = dT_total_CO2 * q = 6C * .57 = 3.4C**

**dT_5doubles_CO2/5 = dT_1double_CO2 = 0.68C**

Next, we check the previous 7 doublings.

RF2.5 = 311.237

**(RF2.5-RF320)/(RF0-RF320) = q = .76**

**dT_7doubles_CO2 = dT_total_CO2 * q = 6C * .76 = 4.6C**

**dT_7doubles_CO2/7 = dT_1double_CO2 = 0.65C**

So this derivation is fairly robust even when the number of doublings changes. It is not tied to an arbitrary starting point. Lansner had an interesting insight and I find this form of the derivation charming in its simplicity.

Note that my derived climate response is 0.67C while Lansner’s is 0.54C. That’s largely due to our different values for the amount of current warming due to CO2 – I have used 6C and he used 5C. We can nearly recover his climate response simply by multiplying mine by 5/6ths: 0.67C * 5 / 6 = .56C.

All forcings were calculated with MODTRAN web interface set to the default values except for the CO2. That includes alt = 70km, sensor facing down.

A ppt presentation addressing some of these issues:

http://geotest.tamu.edu/userfiles/88/greenhouse_effect.pdf

That GE pdf is good. Do you know who wrote it? I hadn’t seen the log derivation presented so clearly as on p 35.

But that shows the limits of the theory. He integrates from OD 1 to infinity. The 1 is rather arbitrary, and the notion that the log result for a single freq makes a log result when the freqs are combined is rough. But it does say something about how many doublings down you can do. The fact that the answer depends on how many you assume is a problem with FL-type thinking.

The log notion fails when OD 1 hits the ground. And for some CO2 IR absorbing bands, that’s happening now. Their effect is a small proportion, but grows rapidly as you double down. I think you’d lose the log property even at 80 ppm, judging from the OD plot on p 15.

The PDF looks like it comes from Gerald North.

Thanks for this, I found Lasner’s description completely mystifying.

Mike

some strange circular thinking here.

The no-feedback sensitivity is about 1.2 C per doubling of CO2 (IPCC AR4 Chapter 8).

Using the MODTRAN app in the most simplistic way, using default settings, I get 0.89 C, +/- 0.01 C, for a range of doublings (50 to 100 ppm, 280 to 560, 320 to 640, 375 to 750). This isn’t the best calculation, but it’s ballpark, no? While I’m at it, if I do an extreme simplistic version of the water vapor feedback (hold rel. humidity constant, leave everything else be), I get it up to 1.5 C for a doubling from 375 to 750.

As for Lasner, the only thing he’s getting out of Modtran is the change in forcing for some change in CO2. I don’t think this is quite the textbook definition of forcing (at the tropopause, and the stratosphere is allowed to respond). But if you accept for the moment that 5.35 ln (C/Co) gives a good enough approximation to the results of the current best radiative transfer calculations, you can just use 5.35 ln (C/Co) to find the change in forcings, and put Modtran aside. Which brings us back to the problematic starting point:

“CO2 accounts for about 18% of the Greenhouse effect.”

This is an ill-defined thing, and used in a way that you can’t. Chase down where this value came from – I think you get something like this if you remove all CO2, and leave everything else the same. There are also higher estimates out there, done other ways. But because of overlaps, if you do that for each greenhouse gas, it won’t add up to 100. Plus, things interact with each other, meaning it’s physically impossible to have all else held the same, that’s only a mathematical construct. In any which case, it’s a strange place to start this calculation.

to the other comments – yes, the 5.35 ln rule doesn’t work at all concentration ranges.

I think the paper you want is from Myhre et al, late 90s sometime, I’ll check.

In North’s presentation, he calculates a temp after removing all the greenhouse gasses and then adds back CO2 first, which gives us a 6C greenhouse effect. 6/33 ~= 18%

If we do it the other way around, I notice that in the numbers above,

an atm with 320ppm CO2 has a flux = 288.597

an atm with 0ppm CO2 has a flux = 318.396

(RF0 – RF320)/RF320 ~= 10%

The reason for the two different results (10% and 18%) is due to the order in which CO2 is added/removed from the atm. The order is important because of the overlapping absorption bands. Remove CO2 last, and some of the CO2 bands are still covered by water vapor. Add CO2 in first, and you get the impact of CO2 but overstate its role in fully mixed atmosphere. I’ve known this, but this is the first time I’ve bumped into the issue directly.

CE … screwing around with the MODTRAN web interface, I noticed that the RF for a doubling was roughly 3 W/m^2, notably lower than the commonly accepted 3.7 W/m^2. I don’t know why. (Wet -v- dry atm?)

Lasner’s main remaining error in the comments seems to be his assertion that the magnitude of the feedback response has to be the same for all doublings of CO2.

I wonder if someone could do a simple derivation of the WV feedback response (simply holding rel humid constant) for each doubling from 5 to 640 or so? Would show the point rather well.

I think this paper is the originator of the 5.35 ln (C/Co) approximation.

http://www.agu.org/journals/gl/v025/i014/98GL01908/98GL01908.pdf

At a glance, it doesn’t look like they go into the absurdly low CO2 range. That said, I don’t think that’s Lasner’s biggest mistake here. The whole approach is just flawed.

Ron, I think both of us found the same discrepancy between simplistic use of Modtran, and the usual rules of thumb. Just expressed in different ways. I don’t know if that’s coming out of not doing a proper global mean over relevant conditions, or a modtran thing; modtran is a bit dated I think.

AGU paywall:

http://www.agu.org/journals/gl/v025/i014/98GL01908/98GL01908.pdf

See instead:

http://folk.uio.no/gunnarmy/paper/myhre_grl98.pdf