# Snow+Ice (by request)

Several readers requested an approximation of the total climate forcing caused by the loss of ice and snow during the satellite era. OK

As before, this is a crude calculation but it should be correct to first order. The simplifying assumptions are that the combined snow+ice area is a circle around the pole, and that the insolation is that falling on the mid-latitude of the snow/ice pack on the mid-day of each given month. What’s computed is the total solar power falling on parts of the earth that are ice/snow covered. As more than one reader pointed out this isn’t really “insolation,” which is solar power per unit area — this is the total solar power (energy per unit time) hitting the top of the atmosphere over snow/ice regions. And the calculation is only for the northern hemisphere. I don’t have snow data for the southern hemisphere (but there’s far less snow-covered area in the south, since there’s far less land than in the north, especially at extreme latitudes, except for Antarctica).

First, here’s the total change in snow+ice covered area over the satellite era in millions of square kilometers, as estimated from the net change of the trend line:

The change has been greatest during exactly the months it really counts — during summer, when the incoming solar power is greatest. And the overall change has been a decline. An extreme decline. During June, the total snow/ice cover in the northern hemisphere has decreased by over 6 million km^2. The increasing trend during winter doesn’t reach statistical significance, but the decrease during summer does.

Here’s the annual average solar power falling on snow/ice covered areas, together with a linear trend (note this does not include the dramatic change in 2012, which isn’t over yet):

The net change estimated as the difference of the beginning and ending values of the trend line is about 880 TW. If spread over the entire surface of the earth, and if the difference in TOA albedo between snow/ice-covered and uncovered regions is 0.2, this accounts for a total climate forcing of about 0.34 W/m^2.

We can instead estimate the net change using a lowess smooth rather than a linear trend:

In this case the net change is about 1150 TW. If spread over the entire surface of the earth, and if the difference in TOA albedo between snow/ice-covered and uncovered regions is 0.2, this accounts for a total climate forcing of about 0.45 W/m^2.

These numbers are substantial, and as I say they’re only a first-order estimate based on some simplifying assumptions. But they’re in good agreement with other estimates from the literature, and certainly in the right ballpark.

I’ll close with a shameless self-promotion. If you want to support your local global climate blog, feel free to use the “donate” button in the right-hand column near the top of the page. Any amount is welcome, 5 bucks, 10 bucks, a million (in case Oprah is reading).

### 82 Responses to Snow+Ice (by request)

1. Bruce

Thanks for doing this. Your number is pretty consistent with Flanner et al. (2011)(through 2008).

I think this factor needs much more attention. It is nearly one-third of the net anthropogenic forcing (and possibly considerably more), and isn’t included in the IPCC’s breakdown:

http://www.ipcc.ch/publications_and_data/ar4/syr/en/figure-2-4.html

Since this effect has really started to kick in in earnest, and has been previously underestimated (or overlooked entirely), I think we’re going to have to downwardly revise the amount of CO2 we can “safely” add to the atmosphere. Given the new research on permafrost CO2, I suspect we’re completely out of wiggle room, and the 2 degree C target is hopeless.

2. By the time Bob Watson is publicly throwing in the towel on 2 degrees (BBC: Science advisor warns climate target ‘out the window’) we can be reasonably certain it’s been out the window for some time.

Kevin Anderson, former Director of the Tyndall Centre in the U.K. has been saying since before Copenhagen what Watson is saying now.

For a sample of Anderson’s thought, listen to “Are climate scientists the most dangerous climate skeptics?”

Anderson believes all the talk that civilization can continue expanding the numbers of humans who experience prosperity while maintaining net economic growth for all AND limit global warming to 2 degrees given the size of the fossil fuel infrastructure and given what is known about how quickly that fossil infrastructure could possibly be phased out is a fantasy many scientists he knows believe is impossible. He says silence gives consent. “We have sat quietly by while this litany of nonsense has gone before us, in terms of the scenarios that have been generated, and we’ve said nothing…. We are culpable as a research community in terms of a conspiracy of silence…. We don’t agree with what’s been going on, but we don’t want to bite the hand that feeds us”.

Moving along….

“Gavin” comments at Stoat, i.e. under the post “Wadhams on seaice, again” regarding back of the envelope calculations of the net forcing effect of changes in sea ice etc., this: “that isn’t even the biggest problem which is that this is a feedback and not a forcing. Since the feedback was presumably present in any observationally contstrained estimate of the climate sensitivity, this effect can’t possibly be used to argue for an increased sensitivity over expectations.” to which blog owner Connelley responded “…really its just an illustration of how big a radiative perturbation it is, compared to CO2 which is probably quite a useful idea”.

There is the Figure 2 in Hansen et.al. Paleoclimate Implications for Human-Made Climate Change which shows a 3.5 +/- 1 w/m2 forcing coming from ice sheet and vegetation change that was about as significant as the 3 +/- .5 w/m2 forcing coming from GHG change in the total forcings present as the Earth was driven out of the last ice age by the relatively tiny Milankovitch trigger forcing of about 0.2 W/m2.

Civilization, which has replaced the Milankovitch forcing as the net primary driver of climate change is having a net effect already three times the size of a Milankovitch forcing (as calculated by Hansen et.al. Earth’s Energy Imbalance), i.e. 0.6 W/m2.

So, we should expect very large forcings to materialize as feedbacks, i.e. from changes in Earth’s albedo or GHG emanating from melting permafrost such as what is being discussed in the latest posts on this blog….

And keep in mind Hansen’s estimate for what anthropogenic aerosols are doing to restrain other anthropogenic effects is -1.6 W/m2….

As the hope that civilization might limit global warming to 2 degrees fades away, it is worth reconsidering how serious Hansen believes the consequences of achieving that target would be:

“The paleoclimate record reveals a more sensitive climate than thought, even as of a few years ago. Limiting human-caused warming to 2 degrees is not sufficient. It would be a prescription for disaster”. (see Paleoclimate Record Points toward Potential Rapid Climate Changes)

I wonder what the best assessment for the trend for heat content and salinity of the Arctic ocean is. I hear there are no Argo floats up there.

3. JCG

I would be interested in seeing a similar analysis of the Southern Hemisphere even with estimated snow + ice areas.

[Response: As I mentioned, I don't have snow cover data for the southern hemisphere. But we know that there's very little land to put snow cover on except in Antarctica, which hardly can have changed its snow cover status. As for the sea ice impact in the southern hemisphere, I already computed that here.]

4. KR

Attempted a donation, no success – We cannot process this transaction because there is a problem with the PayPal email address supplied by the seller. Please contact the seller to resolve the problem.

[Response: You're the 2nd to mention this. We've taken steps -- and would appreciate if you'd try again. Thanks.]

5. Bruce

Thanks for the links. I’ve been saying for some time — as have others — that climate scientists, being scientists, make conservative assumptions. But when you systematically make conservative assumptions, then use the resulting conservative results as the basis for more conservative research, the net result is an almost-certain gross underestimate of consequences.

Since we’re depressing ourselves, Bill McKibben’s Rolling Stone article makes it hard to envision a solution:

http://www.rollingstone.com/politics/news/global-warmings-terrifying-new-math-20120719

“I hear there are no Argo floats up there.”

There’s one:

http://www-hrx.ucsd.edu/www-argo/statusbig.gif

;)

6. Are you sure you’re not building your own climate model? ;-) Thanks for this, I’ve tried this with relative amounts (=assume solar constant is uniform over a grid cell the size of earth’s surface and try to estimate the effects of clouds, ice and snow and how they change over a year (i.e. one box model). It could indeed be better to use the values for energy rather than estimate the total TOA albedo resulting from various subcomponents since they overlap way too much. a(total)=a(uppercloud)+a(lowcloud)+a(snow)+a(ice) certainly won’t do since albedo is in range 0 – 1 and the interplay between humidity and the clouds that result in a relatively lower effect from the ground-based albedo subcomponents. (f.e. when the arctic summer gets more humid after summer solstice).

Is PayPal still preventing transfers to Bradley Manning supporters who want free speech? Might buy a beer or three to you if we’d somehow met in person.

7. KR

Tried a donation again – same results.

[Response: Others have succeeded, I'm unable to explain it. I'll keep investigating. Thanks for trying.]

8. kevjohnno

Donation page worked fine for me on first attempt last night so it seems strange.
Well worth it too.

9. Tamino,

I’ve tried to calculate the change in absorbed solar energy in the arctic sea using solar radiation data from ECMWF and sea ice concentration data from University of Illinois at Urbana-Champaign (the Cryosphere Today site).

From this I’ve found a change in total power absorbed by the northern sea equal to a global forcing of 0.052W/m2 (change from 1979 to 2011). The result is very preliminary, and I would very much like to have feedback. I’ll try to write the method up in the next couple of days, but briefly what I did was:

I got solar radiation from the ECMWF ERA-interim reanalysis:

http://data-portal.ecmwf.int/data/d/interim_full_mnth/

I chose Surface solar radiation downwards from the synoptic monthly means. The data are monthly means for every 3-hour slot, from 1979 to July 2012 (at the time of writing). The data are on a lat/lon grid with resolution 0.75 degree.

I imported these into GRASS GIS. This gives monthly maps of average solar irradiation , measured in J/m2 per day.

I got the sea ice concentration data from UIUC at:

http://arctic.atmos.uiuc.edu/SEAICE/ice.1870-2011.gz

These data are monthly mean sea ice concentrations form 1870 to 2011, on a 1 degree lat/lon grid. I imported these into GRASS as well.

For each month I calculated for each pixel in the sea ice concentration maps the fraction of incoming solar radiation that would be absorbed. I used a sea ice albedo of 0.4 and a sea water albedo of 0.1 (are these roughly correct?). The fraction of energy absorbed is then

R=(1-0.4)*C+(1-0.1)*(1-C)

where C is the sea ice concentration.

The total energy absorbed in a pixel in the map in a given month is then given by:

E=D*R*A*H

with

D number of days in month
R fraction of energy absorbed
A Area of pixel
H Monthly average of solar irradiation per day

The area of the pixel is calculated as cos(lat)*(0.75*111120)^2 m2, where lat is the latitude of the centerpoint of the pixel.

In the calculation of the total absorbed energy I excluded land areas and areas of the sea that were never (1979-2011) even partially ice covered. This means that I’ve included areas that were only rarely and only partially ice covered. A more restricted area could give different results. The chosen area is ~19000000km2.

The total yearly absorbed solar energy and incoming solar energy in the chosen area are (in Petajoule):

Year Absorbed Incoming at surface
1979 4.362E+07 5.820E+07
1980 4.398E+07 5.865E+07
1981 4.328E+07 5.739E+07
1982 4.379E+07 5.864E+07
1983 4.367E+07 5.847E+07
1984 4.385E+07 5.844E+07
1985 4.415E+07 5.889E+07
1986 4.363E+07 5.793E+07
1987 4.368E+07 5.846E+07
1988 4.415E+07 5.856E+07
1989 4.326E+07 5.733E+07
1990 4.390E+07 5.745E+07
1991 4.396E+07 5.820E+07
1992 4.411E+07 5.875E+07
1993 4.394E+07 5.794E+07
1994 4.398E+07 5.835E+07
1995 4.365E+07 5.725E+07
1996 4.432E+07 5.850E+07
1997 4.418E+07 5.831E+07
1998 4.383E+07 5.797E+07
1999 4.386E+07 5.846E+07
2000 4.437E+07 5.888E+07
2001 4.472E+07 5.944E+07
2002 4.424E+07 5.845E+07
2003 4.468E+07 5.883E+07
2004 4.413E+07 5.831E+07
2005 4.408E+07 5.784E+07
2006 4.373E+07 5.711E+07
2007 4.503E+07 5.805E+07
2008 4.481E+07 5.875E+07
2009 4.471E+07 5.823E+07
2010 4.389E+07 5.670E+07
2011 4.407E+07 5.673E+07

For the yearly values I then used the fit function in GNUPLOT to fit straight lines to the data. The result is:

E_a=25733*(t-1979)+4.36004e+07 PJ, Asymptotic Standard Error on slope is +/- 6116

E_s=-15114*(t-1979)+5.84223e+07 PJ, Asymptotic Standard Error on slope is +/-11640

The total incoming solar energy shows a small negative trend (I guess not significant), while the absorbed energy shows a positive trend that is about twice as large.

The total change in yearly absorbed energy is 32 years * 25733PJ. Distributed over 5e14m2 and 86400*365seconds gives around 0.052W/m2

There are lots of sources of uncertainty in this. I don’t know how accurate the ECMWF data are over the sea generally and the arctic in particular. The coarse grid and the fact that there are two different grids (0.75deg. and 1.0deg) mean that some of the coastal areas could be included as well, or near-coast sea areas excluded. Solar radiation can change quite a lot from land to sea.

I don’t know how realistic the difference in albedo is. Judging by the graph in your post on sea ice insolation, when the sun is low in the sky as in the arctic, the difference might be greater than I’ve assumed.

Plus, of course, I could have made mistakes.

I’d be most grateful for comments. I will try to put all the scripts and software on the web, if anybody would like to have a look.

• Rob Dekker

Thomas, that’s awesome work !

One note for the moment : I think your estimate of snow albedo (0.4) is too low. I’ve seen estimates between 0.5 (for old snow) and 0.9 (for fresh snow). I think a average of 0.6 or 0.7 is more accurate, which would also bring the resulting forcing to about 0.1 W/m^2, similar to what Hudson had estimated.

More later.

• The calculation I’ve done is so far only for sea ice, snow albedo might be different. I don’t know if it is possible to find gridded data on snow, but if that is available it would be possible to repeat the calculation for snow-covered land areas.

My suspicion is that the relatively low forcing is due in part to clouds. The actual surface insolation is quite a lot lower than the theoretical clear-sky values. I’ll try to quantify that ASAP (but not today, exam’s coming up). Also, the decreasing overall insolation, though not statistically significant still accounts for quite a lot of the overall change. I’ll try to see if I can separate the effects.

• Rob Dekker

Thomas, thanks for your response. Two notes :

First, on albedo of sea ice, please note that sea ice is almost always snow covered.
All-wave albedo varies depending on the state (phase) of the snow cover which changes over the melting season.

Perovich et al 2000, here :
http://www.iup.uni-bremen.de/seaice/amsredata/modis/sea_ice_papers_database/ice_in_situ_meas/Perovich_etal_2002_SHEBA.pdf
is reporting sea ice cover albedo measurements from the SHEBA project in the Arctic, and finds the following surface albedos, depending on the phase that the snow cover over sea ice goes through.

- 0.8 – 0.9 during April (dry frozen snow values)
- 0.7 – 0.75 during June (wet/melting snow values)
- 0.65 for bare, white ice (all snow melted) in July/August.

Similar numbers obtained from Antarctic sea ice phases in Brandt et al 2005.

So your albedo number for full ice concentration (0.4) is almost certainly (much) too small. If you fix the number to one value, then please set it to 0.7 or so, but if you want to be even more accurate, you could let it slide from 0.85 in April down to 0.6 in August.
That way you will match Perovitch’ albedo observations much better.

Could you please adjust that albedo number, and re-run the calculations ?

Rutgers has gridded snow cover (concentration) numbers,
http://climate.rutgers.edu/snowcover/index.php
I’m not sure about the grid and the file formats etc.

But doesn’t ECMWF already have an entry for snow cover ? I recall I saw one in your link….
If so, and if that data is trustworthy (more or less matches Rutgers), then you should be able to do the calculations in the same grid as the surface downward solar flux data from ECMWF.

Thomas, once again, THANK YOU for going this far in your analysis. This is pretty exciting stuff, and we have not even gotten to discuss the moderate drop in incoming solar flux during years with low sea ice (I agree that this suggests at changes in cloud cover).

• Rob Dekker

Thomas, are you still there ?

10. Aaron Lewis

In the old days, incoming solar energy was all that mattered at the poles because latent heat condensed out before it approached the polar regions. Thus, in the past, there was no mechanism to rapidly transport large amounts of heat from the temperate regions to the polar regions. (Ocean currents transported huge amounts of heat, slowly.) Now, through the magic of polar amplification, the Arctic summers are warm enough to allow latent heat from the south to intrude deeply into the Arctic.

This is a very significant effect. Here we are deep into October and there are places on Greenland that are still above 40F. In the old days, when Greenland was dry, it froze when the sun went down for a few hours. And that layer of frost reflected the morning sun.

The new latent heat from the south drives early loss of frost, snow cover, and sea ice. Now, the Arctic can start warming and melting in the spring, even before the sun comes up. This expands and extends the Albedo effect.

11. Russell Seitz

Things look bad for reindeer, but in the light of Rep. Broun’s recent remarks , it may be unwise for climate scientists to wear antler hats to Georgia Tech Christmas parties.

12. Norman

You have demonstrated how less ice and snow in the Northern Hemisphere will lead to more solar energy heating the surface during the summer months. I am wondering what happens during the winter months with that increased energy.

Warmer bodies radiate more energy. Less snow will warm the land in summer but less winter snow will allow more heat to be lost in winter. Snow acts not only as a solar reflector but also as an insulator keeping the ground below warmer than if there is no snow.

Arctic sea ice would have similar effects. In the summer the ocean would absorb more solar energy with less ice. Now it is winter and no sun above the Arctic circle. The warmed Arctic water is radiating away its energy with no incoming solar energy to balance the loss. Ice acts as a powerful insulator keeping the water below warm in winter. Now with less ice (before it reforms) the water will continue to radiate away the heat it collected during the summer months.

The question I am bringing up is how does all this balance. Less ice causes more warming in summer but how much more cooling takes place during the winter without the insulating property of ice to keep the water below warmer?

• Uh, Dude, what do you think happens to the seawater when the sun goes below the horizon for a couple of weeks?

• I believe that this has been discussed as a real or potential negative feedback. But for the time being it seems likely that refreeze will tend to be pretty rapid, limiting the effect. If that is coupled with increased snow on the sea ice (which I think is a possibility as far as is known right now), then the insulation between ocean and atmosphere might actually improve, either overall, or in specific high snow-burden regions of the ice.

Bottom line, I don’t think there’s any reason to expect a negative feedback of 1.

13. Rob Dekker

Sorry Tamino, but you donation page sucks. I tried to donate via PayPal, but twice it did not accept my entry. Then I tried direct CC entry, filling out all info, only to find out that it thought I did not enter a donation amountt (even though I did). Second try, I first pressed “Update Total”, reentered CC info, and pressed “Review donation and continue”. That directed me to a page that told me that I needed to log into my PayPal account before I could access that page. When I did, it told me that there were no transactions.
Now, I’[m not sure if anything happened, or if I donated three times.

[Response: According to my account info, none of your attempts worked.

I'm very grateful for your repeated attempts, and I'm sorry for the frustration. Something is very weird, because I know that others succeeded with no problem. Paypal will be getting an earful.]

• KR

That matches much of my experience – repeated requests for login, wiping transaction amounts when I did, and finally (after 3rd try) telling me that “…there is a problem with the PayPal email address supplied by the seller…”

• Rob Dekker

Tamino, I tied it again, and this time did a full debug run (I am an engineer after all) of the entire process :

First, I choose the “Donate quickly with PayPal” route :

- If you enter the amount but DO NOT press the “Update Total” button before you enter PayPal userid/passwrd, then when you press “Log In” you get this message : “Please enter an amount greater than zero”.

- If you DO press “Update Total” button, and THEN enter PayPal userid/passwrd and click “continue” then it pops up a page with the header “Peaseblossom’s Closet” and the message :

“We cannot process this transaction because there is a problem with the PayPal email address supplied by the seller. Please contact the seller to resolve the problem…bla bla bla”
and in a second box :
“The seller will only ship to confirmed addresses. To complete this transaction, you will need to enter your information again.”

Second, if you go the “Don’t have a PayPal account” route, and enter “continue” for CC payment info, then :

- Enter all billing info on the page, and press “Review donation and Continue” then it pops up a paypal page that says :
Of course that is silly, since I went the “Don’t have a PayPal account” route.
Any way, if I log into PayPal any way, I get to my PayPal account, and no transactions are mentioned.

Since in the “second” route I entered the CC that matches with my PayPal account. So : I tried one last thing :

Try the “Don’t have a PayPal account” route. Don’t forget to press “Update Total”. Enter a different CC number than the one that PayPal has on file. And….TADA !!! It went through !
Donation to Missletoe now complete !

So it seems that IF you try to pay with your PayPal account, then there is a problem with the “seller address”.
If you enter CC info directly, and you have a PayPal account for that CC number, then you end up in never-never land.

ONLY if you do NOT have a PayPal account, or you enter a CC number that is not on PayPal file, then your donation goes through without problem.

OK, Tamino, with this info you can go beat up these PayPal guys…

[Response: Thank you so much for the detailed information. I'm impressed by your thoroughness and persistence. And yes, this will make it easy to communicate the problem. Frankly, it's flattering to have such high-quality readership.]

14. Jim Eager

Norman wrote: The warmed Arctic water is radiating away its energy

Only until it freezes over with new ice, which is happening pretty fast, as it always does, maybe even faster since surface water salinity has been reduced more than normal. It’s already nearly caught up with 2007 area and extent:
http://arctic-roos.org/observations/satellite-data/sea-ice/ice-area-and-extent-in-arctic

Ice acts as a powerful insulator

Not really, it acts more as a lid, sealing off the sea surface from the atmosphere, thus preventing transfer of heat through the latent heat of evaporation and convection.

15. Norman

Jim Eager,

If ice did not act as an insulator to radiation then why wouldn’t the warm ocean water below the ice continue to radiate away its energy through the ice and freeze much thicker than the thin ice problem we have had?

16. Norman, the ice is no more an insulator than the lid on a pan is–yes, the water boils faster with it, but you’ll still burn your hand if you touch it.

17. Norman

snarkrates,

Hate to disagree with you but the experts in the field do state ice and snow are both insulators.

http://nsidc.org/cryosphere/seaice/processes/thermodynamic_growth.html

I was asking for information about the balance of energy. It is obvious that lower albedo surfaces (water, ground) will absorb more solar radiation in summer. How much more energy will be lost in winter without snow and ice cover? Tamino calculated the total forcing because of the loss of snow and ice in the summer was 0.45 watts/square meter. This is the summer version of things. I am asking how much more energy is lost in winter without the insulation of snow and ice. The Arctic ice is building back but slowly leaving lots of water to continue to radiate away.

• Rob Dekker

Norman,
Your question about heat balance, and what happens to the heat accumulated in summer after the sun sets in the Arctic, seems valid to me.
In fact, I believe it actually touches on an important difference between climate models and reality.

The common scientific explanation is that heat accumulated in summer leads to open ocean, which makes fall temperatures higher than they would be with ice cover. These higher temperatures in fall causes increased radiation to space, and, because thin ice insulates less than thick ice, ice growth during fall is faster than ‘normal’.

These higher temperatures in the Arctic affect fall/early winter in the Northern Hemisphere, and increased ice growth can last into winter are a clear “negative feedback” mechanism, but models suggest that this effect starts to “mellow” out as ice thickness approaches what would be “normal” late in winter. The common understanding is that probably most of past summer’s heat has dissipated by end of winter, and there is little “memory” of past summer melt by the next year.

Papers like Tietsche et al confirm this scenario. Tietsche “removed” all ice in summer and then let CMIP3 models run forward and see what happens. They suggest that 2 or 3 years down the road, ice cover “recovers” to a certain “mean” that is fairly independent of weather evens (and albedo change) in summer.

The problem with these model runs is that they do not match observation. Reality seems to be much more sensitive than our models suggest. Here is a plot of model runs versus reality, for Arctic sea ice extent :

So there is a real issue explaining why Arctic sea ice seems to be declining much faster than our models suggest.
One possible explanation comes from the ice volume data from PIOMAS :

In this graph, you see that ice melt (during the summer) significantly increased from about 16000 Gton/season before 2000, to about 19000 Gton/season in the past couple of years. The negative feedback mentioned above is also apparent, since the volume of ice freezing up in the fall/winter increased as well (from about 19000 Gton/season before 2000 to about 17000 Gton/season in the past couple of years). Needless to say that overall, the negative feedback of increased freezing is simply NOT keeping up with the losses in summer, and the Arctic is loosing close to 1000 Gton on an annual basis.
Since there is only 3000 Gton ice left over at the minimum now, and its loosing 1000 Gton/year one may wonder where the “mean” really is that Tietsche et al and GCMs were referring to or if we are simply heading for a “crash” landing….

So far, reality suggest the latter, and do NOT support IPCC GCM projection. At all.

• 19,000 & 17,000 GT switched in 2nd last para, maybe?

• Rob Dekker

Thanks Kevin.
Point is that summer ice loss increased ~3000 Gton/season over the past decade while ice freeze increased ~2000 Gton.
Result is ~1000 Gton annual loss of sea ice volume.
With only 3000 Gton left over there at the minimum, one may wonder how long there will be ice in summer in the Arctic.

18. Jim Eager

“why wouldn’t the warm ocean water below the ice continue to radiate away its energy through the ice”

Oh my, you are confused, Norman. Sea water has to cool to ~ -2C before ice can form in the first place, and at -2C surface sea water is not radiating very much energy. It gives up a last gasp as the heat of fusion and then cools no more.

19. Bruce

I think Norman asks an interesting question. Some of the extra heat will be lost back to space by the additional radiation due to the loss of insulation (not to mention warming the air, which then radiates to space as well). It’s even possible that by stripping away the insulating layers you actually lose more heat than you take on through the decreased albedo. It would be ironic if the ice cap actually worked to warm the planet, and by removing it the planet cooled.* A counter-intuitive negative feedback, for sure. (Yes, it results in all kinds of chaotic weather, but since when has the planet ever cared about the weather?) But even if the feedback does exist, we’re likely to overwhelm it with carbon (if we haven’t already).

*I seriously doubt this is true. Most of the heating is local to the arctic, so you’re mostly re-radiating the extra energy you’ve absorbed — a process which is thermodynamically guaranteed to leave some heat behind. Still, the ocean is a big circulating heat repository and leaving it exposed has the potential to dump a lot of heat. Someone with a GCM should look into this.

20. David B. Benson

SOmewhere on the web are actual lecture series about ice formation on lakes — by extension in the Arctic Ocean. As I recall these, so call as the surface air temperature is cold, the ice continues to thicken from below. Used to reach considerable thickness over winter in the Great Lakes.

21. Again, we are not going to have a winter ice-free Arctic for a while yet. And while the seasonal asymmetry in ice cover exists–it’s actually growing at the moment, since summer extents are falling much faster than winter ones–you’re going to have a net heat gain, I should think. Over the longer term this will be a negative feedback, I’m sure. But yeah, there’s lots of ground for model investigations.

• Bruce

Agreed. And I suspect that it is always going to be a net gain — especially on land where permafrost and snow are pretty much the same temperature, so you can’t really give up more heat than you gained over the summer. But there should be some mitigation of the warming from less ice and thinner ice. On the flip side of that, the warmer air over open water will have more moisture in it, so that will slow the radiation to space.

22. Jim Eager

Yes, I know that, Bruce. The point is once sea ice forms the seawater beneath it can no longer radiate directly to the atmosphere.

• Phil.

Correct and the seawater can now only lose heat via conduction through the ice, which is a lot slower than radiation and convection. The surface of the ice will cool further due to radiation and convection, ice will form on the lower surface of the ice at an ever decreasing rate as the ice gets thicker and the conduction correspondingly decreases.

• Bruce

Again, the sea water is still radiating into the ice. The ice slows, but does not stop the radiation to space. And the thinner the ice, the less effective it is at slowing the radiation.

23. John Cartmill

The CICE model documentation (
http://oceans11.lanl.gov/trac/CICE) has a good technical discussion of the thermodynamics of sea ice growth and melting.

24. > sea water is still radiating into the ice.
What’s the mean free path for infrared radiation through sea water?

Through biological tissue, at least, the energy is mostly dissipated as heat by contact, not staying with a molecule long enough for a photon to be emitted; see e.g. http://www.medphys.ucl.ac.uk/research/borg/homepages/veronica/thesis/chapter2.pdf

• Bruce

It’s exactly the same whether it is covered by ice or not.

25. Norman

I found this webpage that shows the radiation budget for a 4 year period 1985 to 1989 (Scroll down to heading Net Radiation in the Middle East).

http://www.science.co.il/Weather/Middle-East-Climate/

This shows in the 4 year period the Arctic region had a negative energy balance (radiatiated away more energy than it received). It gives amounts. If one had a raditation budget for the last 4 years it would be easy to see if the Arctic has shifted postive in it radiation budget as compared to the earlier period.

Also interesting in this same heading is that the Sahara desert radiates away more energy than it receives. Even though very hot at times, it acts to cool down the earth in the larger picture.

• Rob Dekker

Norman said

This shows in the 4 year period the Arctic region had a negative energy balance (radiatiated away more energy than it received).

The graph you refer to shows that the Arctic radiates more (IR) energy away than it received in (solar) radiative input.
I hope you understand that it will do that until the end of times (as long as this planet has an atmosphere).
You do, right ?

If one had a raditation budget for the last 4 years it would be easy to see if the Arctic has shifted postive in it radiation budget as compared to the earlier period.

No, Norman, There is not any chance at all that the Arctic will obtain a “positive” radiation balance.
Not unless it becomes warmer in the Arctic than in the tropics.

Are you playing ignorant ?

• Norman

Rob Dekker,

When I said shifted positive, that was poor choice in wording to convey proper meaning. I meant more positve than what it was in the 1980′s (around -100 watts per meter). By more positive I meant a shift to less overall negative balance so instead of -100 you might see -90 meaning it is not cooling the Earth as much as it did relative to 1980′s timeframe. If the total radiation balance for the Arctic is less negative (relative to earlier years) it would cause warming (since it is cooling less). If it is more negative (because there is less ice to stop the heat loss in winter) than the overall Arctic will help cool the Earth even if it is warmer in the Spring and Summer months. Hope that clears it up.

• The phrasing is a bit ambiguous, but FWIW, I took Norman’s “as compared to the earlier period” to suggest that the intended meaning was “a change in the positive direction,” rather than “a change to a positive balance.”

26. Thanks for a provocative comment, Norman.

“Even though very hot at times, [the Sahara] acts to cool down the earth in the larger picture.”

Sure. The huge diurnal temperature difference is one measure of this process: it couldn’t cool down as much as it does if the various cooling modalities weren’t working very efficiently. And the warmer the Sahara (or anything else) is, according to Stefan-Boltzman, the more it radiates.

I wonder how the math works out with respect to the Arctic in that regard. Clearly, the Arctic has warmed a lot; it’s ‘trying’ (if I may be forgiven the anthropomorphic language) to radiate away a big chunk of the planet’s increased radiative forcing. To what extent has the OLR increased? (There must be numbers on that somewhere.) I’m thinking that yes, the Arctic radiation budget must have moved in the positive direction. But how much? Maybe not as much as one would think at first blush?

Reasoning by analogy is always dangerous, but can sometimes help open up new avenues of thought. Is a possible analogy between the Saharan diurnal swing and the Arctic seasonal sea ice swing helpful?

We certainly know that the latter has increased–since summer extents decline much faster than winter ones.

27. Norman, if you have ever camped in the desert. you know that it gets nippy pretty quick. The absence of H2O vapor means cooling is rapid. It also means the effect of added CO2 in these regions is much greater.

28. Phil.

Bruce | October 11, 2012 at 6:01 pm |
Again, the sea water is still radiating into the ice. The ice slows, but does not stop the radiation to space. And the thinner the ice, the less effective it is at slowing the radiation.

No the seaice is opaque to the IR from the underlying seawater. This why when observing IR from space you can distinguish colder ice from surrounding seawater. The IR from the seawater is almost immediately absorbed in the bottom surface of the ice and then the heat is transferred through the ice to the surface, where it can emit, convect and conduct to the air. As a result the top surface is much colder than the bottom and the temperature profile of the ice is approximately linear.
See below for an example.

• Bruce

Phil, your statement to which I was responding was: “Correct and the seawater can now only lose heat via conduction through the ice [...].” That statement is untrue, as you’ve just confirmed above. Conduction is not the only way the seawater can lose heat. The seawater also radiates heat into the ice.

You say “No the seaice is opaque to the IR from the underlying seawater” (note that this in no way contradicts my assertion: “Again, the sea water is still radiating INTO the ice.”) You then immediately give a description of how the radiated heat from the seawater moves through the ice and then either radiates directly to space, or moves into the atmosphere (where it radiates to space). Which is exactly what I said: “The ice slows, but does not stop the radiation to space.”

29. Norman

Phil,

From your temperature profile of ice, would that mean the thin ice that has been forming in the Arctic the last few years will continue to let out more heat than the thick multiyear ice? Does thin ice actually make the Arctic a heat sink overall?

It is possible the arctic ice is having the large melts because of warmer water entering in the system from the southern areas. This may be a way the Earth actually can balance the increased heat from AGW. Warmer water moves into the Arctic sea melting away the ice. Then in the winter the protective ice blanket is not so effective in holding in the heat and more energy is actually released in winter than gained in summer by the lower ice area.

• Phil.

FYI has a conductivity of ~2.1 W/m.K and MYI ~1.9 W/mK. Virtually all of last years new ice melted this summer so ~2.5×10^12 m^2 of this winter’s ice will be MYI and the rest FYI. So the heat lost/m^2 of 1m thick MYI with a 20ºC differential is 1.9 W/m^2 compared with 4.2 W/m^2 through 0.5m thick FYI. The FYI will thicken more rapidly in response.

30. Norman

Thanks Phil,

I did find two contradictary articles on the subject of energy lost in the Arctic during winter.

One claims that the Arctic will warm in the winter and increase Arctic Amplification.

http://www.misu.su.se/~gsven/PBL/Bintanja_etal_2011.pdf

This one shows that the Arctic has not been warming during the December, January, February winter months but actually cooling. I do not think the evidence available supports a winter warming in the Arctic region.

http://www.hindawi.com/journals/amet/2012/505613/

• Rob Dekker

Norman,

I did find two contradictory articles on the subject of energy lost in the Arctic during winter.

Since you assert that these two articles are “contradictory”, which one did you find more convincing and why ?

Also, since this post by Tamino is about albedo effect during Arctic summer, to the account of +0.45 W/m^2, how much colder does the Arctic winter need to get to prevent a collapse of Arctic sea ice during the summer ?

• Norman

Rob Dekker,

I would find the latter one more convincing for this reason. The first article is a model forecast of what will happen with a doubling of CO2. The latter is based upon actual empirical satellite radiation measurements (as described in the first line of the article).

If the empirical evidence is wrong in the second one with better or improved instrumentation proving this, the first could be correct. If the tested evidence is correct, the first article is proven incorrect. We have increased the amount of CO2 in the atmosphere. The first article predicts a 10 C warming in the Arctic winter (that is the greatest warming predicted by the model runs) with a doubling of CO2. If the second is correct the Arctic winter has been cooling for the last 20 years showing no warming trend at all. The first article has the it warming most at the north pole. The cooling trend is greatest in winter above 70 degrees in the second.

Also I checked the work of the lead author of the second. He has quite a few peer reviewed articles to his name concerning Arctic conditions.

Links to the graphs in the second article. There are two graphs that fit together. One shows cooling in the Arctic winter. The other shows greater overall radiation loss in winter (DJF).

http://www.hindawi.com/journals/amet/2012/505613/fig3/
Surface Temperatures

• Norman

Rob Dekker,

“how much colder does the Arctic winter need to get to prevent a collapse of Arctic sea ice during the summer ?

The original point I was making was that an ice free summer in the Arctic could possibly cool the Earth in the long run. It would warm considerably in the summer, but with no ice insulation for a large part of winter, it would lose all that heat and more during the winter causing an overall cooling effect (as the Sahara does in the tropical area, which absorbs vast amounts of heat during the day but radiates away more at night acting as a cooling mechanism).

• Yes, with “could possibly” being very much operative. We can expect a negative feedback, but it is far from clear what sort of magnitude we can expect. The last few sentences of this paper seem relevant:

• Jim Eager

“This one shows that the Arctic has not been warming during the December, January, February winter months but actually cooling.”

That’s not what Tamino finds in his most recent post:
http://tamino.wordpress.com/2012/10/13/arctic-amplification/

Slower warming in those months, to be sure, but warming none the less, and certainly not cooling.

• Norman

Jim Eager,

When you have two very different outcomes how do you go about determining which one is correct?

Tamino shows warming in winter, Wang shows cooling. Wang’s article is in a peer reviewed journal. The temperature graph shows winter cooling in the article but also the overall radiation flux in winter is more negative. Why would the Arctic in winter warm when the overall radiation flux is more negative? (which is an empirical measurement from satellites).

http://www.hindawi.com/journals/amet/2012/505613/#B44

I also notice Tamino has large error bars and chooses the middle point for his analysis. Perhaps the actual value is at the bottom of the error bar for the winter months and they are actually cooling rather than warming.

Quote from Wang Article. “For the Polar Cap, which is the area north of 70°N, the surface temperature has decreased by −0.125°C per year with an SD of 0.042°C.”

• Rob Dekker

Norman,
You are picking the exact data that supports cooling in the Arctic, while ignoring the bulk of data and not providing any context information. Why ?

Let’s look at Wang et al for a second, and their reported cooling in winter.
http://www.hindawi.com/journals/amet/2012/505613/fig3/

First, you see that all three other seasons show increase in temperature.
Why did you pick winter only ?

Second, you see that data only goes to 2004. Do you have data for the last decade, when ice losses started to really be significant ? Does the winter cooling trend (and reduced cloud winter trend) continue, or is it starting to turn around ?
Holland et al suggest that it is. Note this report :
http://www.arctic.noaa.gov/reportcard/temperature_clouds.html
which shows that cloud cover in fall and even winter is starting to increase :
“in 2011, Arctic cloud cover was somewhat higher than the average of the last ten years (2002-2011) in winter”, which is consistent with the notion that “feedback analysis of data from 2000 to 2010 indicates that a 1% decrease in ice concentration leads to a 0.3-0.4% increase in cloud amount”

Third, Wang et al reports data for 60 N and up.
That means that their data is NOT restricted to the Arctic, but in fact is dominated by the entire Siberian and Canadian Boreal forest area. which we KNOW cooled in winter because of more extensive snowfall (presumably because of more preciptable water in the atmosphere, as also reported by Wang et al)
http://www.hindawi.com/journals/amet/2012/505613/fig6/
This explains why Tamino finds a clear polar amplification effect in the data, while you are still holding on to a trend in winter in the Boreal forest area, in a data set that ends in 2004.

Next time, could you please be a bit more skeptical of the data you present, answer some questions rather than stating them, and also provide a bit of context to the data you present ? Thanks !

• Rob Dekker

Norman said, quoting Wamg et al :

For the Polar Cap, which is the area north of 70°N, the surface temperature has decreased by -0.125°C per year with an SD of 0.042°C.

This is data until 2004, from the AVHRR data set. It this number confirmed by other data sets (such as NCEP/NCAR), Norman ?
Or is there is conflict in data sets ?

And what about the past 8 years ? Did the data, and trend, change since then ?

• Rob Dekker

And of course, the question that we all want to have answered :
If your pick of Wang’s report of winter trends until 2004 is truely significant in the overall picture, then why did summer ice extent reduce to record minuma in 2005, 2007 and a whopping record in 2012 ?
Maybe Wang’s winter trend until 2004 is not all there is ?

31. aside; checked again just now, still seeing
“We cannot process this transaction because there is a problem with the PayPal email address supplied by the seller.”

• Rob Dekker

Norman,

NCEP/NCAR data set is available on-line, and you sure should be able to extract the radiative energy balance over the past decade. When you find your data there, please keep in mind that summer albedo effect (as reported here by Tamino) is something like 0.45 W/m^2 globally/annnualized. In other words, the reduction of summer ice and snow cover inserts 7.2 * 10^21 J or so of extra energy into the summer melt season in the Nothern Hemisphere.

PIOMAS reports that this heat causes an additional 3000 Gton sea ice (some 1 * 10^21 J) to melt, which is about 14 % of the extra summer heat caused by albedo effect. Apparently, the other 86 % goes to immediate warming in summer of the Northern Hemisphere.

But the 14 % of summer heat will get carried into autumn and winter, which means that 1 * 10^21 J latent heat will/should be released during freeze-up just to break even. To release 1 * 10^21 J during a 6 month period, over 10 Million km^2 ice cover implies a 6.4 W/m^2 forcing. This means that EVEN if autumn and winter show 6.4 W/m^2 increased TOA heat loss due to IR radiation, that it is just getting rid of it’s summer heat (due to albedo effect), and thus does NOT yet add to winter ‘cooling’ of lower latitudes.

Just keep that in mind when you check NCEP/NCAP on radiation budget over the past decade.

• Norman

Rob Dekker,

I did visit the NCEP/NCAP website you suggested and made a few graphs. It is beyond my available time to make a detailed accounting of radiation. The radiation flux is complex and nonuniform. That is why we pay certain people to have the time to complile the data and hope they do an excellent job to generate the most representitive data possible.

I made two graphs. Global for Clear Sky upwelling longwave radiation and another for Clear Sky downwelling longwave radiation (greenhouse effect). I chose a monthly average of winter months (December, January and February) for the year 2009-2010. Since there is no solar energy available most the radiation energy will be these two. I did check cloudiness in this same time frame and the graph showed just a trace.

Near the North Pole the upwelling radiation was 180 watts/meter while the downwelling radiation was 135 watts/meter. This would indicate that the North Pole area was losing around 45 watts/meter of energy during these months.

32. Norman

Rob Dekker

There does not seem to be much data on arctic warming on internet that is current. Most seem to end at 2003. I did find this article however and it does show that the surface temperature of land in the Northern Hemisphere has not warmed much and is way below the model projections.

[Response: Isn't their claim of "not warmed much" only for winter trends, not for the entire year?]

I would be more interested in the radiation budget to see how much energy is leaving the system in the winter.

• Rob Dekker

Sorry Norman, I accidentally attached my reply to your post to the previous comment.

• Rob Dekker

Norman, FYI, I carried your comment about warming winter trends in the Arctic (as per Wang et al 2012) into Tamino’s “arctic amplification” thread,
http://tamino.wordpress.com/2012/10/13/arctic-amplification/#comment-71456
since at least at first sight, it seems that there is a mismatch between the Wang et al 2004 AVHRR data set and the NCEP/NCAR, UAH, and RSS data sets regarding that alleged warming.

33. AB

This is a highly relevant and sometimes overlooked region of the world that is most affected by a warmer world. Adaptation is not an option for the snow and ice dependent organisms thus this climate change consequence should be made top priority.

34. Phil.

Bruce | October 15, 2012 at 4:35 pm | Reply
Phil, your statement to which I was responding was: “Correct and the seawater can now only lose heat via conduction through the ice [...].” That statement is untrue, as you’ve just confirmed above. Conduction is not the only way the seawater can lose heat. The seawater also radiates heat into the ice.

No Bruce only to a minuscule degree. At the bottom surface of the ice the ice surface temperature is the same as the top layer of the water in contact with it. The seawater surface radiates into the ice depending on T^4 the ice radiates into the water dependent on T^4, the net flow depends on the difference in the emissivity of the two surfaces which is very small. Conductivity through the ice is the dominant heat loss mechanism.

• Bruce

If the temperature difference between the water and the ice is zero, then then the heat conducted from one to the other is also zero. It’s basic thermodynamics: Without a temperature gradient, there is no heat flux; with a temperature gradient, both conduction and radiation produce a flux.

Also, your current assertion that there is no net radiative flux from the water to the ice is in complete contradiction to your earlier assertion: “The IR from the seawater is almost immediately absorbed in the bottom surface of the ice and then the heat is transferred through the ice to the surface, where it can emit, convect and conduct to the air.”

• Phil.

Also, your current assertion that there is no net radiative flux from the water to the ice is in complete contradiction to your earlier assertion:
I did not make the assertion “that there is no net radiative flux from the water to the ice”, go back and read again.
” The seawater surface radiates into the ice depending on T^4 the ice radiates into the water dependent on T^4, the net flow depends on the difference in the emissivity of the two surfaces which is very small.”
The difference in emissivities is small but not zero, the emissivity of the ice is less than the emissivity of the water so there is a net transfer to the ice (where it is absorbed in a few microns). Since the radiative loss from the top of the ice is larger than the heat transfer through the ice by conduction. Heat removed from the water boundary layer leads to more ice forming and conduction through the thicker ice gets slower ultimately a thermodynamic equilibrium thickness is reached (in the Arctic about 3m, takes several years).

35. Bruce

I think you need to review your basic thermodynamics. If the temperature difference between the ice and water is zero, the net heat flow is also zero. Exactly zero.

No amount of emissivity difference would change that fact.

• Phil.

Bruce you need to review radiation heat transfer, try Hottel & Sarofim.

• Bruce

If you believe that Hottel & Sarofim let you violate the second law of thermodynamics, you need to reread it.

• iceman

Phil, thanks for your explanation of IR heat transfer. You left out one detail that would clarify difference in emissivities: the thin bottom layer of ice emits more into the colder ice above than into the nearly-same temperature water below.

36. I think that there is a problem of formulation here. I’m not competent to diagnose it properly, but consider: the ocean–even the surface layer of a stratified Arctic Ocean–is a very large heat sink. And the ice/water interface surely must be darn close to a zero differential.

Yet we know that in winter the temperature of the ice will sink dramatically as we move (in imagination, that is) toward the ice surface. That means a heat flow via the ice from water to air.

So how do we properly model/describe this state of affairs? Is there a temperature differential across the ice/water interface to help ‘drive’ this flow? Or is Phil’s description correct? I don’t know, but we have a paradox to avoid…

• gnumb

My guesses:

Heat flow within sea ice is almost all via conduction (molecular collision); very little via radiation (photon emission and absorption) or convection. (There would be a little convection inside pores in the ice that are filled with either air or high-salinity liquid water.)

Heat flow within sea water at -2 degrees C is mainly via convection, secondarily via conduction, and very little via radiation. Because the average time until a molecule collides with another molecule and transfers energy in that way is shorter than the average time until the molecule loses energy by emitting a photon. See above: Hank Roberts October 12, 2012 at 12:32 am.

For the same reason, heat transfer at the liquid/ice interface is almost all via conduction and very little via radiation.

I seem to see a line of reasoning that goes:
Where liquid water touches ice they must be at exactly the same temperature.
By 2nd law of thermodynamics no net heat flow can occur where substances are at equal temperatures.
Therefore no net heat conduction occurs from the liquid to the ice.

But this same reasoning would lead to the conclusion that no heat flow ever occurs by conduction, even within a homogeneous substance. (Because at any plane you imagine within the substance, molecules on one side are at the same temperature as adjacent molecules on the other side.)

Temperature is a measure of the average kinetic energy of a large number of molecules. The velocity of any one molecule is rapidly varying as it collides with others. Most of the time the kinetic energy of a molecule is not equal to that of any of the nearby molecules.

If there is a temperature gradient in a material, for example if the region to the observer’s left is warmer than the region to the right, then molecules to the left have, on average over time, a slightly higher kinetic energy than adjacent molecules to the right. As the speed of adjacent molecules varies, collisions will transfer energy in both directions, but over time there will be a net transfer from left to right.

37. Phil.

I think you need to review the 2nd Law.

38. Kevin, think of the water as a heat reservoir. The ice loses heat to the cold Arctic air blowing over it (convection, conduction and radiation). The water maintains the temperature of the ice. At least that is how it looks to my simple physicist’s mind.

• Yes, that’s what I’m thinking. Where it gets less clear (to my non-physicist’s mind) is right at the water/ice interface–IOW, the nitty-gritty of what happens there as “the water maintains the temperature of the ice.” I would have guessed that in reality the temperature differential at the interface is very small but non-zero, but thinking about this, I suppose that depends upon the relative rates of heat transfer through the ice and from the water to the ice. If the latter were much faster, then the interface should be “clamped” very closely indeed to the water temperature. That sounds a lot like Phil’s formulation, here:

The difference in emissivities is small but not zero, the emissivity of the ice is less than the emissivity of the water so there is a net transfer to the ice (where it is absorbed in a few microns). …the radiative loss from the top of the ice is larger than the heat transfer through the ice by conduction.

By this view, the water is cooling, but the ice acts to limit the rate at which this can occur. (And hence, Norman’s point–which had already been discussed at some length at Neven’s ASI blog months ago, BTW–since when the ice goes away, the water can radiate its heat away directly, a much faster process.)

At the moment, I’m thinking that formulating the question purely in terms of temperature differential at the interface is a bit reminiscent of Zeno’s paradox, which ‘freezes’ motion by reifying the abstract notion of the geometrical point: Bruce’s formulation is ‘freezing’ heat flow–boy, that’s a rotten unintentional pun!–by reifying the abstract notion of a geometrical plane. (The ice/water interface.)

Apologies to any who found this ‘thinking out loud tedious,’ yet felt obliged to read it anyway…

39. Bruce

“I think you need to review the 2nd Law.”

You mean the one that says that the entropy of an isolated system never spontaneously decreases (as required by your “explanation”)?

You guys need to learn some basic heat transfer. The emissivity only affects the rate of heat transfer, it NEVER affects the direction.

Consider taking a black (high emissivity) object, and placing it within a shiny (low emissivity) chamber. The object and the chamber are at the same temperature, and the system is thermally isolated. By Phil’s logic, heat would flow from the high emissivity object into the low emissivity chamber. Eventually the high emissivity object would cool to absolute zero, and the low emissivity chamber would contain all the available heat in the system. The system would have spontaneously moved to a minimum entropy state, which is a violation of the second law.

Your confusion seems to be (aside from an inexplicable failure to recognize that two objects at the same temperature are in thermal equilibrium) that you do not account for the fact that good emitters are also good absorbers, and poor emitters are also poor absorbers — and they are so to the exact same degree. So while the better emitter radiates more, it also absorbs more; and while the poorer emitter radiates less, it also absorbs less. And these effects exactly cancel one another. If they didn’t, you could easily build a perpetual motion machine. Which, hopefully, everyone can agree is epistemically impossible.

40. Kevin,
think of it in the limit of perfect black body layers–then emissivity doesn’t enter into it. Each layer radiates into the layer above it and below it. If all layers have the same temperature, then each layer receives the same amount of radition it emits, and equilibrium is maintained. Now, imagine that the top layer is in contact with a cold layer above it. It will radiate the same amount but receive less radiation from above. It may cool ever so slightly in response, and the process continues all the way down to the water. The gradient will be arbitrarily small, but you still have flow of heat across the ice.

Emissivity complicates things, but doesn’t change them appreciably.

• Yes, I’m with you. Emissivity or not, the colder upper layers of ice establish a temperature gradient and hence heat flow.