Experiments with a grey atmosphere

So... how does adding CO2 affect the atmosphere? Lets try an answer that for a "grey" atmosphere with only radiative transport of energy. How to interpret the "atmosphere" in this model? Perhaps one could pretend that it represents the statosphere only, in that the statosphere is the bit above convective adjustment. OTOH in reality the statosphere is heated by ozone aborption of UV, which isn't in this model. So its hard to say exactly *what* my model represents.

The immediate interest is "why does the statosphere cool in a GH scenario"? I thought the answer was simple: increased CO2 => increased absorption/emission => increased radiation => cooling high up. However, this is not true: see the results below. In a grey atmosphere that goes to equilibrium, the entire atmosphere heats up (assuming the calculations below are 1) relevant and 2) correct). In the "instantaneous" case then we do get cooling in the upper atmosphere. Fine. However, cooling is a prediction of equilibrium GH studies too: see IPCC '90 section 5.2.2.1 (page 139) "The statospheric cooling is due to enhanced radiative cooling to space and increases with height". This is essentially my explanation above (which I now believe to be wrong; how interesting). HKR claims that it is essentially a feature of a non-grey atmosphere. I winder how I might include that?

All the calculations are based upon my IDL program "grey_atmos". If you think there is a bug in it, all is lost!

Equilibrium

Use grey_atmoss to draw pictures of the equilibrium state, as the number of levels varies.

From the picture (click for a closeup) we see that

  1. As we increase the number of levels, but maintain the total atmosphere "contents" the same (ie keep nlevels*emiss-of-each-level fixed) the temperature profile asymptotes to a solution quite quickly.
  2. The surface is always substantially warmer that the levels above - presumably because the solar radiation is absorbed there. In reality, this would not be a stable situation.

Varying the number of level and the thickness

Use grey_atmoss1 to look at variations in the number of levels and the total atmospheric thickness. "total atmospheric thickness" = "total optical depth" = number-of-atmospheric-levels * absorption-of-each-level. This is not a realistic model of the real atmosphere, but insofar as it applies I suspect that the "real" atmosphere has a total depth >>1, maybe ~10.

Make this picture by: 

!p.multi=[0,1,3]
grey_atmoss1,nl=100,npe=[0.01,.1,1,10]
grey_atmoss1,nl=10,npe=[0.01,.1,1,10] 
grey_atmoss1,nl=3,npe=[0.01,.1,1]     	; Can't use 10 for nl=3, 'cos 0 ge e le 1

We see that

  1. If the number of levels is very large, ie we are asymptoted to the "true" solution, then the TOA temperature does not depend on the total optical depth
  2. The fewer levels there are, the more the TOA temperature changes with changing total optical depth
  3. The greater the depth of each layer, the less the gap between surface and first layer temperature.

Non-equilibrium

Suppose now we consider "instantaneous" forcing: we alter the "amount of CO2" (ie, the emissivity) but don't allow the surface temperature to change. Use grey_atmoss2 to do this.

This picture shows total depths of 10,1 and 0.1 (black, green, yellow). 

grey_atmoss2,n=12,es=[10,1,.1] 
grey_atmoss2,n=120,es=[10,1,.1]

So we see that:

  1. The atmospheric tmperatures do change - the top cools with more CO2, the bottom warms
  2. The temperature in the middle (0.5) appears to be perfectly constant
  3. (not shown) Only if there is only one atmosphereic level (n=2 in my language) is there no forcing in the (single) atmospheric layer.

Thanks: Howard Roscoe, D. Archer and R. Pierrehumbert, and sci.environment who provided the inspiration and ideas for doing this stuff, but the mistakes are all my own...

Some messages from sci.environment

Note that I have edited them for brevity where appropriate but have not changed the sense of anything.

Try this thread archived by Dejanews. 

Subject: Warmer ground, cooler stratosphere?? was Re: I erred, and retract my previous post
Date: 17 Jul 1998 00:00:00 GMT
From: irv115@aol.com (Irv115)
Organization: AOL http://www.aol.com

Craig  wrote:
>Date: Fri, Jul 17, 1998 11:20 EDT
>Message-id: <35AF6BA9.C6C36BC8@removethis.graymills.com>

> To me, that data is interesting.  How could the
>surface be getting
>warmer whilst the atmosphere right above the surface is cooling?  Myself, I
>think that it
>might be a heat sink effect.  It would be interesting to find people who know
>more about
>this data and can inform us of other possible errors.  The point of this
>thread should be
>to find out the facts and to challenge the conventional wisdom.

There could be ways.  A gasseous medium radiates throughout its optical depth. 
Its effective radiation level is at unit optical depth.  Our atmosphere
effectively radiates over a considerable height range, with the effective
height a function of the wavelength & the absorbing species.

If we add to a "greenhouse gass" its mixing ratio increases, & its effective
radiation height increases.

To first approximation, the radiation temperature stays constant, so the
temperature , so any radiation that takes above the average radiation height
takes place at a lower temperature.

Other gasses, that did not have an increase in mixing ratio have to take up the
slack, so their radiation temperature increases a bit.

the net effect, the troposphere gets a little thicker.  The lower troposphere
gets warmer,  the overall radiation temperature stays the same, the upper
troposphere & stratosphere get cooler.

That is the way I understand it.

Irv @ Webster

Here is one that is definitely wrong: 

Subject: Re: Warmer ground, cooler stratosphere?? was Re: I erred, and retract my previous post
Date: 17 Jul 1998 00:00:00 GMT
From: jim blair 
Organization: University of Wisconsin Madison

Wouldn't it be expected that if the CO2 level rises, the lower
troposphere would absord more IR energy, thus get warmer; and that less
energy would get up to the higher stratosphere, so it would cool?  Just
"common sense?

I said the following; sadly I now think my explanation for the stat. cooling is wrong. 

Author: William Connolley
Date:   1998/08/19
Organization: The Natural Environment Research Council

1808981211410001@ampogryz1.chem.ubc.ca, ogryzlo@despamchem.despamubc.ca (Elmer) writes:

What you are trying to understand is actually very difficult if you want to know
the details. Very few people actually do understand it (I don't know the details
myself). Many people think they know but become hopelessly confused when they
try to explain it... I will do my best below, but we really need a radiative
physicist in here.

>      Correct me if any of the following assumptions are not true:
>(1) Above the level that CO2 radiates to space, we have a non-radiating
>atmosphere (composed of  N2 and O2 that don't absorb or radiate IR).

This is wrong. Firstly, there is no "level that CO2 radiates to space".
That entire idea is an oversimplification designed to make things easier to 
understand. Secondly, CO2 is well mixed throughout ?all of the radiatively
important atmosphere?

>(2) Below this level the temperatures are governed by adiabatic cooling
>(fighting gravity) and not by radiative equilibrium.

The *troposphere* is more-or-less turbulently mixed by heating
from below (this is oversimplified), and above this level the stratosphere
comes in where temperatures increase due to absoption of uv by ozone. 
[Basically, this defines the trop and strat]. However,
radiative physics are important in the troposphere structure too. Also,
the "level that CO2 radiaties" (insofar as that exists) doesn't coincide with the
tropopause (trop-strat boundary).

>(3) Since the CO2 below the altitude which finally radiates into space
>does not directly affect the radiation balance, it just serves to get the
>surface heated up to a new level that now radiates more into space at all
>wavelengths.  

Here you touch on an important point - radiation at all wavelengths matters.
The atmosphere is not "grey" (and is not treated as such by GCMs). CO2
has certain bands, which coincide with windows in the bands of other radiatively
important gases. As you go up through the atmosphere the importance of radiation
in different bands changes, because the distribution of water vapour changes.
Given the importance (in the stratosphere) of heating by UV-absorption, the 
stratospheric CO2 is not in radiative balance at IR wavelengths - it is gaining
energy from collisions with O3.

>(4) The whole phenomenon is therefore (a) an increase in the altitude from
>which CO2 radiates into space, and hence  (b) a decrease in that radiation
>into space, because it is colder at that altitude, (c) a compensating
>increase in the entire atmosphere, and Earth's surface temperature while
>retaining the adiabatic lapse rate, (d) increased radiation from the
>earth's surface at all wavelengths to just compensate for the decrease in
>radiation from the top layer of the CO2.

I don't think I like your summary. Try this:
- suppose we instantaneously increase (say, double) CO2
- then 2 things happen:
  - more IR is radiated to space (since the atmos radiates better)
  - more IR is radiated downwards (ditto)
- instantaneously, therefore, the upper atmosphere tends to cool and the
  lower atmos/sfc complex (*) tends to warm
- errm, then add in feedbacks

(*) It can be slightly helpful to think of the lower atmosphere and the sfc as
one level, sometimes, since they are quite tightly bonded by convection, though
this varies with latitude (almost none in the polar regions).

>   ------but I still don't see a mechanism for a drop in temperature of
>some part of the troposphere.

Well, if you replace "troposphere" with "atmosphere", its easy to see (from radiative
balance at the top-of-atmosphere) that *some* of the atmosphere has to cool
(see my last post); since the CO2 has increased making that atmosphere a better
radiator, there is a mechanism too.

I also said, which I still think is correct: 

1908981105560001@ampogryz1.chem.ubc.ca, ogryzlo@despamchem.despamubc.ca (Elmer) writes:
><35da9a28.0@wltss01.nerc-wallingford.ac.uk>, wmc@bsfiles.nerc-bas.ac.uk wrote:
>> - suppose we instantaneously increase (say, double) CO2
>>   - more IR is radiated to space (since the atmos radiates better)
>>   - more IR is radiated downwards (ditto)
>> - instantaneously, therefore, the upper atmosphere tends to cool and the
>>   lower atmos/sfc complex tends to warm

>      In view of what I have said above, I would conclude that "Instantly
>increasing the amount of CO2" in the atmosphere will increase the altitude
>from which the CO2 "radiates to space" ,and since the temperature there is
>lower , less radiation at this wavelength is being emitted to space.....
>until this effect heats up the planet.... which we call "global warming"
>resulting from the increase of  "greenhouse gases".  It is then that the
>whole troposphere should increase its tempearature.

I think you are right to say that the radiation-altitude increases, and so the
atmosphere tends to warm up. But the degree to which that occurs is then not 
clear from these descriptive arguments. Your arguement indicates that
increasing CO2 leads to a radiative inbalance which will lead to T increases, 
but it doesn't tell you where these will occur. To come back to where we began, you 
asked why the upper atmosphere is predicted to cool. Well, its easy to see why this
instantaneously tends to occur, and your argument then tells us that some parts
of the atmosphere will tend to warm, but we can't easily combine these 2 to say
what happens to the upper temperatures (though other arguements of total radiation
balance can be used to show that is some bits warm, others have to cool).