A call for Mathematicians - The Greenhouse Effect, not so Simplified

This is extremely frustrating. Since there are errors in the math for the K&T drawing, I cannot explain the solution until I solve several other problems. So I am going to have to take much more time than I think is reasonable.


While Energy is Fungible, the Work is Not
Something so simple is so hard to explain.


Explaining the Kimoto equation unfortunately requires completely changing the way one looks at the atmospheric effect. In general, outgoing radiation balanced by down welling radiation of equal amounts have no net impact. The atmosphere is different. While the energy fluxes are just flows, they represent energy at a point in space and time. So while a surface flux may be matched, it has more impact at the surface. It creates more collisions in the denser environment. This is an increase in conductivity. So while two flows may pass, they are maintaining conductive conditions. Energy must be conserved, but the location of that energy is important.

In the atmosphere, there are changes in the type of energy flowing. There is latent heat from water warmed a the surface and carried aloft which does not lose its latent energy until it change phase. Conductive energy which is the number of collisions between molecules and radiant every which does not require a collision to release energy.

All three interact in the atmosphere until they revert to radiant energy and escape to space.

Each is better at transferring potential energy of a given type at a give efficiency. How easy it is to transfer heat from one boundary layer to another is extremely important. For example water is very efficient transferring energy via conduction to air that air is transferring energy to water. In an air to water heat exchanger, a conductive metal is used for the water and air boundary. Smooth tubes can be used on the water side without much loss of efficiency. On the air side, undulations are used to cause more turbulence in the air to increase the rate of collisions of the air molecules. On our oceans, heat flow from the surface increases with wave action and wind speed, more molecules contact the water's surface so more heat can be transferred. Water to air transfer of conductive heat is 1000 times more efficient that air to water, because the energy contain in every water molecule is much higher and the contact with other water molecules is much more efficient for transferring energy with in the water.

Air being orders of magnitude lower in density, takes longer to transfer energy between air molecules. Pure gases have different coefficients of heat transfer. Argon and Xenon are noble gases selected for their insulation qualities. Water vapor and carbon dioxide are not because they are better conductors. They are far from idea, but better

For example gases at atmospheric pressure separated by metal have heat transfer coefficients of 5 to 35 W/m^2K under pressure that changes to 100 to 500W/m^2K.

Under normal atmospheric conditions, water vapor and CO2 enhance the heat transfer because they can absorb energy both by conduction and through radiation. Under pressure they can transfer more and at a higher temperature they can transfer more than under lower temperature and pressure. The addition of more carbon dioxide increases the radiate energy absorbed, the number of collisions which increases the rate of conduction and increase the rate of evaporation. If there is a window for radiant heat transfer, some of this energy is transferred further from the water's surface which tends to decrease the rate of heat transfer because radiate heat transfer is not as efficient as conductive.

In the atmosphere, the adaptions of the Kimoto equation tells me, that while energy is fungible, the impact of the flows are very different, 24Wm-2 up of conductive flux is balance by 24Wm-2 down of radiative flux down from one point in the atmosphere, is not the same as 24Wm-2 up 100 meters and matched by a counter flux of 24Wm-2 at the TOA.

Below I was a brief derivation of the initial greenhouse effect. It is difficult to follow, because the assumptions, while valid are a bit unusual in the computer age. Now it is no longer brief.

For a temperature radiative flux relationship, dF/dT=4F/T This basic equation is part of the problem. while it is perfect for a radiative only solution it gets involve proving it can be used for a mixed heat flux solution. When I set the equation up for the Earth I get.

dF/dT= 4*(aFc + bFl +cFr)/T, a=conductive heat flow coefficient, b=latent heat flow coefficient and c=radiative heat flow coefficient aka emissivity. With the separate coefficients, this is totally valid, however with the proper valid assumptions I can make a and b disappear. And not by magic, by logic.


If dT/dt=0, i.e. it is in equilibrium with space and surface temperature does not change, then

Equilibrium can be assumed for a steady state, equilibrium may never truly exist, it is a numerical concept. It is very important that this equilibrium be understood. I am going to apply a radical impulse.

If the Earth was floating in space all by itself at a temperature of 288K and with no atmosphere, T=390W.m-2, aFc=0, dFl=0 and C=1 Fr=390 Do I have to explain how I got that? I hope not, with no atmosphere there would only be radiant heat loss.

Now you need a little imagination,

If we suddenly add an atmosphere, T=4*(aFc+bFl+0.825Fr), 0.825 is an impulse change in emissivity from 1 to 0.825, and following happens, aFc=24, bFl=76 and cFr=290 as it leaves the surface of the Earth. By selecting the approximate final value, I eliminate the need to solve three simultaneous equations.

Note: This value of 0.825 for the coefficient of Fr was carefully selected based on the current estimate of climate sensitivity. While there are many that may disagree it is valid, it simplifies the solution. If it is wrong, that will be obvious once we get to the final solution. I will try to clean up the methods I used to arrive at that value, but not until this stage has been simplified enough for others to understand.

By selecting the final values of aFc and bFl I am in effect assuming a solution for a and c. This is a little bit of mathematical trickery not common place any more. Since I am assuming that the responses of aFc and bFl are low and slow with respect to eFr, Fc and Fl would stabilize prior to Fr. Since the impulse is applied only to Fr via the change in emissivity and the stable values of Fc and Fl assumed correct, there would be insignificant harmonic interference with Fr's approach to the new equilibrium state. This is a very effective simplification.

Okay, a little more imagination, once the surface and the atmosphere reach a steady state, equilibrium, for the briefest moment in time, we get aFc=24, bFl=76 and c becomes 0.825 which changes cFr to 0.825*290=239.5, so for that moment all hell breaks loose. The Earth would attempt to maintain the 390Wm-2 total flux, it would overshoot, and gradually seek equilibrium. Before it finds equilibrium, the atmospheric resistance to flux with the impulse change in emissivity would over shot a peak value. Knowing the final value at the TOA, 237 and the final temperature and flux at the surface, we can determine the initial green house response by artificial selecting a high but very close to reality, impulse flux from the surface. This must be trickier that I thought because no one understands this trick. Then,

T=4(aFc+bFc+cFr)+4GHe, that should be fairly simple to follow, do I need to expand that? Since I will be solving for the impulse value of GHe, I have to modify the equation. Should not be that hard to follow.

4GHe=T-4(aFc+bFc+cFr)

GHe=(T-4(aFc+bFc+cFr))/4

So the initial Greenhouse effect(GHe) is,

GHe=(288-4*(24+76+239.5)/4 = (288-4*339.5)/4 = -267.5

I selected 239.5 for convenience. I could have picked 400 or infinity, but the closer it is to reality the more information we can obtain.

That is the end of the first moment in time, a sudden impulse to a BB in equilibrium at the average temperature of Earth. Still no sun, that will be at the end.

GHe=-267.5 which has to be matched by Outgoing Longwave Radiation (OLR) in the second moment in time for the surface to regain equilibrium. The assumption of equilibrium at the surface is critical. The approximate equilibrium value at the TOA will be assumed for this stage.

390-267=122.5, the initial surface response to the GHe.

Had I selected a higher value for Fr, both the initial -267.5 and the resulting value 122.5 would have been different, but proportionally different. That is the trick by selecting 239.5 up to 390 I avoid the confusion of the sign change. The sign of GHe is very important, the value after subtracting from 390 not so important because we know that it will return to 390. It is the proportion and sign of GHe that matter. In the next step you can see why I chose a value close to the equilibrium value of Fr.

As the surface of the Earth obtains equilibrium with the atmosphere,

Surface (24+76+239.5+122.5)=462 is the initial peak radiation which will decay as equilibrium is approached. -267.5 is the initial value of the greenhouse effect which will decay as equilibrium is approached. These are impulse conditions. At the TOA, the flux has dropped to near zero and will stabilize to a value of 237Wm-2.

As you can see the peak 462 is a manageable number. It is a totally fictitious number, but it is proportional to the -271 GHe impulse which is also fictitious. As long as they are proportional, greater that the final value and the signs are correct, these values will work. But, since I selected 239.5 for the value, which is steady state, there is some meaning in -271 and 462.

At this point we release Temperature so that it actually change. The surface will reduce its emitted radiation by 462-390=72Wm-2 The radiation at the top of the atmosphere will decay to 239.5Wm-2. This is the start of the third layer solution. By selecting 239.5, I get 462 peak at the surface minus 271 in the atmosphere. 462+(-271)=191 at the TOA. The TOA has to increase to 239.5 and 462 has to decrease to 390. GHe cannot increase if 462 is decreasing and 191 is increasing. So both have to decay, or reduce in magnitude, to a stable value while TOA is increasing. This must be part of my trick that is very hard to f0llow.

At the surface, 462 decreases as 122.5 decays to 72 yielding 50.5, the amount of radiation absorbed by the atmosphere. The GHe -267.5 proportionally with the 122.5 decays to -267.5 + 50.5 = -217 The 50.5 is a verification of the method, but not if you were using the Trenberth values. 51 is the determined by NASA, 50.5 tends to confirm their calculations.

This final step may be difficult, but not so bad if you have followed me so far.

There is still no sun in the sky, this is the third moment in the existence of the atmosphere. With no solar input at all, the problem is super simplified. This equilibrium condition would exist on for the briefest period of time.

The TOA value of 237 is assumed to be the momentary equilibrium value. Trenberth added the 72 to 267.5 to get 339 and then decayed that to 321, 339-321=18, the amount he mistakenly has for atmospheric absorption.

In order to solve for the solar input, I have to use the same methods, only the it is not really needed. Since the equilibrium values are very close to the real world values, adding the sun only requires minor accounting to resolve the atmospheric balance. Think of should I let the surface flux decrease to 390 minus any small value, then slowly increase the solar values to maintain the 390. Instead of adding an impulse, the slow matching input maintains the surface balance and the TOA balance, then it is just minor accounting to solve the atmospheric balance. Since the upeard flux does not change, the down welling will not change since it is created by resistance to outgoing flow. The values selected for Fc and Fl are based on the steady state with solar, so only the upper troposphere needs to be tweaked.

Setting up this solution was most of the battle. It took me a while to confirm that 0.825 was a valid estimate, I had to determine a reasonable estimate of the DWR from comparing both K&T drawings to the NASA, I had to basically solve the problem before I could solve the problem. So this is not the best or only way, it is only A way.

Now I have confirmed the Trenberth error several ways, but it is difficult to unseat the champion. Unfortunately, there are probably not many people that can appreciate the simple elegance of this mathematical trickery. Unfortunately, this method will probably not be accepted even though all the assumptions are valid, because they push the limits pretty severely. Until this first step is understood and at least somewhat accepted I cannot advance to calculating climate sensitivity with greater accuracy using the simple formula dF/dT=4F/T with the proper coefficients.

I think I will call this proof by discovery of error. If you look closely at the K&T you should confirm the DWR and atmospheric absorption errors which amplify each other in opposite directions. In the old K&T, 26 was the atmospheric absorption of net OLR and the new it is 18 in the old, both require a little snooping to decipher, but they are there. 51, or 50.5 calculated here, appear to be the appropriate values. The correct DWR value is fairly obvious also in the NASA with a little looking. If that is not enough, Angstrom's turn of the 20th century estimate was closer than the K&T drawing. Check Eli Rabbet's blog.

I would like to thank Fred Moolton for helping me figure out what was so hard to understand.