Kevin Kilty
Very recently a link to the CO2 Coalition and a paper by Wijngaarden and Happer (henceforth W&H) offered a series of models of Earth’s atmosphere, with supplemental information, as a tutorial (primer they called it) on the atmosphere and greenhouse effect. I am unsure what the target audience is. It seems far too complex to serve as a tutorial for the general public. Besides presenting challenging material, at 32 pages in length it takes some determination to power through it.
The models were each one example of a series of increasing complexity. The thread led to many interesting comments, objections to the paper and objections to comments, counter objections, and on and on. The sub threads make for lively reading, but they become long, making arguments hard to follow, encourage people miscommunicating, and sometimes simply end without any resolution of the argument. Some people claimed the paper presented numerous errors. One person gave it only a C+ grade. The paper gave me much material for thought, including thoughts on disagreements that seem to me pertinent to topics at WUWT.
W&H adopted a per molecule point of view for their development of models. I think this leaves most people, even technically savvy people, somewhat cold. Our physical experience is with bulk quantities like temperature, pressure, volume and so forth. In fact, it’s difficult to connect molecules, which have energy levels, with bulk concepts like temperature and pressure, or even with a concept like work without statistical mechanics. A couple of the models seemed beyond peoples’ experience. My personal view is it might have been based on thermodynamic state variables.[1] Entropy in particular is far easier to understand and use if viewed as simply a state variable conjugate to temperature.
On page 20 W&H summarized what they’d hoped readers had absorbed from their simple models (isothermal, adiabatic, and gray) of atmospheres. Let’s go through their points one by one.
- 1 Greenhouse gases [sic, consistently below] cool the upper atmosphere by radiating heat to space.
- 2 For a gray atmosphere, surface radiation to space is attenuated by factor e−τo where τo is the vertical optical depth from the surface to outer space. Absorbed surface radiation is replaced by radiation emitted by greenhouse gases higher in the atmosphere.
- 3 For optical depths, τo >> 1, atmospheres develop a lower troposphere where convection by parcels of warm air floating upward and parcels of cold air sinking downward transport most of the solar energy absorbed by the surface. Greenhouse gases radiate heat to space from altitudes close to the tropopause. Radiation to space from the surface and from greenhouse molecules in the lower troposphere is negligible.
- 3a Radiative heat transport is negligible compared to convective heat transport below the tropopause.
- 3b Convective heat transport is negligible compared to radiative heat transport above the tropopause.
- 4 Without greenhouse gases, the adiabatic temperature profile of the troposphere would evolve into an isothermal profile with the same temperature as the surface.
- 5 Greenhouse gases increase the temperature of the surface, compared to the surface temperature in an atmosphere without greenhouse gases.
Analysis:
Item 1 is not controversial.
Item 2 simply means that a “pencil”[2] of radiation in a gray atmosphere originating at the surface will have its original intensity decline by the factor e−τo by the time it exits at top of atmosphere. This shouldn’t be controversial because it simply states Beer’s law – a well established principle. However, that pencil of radiant intensity will have a value as it exits the atmosphere that includes contributions from the atmosphere above the ground surface. It is a statement of the transport equation which W&H state on page 24 as Equation 69. There should be no controversy over this statement at all.
Items 3, 3a, and 3b are all one issue. The crux of this issue is this: “τo >> 1”. This is an indeterminate condition. To fully see that what W&H said is substantially correct but incompletely stated, we need a fact that I do not recall ever having read at WUWT or any internet site for that matter. W&H’s Equation 69, the equation of transfer, does not provide a solution of the temperature distribution in a medium. It only describes radiant intensity. It has to be augmented with what is called the energy equation and the energy equation itself only rarely includes provisions for factors other than heat.[3]
What about this “energy equation”? If a person considers the three modes of heat transport, the energy equation involves a vector quantity for each describing the magnitude and direction of heat transport rate. Setting the divergence [4] of the sum of these vectors plus local heat generation (latent heat and absorption of radiance) equal to the change of temperature with time multiplied by the heat capacity of the medium, is a partial statement of the First Law of Thermodynamics – partial only because it does not address the additional impact of work. If work input (free convection) is involved, an equation describing fluid dynamics might have to augment the energy equation.
Solving the energy equation simultaneously with the transport equation of Schwarzchild (Eq. 69) is a daunting task made even more so by including fluid dynamics. Rather than pursue such a task, people (engineers typically) make an analysis based on dimensional analysis.
- The rate at which heat moves by thermal conduction is related to the local temperature gradient and a material property called the thermal conductivity. Few gasses possess enough thermal conductivity to move significant amounts of heat in a planetary atmosphere – exceptions occur near the ground where there are large temperature gradients.
- The rate at which heat is transported by convection, bulk material flow, depends on the heat capacity of the medium and the correlation (dot product more properly) of the local temperature gradient and local velocity vector.
- The rate at which thermal radiation transports heat away from a surface is related to the Stefan constant, temperature raised to the fourth power, index of refraction, and the emissivity. However, radiation beyond the emitting surface must also depend on the extinction coefficient of the gaseous medium.
Ratios of these various constants combined with characteristic distances of a problem provide a method to evaluate the relative contribution to total heat transfer by each means. Unfortunately there are two possible length scales in the energy equation. In a free atmosphere the extinction coefficient, which has units of inverse length, provides a natural scale factor for radiation; alternatively the scale height of the atmosphere provides a characteristic length for free convection. Details regarding this are unhelpfully complex for discussion here and are available elsewhere[5], but for optically thick atmospheres the ratio of bulk transport to radiation transport is proportional to optical depth.[6] As the optical depth tends toward infinity convection dominates, and as it tends to zero radiation dominates. Thus, τo >> 1 is the true consideration involved, but only in the context of the energy equation, which W&H never mentioned in their tutorial.
We might wonder if (τo =5), which pertains to their model, is far enough beyond 1 to qualify as a convection dominated model. An isothermal atmosphere (τo =0) is a simple place to start this discussion. Outgoing infrared irradiation equals the irradiance absorbed at the surface for the simple reason there are no other means for heat transfer. That’s it.
A gray atmosphere analysis is best begun with a comparison to some real Earth atmosphere. Let’s use Equation 69 to compute radiative transport in a gray atmosphere and compare the results to a real Earth atmosphere as MODTRAN sees it. I wrote a numerical solution to Equation 69 for a gray atmosphere using Excel and a second order Runge-Kutta method.
Keep in mind that the gray atmosphere model of W&H has a specified temperature profile (specified lapse rate). Thus no solution of the energy equation is required, but without the energy equation’s involvement there is every possibility that the assumed temperature profile is not consistent with assumed irradiance.
A midlatitude summer atmosphere in MODTRAN is calculated as having 19% transparency out to TOA which translates to an optical depth of 1.69. At midsummer, solar irradiance at midlatitudes would amount to 320W/m2 assuming an albedo of 0.3. This same emitted power would come from a black surface (e=0.97) at a temperature of 276K. Surface temperature in the gray atmosphere is 296K while the MODTRAN model is 294K.
Figure 1. Comparison of radiant flux (W&H Equations 69 and 71).
What leaves the atmosphere in the MODTRAN calculation (TOA) is only 255W/m2 which is inadequate to remove the 320W/m2 input locally. The midsummer atmosphere is free to transport heat laterally (net northward) in order to maintain steady temperature. At the ground surface upward directed irradiance (411W/m2) exceeds downward (326W/m2) but this difference is inadequate to maintain steady surface temperature, so a large proportion of heat must be conveyed from the surface by diffusion, convection, and latent heat.
The gray model (τo =1.69) mimics this pattern with obvious departures. First, the upward minus downward directed irradiance is (435-84) 351W/m2 which is more than adequate to remove surface emitted power and maintain a steady ground temperature. At TOA, however, the difference is 257W/m2 net upward which is not quite adequate to maintain steady model temperature; so the gray atmosphere must transfer heat vertically by other means or its assumed temperature profile is inconsistent. One very apparent difference between the models is that net upward irradiance in the gray model decreases with height while in the real atmosphere it grows with height. This is another indication of the need for convection and latent heat to make a consistent picture.
The MODTRAN calculated irradiances are far more curved with height than that of the gray model which is mainly a reflection of water vapor being far more concentrated near the surface than the atmospheric pressure gradient which is what determines the vertical profile of “grayness”.
Figure 2 shows what happens when the optical depth approaches (τo = 5), which corresponds with W&H’s model. The difference at the surface is (435-157) 278W/m2 which cannot maintain a steady surface temperature; At TOA the outgoing irradiance is only 137W/m2 which is about a third of what is needed to maintain a model temperature.
Figure 2. Gray model of optical depth equal to 5. I used less assumed surface irradiance than W&H did and a sub-adiabatic lapse rate, to make the figures more easily compared.
What is apparent is that as optical depth increases radiation becomes less important as a transfer mechanism. What occurs in detail, however, can’t be settled without letting our temperature distribution free to vary and bringing the energy equation and other phenomena into the analysis. This is an instance of Einstein’s dictum “Make things as simple as possible, but no simpler.” The statement W&H made is undoubtedly correct, but without complete context.
Item 4 is of interest because of something from item 3. The non-greenhouse atmosphere is nothing more than bulk matter, albeit low density, which cannot interact through radiation, but can exchange heat only at its boundary with the surface. It can redistribute heat through diffusion. Thermal diffusion in this simple situation involves Poisson’s equation[7] and a result from potential field theory is that the steady temperature distribution in the domain of such a problem cannot be greater than the boundary maximum nor less than the boundary minimum. Since the ground surface in this case has only one temperature, the only steady solution is an isothermal atmosphere. Figure 3, which is from a paper by NASA scientists[8], shows the remarkable resemblance of every known “thick” planetary atmosphere. Each shows a sort of adiabatic atmosphere up to a height at which energy input to a mesosphere becomes significant. This resemblance persists over large variations in surface pressure and over variations in composition and gravitational acceleration. None, despite billions of years to evolve, has become isothermal because each contains some amount of greenhouse gas.
Item 5 is not, or shouldn’t be, controversial.
Figure 3. Temperature with height (pressure) in a number of atmospheres in the solar system. From [8].
Conclusion
I found the tutorial by W&H interesting because its foundation was molecular physics rather than the usual foundations from classical thermodynamics. Unfortunately, I can’t figure out its intended audience because it is a heavy lift intellectually. One can only conclude that such simple models end their usefulness fast because, as W&H say,”…no one knows just how the complicated climate system of Earth’s atmosphere will respond to the small forcings [of GHGs].”
References:
1- Thermodynamic state variables for an ideal gas include temperature, entropy, pressure and volume.
2- By pencil of radiation I mean a very slim cone of solid angle originating at a point and oriented in a particular direction. All such pencils integrate to a value of 4 pi steradians area in a sphere of unit radius.
3-The First Law concerns changes to the internal energy of a system, which in this case means an atmosphere of nearly ideal gasses. dU=dQ-dW translated means that changes in internal energy (U=heat capacity times temperature in this case) are equal to inputs of heat (Q) less outputs in work (W). Work is rarely included in the energy equation, but might be an essential part of the energy equation as applied to an atmosphere in free convection.
4-Divergence is the net transfer in (negative divergence) or out (positive divergence) of a small volume centered at a point.
5-See for example M. Necati Ozisik, 1973, Radiative Transfer, Wiley. Chapters 12 and 13. A very large collection of dimensionless groups are used in solutions of transport or energy equations by combined phenomena – Peclet number, Nusselt number, Thring number, etc. Quite a collection of such are found in the CRC Physics and Chemistry Handbook.
6-There are many ways to think about this. One is that the product of extinction coefficient and the atmosphere’s scale height forms a characteristic transport distance. If this is a large value, then the transport length of convection becomes much greater than that of radiation, which is taking place through many small jumps. A sequence of many small jumps makes radiation behave like a diffusion process. Convection looks far more organized in comparison and dominates heat transport. If, on the other hand, the product is quite small then the transport distance for radiation is large and eventually becomes so large that it is impossible to beat radiation by any other means as it occurs at nearly the speed of light (gasses have an index of refraction nearly equal to 1); or as W&H say it is “ballistic” transport to space.
7-Poisson’s equation thus has some consistent connection to the Second Law of Thermodynamics.
8-T.D.Robinson, D.C.Catling, 2013, Common 0.1 bar Tropopause in Thick Atmospheres Set by Pressure-Dependent Infrared Transparency, https://www.researchgate.net/publication/259445396