## Robinson and Catling model closely matches data for Titan’s atmosphere

Posted by Peter Morcombe, April 2014

“Climate Science” is a strange discipline that makes all kinds of claims that lack any valid mathematical basis. For example, the Arrhenius hypothesis:

“The selective absorption of the atmosphere is……………..not exerted by the chief mass of the air, but in a high degree by aqueous vapor and carbonic acid, which are present in the air in small quantities.”

I write about “Climate Science” because it is used to justify a political agenda that aims to “mitigate” CO2 regardless of negative impacts on billions of people. Even though most “Climate Scientists” get their pay checks from governments they are reluctant to engage with members of the public who express doubt about the need to reduce the atmospheric concentration of carbon dioxide.

Before making a post I reach out to experts in the field. For example, Tom Peterson (GHCN), Albert Klein Tank (KNMI), Richard Alley (Penn State) and several scientists at the DMI helped with my posts on Greenland (here and here). When I wrote about the Unified Theory of Climate (here and here) Ned Nikolov provided many helpful comments. Scott Denning took an interest in my doubts about the Arrhenius theory (here and here). I think of these folks as “Good Sports” who don’t get offended when their ideas are challenged. They behave as real scientists should.

I was building a model of planetary atmospheres using FEA (Finite Element Analysis) when I came across this letter in “Nature”:
http://faculty.washington.edu/dcatling/Robinson2014_0.1bar_Tropopause.pdf

Robinson and Catling (R&C) have constructed physical models to explain temperatures at all heights for bodies that have significant atmospheres. Furthermore they have made their model available to the public. There are many possible uses including predicting temperatures on exo-planets.

I wrote a draft post and sent it to David Catling for comment. He was kind enough to answer in detail and with such effect that this post had to be completely rewritten. My expectation is that this is the first of many that will be based on the Robinson & Catling model. Enough of the preamble, let’s go kick the tires!

TITAN
The R&C model is designed to run in “IDL” a powerful program marketed by the Exelis corporation. At first this seemed like a show stopper as my wife insists that her kitchen upgrade comes before my hobbies. Fortunately there is a freeware alternative so it was just a matter of minutes to install GDL (gnudatalanguage) that comes with my operating system (Linux “Mint”). I followed the instructions included in these files published by R&C:
AN_RC_MOD.pro and EXAMPLE.pro

The output was a plot of temperature vs. pressure in Titan’s atmosphere. Fortunately I had a copy of the HASI probe data handy so I compared the EXAMPLE model output with it and also with my calculations based on Nikolov & Zeller’s equations [=”N&K” in the chart below, vj].

The blue curve marked “HASI” is the published data from the Huygens probe. The R&C model matches the probe data really well whereas the N&K plot does not. So how does the R&C model work? It uses eight parameters to calculate the temperature against pressure. A critic might say he could match any curve given eight parameters to play with but most of R&C’s parameters are not under the modeler’s control:

• Ttoa Temperature at top of atmosphere (assumed as stratopause) [K]
• F1 Flux absorbed in channel ‘1’ (assumed to be “stratosphere”) [W/m2]
• F2 Flux absorbed in channel ‘2’ (assumed to be “troposphere”) [W/m2]
• F20 Flux absorbed in channel ‘2’ down to reference level p0 [W/m2]
• Fi Internal heat flux [W/m2]
• gamma Ratio of specific heats, Cp/Cv (see Eq. 8 of RC12)
• alpha Empirical adjustment to dry adiabatic lapse rate (see Eq. 10 of RC12)
• n Scaling parameter that relates tau ~ pn, where tau is gray thermal optical depth
(see Eq. 6 of RC12)

The only fudge factor I can see is the “alpha” used to tweak the lapse rate. Clearly this is necessary on Earth with oceans of water and on Titan with oceans of methane.

The R&C model is a fine achievement that provides a mathematical basis for gaining insights into atmospheric physics. The model shows three regions namely the stratosphere where radiative processes dominate, a transition region that occurs at a pressure of ~0.1 bar (tropopause) and below that a convective region (troposphere) where the lapse rate is defined as -g/Cp or less when vapors are present. It appears to correspond well with observations on all seven bodies in our solar system that have significant atmospheres.

The model also includes the effect of internal heat sources that are important in the case of the gas giants and it provides an explanation for the anomalous (negative) lapse rate in the upper Venusian atmosphere. When time allows I plan to become proficient enough in its use to be able to understand the work that R&C have already done, with the aim of applying the model to some questions that have puzzled me over the years. I hope that other amateurs will do the same.

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### 11 Responses to Robinson and Catling model closely matches data for Titan’s atmosphere

1. Ron C. says:

Though it is implicit, the bottom line here is: Infrared transparency in earth’s atmosphere depends upon pressure, not the composition of gases. CO2 is the agent of cooling in the stratosphere. The bulk gases, O2 and N2, under pressure delay the surface from cooling and our mild surface temperatures are the result.

• Thanks to collision broadening the ability of complex molecules to absorb radiation in the lower atmosphere is greatly enhanced but most of the energy they collect is given back to the surrounding “chief mass of the air” as Arrhenius would say.

As the pressure rises the mean time between collisions falls until the collision time constant is much smaller than the radiative decay time constant.

In other words radiative transfer in the lower atmosphere has essentially the same effect as convection. Robinson and Catling’s equations confirm this numerically, so I agree with you when you imply that total atmospheric pressure rules.

• Robert Brown says:

The problem with pressure broadening in e.g. Modtran is this: Pressure (mostly collision) broadening is governed by the fourier transform of a continuous wave train with delta-correlated phase shifts caused by phase-interrupting collisions. The result is the familiar Lorentzian line shape which in turn contributes to the integrated absorptivity when one sums over lines and integrates over the spectrum. van Vleck and Weissskopf wrote the seminal paper on computing this shape from a comparatively simple quantum description, where the governing parameter is the mean free time between collisions. Petty’s excellent book walks one through much of this.

That doesn’t mean that broadening doesn’t depend at all upon the species colliding, only that it is a less important factor, usually, than the MFT itself. As Modtran correctly notes, same-species e.g. CO_2 on CO_2 collisions can have a slightly different lineshape than CO_2 on N_2 or CO_2 on O_2. However, this isn’t really likely to be an order of magnitude effect, as the bulk of the lineshape depends on the properties of the line itself to first order, not second order effects in a short, impact approximation interruption of an effectively slowly-varying oscillation.

Still, Modtran has code to correct the overall absorptivity by separately counting e.g. CO_2-N_2 broadening at concentration (1-q) vs CO_2-CO_2 broadening at concentration q, where q < 0.001. It is reasonable to expect that broading due to doubling from q = 0.0003 to q = 0.0006 would have no more than a 0.001 relative effect on the total atmospheric absorptivity computed — almost certainly completely negligible as far as the effects of line broadening are concerned! That is, Beers-Lambert might change from the direct reduction of the mean free path of IR photons, but the changes in the integrated spectral absorptivity would hardly change at all.

Even this seems like it would be an egregious overestimate. The collision time in van Vleck and Weisskopf that could lead to the same-species increase in spectral line width isn’t the general mean free time between any old collisions, it is the mean free time between same species collisions. This means that the lines are sharpened relative to what they might be from the ordinary MFT by a factor of q, or would be up to the limit of pure spontaneous emission (one cannot sharpen a line any farther than permitted by the spontaneous emission lifetime). When I read the Modtran documentation on the subject, it appears that this additional suppression of the same-species lineshape is neglected — although I have not looked at the source code to be sure. Either way, this is an additional factor of 0.001 (or less), more than enough to completely suppress any additional broadening of CO_2 lines other than what they already have not from the partial pressure of CO_2 but from the absolute pressure of the bulk atmosphere. Partial pressure induced variations in atmospheric absorptivity should be literally indetectably different from the general variability in absorptivity brought about by baseline atmospheric pressure that varies locally by several orders of magnitude more than total CO_2 partial pressure at any location on as little as an hourly basis, plus the overall large scale modulation due to water vapor.

This doesn’t mean that CO_2 is not a greenhouse gas — quite the opposite. It does mean that there is very, very little variation in its functioning as a greenhouse gas with partial pressure as long as the partial pressure is less than perhaps 1% of the total, at least as far as its base radiative properties (absorptivity of its bands) are concerned. Those bands are utterly dominated by CO_2 colliding with N_2, O_2, H_2O and Argon, in order and almost all of the lineshape is due to the time between impact-approximation delta-correlated collisions with any of these species, not any sort of “slow” species-species interaction.

As a consequence, I’ve simply never understood what people mean when they assert that there is some sort of pressure broadening contribution to the expected GHE due to increasing CO_2. No, there is not. There is an effect due to increased concentration and a reduced mean free path of IR photons, but this effect is known to be extremely weak as it is long since saturated. I’m curious as to just how much R&C predict that the pressure of the tropopause should change if integrated linewidths do not change but concentration does, as to me it seems likely that pressure broadening changes due to increasing CO_2 concentration is utterly negligible, and would still almost certainly be negligible as concentration approached 1% (as 0.01^2 = 0.0001 — a 1% effect from same species absorptivity as a fraction of all absorptivity, suppressed by a factor of 0.01 due to the fact that only one in a hundred collisions is between the same species).

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2. Ron C. says:

Another way to put the issue.

The CO2 hysteria is founded on a false picture of heat flows within the climate system. There are 3 ways that heat (Infra-Red or IR radiation) passes from the surface to space.

1) A small amount of the radiation leaves directly, because all gases in our air are transparent to IR of 10-14 microns (sometimes called the “atmospheric window.” This pathway moves at the speed of light, so no delay of cooling occurs.

2) Some radiation is absorbed and re-emitted by IR active gases up to the tropopause. Calculations of the free mean path for CO2 show that energy passes from surface to tropopause in less than 5 milliseconds. This is almost speed of light, so delay is negligible.

The bulk gases of the atmosphere, O2 and N2, are warmed by conduction and convection from the surface. They also gain energy by collisions with IR active gases, some of that IR coming from the surface, and some absorbed directly from the sun. Latent heat from water is also added to the bulk gases. O2 and N2 are slow to shed this heat, and indeed must pass it back to IR active gases at the top of the troposphere for radiation into space.

In a parcel of air each molecule of CO2 is surrounded by 2500 other molecules, mostly O2 and N2. In the lower atmosphere, the air is dense and CO2 molecules energized by IR lose it to surrounding gases, slightly warming the entire parcel. Higher in the atmosphere, the air is thinner, and CO2 molecules can emit IR and lose energy relative to surrounding gases, who replace the energy lost.

This third pathway has a significant delay of cooling, and is the reason for our mild surface temperature, averaging about 15C. Yes, earth’s atmosphere produces a buildup of heat at the surface. The bulk gases, O2 and N2, trap heat near the surface, while CO2 provides radiative cooling at the top of the atmosphere.

• One of the consequences of what you say is that Michael Mann’s single layer model is wrong.
http://wattsupwiththat.com/2011/12/29/unified-theory-of-climate/
Mann’s model is shown under the title:
2. The Greenhouse Effect: Re-examining the Basics”.

How can half of the absorbed energy be re-radiated downwards when the majority has been absorbed by nearby molecules via collisions long before a photon can be emitted?

• Robert Brown says:

The single layer model is explored in detail in Petty’s “A First Course in Atmospheric Radiation”. This is a intermediate to advanced physics textbook, basically, and should be read in some detail by anyone who has any interest in talking about the subject of atmospheric radiation (provided that they have a decent background in the underlying physics as in pretty much a bachelor’s degree or better — otherwise it would be rather inaccessible).

The point of the single layer model — which is in absolutely no possible sense of the word “Mann’s” — is that an interpolated perfect absorber (unit emissivity) blackbody shell would, in equilibrium, radiate 100% of the power incident upon it on both sides away on only the outside. Because it is a thin shell, it has to radiate an equal amount of power inward and outward — locally it looks like a flat sheet at a fixed temperature and radiates isotropically away from both faces. That means that it must absorb twice as much radiation on the inner shell from the contained sphere as it radiates away in the outward direction from the outer shell. This means that the sphere in the middle is at a temperature $2^{1/4}T_0 \approx 1.19 T_0$ warmer than $T_0$, the temperature of the surrounding shell, when the system is in any sort of detailed balance.

There is nothing particularly complex about this — it is the idea behind the “space blanket”. It is also an idea that is almost completely independent of the nature of the shell and only weakly dependent on the geometry.

Petty provides a much more detailed and variable description of the single layer model including differential absorptivity in shortwave (visible) and longwave (IR) light, variable albedo and emissivity, etc. But in the end, in the $\alpha_{sw} = 0$, $\alpha_{lw} = 1$ limit, it leads to this precise result, which has also been independently derived and discussed by a cast of thousands, including Dick Lindzen and myself, because it is really pretty obvious.

In reality there are many places that this model is not particularly faithful to a real atmosphere. A real atmosphere doesn’t have two crudely partitioned absorptivities, it has $\alpha(\lambda)$, absorptivity that is a continuous function of the wavelength involved. It isn’t a single layer or multiple layers, it is a structured continuum with nontrivial changes in $\alpha$ with height, humidity, temperature/pressure, with variable albedo. It isn’t a “perfect conductor of heat” and hence isothermic as it is assumed to be in the single layer model, it actually transport heat via conduction, convection, latent heat, and radiation — all four at once.

In the end it still radiates energy down from greenhouse gases, as the spectrographs reproduced in Petty make absolutely clear. Anybody armed with a halfway decent IR thermometer can directly observe this radiation, and anybody with a good full-spectrum spectrometer can measure the entire spectrum and verify the thermal peaks in LWIR bands associated with e.g. CO_2, H2O, and Ozone (and with LWIR holes where they belong as well). These peaks match thermal holes observable in spectrographs made at the TOA looking down. This is direct, incontrovertible evidence of the greenhouse effect to anybody that can read a spectrograph.

None of this answers the question of precisely how much varying CO_2 concentration up from 300 ppm to 600 ppm will affect global “equilibrium” (if such a term can be applied to an open, highly variable system that is basically never in equilibrium) temperature or affect the oscillations of the actual climate as it seeks this unattainable, near-mythical climate “set point” with various positive and negative feedbacks and many sources of amplified and damped noise. The simple single-layer models are inadequate to describe either Venus or the Earth or pretty much any planetary climate we see, and the Earth’s climate is particularly pernicious given that it involves two fluids, not one, where one of the two absolutely dominates the surface area of the planet, the heat content of the planetary surface, the source of all feedback associated with the strongest greenhouse gas, the source repository of the vast majority of the second strongest greenhouse gas, and a major fraction of both the latent heat transport within and albedo variation of the other fluid — and it isn’t the atmosphere and is rather poorly understood or incorporated into climate models.

I just didn’t want you to continue calling this Mann’s model. It is not. Nor is it an unreasonable model, useful for helping people understand how the atmosphere can increase the dynamic equilibrium temperature of the surface by forming an interpolant absorber layer between it (fundamentally warmed by the Sun as a “source”) and the “sink” of 3 degree Kelvin outer space.

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3. rgb,
Many thanks for taking the time to comment here.

For the benefit of folks who do not know you let me publicly thank you for the help you provided over the twelve years we were colleagues in the Duke university physics department. I can’t remember ever helping you with anything. When it comes to physics and computers I respect your opinions above my own. Even so I want to persuade you to agree with the point I was trying to make.

It is true that Michael Mann did not invent the single layer radiative model. I found it convenient to refer to Figure 1 in the Nikolov & Zeller poster on a “Unified Theory of Climate” that has been discussed here on several occasions.

CO2 has many absorption lines but my personal favorite is the one at 15 microns (wave number 667.7). It takes many microseconds for an excited CO2 molecule to release a photon while the mean time between collisions with nitrogen/oxygen molecules near sea level on Earth is <0.2 nano-seconds. Thus it is that the vast majority of exciited CO2 molecules give up their energy to the "bulk of the atmosphere" (as Arrhenius would say) by collision before they have time to radiate a photon.

The collision time constant is directly proportional to pressure, other things such as temperature being equal. In contrast, CO2 molecules in the stratosphere are much more likely to radiate a photon isotropically so half the radiation will return to the surface or the cloud tops. Will it matter? No it won't.

4. JT says:

The assumption that radiation from excited co2 molecules is isotropic depends on the antecedent assumption that it is spontaneous; but there is a second kind of radiation, to wit: stimulated; as in light amplification by stimulated emission of radiation. Co2 is one of the gasses that can be made to lase. Every excited co2 molecule is immersed in an upwelling flux of photons of the same frequency as the frequency of the photon it would emit spontaneously, which is a situation which could give rise to stimulated emission. Stimulated emission is not isotropic, it is biased in the direction of the stimulating flux. One might wonder if the emission of infrared photons from co2 molecules is biased upward whenever the flux upward from below the molecule exceeds the flux downward from above the molecule; and whether such a bias might be large enough to cause a measurable reduction in downwelling flux which would reduce the magnitude of the greenhouse effect which is otherwise calculated assuming isotropic re-radiation.

• As you say, an excited CO2 molecule can emit radiation when stimulated by a photon at (for example) 10.6 microns. The stimulated emission would occur in the the same direction as the incident photon.

In my opinion, at sea level on Earth CO2 molecules will collide with other molecules every 0.2 nano-seconds which means that the vast majority of excited CO2 molecules will give up their excess energy via collisions before they have time to spontaneously emit a photon or be stimulated to emit a photon.

Quantum electro-optics is my field; I have been building lasers for fun and profit since 1970 but I would feel much better if my esteemed ex-colleague Robert G. Brown would agree that there simply is not time for an excited CO2 molecule in the lower atmosphere to radiate a photon.

5. gallopincamel said:

“This means that “Upwelling Radiation” from the Earth’s surface (in this band) cannot escape into space. Instead it is trapped in the lower atmosphere where it works to establish a thermodynamic equilibrium in the troposphere along with convection and conduction.”

Yes.

The adiabatic process of convective uplift and descent returns energy to the surface in the form of heat that was previously taken from the surface.

In the interim, whilst off the surface that energy is in gravitational potential form which is not heat.

The thing is that if one recognises downward air movement as returning heat to the surface then that is the reason for the higher than S-B surface temperature and not DWIR.

Indeed, the idea of surface heating or reduction of surface cooling from DWIR becomes double countng.

I was mocked at WUWT for exploring the logical implications of downward winds heating the surface but it does indeed happen.

For every molecule that rises in convective uplift another descends by exactly the same amount. At any given moment a full 50% of the atmosphere is rising and 50% is falling.

The trhermal implications of that have been completely missed out from the Trenberth cartoon and the resulting energy budget discrepancy has been incorrectly dealt with by attributing a surface warming effect from DWIR.

Then, all one needs to do to deal with the radiative properties of GHGs is alter the height of convection (faster convection always goes higher) to make more energy stay as gravitational potential energy for longer and there is the equal and opposite thermal response to more GHGs.

No effect on surface temperature.

The additional energy in the atmosphere from GHGs does not need to be in the form of heat at the surface if it can be in the form of potential energy off the surface. And it is.

If it were to be in the form of heat at the surface then the surface plus atmosphere would lose more energy to space than is being received and the atmosphere could not be retained.

GHGs make an atmosphere more emissive to space than a non GHG atmosphere so if balance is to be maintained you cannot raise the surface temperature to make the emissivity of the surface higher as well.

If atmospheric emissivity increases then surface emissivity must drop in an equal and opposite system response and vice versa.

It is ackieved by altering the vigour of convection instead of changing surface temperature.

That changes the length of time that energy in the atmosphere stays in the form of gravitational potential energy (not heat) rather than kinetic energy (heat).