Saturday, March 5, 2016

Lapse Rate on Venus, Part 1

... does it explain why the surface is so hot?


One ... occupational hazard ... of invoking Venus as the Solar System's poster-child example of (runaway) CO2-induced global warming is that some folks (properly) don't take the proposed mechanism of its lead-melting surface temperature lying down.  Sometimes, they even raise (annoyingly) good questions.  Here's Chic Bowdrie from a comment in the Competing Mechanisms article:
The surface of Venus is 740K. The temperature is 310K at 53 km. That makes the lapse rate 8.2 K/km. This is pretty close to a value of 7.4 K/km calculated from g/Cp using 8.87 g/sec2 for the gravity on Venus and 1.2 J/g-K for the heat capacity of CO2. Radiative forcing is not required to explain the temperature profile of Venus.
 A visual may be of some help:

Figure 1 - Temperature profiles for Mars, Earth and Venus taken from Astronomy Notes by Nick Strobel.

Yup, damn hot.  Eyeballing the thing, the temperatures Chic cited for the surface and 53 km altitude check out.  In a subsequent comment, he amended his lapse rate calculations:
From above: I made a mistake with my calculation of the lapse rate for Venus based on –g/Cp. The 8.86 for g is correct, but Cp is not 1.2 other than at the surface. Cp varies with pressure. At 50 km, Cp for CO2 is only 0.9. At 25 km, the heat capacity of Venus is 1.05 and that makes the theoretically calculated value of the lapse rate, 8.4 K/km, exactly equal to its measured value at the same altitude. Unlike Earth, there isn’t much diurnal temperature variation or water evaporation to cause the lapse rate to deviate much from the theoretical value due to convection.
It's never simple, is it.  Sigh.

Thus far I have not attempted to replicate his calculations, having been more engaged with my MODTRAN experiments.  I decided today was the day that I could no longer put this off.  So with the table set ...

How to Estimate Dry Adiabatic Lapse Rate

One way is to use a web-based calculator.  This one helpfully includes the formula ...

... but not much explanation, and no way to vary the g and Cp parameters for planets other than Earth.  No biggie, 'tis a simple formula, surely one can look up the proper values and plug them into the slide-rule of choice.  Before I do that, calling the Cp term "specific heat" threw me for a loop because the implied units (m2 s-2 K-1) are wrong ... when I read "specific heat", I expect J g-1 K-1 as the units.  Maybe this naming is a convention of meteorologists and atmospheric physicists, but I kinda doubt it.  Doesn't matter, that calculator gives the "correct" result for Earth.  But enough pedantic digression ...

... this class notes webpage goes through what Cp is, and how to derive it from first principles:
dT/dz = -g/cp

g= gravitational acceleration
cp = Specific Heat Capacity at Constant Pressure

cp = squiggle R / M

M = mean molecular mass =<amu> 0.001 - e.g. air = 29 x 0.001 kg / mol

R = gas constant = 8.31 J K-1 mol-1

squiggle = 5/2 for monoatomic gas, = 7/2 for diatomic gas, = 9/2 for triatomic gas

UNITS: cp - (kg m2 s-2 K-1 mole-1 )/(kg mole-1) = m2 s-2 K-1

dT/dz = g/cp - m s-2 /(m2 s-2 K-1) = K m-1
"Squiggle" is a new one, but Imma skip it for now.  "Cp" is apparently the abbreviation for constant pressure.  Just below that block of text, the page tells us that a reasonable approximation of Cp for Venus is 850 m2 s-2 K-1, or dividing by 1,000, 0.85 km2 s-2 K-1.  This compares favourably to the range 0.9-1.2 Chic presented, albeit a bit outside the range on the low end.

Be that as it may, went ahead and digitized the temperature profile shown above, threw the results into Gnumeric and plugged in the theoretical lapse rate for comparison.  Here are the results:

Figure 2 - Ah, Houston, we have a problem ... temperature readings above 90 km are coming back below absolute zero.
I could have set the x-intercept to 740 K, which is the surface temperature (as that seems to be the convention), but there's a method to my madness -- I want the calculated lapse rate curve to go cross the observed temperature profile at the altitude that I think convection from the surface tops out ... which I'm calling 60 km.  Why?  From the class notes:
- if cools faster than adiabatic - stable (cool air sinks)

- if cools slower than than adiabatic - unstable (warm air rises)
As you can see by how I helpfully plotted the lapse rate curve, Venus' atmosphere clearly begins cooling faster than the estimated adiabatic rate at 60 km.

Update 3/6/2016 3:30 PM: On review, I may have the correct conclusion, but for exactly the wrong reason.  My current understanding (which may also be wrong) is that the -g/Cp curve should predict a lower temperature than observed in the convective regime, and a higher temperature as a function of altitude in the "stable" non-convective (or less-convective) regime.  I am addressing this first thing in Part 2 of this series, which I shall link to here when it's published.

Update 3/6/2016 10:45 PM: As promised, Part 2 has been published and it addresses my above goof.  I'd like to say it solves all the mysteries being explored here, but I'm sorry to report that I may have only deepened them.  The main thing to look for is the modified version of Figure 3 below -- the -g/Cp curve now intercepts the x-axis at the actual surface temperature, which makes things slightly more sensible.

Not only is it physically ridiculous for a negative absolute temperature prediction ANYWHERE I choose to put the x-intercept, IF there's no convection, there's no adiabatic lapse rate as a function of a pressure gradient.  That bit in bold because I get the sense that this principle isn't well-understood in some circles.

There could be some convection 60 km above Venus' surface, and there probably is a bit.  But here we're interested in what goes on between the surface and the Venusian equivalent of Earth's tropopause.  So ...

A More Reasonable Simple Model of Venusian Temperature Profile

Figure 3 - Huston here, ahh ... roger your last.  We suggest you interpolate.

Speaking of NASA, their fact sheet has this to say about Venus:
Bond albedo                   0.9
Visual geometric albedo       0.67
Visual magnitude V(1,0)      -4.4
Solar irradiance (W/m2)    2601.3
Black-body temperature (K)  184
The calculated blackbody temperature comes from the Stefan-Boltzman law of radiative power ...

... solved for temperature; where j* is solar irradiance divided by four, ε (emissivity) is unity (any value < 1 is considered a "graybody", with zero being a perfectly reflective "whitebody"), and σ is the Stefan-Boltzmann constant:

Upshot is, 184 K is Venus' so-called effective temperature ... or the temperature it would "look like" when viewed externally by a hypothetical broadband radiative sensor.  The vertical portion of the yellow dashed line in the plot above is at x = 184 K because it is reasonable to assume that some point of Venus' temperature profile is going to coincide with its theoretical blackbody temperature.  And so it does, somewhere between 83 and 84 km above the surface, and again between 184 and 185 km.  There's nothing super-significant about those two crossing points.  Mainly, the purpose of the vertical portion of the yellow curve is as an easy visual reference that, yes, the S-B approximation does tell us something about reality.

The dotted light-blue curve is, well, curve-fitting bollocks for the most part.  Above 60 km it's a naive guesstimate of what I think the temperature profile might be like if the upper atmosphere were more uniform in composition than it actually is.  Mostly, it just looked right to my eyeballs when I set the inflection point at (140 K, 100 km) for the cubic spline interpolation.  All other points are actual values.  Below 60 km, it follows the linear lapse rate prediction exactly.

There's a method to this madness.  At least I hope there is.

Noting that the lapse rate prediction of surface temperature, starting from 60 km up, is 887 K -- a full 148 K hotter than observed -- the thing I imagine Chic would say is pretty much what he's already said:
Radiative forcing is not required to explain the temperature profile of Venus.

Oh, But I Disagree ...

... of course.  And I have in very broad terms told him why:
Chic: The ideal gas law applies to ideal gases and predicts temperature as a function of pressure and density.

Me: No, it predicts change in temperature as a function of change in pressure. When you inflate a tire, its temperature surely rises. When you stop pumping, its temperature returns to ambient over time as the heat imparted by the work done by compressor dissipates.

He didn't like that:
Chic:What is that supposed to mean? That the temperature of Venus isn’t a result of the pressure and density caused by gravity?

Me: Just what it says. I defy you to predict the absolute temperature of a bicycle tyre inflated to 100 psi without knowing anything about ambient conditions or how much time has elapsed since it was inflated. You would have much better luck predicting temperature change as it's being inflated at "normal" rates b/c you could reasonably assume an adiabatic process. But you still wouldn't know the final absolute temperature without knowing the initial absolute temperature.

Chic:IOW, you think that the temperature of the surface of Venus results totally from radiation and the pressure and density just happen to coincidentally satisfy PV=nRT?

Me: Chicken-egg problem it seems.

No I don't think its coincidence because a similar thing happens on Earth. Review this plot from MODTRAN results posted above. Note how similar in slope the yellow and light blue curves are until 12 km when the LW gradient all but disappears and that lapse rate continues on until 16 km.

I submit to you that without the LW gradient, convection would top out at a much lower altitude. As well, absolute temperatures would be much lower.
Emphasis in original EXCEPT for that final sentence, because really, this is where the whole ball of wax melts as I see it:
Me (to Chic): Your argument, as stated, implies that the temperature of the surface of Earth, Venus, whatever planet, would be the same if [they] occupied any orbit from Mercury to [the] Oort Cloud.
One final piece comes from reader BBD:
How about picturing the Earth looking down from space, viewed in IR. It's a slightly fuzzy ball. Zoom in and the fuzziness is the altitude of effective emission. Increase the atmospheric concentration of an IR absorber like CO2 and it gets fuzzier (more opaque) still. It is impossible for this not to raise the altitude of effective emission. Yes?

But as the altitude of effective emission rises, temperature at altitude falls and this begins to inhibit the radiative transfer of energy to space. Yes?

So, inevitably, an energy imbalance piles up in the climate system until the troposphere is warm enough for equilibrium with solar flux at TOA to be restored.
Ah yes, my good friend "altitude of effective emission" (AEE).  I have previously thought of this as just a curiosity value as, AFAIK, most atmospheric radiative-convective models don't parametrize it.  I guess I've mostly thought of it as an emergent property of the real system, and reasonably representative models of it.  IOW, "raising the roof" isn't the mechanism, it's the result.

Because Chic has made some awfully good points which feels like I'm somewhat flailing to answer, and because of things I've been reading elsewhere, and things I've written elsewhere, it's clear to me that I have to rethink the concept of AEE as a mechanism, not just an outcome.

The Key Is ...

... I think ... that convection doesn't begin with gas falling from altitude -- it begins with gas being heated from the surface.

In the interest of starting simple, Cp can be treated as a constant by assuming atmospheric composition is uniform.  It's a physical property mainly determined by the specific heat content of its constituent components, after all.

Therefore, a way to think about lapse rate in this context is that the difference in temperature from the surface to altitude is determined by how high convection from the surface goes until convection "stalls", and a cooled air parcel begins to want to sink instead of rise.  Working out how, where, when and why that happens will be the subject of Part 2.

But for now, the Eureka moment for me is that Cp determining a constant slope, temperature mainly varies as a function of altitude.  If convection goes to a higher altitude, the difference in temperature between surface and tropopause MUST be higher that it would be otherwise.

One troublesome aspect of thinking about it this way is that it implies that one end or the other of the "ramp" described by the lapse rate slope is going to want to be fixed at some absolute temperature or reasonably close to it.  In my musings about how to determine which (I of course assert now that it's the point at the surface which is the freer parameter) I kluged together this plot:

Figure 4 - Approximate blackbody fluxes derived from the temperature profiles in Figure 3.  Note the log scale for the x-axis, and units in kilowatts per square meter -- visual conveniences only.

Assuming the differences in flux between the actual temperature profile and my theoretical profile (the light-blue curve) are reasonably representative of reality, radiative transfers are a prime candidate to explain much of them.  It rains on Venus you know, sulfuric acid mainly, but possibly other things we wouldn't normally associate with precipitation.  Diffusive transfers by conduction are a possibility as well, but I think they can be ruled out as the relevant gasses are poor thermal conductors.

That's it for now.


  1. Another nice one, Brandon. Are you still taking the class? You should get extra credit for this work.

    I have three comments.

    1) Cp varies with both temperature and pressure. I found my values at this website:

    2) “Radiative forcing is not required to explain the temperature profile of Venus.”

    Someone probably has a corollary of ‘never say never’ that would apply to that statement. I basically meant that the linear drop in temperature below the AEE, corresponding to the temperature of Venus as observed from space calculated from the SB equation, is not determined by radiative forcing. Above the AEE radiative factors dominate what determines the temperature profile.

    3) You stated above that the ideal gas law “predicts change in temperature as a function of change in pressure.”

    Standard thermodynamics says the idea gas law express the relationship between pressure, volume, and pressure by combining several other laws applying to ideal gases. The lapse rate expresses a change in temperature with respect to a change in the height of a theoretical column of an ideal gas exposed to a gravitational force. The derivation of this gradient uses the hydrostatic equation relating a change in pressure with height of a column of gas influenced by gravity. So indirectly temperature can be expressed as a function of pressure, but it’s a log function. In either case, the relationship is not a coincidence. It’s a natural phenomenon that applies to the lower tropospheres of large planets.

    1. Thanks, Chic.

      Extra credit, hell. After writing Part 2 of this fiasco, I'm feeling like I should take a class.

      1) Ayup. I finally looked into that. The units are wrong. Does give a seductively close answer though, doesn't it? Part 2 ends with the beginnings of discussing this.

      2) I think I've understood what you've been driving at there for some time now. I don't completely disagree, but articulating how I think it actually works still presently eludes me. In my meandering way, I'm working on it.

      3) Again, none of that is in dispute here in terms of temperature change. Absolute temps are the dispute, namely where when and how they're set. I'll have more to say as I do more study and playing with my toy models. Hang tight.

    2. 3) There seems to remain considerable dispute about something on the competing mechanisms thread. Going there now to try to sort it out.