Lapse Rate on Venus, Part 2

Review of Part 1

In the first post of this series I made two plots which warrant being shown sequentially without any intervening verbosity:

Figure 1 - Observed and (simplistically) modelled temperature profiles of Venus from the surface to 150 km altitude.

Figure 2 - Calculated blackbody radiant fluxes of the temperature curves shown in Figure 1.

As I see it, some takeaways from Figure 1 are:

1. The dry adiabatic lapse rate as modelled by -g/Cp explains most of the observed temperature profile up to 60 km with reasonable fidelity.
2. Extending the lapse rate curve much past 80 km altitude results in a physically ridiculous negative absolute temperature prediction and ..
3. ... it's obviously inconsistent with observation above 60 km.
4. As drawn, the dry adiabatic lapse rate curve intercepting observation at (261 K, 60 km) predicts a surface temperature of 887 K, or 148 K higher than the observed value.
5. According to some class notes I obtained from a quick Google session, dry adiabatic lapse rate should be less than observed lapse rate in a convective atmospheric regime.  In Part 1 I goofed: the other way of saying this is that when actual absolute temperature is greater than what -g/Cp would predict, the atmosphere is destabilized from below by the relative buoyancy of the warmer air, and convection is the result.
6. Above 60 km, stuff happens which -g/Cp cannot explain, but neither should it be ignored.

Point (5), being an egregious error on my part, needs to be sorted first ...

Revisiting g/Cp Absolute Temperature Predictions on Venus

So, not only is it physically invalid for lapse rate to predict negative absolute temperatures, it is my present understanding that a lapse rate that predicts a higher absolute temperature throughout a known convective atmospheric regime is also nonsense.  There is a graphic in the class notes I've been referencing which makes this clear (unless, like me, one first misinterprets it):

Figure 3 - stable (non-convective) vs. non-stable (convective) atmospheric conditions.  Supporting text from original:

- - - - - = adiabatic lapse rate, White line = actual lapse rate

- if cools faster than adiabatic - stable (cool air sinks)

- if cools slower than than adiabatic - unstable (warm air rises)

..... which leads us to circulation patterns - next class.

Where I cocked this up is in shifting the x-axis intercept in my Figure 1 above by forcing the -g/Cp curve to intercept the observed profile at 60 km.  It makes for a visually easy way to find the point at which convection stalls and temperature profile above that point ceases to track adiabatic lapse rate.

The part I got right is that when the slope of the actual temperature profile is greater than adiabatic lapse rate, we expect instability and upward convection.  Stability and no convection for the converse.

Unfortunately, my visual convenience lead to making physically untenable (i.e. dead wrong) predictions about what absolute surface temperature "should be" at the surface were it not for convection.

It's now abundantly clear to me one way in which there is such confusion out there when lapse rate is invoked to explain absolute surface temperature ... quite obviously I am not immune to mucking it up either.

The above plot takes the surface temperature as the known given, and sets the x-intercept there.  And why not: we have many thermometers on the ground, not so many at altitude.  And as I keep emphasizing: convection begins as a result of solar SW being absorbed by the surface over land, below the surface over bodies of water.

This calls for a revised -g/Cp model for Venus, thus:

Figure 4 - Venus lapse rate model with same -g/Cp slope used in Figure 1, but with x-intercept set to observed surface temperature.

 

This makes much more sense than the way I plotted it originally because now temperatures are warmer at all altitudes than -g/Cp would predict.  And why not?  It really should be beyond dispute that:

1. The main energy source for pretty much any planet in this system is the Sun.
2. Unless astrophysicists have missed something major, radiative loss is the main mechanism by which all Solar System bodies dissipate absorbed solar energy back into the dark, quite cold, void of outer space.
3. It therefore follows that gaseous planetary atmospheres being compressed by gravity cannot reasonably sustain higher temperatures than the sum of radiative gains less radiative losses.

Be that as it may, the above IS disputed.  This wants examples, but they'll have to wait for next post; I have different fish to fry for today's discussion.

First one to batter up and hold over the hot oil is that silly light-blue cubic spline curve fit I added in as the -g/Cp ramp runs impossibly toward negative Kelvin territory.  Again, I think it looks plausible, but there's no real physics behind it except for the two anchoring points in the middle, which you will note have taken up different residence from where they were placed in Figure 1.

The rightmost one at (428 K, 30 km) is based on the Wikipedia Article on the Atmosphere of Venus, which includes this visual:

Figure 5 - Schematic of the Venusian atmosphere giving some hints about its vertical composition.  It's also gratifying to note I'm not the only one fond of smoothed temperature profiles.

Where the clouds and haze are will be important later.  For now, note that the 10 bar layer corresponds to about 30 km, which for now I am calling the equivalent of Earth's tropopause; specifically, the point at which deep convection stalls and radiative transfers come into their own to explain temperature profiles.  Annoyingly, lapse rate continues to follow what -g/Cp would suggest right up to about 60 km as shown in my Figures 1 and 4.  Hold that thought for now.

The (184 K, 84 km) anchor point in Figure 4 goes back to a statement from Benestad (2016):

On average, the net short-wave energy flux (visible light) on earth is roughly 240 W/m2, which must balance an equal upward energy flow for a planet in equilibrium.

For review, 240 W/m2 assumes Earth's solar constant is 1,371 W/m2.  Divide by 2 to account for spherical geometry, and by 2 again to account for only half of the planet being sunlit at any given time, which gives 343 W/m^2.  Multiply by 0.7 to account for Earth's albedo of 0.3 and we get 240 W/m2, a value which is in line with the TF&K (2009) energy budget cartoon.

Down the same page, Benestad continues:

In the atmosphere, IR light can be absorbed and re-emitted multiple times before its energy reaches the emission level where it is free to escape to space (Pierrehumbert 2011). The process of repeated absorption and re-emission will result in a more diffuse structure for the OLR at the top of the atmosphere. Hence, for an observer viewing the earth from above (e.g., a satellite instrument measuring the OLR), the bulk IR light source is expected to be both more diffuse and located at increasing heights with greater concentrations of GHGs, as the depth to which the observer can see into the atmosphere gets shallower for more opaque air. This altitude is henceforth referred to as the ’equivalent bulk emission level’ and is the 254 K isotherm ZT254 K.  It represents the mean height for both cloudy and cloud-free regions.

My emphasis.  First one because it's something I find necessary to keep repeating to Chic.  Second one because for me, it was a Eureka moment when I read Benestad's summary of the principle in his RealClimate post about the paper:

The depth in the atmosphere from which the earth’s heat loss to space takes place is often referred to as the emission height. For simplicity, we can assume that the emission height is where the temperature is 254K in order for the associated black body radiation to match the incoming flow of energy from the sun.

Which is not to say that emission height is the ONLY altitude at which the Earth loses energy to space.  It's the average altitude at which average atmospheric temperature predicts a blackbody flux equal to the average solar energy absorbed by the entire system.

That groundwork laid, it follows that the same principle holding true on Venus is as good a hypothesis as any.  Reviewing information provided in the previous post:

NASA's 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

So, 2601.3 / 4 * (1 - 0.9) = 65 W/m^2.

Plugging into Stefan-Boltzmann solved for temperature, (65 / 5.670367E-08)^0.25 = 184 K.  The lowest altitude where that temperature occurs according to the observed profile is 84 km.  Makes sense to me to anchor my spline fit curve to a point theory says should exist.  So I did.

Now.  Review what I said about Figure 5 above:

Annoyingly, lapse rate continues to follow what -g/Cp would suggest right up to about 60 km as shown in my Figures 1 and 4.  Hold that thought for now.

My cubic spline curve fit shows an obvious deviation away from the linear -g/Cp prediction.  The Wikipedia author gives no references or justification for the shape of the temperature profile above the troposphere above 30 km.  Maybe my eyeballs are playing tricks on me, but my plot appears to show a more pronounced departure from the linear lapse rate curve.  Up to 80 km however, Figure 5 does show a decidedly steeper slope; in fact, at 80 km it's just about vertical.  As my curve looks more like observation from a credible source I'm content to not quibble any longer and run with it for now.

The big question now is: does my approximation from 30 km to 84 km make physical sense?

I think so.  Kind of.  We'd expect near the point convection stalls that radiative cooling would become more dominant that adiabatic cooling, and hence, for there to be a departure from -g/Cp as the leading predictor of temperature change as a function of pressure and therefore altitude.  We do indeed see this in the observed profile for Venus.  I previously thought it was happening at 60 km because that's where the slope of the profile starts getting decidedly more vertical, just as happens on Earth at the tropopause:

Figure 5 - Venus, Earth and Mars temperature profile comparisons, redux.  Note that Earth's is vertical (essentially isothermal) between 12 and 20 km because there's very little convection in the lower stratosphere.

On the basis of Earth's isothermal lower stratosphere alone, my naive assumption would actually be that Venus' tropopause was actually around 90 km instead of 30 km as Wikipedia would have me believe.

There's (at least) one more conundrum here; going back to Benestad (2016):

The temperature drops with height due to convective adjustment (standard atmosphere vertical temperature profile with decreasing temperature with height) and the radiative heating profile (Fleagle and Businger 1980; Houghton 1991; Peixoto and Oort 1992; Hartmann 1994), and equals the emission temperature of 254 K at around 6.5 km above the ground (Fig. S10 in the Supplementary Material).

My bold.  Earth's tropopause is about 12 km.  Altitude of effective emission (AEE) is below that at 6.5 km.

Totally the reverse of Venus with tropopause at 30 km and an AEE (as I am calculating it) above that at 84 km.  It's like the mother of all inversions.  I am perhaps onto something there because, from Figure 5 above, note the sulphuric acid cloud decks living above 80 km.

More questions that will have to wait for another time because there's one final thing I wanted to get to in this instalment ...

Does Cp Vary with Altitude?

Chic has previously commented:

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.

And today, he reminded me of same:

You should note that Cp varies with both temperature and pressure, at least according to this website where I got my values: http://www.peacesoftware.de/einigewerte/co2_e.html

So I go to the link above, it's an online calculator.  Cool.  There are several on the page, first one is called "Calculation of thermodynamic state variables of carbon dioxide" and it takes pressure and temperature as inputs for various units, including bar for pressure and K for temperature.  Very cool.  So I plug in the following values and among the things it spits out is Specific isobar heat capacity :cp as follows:

pressure (bar)	temperature (K)	altitude(km)	cp (kJ/(kg K))
90	739	0	1.18444
10	492	30	1.02193
1	350	50	0.89895

I subbed 30 km for 25 as pressure is an even 10 bar at that altitude.  Rounded to the nearest tenth, the calculations for cp at the surface and 50 km come back to Chic's numbers on the button.

But ... look at the units: kJ/(kg K).  For review, the -g/Cp formula is:

As I mentioned previously, Cp in this context is NOT specific heat in the sense of how many Joules it takes to raise a unit mass of some material by 1 K.  The implied units are km2/(s2 K), which is difficult to translate into English so it kind of makes sense that it gets shortened to "specific heat".  As it turns out, this causes all sorts of mischief:

Figure 6 - Eerie coincidences can happen when plugging in physical constants with the wrong units.

So when Chic asked ...

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.

... I really should have said:

Damn straight it's a coincidence -- the units you've used for Cp are wrong.  Further, with the actual temperature profile following the adiabatic lapse rate, there should be little to no convection happening.  And if there's little to no convection, there wouldn't be a lapse rate to begin with.

Update 3/10/2016:

Chic reminds me down in the comments ...

One J is one kg-m2/s2. So if you multiply 850 m2/s2K by 1kg/1000g you get 0.85 Kg-m2/s2/g-K or J/g-K. So don't throw away that Figure 6 that you think is just a coincidence.

... my bold ...

... which means my struck-out conclusions above proceed from the false premise of Cp being given in the "wrong units".  Hence the spooky-perfect correlation of observed lapse rate to the -g/Cp prediction for Venus up to 40 km, and good agreement up to 60 km, could be reasonably defensible.

Part 3 of this series will delve into this further, specifically taking on the notion I've been holding to that convection doesn't happen if the dry adiabat and environmental lapse rate almost exactly coincide -- further readings tell me that it apparently doesn't work that way on this planet.

Wrapping Up

This post got a bit pear-shaped on me, and seems to have raised more questions than it answered.  The next obvious step is to convert those isobaric specific heats into values with the proper units and see what happens.  It's going to have to wait for Part 3 of this (unexpectedly growing) series because my head hurts, I'm hungry, and the dog wants attention.