Friday, May 07, 2010

Hyperventilating on Venus

Venus by Velazquez

Steve Goddard wrote an interesting article about the temperature of Venus. It essentially argues that the "extra warming" by hundreds of degrees that we see on Venus is mostly due to the adiabatic lapse rate - while the greenhouse effect contributes just a small portion (a dozen of percent at most).
Although I find his text somewhat sloppy about various "details", I had to independently agree with the broad and important conclusion - after some checks and self-corrections - and I will try to convince you about the conclusion. This conclusion does mean that people like Carl Sagan, James Hansen, and others who have been using Venus as the model for the Earth's greenhouse effect were wrong even morally.

Most of the warming is caused by things that have nothing to do with the absorption of infrared radiation, indeed. But as you might expect, most of my text will be about some of the "details" because they can be subtle and they include some elementary but interesting physics.

As I argued in an article about the importance of black bodies, the stable absolute temperature at distance "R" from the Sun goes like "1/sqrt(R)". Because Venus orbits 0.72 AU away from the Sun, the surface temperature should be something like 288 K / sqrt(0.72) = 339 K which is 66 °C if the albedo, the composition of the atmosphere, and details of the greenhouse effect were equal to those we know and love.

With a little bit greater albedo (reflectivity), and Venus indeed has a greater albedo than the Earth, Venus' surface temperature could actually be equal to the temperature on the Earth. However, in reality, it is the evil sister of the Earth: that's why Venus is the symbol of almost all women. ;-)

The surface temperature is about 300-400 °C warmer than what we calculated now. Because the atmosphere is almost entirely composed out of carbon dioxide, and this gas dominates all standard processes because it can do whatever the minor gases can do, we can attribute the whole additional surface warming by 300-400 °C to the carbon dioxide.

But what does the CO2 do to make the surface warmer? Is it the greenhouse effect?

The concentration of CO2 on Venus is something like 300,000-500,000 times greater than the same quantity on the Earth (92 times higher total pressure; 3,000-5,000 times higher a percentage, depending on whether we calculate the molar/mass percentage) - but the warming attributed to this gas is only 100-200 times greater than it is on the Earth (at most 3 °C from all the CO2, including the natural one).

Clearly, the warming increases much more slowly than linearly with the amount of CO2 when the concentrations get really large. However, it increases faster than logarithmically when they're large: 300,000 is equal to 2^{18} or so and 18 CO2 doublings should give about 18 x 1.2 °C = 22 °C (no water feedbacks on Venus): that would be a sensible calculation if the greenhouse effect were the cause.

The actual warming is more than 10 times as large, so if you believed that the extra 300-400 °C on Venus' surface are due to the greenhouse effect - a belief that will be addressed below -, you would have to conclude that the "climate sensitivity per doubling" in the context of the Venus (huge concentrations) has to be about 10 times bigger than it is on the Earth, at very low CO2 concentrations that we are familiar with. The logarithm would still be a relatively good enough approximation - much better than the linear curve - but the sensitivity would have to be pretty radically adjusted when we enter completely new physical regimes.

Now, Steve Goddard essentially wants to argue that the greenhouse effect plays no significant role on Venus: it's negligible relatively to the temperature differences from our black-body calculations that are caused by other effects (independent of the chemical composition). We will see that although he doesn't explain much physics of why the warming should be independent of the composition, he is essentially right.

Goddard claims that the surface of Venus would be equally warm if CO2 were replaced by nitrogen, N2. (Goddard only wants to replace 90% of the CO2, to keep most of its greenhouse effect which is created by the 10%, but even with the full replacement, the results won't change much.) That's the statement we will discuss in the rest of this article. Goddard writes:
9000 kPa atmospheric pressure would occur on earth at an altitude many miles below sea level.  No such place exists, but if it did – it would be extremely hot, like Venus. A back of the envelope estimate – temperatures on earth increase by about 80C going from 20 to 100 kPa, so at 9,000 kPa we would expect temperatures to be in the ballpark  of :

20C + ln(9000/(100-20)) *80C = 400C

This is very close to what we see on Venus.  The high temperatures there can be almost completely explained by atmospheric pressure – not composition. If 90% of the CO2 in Venus atmosphere was replaced by Nitrogen, it would change temperatures there by only a few tens of degrees.
When I (and Olda K.) read his text for the first time, I thought he was saying that the calculation was accurate up to "tenths of a degree". That would make me really upset - a "back of the envelope" calculation can't be suddenly claimed to be that accurate. ;-) I apologize to Steve Goddard for criticisms caused by my wrong reading of the word "tens".

"Tens of degrees" sounds better. The "back of the envelope" calculation may be believed to be more accurate than 100 °C which allows him to say that the error is just dozens of degrees. Note that his calculation is nontrivial because he is using the temperature dependence on the altitude that is derived from the numbers measured on the Earth. He extrapolates the formulae to Venus and gets an agreement with the observed Venus surface temperatures - the 400 °C of extra warming.

Fine. So is the lapse rate enough to explain the extra 300-400 °C of warming?

Goddard uses a formula that secretly assumes that the temperature linearly decreases as a function of the altitude - and the altitude may be expressed as the logarithm of the pressure, because of Boltzmann's distribution. That's probably the implicit justification behind his calculation
20C + ln(9000/(100-20)) *80C = 400C.
Is that correct? First, we must understand why the temperature changes with the altitude. The decrease of the temperature with the altitude is called the lapse rate. It only exists between the surface and the tropopause - inside the layer known as the troposphere. By definition, the lapse rate changes sign at the tropopause, which is by definition the boundary between the troposphere and the stratosphere.

Why is there any lapse rate?

The most important idealized and calculable type of lapse rate is the dry adiabatic lapse rate. The word "adiabatic" means that no heat is transmitted from warmer to cooler objects (no entropy increase). As the gas goes up, it occupies a bigger volume, which means that it does some (reversible) work - on pushing the other gas away - and by doing work, it's losing energy (and therefore temperature).

Equivalently, you may look at this process microscopically. The average kinetic energy of the air molecules (which is proportional to the temperature, assuming the ideal gas approximation) drops with the height because the total energy is conserved and the potential energy increases with the altitude.

Fine. How big this universal effect is?

As the explanation on Wikipedia tells us, you get 9.8 °C per kilometer on Earth. (More realistic "saturated adiabatic lapse rate" takes the latent heat of water into account: it's not "dry" and it's just 5 °C per kilometer or so. We won't discuss it here. The most accurate answer is somewhere in between the "dry" and "saturated".)

The formula for the dry adiabatic lapse rate is
Gammad = - dT / dz = g / cp.
Here, "g" is the gravitational acceleration which is 9.8 m/s^2 on Earth but 8.9 m/s^2 on Venus which is slightly lighter (and, less so, smaller).

Also, "c_p" is the specific heat at constant pressure. It's the heat needed to warm the gas by a unit temperature, assuming constant pressure, and "specific" refers to having a unit of mass.

Now, the thought experiment where we replace CO2 by N2 can have various realizations: we may put the same mass of the gas, or the same number of molecules etc. - these are different situations. While the task to calculate is not quite well-defined, it's important to see some "details" about the calculation of the lapse rate, anyway.

The quantity "c_p" is different for different gases because it is the ratio of the molar heat (at constant pressure) "C_p" and the molar mass "M". Both quantities depend on the type of the gas.

Recall that monatomic, spherically symmetric ideal gases have "C_V=3/2 R" (from three "linear" degrees of freedom) and "C_p=5/2 R": the difference "C_p-C_V" can be shown to be equal to one "R" quite universally, from the first law of thermodynamics. However, complicated molecules such as CO2, N2, or N2O have three degrees of freedom for the momentum and two extra degrees of freedom for rotations (because they're still linear, i.e. symmetric around an axis; otherwise they would have three rotational ones). That increases "C_p" to "7/2 R" for CO2 and N2.

Fine. So CO2 and N2, treated as ideal gases, have the same molar heat. It's probably a coincidence because Goddard apparently hasn't paid any attention to the difference between monatomic, diatomic linear, and diatomic nonlinear gases. If he replaced CO2 by helium, it wouldn't work. ;-)

However, it's the specific heat, "c_p", calculated per unit mass that enters the formula for the adiabatic lapse rate. It is "C_p/M". We just decided that "C_p", the molar heat, was equal for CO2 and N2. However the molar masses "M" are different. The molar mass of CO2 is around 44 grams per mole while N2 has around 28 grams per mole. So the lapse rate (cooling per kilometer of height) for CO2 is actually 44/28 = 1.57 times greater than it is for N2.

However, CO2, being heavier, would also tend to sit 44/28 = 1.57 times closer to the surface than N2. The atmosphere where the lapse rate occurs could naturally be thinner - the exponential change of the pressure would be faster - so the total cooling between the surface and the "tropopause" could indeed be equal in this approximation. So he's right that if he replaced CO2 on Venus by a gas with a similar molar heat, the surface temperature would be pretty much equal to what it is for CO2.

Summary

To summarize, the adiabatic lapse rate is a key effect that drives the temperature difference between the tropopause - many kilometers above the surface - and the surface of a planet. In fact, a pre-existing lapse rate is an essential pre-requisite for the greenhouse effect, too (without it, the absorption and emission would be balanced): the greenhouse effect may be understood as a slight change of the pre-existing lapse rate.

The lapse rate has the capacity to add hundreds of degrees Celsius to the surface temperature of Venus, regardless of the composition of the atmosphere.

However, the "back of the envelope" calculation may still produce hundreds of degrees Celsius of errors because Goddard uses the "real" lapse rate from the Earth - which is not "dry" - and extrapolates it to Venus - where the dry rate could be more appropriate. The two lapse rates differ by a factor of 1.5 - 2. Similar issues concerning the different values of molar heat may also play the role.

The fact that Goddard managed to produce a result that seems to roughly agree with the surface temperature on Venus surely doesn't prove that the whole calculation is correct and everything he has neglected can be neglected. ;-) Clearly, the varying molar heats can't be neglected, and they can add a factor of 7/5 to the lapse rate. The condensation of parts of the atmosphere can't be neglected, either.

However, you could apply Occam's razor and argue that the hot Venus' surface can be "largely" explained by the adiabatic lapse rate - and the greenhouse effect could give the warming of 18 no-feedback CO2 doublings which were claimed to lead to 22 °C of warming only. However, even this estimate may be refined if we use more accurate formulae for the no-feedback greenhouse warming.

For example, I discussed some better formulae for the greenhouse warming than the simple logarithm in "Why is the greenhouse effect logarithmic?". The first particular formula I mentioned was
Temperature = Temperature0 + ln(1 + 1.2 x + 0.005 x2 + 0.0000014 x3)
where "x" is the concentration in ppmv. This formula should only be trusted up to 1,000 ppmv or so but let's see how it deviates from the simple logarithmic rule when the concentration gets really large. The constant "Temperature0" has to be chosen as the pre-industrial temperature minus 6.633 °C, so that Temperature [280 ppm] gives the pre-industrial temperatures.

With this choice, 560 ppm (doubled CO2) gives a temperature warmer by 1.186 °C - the well-known no-feedback sensitivity is 1.2 °C or so. But even if we substitute the concentration 390 * 300 000 ppm, to mimic Venus, we only obtain 36 °C - just slightly above the 22 °C obtained from the simple logarithmic formula. Even though the log-polynomial formula can't be trusted at the huge Venusian concentrations, its slow deviation actually indicates that the greenhouse portion of the Venusian warming is only "dozens of degrees" rather than hundreds of degrees.

Even though I was originally critical about Goddard's text, I do think he has demonstrated - or we have demonstrated, assuming that you agree that many things were missing in his text - that the statement that the "extra hundreds of degrees of Venusian heat" are mostly due to the greenhouse effect is simply wrong. In the past, I have parroted this incorrect statement without any verification, too.

1. Lubos,

Thanks for this. One important correction. My text said "replace 90% of CO2 with N2" - not all of the CO2. Big difference. The first 10% accounts for almost all of the greenhouse effect.

I generally write for WUWT at a qualitative level because it reaches a wide audience of people, with a wide range of technical interest and skill. The whole point of a "back of the envelope" calculation is to provide a rough first estimate.

2. Sometimes (not always) when you use the word "moral" I get the feeling that you mean the word in some other sense than "in compliance with rules against evil conduct." I've checked the dictionary for alternate senses (e.g., the "moral of a story") but I'm still not sure what you mean. You said that Sagan and Hansen "were wrong even morally." What do you mean by "morally" there?

3. How do internal temperatures differ and how do they affect surface temperatures? Isn't the "surface" of Venus in effect more "internal" because of all the matter above it?

4. Lubos,
If the issue were similar quantities of CO2 vs only a small amount of greenhouse gasses but similar amount of N2, your argument would be valid. However if there were nothing but N2 (and assuming that there was no absorption of radiation by the N2), then you are wrong.The lapse rate (drop in atmospheric temperature with increasing altitude) due to the very high surface pressure on Venus would exist with a greenhouse atmosphere or one totally transparent to both incoming light and outgoing long wave radiation. The greenhouse gas as you pointed out does only cause a few degree effect directly. However, without any greenhouse gas, the surface is where the outgoing radiation energy has to match the absorbed (by the ground) energy of the incoming radiation, and thus this is where the surface temperature is determined. In that case, the lapse rate results in the atmosphere getting very cold as altitude increases. The actual lapse rate would probably differ some as gas properties vary at very low temperatures. However if there is any reasonable amount of greenhouse gas, the location of much of the effective source of radiation to space moves up in the atmosphere. Even less than 1% CO2 in Venus’s atmosphere would result in almost all of the radiated energy to space occurring at very high altitude. Now the added temperature from the CO2 is still small, but the location of the temperature matching incoming radiation to outgoing is at a very high altitude, and the increasing temperature due to the lapse rate as you go lower still holds. It is the movement of the location of the source of radiation to space from the ground to high altitude along with high pressure that makes the surface hot on Venus, and this is not a runaway effect but straight Physics given the present conditions.

5. Dear Steve, thanks for your correction. But because you have given evidence that the greenhouse effect on Venus is much smaller than the other effects anyway, it's not a big difference to use 90% or 100%.

Dear ForNow, it's an adjective I began intensely use in the same way as Edward Witten who likes it a lot. The point is that sometimes, a statement is technically correct/wrong, when you really interpret it literally, but the broader and more general message that the statement suggests is, on the contrary, wrong/correct, respectively, and the statement is correct/wrong just because of shallow technical reasons. In that case, we say that the statement is wrong/correct, morally.

It's "moral" because sometimes you may lose/win because the rules of the game just make you lose/win, but you would deserve to win/lose if the rules reflected things that are more important in more general situations. In that case, you are a moral winner/loser, even though you are an actual loser/winner. The Witten usage of "morally" is analogous.

6. Dear Harlow,

the pressure is 90 times higher over there but the temperature gradients don't change much.

The lapse rate on Venus is still just 10 degrees per kilometer or so. This quantity is determined by the molar heat which is universal and independent of pressure.

But because the atmosphere is denser, it's also thicker - you must go higher before the pressure really drops and the atmosphere "ends". That's why Venus is "more internal", as you say, but the thickness/height of the atmosphere differs just by a factor of 5-10 or so from the Earth. The full thickness of the Venus atmosphere is usually said to be 250 km.

Leonard Weinstein: good points, but when we compare the two situations, we would also have to decide how the albedo is "kept" in a different context. I agree that if it were strictly constant, the altitude where the energy flows are easily matched would move in the absence of all greenhouse gases. Still, I think it's morally true - as Steve Goddard says - that the greenhouse effect is not the cause of the surprising warmth of Venus.

Best wishes
Lubos

7. Even if Venus absorbed all sunlight it couldn't be more than 55C. The only way it can be warmer than that is through it's atmosphere absorbing infrared.

That means the hundreds of degrees warmer that Venus is, is due to the greenhouse effect.

If you turned off infrared absorption in Venus's atmosphere, the planet would rapidly cool. The mere presence of high pressure isn't going to keep it warm. The easiest way to see why is to note that if atmospheric pressure could keep Venus warm, it would imply a body like Venus could just sit in space generating heat forever, like some inexhaustible mini-star.

So Steve's post is not just slightly wrong, the conclusion is completely wrong. The heat on Venus is caused by the greenhouse effect.

8. Dear Blob,

Neither Goddard nor me or anyone else has ever argued that the energy conservation is violated. So in his picture, Venus is surely not "producing" energy like a little star.

What is the equilibrium temperature inside is a completely different matter, and unless you want to deny that the lapse rate exists, the temperature does increase as you go deeper, regardless of any IR absorption.

Even the Earth is thousands of degrees warm inside - like a "little star" because according to Al Gore, it's millions of degrees hot. ;-)

It's getting hotter at depths even though there's no IR radiation going there. So it's surely possible for planets to preserve hundreds or thousands of degrees above the "equilibrium radiative temperature" inside them for billions of years.

If the opposite were your only argument, well, then your conclusion has clearly been justified by no valid argument.

Best wishes
Lubos

9. Thank you for the link Lumo...just two corrections needed: my name is Morabito, not Moratobi and the posts were from Feb 2008, not 2010 8-)

10. Did it ever occur to anyone how much heat is generated in the Venusian atmosphere by the combination of SO3 with H2O to form the sulfuric acid droplets of the clouds?

That is the exothermic portion of the formation of sulfuric acid clouds. The heat of the endothermic portion of the reaction (the dissociation and reduction of SO3 back to SO2) is provided by volcanism.

This is essentially how, the volcanic heat of Venus becomes a permanent part of the atmosphere

For those who don't believe me - go ahead and calculate that heat effect in the Venusian atmosphere yourself

11. This argument doesn't seem to address the question of why Venus's atmosphere should have the temperature profile it does. It is possible to have temperature lapse rates that are lower than adiabatic or or even negative (the Earth's stratosphere being an example) and with a sufficiently strong source of heat below it is even possible to have lapse rates higher than adiabatic (I think, but haven't checked, that this is the case in regions of the Sun where convection is the primary heat transport mechanism). The presence of gravity and consequent increase in potential energy with altitude in no way mandates a fall of temperature with altitude. You can have a perfectly stable isothermal atmosphere where only pressure decreases with altitude (see the toy atmosphere in Feynman lecture 40, volume 1). The adiabatic lapse rate is just the lapse rate at which convection can be maintained, so at lapse rates above it the atmosphere is much more efficient at transporting heat upwards. It so happens in the Earth's troposphere that the heating is from below (absorbtion of solar radiation by the ground/sea) and of sufficient intensity to maintain a lapse rate close to adiabatic. If the heating was not from below things would be different, if the heating was from below but much less intense we could see lapse rates well below adiabatic, if the heating was much more intense we could see lapse rates above adiabatic (accompanied by powerful convection cells as seen in the Sun, probably just as well for us that it's not like that). If most solar radiation on Venus reaches the surface before being absorbed (does it?) then maybe the situation there is similar. Even then I'm not sure how that explains a high surface temperature, wouldn't a lower surface temperature and low lapse rate just imply that the atmosphere was pretty hot all the way up, and why is that ruled out?

12. Hello Lubos,
You say that the temperature increase of the surface on the Venus is more due to presence of the thick (heavy) atmosphere itself, not due to greenhouse effect.
So let's consider the example of replacing 90% of CO2 by N2 in the Venus's atmosphere. To be more exact, assume the same surface pressure i.e. the same total mass of the atmosphere by assuming the same gravity of the planet. Only the composition would be 90%N2 and about 10% CO2 with all other minor compounds in the same amount as they are on the Venus now. With these assumptions we could assume the same or at least similar clouds layer and the same abedo too.
So what is your estimate of surface temperature change due to such atmosphere's composition change?
My rough estimate is more than 100 degrees of Celsius, but from your text I understood that you estimate the change to be in order of 10 degrees. Did I understand you correctly?