More thoughts on the Lapse Rate

The adiabatic lapse for a planet depends only on its gravity(g) and the type of molecules (diatomic, triatomic + mass) in the atmosphere (cp). Is it that simple ?

I initially thought that a perfectly still non-radiating atmosphere in a gravitational field sitting on a uniform “hot plate” surface at a fixed temperature would produce a dry adiabatic lapse rate for ever. After several discussions across several blogs, I now think that after millions of years heat conduction from the surface would eventually equalise temperatures, assuming zero energy loss to space. A completely still atmosphere is in practice impossible due to any small differential day/night, seasonal/latitudinal solar heating variations. Any movement in the atmosphere will result in air mixing and generate a lapse rate which then dominates conduction. The dry adiabatic lapse rate is exactly that rate at which gravitational potential energy gain/loss is equal to internal kinetic energy loss/gain. In practice isothermal conditions never occur and all planets have a lapse rate. Asking whether the lapse rate causes convection is like asking which came first the chicken or the egg. The equilibrium (dry) lapse rate exists independent of convection.

Nor does the energy source need to originate from the surface to generate convection. On Venus 90% of solar radiation is absorbed in the thick clouds some 50-60 km above the surface. Only a tiny 17 watts/m2 actually reaches the surface. I think it is high atmospheric winds ( like Hadley cells) that circulate and mix the atmospherethat drive convection. This in my opinion is the primary cause of the 700K surface temperatures – not a “run-away” greenhouse effect as such.

Gravity is essential for the lapse rate because it compresses the atmosphere resulting in a hydrostatic pressure gradient. In bulk thermodynamic terms – air that rises up against gravity looses energy by doing work and air that falls gains energy by having work done on it. One interesting observation is that gravity appears at first sight to reduce  entropy by clumping most of the atmosphere close to the surface.

In the 19th century it was originally thought that the atmospheric temperature would be uniform with height, but then it was observed that in high mountains temperatures dropped (see ref[1]). Carnot  proposed that air high up in the atmosphere was colder because it absorbed less IR from the surface, losing energy by radiation to space. Laplace was the first greenhouse gas sceptic dismissing this argument and instead pointed to adiabatic rising and falling of of air over mountains.

Lord Kelvin in 1904 had doubts about what he called the Boltzmann-Maxwell doctrine and especially its application to the isolated atmosphere. He stated “The time integral of the kinetic energy of any atom will be equal to the time integral of the kinetic energy of any other atom. This truism is simply and solely all that the Boltzmann-Maxwell doctrine asserts for a vertical column of a homogeneous monatomic gas”. He argued that the atmosphere cannot be at rest – radiation loss in upper atmosphere causes air to sink and compensate for rising warmer air. He says “an ideal atmosphere, perfectly isolated from absorption as well as emission of radiation, will, after enough time has passed, reach a state of uniform temperature, irrespective of the presence of the gravitational field”.

Ritter (1878-1883) and Emden (1907) accepted the concept of a convective equilibrium in order  to avoid contradiction with the firmly established belief in the isothermal equilibrium of isolated systems. There was no quantitative theoretical treatment. Loschsmidt (1876) believed that an isolated atmosphere, at equilibrium in a gravitational field has a tempertaure gradient given by -g/Cv. His arguement is similar to the derivation I made earlier for the lapse rate.

Basically Lochsmidt’s argument was that the gain in potential energy mgDZ at height Z+DZ is at the expense of kinetic energy (Temperature). We then get DT/DH = -g/Cv (if we assume no adiabatic expansion). Maxwell and Boltzman argued instead  that a thermally isolated atmosphere would obtain a single temperature throughout its height, based on the second law of thermodynamics i.e. no heat can flow from a cold body to a warm body without external work. Another formulation of the  second law also states that  in any thermodynamic process entropy must increase or stay the same. However gravity seemingly always acts to decrease entropy. For example it is gravity that bunches up the atmosphere near the surface which is a  state very low relative entropy. Likewise the sun and the earth both formed under the gravitational collapse of a diffuse dust cloud 5 billion years. In fact the sun’s energy is the result of gravitational collapse of hydrogen gas. So what is going on – could gravity violate the 2nd law of thermodynamics? Could Loschmidt have been right ?

The clumping of molecules near the Earth’s surface means we know more precisely their position. However if as a result of this they are heated then we know less about their velocities. Considering just the molecules it does appear that gravity decreases entropy. However as far as total entropy goes we need to include the whole universe for the second law and we  have forgotten about radiation losses which must lead to a net overall increase in entropy. This is why the sun and even  black holes eventually evaporate and die.

For any realistic planetary atmosphere there must always be some radiation losses to space. Greenhouse gases are one obvious mechanism but so too are dust particles, and even diatomic gases can also radiate. In the extreme case of a gravitationally contracting interstellar hydrogen gas cloud, with no IR radiation loses  temperatures at the core rises sufficiently to eventually trigger thermonuclear fusion and a star is born. Energy balance is restored  and the entropy increase from radiation exceeds the entropy decrease from gravity. The second law is saved.

1. For a discussion on gravity and entropy see : John Baez :
2. Christian Fronsdal, Univ. Calif, “Heat and Gravitation. I. The Action Principle”,

About Clive Best

PhD High Energy Physics Worked at CERN, Rutherford Lab, JET, JRC, OSVision
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11 Responses to More thoughts on the Lapse Rate

  1. Your interactions with others (supposed “experts” — like those who aggressively promulgate the climate consensus theory?) has damaged your thinking. If you won’t trust the utter simplicity of the lapse rate derivation you first believed, then consider at least this little fact: Even if an atmosphere WERE non-radiating (and that’s something no expert has ever demonstrated), the planetary suface IS, so energy would be lost that way, and your (the “expert’s”, who are all simply deluded in their incompetent thinking) imagining of “zero energy loss” is just — silly. My Venus/Earth comparison, which I already informed you of, is the DEFINITIVE EVIDENCE, not the theoretical pretensions of the many self-styled “experts”, who formulated an incompetent consensus and now can only avoid facing the truth to save their egos and their careers.

    And of course, since the atmospheres of both Venus and Earth, over the range of Earth tropospheric temperatures, are warmed by direct absorption of incident solar radiation (just part of the insight from my Venus/Earth comparison), convection is not needed to maintain the lapse rate. BELOW the 1,000 mb pressure level, convection in the lower 50km of Venus’s atmosphere may well be the dominant heat transfer mechanism, as I have myself tentatively considered. But above that level, it is direct absorption of incident solar radiation.

  2. Clive Best says:

    Your analysis shows nicely the similarity in lapse rates between Earth and Venus. The fact that Venus’s atmosphere is 90 times more dense than the Earth is also the crucial factor in it reaching higher temperatures. The incident solar energy is mostly absorbed in the upper atmosphere and heat is pumped down to the surface through convective winds.

    I think the crucial question for the lapse rate is this. Does the lapse rate work at the molecular level or is it a macro thermodynamic effect ? I think this can be resolved by experiment. I hope to write up how this can be done in a few days.

  3. The laps rate is a fascinating thing. We can’t set it up in a laboratory, and yet there it is all around us. I agree that the non-radiating atmosphere will settle down to a single temperature. If you consider that the conductivity of air is 0.024 W/m-K, we see that a 1-km column of air with 10 C across it will conduct 240 uW/m2, which will cool 1 kg of air (heat capacity roughly 1 kJ/kgK) by 10 C in roughly one year. So the thermal equilibrium of the non-radiating atmosphere would be reached in a few years. In answer to people who think that all columns of air automatically develop a laps rate, whether they radiate out of the top or not, their claim violates the Second Law of Thermodynamics, as you can see here.

  4. Richard111 says:

    Baffled layman question again. Kevan says above “I agree that the non-radiating atmosphere will settle down to a single temperature.” and I see that as all molecules in the atmosphere tending towards the same kinetic speed. But gravity will make a density gradient. A basic thermometer will still show a higher temperature in the lower atmosphere?

  5. Clive Best says:

    Temperature is proportional to the average kinetic energy of all gas molecules at a certain height. There will indeed be far more molecules per unit volume near the surface than say 10km above it. It is true that there is far moreTOTAL kinetic energy in the gas at the surface BUT the average kinetic energy of a single molecule can still be the same 10km up. So you can still have the same temperature at different heights.

  6. Richard111 says:

    This is where I fall down. I’m under the impression that the NUMBER of molecular collisions per unit time is what the thermometer is recording. Ah, well. I’ll keep tagging along and maybe some learning might be absorbed over time.

  7. D o u g  C o t t o n  says:

    If a molecule has an upward component in its free path movement between collisions then some of the translational kinetic energy in that molecule (M.Cp.dT) supplies the additional gravitational potential energy (M.g.dH) that it acquires by virtue of its additional altitude. Vice versa for downward motion. Equate the two and you have the temperature gradient dT/dH = g/Cp which should not be hard to understand.

    Because the laws of physics can be used to explain this gravitationally induced temperature gradient, the fact that the surface temperature of a planet is higher than the radiating temperature of the planet is fully explained (and confirmed empirically) by this autonomous temperature gradient.

    There is thus no need for any other explanation as is supposedly presented in the false radiative greenhouse conjecture.

  8. daveburton says:

    Clive Best wrote, “The dry adiabatic lapse rate is exactly that rate at which gravitational potential energy gain/loss is equal to internal kinetic energy loss/gain.”

    and Doug Cotton wrote something which sounds similar, at first glance: “If a molecule has an upward component in its free path movement between collisions then some of the translational kinetic energy in that molecule (M.Cp.dT) supplies the additional gravitational potential energy (M.g.dH) that it acquires by virtue of its additional altitude. Vice versa for downward motion. Equate the two and you have the temperature gradient “

    However, when I did the arithmetic, Doug’s way, to calculate what the lapse rate would be, it didn’t work. It calculates a lapse rate which is way, way too high:

    Upon reflection, I believe that the word “internal” in Clive’s version is the key to understanding at least part of what’s wrong with Doug’s version.

    It is true that when an air molecule rises it slows, due to the effect of gravity, converting kinetic energy to potential energy. But the motion of the whole molecule is just one part of the molecule’s kinetic energy. Diatomic and triatomic air molecules also have internal rotational and vibrational (and for triatomic molecules, bending mode) kinetic energy. Since air molecules are continually colliding with other air molecules and exchanging energy with them (colliding something like 20x per nanosecond at 1 atm, if I recall correctly), the various types of kinetic energy get very quickly exchanged with one another, and averaged out.

    So reducing one mode of kinetic energy (the z-axis component of the velocity of the whole molecule) by converting some of the z-axis component from kinetic energy to potential energy does not reduce temperature (averaged kinetic energy) nearly as much as I (and Doug) first guessed. Basically, since the vertical motion only immediately affects one of the seven degrees of freedom of a diatomic molecule, you only lose 1/7 as much kinetic energy per degree of freedom.

    Plus, in the real atmosphere there are ways in which energy gets moved around; e.g., via the water cycle when air is moist (which reduces the lapse rate), and via radiative transfer due to the radiatively active gases in the atmosphere (which I believe also reduces the lapse rate). Unfortunately, I don’t know how to calculate those.

  9. Thanks. It took me years to realise this. I wish I’d read this post when it came out.

  10. ferdberple says:

    Years ago I built a simple model of ideal atoms in a gravitational field. What I found was that since there were more atoms on the “down” side than the “up” side, the faster moving atoms were preferentially sorted upwards.

    This sorting process exactly matched the energy loss-gain due to gravity and PE-KE conversion, such that the individual atoms at altitude had the same temperature (KE) as the (denser) individual atoms at low altitude.

    The sample at altitude had less KE per cubic meter than the lower sample, but the individual atoms in each sample had the same average KE.

    This result quite surprised me, but I had a colleague confirm it independently. He wrote his model in Matlab, and I wrote mine in Java.

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