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.

References
1. For a discussion on gravity and entropy see : John Baez : http://math.ucr.edu/home/baez/entropy.html
2. Christian Fronsdal, Univ. Calif, “Heat and Gravitation. I. The Action Principle”, http://arxiv.org/PS_cache/arxiv/pdf/0812/0812.4990v3.pdf

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