I am unhappy with the derivation of the dry adiabatic lapse rate given in all textbooks because it seems to me a somewhat circular argument. Are we really to believe that the lapse rate is caused by adiabatic convection? A rising adiabatic parcel of air in hydrostatic equilibrium apparently produces the “correct formula” but it doesn’t really explain anything. I am sure that the lapse rate remains the same even with zero convection. It must be gravity alone that plays the crucial role in determining the lapse rate. Others have followed similar arguments to an extreme and even claimed that gravitational energy alone heats the bottom of the atmosphere. Such arguments have then also been pillored for violating the second law of thermodynamics. However despite all this there is something not immediately obvious about the Earth’s atmosphere. In particular it is never in thermal equilibrium although it is in overall energy balance. The gound is directly heated by solar radiation and through conduction maintains the near surface layer of the atmosphere at a “fixed” average ground temperature of about 15 deg.C. Energy then flows from the surface upwards through the atmosphere. If we ignore greenhouse gases and radiative transfer for the moment, we can still ask how does that energy distribute itself through the atmosphere given that the temperature of outer space is 3K ?
I like to understand things simply at the molecular level. This post tries to derive the lapse rate only using kinetic theory.
Consider a thought experiment as shown in figure 1. Lets imagine all the atmosphere contained within a skin of thickness ~ one meter such that the atmosphere reaches thermal equilibrium of 15 deg.C. Lets also assume a completely flat surface and fixed surface temperature and therefore no convection. The skin is instantaneously removed so the question is how does the atmosphere evolve ? For simplicity we will take air to be a perfect gas and use maxwell-boltzmann distribution for velocities. Molecules rising up in the atmosphere will lose some kinetic energy to the gravitational field. The Boltzmann factor at height z gives a velocity distribution as follows
At the molecular level temperature is related to the average energy per degree of freedom = kT/2. For a “monatomic gas” with 3 translational degrees of freedom.
For M-B gas the rms velocity
Consider now increasing the height level by height . It is assumed that locally for each level thermal equilibrium is reached. Some work is done to fill the new volume DU +DW = DQ. The Barometric formula for pressure is so
This is the usual formula from thermodynamics based on the (constant temperature) hydrostatic equation and rising parcels of adiabatic air – . My first attempt at the derivation simply ignored the work term Pdv and I got -g/Cv for the lapse rate. I am still not fully convinced that the work term is correct, since it is a very long time since I studied statistical mechanics at university!
The conclusion is that you don’t need convection to have a lapse rate on a planet. It is a consequence of gravity and a fixed surface temperature. Molecules high up in the atmosphere have “lost” average kinetic energy (temperature) rising against gravity. Molecules from the surface provide a constant re-supply of energetic molecules from the tail of a fixed temperature Maxwell Boltzmann distribution.
Gravity clearly can heat gas (despite all the howls of violation of the second law of thermodynamics). This is how galaxies and stars form. Gravitational aggregation of hydrogen gas clouds converts gravitational energy into kinetic energy which at sufficient pressure and temperatures ignites nuclear fusion – a Star. On Earth however the energy source is the sun which (via greenhouse effects) keeps the surface at about 288K. This fixed surface temperature maintains the lapse rate, by providing a constant source of high energy molecules which can migrate to the upper atmosphere against gravity.