# Understanding the Lapse Rate

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.

Fig 1: Thought experiment.

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

$N(v,z) = N_{0} e^{-\frac{(mv^{2}+mgz)}{kT}}$

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.

$\frac{mv_{rms}^{2}}{2} = \frac{3}{2}kT$

For M-B gas the rms velocity $v_{rms} = \sqrt{\frac{3kT}{m}}$

So $T = \frac{mv_{rms}^{2}}{3k}$

Consider now increasing the height level by height $\Delta{h}$. 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 $P(z) = P(z_{0}) e^{\frac{-mg(z-z_{0})}{RT}}$   so  $dP = -\frac{mg}{RT}Pdz$

$mv'^{2} + dW = mv^{2} - 2mg\Delta{h}$

$T' = \frac{m(v_{rms}^{2}-2g\Delta{h})}{3k} -\frac{mg\Delta{h}}{3k}$

$\Delta{T} = T'-T = \frac{-2mg\Delta{h}}{3k} -\frac{mg\Delta{h}}{3k}$

$\frac{\Delta{T}}{\Delta{h}} = \frac{-g}{C_{p}}$, where  $C_{p}=\frac{5R}{2}$

This is  the usual formula from thermodynamics based on the (constant temperature) hydrostatic equation and rising parcels of adiabatic air – $\Gamma = \frac{-g}{C_{p}}$.  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.

References

1. Wikipedia : Maxwell Boltzmann
2. Kinetic Theory of Gases.

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### 8 Responses to Understanding the Lapse Rate

1. I heartily agree that it should be clear (but to most it is not) that the lapse rate is basically just -g/c, with c the effective specific heat, and gravity alone is responsible, not “adiabatic” convection. In fact, I emphasize, and have strongly (and so far as I know, originally and uniquely) suggested in my own writing, that the lapse rate should be known as the “hydrostatic lapse rate”, not the “adiabatic lapse rate”, because it follows basically from the hydrostatic condition.

But you might just as well say the lapse rate is maintained by the mass mean tropospheric temperature, instead of the surface temperature. I say this, because in my comparison of temperatures in the atmospheres of Venus and Earth, the Venus/Earth temperature ratio is obviously just that due to the ratio of the two planets’ distances from the Sun, nothing else (and that means, there is No greenhouse effect, of increasing temperature with increasing carbon dioxide, at all). The explanation for this empirical and logical fact is that the two tropospheres must both be warmed by direct absorption of incident solar IR radiation–in fact, they both must directly absorb the same physical portion of that incident radiation (otherwise, the great difference in planetary albedo and physical surface–in addition to the great difference in atmospheric carbon dioxide concentration–would not allow their temperature ratio to reflect only their distances from the Sun). Consensus theorists, if they would dispute this conclusion about the fundamental warming of the troposphere, must explain, within their theory, why the Venus/Earth temperature ratio does in fact depend only upon the ratio of their solar distances, according to the most basic physics understanding–and, to avoid misunderstanding, see my linked article, for the simple Stefan-Boltzmann equations governing the mass mean tropospheric temperatures in the two planets, if they absorb, and are warmed only by, the same fraction f of the incident solar.

From my Venus/Earth findings, it seems clear to me that the surface cannot warm the atmosphere, globally (only locally and transiently), and indeed, at night, it can and does COOL the near-surface atmosphere, so that many locations see a local temperature inversion around dawn.

• The lapse rate does not follow from the hydrostatic balance. If it did, we could create a heat differential out of nothing with the help of a column of air (see here for diagram) and that would violate the second law of thermodynamics.

2. Trick says:

Kevan (and Clive) – I’ve been interested enough in this subject to look up the texts and papers and dig in to the thermo theory. Theory shows nothing unphysical happens, there is a lapse rate in the linked closed column and no perpetual motion.

In Kevan’s diagram link on his site, 1st snip Q, q1, and q2 connections across the adiabatic container. At LTE, approximate theory shows a lapse rate –g/Cp and exact theory shows the lapse rate is T(z)/To = (P(z)/Po)^R/Cp. The various threads at pingback below show the text book and paper references to find the theory, you can dig into them too.

Now connect Kevan’s Q, q1, q2 and yes you can get work out but not forever. Kevan’s machine has been built for deep water & it works so it is not perpetual motion (look up OTEC on wiki). The energy source is the sun, machine runs as long as sun is “on”.

3. Nabil Swedan says:

The derivation assumes that the air parcel expands adiabatically on its way up. This is not the case. The air parcel exchanges solar radiation and sensible heat. You still need to work out the details in order to get the exact and comprehensive formula.

4. nuwurld says:

Clive, I believe your reasoning is entirely valid and agree with Harry DH. However, convection, as defined, is the sum of advective energy transport and diffusion. This takes convection, as a concept, down to a molecular level.
The principle being the pure physical logic wrt this argument that, at equilibrium, every higher state has greater potential energy, therefore every lower state has greater kinetic (thermal for a gas, as you know), such that at each level every state evolves through isentropy to equal total energy. That ‘state’ can be molecular or a convective cell the size of a planetary shell. Physics has no reason to make any concession. At the molecular level every mean free path is modified through gravity and energetically expressed through Cp as a change in temperature due to ‘any’ motion. There is ‘no’ shortcut, ‘no’ exception to this concept. Any vertical motion must relate to work being done or released. Every horizontal motion is part of a curve due to perpendicular acceleration. Gravity is the containment and the ‘raison d’être’.