New: Lapse Rate by Gravitation: Loschmidt or Boltzmann/Maxwell?

Written by Professor Claes Johnson

by Professor Claes Johnson

Will an atmosphere under the action of gravity assume a linear temperature profile with slope equal to the dry adiabatic lapse rate? Loschmidt said yes, while Boltzmann and Maxwell claimed that the atmosphere would be isothermal. Graeff (2007) has made experiments supporting Loschmidt and so it is natural to seek a theoretical explanation. 

Claes Johnson
Consider a horizontal tube filled with still air at uniform temperature. Let the tube be turned into an upright position. Alternatively, we may consider a vertical tube with gravitation gradually being turned on from zero, or a horizontal tube being rotated horizontally starting from rest. During increasing gravitational force the air will be compressed and knowing that compression of air causes heating, we expect to see a temperature increase. How big will it be? Well, the 1st Law of Thermodynamics states that under adiabatic transformation (no external heat source):

  • c_vdT + pdV =0   

where c_v is heat capacity under constant volume, dT is change of temperature T, p is pressure and dV is change of volume V. Recalling the differentiated form of the gas law pV = RT with R a gas constant

  • pdV + Vdp = RdT
and the equilibrium equation in still air with x a vertical coordinate 
  • dp = -g rho dx or Vdp = – gdx
where g is the gravity constant, rho = 1/V is density, we find
  • (c_v + R) dT = -gdx or c_p dT/dx = – g, 

where c_p = c_v + R is heat capacity under constant pressure.

We thus find that stationary still air solutions must have a dry adiabatic lapse rate dT/dx= – g/c_p = – g with c_p = 1 for air, as a consequence of compression by gravitation, using

  1. work by compression stored as heat energy
  2. pressure balancing gravity (hydrostatic balance).
We thus find a family of stationary still air solutions of the form (assuming x = 0 corresponds to the bottom of the tube):
  • p(x) ~ (1 – gx)^(a+1)
  • rho(x) ~ (1 – gx)^a
  • T(x) ~ (1 – gx)
with a >0 a constant. In the absence of heat conduction such solutions may remain as stationary still air solutions. We thus find support of Loschmidt’s conjecture of still air solutions with the dry adiabatic lapse rate, in the absence of heat conduction. In the presence of (small) heat conduction, it appears that a (small) external source will be needed to maintain the lapse rate. Of course, in planetary atmospheres external heat forcing from insolation is present.

Returning the tube to a horizontal position would in the present set up without turbulent dissipation, restore the isothermal case. Turning the tube upside down from the vertical position would then establish a reverse lapse rate passing through the horizontal isothermal case. 

Further, it seems that without heat source, the isothermal case of Boltzmann/Maxwell will take over under the action of heat conduction, with p(x) ~ exp( – cx) and rho(x) ~ exp ( – cx) with c>0 a constant.  
 

For the Euler/Navier-Stokes equations for a compressible gas subject to gravitation, see the Computational Thermodynamics and the chapter Climate Thermodynamics in Slaying the Sky Dragon.

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Comments (10)

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    Dr A Hamilton

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    Claes

    The equations for thermodynamic potentials (such as entropy) are very specifically based on the assumption that molecular gravitational potential energy does not change. So they apply only in a horizontal plane. Of course you can “prove” isothermal conditions in a troposphere if you use such equations that ignore the contribution to entropy made by gravitational potential energy. That’s just a circular argument using a false assumption to get a result on which that false assumption is based.

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    Dr A Hamilton

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    Climate has nothing to do with carbon dioxide, because the whole concept that planetary surface temperatures are determined primarily by direct solar radiation is totally incorrect. The planet Uranus, for example, has no surface at the base of its nominal troposphere where the temperatures is about 47°C.

    A study of this nominal troposphere of Uranus confirms that the temperature gradient is about 95% of the -g/Cp value mentioned below. All tropospheres have slightly less steep gradients because of the temperature-leveling effect of inter-molecular radiation between IR-active molecules. The temperature gradient (aka “lapse rate”) is the state of thermodynamic equilibrium which the Second Law of Thermodynamics says will evolve autonomously as entropy approaches the maximum. At that maximum there can be no unbalanced energy potentials, and so, other forms of energy being equal, there must be a homogeneous sum of molecular gravitational potential energy and kinetic energy. Because PE varies with altitude, and because temperature is proportional to mean molecular KE only, there must be a temperature gradient, which we can calculate to be -g/Cp where Cp is the weighted mean specific heat of the gases.

    Now, because the temperature gradient represents the state of thermodynamic equilibrium, any new thermal energy absorbed in the upper atmosphere from insolation will disturb the equilibrium and lead to a new state of thermodynamic equilibrium evolving with the same temperature gradient, but a higher overall temperature. This means some thermal energy moves downwards to warmer regions, not by radiation, but by convective heat transfer which, in physics, includes transfers of KE in molecular collisions and diffusion.

    All this is explained in more detail in our group’s website and you can read about Uranus on the ‘Evidence’ page therein.

    What happens in the real universe is a complete paradigm shift from what climatologists think about planets cooling off and having surfaces warmed only by solar radiation. All planetary temperatures are determined from the “anchor point” in their atmosphere right down to their cores, and all temperatures en route are supported by downward diffusion and convective heat transfer from the anchor point, that being where there is radiative equilibrium with the Sun. If the Sun’s radiation somehow ceased, then all planets and satellite moons would cool right down, even their cores wherein any energy generation is in reality nowhere near sufficient to maintain existing temperatures, and not necessary anyway. That’s what the Second Law of Thermodynamics gives us reason to say must be the case.

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    Physicist

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    The Ranque Hilsch vortex tube is the best “experimental test” of the fact that molecular kinetic energy is redistributed in a force field, forming a temperature gradient.
     
    Because the temperature gradient in a planet’s troposphere is the state of thermodynamic equilibrium which the Second Law of Thermodynamics says will evolve, the planet’s supported surface temperature is autonomously warmer than its mean radiating temperature, so warm in fact on Earth that we need radiating gases (mostly water vapour) to reduce the gradient and thus cool the surface from a mean of about 300K to about 288K, this being confirmed by empirical evidence (as in the study in my book) which confirms with statistical significance that water vapour cools rather than warms, all these facts thus debunking the greenhouse conjecture.

     

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    Physicist

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    Jeff Conlon on The Air Vent asked: “So you are stating that your idea is not new and are of the belief that Loshmidt had the right answer correct?”

    To this I replied:

    The fact that the temperature gradient forms autonomously at the molecular level (without any specific need for upward convection) was first explained in the 19th century, and has never been correctly rebuked. But this “gravito-thermal effect” has been overlooked by James Hansen et al. Hence 255K is not the right “starting point” and there is not “33 degrees of warming” but more like 10 to 12 degrees of cooling by radiating molecules, mostly water vapour of course, because the radiating properties of these molecules have a temperature levelling effect working against the gravitationally-induced temperature gradient that is not due to any lapsing process..

    But what has not been explained prior to the 21st century is how the necessary energy transfers over the sloping thermal profile just like new rain water falling on a small section of a lake spreads out evenly over the whole lake. This is what happens (and must happen) in planetary tropospheres, and it happens because the Second Law of Thermodynamics is all about thermodynamic equilibrium evolving. Thermodynamic equilibrium has a density gradient (because there must be more kinetic energy per molecule at lower altitudes) and that density gradient thus has a temperature gradient.

    Thermodynamic equilibrium is what it says – an equilibrium state just as much as is mechanical equilibrium which keeps the surface of a lake more-or-less following the curvature of the Earth. Gravity spreads new rain water over the lake, raising the level all around the lake. Likewise, new thermal energy absorbed in a planet’s upper troposphere or elsewhere (such as in and above clouds) spreads out in all accessible directions by convection, where I use the term to mean both diffusion and advection in accord with normal usage in physics. And that’s how the required energy gets down to the base of the Uranus troposphere to maintain temperatures hotter than Earth’s surface. Likewise on Venus and likewise on Earth because solar radiation directly to the surface is nowhere near sufficient and we would freeze on cloudy days if this downward convection were not a reality.

     

    Moderator:

    This comment is being posted on about six other blogs as I don’t like wasting my time on just one blog, unless a blog owner runs an article on the content of my book and agrees not to delete any of my comments replying to comments on that thread. I may do likewise with any future such questions and answers in the interests of disseminating correct physics and gradually wearing down the greatest error ever made since the flat Earth garbage.

     

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    D o u g   C o t t o n  

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    Convection in gases can be diffusion or advection in physics. As diffusion accelerates there comes a point at which actual (very slow) bulk movement can be detected. If there was initially a state of thermodynamic equilibrium (with its gravitationally induced temperature gradient reduced somewhat by inter-molecular radiation) then convection always transfers thermal energy in all accessible directions away from the source of newly absorbed thermal energy which disturbed the thermodynamic equilibrium. What drives the measureable bulk movement is always a source of newly absorbed thermal energy. Remember that the speed of the advection is orders of magnitude less than that of molecules in free path motion which are busily repairing the temperature gradient. All this is so obvious in all planetary tropospheres, but perhaps most convincingly in the troposphere of Uranus.

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    D o u g   C o t t o n  

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    Air molecules move between collisions at a speed of about 500 metres per second. This is orders of magnitude faster than any bulk air movement.

    That should give you an idea of the rate at which gravitational potential energy changes for any particular molecule, because Newton’s laws can be applied for all practical purposes.

    At thermodynamic equilibrium the mean sum ….
    (KE + Gravitational PE) = constant at all altitudes.

    When a molecule with mean KE for its initial altitude rises (passing on average about 30 molecules in its mean free path) its kinetic energy is partly converted to extra gravitational potential energy and so, when it next collides its new KE is the same as the mean KE at that level, and so it has no warming or cooling effect on the molecule it strikes. Hence there is a temperature gradient, because the mean KE per molecule is proportional to the absolute (K) temperature.

    So the “lapse rate” is not caused by any slow convection rising as fast as it can from a heated surface. It’s just there because the Second Law of Thermodynamics says thermodynamic equilibrium is the trend.

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     D J C 

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    So I’m sorry, Claes, but you’re wrong here and in your article on your own blog because the gravito-thermal effect does not require a heat source to trigger it. Furthermore, without understanding that the Second Law dictates that gravity forms both a density gradient and a temperature gradient, you have no explanation or understanding that convection can transfer thermal energy downwards in planetary tropospheres so that thermodynamic equilibrium is restored. Unless it does so, energy does not balance in the tropospheres of Venus and Uranus, for example. So do the right thing and withdraw or modify your articles and explain your error especially to John O’Sullivan so that PSI van get on the right track as in my book.

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     D J C 

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    No, the isothermal state cannot exist as an equilibrium state because it is not the state of maximum entropy. A new heat source disturbs the equilibrium and so thermal energy from that new source disperses in all accessible directions (including downwards) until a new state of thermodynamic equilibrium (with its associated temperature gradient) is established at a higher overall temperature but the same temperature gradient.

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     D J C 

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    The Ranque-Hilsch vortex tube confirms the existence of the temperature gradient in a force field.

    The Second Law of Thermodynamics says thermodynamic equilibrium will evolve. Such equilibrium has no unbalanced energy potentials. Hence (kinetic energy + gravitational potential energy) = constant. Hence there is a temperature gradient.

    If you disagree there’s a $5,000 reward to prove the content of my book “Why It’s Not Carbon Dioxide After All” to be substantially incorrect.

    You cannot explain observed temperatures on Venus, Uranus or Earth without reference to the gravito-thermal effect.

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    Vu Ngoc Han

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    Loschmidt, Boltzmann and Maxwell are all wrong

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