Accurate formula for calculating environmental lapse rate on three worlds

Posted: September 2, 2015 by tallbloke in Analysis, data, solar system dynamics
Tags: , , ,

A commenter on another site with the handle ‘Agent009’ has come up with an interesting formula for calculating the environmental lapse rate on three solar system bodies with atmospheres. Talkshoppers might offer some ideas as to why it works. H/T to Stuart ‘Oldbrew’ for flagging this one up.

I’ve been trying to solve a puzzle… dry adiabatic lapse rate is normally calculated as following:
Γ = g·M/cp
where Γ is lapse rate, g is surface gravity acceleration, M is mole mass and cp is molar heat capacity.
However, if you calculate this for Earth, you arrive at 9.77 K/km, but actual environmental lapse rate, as defined in the ISA, is 6.49 K/km, which is about 9.77 * 0.665. So, I decided to take a look at how this works on Venus and Titan – the only two other worlds in the Sol System that actually have tropospheres.
On Venus (assuming tropopause at 55 km), the average lapse rate is about 7.9 K/km, but the above formula gives you 10.46 K/km, which means that you must multiply the result by 0.756 to get the actual value. On Titan (assuming tropopause at 42 km), actual average lapse rate appears to be around 0.5 K/km, but predicted lapse rate is 1.26 K/km – which gives you the coefficient 0.427. So I’ve been trying to figure what this mysterious coefficient depends upon – and, I think, I’ve found it. The following expression gives you almost exactly those numbers (using SI units, that is):

³√(12·g·M·(1/R – 1/cp))
where R is the ideal gas constant.

Note, that M·(1/R – 1/cp) is what you must divide the temperature value by to get the square of the speed of sound (I call inverse of this number the “speed of sound factor” because it is constant for any arbitrary gas mixture and gives you the best idea how any changes in atmospheric composition would affect the speed of sound) .
I’m not sure why the heck this works, but the following formula predicts the actual mean tropospheric lapse rates for all 3 worlds with astonishing degree of accuracy:

Γ = (g·M/cp)·³√(12·g·M·(1/R – 1/cp))

For Earth this gives you exactly the standard ISA lapse rate – down to 0.01 K/km precision! For Venus, it predicts the average measured lapse rate with nearly the same degree of accuracy! For Titan, there is somewhat less actual data available and the situation seems to be complicated by the fact that mole fraction of methane apparently varies significantly with altitude, but all measurements reported so far seem to correspond with the result predicted by the above formula at least by order of magnitude.
A coincidence? Well, the lapse rate predicted for Titan (based on 95% N₂ / 5% CH₄ atmospheric composition) would be 0.42 K/km. We’ll see if this actually turns out to hold true once more data is available.

Basically, I was just looking at ISA and Venus mean values and it suddenly struck me that this relation was very close to the relation between the “speed of sound factor”. I’ve played a bit with numbers and it turned that the relation between cube root of this number and lapse rate was nearly the same. Then I took a look at Titan data, but the relation did not seem to hold. So, I though to myself: “what might Earth and Venus have in common that could matter for lapse rate relation that is very different on Titan?”. The obvious answer was g, because otherwise, atmospheres of Earth and Venus are radically different. So, I tried to introduce g into the equation and, I was astounded, when it turned out that factoring the relation by ³√g not only made sense in terms of Titan data, but also got me closer to the relation between Earth and Venus values – the relation was proportional by nearly the same number. I was even more amazed when I realized that this factor was almost exactly the cube root of 12! And I was completely shocked when I approximated this factor to ³√12 and result for Earth turned out to be exactly the ISA lapse rate – down to 2 decimal points precision! The results for Venus still appear to be a nearly perfect match of the actually measured values and the results for Titan seam to work reasonably well considering the scarcity and imprecision of the data we have so far. Probably this “factor 12” has some significance I do not understand yet… but since Earth, Venus and Titan are the only worlds we are currently able to study that actually have tropospheres, I cannot do much better than this for now.

Actually, it’s more like I have 2.5 data points (since data available for Titan is still quite controversial and imprecise right now). Unfortunately, it does not seem that we gonna have any more data points in any foreseeable future, so I have to do the best with what what we’ve got.
Regarding the units… g is m/s², M is kg/mol and (1/R – 1/cp) is mol·K/J as both R and cp are J/mol·K. Since J = kg·m²/s² the g·M·(1/R – 1/cp) would be K/m. Thus, as the result of ³√(12·g·M·(1/R – 1/cp)) should to be dimensionless, this mysterious “factor 12” would need to be 12 m/K. This might be some relation I do not yet understand, but it could also be just another universal constant, like R or G, for all I know.
More measurements from Titan and some data from outside the Sol System would certainly help, but I cannot dare to hope for that anytime very soon, I fear.

Comments
  1. ivan says:

    Isn’t this the sort of thing the scientists at the Met office should be using their very expensive super computer to verify? I think they used the ‘it’s necessary for research’ statement when they asked for the money so why aren’t they doing any research?

    This would be a good start. Oh, sorry, I forgot, this isn’t ‘on message’.

  2. tchannon says:

    No mystery over the 12 or cube root.

    If someone can explicitly give me the equation *with* parenthesis setting precedence I’ll set the right form in Maxima which will spit out a good simplification

    \frac{4\,{g}^{2}\,{M}^{2}\,\left( \frac{1}{R}-\frac{1}{cp}\right) }{cp}

    or

    \frac{4\,{g}^{2}\,{M}^{2}\,\left( \frac{1}{R}-1\right) }{{cp}^{2}}

    Or something else.

  3. tallbloke says:

    Thanks Tim. How do we determine parenthesis setting precedence?
    I’m guessing that if I have to ask the question, I’m not going to be able to supply the answer though…

  4. tchannon says:

    Best to assume nothing. Some hard numbers would allow checking by calculation. Doesn’t work so far.

    L=(g*M/cp)*(12*g*M*(1/R-1/cp))^1/3;

    And shiftreturn

    Values can be set using eg. g:9.81;

    http://andrejv.github.io/wxmaxima/

  5. Larry Ledwick says:

    local Speed of sound in the gas mixture makes sense in that it would be the natural unforced speed that pressure could propagate through the gas. Suppose you change the combined net energy by changing either the gravitational potential energy of a parcel or its kinetic potential energy there would be a force needed to restore equilibrium and that force would propagate at the local speed of sound (which by the way is strongly determined by the temperature of the gas).

    That is my first guess about why that factor matters.

  6. Larry Ledwick says:

    If you take that idea just a bit farther, suppose you have a mass of air (parcel) some distance above the planet in equilibrium with the surrounding air both thermally and static with respect to the planet (and the surrounding atmosphere).

    Some forcing instantaneously raises its pressure or temperature slightly so now it has in sum more kinetic potential energy and gravitational potential energy than surrounding parcels. Let’s assume the change was small increase in pressure. Since it is now no longer in equilibrium with its surroundings it would immediately start expanding at the local speed of sound for its conditions to get back in pressure balance with surrounding gas parcels. The molecules moving upward would be losing kinetic potential energy but gaining gravitational potential energy, in the process they would be slowing down (average velocity of the molecules dropping (ie cooling)) Likewise the molecules moving downward would also be losing velocity due to the pressure gradient dropping as the parcel expands but they would also be gaining energy of motion due to conversion of gravitational potential energy to kinetic energy of motion downward as the moved deeper into the gravity well (ie warming slightly as they fell).

    If you have this occurring over an infinite number of very small parcels in the atmosphere, all the time due to slight variances in pressure and temperature or height, the net effect would be a continuous gradient in temperature and pressure from ground level to the top of the atmosphere.

    Since there would always be some slight imbalance due to absorption or emission of radiant energy, or Brownian motion if nothing else, this would be a continuous process which would cool the upper reaches of the atmosphere with respect to lower layers of the atmosphere (if no other forcings existed [ covering the exception of the stratosphere where UV warming is a new forcing], and the lapse rate defining that gradient of temperature/pressure with altitude would logically fit with the expression derived above.

  7. Larry Ledwick says: September 3, 2015 at 4:23 am

    “If you take that idea just a bit farther, suppose you have a mass of air (parcel) some distance above the planet in equilibrium with the surrounding air both thermally and static with respect to the planet (and the surrounding atmosphere).”

    “Some forcing instantaneously raises its pressure or temperature slightly so now it has in sum more kinetic potential energy and gravitational potential energy than surrounding parcels.”

    Can you explain how an atmospheric mass can express (have) “gravitational potential energy”? Any atmospheric mass is free to accelerate in any direction with any force. This same atmospheric mass does not accelerate downward in this gravity!

    “Let’s assume the change was small increase in pressure. Since it is now no longer in equilibrium with its surroundings it would immediately start expanding at the local speed of sound for its conditions to get back in pressure balance with surrounding gas parcels. The molecules moving upward would be losing kinetic potential energy but gaining gravitational potential energy, in the process they would be slowing down (average velocity of the molecules dropping (ie cooling))”

    Why do you claim the ‘average velocity of the molecules” is dropping with free expansion? Is not the increase in mean free path sufficient to lower the rate of change in momentum KE thus resulting in a lower temperature, with no change in average or rms velocity? The gas laws always express such velocity, for a single species contained within a volume. With a different velocity based on molecular mass. An atmosphere is not contained nor a single species Why should the gas laws even apply?
    If the atmosphere were H2 and N2 would each gas have a different velocity at constant pressure density and temperature?

  8. tchannon says: September 3, 2015 at 1:06 am

    “Best to assume nothing. Some hard numbers would allow checking by calculation. Doesn’t work so far.”

    L=(g*M/cp)*(12*g*M*(1/R-1/cp))^1/3; And shiftreturn Values can be set using eg. g:9.81;

    Tim I think this has to do with a shift from 3D to fourspace, so that KE, do to velocity reversals, impacts, intercepts, rather than linear acceleration can be obtained. This atmosphere is not a 3D volume It seems like free space, with pressure, density, and temperature gradients with altitude offsetting all other potentials an isopotential atmosphere as pe Drs. Maxwell and Boltzmann, countering Dr. Loschmidt on available atmospheric work

    Have you tried:
    (𝛿T/𝛿z)³· = Γ³ = (g·M/cp)² · (g·M/cp)² · 12 · (cp/R – 1)
    (𝛿T/𝛿z)³· = Γ³ = (g·M/cp)² · (g·M/cp)² · (12 · (cp – R)/R)
    To blow your symbolic algebra mind, still with no understanding?

  9. Molar Specific Heat-Cp-Cv = R. by one Kelvin at constant pressure is called molar specific heat at constant pressure”. heat supplied at constant pressure is consumed in two purposes: (1) To raise the temperature of gas.

  10. Larry Ledwick says:

    You have taken part of the comment out of context, the upward moving molecules would lose energy fighting gravity and trade velocity for gravitational potential energy as they went up, like a base ball thrown in the air. Therefore the upward moving molecules would cool (slow down converting kinetic energy of motion to gravitational potential energy as they gain height and lose upward velocity) ever so slightly as they moved upward. The exact opposite would happen to the downward moving molecules, they would slightly warm as they moved downward due to the acceleration of gravity, trading gravitational potential energy for increased velocity and kinetic energy. Conservation of total energy demands that they slow down as they rise in the gravity field and speed up as they descend in the gravity field.

    Once the parcel finished expanding so that it had again reached pressure equilibrium with the surroundings the bottom of the parcel volume would be ever so slightly warmer than the top of the parcel volume, even though their pressures were equal there would be a slight temperature gradient vertically. That expansion and equalization process would occur at the local speed of sound for the temperature/pressure of the gas.

    As you note, the whole parcel would cool over all per the ideal gas laws but it would not be uniform cooling, the bottom would cool infinitesimally less than the upper boundary of the volume. The “average temperature” of the whole parcel would follow the gas laws exactly but to maintain the lapse rate there must be a very slight difference in the cooling top to bottom.

    If the dry adiabatic lapse rate is 5.5C°/km, then a one meter cube parcel of air would see 1/1000 of that top to bottom or 0.0055 C° which is basically un-measurable under normal circumstances and gets totally lost in turbulence and measurement errors.

    The complication is that molecules or atoms are not baseballs and do no travel very far without exchanging energy through millions of collisions with other atoms and molecules, so you have to integrate the gain or loss of potential energy over a population of atoms/molecules.

    In theory, if you had two isolated molecules in a perfect vacuum at some point above a planet one heading upward and the other heading downward at the same initial velocity they would behave exactly like two baseballs in the same circumstances. The upward moving molecule would eventually reach its maximum altitude in the gravity field (assuming its initial velocity was below escape velocity) and then it would fall back toward the planet. meanwhile the downward moving molecule would retain its initial velocity and accelerate at the local gravitational acceleration (ie getting hotter “gaining kinetic energy”) until it ran into the planets surface. Both of them would arrive back at the surface at the same terminal velocity determined only by the local gravitational acceleration and their initial altitude.

    Any other result would violate the conservation of energy law. If a particle loses gravitational potential energy as it falls it “has” to be converted to some other form of energy, either kinetic energy of motion or the particle would have to radiate some of the energy away in some other form of energy or convert it to some other form of potential energy (ie spin).

    What works for a brick falling off a building will work for an atom or molecule.
    We know that many molecules radiate according to their temperature in the visible, IR and even microwave frequencies so the final equalization would be so complex it could not be detailed by any real measurement or calculation only by averaging the results over a reasonable volume of gas. Temperature is really a way of measuring average kinetic energy of a population of particles, some going much faster than the average some going much slower than the average, but net the final result is as if all of them were moving at the average temperature velocity.

  11. oldbrew says:

    Look up speed of sound on Wikipedia.

  12. Larry Ledwick says: September 3, 2015 at 7:08 am

    Will Janoschka says: September 3, 2015 at 6:08 am

    (“Can you explain how an atmospheric mass can express (have) “gravitational potential energy”? Any atmospheric mass is free to accelerate in any direction with any force. This same atmospheric mass does not accelerate downward in this gravity!”)

    “You have taken part of the comment out of context, the upward moving molecules would lose energy fighting gravity and trade velocity for gravitational potential energy as they went up, like a base ball thrown in the air. Therefore the upward moving molecules would cool…. What works for a brick falling off a building will work for an atom or molecule.”

    Really now! Have you any concept of an Atmosphere?
    At STP Kick 28 gm of atmosphere,1 mole , 22.4 litre from the top of a building Just watch that puppy accelerate downward at 9.8m/s². Why not? Atmospheric molecules are and act just like a brick!

  13. Larry Ledwick says: September 3, 2015 at 7:08 am

    “In theory, if you had two isolated molecules in a perfect vacuum at some point above a planet one heading upward and the other heading downward at the same initial velocity they would behave exactly like two baseballs in the same circumstances. The upward moving molecule would eventually reach its maximum altitude in the gravity field (assuming its initial velocity was below escape velocity) and then it would fall back toward the planet. meanwhile the downward moving molecule would retain its initial velocity and accelerate at the local gravitational acceleration (ie getting hotter “gaining kinetic energy”) until it ran into the planets surface. Both of them would arrive back at the surface at the same terminal velocity determined only by the local gravitational acceleration and their initial altitude.”

    Who has such a theory? Is an atmosphere a vacuum? Any object in elliptical orbit about is primary does this with no change in temperature. Molecules in our stratosphere act in this way but go to lower temperature as they accelerate downward. Now consider Earth’s troposphere where at lower levels “only” increasing density, (decreasing mean free path), spontaniously increases atmospheric Kinetic energy and temperature. How does such theory operate there? How is gravitational potential energy ever expressed in this troposphere?

  14. Will Janoschka says:
    September 3, 2015 at 6:44 am

    Molar Specific Heat-Cp-Cv = R. by one Kelvin at constant pressure is called molar specific heat.
    Re-aranging Cp-R = Cv
    (𝛿T/𝛿z)³· = Γ³ = (g·M/Cp)² · (g·M/Cp)² · (12 · (Cp – R)/R)
    (𝛿T/𝛿z)³· = Γ³ = (g·M/Cp)² · (g·M/Cp)² · (12 · Cv/R)
    Splitting the dimensionality back to orthogonal directions:
    𝛿T/𝛿z)³/3· = Γ³/3 = 2(g·M/Cp)² · 2(g·M/Cp)² · Cv/R
    Or as p is 2D and v is 3D:
    𝛿T/𝛿z)³ = Γ³ = 2(g·M/Cp)² · 2(g·M/Cp)² · 3(Cv/R)
    To blow your ever lovin’ symbolic algebra mind, still with no understanding?
    To be scientific all must be expressed as symbolic algebra. Any symbolic algebra will do!!! Just like with “back radiation”, if the value is correct, call it net, and, feel free to turn symbolic algebra to mush! Parenthesis just disappear.😦

  15. Roger Clague says:

    No need for a magic formula to calculate lapse rate.
    If the specific or molar heat of the right gas is used, the answers for Earth and Titan are much better. They are about 10% away from theoretical calculation
    .
    The absorbing gas for Earth is H20 (cp = 1.9J/kgK)
    g/cp = 9.8/1.9 = 5.2K/km. measured ~6K/km

    The absorbing gas for Titan is methane (cp = 2.2J/kgK)
    g/cp = 1.4/2.2 = 0.64K/km. measured 0.5K/km

  16. Roger Clague says: September 3, 2015 at 1:11 pm

    “No need for a magic formula to calculate lapse rate.
    If the specific or molar heat of the right gas is used, the answers for Earth and Titan are much better. They are about 10% away from theoretical calculation
    .
    The absorbing gas for Earth is H20 (cp = 1.9J/kgK)
    g/cp = 9.8/1.9 = 5.2K/km. measured ~6K/km

    The absorbing gas for Titan is methane (cp = 2.2J/kgK)
    g/cp = 1.4/2.2 = 0.64K/km. measured 0.5K/”

    Lapse rate has no average, however gravity maintains the correct local rate at near the speed of sound, even during a Solar eclipse. Lapse rate has a “standard” used to calibrate barometric altimeters so all aircraft at the same location give the same reading. Useful for maintaining vertical separation!

  17. Larry Ledwick says:

    Who has such a theory? Is an atmosphere a vacuum? Any object in elliptical orbit about is primary does this with no change in temperature.

    You are missing the point. Yes if you had a single planetary body with an atmosphere of 2 molecules of gas you would have a simple 3 body orbital problem and in the frame of reference of the two molecules in orbit around the planet, they would be orbiting in a nearly perfect vacuum.

    The gas molecules would orbit (in a vacuum) just like tiny moons and their temperature would not change as they orbited. They would gain kinetic energy of motion as they approached perigee and lose kinetic energy of motion as they approached their apogee with no change in their individual temperature – – – – – until they ran into either the planetary body or each other. If they collide with something that would be expressed as pressure of this very tenuous gas.

    In that impact all that kinetic energy would be converted to heat energy and remaining kinetic energy of motion following the collision. They would move off with a different individual temperature and a different velocity (kinetic energy of motion) in a new orbit.

    Our concept of the temperature of a gas is just a statistical construct that describes this result of endless collisions between the atoms/molecules and in this case the planetary body. It is constantly moving energy around and changing its form from kinetic energy of motion or spin or vibration or thermal energy of the atoms of the gas as a whole. Pressure is a result of these innumerable collisions. Under conservation of energy laws the sum total of all these forms of energy must be conserved for the whole. All the relationships we use to describe behavior of gasses are simply different ways of describing these energy transformations. Specific heat etc. are just ways to describe how energy gets partitioned in collisions between orbital bodies over very short mean free paths.

    In an isolated system like the model above of one planetary body and two gas molecules you would find it easiest to treat it like a 3 body orbital problem and you would clearly see that as the molecules moved closer to the body they were orbiting, they would gain kinetic energy of motion as they “fell” toward the body to perigee. That kinetic energy of motion does not get thermalized and converted to heat until the molecule bumps into something. Then the internal energy of the molecule gets re-partitioned into spin, vibration etc. as the velocity changes. The temperature of the individual molecules and the gas as a whole are different measures of energy. This is how you can have a very rarefied gas with a “temperature of several thousand degrees” but almost no heat content because the sum of all those individual molecular energies are very small due to the low gas density.

    In a real gas atmosphere, the mean free path of the individual molecules orbits are very very short as they are constantly colliding with other molecules. Then it is much easier to deal with the behavior of the gas as a statistical value of the whole in terms of temperature and pressure in this case the temperature is a measure of the average speed of motion in the cloud of gas (plus all the other factors like spin and vibration), but the laws of motion in a gravitational field must still apply even if they are only over very short distances between collisions. The inescapable conclusion of that is that as a gas parcel falls in a gravitational field its energy increases (increases speed (pressure), spin ie temperature) as that gravitational potential energy is thermalized through collisions that increased energy is expressed as either higher pressure or higher temperature.

    As a result in a real atmosphere you get a gradient in “both” pressure and temperature due to gravitational effects (excluding external inputs like absorption of external radiation or loss of energy through radiation). Both pressure and temperature of the atmosphere will be higher near the planetary body than they will be at a great distance from the surface. Both temperature and pressure must gradually change with altitude.

  18. wayne says:

    “You have taken part of the comment out of context, the upward moving molecules would lose energy fighting gravity and trade velocity for gravitational potential energy as they went up, like a base ball thrown in the air. Therefore the upward moving molecules would cool (slow down converting kinetic energy of motion to gravitational potential energy as they gain height and lose upward velocity) ever so slightly as they moved upward. ”

    Yes Larry, but due to mass conservation there is always an equal mass of the “other” atmosphere not being addressed that is doing exactly the same in the opposite direction… Newton’s Third Law that is pretty well established. Seems the overall net change is zero unless you want to just look at what is happening to one isolated “parcel”.

    Everything you are saying of molecules, just being mass like any other mass, operating in vacuum in a gravitational field seems to ring’s true.

  19. Larry Ledwick says:

    I agree with that, you have atoms at the top of atmosphere which are very cold, and have little kinetic energy and due to collisions with colder atoms have also lost thermal energy and an equal mass of atmosphere near the surface at our average (teeth gritted as I use that term) surface temperature for that elevation. I would bet that if you summed it all up and averaged the total energies of that “column of air” its average thermal energy would be exactly the effective radiation temperature of the earth seen from space (assuming you also accounted for energy deposited by radiation from the sun at top of atmosphere).

    Suppose you had a weightless balloon full of one mole of air motionless with respect to the planet and let if fall freely through a tube with a perfect vacuum in it. When it slammed into the earth’s surface at terminal velocity, how hot would it get as it converted all its kinetic energy to heat on impact? That would be the heat energy contained in the entire column of air that tube represents in the normal atmosphere.

    It’s increase in kinetic energy as it fell in this highly simplified situation would (should) according to conservation of energy, exactly equal the sum of the pressure/temperature and gravitational potential energy it would have in our atmosphere at any similar level in its fall.

  20. Larry,

    Thanks for your contribution.

    I’ve been saying that here (and elsewhere) for some time but it felt like banging my head against a brick wall🙂

  21. wayne said:

    “Seems the overall net change is zero ”

    Of course it is but you still have constant motion and constant motion needs an energy source.

    Thus that ‘extra’ 33K (or whatever) at the surface over and above the S-B prediction.

  22. Larry Ledwick says: September 3, 2015 at 4:39 pm
    Will Janoschka says: September 3, 2015 at 9:43 am

    (“Who has such a theory? Is an atmosphere a vacuum? Any object in elliptical orbit about is primary does this with no change in temperature.”)

    “You are missing the point….”

    You have not stated a point! What is your point With vacuum rather than atmosphere?
    You seem ti be insisting that energy be conserved in an atmosphere. Why? Where is any evidence that atmospheric energy should be conserved? Where is any evidence using the term “energy” is even applicable to an atmosphere? Use power (kT/t) the power expressed by each gas molecule! The accumulation of power by mass is but part of accounting for power and if that is stored as sensible heat, structural (latent) heat, or as pressure. all comprising the “all” of internal energy of mass.
    My point is that the use of energy to describe an atmosphere is pointless, useless,and a distraction to the “is” of an atmosphere. A molar concept is much more informative. If their is any mass the aggregation of power is easily determined, after the ‘is’ is known.

    “In that impact all that kinetic energy would be converted to heat energy and remaining kinetic energy of motion following the collision. They would move off with a different individual temperature and a different velocity (kinetic energy of motion) in a new orbit.”

    In a troposphere their is no orbit density way to high, MFP too small. If molecular power is correct (kT/t) they all have the same temperature, the lapse rate with altitude. Energy, energy, energy, Why is energy important in a constant6ly moving atmosphere?

    “Our concept of the temperature of a gas is just a statistical construct that describes this result of endless collisions between the atoms/molecules and in this case the planetary body. It is constantly moving energy around and changing its form from kinetic energy of motion or spin or vibration or thermal energy of the atoms of the gas as a whole. Pressure is a result of these innumerable collisions.”

    Whose concept? “Who has such a theory? Is an atmosphere a vacuum? Temperature of an atmosphere is a local aggregate (average) of T = t/k for each molecule in the aggregate and (t) being the statistical average of time between intercepts of the molecules in aggregate.

    “Under conservation of energy laws the sum total of all these forms of energy must be conserved for the whole.”

    Where are these conservation laws for an atmosphere? The secondary purpose of an atmosphere is to dispatch entropy to space as effectively as possible via EMR!

    wayne says: September 3, 2015 at 6:17 pm

    “Everything you are saying of molecules, just being mass like any other mass, operating in vacuum in a gravitational field seems to ring’s true.”

    Why do you insist on ever considering mass in an atmophere, not a vacuum.

    Larry Ledwick says: September 3, 2015 at 6:54 pm

    “It’s increase in kinetic energy as it fell in this highly simplified situation would (should) according to conservation of energy, exactly equal the sum of the pressure/temperature and gravitational potential energy it would have in our atmosphere at any similar level in its fall.”

    What fall in an atmosphere? If you would check the actual numbers for such an extravagaint claim, you would find the your fantasy tropospheric KE and PE do not sum to a constant! No conservation of energy in an atmosphere. Atmospheres do not fall, they express no weight, and no potential expresses in an atmosphere. Atmospheres have self-buoyancy and are isopotential to themselves. Why that lapse rate? The proper expression of a tropospheric, IGL is PV (joules) = kT (joules) for each molecule or better for a moving “tropo”-sphere, PV/t (watts) = kT/t (watts) for each molecule. Get rid of joules, get rid of mass, they only there to confuse the issue of atmosphere. Suppose a 10 cm^2 atmospheric cloumn of moles (1Kg on earth) and sufficient moles of earth to produce 9.8 meters/s^ acceleration on 1Kg of (not atmosphere). All can be done with molar mass, molar density, and diferential temperatures.
    All the best! -will-

  23. Thanks for posting this Roger. I mentioned the find on HockeySchtick and here. I thought it was interesting. The speed of sound in natural gas (methane) is a little higher than in air at close to 400 m/sec. For natural gas burners operating at almost zero excess air and having a high emmissivity due to creation of internal soot (cracking of CH4 molecule) I used pressures (300+kPa) to give the speed of sound in the nozzle and have some additional pressure in the jet to entrain more air at the exit. Could get emmissivities close to 0.8 compared to a nozzle mixed gas flame of about 0.45.

  24. wayne says: September 3, 2015 at 6:17 pm

    “Everything you are saying of molecules, just being mass like any other mass, operating in vacuum in a gravitational field seems to ring’s true.”

    What a change from while ago when attempting to figure lapse rate!
    https://tallbloke.wordpress.com/2012/11/25/wayne-jackson-new-identity-linking-meteorological-phenomena/
    In that thread there are hints of speed of sound and a hints of optical depth, (understanding). Both are more pertinent to atmosphere than lapse rate! On the optics end, it is not 2/3 it is 0.68012, the emissivity of one optical depth at any frequency.
    An optical depth is defined as the distance where transmittance by the intervening media lessens to 1/e that of zero depth, at each frequency. Emissivity at 2 Tau is 0.865, transmittance 0.135. Earth’s atmosphere is 100km, and has sufficient Tau at every frequency to calculate total EMR exitance to space. However this Tau is highly dependent on airborne water condensate, (clouds).
    I use emissivity (capability) rather than emittance (flux), as emittance is highly dependent on opposing radiance (high within the atmosphere), very low to space!
    All the best! -will-

  25. Aircraft operate, not in a gas, but in an atmosphere, Many aircraft folk demonstrate competence of atmosphere, The aircraft work!!! Would you actually get on an aircraft designed by an academic meteorologist, climatologist, or the new atmospheric physicist? Please please give credit to competence, and flush the overwhelming incompetence of the 97%. Before December!!

  26. tallbloke says:

    CF: “Could get emmissivities close to 0.8 compared to a nozzle mixed gas flame of about 0.45.”

    Do you mean “efficiencies” rather than “emissivities”?

  27. No Roger I mean emissivity of the flame as used in the modified form of the S-B equation determined by Hoyt Hottel Professor of Chemical Engineer at MIT (see Perry’s Chemical Engineering Handbook -I think also in Mark’s Mechanical Engineering Handbook -section on heat transfer by radiation) I should also mention that of the 0.45 some 0.42 is contributed by water vapor. The emissivity of a coal flame is close to 1.0 due to the coal and ash particles in the flame. Further, in the section on heat transfer in a table (5-9) and the text you will find that the emissivity of the gas methane (CH4) is less than CO2 and not significant contrary the IPCC BS of 21 times CO2. Boilers operated by burning natural gas are less efficient than on coal.

  28. Roger Clague says:

    Larry Ledwick says:
    September 3, 2015 at 4:39 pm

    The inescapable conclusion of that is that as a gas parcel falls in a gravitational field its energy increases (increases speed (pressure), spin ie temperature) as that gravitational potential energy is thermalized through collisions that increased energy is expressed as either higher pressure or higher temperature.

    1. What is a gas parcel?
    2. What is the evidence that a gas parcel exists?
    3. How big is a gas parcel?
    4. What separates one gas parcel from another?

    There are no answers to these questions. The idea if an air parcel was invented, by fluid dynamics theory so that molecules can be ignored. Air is a gas not a fluid like water. Ignoring molecules will prevent understanding of the atmosphere, not help.
    Explanations using molecules and air parcels are inconsistent and unhelpful.

  29. tallbloke says:

    CF: Thanks for that. I know little of the emissivities of flames I must say. I’ve done efficiency calcs for the backpacking stoves I’ve made based on figures for ‘heat of combustion’ I’ve found online for various fuels though.

  30. Larry Ledwick says:

    1. What is a gas parcel?
    2. What is the evidence that a gas parcel exists?
    3. How big is a gas parcel?
    4. What separates one gas parcel from another?

    1 – any arbitrary mass of air, commonly used term in meteorology
    2 – it is a defined concept it needs no evidence other than common usage by thousands of meteorologists for over a hundred years
    3 – how ever big you want it to be for the case at hand, could be a cube 10 cm by 10 cm by 10 cm or a kilometer on a side
    4 – nothing it is a defined concept of a volume of a fixed mass of air which has a weightless container which does not restrict its expansion or contraction or interfere with energy transfer.

    http://www.erh.noaa.gov/er/box/glossary.htm
    Air Parcel- An imaginary small body of air that is used to explain the behavior of air. A parcel is large enough to contain a very great number of molecules, but small enough so that the properties assigned to it are approximately uniform throughout.

    search is your friend

  31. Larry Ledwick says: September 5, 2015 at 12:55 am
    Roger Clague says: September 4, 2015 at 10:06 am

    (“1. What is a gas parcel?
    2. What is the evidence that a gas parcel exists?
    3. How big is a gas parcel?
    4. What separates one gas parcel from another?”)

    From the Religious Meteorologists handbook of criminal nonsense:

    “1 – any arbitrary mass of air, commonly used term in meteorology”

    Ah yes often used in criminal meteorology! We all know that. Why do “you” use the term that only demonstrates intent to defraud?

    “2 – it is a defined concept it needs no evidence other than common usage by thousands of meteorologists for over a hundred years”

    So do you agree there is no such thing as a physical atmospheric parcel? There is no defined concept, only undefined fantasy!

    “3 – how ever big you want it to be for the case at hand, could be a cube 10 cm by 10 cm by 10 cm or a kilometer on a side”

    So you claim an undefined volume of any size. How is your parcel delimited in your fantasy?
    What case in hand? another fantasy?

    “4 – nothing it is a defined concept of a volume of a fixed mass of air which has a weightless container which does not restrict its expansion or contraction or interfere with energy transfer.”

    There is no defined concept, only undefined fantasy! What is you container made of? How do you detect such a boundary? Does your fantasy parcel promote or deny mass diffusion and mixing always present in every atmosphere?

    http://www.erh.noaa.gov/er/box/glossary.htm
    ‘Air Parcel- An imaginary small body of air that is used to explain the behavior of air. A parcel is large enough to contain a very great number of molecules, but small enough so that the properties assigned to it are approximately uniform throughout.’

    Leave it to NOAA! Why do you defend the preachings of the highest level proponents of CAGW fraud, while at the same time demonstration no competence of atmosphere? As you illustrate.

    Air Parcel- An NOAA imaginary small body of air that is used to confuse any possible understanding of the behaviour of atmosphere. Such never exists in any physical atmosphere.
    A atmospheric ‘particle’ is large enough to contain a very great number of molecules 10^9, the smallest visible part of a cloud, but small enough so that the properties assigned to it are approximately uniform throughout. The H2O molecules are starting to hydrogen bond, but the particle is very deformable, allowing non elastic collisions with other H2O molecules/particles. Such structure reversibly releases the power used to transform liquid to gas. This power is always dispatched to space via EMR.There is no work done in this power/energy re-arrangement into structure. The power dispatched to space results in a contraction of the cloud volume.
    All the best! -will-

  32. Larry Ledwick says: September 5, 2015 at 1:05 am

    http://fas.org/irp/imint/docs/rst/Sect14/Sect14_1b.html

    Federation of American Scientists

    More preprogrammed CAGW nonsense from Washington lobbyists!

  33. Larry Ledwick says:

    Will:
    You clearly have no interest in communicating or working out the validity of the original post.
    Nor do you have any understanding of common science terminology used in teaching elementary concepts in atmospheric behavior which long predate the current pathology of global warming nonsense. I heard the term “parcel of air” in 7th grade science back in the 1960’s and it had long been in use at that time, and has absolutely nothing what ever to do with the current deceptive practices in the global warming / climate change dog and pony show.

    I suggest you go to a large library and find the oldest book on meteorology or basic atmospheric science you can find that discusses why a thunderstorm updraft develops and issues about convection in the atmosphere and stability in the atmosphere. You will find use of that term long predates both of us, and like other conventions used in science analysis and discussion such as free body diagram, friction free motion, motion of a projectile with air drag disregarded, infinity, a point with no dimension, adiabatic heating, perfect black body etc. is very useful for discussion and is common practice in almost all scientific specialties.

    Since you are no longer even attempting to discuss this topic in good faith, – have a nice day, I have better things to do than try to discuss a topic with someone who’s only interest is in dragging the discussion off to some meaningless tangent instead of trying to explore the original post and its implications.

  34. gallopingcamel says:

    I am impressed by the work of Robinson & Catling. They can explain the temperature from a planet’s surface to the top of its stratosphere with remarkable precision:
    http://www.nature.com/ngeo/journal/v7/n1/abs/ngeo2020.html

    They can even explain the temperatures on Titan:
    https://diggingintheclay.wordpress.com/2014/04/27/robinson-and-catling-model-closely-matches-data-for-titans-atmosphere/

    However, R&C don’t have much to say about why the DALR (Dry Adiabatic Lapse Rate) needs to be reduced on planets with oceans such as Titan and Earth. They simply postulate a fiddle factor which they call “Alpha” that agrees closely with the figures you mention above.

    Venus does not have a well defined tropopause and it has an anomalous (negative) temperature gradient in its stratosphere. Robinson & Catling have plausible theory to explain this.

  35. Larry Ledwick says: September 5, 2015 at 4:08 am

    “Will:
    You clearly have no interest in communicating or working out the validity of the original post.”
    accurate-formula-for-calculating-environmental-lapse-rate-on-three-worlds

    How does your fake religious meteorological fantasy have anything to do with an ‘environmental-lapse-rate’. It seems like you only wish to hijack the thread for your own political interest. Do you have anything to offer that may help with understanding of a atmosphere?

    “Nor do you have any understanding of common science terminology used in teaching elementary concepts in atmospheric behavior which long predate the current pathology of global warming nonsense.”

    All of the ‘current pathology of global warming nonsense’ is based on the same religious meteorological fantasy that you promote. There is absolutely no science in meteorology! Meteorologists make astrologists, tarot card readers, and witch doctors seem competent in understanding this world.

    “I heard the term “parcel of air” in 7th grade science back in the 1960’s and it had long been in use at that time, and has absolutely nothing what ever to do with the current deceptive practices in the global warming / climate change dog and pony show.”

    The exact same terms, with no understanding, are used in the current deceptive practices in the global warming / climate change dog and pony show. It is the inherent meteorological deceit of trying to fake competence that is the fraud.

    “I suggest you go to a large library and find the oldest book on meteorology or basic atmospheric science you can find that discusses why a thunderstorm updraft develops and issues about convection in the atmosphere and stability in the atmosphere.”

    You spent a day on the internet getting brainwashed about meteorology, and now wish to suggest things?
    You started with rubbish of conservation of KE and PE in an atmosphere, and now are an expert. Your spoutings have nothing to do with accurate-formula-for-calculating-environmental-lapse-rate-on-three-worlds!

    “You will find use of that term long predates both of us, and like other conventions used in science analysis and discussion such as free body diagram, friction free motion, motion of a projectile with air drag disregarded, infinity, a point with no dimension, adiabatic heating, perfect black body etc. is very useful for discussion and is common practice in almost all scientific specialties.”

    What term “Air parcel”? Such abstractions are useful abstractions in context to assist in understanding of a particular POV. Such are always disasters when applied to the physical with no understanding. The application of fantasy to the physical, is is exactly what meteorology has been brainwashing into innocent children for the last 70 years. It is high time to stop with the nonsense and listen to those that have demonstrated competence of atmospheres. Aerodynamic engineering!
    All the best! -will-

  36. GC The R&C pper is one I have been looking for any chance of a link to the full paper. i can not afford the price. These type of papers should be open source.
    Slightly off topic in a comment on the post about the paper and Titan (I have bookmarked the site) you mention photons. I put the following comment at physicsworld.com which is a publication of Institute of Physics (IOP) but there have been no replies -maybe you would comment
    Comment at physicsworld.com 290815
    http://physicsworld.com/cws/article/news/2015/aug/25/single-photons-see-the-light
    cementafriend Aug 29, 2015 6:44 AM
    Photon
    Does the paper define a photon? has it a frequency range say 0.5-0.7 micron or a single frequency say 0.7 microns (then there would be infinite types of photons)? How does a photon used in this post’s concept differ from a pulse of light with a limited frequency range?
    Some physicists claim there are photons in the long wave infra red range emitted or absorbed by the molecule CO2, Are these different photons than light photons?
    W E Lamb jr who won the 1955 Nobel prize for physics has written that there are no photons (I think he titled a paper “anti-photon”)
    Is it not time people talked about electro-magnetic energy waves and pulses of defined frequency range?

  37. cementafriend says: September 5, 2015 at 7:10 am

    “GC The R&C pper is one I have been looking for any chance of a link to the full paper. i can not afford the price. These type of papers should be open source.
    Slightly off topic in a comment on the post about the paper and Titan (I have bookmarked the site) you mention photons”

    I wanted to give Peter time to respond. Anyhow:

    Photons is likey the most misused and misunderstood term ever! Current photon is the name given to the conceptual gauge boson that mediates electromagnetic power transfer. A. Einstein 1905 I think developed the concept of photoelectric effect where it took sufficient EMR power density in four-space to emit an electron from a nickel surface or energy of 5 electron volts wavelength 200 nanometers. Nickel now has an electron work function of 5.01 ev. About the same time Max Planck was wrestling with the concept of quantum for EMR especially thermal EMR.
    That solution was his Planck constant (h) a unit of action (J x s) When (k) so combined with frequency (hv) that results on the energy for each cycle of EMR power, an indivisible quantity, (so far). Sometimes folk think that is ‘a’ photon, the quantum.
    How this all got turned around to little bullets of EMR energy rushing off in every direction independent of the opposing EMR environment, and only dependent on T^4 of a possible emitter is anyone’s guess! Seems to be deliberate misrepresentation, about the time of Carl Sagan and his trash! No one with even the slightest understanding of electromagnetic field theory, so very well tested, would ever accept such nonsense.
    What your reference at physicsworld.com means by photon is perhaps all of the above, signifying nothing but incompetence at PR.
    All the best! -will-

  38. Will, thanks for a reply.. No one at physicsworld.com has commented, I suggest they are too embarrassed that they do not understand their own physics but then they are not engineers.

  39. tchannon says:

    Arrives back at the irresolvable particle/wave conundrum.

    Might have been a poor idea to say frequency and size, trips the amateur flag.

    http://www.photonics.byu.edu/fwnomograph.phtml

  40. tchannon says: September 6, 2015 at 5:40 pm

    “Arrives back at the irresolvable particle/wave conundrum.”

    Sparkies don’t have a problem!

    “Might have been a poor idea to say frequency and size, trips the amateur flag.”

    Only with no understanding if the inverse complex conjugate!🙂

    http://www.photonics.byu.edu/fwnomograph.phtml

    Looks like a cps to Hz conversion table!

  41. dscott says:

    The speed of sound is dependent on atmospheric density. Earth 1 bar, Venus 90 bar and Titan 2 bar. Boyles and Charles Laws work the same no matter what the composition and gravity so it isn’t surprising there is a Constant that governs planetary heat content.

  42. dscott says: September 8, 2015 at 3:32 pm

    “The speed of sound is dependent on atmospheric density. Earth 1 bar, Venus 90 bar and Titan 2 bar. Boyles and Charles Laws work the same no matter what the composition and gravity so it isn’t surprising there is a Constant that governs planetary heat content.”

    planetary heat content?
    planetary global average temperature?
    atmospheric heat content?
    atmospheric lower level global average temperature?
    atmospheric lapse rate?

    Which? What can you possibly mean?

  43. cementafriend,
    Something is wrong when research that taxpayers paid for is hidden behind “Paywalls”.

    I am a proverbial “Leaky Bucket” so I consider hiding research to be repugnant.

    Here is a link to the full paper which is not paywalled:
    http://arxiv.org/pdf/1312.6859v1

    Here is something that may be relevant:
    http://astrobites.org/2013/12/31/unifying-planetary-atmospheres/

    You mention a paper where photons are interacting in cold condensed matter. Some of my colleagues at Duke were using similar techniques to slow light down to bicycle speed. As a mere old school laser jock, this is way beyond my pay grade so the best I can do is refer you to an article by Dan Gauthier. There is a related 20 page paper loaded with equations but you might want to start at the shallow end!
    http://www.phy.duke.edu/~qelectron/pubs/PhotonicsSpectraFastLightSlowLight2007.pdf

  44. gallopingcamel says:
    September 9, 2015 at 4:33 am

    (‘cementafriend,
    Something is wrong when research that taxpayers paid for is hidden behind “Paywalls”.
    I am a proverbial “Leaky Bucket” so I consider hiding research to be repugnant.
    Here is a link to the full paper which is not paywalled: http://arxiv.org/pdf/1312.6859v1
    Here is something that may be relevant:
    http://astrobites.org/2013/12/31/unifying-planetary-atmospheres/‘)

    Thank you Peter,
    In the paper I find way to much emphasis on the ‘radiative’ determining tropopause. No-mater 10kPa or 20kPa or perhaps the pressure for an atmosphere of UF6 that would be different. The atmosphere is not in a volume. The ‘bottle’ is inside the atmosphere. The rate/m^2 of molecules banging into the ‘bottle’ gives both surface pressure and surface temperature. Who knew?
    The atmospheric tropopause does begin at the ‘pressure/density’ where the molecules stop being constrained by mean free path (neighbours), with various degrees of freedom. At higher altitudes but below this pressure in a gravitational field the molecules or clumps of molecules are constrained only by the gravity of Kepler, i.e. they are in orbit almost never interaction with others.
    Here temperature becomes nebulous and mainly devoid of meaning. I go to 30+km measure normal to nadir radiance I am still measuring the combined emissivity of 200km of thin stuff. I check emission bands for each species, measure there. I convert that to some spectral brightness temperature and go on to the next species. If they all have the same brightness temperature independent of wavelength only then I am justified in calling that a temperature at that altitude.
    None of these ClimAstrologists would go to that trouble, they just pull a number from the toilet.
    There is simply not enough sensible heat power to influence anything. OTOH a magnitude of very high energy ‘stuff’ would still be absorbed in 200km. Can you tell us about the intense fascination of the stratospheric temperature profile by ClimAstrologists?
    All the best! -will-

  45. dscott says:

    You do recognize the irony of your avatar? Maybe it’s an expression of irony that’s only known Americans? There are cultural differences.

  46. tchannon says:

    Observation: –

    ” For  Earth,  for  example,  T eff   = 255  K  and  the  skin  temperature  is  214  K,  which  is  within  a  few  percent  of  the  observed global  mean  tropopause  at ~208  K  (ref.  27). ”

    and
    “Equator Average Temperature (K) ~206K

    Former is R&C
    Latter is Diviner web site

    Earth has a servo loop at altitudes below the variable altitude tropopause, the moon does not.

  47. dscott says: September 8, 2015 at 3:32 pm

    “The speed of sound is dependent on atmospheric density. Earth 1 bar, Venus 90 bar and Titan 2 bar. Boyles and Charles Laws work the same no matter what the composition and gravity so it isn’t surprising there is a Constant that governs planetary heat content.”

    “Governs planetary heat content.”? Do you mean apmospheric heat content.”?

    dscott says: September 10, 2015 at 3:54 pm

    “You do recognize the irony of your avatar? Maybe it’s an expression of irony that’s only known Americans? There are cultural differences.”

    dscott, Do you have a point? Or just a NOAA troll?