# What is the adiabatic lapse rate of air?

When explaining the greenhouse warming effect, I’ve avoided going into the cause of the adiabtic temperature change of the atmosphere as we get higher and instead used a hand waving argument that expanding air is cooler which has been enough to explain the necessary temperature gradient up through the atmosphere. (link, other articles) However, this isn’t really the mechanism behind the adiabatic lapse rate.
Then I came across a comment on Roy Spencer’s blog to the effect that the adiabatic lapse rate was caused by the greenhouse effect – indeed driven by it. That seemed to be counter to my  understanding. Then another commenter on my blog said it wasn’t due to loss in potential energy as I suggested. So, I decided to think about it more.
First, I should explain that “adiabatic” just means that we are assuming the air does not gain or lose heat to its surroundings (by IR or conduction).
As we know gases are governed by the rule that:

PV = nRT

So, this would suggest that when pressure is reduced, so is temperature. However this is only true of a fixed volume. But when air rises, it expands, so both its pressure goes down and its volume goes up.

### Work done

So, another way that is used to attempt to explain why the temperature of the gas goes down, is to suggest that it is due to “work done”: because whilst it is expanding it is doing work, this is why it loses energy and therefore heat. That again sounds superficially attractive, but in actual fact when the parcel of air rises it is also gaining potential energy. And moreover, all the surrounding gas is also “doing work” on the other bits of gas around it. So, not only is the parcel of air we are considering doing work against other parcels, but other parcels are doing work against the parcel we are considering. So where is all this “work” going – it can only go into other parcels of air. There can be no net change in energy!

### Adiabatic lapse rate is just loss of potential energy

However, once we start thinking about potential energy we have the solution. Imagine if you will a balloon of gas slightly lighter than air (assume the balloon itself is negligible weight). Because the balloon is lighter, it will rise upward. But the gas within the balloon as a finite mass, so in order to make this rise, there has to be a gain in potential energy. But also as the balloon rises, the pressure drops, the balloon expands and the air within the balloon does work (force x distance = work). This work comes from the thermal pressure of the gas. So, as the air molecules expand they lose heat. And so this is where the potential energy is coming from. If we assume the balloon is big enough to contain 1kg of air, then the transfer of energy is as follows:

Potential energy = energy lost from air = temperature change x specific heat capacity
mgh = T Cp m
(where m is mass in kg, g is gravitational constant 9.8, h is height m, T is temperature C, Cp is specific heat capacity in Joules Centigrade-1 kg-1)

Simplifying we get:-

T/h = g/Cp

The specific heat capacity of air at -50 to 40C is 1.005 kJ Centigrade-1 kg-1

So lapse rate T/h = 9.8/1005
= 9.75C/km

This is the lapse rate of dry air. (Moist air has a different lapse rate because it condenses out water droplets which effectively makes Cp much larger)

### Breaking the “second law” that heat flows from hot to cold

The second law of thermodynamics is wrongly interpreted as meaning that heat MUST flow from hot to cold (that’s not what the actual “law” says). Therefore some argue that because air gets cooler as we get higher, that this must mean there is a heat flow.

This is false.

To explain why imagine the situation shown to the right. Here the molecules of the atmosphere are free to move up and down, but they cannot move from side to side. Imagine first a column without gravity (there is now really no “up” or “down”, but I will use these terms). An atom impacts a lower one at say height h1 and it moves “up” to point h2 where it hits another molecule. On average the energy it gets from the “lower” one will all go to the “upper” one. If we then assume some variation in the size of molecules to disperse energy through the column, eventually the energy will tend to equalise so that they are all effectively vibrating with the same temperature. This means there should be a uniform temperature “up” the column.
However, now imagine a column with gravity. Now if the molecule of weight m hits the lower one at h1 and then hits the higher one at h2, then because of the difference in height between h1 and h2 it will lose potential energy equivalent to:

m g (h2 – h1)

Therefore slightly less of the vibrational energy is given to the molecule above than was received from the molecule below. Therefore as we rise h meters up the column, the energy drops as:

Vibrational Energy drop  = m g h

This vibrational energy is another name for heat, so, the temperature drop is equal to T  Cp m. In other words, the heat conducted from the bottom of the gas upwards drops at the lapse rate. That is to say, heat energy is gradually lost both to conduction and potential energy as it is transferred up the column. So, in the absence of any other heat flows, the column will eventually stabilise with a lapse rate of g/Cp thus we have a thermal gradient which is stable with no heat flow.
However if we introduce other heat flows, the adiabatic lapse rate changes the behaviour of the air.

### Rising hot air

If air lower in the column is heated by e.g. incoming solar energy, its temperature will rise. As it is surrounded by air at the same pressure, and PV = nRT, that temperature rise will cause an expansion of the air. It will then become less dense than the surrounding air and so it tends to rise. However, so long as the air profile drops per the adiabatic lapse rate, this packet of air will always be lighter than the surrounding air (another way to view it, is that it has more energy than the adiabatic lapse rate) and so it will continue upward, either until it encounters a layer that has less energy than the adiabatic lapse rate, or it loses the excess energy by conduction to the surrounding air, IR heat loss or condensing water droplets.
In theory, if there is no way to lose energy but conduction, it will continue upward effectively heating the whole column marginally until it replaces the very topmost molecules of air.

### Infra-red breaking the lapse rate

However, whilst there is a lapse rate for conductive heat, so that the temperature drops as me move upward as heat is lost to potential energy, the same is not true of infra-red which is carried by electro-magnetic waves which have no weight and so energy is not needed to rise up against gravity.
So infra-red energy will tend to equalise the temperature causing the higher levels to warm more than the adiabatic lapse rate and the lower levels to cool to be lower than the adiabatic lapse rate. When this happens, the lower air becomes denser than the adiabatic lapse rate and less likely to rise and conversely the upper air becomes less dense than the adiabatic and more likely to float higher up. So, now, even if some air is slightly heated, it may initially rise and expand, but because the IR has broken the lapse rate profile, it quickly reaches a layer of air which is less dense at that temperature.
And the more IR is a significant conveyor of heat, the more likely it is that IR will break the adiabatic lapse curve.
So, as long as there is heating of the lower atmosphere large enough to cause rising hot air to carry enough heat to offset the infra-red transfer, the atmosphere will retain the lapse rate. However, this only happens when the atmosphere is dense enough to block the movement of IR. Eventually as we rise up through the atmosphere, the rate of transfer of IR become greater than the potential to transfer heat through convection and the adiabatic lapse rate breaks down and the IR heat transfer breaks the adiabatic lapse rate, equalises the thermal gradient and then effectively blocks upward movement.
This is effectively happens at the top of the troposphere (or at least once the air loses its moisture).
But also there is IR heat loss to space. This is stronger the higher we go, and so this should tend to reduce the temperature of the upper atmosphere. But then other effects come into play and it is far too complicated to work out what is happening from theory.

### Greenhouse gases and adiabatic lapse

From this we can conclude:

1. The lapse rate is the rate of loss of energy to potential energy as a packet of air rises.
2. The adiabatic lapse rate will occur naturally in a transparent atmosphere which has no interaction with IR or visible light.
3. If lower layers of the atmosphere are heated, they tend to rise until they encounter a layer at a lower temperature than the lapse rate. So, this mechanism tends to cause convection from the ground up to where the lapse rate breaks down.
4. The lapse rate breaks down because of IR heat loss (or gain).

Far from the lapse rate being a result of IR interactive gases (aka greenhouse gases), as has been suggested, greenhouse gases tend to break down the lapse rate. In the bulk of the atmosphere, they  tend to move the thermal gradient away from the lapse rate and toward an isothermal region which tends to repress convection, but near the top the IR heat loss to space tends to make the upper layers with more of a “window” to the cold of space, colder, this tends to make convection easier. So,  what creates the lapse rate is the loss of potential energy. And far from greenhouse gases creating the lapse rate, the effect of “greenhouse” gases is complicated causing both and increase in gradient as IR is lost to space AND a decrease in gradient as IR tends to equalise the temperature within the atmosphere.

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### 29 Responses to What is the adiabatic lapse rate of air?

1. oldbrew says:

Reblogged at Tallbloke’s Talkshop:
This ties in with recent discussions about convection at the Talkshop.

• Scottish-Sceptic says:

Thanks.

2. I find the logic convincing. That is:
1. The gas law PV =nRT is not helpful because V is not fixed.
2. As gas rises and cools where does the energy go?
3. Work is done against gravity not pressure.
4. Heat lost = gravitational potential gained
mcT = mgh
T/h = g/c
for dry air 10K/km
Pressure and mass are the NOT the cause of T/h
T/h for moist air
Moist air c = 2J/kgK
T/h moist air = 10/2 = 5J/kgK , the measured value
Rising hot air (convection)
Convection is cause by the lower density of moist air
Convection is not caused by expansion of air.

• In the absence of water vapour uneven surface heating will cause density differentials in the horizontal plane which causes dry convection.
Water vapour just helps the process along because water vapour is lighter than air.

3. Scottish-Sceptic says:

I like the way you phrase it: “work is done against gravity not pressure”.

4. Mike M. says:

Scottish Sceptic,
The second law of thermodynamics states that heat MUST flow from hot to cold. That is not an interpretation, that is what the actual law says. Although there are several formulations of the law, all are ultimately equivalent.
At equilibrium, a gas must have a uniform temperature. Consider an imaginary horizontal surface through an isothermal gas. The molecules moving upwards across the surface are losing kinetic energy. So when such a molecule undergoes a collision at the “end” of its flight (above the surface) it has less kinetic energy than it did at the “beginning” of its flight (its last collision before crossing the surface). However, molecules with a very small initial upward velocity will soon stop moving upward and fall downward; such molecules will never make it across the surface (that is, their next collision will be below the surface). As a result, the upward moving molecules that end up colliding above the surface must initially have had higher than average kinetic energy. That extra initial energy compensates for the energy changed into potential energy. A similar argument applies to the downward moving molecules. This does not prove that the energy flow is zero when the system is isothermal, but it does prove that your argument is inadequate.
There are at least two ways to get a correct result. One is to do a detailed calculation that takes into account all relevant aspects of the molecules motion. The other is to make use of a more general principle, such as the second law. That gives an incontrovertible result: a system at equilibrium must be isothermal.

• Scottish-Sceptic says:

If that was what the “law” says, then it is clearly wrong as I have detailed. However, the actual law deals with entropy and not heat.
Also, in a real atmosphere, the molecular spacing reduces so that the molecules are in contact, so that on average molecules hit other molecules both above and below. And it doesn’t matter how far they move, they lose potential energy at the rate of mgh. Therefore the heat energy decreases by mgh IN A STABLE ATMOSPHERE.
So, in a stable atmosphere, it is not iso-thermal but instead there is a balance between potential and thermal energy so that the sum of these is the same across the atmosphere when it is stable.

• Mike M. says:

The second law can be stated in multiple equivalent ways: in terms of heat flows, entropy, heat engines, perpetual motion. Most thermodynamics boos deal with the equivalence of these.
For instance, it can be proven that if you have a T difference at equilibrium, then you can build a perpetual motion machine. I have shown how Scottish Sceptic’s claims leads to perpetual motion at: http://scottishsceptic.co.uk/2015/06/28/greenhouse-warming-test-cases-do-you-agree/#comment-34276

• Mike M. says: 3rd July 2015 at 7:48 pm
“Scottish Sceptic,
The second law of thermodynamics states that heat MUST flow from hot to cold. That is not an interpretation, that is what the actual law says. Although there are several formulations of the law, all are ultimately equivalent.”
Nowhere does the second law of thermodynamics states that heat MUST flow from hot to cold. The second is based on long term observation of spontaneity! That’s all! If Clausius had considered gravitational potential and electromagnetic radiative potential the thing would read “Spontaneous energy transfer cannot occur in a direction of higher total potential.”.

• Mike M. says:

I oversimplified my statement of the second law in a way that does not matter for this discussion. A proper statement of the Clausius formulation is: “Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.” The exception is for the case where some other form of energy is consumed, such as using electricity to run an air conditioner or heat pump. It was sloppy of me to not include the exception, but it does not matter for this discussion since the claim made by Scottish Sceptic is that no expenditure of work is needed to maintain the gradient.
The mere existence of a potential does not affect the flow of heat.

• Mike M. says: 4th July 2015 at 10:17 pm
“but it does not matter for this discussion since the claim made by Scottish Sceptic is that no expenditure of work is needed to maintain the gradient. The mere existence of a potential does not affect the flow of heat.”
Much that Mike says is a copycat from Met. However you will have to demonstrate that the pressure, density, temperature “require” some work to maintain! These gradiants are but the expression of gravitational potential in a compressable fluid They need no more work than the force of gravitational attraction between masses need work. Air being elevated in this atmosphere gains no potential energy “because” of these gradients!
The mere existence of a potential does indeed affect, but not demand, the spontaneous flow of energy. Neither the spontanious mass flow nor heat conduction upward to a lower pressure/temperature is allowed! “Lifting” a rock to a place on top of a post requires work, but the rock has no potential energy while resting upon the post. Energy is not conserved, except for the bookkeeping done by imaginary entropy!

• Mike M. says:

Will Janoschka,
“Energy is not conserved”.
In other words, you are either ignorant of science or choose to reject science. So no chance of reasoning with you.
For others who might read this, here is why Janoschka is wrong. There are exactly two fundamental ways to transfer energy: heat and work. Heat is the transfer of energy due to a difference in temperature, work is everything else. All forms of work are, in principle, interchangeable. But work and heat are not interchangeable. There is a one-way aspect: work can be fully converted to heat, but heat can not be fully converted to work.
Because of of the interchangeability of various forms of work, potential gradients associated with different forms of work can interact symmetrically. So, for example, a gradient in gravitational potential can maintain a gradient in pressure at equilibrium and vice versa; no continuous expenditure of work is required. But the asymmetry between work and heat means the same is not true for temperature (the potential for heat). You can not maintain a temperature gradient without a continuous expenditure of work. As a result, a system at thermodynamic equilibrium must be isothermal.
Note: You might ask “What about a dewar flask (thermos bottle)?” That will make it take a long time to reach equilibrium, but it won’t stop it. You coffee will eventually get cold.

• No net expenditure of work is required to maintain the gradient but there is still work done in that work done against gravity in the ascent is matched by work done with gravity in the descent at equilibrium.
Convective overturning locks 33K in at Earth’s surface and constantly recyles it up and down thus preventing it from radiating to space.
The ‘engine’ is uneven density differentials at the surface in the horizontal plane.
It is not a perpetual motion machine because the process is simply using an ever changing portion of solar energy temporarily as it passes through the system. Stop continuing insolation and the cycle will stop and the mass of the atmosphere will fall to the ground.
The necessary power is derived from the delay to the throughput of solar energy caused by conduction and convection being slower processes than radiation so that the necessary 33K of ‘surplus’ energy is allowed to accumulate at the surface to drive convective overturning and thereby maintain hydrostatic balance.
That 33K of ‘heat’ flows in and out of potential energy as necessary to adjust for radiative imbalances.
http://www.public.asu.edu/~hhuang38/mae578_lecture_06.pdf

5. Mike M. says: 5th July 2015 at 2:06 pm
Will Janoschka, “Energy is not conserved”.
“In other words, you are either ignorant of science or choose to reject science.
So no chance of reasoning with you.”
I reject your non-science. If I plough a field with a horse for one hour I use up one horsepower hour of stored horsepower but I never get one horsepower hour of plowed field, except by your fake entropy bookkeeping! Energy converted to work is never conserved unless that conversion is isentropic! Keplerian orbital mechanics.

6. What is the adiabatic lapse rate of air?
The adiabatic lapse rate of air is “a fictitious construct promoted by meteorologists to fool folk”!! The adiabatic lapse rate of air is also the temperature component of a non condensing atmospheric radial pressure, density, temperature profile determined completely by the gravitational force on the mass of the compressible fluid called atmosphere, whatever the density of any atmospheric component.
The adiabatic lapse rate of air has no thermodynamic meaning whatsoever!
The pressure lapse rate of air has no thermodynamic meaning whatsoever!
Both the concepts are but the expression of gravitational PE within its surrounding fluid.
Both the concepts were invented by meteorologists with the intent to confuse.

• The one thing I’ve learnt, the longer I’ve been looking at climate is how appalling even some of the most basic concepts are. Whenever i look deeply into the subject I seem to find concepts which are very poor science being accepted as “standard”.
And so, anyone with a decent degree in physics can usually look at the subject and find something that is “not quite right”.
Originally I put that down to “dumbing down” of the subject to make it more understandable. But I’ve now grown to understand that many prominent climate “scientists” really don’t know much physics at all.
It really is as if all the people in my class who flunked the exams, got together and formed a new subject.
So, e.g. the “greenhouse theory” with IR fluxes is an appalling mess. I just assumed it was intended as a “outline of flows” and not actually being suggested as a real theory. But the more I see the kinds of things being written, the more I now think that prominent climate “scientists” actually believe the greenhouse “trapping” theory and have no actual understanding how greenhouse gases really work.

• Mike,
I agree with your take on the current state of climate discussion. It seems that the internet has spawned 7 billion religions, all struggling for converts. The academics have been brainwashing children since 1950 with not a care for any science or proof, only you must believe of fail! WE are learned and we believe!
I have noted 6 observations of comments here and tallblokes on this subject. Would you care to point out your objections to my answers. 🙂

• Scottish-Sceptic says:

[snip] Just read your comment below and I’ve either got you confused with another or I didn’t understand your posts. [/snip]
I’m now reasonably confident I understand why we have a convective troposphere. (Heating from surface creates a steeper gradient than lapse rate causing convection).
I can now see how IR radiation from the top of the troposphere tends to put a cap on that layer and create a layer above which is relatively static (because it is warmer at the top than the lapse rate so it counters convection).
I’ve also realised that the top of the troposphere is likely a point where IR radiation balances in some way – so physical IR properties define the top, which explains why the “skydragon” adiabatic heated atmosphere model works. It’s probably not that there is no IR, but instead that IR defines the “top” – so they are actually unwittingly including the IR effect in their model when they assume there is a “top” at the top of the atmosphere.
I suppose thinking about it now … because the height of the troposphere changes as we head poleward … this must mean the “balance” defining the top has changed in some way.

• wildeco2014 says:

It is a shame that Will keeps distracting from the reality with such paranoid assertions about the indispensible discipline of meteorology.
Meteorology is all about the way the rules of physics play out in a real atmosphere, often in a counterintuitive fashion.
Climatologists seem oblivious to it

• Stepen Wilde keeps spouting his KE conversion to/from PE (gravitational) in this atmosphere, when there are no such gravitational forces acting on this fluid. All has been replaced by the pressure/density gradient in this compressible fluid!
Steven, point out “any” error in my 6 observations above! I do not accept your lawyerly POV.

• How does one get a density / pressure gradient without gravity ?
The KE/PE conversion of internal energy for gases moving up and down within a gravitational field is a matter of established science.
Gases are different from liquids and solids because their compressibility is magnitudes greater. As a consequence, PE is derived not only from height above the surface but also from increasing the distance between molecules against the attracting molecular forces.
That is why temperature falls so much with height (far more than for liquids or solids) as one moves gases up from a surface within a gravitational field. The amount of space available for the gases to expand into increases exponentially with height and so the conversion of the molecular attractive forces to PE instead of KE becomes a substantial factor.

• Stephen Wilde says:6th July 2015 at 10:52pm
“How does one get a density / pressure gradient without gravity ?”
For any gas, a centrifuge will do nicely!
“The KE/PE conversion of internal energy for gases moving up and down within a gravitational field is a matter of established science.”
That is but a fantasy of meteorological religion according to temperature. Such has no scientific basis. Show any!! This atmosphere is not even a gas. It is always a composite of gas; and airborne liquids, solids and colloids. All except gas pay no attention to any gas law. By adjusting the pressure/density, gravitational force can spontaneously adjust for any weight/volume local differences. There is no other expression of gravitational force in this atmosphere.
This demonstrates the meteorologists have not a clue to their religious spouting!

• “Meteorology is all about the way the rules of physics play out in a real atmosphere, often in a counterintuitive fashion.”
The only useful part is weather forecasting, as poor as that may be.
The skill to select the deterministic, from the chaotic and statistical is amazing. To be able to watch a doppler radar display and decide when to give a recommendation to dive for cover NOW!, is worth all that the public pay for it.
It is the academic religion of meteorology, that fantasizes reasons for that tornado, and all weather, that has no value, scientifically or publicly.

7. SC,
I’ve been arguing these very points for several years and have a new article that may be of interest to you.
Please email me and I’ll send it to you.

8. SS, you said this:
“not only is the parcel of air we are considering doing work against other parcels, but other parcels are doing work against the parcel we are considering”
That isn’t actually correct for an adiabatic process. Such work would be part of a diabatic energy exchange between the parcel and the surroundings with heat moving in or out.
Since the density and pressure of a rising or falling parcel changes at the same rate as the density and pressure of the surroundings no work needs to be done by the rising or falling molecules on the surrounding molecules.
Instead, kinetic energy is transformed to potential energy in ascent and the opposite in descent.
A rising parcel simply expands into the additional space made available to it by the fall in density and pressure of the surroundings at the higher levels.
A falling parcel simply contracts into the reduced space made available to it by the rise in density and pressure of the surroundings at the lower levels.
For a purely adiabatic convective process all the work done is done with or against gravity alone hence the transformation of energy between KE and PE instead of an exchange of heat or energy with the surrounding molecules.
That is a point that everyone seems to have overlooked.

• Scottish-Sceptic says:

I like your description, but there’s a lot of complexity hidden beneath the description.

• “Instead, kinetic energy is transformed to potential energy in ascent and the opposite in descent.”
Just what kind if potential energy of this atmospheric mass is gained in ascent? Gravitational potential of air mass is always cancelled by the underlying atmospheric pressure. Pressure potential decreases with altitude. Just how are your (packets) prevented from exchanging energy with the continuum and space? How is some non existent PE be transformed back to so called KE on descent. How can this ever be “assumed” to be adiabatic when it is clearly not? The continuum continuously transfers energy to space that has no place in thermodynamics or hydrodynamics, but always a loss of energy that you frantically wish to conserve. Yet you fail to even acknowledge that you do not know! Your religious fantasy makes assumptions of no value about this atmosphere that are intended to confuse folk into thinking you have the knowledge of God! Please stop trying to conserve energy within a gravitational field yet the open to insolation and space. Such a concept was formally falsified by Emmy Noether in 1917.
“For a purely adiabatic convective process all the work done is done with or against gravity alone hence the transformation of energy between KE and PE instead of an exchange of heat or energy with the surrounding molecules. That is a point that everyone seems to have overlooked.”
We have all carefully looked and have decided that no air mass movement within the Earth’s atmosphere can ever be adiabatic! This claimed illusion has only the intent to confuse. Where is the evidence that such motion is ever adiabatic

9. Stephen Wilde says: 6th July 2015 at 10:13 pm
“SS, you said this:”
(“not only is the parcel of air we are considering doing work against other parcels, but other parcels are doing work against the parcel we are considering”)
“That isn’t actually correct for an adiabatic process. Such work would be part of a diabatic energy exchange between the parcel and the surroundings with heat moving in or out.”
Please identify any of your illusionary parcel. such a parcel must be a contradiction to all continuum theory that actually describes this compressible non gas fluid that surrounds this Earth!