# Lapse Rate

Gases are governed by the rule that:

PV = nRT

So, this would suggest that when pressure (P) is reduced, so is temperature (T). However this is only true of a fixed volume (V). But when air rises, it expands, so both its pressure goes down and its volume goes up. (n & R are constants)

### 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!

### 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 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 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 lapse rate) and so it will continue upward, either until it encounters a layer that has less energy than the 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 lapse rate and the lower levels to cool to be lower than the lapse rate. When this happens, the lower air becomes denser than the lapse rate and less likely to rise and conversely the upper air becomes less dense than the lapse rate 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 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  lapse rate breaks down and the IR heat transfer breaks the 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 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 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.