Lapse Rate

As we have already seen, as one rises to higher altitude, atmospheric pressure decreases. If a parcel of air rises, because of the reduced pressure the parcel will expand. It generally takes much longer for a parcel of air to absorb/emit heat from/to its surroundings than to expand/contract, so during its expansion it will, for all practical purposes, exchange no heat with its surroundings; in other words, the expansion of the parcel of air will be adiabatic.

We’ve already derived the change in temperature when a parcel of air expands or contracts adiabatically to a new pressure :

.

This is for ordinary air, which is almost entirely made up of the diatomic gases nitrogen N₂ and oxygen O₂ Recall that in thermodynamics we must use a temperature scale where the zero point is at a temperature absolute zero; the universal choice is to measure temperature in degrees Kelvin, usually called simply “Kelvins” and signified simply by “K”. The relationship between temperature and pressure given by this formula is called the dry adiabat.

When pressure goes down (as it does with altitude), the adiabatic expansion causes the temperature to go down; the air parcel cools. So we expect that atmospheric temperature will decrease with height. This turns out to be the case, until one gets to an altitude which is the boundary between the two primary layers of earth’s atmosphere, the troposphere and the stratosphere. This boundary is called the tropopause; its height is different at different latitudes, about 8 km (5 miles) at the poles and roughly 18 km (11 miles) at the equator. The pressure at the tropopause is about 100 mb (millibars), whereas at the surface it’s 1013 mb (roughly 1000 mb). Once one gets into the stratosphere, temperature increases with height. By far most of our atmosphere is in the troposphere; the stratosphere is very thin by comparison.

Now suppose that a parcel of air rises to a slightly higher altitude at which the pressure is , and expands adiabatically; then its temperature will be determined by adiabatic expansion. If the ambient temperature at that height is lower than this, then the parcel of air will be hotter than its surroundings, so it will be bouyant (hot air rises!) and will rise even further. In this case, the atmosphere is unstable against small vertical movements of air parcels. If, on the other hand, the ambient temperature is higher, than the air parcel will be colder than its surrounding and will tend to sink (cold air sinks). In this case, the natural forces of the atmosphere will tend to restore the air parcel to its previous altitude, so the atmosphere is stable against small vertical movements of air parcels.

The upshot is that in order for the atmosphere to be stable against vertical movements, the temperature as a function of height must be as great, or greater than, that given by adiabatic expansion. If temperature at altitude is cooler, small vertical movements will be magnified and the air will be in constant turmoil, but if temperature at altitude is hotter than that given by adiabatic expansion, it at least can be stable. It can still show movements, but it will not necessarily be in constant turmoil.

In fact, as a parcel of air rises or falls we expect its temperature to change according to adiabatic expansion. So, as a first approximation at least, we might expect the dry adiabat to give us the change in atmospheric temperature as a function of height. Also, if the temperature is very much hotter than adiabatic as height increases, the atmosphere will be too stable against vertical movements, so such movements will be strongly damped and almost never observed, when in fact we do note vertical movements of air. Since the atmosphere is reasonably stable against vertical displacement (it’s not in constant turmoil), but not so stable as to make them rare, we can conclude that the temperature profile must be close to that given by adiabatic expansion.

We can use our formula for the dry adiabat to compute what the temperature at any altitude is according to this idea, if we know the “starting” temperature (the temperature at the surface). Using a surface temperature of 300 K, I will follow tradition by expressing it not as a function of height but as a function of pressure (which is a perfectly good “proxy” for height). I will also use a logarithmic scale for pressure (which will be a roughly linear scale for height):

We note that according to this formula, by the time we get to the tropopause (at a pressure of about 100 mb), the temperature will have dropped to about 155 K. Here’s the observed temperature-vs-pressure graph at a point over the midlatitude Pacific ocean:

We see that the actual temperature profile is considerably warmer than the dry adiabat. In fact by the time we get to the tropopause at a pressure of 100 mb, the temperature has not dropped to 155 K, but is considerably warmer than that, about 190 K.

How can this be? If the temperature profile is that much hotter at altitude than given by adiabatic expansion, should it not be over-stable against vertical displacements, and therefore strongly stratified? The answer is that we have left one factor out of our expression for adiabatic expansion: water vapor.

As a parcel of air rises, it expands and cools. Cooler air cannot hold as much water vapor as warmer air, so if there’s a significant amount of moisture in the air, some of it will condense to form water droplets. When gaseous water condenses to liquid, it releases heat: the latent heat of vaporization. This released heat warms our air parcel, partly mitigating the cooling usually associated with expansion.

The air parcel can still be treated (for all practical purposes) as though it exchanges no heat with its surroundings, so the expansion is still adiabatic. But the latent heat of condensation comes from the water vapor in the air parcel itself. So, the cooling which a moist air parcel experiences as it expands is less than that which a dry air parcel experiences when it expands. Instead of temperature-vs-height following the dry adiabat, it tends to follow the wet adiabat, and this is what is actually observed in earth’s atmosphere.

The lapse rate has an important consequence for global warming. At the wavelengths at which CO2 absorbs and emits infrared radiation, the atmosphere is reasonably opaque to this radiation. The infrared in this wavelength region which actually escapes to space (and therefore cools the earth) comes from higher in the atmosphere, where it has a realistic chance to escape the planet.

As we add more CO2 to the atmosphere, it becomes optically thicker for those wavelengths of infrared. This means that to escape the atmosphere, it must be radiated from ever-higher altitudes. The temperature at the altitude at which the infrared actually escapes to space is determined by the incoming energy from the sun, and so does not change as we add more CO2 to the atmosphere. But the temperature at the surface will be higher than the temperature of this high-altitude air, by an amount which is governed by the lapse rate (which doesn’t change much) and by the height of that radiating layer.

So, even if the atmosphere has so much CO2 that it is saturated in those infrared wavelengths, adding more CO2 raises the altitude at which those wavelengths of infrared escape to space, increasing the distance to the ground, and therefore increasing temperature difference between the radiation layer and the ground (which is the product “lapse rate” x “distance”).

That’s one of the reasons that arguments that CO2 absorption of infrared is saturated, and hence adding more CO2 won’t increase global warming, are mistaken. Because of the lapse rate, raising the altitude of CO2 radiation escaping to space will still warm the surface.

The other reason such arguments are mistaken is that CO2 absorption of infrared is not saturated. Then absorption bands are near-saturated, but at their very edges are not quite so. Hence adding more CO2 to the atmosphere really does increase the net absorption of surface-infrared by CO2.