Imagine a packet of air which is being swirled around in the atmosphere. We would
expect it to always remain at the same pressure as its surroundings, otherwise it
would be mechanically unstable. It is also plausible that the packet moves around
too quickly to effectively exchange heat with its surroundings, since
air is very a poor heat conductor, and heat flow is consequently quite a
slow process. So,
to a first approximation, the air in the packet is *adiabatic*.
In a *steady-state* atmosphere, we expect that as the packet moves upwards,
expands due to the reduced pressure, and cools adiabatically, its temperature
always remains the same as that of its immediate surroundings.
This means that we
can use the adiabatic gas law to characterize the cooling of the
atmosphere with increasing altitude. In this particular
case, the most useful manifestation of the adiabatic law is

(329) |

(330) |

(331) |

Now, the ratio of specific heats for air (which is effectively a diatomic gas) is about 1.4 (see Tab. 2). Hence, we can calculate, from the above expression, that the temperature of the atmosphere decreases with increasing height at a constant rate of centigrade per kilometer. This value is called the

According to the adiabatic lapse rate calculated above, the air temperature at
the cruising altitude of airliners ( feet) should be about
centigrade (assuming a sea level temperature of centigrade).
In fact, this is somewhat of an underestimate. A more realistic value is about
centigrade.
The explanation for this
discrepancy is the presence of
water vapour in the atmosphere. As air rises, expands, and cools, water
vapour condenses out releasing latent heat which prevents the temperature
from falling as rapidly with height as the adiabatic lapse rate would indicate.
In fact, in the Tropics, where the humidity is very high, the lapse rate of
the atmosphere (*i.e.*, the rate of decrease of temperature with altitude)
is significantly less than the adiabatic value. The adiabatic
lapse rate is only observed when the humidity is low. This is the case in deserts,
in the Arctic (where water vapour is frozen out of the atmosphere), and, of course,
in ski resorts.

Suppose that the lapse rate of the atmosphere differs from the adiabatic value.
Let us ignore the complication of water vapour and assume that the atmosphere
is dry. Consider a packet of air which moves slightly upwards
from its equilibrium height. The temperature of the packet will
decrease with altitude according to the adiabatic lapse rate, because its
expansion is adiabatic. We assume that the packet always maintains pressure
balance with its surroundings. It follows that since
,
according to the ideal gas law, then

(333) |

Let us consider the temperature, pressure, and density profiles in an
adiabatic atmosphere. We can directly integrate Eq. (332) to
give

(335) |

Consider the limit . In this limit, Eq. (334) yields independent of height (

(337) |

which, not surprisingly, is the predicted pressure variation in an isothermal atmosphere. In reality, the ratio of specific heats of the atmosphere is not unity, it is about 1.4 (

(339) |

(340) |

Note that an adiabatic atmosphere has a sharp upper boundary. Above height
the temperature, pressure, and density are
all zero: *i.e.*, there is no atmosphere. For real air, with ,
kilometers. This behaviour is quite different
to that of an isothermal atmosphere, which has a diffuse upper boundary. In reality,
there is no sharp upper boundary to the atmosphere. The adiabatic gas law
does not apply above about 20 kilometers (*i.e.*, in the *stratosphere*) because
at these altitudes the air is no longer strongly mixed. Thus, in the stratosphere
the pressure falls off exponentially with increasing height.

In conclusion, we have demonstrated that the temperature
of the lower atmosphere should fall off
approximately *linearly* with increasing height above ground level, whilst the
pressure should fall off far more rapidly than this, and the density should
fall off at some intermediate rate. We have also shown that the
lapse rate of the temperature should be about centigrade per kilometer
in dry air, but somewhat
less than this in wet air. In fact, all off these predictions
are, more or less, correct. It is amazing that such accurate predictions can
be obtained from the two simple laws, constant for an isothermal gas, and
constant for an adiabatic gas.