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    The adiabatic atmosphere

Of course, we know that the atmosphere is not isothermal. In fact, air
temperature falls quite noticeably with increasing altitude. In ski
resorts, you are told to expect the temperature to drop by about 1
degree per 100 meters you go upwards. Many people cannot understand why
the atmosphere gets colder the higher up you go. They reason that as
higher altitudes are closer to the Sun they ought to be hotter. In fact,
the explanation is quite simple. It depends on three important
properties of air. The first important property is that air is
transparent to most, but by no means all, of the electromagnetic
spectrum. In particular, most infrared radiation, which carries heat
energy, passes straight through the lower atmosphere and heats the
ground. In other words, the lower atmosphere is heated from below, not
from above. The second important property of air is that it is
constantly in motion. In fact, the lower 20 kilometers of the atmosphere
(the so called /troposphere/) are fairly thoroughly mixed. You might
think that this would imply that the atmosphere is isothermal. However,
this is not the case because of the final important properly of air:
/i.e./, it is a very poor conductor of heat. This, of course, is why
woolly sweaters work: they trap a layer of air close to the body, and
because air is such a poor conductor of heat you stay warm.

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

\begin{displaymath} p^{\,1-\gamma} \,T^{\,\gamma} = {\rm constant},
\end{displaymath} 	(329)


giving
\begin{displaymath} \frac{dp}{p} = \frac{\gamma}{\gamma -1}
\frac{dT}{T}. \end{displaymath} 	(330)


Combining the above relation with the equation of hydrostatic
equilibrium, (326 <node54.html#e6.66>), we obtain
\begin{displaymath} \frac{\gamma}{\gamma-1}\frac{dT}{T} = -\frac{\mu
\,g}{R\,T} \,dz, \end{displaymath} 	(331)


or
\begin{displaymath} \frac{dT}{dz} = -\frac{\gamma -1}{\gamma}
\frac{\mu\, g}{R}. \end{displaymath} 	(332)


Now, the ratio of specific heats for air (which is effectively a
diatomic gas) is about 1.4 (see Tab. 2 <node51.html#tab1>). Hence, we
can calculate, from the above expression, that the temperature of the
atmosphere decreases with increasing height at a constant rate of
$9.8^\circ$ centigrade per kilometer. This value is called the
/adiabatic lapse rate/ of the atmosphere. Our calculation accords well
with the ``$1$ degree colder per 100 meters higher'' rule of thumb used
in ski resorts. The basic reason why air is colder at higher altitudes
is that it expands as its pressure decreases with height. It, therefore,
does work on its environment, without absorbing any heat (because of its
low thermal conductivity), so its internal energy, and, hence, its
temperature decreases.

According to the adiabatic lapse rate calculated above, the air
temperature at the cruising altitude of airliners ($32,000$ feet) should
be about $-80^\circ$ centigrade (assuming a sea level temperature of
$15^\circ$ centigrade). In fact, this is somewhat of an underestimate. A
more realistic value is about $-60^\circ$ 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 $\rho\, T
\propto p$, according to the ideal gas law, then

\begin{displaymath} (\rho\, T)_{\rm packet} = (\rho \,T)_{\rm
atmosphere}. \end{displaymath} 	(333)


If the atmospheric lapse rate is less than the adiabatic value then
$T_{\rm atmosphere} > T_{\rm packet}$ implying that $\rho_{\rm packet} >
\rho_{\rm atmosphere}$. So, the packet will be denser than its immediate
surroundings, and will, therefore, tend to fall back to its original
height. Clearly, an atmosphere whose lapse rate is less than the
adiabatic value is /stable/. On the other hand, if the atmospheric lapse
rate exceeds the adiabatic value then, after rising a little way, the
packet will be less dense than its immediate surroundings, and will,
therefore, continue to rise due to buoyancy effects. Clearly, an
atmosphere whose lapse rate is greater than the adiabatic value is
/unstable/. This effect is of great importance in Meteorology. The
normal stable state of the atmosphere is for the lapse rate to be
slightly less than the adiabatic value. Occasionally, however, the lapse
rate exceeds the adiabatic value, and this is always associated with
extremely disturbed weather patterns.

Let us consider the temperature, pressure, and density profiles in an
adiabatic atmosphere. We can directly integrate Eq. (332 <#e6.73>) to give

\begin{displaymath} T = T_0 \left(1 - \frac{\gamma -1}{\gamma}
\frac{z}{z_0} \right), \end{displaymath} 	(334)


where $T_0$ is the ground level temperature, and
\begin{displaymath} z_0 = \frac{R \,T_0}{\mu\, g} \end{displaymath} 	(335)


is the isothermal scale-height calculated using this temperature. The
pressure profile is easily calculated from the adiabatic gas law
$p^{\,1-\gamma} \,T^{\,\gamma} =$ constant, or $p \propto
T^{\,\gamma/(\gamma -1)}$. It follows that
\begin{displaymath} p= p_0 \left(1 - \frac{\gamma -1}{\gamma}
\frac{z}{z_0} \right)^{\gamma/(\gamma-1)}. \end{displaymath} 	(336)


Consider the limit $\gamma\rightarrow 1$. In this limit, Eq. (334
<#e6.75>) yields $T$ independent of height (/i.e./, the atmosphere
becomes isothermal). We can evaluate Eq. (336 <#e6.77>) in the limit as
$\gamma\rightarrow 1$ using the mathematical identity
\begin{displaymath} ~_{ {\rm lt}\, m\rightarrow 0} \left(1 +
m\,x\right)^{1/m} \equiv \exp(x). \end{displaymath} 	(337)


We obtain
\begin{displaymath} p = p_0\exp\left(-\frac{z}{z_0}\right),
\end{displaymath} 	(338)


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 (/i.e./, the ratio for diatomic
gases), which implies that in the real atmosphere
\begin{displaymath} p= p_0 \left(1 - \frac{z}{3.5\,z_0} \right)^{3.5}.
\end{displaymath} 	(339)


In fact, this formula gives very similar results to the exponential
formula, Eq. (338 <#e6.79>), for heights below one scale-height (/i.e./,
$z<z_0$). For heights above one scale-height, the exponential formula
tends to predict too low a pressure. So, in an adiabatic atmosphere, the
pressure falls off less quickly with altitude than in an isothermal
atmosphere, but this effect is only really noticeable at pressures
significantly below one atmosphere. In fact, the isothermal formula is a
pretty good approximation below altitudes of about 10 kilometers. Since
$\rho\propto p/T$, the variation of density with height goes like
\begin{displaymath} \rho= \rho_0 \left(1 - \frac{\gamma -1}{\gamma}
\frac{z}{z_0} \right)^{1/(\gamma-1)}, \end{displaymath} 	(340)


where $\rho_0$ is the density at ground level. Thus, the density falls
off more rapidly with altitude than the temperature, but less rapidly
than the pressure.

Note that an adiabatic atmosphere has a sharp upper boundary. Above
height $z_1 = [\gamma/(\gamma-1)]\,z_0$ the temperature, pressure, and
density are all zero: /i.e./, there is no atmosphere. For real air, with
$\gamma=1.4$, $z_1 \simeq 3.5\,z_0 \simeq 29.4$ 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 $10^\circ$ 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, $p\,V =$ constant for an isothermal
gas, and $p\,V^{\,\gamma}=$ constant for an adiabatic gas.

------------------------------------------------------------------------
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*Next:* Heat engines <node57.html> *Up:* Classical thermodynamics
<node48.html> *Previous:* The isothermal atmosphere <node55.html>
Richard Fitzpatrick 2006-02-02