As explained on the hydrostatic equilibrium page, the problem in integrating the hydrostatic equation is that the equation of state for air involves both the pressure and the density (or, if you prefer, the pressure and the temperature), so that an additional variable is thrown into the problem. To get rid of this new variable, some additional information is needed.

It's convenient to “parameterize our ignorance” here by
*assuming* some convenient (read: easy-to-integrate) relationship.
The polytropic assumption is one way of doing this.
It's just a convenient, arbitrary assumption; but its free parameter
provides enough flexibility to represent reality reasonably well.

The idea of polytropes arose historically from considerations
of fully convective gases.
In this picture, the gas is “turned over” repeatedly by convection;
hence the term *polytropic*.
Convection is assumed to be rapid enough
that the rising and falling parcels of gas don't have enough time to
exchange appreciable amounts of heat with their surroundings, so their
changes are adiabatic. This condition leads to a power-law relation
between the pressure and the density:

For adiabatic expansion or compression, the exponent **γ**
turns out to be the ratio of specific heats, c_{p}/c_{v}.

However, in general we are *not* dealing with free convection and adiabatic
processes, and the exponent **γ** has no physical significance.
It's just a “fudge factor” — an adjustable parameter, whose
numerical value is chosen to approximate actual conditions.

One of the most useful properties of polytropic atmospheres is their lapse rate — that is, the temperature gradient. Here's a derivation to show the most interesting feature.

Let's begin with the hydrostatic equation:

(Here, the proportionality constant would be **−g**, but it will be
omitted to avoid distracting attention from the important functional
dependences.)

Next, there's the ideal-gas equation of state:

(Here, the omitted proportionality constant is the gas constant.)

As mentioned on the hydrostatic-equilibrium page, the problem is that we have more variables than equations. The hydrostatic equation involves height, pressure, and density; the equation of state introduces temperature as well, so we're no better off. We need another relation to remove the indeterminacy, and the polytropic assumption does the trick:

We can differentiate (3) to get a **dP,** which can be equated to eq. (1),
thus eliminating **P**:

so

or

This gives us **dh** as a function of just **ρ**. Now we'll play some
games with the equation of state to get **dT** as a function of **ρ**
alone, and divide that result by (6) to get **dT/dh**.

First, equate the expressions for **P** in (2) and (3):

or

Now differentiate this to get **dT**:

Finally, divide (9) by (6) to get

because they both contain **ρ ^{(γ−2)}**,
which cancels out
— in other words,

We don't know what the actual structure of the atmosphere is at a given place and time. However, there is a vast body of empirical data, obtained originally from instrumented kites and balloons, as well as from observations at mountain stations. Even in the early 19th Century, it was clear that the lower atmosphere becomes cooler with height, at a fairly steady rate (called the “temperature lapse rate” originally, but usually shortened to just the “lapse rate” today.)

The simplest assumption is that the lapse rate is constant. This is, in
fact, entirely arbitrary; but it is a fair approximation to the *average*
state of the lower atmosphere.
The constant lapse rate of polytropes makes them attractive candidates for
model atmospheres.

One such model corresponds to the dry adiabat; that is, a freely convecting atmosphere without condensation. However, only a very limited portion of the real atmosphere ever is in such a state, because strong convection is usually driven by the latent heat released in clouds, and the wet adiabatic atmosphere is not polytropic.

It should be clear that adopting a particular polytrope is a completely arbitrary assumption. There's no physics in it; but this mathematical form makes the integration of the hydrostatic equation possible. The polytropic index is just a “fudge factor.”

On the other hand, atmospheric scientists find piecewise-linear
temperature profiles convenient to use. In fact, the
Standard Atmosphere
is a sequence of such pieces; that is, it is a *piecewise-polytropic*
model atmosphere.
And if you use enough pieces, you can approximate any realistic
temperature profile arbitrarily closely.

However, the drawback is that real temperature profiles must be *smooth*
as well as continuous, because of heat transfer.
This means that “enough”
pieces may be very many indeed, depending on how much accuracy is
required. “Enough” may turn out to be thousands, or even tens of
thousands.

Fortunately, computers allow us to handle even these complicated situations. So we really can approximate realistic temperature profiles very accurately.

The idea of polytropes was established by Robert Emden in 1907, though such
models had actually been used much earlier. In 1923,
**Emden**
applied the idea to refraction in the Earth's atmosphere, and pointed out
earlier examples.

Emden introduced a new parameter to characterize polytropes. Instead of
the exponent **γ** he used the **polytropic index n**, where

The polytropic index can, in principle, be any number. However, polytropes with index less than about 2.5 are convectively unstable, and so cannot represent any appreciable portion of the real atmosphere. A polytropic index of zero corresponds to Cassini's uniform atmosphere, which is extremely unstable against convection. A polytropic index of infinity corresponds to an isothermal atmosphere, stably stratified. Regions in which the lapse rate is inverted have negative values of the polytropic index. The average lapse rate in the troposphere (6.5 K/km) corresponds to a polytropic index of about 4.26.

Because a polytropic atmosphere has a constant lapse rate, the temperature
decreases monotonically with height, together with the pressure and
density. At the top of the atmosphere, all these state variables reach
zero. This limiting height turns out to be **(n+1)** times the
height of the homogeneous atmosphere, because the polytropic lapse rate is
just (**γ**−1)/**γ** × **T/H,** if
**T** is the temperature at the surface, and **H** is the
height of the homogeneous atmosphere.

As the height of the homogeneous atmosphere is about 8 km, these limiting
heights were an embarrassment to the people who calculated refraction in
the 19th Century. They knew, from observations of
aurorae, twilight, and meteors, that the atmosphere extended well above
the height of about 40 km calculated from models with fixed lapse rates
that gave the observed refraction at the horizon. However,
**Ivory**
noticed that the lapse rate could be flattened out in the
upper atmosphere (thus making it more nearly isothermal, i.e., more nearly
infinite in extent) without markedly affecting the calculated refraction.
Such efforts were the first hints of the existence of the (nearly
isothermal) stratosphere.

Copyright © 2003 – 2007 Andrew T. Young

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