# Polytropes

## Atmospheric structure

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 polytropic exponent

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:

P ∝ ργ .

For adiabatic expansion or compression, the exponent γ turns out to be the ratio of specific heats, cp/cv.

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.

## Lapse rate

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:

dP ∝ ρ dh       (1).

(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:

P ∝ ρT       (2).

(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:

P ∝ ργ       (3).

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

dP ∝ ρ(γ−1)       (4)

so

ρ dh ∝ ρ(γ−1)       (5)

or

dh ∝ ρ(γ−2) .       (6)

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):

ρT ∝ ργ       (7)

or

T ∝ ρ(γ−1).       (8)

Now differentiate this to get dT:

dT ∝ ρ(γ−2).       (9)

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

dT/dh ∝ 1       (10)

because they both contain ρ(γ−2), which cancels out — in other words, dT/dh is a constant. The polytropic atmosphere has a constant lapse rate.

## Polytropic models

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.

## Real-world complications

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.

## Emden and the polytropic index

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

n  =  1/(γ−1) .

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.

## Useful results

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.