As noted on the introductory polytropes page, Robert Emden introduced the word “polytrope” — first (1907) in the context of stellar structure, and later (1916) in atmospheric physics.
In 1923, he finally applied his model to astronomical refraction in two papers. The one in Meteorologische Zeitschrift seems to be a shortened version of the full paper, which appeared in Astronomische Nachrichten.
I'll summarize that work here, using a slightly simpler notation than Emden's. It's similar to the treatment on the previous page, but includes the proportionality constants, so as to provide actual numbers.
Instead of developing the theory in terms of the ratio of specific heats, which is the natural parameter to use in problems of convection, Emden used the polytropic index n, which let him avoid the restriction to adiabatic convection, and made the theory more general.
A nice feature of his 1923 papers is to simplify all the relations by expressing the actual values of state functions in terms of a characteristic length: the height of the homogeneous atmosphere, $$ \mathbf { H_{\scriptstyle 0} = \frac{R\, T_{\scriptstyle 0}}{m \,g} } $$ where: R is the universal gas constant; T₀ is the temperature at the base of the atmosphere; m is the mean molecular weight; and g is the (assumed constant) acceleration of gravity. [If you've forgotten how this relation is derived, see the details on the hydrostatic equilibrium page.] On pp. 354–355 of the Met. Z. paper, Emden shows that the total height H of the whole polytropic atmosphere of index n is just (n + 1) times this. Notice that if n is zero, (n + 1) is exactly 1. That's the case for the homogeneous atmosphere.
So, in general, the height of a simple polytropic atmosphere is $$ \mathbf { H = ( n + 1 ) \cdot \frac{R\, T_{\scriptstyle 0}}{m \,g} } $$ At this height, the pressure, temperature, and density of the model all become zero. (This is physically impossible, but is a property of such simple mathematical models.)
Emden also shows that the vertical temperature gradient dT/dh is constant. Its magnitude is T₀ divided by the whole height (n + 1)(R T₀)/(m g), and its sign is negative. That's \(\mathbf{-m\,g/[(n+1)\!\cdot\!{R}] }\) . (The temperature cancels out here, as it must, because the gradient is constant, while the temperature varies with height.) The conventional lapse rate Γ is just the negative of this: $$\mathbf {\Gamma = -\, \frac{d T}{d h} = \frac{m \,g}{ (n + 1 ) R } } $$ which of course is independent of height or temperature, if g is constant.
The isothermal atmosphere has \( \mathbf {\Gamma = 0} \), which is possible only if n is infinitely large. Emden showed that the polytropic expressions for large integer values of n are a well-known series that converges to the exponential function, so that the density and pressure decay exponentially with height in this case (neglecting the slight change in g with height). If the variation of g with height is considered, Emden showed in his 1923 Met. Z. paper that a fixed polytropic index corresponds to a constant ratio of lapse rate to g. In modern terms, that means a constant geopotential lapse rate.
In both of the 1923 papers, Emden included a convenient table of astronomical refractions for eight values of n from 1 to infinity. The A. N. paper also has a table of model heights and lapse rates for the same eight polytropic indices. (This is abbreviated from a longer table with 14 values of n in the 1916 Met. Z. paper.) He found that the average atmospheric structure was nearly a polytrope of index 5, for which the total height would be six times the homogeneous height, or about 48 km. (The “Standard Atmospheres” used today all have a tropospheric lapse rate of 6.5°/km, corresponding to a polytropic height of 44.330769230769 km. [The ridiculous number of digits is due to the particular values adopted for the physical constants used.]) The constants adopted by the United States Standard Atmospheres of 1962 and 1976 correspond to n = 4.25587611327849; the more recent (1993) ICAO model has n = 4.25587974247581744045... because it uses a slightly different value for the gas constant of dry air. These values for n differ by less than 1 part per million.
The range of observed lapse rates in the real atmosphere far exceeds the range in Emden's table; in particular, on clear nights at temperate latitudes there is normally at least one thermal inversion in the lowest few hundred meters. Emden was aware of nocturnal inversions, but did not treat them in detail. However, he did point out that a thermal inversion corresponds to a value of n more negative than − 1.
These power-law profiles are the basis for practical calculations in polytropic layers when a real temperature profile is approximated by a series of linear segments, so that each layer has its own polytropic index. The height in every layer is measured from the base of the layer. Such atmospheric models are often called piecewise polytropic.
Because the equation of state is a simple power law, the basic polytropic condition that p is a power of ρ also makes each of them a power of T. For example, it's obvious from the equations above that p is just proportional to \( \mathbf { T^{ (n+1) } } \). If we were to plot p as a function of T on log-log graph paper (as was done in the days before computers), we would get a straight line with a slope of \( \mathbf { (n+1) } \) .
Here's such a graph for just the lowest 1500 meters of the U.S. Standard
Atmosphere. The black line uses the tabulated values at every 50 meters
of height, and the red dashed line is calculated from just the 1500-meter
point and the value at the surface, which is circled. The two lines
coincide perfectly.
So, if we have T and p values for two heights, we can get that slope very simply. Let the data for the two heights be labelled with subscripts 1 and 2. Then the slope of the line is just $$ {\mathbf { n+1 = { \left( \frac{ \log p_{\scriptstyle 1} - \log p_{\scriptstyle 2} } { \log T_{\scriptstyle 1} - \log T_{\scriptstyle 2} } \right) } } } $$ — but of course, \(\mathbf {( \log a - \log b)}\) is just \( \mathbf {\log (a/b)}\). So we can replace two expensive logarithm evaluations with cheaper divisions: $$ {\mathbf { n+1 = { \left( \frac{ \log ({{p_{\scriptstyle 1}}/{p_{\scriptstyle 2}}) } } { \log ({{T_{\scriptstyle 1}}/{T_{\scriptstyle 2}}) } } \right) } } } $$ and we can obviously get n itself by subtracting 1 from that expression. And, as the height of the whole polytrope is just \( \mathbf { (n+1) } \) times the height of the homogeneous atmosphere, once we have that, we have H.
In making the plot, I used the ratios of temperatures and pressures at the upper point to their values at the lower one. But it's immaterial whether point 1 lies above or below point 2. And it also makes no difference whether we choose natural or common (base-10) logarithms, as long as we use the same kind of logs for both numerator and denominator.
If a layer is thin, the ratio of the pressures is close to unity, so its logarithm is close to zero. The same is true for the temperatures; so n isn't very accurately determined. Then the polytropic height H is also uncertain. Partly for this reason, the exact polytropic determination of heights never became standard practice in meteorology or aviation. Unfortunately, the practical need to do something to get roughly realistic heights allowed an empirical procedure to become entrenched, long before better physical (i.e., polytropic) models were available.
So the traditional approach to finding heights from pressures uses a systematically incorrect method that goes back two centuries to Laplace's Mécanique Céleste (1805) — where the temperature was assumed constant. Various artificial and unrealistic modifications of the constants involved, such as the acceleration of gravity and the gas constant, have become fixed in this process, which is described in meteorology textbooks as “the” hypsometric equation.
Laplace's hypsometric equation had been used for more than a century to estimate the heights of mountains, despite its weaknesses, before polytropes were invented. After just a few years of use, it became evident that the calculated heights had both diurnal and seasonal systematic errors that varied from place to place. Today, it's obvious that these effects were the natural result of changes in the lapse rate from day to night and from summer to winter; but in the 19th Century this was not understood.
The physically incorrect assumption of an isothermal atmosphere forced the adoption of a fictitious “mean temperature”. But the mean adopted has always been the simple arithmetic mean, which is physically wrong for any realistic model — as was pointed out by Diehl in 1925. (He favored the harmonic mean.) However, Laplace's isothermal rule was by then too firmly established to be replaced.
The use of adjusted constants to get answers closer to real-world measurements is similar to the procedure used by Simpson (1743), Bradley [as published by Maskelyne in 1763 and 1764], and Tobias Mayer [again, as published posthumously by Maskelyne in 1770, and then explained in more detail by him in 1787], to obtain better refraction tables near the horizon from the assumed linear decrease in density with height. Such artifices were needed before the gas laws were fully understood.
With the lapse rate specified, it's trivial to find the polytropic index, because the lapse rate is just the temperature at the base divided by the total height H of the polytrope; and the height of the polytrope is exactly \( \mathbf { (n+1) } \) times the height of the homogeneous model. So from $$ \mathbf { \Gamma = \frac{ T_{\scriptstyle 0}}{H} } $$ and $$ \mathbf { H = ( n + 1 ) \cdot \frac{R\, T_{\scriptstyle 0}}{m \,g} } $$ we have $$ \mathbf { ( n + 1 ) = \frac{m \,g }{R\, \Gamma} \; }. $$ This is the relation between the lapse rate \( \Gamma \! \) and the polytropic index, n. It does not depend on height, or on conditions at the base of the polytrope. The only other parameters used here are the mean molecular weight and the acceleration of gravity. However, adopting a different value for the gas constant or the molecular weight changes the numerical value of n.
Notice that an isothermal layer has zero lapse rate, and an infinite polytropic index. The large range in actual lapse rates near the ground produces a large range in n, and the systematic variations in conventional barometric hypsometry.
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