GENERALIZING THE FORMULA

In the present section three generalization of the barometric formula are presented that allow to amplify the range of applicability of the formula.

2.4.1 THE ATMOSPHERE IS NOT ISOTHERMAL

Temperatures throughout the troposphere, the lowest layer of Earth’s atmosphere, cover a wide range of values. This layer is heated from below, and it is wannest at the bottom near Earth’s surface (average temperature = 15°C), and coldest at its top (-57°C), and, typically, the temperature drops about 6.5°C per km (= p) increase in altitude. The troposphere extends from Earth’s surface up to a height of 6 km (in polar regions in winter) to 17 km (middle latitudes) above sea level, the average altitude being 13 km. Most of the mass (about 75%-80%) of the atmosphere is in this region, and almost all weather occurs within it. The sunlight heats the Earth’s surface that radiates the heat back into the adjacent atmosphere. Temperatures in the troposphere, both at the surface and at various altitudes, do vaiy based on latitude, season, time of day or night, regional weather conditions, and in some circumstances, the temperature at the top of the troposphere can be as low as -80°C. The phenomenon known as “temperature inversion” means that the temperature in some part of the troposphere gets wanner with increasing altitude, contrary to the normal situation.³² Consider the case of uniform gravitational field, g, with a vertical temperature gradient, where the temperature changes linearly with height, as in the following eq (2.14), which is a good approximation for the troposphere.

Here p [K/km] is a positive constant. Now, by the aid of eq (2.3) or (2.5) we obtain,

Equation (2.15) is a good model for the pressure dependence on altitude up to 11 km with: g₀ = 9.8 m s-²,/>₀ = 10³ Pa, T_Q = 288 К (15°C), and p = 6.5 K/km.

2.4.2 NONCONSTANT G

This constant varies with latitude, longitude, and elevation. These variations are rather small and lead to a minor error. The general case for an isothermal atmosphere where the acceleration g depends on altitude z, is given by the following equation.

According to the law of gravitation, and noting that the mass of the atmosphere is quite small compared to Earth’s mass we have,

Here, M = mass of the Earth, and at the Earth’s surface: g₀ = GM/R². Now, inserting eq (2.17) into eq (2.16), we have eq (2.18) that integrates into eq (2.19),

Now, with 1 IH=mgJkT we obtain,

Calculations show that a noticeable difference between data obtained from eqs (2.1) and (2.20) shows up only for г > 0.017? ~ 64 km (~ 8%, T_Q = 288K), that is, well above the stratosphere. Following eq (2.20) pressure approaches a nonzero value for г —» oo, which is physically questionable. This aspect shows that a static and isothermal atmosphere is intrinsically unstable. A deeper analysis of this problem is discussed in Ref. 1.

2.4.3 EARTH ROTATION

Suppose that Earth atmosphere and solid Earth rotate with a unique angular velocity со (for the troposphere this is quite appropriate), this means that due to this rotation, the weight of gas is not the same at the pole than at the equator. At the pole, acceleration of gravity is described by eq (2.17), that is, g(z) = g₀(+:/R)-². At the equator, instead, the weight is decreased by the centrifugal force ?nco²(R +c), that is, the effective acceleration is smaller than the acceleration of gravity and obeys eq (2.21), where Ф(г) is given by eq (2.22). Assuming, со = 7.27-10⁵ rad/s, g_g = 9.8 m s², and R = 6.4-10⁶ m, we have co²R/ g₀ ~ 3.45-10'³. It is clear from eq (2.21) that the centrifugal force becomes important (change in gravity acceleration > 1%) for (:) >0.01, that is, for :/R > 0.43, which leads to a large г = 2700 km.

If it is assumed that atmosphere and Earth rotate as a whole, irrespective of height, the upper limit of the atmosphere at the equator can be obtained from eq (2.21) by setting g(z) = 0. Then, Ф(:) = 0 —*■ :/R = 5.6, and the result is a meaningless z = 36,000 km. In fact, the density of outer space is attained for altitudes lower than 1000 km. Escape of molecules, atoms, and ions from the upper atmosphere occurs by thermal and photochemical mechanisms still in the presence of a significant inward force. Combining eqs (2.21) and (2.16), and reminding that Я¹ = mgJkT we obtain for the barometric equation in the case of an isothermal atmosphere,

Integrating this expression we finally obtain,

Calculations show a noticeable difference (~ 2%) between data obtained from eqs (2.24), and (2.20) for r > 0.017? « 64 km, that is, well above the stratosphere. The major problem in applying the barometric formula to the real atmosphere, however, derives from the fact that the atmosphere is not in equilibrium.

PRESSURE INSIDE THE EARTH

Suppose a shaft is drilled down to the center of the Earth, notwithstanding the technical impossibility of this feat, namely, owing to the iimnense pressures and temperatures that exist inside the Earth, and to the physical state of its inner layers, it is interesting to imagine what would be the depth dependence of air pressure within this imaginary shaft. Let us first have a short excursus throughout history about this topic (a more detailed excursus with appropriate references see Ref. 2).

2.5.1 A BIT OF HISTORY

The motion of an object (neglecting drag) dropped in a bottomless shaft was again considered by Hooke in 1679. The main point under discussion was the effect of Earth’s rotation on the trajectory. Hooke obtained the correct result qualitatively: The object should oscillate like a pendulum, describing an ellipse, hi fact, an object dropped in a shaft connecting the poles of a homogeneous and spherical Earth behaves as a one-dimensional harmonic oscillator and strictly obeys Hooke’s “law,” although this is not the present standard pedagogical example. In 1882 the respected French civil engineer and applied mathematician Ё. Collignon (1831-1897) speculated on the possibility of travel between cities by means of long linear tunnels inside the Earth, in a kind of partial free-fall planetary subway, for which the transit time in the absence of drag is 42 min, independently of the location of the two cities. An account of his ideas, published on a semi-humorous tone in the scientific periodical La Nature, is suggestively entitled “From Paris to Rio de Janeiro in 42 minutes and 11 seconds” In it, the effect of pressure is discussed, and it is considered an insurmountable problem. Numerical estimates of the enormous pressures at several depths (but provided with no computational details) are given, but differ from the calculations given below by several orders of magnitude.

2.5.2 THE MATHEMATICS

Assuming for simplicity that air temperature and Earth’s density are both uniform, eq (2.18) applies, where the acceleration of gravity now is,

Equation (2.18), with -R < : < 0, and with eq (2.25) now, after integration, becomes (reminding that, g₀ = GM/R^:, and 1 /Н = mg₀/kT₀),

As the reader can notice the dependence is similar to eq (2.1), apart from the multiplicative factor (<1) in the argument of the exponential that slightly reduces the variation, owing to the decrease of g with depth. The deepest gold mines in South Africa attain a depth of 3.9 km, ca., for which one obtains p = 1.6 atm, in good agreement with the observations. While eq (2.26) predicts a pressure of 1000 atm for : = -58 km, for г = -R, instead, the calculated pressure becomes: p(-R) = p₀exp (R/2H) = 10¹⁶⁵p₀, clearly, a meaningless value, as the air for pressures of few tons of atm ceases to behave as an ideal gas. A more detailed calculation with the van der Waals equation is discussed in Ref. 2.

LAST REMARKS

Whenever we take a formula that is valid only for an ideal gas under conditions of equilibrium and apply it to real cases, it is not a surprise if it does not fit the data perfectly. Nevertheless, pressure seldom departs from the average value by more than a few percent, and within this restriction the barometric equation given by eq (2.1) does its job. The nonideality of a gas has instead a much more dramatic influence on the barometric equation at negative heights.

It should be remarked that the atmosphere as a whole is never in a state of equilibrium, as it continuously exchanges mass and energy with its surroundings and this is why there is weather. To better understand this last topic, we should know something about the Bernoulli’s principle of fluid dynamics³³ that is though valid only for ideal fluids. It states that an increase in the speed of a fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid’s potential energy, it can be applied to various types of fluid flow, and there are different forms of Bernoulli’s equation for different types of flow. In Ref. 2 and 5 a full discussion of this principle is given.

KEYWORDS

• • barometric formula
• • pressure
• • mathematics
• • generalization
• • ideal gas

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