Note 1.G The hydrostatic balance

This equates the fall of air pressure Dp as one ascends by Dz (metres) with the factors responsible, thus -

Dp = r.g.Dz Pascals

where r is the air's density (kg/m3) at that level, and g is the acceleration (m/s2) due to gravity, i.e. 9.81m/s2. The equation states that the downwards pressure due to the weight of air above, given by the right-hand side, is matched by the upwards pressure Dp of the compressed air. In the lowest 5km of the atmosphere, the air's density is about 1kg/m3, and so Dp/Dz is 10kg/m2.s2, or 10Pa/m or 0.1hPa/m or 1hPa every 10m in that layer, on average.

If this equation is combined with the Ideal Gas Law (Note 1.M), the result is this -

Dp/p = - g.Dz / (R.T)

where R is 287J/kg.K for air and T is the Kelvin temperature.Integration of that equation yields the following-

ln p/po = - g.z/ (R.T)

where the logarithm is to base e (the exponential constant 2.718), and po is the pressure at sea level. Assuming a uniform air temperature of 288K and a gravitational acceleration of 9.8m/s2, the equation simplifies to this -

ln p/po = - 0.12.zk

where zk is the height in kilometres. At the scale height H (ie 8km), the right-hand side of this equation is unity, which equals ln p/po, which equals g.H/(R.T). Hence H is the height equal to R.T/g metres, and ln p/po equals - z/H. So the altitude is proportional to the logarithm of the pressure. Also, the pressure p at height z equals po times exp (-z/H), where 'exp' refers to the exponential constant equal to 2.7.1828, and exp (-z/H) means 2.71828 multiplied by itself z/H times, and then divided into unity.

hypsometric equation

The following relationship between altitude and pressure (ie hypsometric equation) is a convenient rearrangement of that above, except that the logarithm involves the base 10, instead of the exponential constant -

z = 67.5 .Tm.log10 (po/p) metres

where Tm is the mean temperature (Kelvin) of air between height z and sea level, where the pressure is po. Thus, if p has a value of 700hPa, and Tm is 288K (ie 15oC), the height is 3.1km. The expression shows that the level midway through the atmosphere's mass (where pz is half of 1013hPa) lies at an altitude of 5.5km, assuming the mean temperature Tm to be -3oC. If the temperature were only -5°C (or 168K), the height would be less (2.9 km), showing that colder air is more compact.

Evaluation of the equation for the temperatures at various levels specified in the ICAO standard atmosphere (Section 1.6) gives a pressure difference on rising 8.3 metres at sea level (i.e. at 1013hPa), 9.2m at 1000hPa, 10.2m at 2km, 12.5m at 4km. So it is about 10m/hPa in the lower troposphere.

consequences

One consequence of the hypsometric equation is the possibility of deducing elevation from boiling point, as explained in Chapter 4. For example, if the boiling point of pure water is measured as 76.5oC, a table in Chapter 4 gives the ambient pressure as 410hPa. Then the hypsometric equation gives the elevation as 7,015m, assuming -8oC as the mean air temperature. That is about the height of South America's highest peak, Aconcagua (33oS), which reaches 6,960m.

Charles Darwin recorded his experience in boiling potatoes at 11,00 feet in the Andes (ie at 3,400m) in 1835. They were boiled for some hours but were still hard by evening. The pot was left on the fire overnight, and then boiled again, and still they were not cooked.