1.1 The properties of the atmosphere

The atmosphere’s characteristics are important in various aspects of engineering such as the design and operation of aircraft, road and rail vehicles, buildings and other structures, and the all-important weather forecasting. The atmosphere has also provided a source of mechanical power down the ages – for example for windmills and wind pumps, sailing vessels, etc., and now of course a source for electrical power generation with wind-driven turbines and wind farms. The atmosphere’s properties and behaviour of interest in these fields include density, pressure, temperatures, wind speeds, accelerations and turbulence. These are very rarely stable being in a constant state of flux because of the rotation and other motions of the Earth with respect to the Sun and the intermittent heating and cooling cycles which result.

Although vital, the atmosphere height-wise is relatively very thin in relation to the diameter of the Earth. It has been likened to the thickness of the skin on an apple, but there is no real edge or boundary at the upper level. A common rule of thumb is for the upper limit (the Kármán line) to be 100 km from sea level, but in reality the density and pressure continue to diminish with height, with traces of atmosphere being detected at many hundreds of kilometres further up. Even at the height of some of the lower-orbit satellites (say around 150 km), there is a discernible atmosphere which will ultimately slow them down enough for them to fall back to Earth, and all but the largest will burn up before they hit the ground. The largest ones are decommissioned carefully so as to return to Earth in specified safe areas.

The density of the atmosphere at ground/sea level on a still day is taken as 1.225 kg m⁻³. The mean pressure at this level is stated as 101.325 kPa, which can be read as a mass of air of just over 10 tonnes on each square metre on the Earth’s surface. However, because the pressure reduces with height above ground level, the density decreases in proportion, hence the gradual diminishing with no definite boundary.

Figure 1 Temperature variation with height in standard atmosphere

Show description|Hide description

This figure is a graph of height (km) on the vertical axis against temperature (degrees C). The region below 11 km height is the troposphere and above it the stratosphere. The boundary at 11 km is labelled ‘isothermal height, tropopause’.

The graph itself is a straight line from 15 degrees C at the surface, sloping upwards to the left to the tropopause at 56.6 degrees C. Above the tropopause the graph is a vertical line up to 16 degrees C. The gradient of the line in the troposphere is – 6.5 degrees C per km.

Figure 1 Temperature variation with height in standard atmosphere

Note that from sea level (zero on the vertical axis) the temperature reduces with height in a directly linear manner up to about 11  km altitude. This lower region (0–11  km) is called the troposphere. Above 11  km the temperature stays the same at about –56.5  °C for increasing heights. This region is the stratosphere, and the height at which the constant temperature starts is the isothermal height (sometimes isothermal level), also known as the tropopause. At far greater altitudes there is more variation but because the air is so thin by then the concept of atmospheric temperature does not mean very much.

The values describing the graph in Figure 1 vary a little around the Earth, but typically the sea-level mean (average) temperature is assumed to be 15 °C and the slope or gradient of the graph from zero to 11 km height in degrees per km change in height is −6.5 °C km ⁻¹. Above 11 km, the gradient is of course zero as the temperature stays constant.

Other properties of air have been deduced or derived from known relationships with temperature and some of these are presented in Figure 2 in non-dimensionalised form so as to fit them all on one graph.

Maximise

Figure 2 Properties of the standard atmosphere. Note that kinematic viscosity increases with altitude, so the inverse ratio is shown.

Show description|Hide description

This is a graph with a dimensionless vertical scale from 0 to 1, against height (km) from 0 to 20 km. 6 lines are shown. They all start from (0 km, 1) i.e. maximum value at the surface.

Straight line sloping down from the start to 0.85 at 11 km then horizontal for remaining heights. It is labelled c/c subscript 0 where c subscript 0 is 340 m per s.

The second line is sloping down to 0.8 at 11 km and labelled Greek letter eta/eta subscript 0, where eta subscript 0 is 18 x 10 to the power of -6 N s per m squared.

The third line is sloping down to 0.6 at 11 km and labelled T/T subscript 0, where T subscript 0 is 288.15 K.

The remaining 3 graphs are curves that continuously fall with decreasing slope to low values (around 0.1) at 20 km height. They are close together and the top and middle curves have slight kinks at height 11 km.

The top of these is Greek letter nu subscript 0/nu, where nu subscript 0 is 14.7 times 10 to the power of -6 m squared per s.

The middle one is Greek letter rho/rho subscript 0 where rho subscript 0 is 1.225 kg per m cubed.

The lower one is P/P subscript 0 where P subscript 0 is 101.3 kPa.

Figure 2 Properties of the standard atmosphere. Note that kinematic viscosity MathJax failure: MathML - Unexpected text node: '

The constants used to non-dimensionalise the properties are depicted on the graph with the subscript ‘0’ and correspond to the values at sea level. Note that the height is now on the horizontal axis with the 11 km height marked as a vertical line. The properties shown are the local speed of sound (sonic velocity), , the dynamic viscosity, , the kinematic viscosity, , the density, , and the pressure, . For instance, it can be seen from the graph that the speed of sound  = 340m s⁻¹ at sea level and decreases linearly through the troposphere; above the tropopause it remains constant, given approximately by

In the troposphere two properties, pressure and density, can be modelled by simple expressions as follows.

For pressure:

where is the absolute pressure, (= ) is the standard sea-level value of 101.3kPa, is the altitude under consideration and is a constant value of 44 300 m.

For density:

where is the required density, is the standard sea-level value of 1.225kg m⁻³, is the altitude under consideration and is a constant value of 44 300 m.

Above the isothermal level of the tropopause, the pressure and density are based on the values at this isothermal level, as:

where is a constant value of 6377 m and , and are the values of pressure, density and altitude at the isothermal level, and which will be, respectively, 22.6 kPa, 0.364 kg m⁻³ and 11000 m.