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# 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.

## An illustrative statistic for the Earth’s atmosphere

As a matter of comparison, if the density of the atmosphere at sea level did remain constant all the way up, what would be the height or thickness of the atmosphere to create the sea-level pressure? Give your answer in km to 3 significant figures.

### Solution

From the fundamental law of hydrostatics

so

Therefore the height of the atmosphere would be 8.43 km (to 3 s.f.).

## Activity 4

If the height of the atmosphere was 100.0 km, what would be the atmospheric pressure at sea level if the density was constant at 1.225 kg m−3? Express the answer as a comparison with the standard figure of 101.325 kPa. Give your answer to 3 significant figures.

### Answer

From the fundamental law of hydrostatics

therefore the atmospheric pressure will be

Comparing this to the standard figure,

which is 11.9 times (to 3 s.f.) greater than the standard figure.

In reality, the atmosphere’s height and thickness are many times what they would be if the density was constant. The real density at sea level varies from a maximum of approximately 1.4 kg m−3 to a very low density at and above about 6 km height. The reduction of density of the atmosphere is evident at quite low levels; this limits the heights at which aircraft can generate sufficient lift and is why they have to fly so fast to climb high. Figure 1 is based on previously published data from the United Nations International Civil Aviation Organization (ICAO) regarding temperature variation with height. Other properties of interest can be deduced from their relationships with temperature.

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.

## Calculating the height of the atmosphere at 0 °C

From the above data determine the height at which the air temperature reaches 0°C. Give your answer to 2 significant figures.

### Solution

In Figure 1, studying the proportions of the slope part of the graph by similar triangles of height (vertical) divided by temperature (horizontal) gives

so

Therefore the height at which the air temperature reaches 0 °C is 2.3 km (to 2 s.f.).

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.

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

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:

Equation label:(Equation 1)

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:

Equation label:(Equation 2)

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:

Equation label:(Equation 3)

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.

## Reductions in atmospheric pressure with height

What is the percentage reduction in atmospheric pressure at the height when the air temperature drops to 0 °C? Give your answer to 3 significant figures.

### Solution

Rearranging Equation 1 to find pressure gives

From Example 2, the height of the atmosphere at which the air temperature reaches 0 °C was found to be , so

Therefore the percentage reduction in pressure is

## Activity 5

What is the percentage reduction in atmospheric density at the height when the air temperature reaches 0 °C, compared with the value at sea level? Give your answer to 3 significant figures.

### Answer

Equation 2,

can be rearranged to find density

Therefore the percentage reduction in density is

## Atmospheric model equations

The troposphere denotes the part of Earth’s atmosphere from an altitude of zero to 11 000 m. In this region the local atmospheric pressure can be evaluated from the expression in equation (4) as

where is the standard sea-level value of 101.3 kPa, is the altitude in metres under consideration and is a constant value of 44 300 m.

Also in the troposphere, the local air density can be evaluated from the expression in equation (5) as

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 44300 m.

The top level of the troposphere is known as the tropopause, and the height of the lower boundary of the tropopause, 11 000 m, is known as the isothermal height. Above this height, the pressure and density are both related by the expression in equation (6) as

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

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