Variation of Pressure with Height in the Atmosphere

Let us derive the equation governing the pressure in the static atmosphere. Imagine a volume element of the atmosphere of horizontal area dA between two heights, z and z + dz. The pressures exerted on the top and bottom faces are p(z + dz) and p(z), respectively. The gravitational force on the mass of air in the volume = pgdA dz, with p(z) p(z + dz) due to the additional weight of air in the volume. The balance of forces on the volume gives 

The temperature in the atmosphere varies by less than a factor of 2, while the pressure changes by six orders of magnitude (see Table A.8). If the temperature can be taken to be 

Since the temperature was assumed to be constant in deriving (l.S), a temperature at which to evaluate H must be selected. A reasonable choice is the mean temperature of the troposphere. Taking a surface temperature of 288 K (Table A.8) and a tropopause temperature of 217 K, the mean tropospheric temperature is 253 K. At 253 K, H = 7.4 km. 

Throughout this book we will need to know the number concentration of air molecules as a function of altitude. We can estimate this using the average scale height H = 7.4 kin and 

---

The variation of pressure with height in the atmosphere can be addressed with the hydrostatic equation, ... 

The Earth s atmosphere is characterized by variations of temperature and pressure with height. In fact, the variation of the average temperature profile with altitude is the basis for distinguishing the layers of the atmosphere. The regions of the atmosphere are (Figure 1.1) ... 

Concentrations of contaminants in the atmosphere may vary significantly from time to time due to seasonal climatic variation, atmospheric turbulence, and velocity and direction of wind. The most important meteorological factors are (1) wind conditions and the gustiness of wind, (2) the humidity and precipitation, (3) the temperature, which varies with latitude and altitude, (4) barometric pressure (varying with the height above the ground), and (5) solar radiation and the hours of sunshine, which vary with the season. 

The barometer formula can be derived by elementary methods, thus checking this part of the Maxwell-Boltzmann distribution law. Consider a column of atmosphere 1 sq. cm. in cross section, and take a section of this column bounded by horizontal planes at heights ft and ft + dh. Let the pressure in this section be P we are interested in the variation of P with ft. Now it is just the fact that the pressure is greater on the lower face of the section than on the upper one which holds the gas up against gravity. That is, if P is the upward pressure on the lower face, P + dP the downward pressure on the upper face, the net downward force is dP,... 

This equation, when integrated, approximates the pressure poorly because it ignores the change in temperature with height Figure4.5 illustrates the variation of atmospheric temperature with elevation. The temperature decreases linearly with elevation in the troposphere and mesosphere but it increases with elevation in the stratosphere and thermosphere ... 

Let us now consider the sum of the forces upwards. F is designated as the force, downwards, needed to hold the balloon in place. Another force is due to the pressure distribution of the cold atmosphere on the balloon. Assuming the atmosphere is motionless, fluid statistics gives the pressure variation with height x as... 

One way of varying the the Grashof number in experimental studies involving air is to keep the model size fixed and to vary the air pressure. Consider a situation in which the model height is 5 cm. the model surface is at a temperature of 60°C, and the air at a temperature of 20°C. Plot the variation of Grashof number with air pressure for air pressures between 0.1 and 10 times standard atmospheric pressure. 

Illustration 3.—The variation of atmospheric pressure with altitude above sea-level can be shown to follow the compound interest law. Let pQ be the pressure in centimetres of mercury at the so-called datum line, or sea-level, p the pressure at a height h above this level. Let p0 be the density of air at sea-level (Hg = 1). Now the pressure at the sea-level is produced by the weight of... 

The pressure of the air, as already mentioned, varies with altitude indeed, at one and the same place it does not remain constant in consequence of variation m composition, the influence of wind, etc. A standard pressure, known as an atmosphere, has been chosen. The British unit is a column of mercury 29-905 inches in height, measured at 82° F. m London, and is equivalent to a pressure of 14-78 lb. per square inch. 

---