# Lecture 2: Air Pressure, Density and Temperature

## Introduction

The atmosphere is a mix of gases. The most important components are:

Constant gases (% of dry air):

Nitrogen (N2)            78.08%
Oxygen   (O2)           20.92%
Argon   (Ar)                0.93%

Variable gases:

Water vapour (H2O)       0-4%
Carbon dioxide (CO2)    0.035%
Methane (CH4)               0.00017%

This composition has been profoundly influenced by Life, especially photosynthesis by plants. The early atmosphere was created by outgassing from volcanoes: 4 billion years ago it was predominantly Nitrogen and Carbon Dioxide. The atmosphere became gradually enriched in oxygen by weathering, the deposition of carbonate rocks, and photosynthesis, first by algae, then by higher plants. The present atmospheric mix is maintained by ecosystems, and the level of oxygen is optimum for life: a smaller proportion would mean less available for animal respiration; and a larger (c. 25%) would result in very extensive wildfires.

## Air Density and Pressure

Air consists of molecules in constant motion, colliding with each other and with adjacent surfaces: about 1010 collisions per molecule per second at 10o C near sea level. This is part of the kinetic energy of the atmosphere (Lecture 1). We can view the mass of moving molecules in terms of three interrelated properties: pressure, density, and temperature.

Density is the mass of air molecules per unit volume. Mass is measured by the kilogram (kg), and volume in cubic metres (m3), so the unit of density is kilograms per cubic metre (kg m-3). We think of kilograms in terms of weight, but weight is the result of a mass subjected to gravity. In classical physics, mass is independent of gravity, and the kilogram measures a set mass of atoms, regardless of whether these are on the Earth, on the moon, or freely floating in space. At sea level, air density is around 1.2 kg m-3. Air density decreases with altitude, for reasons discussed below.

Pressure is the cumulative effect of the push exerted by each molecular collision on its surroundings. Pressure is defined as force per unit area. Force is defined as a mass subjected to an acceleration, or mass x acceleration. The unit of mass is the kilogram (kg), and an acceleration is a change in velocity through time, measured in metres per second (velocity) per second (m sec-1 sec-1, or m sec-2). Thus the unit of force is kg m sec-2 or Newton, named after Sir Isaac Newton, who formulated the basic laws of force and motion (alongside major contributions to many other aspects of science and mathematics). Pressure is a force distributed over an area. The unit of Pressure is N m-2, or Pascal. Atmospheric pressure is often measured in millibars, 1 Millibar = 100 Pa (100 Pa = 1 hecaPascal or hPa). Mean pressure at sea-level is 1013 mbar or approximately 100 kPa. This pressure is exerted in all directions: up, down, and to all sides. For equilibrium, the pressure exerted by an air parcel is exactly balanced by the downward force exerted by the overlying air pulled by gravity. This balance is known as hydrostatic equilibrium.
A pressure of 1000 mbar (100 kPa) results from the weight exerted by 10,000 kg of air overlying one square metre of surface, accelerated by gravity (c. 10 m sec-2). The huge pressure does not crush us because it is exactly balanced by outward pressure from the inside of our bodies. Ears popping due to change in altitude are the result of the pressure difference between the inside of our heads and the surrounding air. Pressure decreases with altitude, due to the reduction of the mass of overlying air with height. Pressure is c. 700 mbar at 3,000m; 500 mbar at 5,500m; 300 mbar at 10,000m.

Temperature is a measure of the average speed of the moving molecules. In meteorology and other branches of physics, temperature is measured on the Kelvin scale, which begins at absolute zero, where there is no molecular kinetic energy.

0°K = - 273° C.

The temperature of a mass of air depends on the average velocity of the air molecules and their mass, and so temperature generally increases with air density.
The atmosphere is bombarded by shortwave radiation (including UV) from above, and shortwave + longwave radiation from below, so the atmosphere receives energy from above and below. Combined with the dependence of temperature on air density, this gives the atmosphere a distinctive temperature profile. From the surface, temperature decreases with altitude, as the air becomes thinner. This zone is called the troposphere (from the Greek tropos meaning 'change'): it is in this layer that the world's weather happens. Then at altitudes of c. 10 - 15 km, the temperature stabilises, then rises with altitude. This zone of rising temperature with altitude is called the stratosphere. The high temperatures in the stratosphere are due to the absorption of UV radiation by ozone. The stratospheric temperature inversion creates a stable lid on the lower atmosphere (the tropopause), limiting the maximum thickness of weather systems to about 10-15 km. Above the stratosphere (c. 50 km) the temperature again declines (the mesosphere), then again rises above c. 85 km (the thermosphere). These higher levels of the atmosphere do not concern us in this course.

The average temperature at the Earth's surface c. 15o C (288o K), and generally decreases with height up to top of the Troposphere. The average value of this vertical lapse rate is 6.6o C km-1. However, temperature profiles in the lower atmosphere are actually very variable, due to heating and cooling from the Earth's surface (sensible, latent, and radiative heat transfer).

## Relations between density, pressure and temperature

These three quantities are related in this way:

p = R r T

p is pressure,
r (Greek rho) is density
T is temperature (in degrees Kelvin),
R is the specific gas constant, which varies from gas to gas.
For dry air, R is 287 J K-1 kg-1.

This very important relationship is known as the Equation of State, and simply means:

(1) for constant density, pressure increases with temperature (that is, if the molecules have a higher average kinetic energy, they exert a greater push on their surroundings);
(2) for constant temperature, pressure increases with density (the more molecules per unit volume, the greater the push exerted by collisions);
(3) for constant pressure, temperature and density are inversely related (that is, if there are fewer molecules in a given volume, they need to be travelling at a greater average speed to exert the same pressure).

Therefore, we can see that any change in any one variable is likely to cause changes in the others. For example, if we heat a mass of air, we increase its pressure, if the air is allowed to expand to equalise the pressure difference with the surrounding air, the density will decrease. When this happens, it will be lighter than the same volume of surrounding air, and will rise.

## Influence of Pressure and Density on Temperature: The Demijohn experiment:

The relationship between density and pressure can be demonstrated in a corked demijohn combined with a bicycle pump. As air is pumped into the demijohn, the temperature (measured by a thermistor probe) increases. The increase in temperature is due to an increase in density (more air is pumped into the jar, increasing its mass compared to an equivalent volume of air in the surrounding room), and an associated increase in pressure. When the jar is depressurised, the temperature falls to its initial value.
This is described by the equation p = R r T  in the following way:

• Rearranging the equation to examine the effect of variations of pressure and density on temperature, we write: p/(r R) = T
• This says that pressure divided by (density multiplied by a constant) equals temperature
• In the demijohn experiment, we increase the air pressure and the density. As it turns out, the % increase in pressure is greater than the % increase in density, so the term p/rR increases. Hence, temperature increases.
• This explains the general reduction in temperature with altitude (a reduction in temperature due to decreasing pressure)

## Potential Temperature

The dependence of temperature on pressure makes it difficult to compare the amount of energy contained in two air masses. Is a temperature difference due to differences in pressure, or to real differences in the amount of energy in the air? To overcome this problem, meteorologists use the concept of Potential Temperature, which is defined as the temperature of the air at a pressure of 1000 mbar. This standardises temperature to a fixed base level, and allows ready comparison of air masses at different air pressures, whether due to differences in elevation or other reasons. We shall return to the concept of potential temperature in later lectures, so it is worth taking time to ensure you understand it.

## Vertical changes in pressure and density

As noted above, air pressure reduces with altitude due to the reduction in the mass of overlying air. This reduction in pressure is associated with a reduction in air density. We can use the equation of state to explore why this is so. First, we can rearrange the equation to isolate air density:

r = p / R T

This simply says that the air density is given by the pressure divided by (temperature x a constant). Therefore, for any given temperature, as pressure decreases, so does the air density. We have seen, however, that temperature decreases with reductions in pressure. A decrease in temperature will actually have the opposite effect on density, since density and temperature are inversely related in this equation. However, it turns out that this effect is outweighed by the pressure-density relationship, and as pressure decreases with height, so does air density.

We can see this effect with a worked example.

(1) The air pressure at sea-level is c. 1000 mbar, or 100,000 Pascals. If the air is at 25o C (298o K), then:

r = 100,000 / (287 x 298) (recall that the gas constant for dry air is 287)
= 1.17 kg m-3 (this figure is very close to the mean figure for sea-level density quoted above)

(2) For 500 mbar (the air pressure at around 5,500 metres above sea-level), the air temperature is typically -30o C (243o K)
thus:

r = 50,000 / (287 x 243)
= 0.72 kg m-3

This is a little over half of the value for sea-level, showing that the pressure is the overwhelming influence on the change in density.

## Hydrostatic Equilibrium

We have seen that there is a decrease in air pressure with altitude, i.e. air pressure is greater at lower elevations than higher elevations.

Q: If there is a pressure gradient up though the atmosphere, why are there not constant upward-blowing winds?

A: Because the upward-directed pressure gradient force is exactly balanced by the downward force of gravity acting on the air. This is called the hydrostatic equilibrium, and is expressed thus:

Dp / Dz = - r g

Dp (Greek delta p) is the difference in pressure, and Dz is the change in height, so Dp / Dz is the vertical pressure gradient. r is the local air density, g is gravity (9.8 m sec-2), and the minus sign on the right-hand side shows that gravity is directed downwards.

## Disturbing the Hydrostatic Equilibrium: The effect of heating on a small mass of air

The relationship between pressure, density and temperature explains what happens when a small mass of air is heated above the temperature of its surroundings. When air is heated (and this may be due to the transfer of sensible heat, radiative energy, or latent heat), the molecules in the air move more rapidly. They therefore exert a greater push on the surrounding, cooler air. In other words, they exert a slightly greater pressure on the surrounding air than the surrounding air exerts on the heated air. On small spatial scales, there this little to resist this excess pressure, with the result that the heated air expands to locally restore the pressure balance. That is, the heated air attains a lower density as the result of the initial heating.

This is described by the equation p = R r T in the following way:

• The rapid equalisiation of pressure means that we can regard the pressure as constant.
• Since the term R is also a constant for any gas, we can write: r x T = constant
• In turn, this means that for higher temperatures, the density is lower.

The lower density of a heated mass of air means that the downward force - r g (the mass of air accelerated by the downward force of gravity) is reduced, relative to that exerted by the vertical pressure gradient. As a result, there is a net upward force on the air mass, and it rises.  Thus, the increase in temperature and reduction in density disturbs the local hydrostatic equilibrium and the air mass rises. The opposite happens when an air mass is chilled relative to the surrounding air. The reduction of temperature reduces the average velocity of the consitituent molecules, reducing the force they exert on the surrounding air. As a result, the chilled air mass contracts, increasing its density. The consequent increase in downward force upsets the local hydrostatic balance, and the air mass sinks.

The magnitude of the buoyant force is given by a minor modification to the right-hand side of the hydrostatic equation:

buoyant force = -((r0 - rf)/r0)g

where r0 is the density of the air parcel, and rf is the density of the surrounding air. Thus if r0 = rf  the air is neutrally buoyant, if r0 > rf  the buoyant force is more negative and the air sinks, and if r0 < rf  the buoyant force increases and the air rises.

## The Spectacular Effect of Air Pressure Differences

The rapid equalisation of pressure over small spatial scales can be demonstrated by bursting a balloon. The air in an inflated balloon is at higher pressure than in the surrounding air, due to the stress exerted by the stretched rubber. Bursting the balloon removes the rubber almost instantaneously, and the excess pressure is released in a pressure wave moving at close to the speed of sound. Within a second, there will be no detectable excess in pressure remaining at the original location. An even more spectacular example is a compressed air rocket

Rocket experiment: the dramatic effects of air pressure differences are demonstrated by a compressed-air powered rocket! Pumping air into the rocket increases its pressure. The seal fails when the pressure inside the bottle is c. 180 kPa, or 180% of atmospheric pressure. This rapidly forces water out of the base, sufficient to cause spectacular take-off.

Where to get a rocket:

Rokit, Hinterland Ltd. Stanstead Rd., Hertford SG13 7AY, UK.

## Spatial variations in pressure

Air pressure at any given height will vary spatially due to:

(1) differences in potential temperature
(2) air motions, especially convergence and divergence.

The influence of potential temperature can be understood in terms of the equation of state discussed above. Air with lower potential temperature will become more dense, reducing the vertical thickness of a given mass of atmosphere. Thus a given pressure (say, 500 mbar) will occur at lower altitude than for air with a higher potential temperature. Conversely, air with a higher potential temperature will tend to expand, so in regions where the air has a relatively high potential temperature, a given air pressure (again, say, 500 mbar) will occur at a higher altitude. The altitude at which the pressure of 500 mbar is encountered will thus be higher for warm air masses than for cool air masses. We therefore say that the 500 mbar surface is higher or lower, depending on the potential temperaure of the atmosphere. The 500 mbar surface is lower at high latitudes than nearer the equator. In weather forecasting, the altitude of the 500 mbar surface is a valuable indicator of the position of different air masses, and in the mid-latitudes it varies on a daily basis with the passage of weather systems. Thus the instantaneous position of the surface is highly irregular, due to the presence of waves in the atmosphere. These will be discussed in greater detail later in the course.

The 500 millibar surface on 17th January 2001. The surface altitudes are in tens of metres (dam). Note the decline in altitude of the surface with latitude: over the Sahara, it is at 5,880 metres, whereas over Greenland it is only 4,920 metres: almost 1 kilometre lower. The red and blue coloring denotes vorticity, or rotation of air. This will be discussed in Lecture 6.

The influence of divergence and convergence on air pressure arises because convergence imports air into a region, increasing the local mass of the atmosphere, and conversely, divergence exports air from a region, reducing the mass of the atmosphere.

## Internet resources:

Meteorological maps for the all parts of the world (current analyses and forecasts) can be viewed at the COLA-IGES web pages. Of special relevance to this lecture are the "500 mbar geopotential height" maps in the Northern Hemisphere MRF forecasts. Also well worth checking out, if you are using a reasonably fast computer, is the one-month Java animation of Northern Hemisphere circulation: viewing the 500 mbar surface in motion gives a vivid impression of the varying pressure waves on the planet.