PRESSURE AND DENSITY IN THE ATMOSPHERE AND OCEAN
BY HANS ROSENDAL, WFO, HONOLULU, HAWAII
(submitted to the Mariners Weather Log)
The atmosphere, at the bottom of which we live, is the envelope of air surrounding the Earth and held in place by gravity. Air is considered to be a fluid. By fluid we mean a substance, such as a liquid or gas, that is capable of flowing or readily changing its shape. A fluid exerts pressure in all directions, but the pressure is a function of the mass of fluid above a given surface. Pressure is measured in force per unit area. In other words, atmospheric pressure is a measure of the weight of the column of air above a point.
What we call air is a mixture of gases, mainly nitrogen (78%) and oxygen (21%), plus trace amounts of a few other gases, and variable amounts of water vapor in relatively small quantities. Since there is a constant number of molecules in a given volume of air, adding the water vapor molecules means that these lighter H2O molecules will displace heavier nitrogen (N2) and oxygen (O2) molecules and the resultant mixture of gases will be lighter than dry air.
Air, unlike water, is a compressible fluid so the rate of pressure decrease as we move up in the atmosphere is not as simple as in water where pressure varies very simply with depth at a rate of one atmosphere of pressure per 33 feet or 10 meters of water. Thus the pressure measured at the depth 50 meters of water would be 5 atmospheres of pressure due to the weight of the overlying water plus one atmosphere's pressure due to the weight of the air in the overlying atmosphere for a total of 6 atmospheres of pressure.
In meteorology we use the unit of millibars for measuring pressure where 1013 mb (or 1013 hPa) equals roughly one atmosphere's pressure. Atmospheric pressure is measured by a barometer. As mentioned earlier, the pressure of the atmosphere will balance about 33 feet (10 m) of water or 30 inches (76 cm) of the much denser liquid of mercury. The ratio of the densities of the two liquids, mercury to water, is about 13.6 to 1.
From the early days of meteorology, the relationship between atmospheric pressure and height above sea level has been studied very closely. The pioneering work by the French scientist Pascal is particularly important and that is why he is honored by the naming of the unit of pressure after him. The hectoPascal (100 Pascals) will soon be used worldwide by meteorologists instead of millibars. The early meteorologists or aviators ascended mountains or used balloons or aircraft to measure the change in pressure with height. Some rough standards were decided on, such as the 5000 foot level for 850 mb, 10,000 ft for 700 mb, 500 mb at 18,000 ft, 250 mb at 34,000 ft, and 150 mb at 45,000 ft. Rocket scientists and space explorers also entered the picture in more recent times as knowledge about the outer fringes of the atmosphere became important.
The pressure in the atmosphere varies with the density and temperature of the air. There is a simple equation called the Equation of State that explains this relationship
p= rho*R*T
If we work with meters, tons and seconds as basic units (the socalled m-t-s units of length, mass and time), then pressure must be given in centibars or kilopascals, where 1 cb=10 mb. If we plug in some numbers for sea level pressure of 100 cb and a density (rho) of 1.2 *10-3 tons/m3 (1.2 kg/m3), and a temperature of 27C=273+27=300K (Kelvin degrees), and the gas constant (R) of 278 kJ/t*K, we will find the equation in rough balance.
We can perform the same calculation for a point half way up in the atmosphere with respect to pressure at 500 mb or 50 cb. This is at a height of about 18,000 ft or 5.5 km where the temperature typically is -5C (268K). If we solve for density at that level, we arrive at .67 * 10-3 tons/m3 or roughly half the density of the air at sea level.
If we do the same calculation for a pressure of 250 mb, or 25 cb, and a typical temperature of -43C or 230K, we arrive at a value of density of air of .39*10-3 t/m3 or slightly more than half of the value we found at 500 mb. 250 mb is at a height of roughly 36,000 ft or 11 km where modern jet aircraft fly. As we saw earlier, the standard height for the 250 mb pressure surface is closer to 34,000 ft so we see our approach is only a rough approximation of conditions in the actual atmosphere.
Ascending another 18,000 feet or 5.5 km to 54,000 feet or 16.5 km we are at an elevation where the supersonic aircraft operate. We are now at a pressure of about 125 mb (12.5 cb) and a temperature of -65C (208K) and we find an air density of .22*10-3t/m3. Cutting the pressure in half once more at 62.5 mb we find ourselves in the stratosphere at 22 km height where temperatures change very little with height and there may actually be some warming.
There is another simple equation, the socalled Hydrostatic Equation, that relates changes in pressure to changes in height. This equation is as follows
^p=-rho*g*^z
where ^p is the change in pressure and ^z is the corresponding change in height in m. Rho, as above, is air density and g is gravity held constant at 9.8 m/s2. If we want to make a quick calculation of what vertical distance in meters that corresponds to a change in pressure of 1 millibar (0.1 cb) near sea level where the density is 1.2*10-3 t/m3, we get the following result:
0.1 cb= - 1.2*10-3 t/m3 * 9.8 m/s2 * ^z
Solving for ^z, we get 8.5m. In other words, a change in height of 8.5 meters near sea level, as you go up in the atmosphere, results in a reduction (thus the minus sign in equation above) in pressure of 1 millibar. If your location is 34 meters (111 feet) above sea level you will have to add 4 millibars to your barometric pressure reading to get an estimate of the sea level value. Thus relatively small changes in elevation show up on your barograph trace if you move the instrument as, for example, from the bridge down to a cabin on the lower deck. Likewise, the barogram from a ship entering the Great Lakes along the St. Lawrence Seaway as it moves from the St. Lawrence River to Lake Ontario via the locks at Massena and from Lake Ontario to Lake Erie via the Welland Canal will show the typical stairway steps as it moves up in the atmosphere within the locks. We are here talking about a change in elevation from sea level to the level of Lake Erie at 174 m above sea level or a reduction in pressure of about 20 millibars.
At 500 mb we found out in our earlier computations that the density of air was roughly half of that at sea level, namely 0.67*10-3 t/m3. We should therefore expect that the height difference (^z) that corresponds to a 1 mb difference (^p= .1 cb) in pressure at the 500 mb level, say going from 500 mb to 501 mb, will be roughly twice the height increments for the same pressure change at 1000 mb. Plugging in the above values for density, we arrive at a value of 15.2 m.
What all this tells us is that as we ascend in the atmosphere, the pressure is roughly cut in half every 5.5 km or 18,000 feet. It is not an exact presentation of the variation in pressure with height since both density and temperature vary, but it is a rough approximation of the exponential loss of atmospheric mass or pressure with height shown in graphical form in figure 1. In comparison the very simple variation of pressure with depth in the ocean is depicted in Figure 2.
The mariner cruises the ocean at the interface between the two fluids of air and water. As we have seen water is about three orders of magnitudes denser than air. By an order of magnitude we mean a factor of 10. Thus one cubic meter of water weighs a ton or 1000 kg, while the same volume of air weighs 1.2 kg. Motions in the air in response to external forces are thus much swifter than in the oceans. Air currents, such as the jet streams up high in the atmosphere, thus may change their meandering paths in a matter of hours or days whereas similar drastic changes in the ocean perhaps will take years. The oceans have been referred to as the flywheels of climate due to the large inertia or resistance to change inherent in them.
Density differences in the ocean are minor as compared to the dramatic changes we experience in the atmosphere. Water, being a liquid, is not compressible, but it does expand and contract in volume slightly with changing temperatures. Small, but important density increases also occur with increases in salinity. The density of water thus varies only little, but these small variations are nevertheless very important since they determine how ocean currents flow and water circulates. We sometimes see these flows referred to as the thermo-haline circulations of the ocean.
These density differences also have some practical aspects to navigation. For instance, from the load lines painted on the side of the ship, we know that the depth at which a vessel is immersed into the water varies somewhat. The weight of the ship and its cargo displaces an equal weight of water. If the water is fresh and warm, the volume of this water will be larger, so the ship will be immersed deeper into the water. Thus the tropical fresh water in the Panama Canal is somewhat less dense than the cold water in the North Atlantic in winter. So we see that temperature and salinity are the two determining factors. Cold water of high salinity is most dense and will tend to sink to the bottom of the ocean. The deep oceans therefore contain cold water that has been sinking in the North Atlantic and in the Antarctic from where it is spreading equatorward. The vertical stability of the ocean waters is smaller at high latitudes in winter as water masses move up and down or turn over more readily than at lower latitudes. These high latitude waters are well mixed so salinity differences in the vertical are relatively small in most parts of the world.
Ocean currents, besides being caused by wind stresses in the surface layers, are therefore nature's way of tending to even out these differences in density along a horizontal surface. The same laws of motion hold for the oceans as well as the atmosphere.
Ocean surface water temperatures vary between the freezing temperature of salt water (about minus 2C or 28F) along the fringes of the ice pack in polar regions and 30C (86F) in the warmer and more humid parts of the tropics. Salinity varies from brackish or almost fresh in some inlets and estuaries (and the Panama Canal) to the typical open ocean values of about 3.5 % or 35 parts per thousand. Fresh water additionally has the important quality of being most dense at about 3.5 degrees Celsius. This means that fresh water lakes 'turn over' in cold climates and oxygen and nutrients are mixed throughout the volume.
Minor variations in the open ocean salinity values are caused by differences in evaporation and rainfall and outflow of fresh water from rivers. Highest salinity values are found in areas where evaporation greatly exceeds precipitation. These conditions occur underneath the atmospheric subtropical high pressure cells. The Mediterranean Sea and the Red Sea also qualify in this category. The Black Sea, while located in a climate of strong evaporation, experiences heavy runoff from land areas to cause local differences. High water densities due to very high salinity are of course found in the Caspian Sea, which is really a lake, as are the extreme cases of the Salton Sink of California and the Dead Sea of Israel/Jordan.
When fluids are in motion, we have transfer of momentum within a fluid, and we may experience transfers of momentum between different fluids. Transfers of momentum within the fluid of the atmosphere itself become more complex since we have a variation in density with height in air. On the other hand, density differences with increasing depth in the ocean, as we have seen, are small. By transfer of momentum is meant the product of mass times velocity. We are familiar with the concept in billiard balls or automobile collisions. In the case of air, we can use density of the air times its velocity. The usual case in the atmosphere is that we mix momentum downward from the fast flow aloft even though this fast flow consists of a air stream that is not nearly as dense as the air near the surface. In a mid latitude storm or low pressure system, within the western semicircle of this storm we often find northwest winds of 160 kt at 250 mb height (35,000 ft) in the atmosphere near the jet stream level. At 500 mb winds are from the same direction at 80 kt, while at 1000 mb on the ocean's surface winds are northwest at 40 kt. Assuming density proportional to pressure, we find change in momentum with height is close to zero. In other words, as we descend in the atmosphere the wind speed diminishes at the same rate as the density increases. That is, we have the unique condition of a well mixed fluid in almost the entire depth of the atmosphere with no momentum transfers upward or downward.
Ocean surface waves, also called gravity waves as gravity is the restoring force, are a manifestation of momentum transfers from the atmosphere to the ocean as the strong winds in a storm whip up large waves. The amplitude (height, or actually half of the height) of surface waves is relatively small (as gravity waves go) since we have large density differences (three orders of magnitude, we saw earlier) between the two fluids of air and water. Internal waves in the ocean (or within the atmosphere, such as along an inversion) have small density differences across the surface along which these waves are generated and propagate. When talking about internal waves, we are therefore talking about relatively large amplitude waves that may be generated by a relatively slight force.
The case of a ship experiencing 'dead water' may be a manifestation of how the energy of propulsion of a ship is diverted into the development of internal waves in the ocean. The density differences across the surface, along which these internal waves travel, may in this case be caused by a shallow layer of warmer or fresher water overlying colder more saline water.
There are many other interesting aspects of the changes in pressure and density in the atmosphere and the ocean. The temporal and spatial changes in temperature, as we have seen, is another important factor to consider. In the atmosphere, consisting of air which is compressible, temperature takes on a entirely new meaning. That is, as air moves downward in the atmosphere it moves into regions of higher pressures and will be compressed which will result in warming. The opposite effect of cooling occurs as air rises and expands at the lower pressures aloft. Then we also have the complicating factors of water vapor and the change of phase between the gaseous, liquid and solid states of water. Likewise, in the oceans, we have the complicating factors of varying salinity. There is therefore much more to discuss in following articles.
