# 2.1 MEASURING ATMOSPHERIC PRESSURE

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1 12 CAPTER 2. ATMOSPERIC PRESSURE 2.1 MEASURING ATMOSPERIC PRESSURE The atmospheric pressure is the weiht exerted by the overhead atmosphere on a unit area of surface. It can be measured with a mercury barometer, consistin of a lon lass tube full of mercury inverted over a pool of mercury: vacuum A B h Fiure 2-1 Mercury barometer When the tube is inverted over the pool, mercury flows out of the tube, creatin a vacuum in the head space, and stabilies at an equilibrium heiht h over the surface of the pool. This equilibrium requires that the pressure exerted on the mercury at two points on the horiontal surface of the pool, A (inside the tube) and B (outside the tube), be equal. The pressure P A at point A is that of the mercury column overhead, while the pressure P B at point B is that of the atmosphere overhead. We obtain P A from measurement of h: P A ρ h (2.1) where ρ 13.6 cm -3 is the density of mercury and 9.8ms -2 is the acceleration of ravity. The mean value of h measured at sea level is 76.0 cm, and the correspondin atmospheric pressure is 1.013x10 5 k m -1 s -2 in SI units. The SI pressure unit is called the Pascal (Pa); 1 Pa 1 k m -1 s -2. Customary pressure units are the atmosphere (atm) (1 atm 1.013x10 5 Pa), the bar (b) (1 b 1x10 5 Pa), the millibar (mb) (1 mb 100 Pa), and the torr (1 torr 1 mm 134 Pa). The use of millibars is slowly ivin way to the equivalent SI unit of hectopascals (hpa). The mean atmospheric pressure at sea level is iven equivalently as P 1.013x10 5 Pa 1013 hpa 1013 mb 1 atm 760 torr.

2 MASS OF TE ATMOSPERE The lobal mean pressure at the surface of the Earth is P S 984 hpa, slihtly less than the mean sea-level pressure because of the elevation of land. We deduce the total mass of the atmosphere m a : m a 4πR 2 P S 5.2x10 18 k (2.2) where R 6400 km is the radius of the Earth. The total number of moles of air in the atmosphere is N a m a /M a 1.8x10 20 moles. Exercise 2-1. Atmospheric CO 2 concentrations have increased from 280 ppmv in preindustrial times to 365 ppmv today. What is the correspondin increase in the mass of atmospheric carbon? Assume CO 2 to be well mixed in the atmosphere. Answer. We need to relate the mixin ratio of CO 2 to the correspondin mass of carbon in the atmosphere. We use the definition of the mixin ratio from equation (1.3), C CO2 n CO2 n a N C N a M a M C m C m a where N C and N a are the total number of moles of carbon (as CO 2 ) and air in the atmosphere, and m C and m a are the correspondin total atmospheric masses. The second equality reflects the assumption that the CO 2 mixin ratio is uniform throuhout the atmosphere, and the third equality reflects the relationship N m/m. The chane m C in the mass of carbon in the atmosphere since preindustrial times can then be related to the chane C CO2 in the mixin ratio of CO 2. Aain, always use SI units when doin numerical calculations (this is your last reminder!): m C M C m a C CO2 5.2x10 M a x10 ( 280x10 6 ) x x x10 14 k 180 billion tons!

3 VERTICAL PROFILES OF PRESSURE AND TEMPERATURE Fiure 2-2 shows typical vertical profiles of pressure and temperature observed in the atmosphere. Pressure decreases exponentially with altitude. The fraction of total atmospheric weiht located above altitude is P()/P(0). At 80 km altitude the atmospheric pressure is down to 0.01 hpa, meanin that % of the atmosphere is below that altitude. You see that the atmosphere is of relatively thin vertical extent. Astronomer Fred oyle once said, "Outer space is not far at all; it's only one hour away by car if your car could o straiht up!" Mesosphere Altitude, km Stratosphere Troposphere Pressure, hpa Temperature, K Fiure 2-2 Mean pressure and temperature vs. altitude at 30 o N, March Atmospheric scientists partition the atmosphere vertically into domains separated by reversals of the temperature radient, as shown in Fiure 2-2. The troposphere extends from the surface to 8-18 km altitude dependin on latitude and season. It is characteried by a decrease of temperature with altitude which can be explained simply thouh not quite correctly by solar heatin of the surface (we will come back to this issue in chapters 4 and 7). The stratosphere extends from the top of the troposphere (the tropopause) to about 50 km altitude (the stratopause) and is characteried by an increase of temperature with altitude due to absorption of solar radiation by the oone layer (problem 1. 3). In

4 15 the mesosphere, above the oone layer, the temperature decreases aain with altitude. The mesosphere extends up to 80 km (mesopause) above which lies the thermosphere where temperatures increase aain with altitude due to absorption of stron UV solar radiation by N 2 and O 2. The troposphere and stratosphere account toether for 99.9% of total atmospheric mass and are the domains of main interest from an environmental perspective. Exercise 2-2 What fraction of total atmospheric mass at 30 o N is in the troposphere? in the stratosphere? Use the data from Fiure 2-2. Answer. The troposphere contains all of atmospheric mass except for the fraction P(tropopause)/P(surface) that lies above the tropopause. From Fiure 2-2 we read P(tropopause) 100 hpa, P(surface) 1000 hpa. The fraction F trop of total atmospheric mass in the troposphere is thus P( tropopause) F trop P( 0) The troposphere accounts for 90% of total atmospheric mass at 30 o N (85% lobally). The fraction F strat of total atmospheric mass in the stratosphere is iven by the fraction above the tropopause, P(tropopause)/P(surface), minus the fraction above the stratopause, P(stratopause)/P(surface). From Fiure 2-2 we read P(stratopause) 0.9 hpa, so that P( tropopause) P( stratopause) F strat P( surface) The stratosphere thus contains almost all the atmospheric mass above the troposphere. The mesosphere contains only about 0.1% of total atmospheric mass. 2.4 BAROMETRIC LAW We will examine the factors controllin the vertical profile of atmospheric temperature in chapters 4 and 7. We focus here on explainin the vertical profile of pressure. Consider an elementary slab of atmosphere (thickness d, horiontal area A) at altitude :

5 16 horiontal area A pressure-radient force (P()-P(+d))A +d weiht ρ a Ad Fiure 2-3 Vertical forces actin on an elementary slab of atmosphere The atmosphere exerts an upward pressure force P()A on the bottom of the slab and a downward pressure force P(+d)A on the top of the slab; the net force, (P()-P(+d))A, is called the pressure-radient force. Since P() > P(+d), the pressure-radient force is directed upwards. For the slab to be in equilibrium, its weiht must balance the pressure-radient force: ρ a Ad ( P( ) P ( + d) )A (2.3) Rearranin yields P ( + d) P ( ) ρ d a (2.4) The left hand side is dp/d by definition. Therefore dp ρ d a (2.5) Now, from the ideal as law, ρ a PM a RT (2.6) where M a is the molecular weiht of air and T is the temperature. Substitutin (2.6) into (2.5) yields: dp P M a d RT (2.7) We now make the simplifyin assumption that T is constant with

6 17 altitude; as shown in Fiure 2-2, T varies by only 20% below 80 km. We then interate (2.7) to obtain lnp ( ) lnp( 0) M a RT (2.8) which is equivalent to M a P ( ) P( 0) exp RT (2.9) Equation (2.9) is called the barometric law. It is convenient to define a scale heiht for the atmosphere: RT M a (2.10) leadin to a compact form of the Barometric Law: P ( ) P( 0)e ---- (2.11) For a mean atmospheric temperature T 250 K the scale heiht is 7.4 km. The barometric law explains the observed exponential dependence of P on in Fiure 2-2; from equation (2.11), a plot of vs. ln P yields a straiht line with slope - (check out that the slope in Fiure 2-2 is indeed close to -7.4 km). The small fluctuations in slope in Fiure 2-2 are caused by variations of temperature with altitude which we nelected in our derivation. The vertical dependence of the air density can be similarly formulated. From (2.6), ρ a and P are linearly related if T is assumed constant, so that ρ a ( ) ρ a ( 0)e ---- (2.12) A similar equation applies to the air number density n a. For every rise in altitude, the pressure and density of air drop by a factor e 2.7; thus provides a convenient measure of the thickness of the atmosphere. In calculatin the scale heiht from (2.10) we assumed that air

7 18 behaves as a homoeneous as of molecular weiht M a 29 mol -1. Dalton s law stipulates that each component of the air mixture must behave as if it were alone in the atmosphere. One miht then expect different components to have different scale heihts determined by their molecular weiht. In particular, considerin the difference in molecular weiht between N 2 and O 2, one miht expect the O 2 mixin ratio to decrease with altitude. owever, ravitational separation of the air mixture takes place by molecular diffusion, which is considerably slower than turbulent vertical mixin of air for altitudes below 100 km (problem 4. 9). Turbulent mixin thus maintains a homoeneous lower atmosphere. Only above 100 km does sinificant ravitational separation of ases bein to take place, with lihter ases bein enriched at hiher altitudes. Durin the debate over the harmful effects of chlorofluorocarbons (CFCs) on stratospheric oone, some not-so-reputable scientists claimed that CFCs could not possibly reach the stratosphere because of their hih molecular weihts and hence low scale heihts. In reality, turbulent mixin of air ensures that CFC mixin ratios in air enterin the stratosphere are essentially the same as those in surface air. Exercise 2-3 The cruisin altitudes of subsonic and supersonic aircraft are 12 km and 20 km respectively. What is the relative difference in air density between these two altitudes? Answer. Apply (2.12) with 1 12 km, 2 20 km, 7.4 km: ( 2 1 ) ρ( 2 ) e ρ( 1 ) e e The air density at 20 km is only a third of that at 12 km. The hih speed of supersonic aircraft is made possible by the reduced air resistance at 20 km. 2.5 TE SEA-BREEZE CIRCULATION An illustration of the Barometric Law is the sea-breee circulation commonly observed at the beach on summer days (Fiure 2-4). Consider a coastline with initially the same atmospheric temperatures and pressures over land (L) and over sea (S). Assume that there is initially no wind. In summer durin the day the land

8 19 surface is heated to a hiher temperature than the sea. This difference is due in part to the larer heat capacity of the sea, and in part to the consumption of heat by evaporation of water. (a) Initial state: equal T, P over land and sea (T L T S, P L P S ) slope - S L LAND SEA ln P (b) Sunny day: land heats more than sea (T L > T S ) L > S P L () > P S () hih-altitude flow from land to sea Flow S L LAND SEA ln P (c) ih-altitude flow from land to sea P L (0) < P S (0) reverse surface flow from sea to land L S Circulation cell LAND SEA ln P Fiure 2-4 The sea-breee circulation As lon as there is no flow of air between land and sea, the total air columns over each reion remain the same so that at the surface P L (0) P S (0). owever, the hiher temperature over land results in

9 20 a larer atmospheric scale heiht over land ( L > S ), so that above the surface P L () > P S () (Fiure 2-4). This pressure difference causes the air to flow from land to sea, decreasin the mass of the air column over the land; consequently, at the surface, P L (0) < P S (0) and the wind blows from sea to land (the familiar "sea breee"). Compensatin vertical motions result in the circulation cell shown in Fiure 2-4. This cell typically extends ~10 km horiontally across the coastline and ~1 km vertically. At niht a reverse circulation is frequently observed (the land breee) as the land cools faster than the sea.

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