IV. Intersection: what we know, would like to know, will never know, and what can we contribute to the debate. III. Atmospheric & Ocean Biogeochemistry: Second element of climate and environmental science Atmospheric and ocean composition, past and present Human impact, global change Road map to ES 5 Lectures5: ressure, barometric law, buoyancy air water fluid moves II. Atmospheric & Ocean hysics: First element of climate and environmental science Atmospheric structure (T, in "4-D") Winds, Weather, General Circulation, Climate I. hysical rinciples: The foundation & the tools Newton's laws: forces, pressure, motion Energy: Temperature, radiant energy L-2 L-3 Fig. 7.6: ressure in the atmosphere (compressible) and ocean (incompressible). Lecture 5. ES 5: 2 Sep 200 Review discussion of the perfect gas law from Lecture 2. 2.! Further discuss the concept of density! ; with the weight of overlying atmosphere? water Water columns have the same height: ressures equal on both sides. fluid will start to move Cylinder volume = h x A = h x " r 2. Mass =! h " r 2. Weight = Mass x g =! h " r 2 Water columns higher on the left: ressure higher on the left. air Mass of water = volume x density; Which has the greater volume? ressure = Mass x g / A = h! g
Boyle s law V = 2 V 2 ; Charles Law / T = 2 / T 2 erfect gas law (a.k.a. Boyle's and Charles' Laws) V = NkT where is pressure, V volume, N the number of molecules in the volume, and T the absolute temperature (Kelvin; T(K)=T(C)+273.5); k is Boltzmann's constant (.38 x 0-23 Joules/Kelvin). = nkt, where n (= N/V, the number density) is the number of molecules per unit volume. The erfect Gas Law relates pressure to temperature (the kinetic energy of the molecules) and "number density". Lecture 5. ES 5: 2 Sep 200 with the weight of overlying atmosphere? The density (! ) is defined as the mass per unit volume. If m is the mass of one molecule, then! = m n. The pressure, density and temperature of air are therefore related by: =! (k/m) T =! R T, an important form of the perfect gas law. The constant (k/m) is called R (the gas constant), "R" = 287.5/M J kg - K -. Kinetic energy and molecular motion E = /2 m v 2 = 3/2 k T k =.38 # 0-23 Joules/Kelvin; T = 300 K (room temperature) E = 6.2 # 0-2 Joules/molecule; for one mole, N 0 (6.02 # 0 23 molecules): E 0 = 3738.42 Joules/mole. Thus the molecules in only 29 grams of air ( mole) contain 3.78 kj of kinetic energy. Since Watt = Joule/s, this amount of energy fires up the electrical appliances in an average house for second! How fast do molecules move? 3/2 kt = /2 mv 2 m air = 29/N 0 = 4.83 # 0-26 kg per molecule v = (3 kt/ m ) /2 = 500 m s - at T=300 K A more exact treatment gives (8 kt/("m)) /2 = 467 m s -. Lecture 5. ES 5: 2 Sep 200 with the weight of overlying atmosphere? This is the speed of sound! (why is that?)
Atmospheric pressure and temperature Distribution of pressure with altitude: the barometric law. Relationship between density, pressure and altitude atmosphere 2 2 Z 2 Z D 2 Z (altitude) increases upward > 2 at Z < Z 2 D ocean D (depth) increases downward > 2 at D > D 2 Changes in pressure with altitude in the atmosphere (left) and depth in the ocean (right). ressure always increases as the observer moves downward because the weight of the overlying column of fluid (air or water) increases. Note: Altitude is conventionally measured increasing upwards from the surface of the earth, and depth increasing downwards. Therefore pressure decreases with increasing altitude in the atmosphere and pressure increases with increasing depth in the "air is compressible"! density depends on pressure! = / [ (k/m) T ] Net 2!g gravity Z Z 2 2 By how much is > 2? The weight of the slab of fluid between Z and Z2 is given by the density,!, multiplied by volume of the slab) and g weight of slab =!#(area # height) #g. Set the area of the column to m 2, the weight is! g # (Z2 -Z): If the atmosphere is not being accelerated, there must be a difference in pressure (2 - ) across the slab that exactly balances the force of gravity (weight of the slab). - (2 - ) = weight =! g #(Z2 -Z). ressure increases as we descend in the atmosphere because the air at each level must hold up the weight of all the air above it. (Note the minus sign, pressure is lower at 2.) But the atmosphere is compressible, meaning the density! depends on the pressure itself! Use the erfect Gas Law =! (k/m) T to account for this fact. Obtain the barometric law:! = av /( ( k/m )T ) $ = - av [ mg / (kt) ] $Z. The quantity kt/mg = H has units of length. It is a property of the atmosphere, and it is the most important length in atmosphere, given a special name, scale height. $ = - av $Z/H. or $/ = % $Z/H If we had an atmosphere where the temperature did not change with altitude, the barometric law would have a very simple form in terms of the exponential function exp(), which appears on most hand calculators, exp(x) = e x, where e=2.78282., (Z) = o e -Z/H. o is the pressure at the ground (00,000 N/m 2 or atm = 000 mb). This is the simplest form of the Barometric Law describing the change of pressure with height in the atmosphere. How big is H: H = kt/mg=.38 e -23 x 273 /(4.8e -26 * 9.8) H ~ 8000 ( 8 km) The scale height H is the measure of depth in the atmosphere. total #mol/m 2 N = Hn o H x local n/m -3 kg/m 2 : M = H x!
Lecture 5. ES 5: 2 Sep 200 with the weight of overlying atmosphere? The distribution of masses (densities) in the atmosphere will adjust until this balance is reached, because, if there are unbalanced forces, masses of air will accelerate air parcels towards the balanced distribution. ressure vs depth in the ocean The weight of a slab of ocean with unit area ( m 2 ) is [mass of the slab]#g =! g (D - D2) which gives the pressure difference between the top and bottom of the slab, $ =! g $D. Since! is essentially constant for water (water is incompressible), the change in pressure across the slab is proportional to the thickness of the slab but not proportional to pressure itself (contrast to atmosphere). The pressure changes by the same increment for a given depth change, and pressure increases linearly, not exponentially, with depth in the ocean, = o +! w g D where! w is the mean mass density of seawater. Since the mass density of liquid water is about 000 times greater than the density of air, the pressure becomes very large in the deep ocean 2 D 2 D ocean atmosphere bar = 0 5 N/m 2 Lecture 5. ES 5: 2 Sep 200 ressure in the atmosphere (compressible) and ocean (incompressible) [Fig. 7.6 (McElroy)].
sealed vacuum (vapor pressure) Diagram of a barometer: measures atmospheric pressure Buoyancy Buoyancy is the tendency for less dense fluids to be forced upwards by more dense fluids under the influence of gravity. Buoyancy arises when the pressure forces on an object are not perfectly balanced. Buoyancy is extremely significant as a driving force for motions in the atmosphere and oceans, and hence we will examine the concept very carefully here. The mass density of air! is given by mn, where m is the mean mass of an air molecule (4.8#0-26 kg molecule - for dry air), and n is the number density of air (n =2.69 # 0 25 molecules m -3 at T=0 o C, or 273.5 K). Therefore the density of dry air at 0 C is! =.29 kg m -3. If we raise the temperature to 0 C (285.5 K), the density is about 4% less, or.24 kg m -3. This seemingly small difference in density would cause air to move in the atmosphere, i.e. to cause winds. Densities ( atm pressure, 300K) water: 000 kg/m 3 air:.6 kg/m 3 Hg (mercury): 3,600 kg/m 3 x Buoyancy force: Forces on a solid body immersed in a tank of water. The solid is assumed less dense than water and to have an area A (m 2 ) on all sides. is the fluid pressure at level, and x is the downward pressure exerted by the weight of overlying atmosphere, plus fluid between the top of the tank and level 2, plus the object. The buoyancy force is x (up &) per unit area of the submerged block. x Buoyancy force: Forces on a solid body immersed in a tank of water. The solid is assumed less dense than water and to have an area A (m 2 ) on all sides. is the fluid pressure at level, and x is the downward pressure exerted by the weight of overlying atmosphere, plus fluid between the top of the tank and level 2, plus the object. The buoyancy force is x (up &) per unit area of the submerged block. Net Force (Net pressure forces Gravity)
Lecture 5. ES 5: 2 Sep 200 Road map to ES 5 Lectures5: ressure, barometric law, buoyancy air water fluid moves Fig. 7.6: ressure in the atmosphere (compressible) and ocean (incompressible).