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Chapter 2 Fundamentals of Atmospheric Modeling

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Overhead Slides for Chapter 2 of Fundamentals of Atmospheric Modeling by Mark Z. Jacobson Department of Civil & Environmental Engineering Stanford University Stanford, CA 94305-4020 January 30, 2002 Scales of Motion Table 1.1. Scale Name Scale Dimension Examples Molecular scale « 2 mm Molecular diffusion Molecular viscosity Microscale 2 mm - Eddies 2 km Small plumes Car exhaust Cumulus clouds Mesoscale 2 - Gravity waves 2,000 km Thunderstorms Tornados Cloud clusters Local winds Urban air pollution Synoptic scale 500 - High / low pressure systems 10,000 km Weather fronts Tropical storms Hurricanes Antarctic ozone hole Planetary scale > 10,000 km Global wind systems Rossby (planetary) waves Stratospheric ozone loss Global warming Processes in an Atmospheric Model Figure 1.1 Dynamical / thermodynamical Gas processes processes Wind speed Gas photochemistry Wind direction Gas-to-particle conversion Air pressure Air density Air temperature Soil temperature Turbulence Radiative processes Transport processes Optical depth of Emissions gases / aerosols / Transport cloud drops Gases / aerosols / cloud drops / heat Visibility Dry deposition Infrared radiative transfer Gases / aerosols / Solar radiative transfer cloud drops Sedimentation Aerosols / cloud drops / Aerosol / cloud raindrops processes Nucleation Freezing / melting Coagulation Reversible chemistry Condensation / evaporation Irreversible chemistry Dissolution / evaporation Heterogeneous chemistry Deposition / sublimation Pressure Versus Altitude Figure 2.1 a. 100 80 60 1 mb (above 99.9%) 10 mb (above 99%) 40 Altitude (km) 100 mb (above 90%) 20 500 mb (above 50%) 0 0 200 400 600 800 1000 Pressure (mb) Density Versus Altitude Figure 2.1 b. 100 80 60 40 Altitude (km) 20 0 0 0.4 0.8 1.2 Density (kg m-3) Composition of the Lower Atmosphere Table 2.1. Volume Mixing Ratio Gas (percent) (ppmv) Fixed Gases Nitrogen (N2) 78.08 780,000 Oxygen (O2) 20.95 209,500 Argon (Ar) 0.93 9,300 Neon (Ne) 0.0015 15 Helium (He) 0.0005 5 Krypton (Kr) 0.0001 1 Xenon (Xe) 0.000005 0.05 Variable Gases Water vapor (H2O) 0.00001-4.0 0.1-40,000 Carbon dioxide (CO2) 0.0360 360 Methane (CH4) 0.00017 1.7 Ozone (O3) 0.000003-0.001 0.03-10 Fluctuations in Atmospheric CO 2 Figure 2.2. 360 350 340 330 mixing ratio (ppmv) 2 320 CO 310 55 60 65 70 75 80 85 90 95 Year Specific Heat and Thermal Conductivity Specific heat Energy required to increase the temperature of 1 g of a substance 1 oC Thermal conductivity Rate of conduction of energy through a medium Thermal conductivity of dry air (J m-1 s-1 K-1) - 5 k d » 0.023807 + 7.1128 ´ 10 (T - 273.15) (2.3) Table 2.2. Substance Specific Heat Thermal (J kg-1 K-1) Conductivity at 298 K. (J m-1 s-1 K-1) Dry air at constant pressure 1004.67 0.0256 Liquid water 4185.5 0.6 Clay 1360 0.920 Dry sand 827 0.298 Conductive Heat Flux Equation DT H = -k (J m-2 s-1) c d Dz Example 2.1. -1 -1 -1 Near the surface (T = 298 K, kd = 0.0256 J m s K ) DT = 12 K Dz = 1 mm ----> -2 Hf,c = 307 W m -1 -1 -1 Free troposphere (T = 273 K, kd = 0.0238 J m s K ) DT = -6.5 K Dz = 1 km ----> -4 -2 Hf,c = 1.5 x 10 W m Consequently, air conductivity is an effective energy transfer process only at the immediate ground surface. Daytime Boundary Layer Figure 2.3 a. Free troposphere Entrainment zone / Cloud Inversion layer layer Subcloud layer Altitude Neutral convective Boundary layer mixed layer Surface layer Daytime temperature Nighttime Boundary Layer Figure 2.3 b. Free troposphere Entrainment zone / Inversion layer Neutral Altitude residual layer Boundary layer Stable boundary layer Surface layer Nighttime temperature Temperature Structure of the Lower Atmosphere Temperature 4 1 k T = M v 2 (2.2) p B 2 a Figure 2.4 100 0.00032 Thermosphere 90 Mesopause 0.0018 80 0.011 70 Mesosphere 0.052 Pressure (mb) 60 0.22 50 Stratopause 0.8 40 2.9 Altitude (km) Ozone 30 Stratosphere 12 layer 20 55 10 Tropopause Troposphere 265 0 1013 180 200 220 240 260 280 300 Temperature (K) Zonally-/Monthly-Averaged Temperatures Figure 2.5 a January 100 160 180 200 80 220 60 240 280 260 40 240 Altitude (km) 220 210 210 20 200 220 0 -80 -60 -40 -20 0 20 40 60 80 Latitude Zonally-/Monthly-Averaged Temperatures Figure 2.5 b July 100 180 160 200 140 80 220 60 240 260 260 280 40 240 Altitude (km) 220 20 200 210 220 0 -80 -60 -40 -20 0 20 40 60 80 Latitude Ozone Production / Destruction in the Stratosphere Natural ozone production 1 O2 + hn O( D) + O l < 175 nm (2.4) O2 + hn O + O 175 < l < 245 nm (2.5) 1 O( D) + M O + M (2.6) O + O2 + M O3 + M (2.7) Natural ozone destruction O + hn 1 l < 310 nm 3 O2 + O( D) (2.8) O3 + hn O2 + O l > 310 nm (2.9) O + O3 2O2 (2.10) Equation of State Boyle's Law 1 p µ at constant temperature (2.12) V Charles' Law V µ T at constant pressure (2.13) Avogadro's Law V µ n at constant pressure and temperature (2.14) Ideal gas law (simplified equation of state) nR*T nA æR * ö p = = ç ÷T = NkBT (2.15) V V è A ø Equation of State nR*T nA æR * ö p = = ç ÷T = NkBT (2.15) V V è A ø Example 2.2 . Surface p = 1013 mb T = 288 K -19 3 -1 kB = 1.3807 x 10 cm mb K ----> N = 2.55 x 1019 molec. cm-3 At 48 km altitude p = 1 mb T = 270 K ----> N = 2.68 x 1016 molec. cm-3 Dalton's Law of Partial Pressure Total atmospheric pressure equals the sum of the partial pressures of all the individual gases in the atmosphere. Total atmospheric pressure (mb) pa = å pq = kBT å Nq = NakBT (2.17) q q Partial pressures of individual gas (mb) pq = NqkBT (2.16) Dry and Moist Air Total air pressure (mb) pa = pd + pv Number concentration air molecules (molec. cm-3) Na = Nd + Nv Equation of State for Dry Air * æ * ö æ * ö nd R T ndmd R nd A R pd = = ç ÷T = r d R¢ T = ç ÷T = Nd kBT V V è md ø V è A ø (2.18) Dry air mass density (g cm-3) n m r = d d (2.19) d V Dry air number concentration (molec. cm-3) n A N = d (2.19) d V Dry air gas constant (Appendix A) R* R¢ = (2.19) md Equation of State Examples Examples 2.3 and 2.4 Dry air, at sea level * æ * ö nd R T ndmd R pd = = ç ÷T = r d R¢ T (2.18) V V è md ø pd = 1013 mb T = 288 K R' = 2.8704 m3 mb kg-1 K-1 ----> -3 r d = 1.23 kg m Water vapor, at sea level * æ * ö nvR T nvmv R pv = = ç ÷T = r vRvT (2.20) V V è mv ø pv = 10 mb T = 298 K 3 -1 -1 Rv = 4.6189 m mb kg K ----> -3 -3 r v = 7.25 x 10 kg m Equation of State for Water Vapor * æ * ö æ *ö nvR T nvmv R nvA R pv = = ç ÷T = r vRvT = ç ÷T = NvkBT V V è mv ø V è A ø (2.20) Water-vapor mass density (kg m-3) n m r = v v (2.21) v V Water-vapor number concentration (molec. cm-3) n A N = v (2.21) v V Gas constant for water vapor R* Rv = (2.21) mv Volume and Mass Mixing Ratios Volume mixing ratio of gas j (molec. gas per molec. dry air) Nq pq nq c q = = = (2.24) Nd pd nd Mass mixing ratio of gas q (g of gas per g of dry air) r q mq Nq mq pq mqnq mq wq = = = = = c q (2.25) r d md Nd md pd md nd md Example 2.5. Ozone c = 0.10 ppmv -1 mq = 48.0 g mole ----> w = 0.17 ppmm T = 288 K pd = 1013 mb ----> 19 -3 Nd = 2.55 x 10 molec. cm ----> 12 -3 Nq = 2.55 x 10 molec. cm ----> pq = 0.000101 mb Mass Mixing Ratio of Water Vapor Equation of state for water vapor æR ö r R ¢T p = r R T = r ç v ÷ R¢ T = v (2.22) v v v v è R ¢ø e * R¢ R æm v ö mv e = = ç * ÷ = = 0.622 (2.23) Rv md è R ø md Mass mixing ratio of water vapor (kg-vapor kg-1-dry air) rv mv pv pv epv wv = = = e = = ecv (2.26) r d md pd pd pa - pv Example 2.6. pv = 10 mb pa = 1010 mb ----> -1 wv = 0.00622 kg kg = 0.622%. Specific Humidity = Moist-air mass mixing ratio (kg-vapor kg-1-moist air) pv R¢ pv r v r v RvT Rv epv qv = = = = = r r + r pd pv R ¢ p + ep a d v + pd + pv d v R¢ T RvT Rv (2.27) Example 2.7. pv = 10 mb pa = 1010 mb ----> pd = 1000 mb ----> -1 qv = 0.00618 kg kg = 0.618%. Equation of State for Moist Air Total air pressure r d + r vRv R ¢ pa = pd + pv = rd R ¢T + r vRvT = r aR¢ T r a (2.28) Gather terms, multiply numerator / denominator by density r d +r v e 1+ r v (r de) 1+ wv e pa = r aR¢ T = r a R¢ T = ra R¢ T r d + rv 1 + r v r d 1 + wv (2.29) Equation of state for moist (or total) air pa = r aRmT = r aR ¢T v (2.30) Gas constant for moist air (2.31) 1+ wv e æ 1- e ö Rm = R ¢ = R¢ ç 1 + qv÷ = R¢ (1 +0.608qv) 1 + wv è e ø Virtual temperature (2.32) Temperature of dry air having the same density as a sample of moist air at the same pressure as the moist air.
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