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UW ATMS 587 - Chapter 3 Notes

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Chapter 3Atmospheric ThermodynamicsRevised: 26 May 2005NOT FOR DISTRIBUTIONiiContents3 Atmospheric Thermodynamics 13.1TheGasLaws ....................... 23.1.1VirtualTemperature................ 73.2TheHydrostaticEquation................. 93.2.1Geopotential .................... 113.2.2 Scale Height and the Hypsometric Equation . . . 133.2.3 Thic kness and heights of constant pressure surfaces 163.2.4 Reduction of pressure to sea level . . . . . . . . . 173.3TheFirstLawofThermodynamics............ 193.3.1Joule’slaw ..................... 223.3.2 Specificheats.................... 233.3.3Enthalpy ...................... 253.4AdiabaticProcesses .................... 263.4.1Conceptofanairparcel.............. 273.4.2 The dry adiabatic lapse rate . . . . . . . . . . . . 283.4.3Potentialtemperature ............... 283.4.4Thermodynamicdiagrams............. 303.5WaterVaporinAir .................... 323.5.1 Moisture parameters . . . . . . . . . . . . . . . . 333.5.2Latentheats .................... 413.5.3 Saturated adiabatic and pseudoadiabatic processes 423.5.4 The saturated adiabatic lapse rate . . . . . . . . 423.5.5 Equivalent potential temperature and wet-bulbpotentialtemperature .............. 433.5.6Normand’srule................... 463.5.7 Net effects of ascent followed by descent . . . . . 47iii3.6StaticStability ....................... 483.6.1Unsaturatedair................... 483.6.2Saturatedair .................... 543.6.3 Conditional and convective instability . . . . . . 543.7 The Second Law of Thermodynamics and Entropy . . . 583.7.1TheCarnotcycle.................. 583.7.2Entropy ....................... 633.7.3 The Clausius—Clapeyron Equation . . . . . . . . 663.7.4 Generalized Statement of the Second Law ofThermodynamics................. 71EXERCISES .......................... 74IChapter 3A tmospheric Thermodynam icsThe theory of thermodynamics is one of the cornerstones and crown-ing glories of classical physics. It has applications not only in physics,chemistry and the Earth sciences, but in subjects as diverse as biol-ogy and economics. Thermodynamics plays an important role in ourquantitative understanding of atmospheric phenomena ranging fromthe smallest cloud microphysical processes to the general circulationof the atmosphere. The purpose of this chapter is to introduce somefundamental ideas and relationships in thermodynamics and to applythem to a n umber of simple, but important, atmospheric situations.Further applications of the concepts developed in this chapter occurthroughout this book.In the firstsectionweconsidertheideal gas equation and itsapplication to dry air, water vapor, and moist air. In Section 3.2an important meteorological relationship, kno wn as the hydrostaticequation, is derived and interpreted. The next section is concernedwith the relationship between the mechanical work done by a systemand the heat the system receives, as expressed in the First Law ofThermodynamics. There follow several sections concerned with ap-plications of the foregoing to the atmosphere. Finally, in Section 3.7,the Second Law of Thermodynamics and the concept of entropy areintroduced and used to derive some important relationships for at-mospheric science.12 Chapter 3 Atmospheric Thermodynamics3.13.1 The Gas LawsLaboratory experiments show that the pressure, volume, and tem-perature of any material can be related by an equation of state overa wide range of conditions. All gases are found to follow approxi-mately the same equation of state, whic h is referred to as the idealgas equation. For most purposes we may assume that atmosphericgases, whether considered individually or as a mixture, obey theideal gas equation exactly. In this section we consider various formsof the ideal gas equation and its application to dry and moist air.The ideal gas equation may be written aspV = mRT (3.1)where p, V, m,andT are the pressure (Pa), volume (m3), mass (kg),and absolute temperature (in kelvin, K, where K =◦C + 273.15) ofthe gas, respectively, and R is a constant (called the gas constant)for 1 kg of a gas. The value of R depends on the particular gas underconsideration. Since m/V = ρ,whereρ is the density of the gas, theideal gas equation may also be written in the formp = ρRT (3.2)For a unit mass (1 kg) of gas m =1and we ma y write (3.1) aspα = RT (3.3)where α =1/ρ is the specificvolumeof the gas, that is, the volumeoccupied by 1 kg of the gas at pressure p and temperature T .If the temperature is constant, (3.1) reduces to Boyle’s law1,which states if the temperature of a fixed mass of gas is held con-stant, the volume of the gas is inversely proportional to its pressure.Changes in the physical state of a body that occur at constan t tem-perature are termed isothermal. Also implicit in (3.1) are Charles’two laws.2The first of these laws states for a fixed mass of gas atconstant pressure, the volume of the gas is directly proportional to1The Hon. Sir R ob ert Boyle (1627—1691) Fourteenth child of the first Earl ofCork. Physicist and chemist, often called the “father of m o dern chemistry.” Discoveredthelawnamedafterhimin1662. Responsibleforthefirst sealed th erm om e ter m a dein England. One of the founders of the Royal So ciety of London, Boyle declared: “T heRoyal So ci ety value s no kn owle d g e bu t as it h a s a t e n d e n cy to use it!”2JacquesA.C.Charles(1746—1823) French p hysical chemist and inventor. Pioneerin th e use of hydro g e n in m a n -c a rr y in g balloon s . W hen B e n ja min Fran k lin ’s experimentswit h lightnin g beca me kn ow n , Charles repea te d the m w it h his ow n inn ovatio ns. Fran klinvisited Charles and congratulated him on his work.Section 3.1 3.1 The Gas Laws 3its absolute temperature. The second of Charles’ laws states for afixed mass of gas held within a fixed volume, the pressure of the gasis proportional to its absolute temperature.begin box BOX 3.1: GAS LAWS AND THE KINETICTHEORY OF GASES: HANDBALL ANYONE?The kinetic theory of gases pictures a gas as an assem blage ofnumerous identical particles (atoms or molecules)3that move in ran-dom directions with a variety of speeds. The particles are assumedto be very small compared to their average separation, and they areperfectly elastic (i.e., if one of the particles hits another, or a fixedwall, it rebounds, on average, with the same speed that it possessedjust prior to the collision). It is shown in the kinetic theory of gasesthat the mean kinetic energy of the


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UW ATMS 587 - Chapter 3 Notes

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