Lecture 6DefinitionWork and Temperature (General)Work and Temperature (Ideal Gas)Relationship between T and pThusAdiabatic TransitionIntegrateContinueReviewThus,Slide 12Dry AirExerciseSlide 15Potential TemperaturePhysical MeaningSlide 18AdiabatsDry Adiabats = 290 KReduce pressure to 900 hPaSlide 23Reduce pressure to 700 hPaSlide 25Adiabatic ProcessesSlide 27Slide 28Slide 29Slide 30Slide 31Slide 32Slide 33Slide 34Slide 35Slide 36Slide 37Slide 38The Parcel ModelRising & Sinking ParcelsMovie: “The Day After Tomorrow”Problem With PremiseSolutionHeight Dependence of TParcel TemperatureStep 1Step 2: DifferentiateStep 3: Hydrostatic EquationStep 4Slide 50ResultSlide 52ConclusionDry-Adiabatic “Lapse Rate”Important NoteLecture 6Lecture 6Adiabatic ProcessesAdiabatic ProcessesDefinitionDefinitionProcess is Process is adiabaticadiabatic if there is no if there is no exchange of heat between system and exchange of heat between system and environment, i.e.,environment, i.e., dq = 0dq = 0Work and Temperature (General)Work and Temperature (General)First law: dU = dQ – dWFirst law: dU = dQ – dWAdiabatic process: dU = -dWAdiabatic process: dU = -dWIf system If system does workdoes work (dW > 0), dU < 0 (dW > 0), dU < 0 system coolssystem coolsIf work is done If work is done on on system (dW < 0), dU > system (dW < 0), dU > 00  system warmssystem warmsWork and Temperature (Ideal Gas)Work and Temperature (Ideal Gas)Adiabatic process: cAdiabatic process: cvvdT = - pddT = - pdExpansion (dExpansion (d > 0) > 0)  dT < 0 (cooling) dT < 0 (cooling)Contraction (dContraction (d < 0) < 0)  dT > 0 (warming) dT > 0 (warming)Relationship between T and pRelationship between T and pdpdqdTcpdq = 0 dpdTcpBut, pRT[Eq. (4) from last lecture.]ThusThusdppRTdTcppdpcRTdTpAdiabatic TransitionAdiabatic TransitionSuppose system starts in state with Suppose system starts in state with thermodynamic “coordinates” (Tthermodynamic “coordinates” (T00, p, p00))System makes an adiabatic transition to System makes an adiabatic transition to state with coordinates (Tstate with coordinates (T11, p, p11))IntegrateIntegrate 1010TTppppdpcRTdT11lnlnpppTToopcRT (1)ContinueContinue(1)  0101lnlnlnln ppcRTTp0101lnlnppcRTTp(2)ReviewReview xaxalnln Thus,Thus,(2) becomes(3)pcRppTT0101lnlnThus,Thus,(3) becomes(3) becomespcRppTT0101Poisson’s Equation(4)Dry AirDry Air2859.010040.2871111KkgJKkgJcRp2859.00101ppTTExerciseExercisepp00 = 989 hPa = 989 hPaTT00 = 276 K = 276 Kpp11 = 742 hPa = 742 hPaTT11 = ? = ?Answer: TAnswer: T11 = 254 K = 254 KExerciseExercisepp00 = 503 hPa = 503 hPaTT00 = 230 K = 230 Kpp11 = 1000 hPa = 1000 hPaTT11 = ? = ?Answer: 280 KAnswer: 280 KPotential TemperaturePotential TemperatureLet pLet p00 = 1000 hPa = 1000 hPaRemove the Remove the subscripts from psubscripts from p11 and Tand T11Denote TDenote T00 by by pcRphPaT/1000 is called the potential temperaturepcRpPT/0Physical MeaningPhysical MeaningInitial state: (T, p)Initial state: (T, p)Suppose system makes an adiabatic transition Suppose system makes an adiabatic transition to pressure of 1000 hPa to pressure of 1000 hPa New temperature = New temperature = Potential temperature is the temperature a Potential temperature is the temperature a parcel would have if it were to expand or parcel would have if it were to expand or compress adiabatically from its present compress adiabatically from its present pressure and temperature to a reference pressure and temperature to a reference pressure level. Po = 1000 mb.pressure level. Po = 1000 mb.Physical MeaningPhysical MeaningRemoves adiabatic temperature changes experienced during vertical motionºC and K are interchangeable; best to convert it to K when making calculations such as differences.is invariant along an adiabatic pathadiabatic behavior of individual air parcels is a good approximation for many atmospheric applications…from small parcels to larger convection.AdiabatsAdiabatsLet Let  be given be givenRe-write last equation:Re-write last equation:pcRhPapT1000In the T-p plane, this describes a curve.Curve is called a dry adiabat.Dry AdiabatsDry Adiabats = 290 K= 290 KInitial state:T = 290.0 K, p = 1000 hPaReduce pressure to 900 hPaReduce pressure to 900 hPaNew temperature:T  281 KExerciseExerciseCalculate T to nearest tenth of a degreeCalculate T to nearest tenth of a degree2859.01000hPapTKK4.28110009002902859.0Reduce pressure to 700 hPaReduce pressure to 700 hPaNew temperature:T  262 KExerciseExerciseCalculate new T to nearest tenth of a Calculate new T to nearest tenth of a degreedegreeAnswer: 261.9 KAnswer: 261.9 KAdiabatic ProcessesAdiabatic ProcessesIn the T-p plane, an adiabatic process can In the T-p plane, an adiabatic process can be thought of as a point moving along an be thought of as a point moving along an adiabat.adiabat.The Parcel ModelThe Parcel Model1.1.An air parcel is a hypothetical volume of An air parcel is a hypothetical volume of air that does not mix with its surroundingsair that does not mix with its surroundings1.1.Parcel is a Parcel is a closed closed system.system.2.2.Parcel moves adiabatically if there is no Parcel moves adiabatically if there is no exchange of heat with surroundings.exchange of heat with surroundings.1 and 2 1 and 2  parcel is parcel is isolatedisolatedParcel doesn’t interact with surroundings.Parcel doesn’t interact with surroundings.Rising & Sinking ParcelsRising & Sinking ParcelsIf a parcel rises adiabatically, its pressure If a parcel rises adiabatically, its pressure decreasesdecreases parcel coolsparcel coolsIf a parcel sinks adiabatically, its pressure If a parcel sinks adiabatically, its pressure increasesincreases parcel warmsparcel warmsMovie: “The Day After Tomorrow”Movie: “The Day After Tomorrow”Premise: global warming produces Premise: global warming produces gigantic stormsgigantic stormsIn these storms, cold air from upper In these storms, cold air from upper troposphere is brought down