Part 3. Atmospheric Thermodynamics The Gas Laws

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1 Part 3. Atmosheric Thermoynamics The Gas Laws Eq. of state V = mrt mass gas constant for 1 kg of a gas For ry air ρ = m/ = ρrt or α = RT where α = 1/ ρ = ρ R T, where R a R M * = uniersal gas constant ( J eg kmol 1 ) 35

2 M mi = i = m / M i i i weighte oer molecular. wt. M i = molecular weight For water aor: Define: ε = R Mw R = M = Dalton s Law: R = 287Jeg kg e= ρ RT 1 1 * R R = = 461Jeg kg M w i 1 1 = = + e m + m V ' ' ρ = = ρ + ρ 36

3 e= ρ RT an = R ρ T ' ' ' ' e e ρ = ρ + ρ = + RT RT e e = + RT R R R e eε ρ = 1 + RT e = 1 (1 ε) RT e Define T = T / 1 (1 ε ) Then T = R ρt Virtual tem: T that ry air must hae in orer to hae the same ensity as moist air at the same ressure. 37

4 Aing moisture to air has the effect of raising T Moist air is less ense than ry air. Hyrostatic Equation Hyrostatic ressure is a result of the weight of an air column. Consier an air slab haing unit cross-sectional area: 38

5 Weight of slab: mg = gρz is a force/area. Therefore, δ reresents a net force acting in the ertical irection, an since ecreases with height, -δ acts uwar. To be in equilibrium: or δ = g ρ δ z z = ρ g hyrostatic equation 39

6 Alternate iew Vertical equation of motion: w t 1 = g + Viscosity ρ z acceleration P-gra graitational in ertical forceacceleration acceleration w Ignoring iscosity, assume 0 t = Then 1 = g ρ z or = ρg z 40

7 To fin at any height z, (0) = 0 = or ( z) z ( ) = gρ z z z gz Geootential φ - is the work that must be one against graitational fiel to raise a mass of 1 kg from sea leel to a gien height. 41

8 φ = g z (Change in otential energy) from the hyrostatic relation = gρz gz = α or φ = α The geootential at height z is We efine geootential height φ ( z) φ( z) = φ = gz φ (0) = 0 0 z φ( z) 1 z z = = gz g g

9 where g 0 is globally-aerage g at earth s surface g0 = 9.8ms Z is what is lotte on our weather mas. 2 Exress Z in terms of T, Eq. Of state: φ = α = gz = R ρt Then or R T α= φ = R T ln Integrating from with φ φ2 φ1 = R T ln 2 1 φ 43

10 Diiing by g 0, we hae 2 R Z2 Z1 = T ln g 0 1 Relace T by aerage T through a layer, RT Z Z ln ( / ) or 2 1 = 2 1 g0 RT Z Z ln ( / ) 2 1 = 1 2 g0 Calle thickeness eq. 44

11 The geootential height of the 500 mb surface is RT Z500 go Z 500 will be low if sea leel is low. ln sea leel 500 Conersely, for a gien sea leel, Z 500 is low, if mean T between surface an 500 mb is low. 45

12 First Law of Thermoynamics Internal Energy = Sum of kinetic energy of molecules. Increases in internal energy in the form of molecular motions is manifeste as increases in temerature. Consier a unit mass of gas which absorbs a certain quantity of heat energy q (in joules) i.e., by raiation or thermal conuction. As a result, the gas may o a certain amount of external work W. The excess of energy sulie to the gas oer the external work one by the gas is q w= u = u2 u1. In ifferential form: q w = u (First Law of Thermoynamics) incremental ifferential ifferential heat elemental internal work energy Consier a gas containe in a cyliner of fixe cross-sectional area with a moeable, frictionless iston. V since area is constant 46

13 The work one by the gas in exaning is equal to the force exerte on the iston (A) multilie by the istance x through which the iston moes. Thus, } V w = Ax = V calle -V work For a unit mass of gas in the free atmoshere we hae or w = α } w q = u + α First law 47

14 Secific Heats Suose that a small amount of heat q is gien to a unit mas of gas an its temerature increases from T to T + δt. The ratio q/t is calle the secific heat. If the olume is hel fixe then q C = T α= const. If α=const. 0 q = u + α 48

15 C u = T α= const. but for an ieal gas, then u = u( T) only or C u = t q = CT + α secific heat at const. q C = T = const. q = CT + α = CT + ( α) α since α = RT q = C T + ( RT ) α 49

16 or q = ( C + R) T α if = const. q q = ( C + R) T = C + R = C T or C = C + R Thus q=c T α Enthaly Define: h= u+ α as enthaly h = u + ( α ) 50

17 Remember u = C T From 1 st Law: q = C T + α = u + ( α) α h q = h α (Rem : q = ct α) Thus or h = C T h= C T with T=0 51

18 Preiously we efine geootential φ = gz = α q = h α = h ( + φ) = CT ( + φ) Calle Montgomery stream function If no heat is ae or taken away from a arcel in a hyrostatic atmoshere, then q = 0 h + φ = const. The Montgomery stream function is consere along isentroic surfaces. 52

19 Potential Temerature It is often conenient to efine a ariable that is conseratie uner aiabatic motion. Consier an aiabatic rocess: 0 q = = C T α Using the eq. of state, α = RT Then, C CT RT = 0= lnt l n R integrating from 00 =1000 mb where we let T=θ, to, C R T nt= T = θ l n l 00 C l nt θ = l n 00 R or T θ C / R = R/ C θ = T( / ) Poisson s Eq. The otential temerature is the tem. that the arcel of air woul hae if it were comresse aiabatically from its initial leel (,T) to sea leel ressure 00 = 1000mb. 53

20 The Aiabatic Lase Rate Consier an infinitely small arcel of air that is: Thermally isolate from its enironment such that heat is not ae or taken away from the arcel (aiabatic). The arcel immeiately ajusts to the hyrostatic ressure at any leel. Its motion is small so that its K.E. remains small. 54

21 For a arcel exeriencing aiabatic transformations, 0 q = = C T α Take eriatie with resect to Z, T C α = 0 z z Since ressure ajusts to hyrostatic at any leel, the hyrostatic equation. Gies us: z or = gρ o Define: γ = g/ C = 9.8 C/ km T T C + g = 0 = g/ C z z 55

22 Parcel Stability The buoyancy of a arcel relatie to its enironment is efine as: ' T θ ' Boy ( g ( g T0 θ0 Vertical accelerations are then, ' ' w θ T ( g = g t θ T 0 0 θ Suose the enironmental lase rate is > 0 z Enironmental lase rate For aiabatic motion, θ=constant, thus ' θ θ θ θ θ A 0 = < < 0 0 w 0an 0. t 56

23 The arcel will return to 0. Conersely, if it is brought own to ' θ θr θ0 B, = > 0 an arcel will return to 0. θ θ θ > the enironment is sai to be stable. z 0 0 Thus, if 0 θ Now suose < 0 z Enironmental lase rate 57

24 Parcel lifte to A will hae buoyancy continue to accelerate uwars. θ θ0 θ A 0 w > 0or > 0. It will t Parcel eresse to B will exerience buoyancy θb θ0 θ 0 < 0. w t < 0 an arcel will accelerate ownwars. The lase rate is unstable. If 0, z θ = then the atmoshere is neutral. 58

25 Stability in terms of Tan Γ = g/ C, Γ = e T z Γ e <Γ Γ e >Γ T At A TA T e Boy = g < 0 Te Stable Boy T At A TA T e = g > 0 Te Unstable 59

26 Summary Stable Neutral Unstable Γ<Γ Γ=Γ Γ>Γ θ θ θ > 0 = 0 < 0 z z z 60

27 Water Vaor Moisture Parameters Mixing ratio r = m ( massof water aor) m ( massof ry air) exresse in g/kg r (Troics marine air) at surface ~20 g/kg. r (hot summer ay in Colorao) ~ 6 g/kg. r (col ry winter ay in Colorao) ~0.1 g/kg. 61

28 Secific Humiity q = m ( massof water aor) m + m ( massof moistair) Saturation aor ressure 62

29 Saturation aor ressure oer ice e < e. Saturation Mixing Ratios si sw r s Since R ms ρs es / RT es = = = = m ρ ( e) / RT R es r s = ε es e s Generally es ( or ε es rs ( 63

30 Relatie Humiity (RH RH r = 100. r s Dew Point Temerature air becomes saturate when it is coole isobarically (at const. P) RH rs( atts, ) = 100 r ( att, ) s Lifting Conensation Leel (LCL) Leel at which a arcel of moist air can be lifte aiabatically before it becomes saturate. 64

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32 During lifting θ an r are constant, but r s ecreases until r s = r (at the surface). Wet-bulb temerature (T w ) Temerature to which a arcel of air is coole by eaoration of water into it at constant until the air is saturate with resect to water. It is measure with a well-entilate moistene wick coere with a thermometer. In general, T T T. w The eaoration of the wette bulb (the wick) as a certain amount of water aor ' r to the air, thus making ' rs.( Tw) = r( T) + r but rs( T) = r( T) Hence, r ( T ) > r ( T ) T > T. s w s w 66

33 If once a arcel of air becomes saturate, we retain all the conensate, the rocess can be consiere reersible since the latent heat liberate uring ascent will be consume by eaoration of water uring escent. Such a rocess is sai to be saturate aiabatic. If all the conensate is roe out of the arcel uring ascent, howeer, the rocess is irreersible since the water is not aailable uring escent for eaoration. The rocess is calle seuoaiabatic. 67

34 Saturate-Aiabatic an Pseuoaiabatic Processes When an air arcel is liften ry aiabatically until it becomes saturature, Conensation on rolets is initiate uon further ascent. Latent heat is release an the arcel cools at a lesser rate (becomes warmer than a rylifte arcel) than the ry arcel. Consier the First law for a system comose of ry air, water aor r, an liqui water r l : R C + r C + r C T + r q R α = ( l l ) 1 68

35 Deriation of seuoaiabatic equation Latent heat ue to conensation or eaoration is q = Lr. Thus if a certain amount of water is conense: r = r. l Since r 3 3 ~1 10 ; rl ~5 10 C Cl R ~1; ~1; r 1 C C R << C 0 0 C l C 1+ r + r T l C C 0 R 1+ r α = Lr R 69

36 Thus, CT α= Lr. Assume arcel is exactly saturate: then r = r CT α= Lrs. s Let α = gz (hyrostatic eq.) Differentiate with resect to z, C or T Lrs + g = z z T L rs T = g/ C. z { C T z Γ 70

37 The seuoaiabatic lase rate is T L rs 1+ = Γ z C T or T Γ Γ s = = z L 1+ rs C T rs Since 0then s. T > Γ <Γ Assumtions No suersaturation. No heat storage on ros. Water roe out of arcel. 71

38 Equialent Potential Temerature θ e We now erie a ariable which is conseratie for seuo aiabatic motions. Return to Let CT α= q RT α = (ignoring moisture) T α q C = = { T { T Remember: Thus, lnt ln θ = T s( entrohy) R/ C 00. lnθ = lnt R/ C l n+ const. 72

39 C lnθ = C lnt Rl n. or Thus, for a seuoaiabatic rocess: q Lr Cn l θ = = T T L Lr L CT CT CT s l nθ = rs = + r s s Can be shown to be small. θe 0 Lrs lnθ θ Lrs CT CT Lrs lnθe / θ = CT = or Lr s θe = θex. CT θ is the otential temerature of a arcel that has been lifte until r 0. e s On thermoynamic iagram, lift seuoaiabatically until seuoaiabat arallels θ 's then comress ry aiabatially to 1000 mb yiels θ. e θ e is consere uring seuoaiabatic ascent an escent. 73

40 Conitional Instability Consier the aboe souning Γ. Suose that the air with early morning surface temerature T 1 is heate to T 2 an lifte ry aiabatically to 0. At 0 the arcel is at the same temerature as the enironment. If it is lifte to A (LCL) it will become saturate an thereafter cool at Γ s when lifte. If lifte to B it will again equal the enironmental lase rate an if lifte further 0 ' T > an the arcel will accelerate uwars. 74

41 Summary The atmoshere is sai to be conitionally unstable wheneer: Γ <Γ<Γ. s 75

42 Conectie (or otential) Instability Consier an inersion layer A-B, which is at the to of a moist layer. Suose the entire layer A-B is lifte. The LCL for A is A. The LCL for B is B much higher than B. After the bottom of the layer near A becomes saturate, it will cool seuoaiabatically, whereas B continues to cool ry aiabatially. Eentually the entire layer is unstable. θ e Conition: < 0. z 76

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