Moule 4 Water Vaour in the Atmoshere 4.1 Statement of the General Meteorological Problem D. Brunt (1941) in his book Physical an Dynamical Meteorology has state, he main roblem to be iscusse in connection to the thermoynamics of the moist air is the variation of temerature rouce by changes of ressure, which in the atmoshere are associate with vertical motion. When am air ascens, it must eventually attain saturation, an further ascent rouces conensation, at first in the form of water ros, an as snow in the later stages. his statement of the roblem emhasizes the role of vertical ascens in roucing conensation of water vaour. However, several text books an aers iscuss this roblem on the assumtion that roucts of conensation are carrie with the ascening air current an the rocess is strictly reversible; meaning that if the am air an water ros or snow are again brought ownwars, the evaoration of water ros or snow uses u thame amount of latent heat as it was liberate by conensation on the uwar ath of the air. Another assumtion is that the ros fall out as the am air ascens but then the rocess is not reversible, an Von Bezol (1883) terme it as a seuo-aiabatic rocess. It must be ointe out that if the roucts of conensation are retaine in the ascening current, the mathematical treatment is easier in comarison to the seuo-aiabatic case. here are four stages that can be iscusse in connection to the ascent of moist air. (a) he air is saturate; (b) he air is saturate an contains water ros at a temerature above the freezing-oint; (c) All the water ros freeze into ice at 0 C; () Saturate air an ice at temeratures below 0 C. However, clous may be comose of suercoole water ros at temeratures as low as 40 C; it also forms an imortant stage in conjunction with the aforementione four stages of moist air. o treat water vaour in the atmoshere, its mass mixe with a certain amount of air in a given volume must be known. he moisture content of air can be exresse in several ways that relate the artial ressure of water vaour to its volume or mass mixing ratio in the atmoshere. 4.2 Moisture arameters We efine here the basic humiity variables, which essentially give an estimate of the amount of water in vaour form resent in a given volume of moist air. hese efinitions coul also be extene to other forms of hyrometeors which are at atmosheric ressure an temerature. he following efinitions are use in for the moist air arcels. (a) Mixing ratio (r): It is efine as the ratio of the mass of water vaour in a certain volume of air to the mass of ry air in that volume, so r m v m ρ v ρ ; m v mass of water vaour m mass of ry air ( kg/kg) (4.1) (b) Secific humiity: It is efine as the mass of water vaour in a given mass of air which is a mixture of air an water vaour; thus, m q v r (4.2) m v + m 1+ r 1
Since r is only a few er cent, the numerical values of r an q are nearly equal. Further, ieal gas law can be alie to each constituent of any mixture of gases, we may write the gas law for water vaour as e ρ v (4.3) where e an ρ v are the ressure an ensity of the water vaour an is the gas constant for 1 kg of water vaour. he gas constant for water vaour is calculate from the universal gas constant an molar mass water vaour as 1000 R * M H2 O 1000 8.3145 18.016 461.51 J kg 1 K 1 ; R 287 J kg -1 K -1 he ratio of gas constants for ry air an water vaour is calculate as ε R M v 18.016 M 28.97 0.622. From Dalton s law of artial ressures of a mixture of gases, the moisture content of the air can also be efine in terms of its artial ressure in the atmoshere. (c) Volume mixing ratio (r v ): r v e / (4.4) () Absolute Humiity: he concentration ρ v of water vaour in air is calle the absolute humiity (g/m 3 ). It can be easily obtaine from the ieal gas law of water vaour (4.3). Examle 1: At 0 C, ρ 1.275 kg/m 3, an ρ v 4.770 10-3 kg/m 3. Fin the total ressure exerte by the mixture. Solution: (1.275 kg m )(287.0 JK 1 kg 1 )(273.2 K) 3 99970.71Pa kg e ρ v 4.770 10 3 m 3 461.51 J K kg 273.2 K ( ) 601.4232 Pa Hence total ressure of the mixture 999.71 + 6.014 1005.72 hpa 4.3 Virtual emerature In meteorology, it is customary to efine a virtual temerature to take into consieration effect of water vaour in the atmoshere. Since M v < M an > R, therefore instea of using the gas constant, it is convenient to use R an fictitious temerature v (because never measure) in the equation of state for water vaour. hus, ρr v (4.5) Here v is the virtual temerature; its exression is now erive starting with the ensity of mixture, ρ ρ + ρ v ; ρ R an e ρ v. he ressure of the mixture is given as, + e or e Since ρ e R ; ρ e v, we can write the ensity of the mixture ρ as 2
ρ ρ + ρ v e R + e R + e 1 1 R ; which can be further written as, ρ R 1+ e R 1 or ρ R 1 e (1 ε) (4.6) Now (4.6) can be rearrange as ρr 1 e an ε R 287 0.622 (4.7) (1 ε) 461.51 If we efine v 1 e then (1 ε) the ressure in (4.7) can be written as given in the equation (4.5) above with ρ an as ensity an ressure of the air resectively. hus for the moist air, actual temerature is relace by virtual temerature in the equation of state; it is in fact use as a state variable in moist air ynamics. Also one may efine the virtual temerature, as that temerature the ry air woul attain so as to have the same ensity as that of the moist air at thame ressure. Examle 2: Given a total ressure 1026.8 hpa, fin the vaour ressure e in a mixture of water vaour an air if the water vaour mixing ratio r 5.5 g/kg. Solution: r ρ v ρ e R e e e. R εe e. Hence e r r + ε Put r 5.5 g/kg an 1026.8 hpa 102680 Pa in the above exression of e, an we get, ( 5.5 10 3)(102680) e 900.70 Pa ; or water vaour ressure e 9.00 hpa. 0.622 + 5.5 10 3 Exercise 1: Starting from the efinition of virtual temerature, show that v (1+ 0.61q). Also fin the exression for q. 4.4 Relation between moisture variables an water vaour ressure Humiity variables can also be efine in terms of water vaour ressure. he relation between mass mixing ratio r an e is erive easily in the following manner. Starting with the efinition of mixing ratio r given in (4.1), we have r ρ v ρ e R e ε e e (4.8) hecific humiity is relate to the mass mixing ratio r as q ρ v ρ ρ v ρ + ρ v r, as efine in (4.2). Now use the relation (4.8) for r to get q as, 1+ r 3
q ε e e 1+ ε e e 1 ε e (1 ε)e (4.9) 4.5 Saturate an unsaturate air he most commonly use hrases for moist air are inee misleaing. Phrases such as air is saturate with water vaour ; or air can hol no more water vaour ; an warm air can hol more moisture than col air are quite commonly use in weather escritions. All these hrases suggest that air absorbs moisture much like a songe, which is not the case with air. Accoring to Dalton s rincile of artial ressures for a mixture of gases, each comonent exerts a artial ressure an sum of the artial ressures is the total ressure exerte by the mixture. Hence, the exchange of water molecules between the li qui an vaour hases is inee ineenent of the resence of other comonents in air. Moreover, an equilibrium vaour ressure is establishe between the liqui an vaour hases when a certain number of water molecules leave the liqui hase at a given temerature an an equal number of water molecules from vaour hase enter the liqui hase. Air is sai to baturate at such equilibrium otherwise it is unsaturate. Both terms are use for characterizing ry an moist air. Unsaturate air: When liqui an vaour hase are imagine to coexist an the rate of evaoration of water from liqui excee rate of conensation, then air is unsaturate at a given temerature an it can be exresse by the air e, ( ). Saturate air: When the vaour ressure in the gas hase reach to a value (say ) such that the rate of evaoration of water from liqui is equal to rate of conensation of vaour in air at temerature of thystem. he air (, ) may exress thtate of thystem. Normally the interface of liqui an vaour hase is regare as a lanurface. 40 Fig. 4.1 Variation of saturation vaour ressure over lanurface of ure water with temerature. Saturate vaour ressure (hpa) 30 20 10 50 40 30 20 10 0 10 20 30 40 50 emerature ( 0 C) 0 Saturate mixing ratio: It is the mass of a given volume of air that is saturate with resect to lanurface of ure water, to the mass of ry air in a given volume; that is, R r s m vs ρ vs m ρ 0.622 Because >>, the eqn. (4.10) can be ut in the following simlifie form 1 ε (4.10) r s 0.622 ε (4.11) 4
hat is, at a given temerature, r s is inversely roortion to. Note that ( ), so the saturate mixing ratio r s is also a function of temerature; this imlies that r s increases with increasing temerature at constant ressure. But at constant temerature, r s increases with ecreasing ressure, which is evient from (4.11). he curve ( ) is shown in Fig. 4.1 as a function of temerature; ecreases with height in the trooshere. Since atmosheric ressure also ecreases with altitue, that is why the rising arcels of unsaturate air turn saturate leaing to clou formation an rains. he rocess is hastene in the trooshere because temerature also ecreases with altitue. Relative humiity: he relative humiity r h is the ratio of mass of water vaour m v in a samle of moist air of volume V, to the mass of water vaour m vs if the air in volume V weraturate. hus, we have r h m v m vs r r s e (4.12) he reorte relative humiity is generally exresse in er cent; i.e. RH r h 100. Isobaric cooling: Imagine that temerature of a arcel of moist air reuces continuously but its ressure remains thame. During this rocess, e will not change but will ecrease as the temerature falls uring cooling. As a consequence, uner isobaric cooling, the relative humiity of the arcel will increase with ecreasing temerature. If temerature continues to cool further own, then hase change of water vaour to liqui or ice may haen when threshol temerature is reache. Dew oint temerature ( ): It is that temerature to which air must be coole at constant ressure (i.e. isobarically) for it to becomaturate with resect to lanurface of ure water. hus, we have r, ressure is given by ( ) r s (, ) an the relative humiity at temerature an RH r s (, ) r s (, ) 100. A humb Rule: For moist air (RH > 50%), ecreases by 1 C for every 5% ecrease in RH. hus, starting at (ry bulb temerature) where RH 85%, then 100 85 5 3 C or 3 C Frost-Point emerature: It is that temerature to which air must be coole isobarically for it to becomaturate with resect to lanurface of ice. Examle 3: What are RH an for air at 1000 hpa an 18 C having a mixing ratio 6 g kg -1. Saturation mixing ratio is 13 g kg -1 at this state. Solution: RH 46%; from the thumb rule 7.2 C. However, to fin the actual value, we use the ehigram. One has to move from right to left along 1000 hpa line to intercet thaturation line of magnitue 6 g/kg; this occurs at 6.5 C; this is the ew oint temerature in this case. 5
On the globe, ew oint is a goo inicator of moisture levels in the air as the ressure tyically varies only by a few er cent satially an temorally. Over warm boies, highest ew oint temeratures are note, though comletaturation is absent in hot area. 4.6 hermoynamics of unsaturate moist air It has been shown that the equation of state for ry air can be alie to moist air when is relace by v, that is ρr v ρ ρ + ρ v ρr m R m 1+ 0.61 q ( ) R (4.13) hese two exressions are equivalent; here R m is the gas constant for moist air. Since moist air is a mixture of ry air an water vaour, thecific heat at constant volume of the mixture is etermine by consiering one kilogram of ry air an ξ kilogram of water vaour. Next, consier aing heat q to thamle which raises its temerature by ; we thus have (1+ ξ)q ( C v ) air + ξ(c v ) va ; volume (V ) const. (4.14) Because C v Since that C v ( ) mix q V, we have (1+ ξ) ( C v ) mix ( C v ) air + ξ (C v ) va ( ) mix q C v V (C v) air + ξ (C v ) va 1+ ξ ; V const. (4.15) ( C v ) air at SP 718 J K 1 kg 1 ; ( C v ) va at SP 1390 J K 1 kg 1, eqn. (4.15) states ( ) mix in a arcel is the mass weighte mean of ( C v ) air an ( C v ) va. In thame manner, we can calculate Since m ( ) mix ( ) mix as a mass-weighte mean of ( ) an C air ( ) as va ( ) + ξ(c air ) va 1+ ξ (4.16) ( ) at SP 1005 J K 1 kg 1 ; C air ( ) 1850 J K 1 kg 1, thecific heat of the va mixture with ressure remaining constant can be calculate from relation (4.16). Knowing thecific heat of the mixture at constant ressure, so long as there is no conensation of water vaour uring the lifting of arcels, we can calculate the lase rate of unsaturate moist air arcel as g Γ m ; Γ m < Γ (4.17) m ( ) g mix 4.7 Lifting Conensation Level (LCL) he lifting conensation level (LCL) is efine as the level to which an unsaturate arcel of moist air shoul be lifte aiabatically in orer to becomaturate. he rocess is exlaine in the Fig. 4.2. During lifting, the mixing ratio r an the otential temerature θ remain constant but r s ecreases until it becomes equal to r at the LCL. Hence the lifting 6
Pressure conensation level is the oint C on Fig. 4.2, to which a arcel has been lifte along the line AC from its initial osition at a oint A(,, r) with ressure, temerature an mixing ratio r. It is require to locate the oint C on a ehigram for issuing the local forecast. Mathematically, one can fin the LCL an the corresoning Z LCL, LCL an P LCL can be calculate from thermoynamic relations using the conition that thaturation mixing ratio of a arcel at oint C is equal to its mixing ratio at A. Note that oint C in Fig. 4.2 is the intersection of two lines: θ const. an r s const. Observations rovie values of,, an r at oint A. haturation mixing ratio line is rawn with r s const. In orer to locate the oint B, the arcel is coole isobarically; that is, the arcel is move horizontally to arrive at intersection oint B, where its temerature is the ew oint temerature ; the air arcel at B is saturate. From oint B, move along the line r s const. u to a oint C where it intersects the ry aiabat through oint A. he LCL is also referre to as the isentroic conensation temerature in the literature. Fig. 4.2 Lifting Conensation Level (LCL): he figurhows the roceure to fin the LCL of a arcel unergoing aiabatic ascent along the line θconstant (isentroe). he isobars are horizontal lines. he arcel is initially at oint A having ressure, temerature an mixing ratio r. he arcel cools isobarically to become saturate r s r, its temerature is, the ew oint temerature. he oint C is the LCL which is the intersection of line r s const., an θconst. Also, note that on this figure, isentroes an the isotherms are erenicular to each other. LCL is also calle the isentroic conensation level. A(,,r) B(,,r s ) C( LCL, LCL,r s ) LCL LCL isotherm θ const. B emerature r s const. C Saturation mixing ratio line isotherm A o calculate the lifting conensation level temerature l from given values of temerature () an the corresoning ew oint temerature ( ), a sufficiently accurate mathematical formula ue to Bolton (1980) is 1 1 +1.266 10 3 ln l l 0.514 10 3 ln (4.18a) In the above formula all temeratures are absolute. However, eq. (4.18a) can only bolve iteratively to obtain l. Bolton has also given a simle formula that exlicitly gives the lifting conensation level temerature, which reas 2840 l 3.5ln lne 4.805 + 55. (4.18b) In (4.18b) too all temeratures are in Kelvin. Wet-bulb temerature: It is measure with a thermometer covere with a moist cloth. he efinition of the ew oint temerature ( ) an the wet-bulb temerature ( w ) both involve cooling of a hyothetical air arcel to saturation, but there is a ifference. At, air has been coole to becomaturate, i.e. its mixing ratio is r s. However, the mixing ratio of air as its temerature tens to attain the wet bulb temerature is r; generally w, equality will aly 7
when air is fully saturate. Usually < w ; the thumb rule is that w is the arithmetic mean of an ; that is, w 0.5 ( + ). Whall be using the following terms frequently. Latent heat of vaorization (L v ) 2.25 10 6 J kg -1 (liqui to vaour) at 1 atm an 100 C Latent Heat of Conensation (L v ) 2.50 10 6 J / kg; (vaour to liqui) at 0 C. Latent Heat of Melting (L m ) 3.34 10 5 J / kg. (ice to liqui) 4.8 Equivalent Potential emerature When air is lifte vertically, its ascent will be just like ry air until it reaches saturation; further lifting of thaturate air will result in water vaour conversion to liqui water with the release of latent heat, which must be ae in thystem. he analysis becomes comlex as water may stay in thystem or the liqui ros may fall out of the system. In the latter case, the rocess is seuoaiabatic, because it is not reversible. Also, heat release as buoyancy to the rising moist saturate air to further lift it, which may again result in converting vaour to liqui. Consier an amount of heat Q release ue to conversion of water vaour to liqui in the first law of thermoynamics in the form given by eqn. (3.24), which can be easily ut in the entroy form for further analysis. he equation (3.14) reas as Q Q, which imlies ρ R R/C ake logarithm on both sies an then ifferentiate, θ o, then we obtain θ θ C R (4.19a) In eqn. (4.19a), the right han sie is the entroy of thystem therefore θ θ Q (4.19b) In view of eq. (4.19b), the otential enthaly ( θ) is taken as a variable instea of θ in the resent ay global moels esecially the unifie moels of atmoshere. It is now require to calculate Q. When an air arcel takes its uwar journey it will first becomaturate an any further rise of the now saturate air in the arcel will conense out r s (kg/kg of water), the latent heat release Q is given by Q L v r s (4.20) Substituting Q from (4.17) in (4.16) we obtain, θ θ L v r s (4.21) (lnθ) L vr s L vr s (4.22) 8
However, it shoul be rove that L v r s L r v s. Inee, L r v s L v r s r s 2 L v r r s s L v r s r s ; if << r s r s then r s >> r s L r v s L v r s L v r (neglecting r s s in the above equation) (4.23) Integrating the equation (4.23), we obtain the following result lnθ L vr s + c he constant c is to be evaluate; at low temeratures as r s 0, θ θ es c lnθ es. hus θ ex L vr s θ es ; an θ θ ex L vr s es (4.24) he quantity θ es is calle thaturate equivalent otential temerature. Accoringly, we can efine the equivalent otential temerature θ e as follows, L θ e θ ex v r θ ex L v q ; q secific humiity. (4.25) he otential temerature θ is conserve uring aiabatic transformations; but equivalent otential temerature θ e is a conserve quantity uring both ry an saturate aiabatic rocesses. We may efine θ e of a arcel as the otential temerature of the arcel when all its water vaour has conense out so that its mixing ratio r is zero. o fin θ e, the air arcel is first lifte aiabatically so that all the water vaour has conense an fallen out uring its uwar journey, an release of latent heat is ae to the arcel; then the air arcel is brought back aiabatically ownwar to 1000 hpa. he temerature thus attaine is the equivalent otential temerature θ e of the arcel. he equivalent wet-bulb otential temerature can be obtaine from wet-bulb temerature using the Poisson s equation. Both θ e an θ w (wet-bulb) rovie equivalent information for a rising or sinking moist arcel. Dry Static Energy (DSE): Sum of enthaly an the otential energy Φ; DSE + Φ (4.26) Moist Static Energy (MSE): Sum of enthaly, otential energy an latent heat content (L v q) of a moist arcel. Secific humiity q is close to the mixing ratio r an MSE is efine as, MSE + Φ + L v q (4.27) Both DSE an MSE are conserve quantities an lay a key role in the vertical motion of a arcel. When unsaturate air is lifte aiabatically, the enthaly ( ) is converte to otential energy (Φ) an the latent heat content remains unchange uring the uwar journey; therefore, so long the arcel remains unsaturate, DSE is the key variable in this 9
rocess. However, when air becomes saturate an further lifte aiabatically, there is an exchange in all the three terms aearing in the exression (2.27) for MSE. hat is, uring the uwar journey of saturate air, the otential energy (Φ) increases, while the enthaly ( ) an the latent heat content L v q ( ) both ecrease in such a manner that their sum remains unchange. Hence, MSE is an imortant conserve variable for the treatment of clous in numerical moels. Let s reresent DSE an h the MSE of a system, then conservation of s an h imlies that uring aiabatic ascent s 0 an uring saturate aiabatic ascent h 0, s 0 ( + Φ) 0 ; which gives z g Γ While for saturate aiabatic ascent of a arcel, h 0 ( + Φ + L v q) 0. Since thecific heat of air changes with the aition of moisture, the lase rate of moist air ( Γ m ) will iffer from Γ in accorance to the water vaour mixing ratio r of the air. For moist air, (1+ 0.87r) with 1005J kg 1 K 1 Γ m Γ 1+ 0.87r Γ [ 1 0.87r] therefore we have 4.9 Entroy of the Dry Air he form of the first law of thermoynamics together with thecon law of thermoynamics are use to erive an exression for the change in the entroy of the ry air when an amount Q of heat is ae to thystem. aking the first law of thermoynamics, Q C R. (4.28) From thecon law of thermoynamics, increase in entroy η of thystem is efine as η Q. (4.29) But, if heat exchanges reversibly then equality in (4.29) will hol an we have η Q. (4.30) hus on combining (4.28) an (4.30), an using (4.19) we get η R C θ θ. (4.31a) hus, eqn. (4.31) relates the change in entroy η to change in temerature an ressure of thystem. On integrating (4.31), we get the entroy of thystem as η lnθ + η 1. (4.31b) Here η 1 is a constant. Hence, the logarithm of the otential temerature θ gives the entroy of the air, an enables us to interret entroy of the ry air in terms of the well-unerstoo concets of the otential temerature. he entroy or θ an constitute the coorinates of a tehigram: θ as the orinate an as the abscissa; thus isentroes an isotherms are erenicular to each other in a tehigram. 10
4.10 Clausius-Claeyron Equation It is one of the most imortant equations, which is necessary while iscussing the water vaour conversion to liqui an its thermoynamic effects in atmoshere. When air is saturate, an equilibrium conition at a given temerature revails; that is, the number of molecules ue to evaoration leaving the liqui hase equals the number of molecules that return to liqui hase ue to conensation from the vaour hase. Evaoration takes lace when molecules from thurface of water are breaking away as vaour molecules; conensation occurs when vaour hase molecules collie with the liqui water surface an stick to it. Since the kinetic energy of vaour molecules is more than those of liqui, evaoration requires suly of heat to articles at the water surface. In other wors, heat is require to change the liqui hase to vaour. o convert a unit mass of water (hase 1) into vaour (hase 2), the amount of heat require is L v joules ( L v 2.25 x 10 6 J kg -1 ) which is, q2 L v Q u + α (4.32) q1 u2 u1 α 2 α1 For saturate air,, an it is constant throughout the rocess; therefore, we have from (4.32) L v u 2 u 1 + (α 2 α 1 ) (4.33 a) Since temerature is also constant uring the hase change rocess, we may also write q2 L v Q Q η L v (η 2 η 1 ) (4.33 b) q1 q2 q1 η2 η1 Equating the exression in (4.33a) with that in (4.33b), we obtain u 2 u 1 + (α 2 α 1 ) (η 2 η 1 ). (4.34) he above exression on rearrangement gives u 1 + α 1 η 1 u 2 + α 2 η 2 (4.35) Let us introuce the Gibbs Free Energy Function (or simly the Gibbs function) G as G u + α η (4.36) With the efinition of the Gibbs function, eqn. (4.35) can be written as G 1 G 2 (4.37) We thus infer from (4.35) an (4.37) that in an isothermal (constant), isobaric (constant) hase change, the Gibbs function G, remains constant. he Gibbs function G eens on temerature an ressure, but it remains constant uring hase transition. In orer to etermine the eenence of G on ressure an temerature, we ifferentiate eqn. (4.36) to obtain, G u + α + α η η (4.38) Now Q u + α an Q η, thus u + α η 0 in (4.38) an it becomes, G α η (4.39) Because G is same for both hases, i.e. G 1 G 2 gives α 1 η 1 α 2 η 2 (4.40) η 2 η 1 α 2 α 1 (4.41) 11
Substituting for η 2 η 1 from (4.33) in (4.41), we obtain the following relation L v (α 2 α 1 ) (4.42) he equation (4.42) is the Clausius-Claeyron equation, which exresses the change in saturation ressure that results from a change in temerature. Uner orinary atmosheric conitions, vaour hasecific volume is much larger than that of the liqui hase, i.e. α 2 >> α 1, an moreover, water vaour also behaves like an ieal gas; hence, we can neglect the term α 1 in the equation (4.42) to obtain the final form of the Clausius- Claeyron equation after substituting for α 2 / as L v α 2 L v 2 (4.43) At temeratures below 0 C, eq. (4.43) exresses thaturation vaour ressure of suercoole liqui water. However, (4.43) nees moification for exressing saturation vaour ressure of ice. For this case, latent heat of sublimation L s is substitute in lace of L v an the change of saturation vaour ressure over ice with temerature is calculate from the following relation ice L s 2. (4.44) At temerature above 0 C, only liqui water an water vaour are in equilibrium. Hence Clausius-Claeyron equation is the key equation in hase change. In eqn. (4.43), 461.55 J kg 1 K 1 is constant an to a first aroximation, if L v is also taken as a constant then thtraightforwar integration of (4.43) yiels, ln o L v 1 o ln o L v 1 1 R v o (4.45) In (4.45), o ( 0 ) is the value of saturation vaour ressure at the temerature 0. It is foun that o 6.112 hpa for the trile oint temerature 0 273.16 K. We therefore write (4.45) as o ex L v 1 1 o ; o 6.112 hpa (4.46) o ex C B ; C19.8313 (4.47a) Ae B/ ; A 2.53345 10 9 hpa ; B 5417.1181 K (4.47b) At subfreezing temeratures, saturation vaour ressures for vaour an ice can be comare from the following exression L f ex o i R o 1 ; L f L s L v (4.48a) 12
Numerically, a goo aroximation to this equation is the following relation, 273 2.66 i ( is in Kelvin) (4.48b) One can comute ( ) starting with o ( o ) an o 273.16 K using the Clausius- Claeyron relation. One can create a look-u table for every δ Τ 1 K increment in temerature in the above formulae to see u the calculations. More accurate comutation of ( ) an i ( ) have been comute by etens formulae (with an 0 in Kelvin) as given below, 0 ex 17.502 0 (hpa) ; over liqui water 32.19 (4.49a) i i0 ex 22.587 0 (hpa) ; over ice + 0.7 (4.49b) ( ) was erive by Bolton (he A more accurate formula for saturate vaour ressure for comutation of equivalent otential temerature. Monthly Wea. Rev., vol.108, 1980). he emirical formula with 0.1% accuracy over the temerature range 30 C 35 C, is as follows: 17.67 ( ) 6.112 ex + 243.5 ( in hpa; in 0 C) (4.49c) 4.11 Entroy of moist air Combining the first an secon law of thermoynamics in Sec. 4.9, the change in entroy of ry air η uring a reversible rocess is exresse as η R. A similar erivation for moist air can also be accomlishe when the referenctate is efine with temerature an ressure. In a arcel of moist air, there can be two isolate hases, viz., the gas an liqui water. In a unit kilogram of mixture, let there be m kg of ry air, m v kg of water vaour an m w kg of liqui water. In the mixture, the total ressure + e is exerte on liqui water when is ressure of ry air an e that of water vaour. Hence the entroy of the moist air mixture is given by η S + S v + S w m η + m v η v + m w η w (4.50) Also, η S + S v + S w (4.51a) o comute η, we nee to comute each comonent of the equation (4.51), thus S m η ( ) m η + η m m η ; m 0 Hence S m R Next, we calculate S v an S w as. S v m v η v + η v m v an S w m w η w + η w m w. 13
Now η v v R e v e, therefore S v m v {v R e v e } + η m v v to get S w m w C w + η m. w w hus an exression for, η ( S + S v + S w ) is the following: η ( m + m v v + m w C w ) m R e m v e + η m + η m. v v w w ; an it is easy (4.51b) Since a arcel is assume to be isolate with its surrounings, an increment m w in the water content of the arcel can only haen at cost of an equal amount of conversion water vaour; therefore, we have a ecrease m v such that, m w m v. From this fact, the final exression for the change in entroy of the mixture (air + WV + liqui), reas as η ( m + m v v + m w C w ) m R e m v e + (η η )m. (4.52) v w v From equation (4.33b), we can calculate the change in entroy ue to heat ae as a result of conversion of water vaour to liqui using η v η w L v in eqn. (4.52) to obtain η ( m + m v v + m w C w ) m R e m v e + L v m (4.53) v Integrating the exression in (4.53) on both sies, we obtain the following result for the entroy of the moist air, η η o + (m + m v v + m w C w )ln o m R ln o m v R ln e v + L v m v (4.54) he exression (4.54) consiers that water hase an vaour arearate in the arcel with e as the vaour ressure. However, such a system coul not be in equilibrium, an liqui water in the arcel will freely evaorate until air in the arcel attains saturation; an at this state the vaour ressure in the arcel is e. From (4.54) we can, therefore, obtain the entroy of a system with air at saturation vaour ressure at the given temerature ; an surely, this case is interesting. Substituting e in the exression (4.54) for η, we obtain an exression for saturate air arcel with m v as water vaour an m w as liqui water, η η o + (m m + m w C w )ln o m R ln o + L v m v (4.55) In (4.55), m as efine in (4.16) is thecific heat of the mixture. Now, consier a saturate air-water mixture in a arcel at ressure an temerature with 1 kg of ry air, ξ kg of liqui water an saturation vaour ressure. he entroy of such a saturate moist air arcel with liqui water resent, can be obtaine from exression (4.55) by setting m 1 kg ; m w ξ kg an ; the esire exression is η (m + ξ C w )ln R ln( ) + L v m v + const. (4.56) 14
he exression for the entroy of thaturate moist arcel given by (4.56) is much more comlicate as comare to that of a ry air arcel. 4.12 Processes that lea to saturation: A samle of moist air may attain saturation in several ways in the atmoshere. But rimarily there are five ways for the air to becomaturate, an these are iscusse in what follows. (i) Cooling of the air: When moist air is coole, it will attain a temerature (tho calle ew-oint temerature) where it becomes saturate. At the ew-oint temerature, the mixing ratio r, ( ) is equal to r s (, ). In view of (4.47b), we have r(, ) r s (, ) ε ( ) ε Aex B he above equation can bolve for obtaining as B ln Aε / (r ) { } (4.57) (ii) Cooling of air by evaorating water within the arcel at constant ressure: he wetbulb temerature w is efine as the temerature to which air must be coole at constant ressure by evaorating water that is loge in the given arcel. One can also erive an exression for w by consiering a samle of one kilogram of ry air an w kilograms of water vaour. hecific heat of the mixture then calculates to m + v w [1+ 0.94w] 1+ w Heat loss ue to evaoration of w mass of water in the arcel is calculate as (1+ w)q ( 1+ w)m ( 1+ w) ( 1+ 0.94 w) which is same as the amount of heat ( L v w ) use in evaorating mass w. Equating the two, we have ( 1+ w) ( 1+ 0.94 w) L v w L v Or, (1+ w)(1+ 0.94w) w (1 1.9w)L w v One may neglect the correction factor (1 1.9w) in the above exression, an we obtain w L v (4.58) At the initial temerature the mixing ratio is w an, at w air attains saturation when w mass of water has evaorate. herefore, for a change in temerature w with a corresoning change in mixing ratio w w s (, w ) w(, ), from (4.58), we have w w s (, w ) w(, ) L v or w w s (, w ) w L v. 15
hen, w is given by w L v [ w s (, w ) w]. haturation mixing ratio w s can be exresse in terms of saturation vaour ressure at the temerature w ; so we obtain the final exression for w as w L v ε ( w ) w (4.59) he exression (4.59) allows us to calculate a temerature if all the water vaour in the samle of air were conense out. In this case, the initial conitions are temerature an mixing ratio w ; an when all water has conense out, thystem attains a temerature e. Hence in (4.59), wubstitute w e an w s 0 an we get e + L vw. (4.60) From eqn. (4.60) a new temerature e is efine which is calle the equivalent temerature. (iii) Aiabatic cooling of moist air: When air arcels at a state ( 0, 0 ) rise in the atmoshere, their temeratures ecrease accoring to ry aiabatic lase rate Γ. If the initial mixing ratio of water vaour is w, then arcels coole aiabatically ue to ascening motions in the atmoshere, will reach to saturation when w s w. In other wors, arcels cool aiabatically until they attain saturation where w s w an c an arcels reach the final state ( c, c, w s ) ue to lifting. Dry aiabats at ressure c, along which arcels are rising, woul intersect thaturation vaour line w s w. Note that c is the isentroic conensation ressure, also known as the lifting conensation level (LCL) relate to temerature c as c ( w, c ), w s w. his allows the temerature c at the LCL to be calculate as, B c (4.61) Aε ln w c Hence, arcels rising aiabatically from their initial surface temerature 0, when become saturate, their temerature is c. At surface, arcels are at the otential temerature θ 0 where 0. he ry aiabatic from 0 will intersect thaturation vaour line w s w at ressure c. But c is still unknown. In orer to calculate c, we use the fact that arcels also have otential temerature θ at this level, because in aiabatic ascents θ is conserve. Now, the otential temerature θ at the LCL is given by θ c o c κ, κ R ; also θ o at o. 16
Since c an 0 are on thame aiabat, so ( c / o ) c / o ( ) ( /R) ( ) κ, which gives c o c / 0. Substituting the value of c in (4.61), we obtain the temerature at the lifting conensation level as, B c (4.62) 1 ln Aε o k w o c he temerature of the arcel at LCL, c (also calle the aiabatic conensation temerature), is obtaine by solving eqn. (4.62) iteratively. On the tehigram, LCL is locate at the intersection of θ const. an θ es const. (iv) Horizontal mixing: his kin of mixing occurs when two air masses of ifferent rovenance travel far an meet over another region. Generally, air masses retain their characteristics with clearly ientifiable bounary as a front, but their mixing rouce comlex weather associate with largcale clouing an wiesrea rains. During winters, the western isturbances visiting northwest Inia, originate over the Meiterranean Sea as extra-troical frontal systems, move over western Himalayas, Pakistan an arts of north Inia with the westerlies. A western isturbance is a low ressurystem which can be ientifie by a well-marke trough in the col westerlies at 500 hpa. he cyclonic circulation in the lower trooshere associate with a western isturbance woul rouce mixing of warm moist an col ry air masses that results in much neee reciitation (winter rains) an relief from severe col. But when temeratures i uring nights in winter, the relative humiity rises an it rouces largcale fog in the Ino-Gangetic Plain along the foothills of Himalayas which sreas eastwar to istances as long as 2000 km. he fog cover in this region is ense an ersistent which is clearly ientifiable on the INSA/KALPANA satellite imageries. he ersistent fog often turns into a ermanent haze ue to the resence of carbonaceous aerosols from the biomass burning in this region. With such meteorological consequences of mixing, not only over Inia but elsewhere also, it is very essential to unerstan the mechanism of horizontal mixing in the atmoshere. Fig. 4.3 wo unsaturate arcels A an B mix in the atmoshere. he curve searates thaturate an unsaturate air. After mixing of the air masses tyically with characteristics of arcels A an B, the mixture becomes suersaturate with resultant temerature at C. hus mixing of air masses of ifferent rovenance is a ominant weather roucing mechanism for various henomena that inclue fog, rainfall an thunerstorms. e Saturate C A B unsaturate Horizontal mixing can easily be exlaine on a thermoynamic iagram. he mixing of two air arcels is shown in Fig. 4.3 at thame ressure but ifferent temeratures an vaour ressures. his kin of mixing in the atmoshere is calle the isobaric mixing. 17
( ) an arcel B at (, 2, e 2 ) mix Assume that arcel A at a thermoynamic state, 1, e 1 on an isobaric surface accoring their masses. o further simlify matters, assume that arcels are of equal masses. After mixing, the arcels A an B havame temerature an vaour ressure (, w) calculate as 1 + 2 2, w w 1 + w 2 2 he vaour ressure of the mixture e woul form, if vaour ressure e when the air mixture is suersaturate, e ; with w 1 εe 1, w εe 2 2 ( ) is calculate to be e e( ) w ε ( ) excees thaturation vaour ressure ( ) > ( ). (4.63) ; Hence Fog ( ); that is, (v) Vertical mixing: Consier mixing of arcels ue to instability of an atmosheric layer; that is, Γ > Γ. Also the transfer of heat by mixing of the arcels woul increase the entroy (relate to θ) of thystem. Let us consier a arcel of air of mass m 1 at temerature 1, ressure 1 an otential temerature θ 1 be mixe with a arcel of air mass m 2 at temerature 2, ressure 2 an otential temerature θ 2. Assume that after mixing the resultant ressure of the mixture is. he mixing of these two arcels coul be assume to have roceee as a two-ste rocess: Ste 1: he arcels are brought aiabatically to ressure. hen, their resective temeratures at ressure can be calculate immeiately from their resective otential temeratures θ 1 an θ 2 as R 1 () θ 1 o an () θ 2 2 o (4.64) Ste 2: We let the two masses in the arcels mix, then the mean temerature m () of the mixture is calculate as m () m 1 1 () + m 2 2 () m 1 + m 2 On substituting for 1 () an 2 () from (4.64) in the RHS of the above equation we get, m () m 1θ 1 + m 2 θ 2 m 1 + m 2 hus the otential temerature θ m () of the mixture is given by o R R (4.65) θ m m 1θ 1 + m 2 θ 2 θ mean (4.66) m 1 + m 2 Since θ is a conserve variablo one can erform the calculations taking unit masses; that is, we can take m 1 m 2 1 kg mass. Hence, it may be remarke that the otential temerature of the mixture of two air masses after their mixing, is the arithmetic mean of otential temeratures of the original unmixe air masses. 18
4.13 Saturate Aiabatic Lase Rate (SALR) he rising arcels becomaturate as temerature in the trooshere ecreases with height. Moreover, temerature of the arcel with height falls at first with the ry aiabatic lase rate ( Γ g / ), but once air in rising arcel becomes saturate, its lase rate changes significantly an the temerature of thaturate air arcel shall change with height at thaturate aiabatic lase rate ( Γ s ). When water vaour conenses out in a rising air arcels, we have Γ s < Γ ue to release of latent heat. Moreover, tyical values of Γ s vary between 4-7 K km -1. In a saturate atmosheric layer near the groun, Γ s ~ 4K km -1 ; whereas Γ s ~ 6-7 K km -1 in the mile trooshere. On the tehigram, curve lines eicting the rise of air arcels uner saturate conitions are calle saturate aiabats (or seuoaiabats). he equivalent otential temerature remains constant along these curve lines. he temerature change of sinking saturate arcels also haens along seuoaiabats. For comuting thaturate lase rate in the atmoshere, thaturation mixing ratio of rising air arcels nee to be consiere in the erivation. Let us begin with the entroy of ry air is given by η lnθ + η 1 ln R ln + η 1 (4.67) Next, consier that an air arcel is lifte vertically with both vaour an liqui water resent in it. he conversion of water vaour to liqui water is accomanie by the release latent heat, which is ae to the rising arcel. If ξ is the total water in a arcel an it assume to remain in the arcel, then air will remain saturate because both evaoration an conensation will maintain equilibrium. If r s is the fraction of water in the vaour form, then ξ r s is the amount of liqui water resent in the arcel. In such a situation, note that r s is thaturation mixing ratio of the arcel; consequently, r s ρ v ε ρ a ( ε 0.622); ρ a R an ρ v When saturate air rises, the latent heat release uring the conensation rocess must be ae to thystem. hus the entroy of water vaour an liqui water is the entroy of ξ amount of liqui water at temerature lus the aitional entroy L v require to convert r s amount of water to vaour hase. he entroy η of saturate air is erive in (4.56) is ( + ξ C w )ln R ln( ) + L vr s + η 1 η. (4.68) he quantity C w reresents thecific heat of water. Also, η remains constant for a saturate air arcel rising aiabatically; hence ifferentiating (4.68), we get ( +ξ C w ) R ( ) ( +ξ C w ) R + L vr s η 0 ; which is written as + R + L vr s 0. he above equation may be further simlifie an ut in the following form, 19
( +ξ C w ) + R e + L vr s R 0 (4.69) he hyrostatic balance also alies for the moist air (vaour + liqui), so we have, z gρ(1+ ξ) R g (1+ ξ)z (4.70) he last term in (4.69) can be relace using (4.70) an we obtain, Since ( +ξ C w ) + R e r s ε r s z ε + L vr s z ε. 2 z From Clausius-Claeyron equation, we have + g (1+ ξ)z 0 (4.71) (4.72) L v 2 (4.73) Using (4.70), (4.72), (4.73) an taking ε R, we may arrange (4.71) to get an exression for thaturate lase rate of temerature, Γ s, if air is just saturate (ξ r s ), as z Γ s g 1+ ε 1+ L v 2 1+ ε C C w + L v { } + ε L 2 v 2 (4.74) Since <<, thecon term in the enominator on the RHS of eqn. (4.74) is negligible in comarison to other two terms. he equation (4.74) can be written in a more convenient form as Γ s z L 1+ v R Γ v 2 1+ ε L 2 v 2 We have use in the above equation Γ g DALR an ε << 1. (4.75) haturate aiabatic ase rate (SALR) coul also be erive from the exression of moist static energy (MSE) given in eqn. (4.24) which reresent thum of enthaly, otential energy an latent heat ( L v r s ) of a moist arcel. he MSE of a saturate air arcel, h s is given as, h s + Φ + L v r s We have alreay ointe out that h s aiabatic ascent, is a conserve quantity; therefore, uring saturate 20
h s 0 ( + Φ + L v r s ) 0 Which gives the following ifferential form, if L v is treate as a constant; that is, + gz + L v r s 0 Now r s ε ; taking logarithm on both sies an ifferentiating the result we get the following exression, r s r s ; or we write it as r s r s 1 + g R z. On using the Clausius-Claeyron equation, the final form 1 L v, the receing exression takes 2 r s L v r s + g z. (4.76) 2 R From (4.76), the exression for r s can bubstitute to eliminate it in C-C equation, an we get the equation, L + gz + L v r v s + g 2 R z 0. he above equation can be arrange as follows, ( + L2 v r s 2 ) + g(1 + L v r s R )z 0 he above equation now gives thaturate aiabatic lase rate, Γ s as follows Γ s z L 1 + v r s R Γ 1+ L2 vr s 2 (4.77) he exression for Γ s in (4.77) is same as that in equation (4.75), but the erivation (4.77) is very fast. However, in eriving (4.77) it has been assume that the air is saturate an there is no liqui water resent in the arcel, hence the revious erivation is much more general. Note that, Γ s < Γ with bouns as 0.3 Γ < Γ s < Γ in the atmoshere. If the actual lase rate or tho-calle ambient lase rate z is given by γ, then we have the following states of stability of moist air: (i) γ < Γ s Absolutely stable atmoshere (iv) γ Γ Dry neutral atmoshere (ii) γ Γ s Saturate neutral atmoshere (v) γ > Γ Absolutely unstable (iii) Γ s < γ <Γ Conitionally unstable 21
When the atmoshere is unstable with resect to seuoaiabatic islacements of arcels, there is a ossibility of conensation leaing to rainfall an thunerstorms. In numerical moels the instability is remove by ajusting the temerature rofile. Starting from the given temerature rofile at a gri oint of the moel, the lase rate is comute by evaluating iscrete erivatives of temerature in the vertical in orer to ientify unstable layers in the atmoshere. In an unsaturate atmosheric layers where γ > Γ, instability is remove by ajusting the lase rate with the ry aiabatic. But in thaturate unstable layers, the lase rate is ajuste with the Γ s as calculate from (4.77) at any level in such a manner that excess moisture rains out an the latent heat thus liberate is use in uating the temerature at that level. his is truly a very simlistic aroach that is harly use in resent ay numerical moels but a clear insight can be gaine from such simle comutations relate to moist arcel movements in the vertical. It may be mentione here that the conitional instability criterion (iii) is also known as the "conitional instability of first kin", but it takes longer times for such a conition to evelo in the atmoshere an, moreover, it can only exlain amlification of isturbances on a scale of a few kilometers. However, rai weather eveloments haen leaing to rain an thunershower even in the ry atmoshere on a scale of 100-1000 km in a short san of time. Such rai eveloments can be exlaine on the basis of "conitional instability of secon kin" that arises over a region ue to moisture convergence force by friction in the lanetary bounary layer in the largcale flow (J. Charney an A. Eliassen, 1964, On the growth of the hurricane eression. J. Atmos. Sci., 21). Later, H.L. Kuo (1974, Further stuies of the arameterizations of the influence of cumulus convection on largcale flow. J. Atmos. Sci., 31) roose a scheme to arameterize cumulus convection where the moisturuly into convective element is ue to the effect of large moisture convergence in the lanetary bounary layer. Inee, CISK escribes the cooeration (rather than cometition) of subgri scale an largcale rocesses in the atmoshere. Numerical weather reiction witnesse rai avances when Kuo s cumulus arameterization scheme was inclue in the forecast moels. It may nevertheless be remarke that moels that use the mass-flux schemes, rouce accurate numerical weather forecasts. hese moels are also use even for climate change rojections. Parameterization of convection ee, shallow or cumulus is a vast toic of current research esecially for their use in high resolution weather reiction moels. 22