ATMO 551a Homework 6 solutions Fall 08

Transcription

1 . A rising air parcl in th cor of a thundrstorm achivs a vrtical vlocity of 8 m/s similar to th midtrm whn it rachs a nutral buoyancy altitud at approximatly 2 km and 2 mb. Assum th background atmosphr is isothrmal at this altitud. How much will th air parcl ovrshoot th nutral buoyancy lvl altitud? a. St th kintic nrgy/mass of th air parcl at th nutral buoyancy lvl qual to th work/mass don by th ngativ acclration of th air parcl abov th nutral buoyancy altitud. Th acclration is proportional to th Brunt Vaisala frquncy squard: Eq s from adiabatic laps rat nots. a = d 2 x dt = "# 2 x = "x g $ $ z "# # "z = $ $ "T "z "z 32 T "T adiabatic #z #z #z 2 w #z 2 = " $ adz = $ 2 zdz = 2 $ 2 zdz = 2 2 b. Solv for th hight abov th nutral buoyancy lvl at which th potntial nrgy pr unit mass quals that of th kintic nrgy and thrfor at which th parcl stops moving vrtically. "z = w # = w g $ = w T + / g T $ z z, T + adiabatic. / z 2 + In this cas, d T /dz = and T ~ 22K and w = 8 m/s 2. # 22 "z = 8 $ [.] / 2 # 22 = 8 $ [.] / 2 / 2 = 3833m This is quit a bit of ovrshoot and can b important for instanc xchang btwn th stratosphr and troposphr. Such larg ovrshoot rquirs a lot of CAPE which can only occur ovr land whr a lot of low lvl moistur is availabl with coolr air aloft. Kursinski 2/7/28

2 2. N 2 collisional crosssction you ll nd this for problm 3 a. Givn that th diffusivity of air for typical surfac conditions is 2-5 m 2 /s, work backwards to dtrmin th approximat collisional crosssctional ara of N 2 molculs colliding with othr N 2 molculs chck units: kg3 / 2 m 3 s s 2 m 2 kg / 2 s 3 m 2 kg m = m2 D = = 8 "m 8 "m k B T 3 / 2 P k B T 3 / 2 DP 3 m =.28 kg/mol /N A = kg/molcul k B = J/K D = m 2 /s P = mb T = 288 K = m Whn th man fr path quals a scal hight, molculs can potntially ovrcom arth s gravity and scap to spac. a. Dtrmin th approximat prssur lvl at which this occurs in Earth s atmosphr? " = N Prssur scal hight H = k B T mg N = k T B mg = k T B P 2 3 P = mg 4 Using a mass of 28 g/mol = kg, yilds a prssur of Pa = mb. 2 Kursinski 2/7/28

3 b. Dtrmin th approximat hight of th xobas for N 2. Assum th avrag tmpratur of th atmosphr up to 9 km is 23K and th air is sufficintly mixd that th man molcular mass is g/mol. Abov 9 km, assum th avrag tmpratur is 5 K and th scal hight dpnds on th slctd molcul, in this N 2. Th numbr of prssur scal hights btwn th surfac and mb is ln/4.5-9 ~ 26. Assuming an avrag tmpratur of 23 K blow 9 km and a man molcular mass of g/mol and a surfac prssur of mb yilds a prssur at 9 km of approximatly.6 mb [P = P surf xp-9/h ] and thr ar approximatly 3.4 scals hights btwn th surfac and 9 km [=ln mb/.6 mb]. This mans anothr 2.7 scal hights = ar ndd to rach th xobas. Assuming a tmpratur of 5 K and using th N 2 mass, th avrag scal hight abov 9 km is 5.2 km. Th xobas altitud for N 2 is thrfor approximatly km + 9 km = 28 km. 4. Using th form whr u = u + u, a. Show that th horizontal moistur flux is " a uq = " a u q + " a uq First ky point: whn w dcompos a variabl, x, into its man and prturbations, th man of th prturbations, x =. Now lts calculat th horizontal moistur flux. " a uq = " a u + u [ ][ q + q ] = " a u q + u q+uq + uq " a uq = " a u q + u q + u q + uq Th rason for th last stp is q = and u =. = " a u q + " a uq b. Dscrib th kinds of fluxs that th two trms, " a u q and " a uq rprsnt. Th first is just th man wind tims th man spcific humidity which is simpl man advction. Th scond trm is an ddy corrlation trm which is nonzro if and only if th variations in vlocity ar corrlatd or anticorrlatd with th variations in spcific humidity. c. Explain why for th vrtical flux " a wq = " a w q + " a wq # " a wq Thr is typically not much of an avrag vrtical vlocity bcaus you can t accumulat mass in th uppr troposphr. As som mass gos up, othr mass has to com down. 3 Kursinski 2/7/28

4 5. SH Flux into surfac Considr a citrus tr grov whr th trs ar about 5 m in hight and th wind at m is blowing at 2 m/sc with a surfac tmpratur of o C and prssur of mb. a. Calculat an approximat ddy diffusivity Assum th tmpratur at m is o C warmr than th surfac tmpratur W ll crudly assum th scal of th ddis is th scal of th trs and w ll us th wind at 5 m hight as th ddy vlocity which w ll tak as half th wind at m hight. So th ddy diffusivity is approximatly D ~ 5 m 2 m/s ½ = 5m 2 /s. Th diffusiv flux is b. Calculat th vrtical snsibl hat flux F dif = -D ρ C p dt/dz ~ -5m 2 /s.2 kg/m 3 4 J/kg/K / K/m = - 6 Wm 2 This is larg and is probably a bit of an ovrstimat bcaus th diffusivity is probably ovrstimatd a bit. c. Is it up or down? Diffusion works in th opposit dirction as th tmpratur gradint. Th tmpratur incrass upwards so th snsibl nrgy flux is dirctd downwards in th atmosphr. 6. Surfac vaporativ flux incras with global warming Suppos th Earth s surfac wr to warm by 4 o C as prdictd by som climat modls. Suppos that th rlativ humidity of th air and th winds wr to rmain th sam. Using th arodynamic formula for latnt hat flux, dtrmin th ratio of th nw surfac latnt hat flux to th prsnt surfac latnt hat flux. Th arodynamic surfac latnt hat flux is F LH = " a C W U q s # q a and th ratio of th nw to th prsnt or old latnt hat flux is F LH "nw F LH "old = # L C U q " q a"nw v W "nw nw s"nw a"nw # a"old C W "old U old W xpand th spcific humidity ratio in trms of what th spcific humidity is 4 Kursinski 2/7/28

5 q s"nw " q a"nw = q s"nw " q a"nw # # m v s"nw $ s"nw # m v s"old $ s"old P a"nw a"nw P a"old a"old a"nw s"nw s"old a"old " m v a"nw " m v a"old q s"nw " q a"nw # $ m v s"nw P a"nw # $ m v s"old P a"old s"nw a"nw s"old a"old " m v a"nw " m v a"old Now w us = s RH and th diffrntial form of th Clausius Clapyron quation d s s = T 2 dt So whn RH is constant, So d = d RH s RH s = RH d s = d s = RH s s T dt 2 so nw old nw = s nw = s old + " s = old old old = RH s RH s s q s"nw " q a"nw # q s"nw " q a"nw # q s"nw " q a"nw # s L s old + s v old T "T 2 old = s"nw $ a"nw RH s s"s"nw s"a"nw = s"old $ a"old RH s s"s"old s"a"old s T 2 "T RH s s"s"old " RH 2 a s"a"old 2 RH s s"s"old s"a"old $ RH s s"s"old s"a"old $ RH s s"s"old s"a"old T T = P $ s"old 2 T T 2 5 Kursinski 2/7/28

6 Notic that with ths simplifications w didn t vn nd to know what th currnt rlativ humidity is. W simply had to assum it rmains unchangd in a warmr climat. This is approximatly what th global climat modls do. Is this corrct? I don t know. Thr ar som obsrvations that ar consistnt with this typ of constant RH bhavior with warming. W thn assum th surfac prssur dos not chang with global warming to simplify this furthr. Thr actually will b a slight incras in surfac prssur as th amount of watr vapor in th atmosphr incrass but w ignor that subtl chang. So q s"nw " q a"nw # 2 W also nd to considr potntial changs in " a#nw C W #nw U nw. W will ignor changs in th latnt hat although thr will b a slight dcras as th tmpratur changs th watr molculs ar mor activ whn thy ar warmr and thrfor it is asir for thm to fly off th surfac. Thrfor thr is not as much nrgy associatd with thm changing phas and flying off th surfac as tmpratur incrass. Th tmpratur dpndnc of thmolar latnt hat, L m, is L m = α T crit - T.375 whr T crit is th critical tmpratur which is th tmpratur whr th distinction btwn vapor and liquid cass to xist K for watr and α is a constant that dpnds on th liquid. From this w s that L m"nw L m"old = T crit " T nw T crit " T = " 292 = old " 288 W will ignor any possibl changs in th wind spd. In som gnral sns th hat ngin concpt says thr should b som incras in th winds with global warming but w ignor this. Th drag cofficint dos not chang if th surfac roughnss dos not chang. If th plants start changing in a givn ara du to global warming thn thr could b a chang. Ovr th ocans, C w dpnds on th wnds spd which crats surfac roughnss ovr th ocans. W hav assumd th winds don t chang so C w ovr th ocans won t chang. Th dnsity of th air changs slightly " air#nw " air#old = P surf #nw P surf #old T old T nw = T old T nw Plugging ths last two quations into th surfac latnt hat flux ratio quation yilds F LH "nw F LH "old = T old = # L C U q " q a"nw v W "nw nw s"nw a"nw # a"old C W "old U old T nw 2 So for a 4 K incras in surfac tmpratur, th latnt hat flux will incras by 24. F LH "nw = 288 # F LH "old =.24 $ 2 6 Kursinski 2/7/28

7 Th prsnt globally avragd latnt hat flux is 78 W/m 2. So a 24 incras would rais it to 97 W/m 2. So vn though watr vapor is a grnhous gas that nhancs and roughly doubls th global warming du to incrasd CO 2 concntrations, th incras in th surfac latnt hat flux with global warming will actually cool th Earth s surfac. Roughly half of th incras in downward IR flux into th surfac is offst by an incras in th latnt hat flux from th surfac. Th othr half is mad up primarily by th incras in IR mittd by th surfac du to th highr surfac tmpraturs. 7. Diffusion scaling: Th tim to cook a hard-boild gg is ~2 minuts. Basd on your undrstanding of diffusion s q. 6 of th diffusion lctur, how long should it tak to cool a watrmlon in a rfrigrator down to th rfrigrator s tmpratur. Stat whatvr assumptions you nd to mak. " # = n" # = n # $ = X v t # 2 # v t = X 2 v t # Th point hr is that th man fr path and vlocity of molculs in th gg and in th watrmlon ar about th sam. Th watrmlon is much largr maning that X is much largr and th tim to cook should thrfor b about th ratio of thir linar dimnsion siz squard. If th watrmlon is about 5 tims largr than th gg in ach dimnsion thn th watrmlon will tak about 25 tims as long to cool down. On can do this mor prcisly by accounting for th fact that th vlocitis in th gg will b highr bcaus of th highr tmpraturs nar boiling in th gg and th coolr tmpraturs in th watrmlon nar frzing. Th ratio of th vlocitis ignoring diffrncs in th masss of th avrag molculs which ar both probably primarily watr will b v gg /v watrmlon = T gg /T watrmlon /2 = 373/273 /2 =.37 /2 =.7. So th vlocity ffct is thr but th snsitivity to th squar of th rlativ sizs of th objcts is much largr. So taking into account th vlocity dpndnc on tmpratur, th watrmlon that is 5 tims largr than th gg in trms of thir linar dimnsions will tak about.7 tims 25 which is 29 tims as long to cool as th gg will tak to cook. Sinc th gg taks 2 minuts or /5 of an hour, th watrmlon will tak 5-6 hours to cool down. You can us this physics cookbook to figur out how long your Christmas turky, ham or roast bast will tak to cook. 6 7 Kursinski 2/7/28