Frictional effects, vortex spin-down

Transcription

1 Chapt 4 Fictional ffcts, votx spin-down To undstand spin-up of a topical cyclon it is instuctiv to consid fist th spin-down poblm, which quis a considation of fictional ffcts. W xamin fist th ssntial dynamics of th poblm and pocd thn to a scal analysis of th quations with th fiction tms includd. 4.1 Spin-down of a otating votx Th classic votx spin down poblm consids th volution of an axisymmtic votx abov a igid bounday nomal to th axis of otation. W will show that th spin down is intimatly connctd to th Coiolis toqus associatd with th sconday ciculation inducd by fiction. Th dict ffct of th fictional diffusion of momntum to th sufac is of sconday impotanc in th paamt gims lvant to topical cyclons. In a shallow lay of ai na th sufac, typically m dp, fictional stsss duc th tangntial wind spd and thby th cntifugal and Coiolis focs, whil it can b shown that th foc associatd with th adial incas of pssu mains lagly unchangd. W call this lay th fiction lay. Th sult of th foc imbalanc is a nt inwad foc that divs ai pacls inwads in this lay. On can dmonstat this ffct by placing ta lavs in a bak of wat and vigoously stiing th wat to st it in otation. Aft a shot tim th ta lavs conggat na th bottom of th bak na th axis as shown in Fig. 4.1: thy a swpt th by th inflow in th fiction lay. Slowly th otation in th bak dclins bcaus th inflow towads th otation axis in th fiction lay is accompanid by adially-outwad motion abov this lay. Th dpth of th fiction lay dpnds on th viscosity of th wat and th otation at and is typically only on th od of a millimt o two in this xpimnt. Bcaus th wat is otating about th vtical axis, it possss angula momntum about this axis. H angula momntum is dfind as th poduct of th tangntial flow spd and th adius. As wat paticls mov outwads abov th fiction lay, thy consv thi angula momntum and as thy mov to lag adii, thy spin mo slowly. 81

2 CHAPTE 4. FICTIONAL EFFECTS, VOTEX SPIN-DOWN 8 Figu 4.1: Th bak xpimnt showing th ffcts of fictionally-inducd inflow na th bottom aft th wat has bn stid to poduc otation. This inflow cais ta lavs to fom a nat pil na th axis of otation. Th sam pocss would lad to th dcay of a huican if th fictionally-inducd outflow w to occu just abov th fiction lay, as in th bak xpimnt. What thn pvnts th huican fom spinning down, o, fo that matt, what nabls it to spin up in th fist plac? Claly, if it is to intnsify, th must b a mchanism capabl of dawing ai inwads abov th fiction lay, and of cous, this ai must b otating about th vtical axis and possss angula momntum so that as it convgs towads th axis it spins fast. Th only concivabl mchanism fo poducing inflow abov th fiction lay is th upwad buoyancy foc in th clouds, th oigins of which w xamin blow. 4. Scal analysis of th quations with fiction W pat now th scal analysis of th dynamical quations (3.1) - (3.3) with th fiction tms K( u, v, w) addd to th ight-hand-sids. W hav assumd a paticulaly simpl fom fo fiction with K an ddy diffusivity, assumd to b constant. W assum that th ai is homognous and that th motion is axisymmtic and dfin vlocity scals (U, V, W ), lngth scals (, Z), and an advctiv tim scal T = U/ as bfo. Now w assum that p is ptubation pssu and tak Δp to b a scal fo changs in this quantity. Th continuity quation, (3.4), yilds th sam lation btwn W/Z U/ as bfo. Thn th tms in Eqs. (3.1) to (3.3) hav nondimnsional scals as shown in Tabl 4.1. As in sction 3.5 w divid tms in lin (a) by V / to obtain thos in lin (b) and divid tms in lin (3a) by UV/ to obtain thos in lin (3b). Th

3 CHAPTE 4. FICTIONAL EFFECTS, VOTEX SPIN-DOWN 83 tms in lin (4a) a dividd by g to obtain thos in lin (4b). Th tms in lins (b) a thn nondimnsional. W dfin a ynolds numb, = VZ/K, which is a nondimnsional paamt that chaactizs th impotanc of th intial to fictional tms, and an aspct atio A = Z/, that masus th atio of th bounday-lay dpth to th adial scal. u-momntum u t +u u +w u z v fv = 1 p ρ +K ( h u u ) +K u z (4.1) U T U W U Z V fv Δp ρ K U K U Z (1a) S S S 1 1 o Δp ρv SA 1 S 1 (1b) v-momntum v t V T +u v U V +w v z W V Z + uv U V S S S S +fu = +K ( h v v ) +K v z (4.) fu K V K V Z (a) S o A 1 1 (b) w-momntum w t +u w +w w z = 1 p ρ z +K h w +K w z (4.3) W T UW WW Z Δp ρz K W K W Z (3a) S A S A S A Δp ρv SA 3 1 SA 1 (3b) Tabl 4.1: Scaling of th tms in Eqs. (3.1) to (3.3) with fictional tms addd. H h =( / )( / ). Typically, th bounday lay is thin, not mo than 500 m to 1 km in dpth and th aspct atio is small compad with unity. Moov, typical valus of K a on th od of 10 m s. Thfo taking V =50ms 1, =50km,andZ = 500 m,

4 CHAPTE 4. FICTIONAL EFFECTS, VOTEX SPIN-DOWN 84 = and A =10. It follows fom lin (3b) in Tabl 4.1 that Δp/(ρV ) max(s A, SA 1 )=4 10 4, assuming that S 1 in th bounday lay. Thus th vtical vaiation of p acoss th bounday lay is only a tiny faction of th adial vaiation of p abov th bounday lay. In oth wods, to a clos appoximation, th adial pssu gadint within th bounday lay is th sam as that abov th bounday lay. Fom lins (1a) and (a) in Tabl 4.1, w s that a vtical scal Z which maks fiction impotant compad with th oth lag tms is (K/f) 1 if o << 1and (K/(V/)) 1 if o 1 o lag. Th fom sult givs us th appopiat scaling fo th Ekman lay. 4.3 Th Ekman bounday lay Th scal analysis of th u- and v-momntum quations in Tabl 4.1 show that fo small ossby numbs (o << 1), th is an appoximat balanc btwn th nt Coiolis foc and th diffusion of momntum, xpssd by th quations: f(v g v) =K u (4.1) z and fu = K v z, (4.) wh w hav usd th fact dducd fom th scal analysis of th w-momntum quation that th hoizontal pssu gadint in th bounday lay is ssntially th sam as that abov it. Thus w hav placd th adial pssu gadint in Eq. (4.1) by th gostophic wind, v g, abov th bounday lay. W assum also that th adial flow abov th bounday lay is zo. Equations (4.1) and (4.) a lina in u and v and may b adily solvd by stting V = v + iu, whi = ( 1). Thn thy duc to th singl diffntial quation K d V dz ifv = ifv g, (4.3) wh V g = v g. This quation has th paticula intgal V = V g and th a two complmntay functions popotional to xp(±(1 i)z/δ), wh δ = K/f. Sinc w qui a solution that mains boundd as z w jct th solution with a positiv xponnt so that th solution has th fom V = V g [1 A xp( (1 i)z/δ), ] (4.4) wh A is a complx constant dtmind by a suitabl bounday condition at z =0. Fo a lamina viscous flow, th flow at z = 0 is zo giving A =1. Thisisthclassical Ekman solution. Th pofils of u and v a shown in Fig. 4.a and th hodogaph thof is th Ekman spial shown in Fig. 4.b. Two intsting fatus of th

5 CHAPTE 4. FICTIONAL EFFECTS, VOTEX SPIN-DOWN 85 solution a th fact that th bounday-lay thicknss, masud by δ is a constant and th a gions in th bounday lay wh th flow is supgostophic, i.. v > V g. Ths a unusual fatus of bounday lays which a typically gions wh th flow is tadd and which gow in thicknss downstam as mo and mo fluid is tadd. In th Ekman lay, th fluid that is tadd in th downstam diction, but is -ngizd in th coss-stam diction by th nt pssu gadint foc in that diction. Th latt occus bcaus fiction ducs th Coiolis foc in th bounday lay whas th adial pssu gadint mains ffctivly unchangd. Th hight ang wh th flow is supgostophic is on in which th nt adial flow is outwads. Th fact that th adial flow is inwads ov this ang is a sult of th upwad diffusion of u-momntum, which is lagst at low lvls. Th xistnc of a gion of supgostophic winds may b mad plausibl by considing th spin-down poblm in which th is initially a unifom gostophic flow, V g, at lag adius. If th fictional stss is switchd on at th sufac at tim zo, th tangntial flow will b ducd na th sufac lading to a nt pssu gadint in th adial diction. This will div a adial flow that is lagst na th sufac wh th nt pssu gadint is lagst. Th Coiolis foc acting on th adial flow will tnd to acclat th tangntial flow as pat of an intial oscillation. Whth o not th tangntial flow bcoms supgostophic aft som tim will dpnd subtly on th vtical diffusion of momntum in th adial and tangntial dictions, but th Ekman solution indicats that it dos. Th classical Ekman solution with a no-slip bounday condition at th sufac is a poo appoximation in th atmosphic bounday lay. A btt on is to pscib th sufac stss, τ 0, as a function of th na-sufac wind spd, nomally takn to b th wind spd at a hight of 10 m, and a dag cofficint, C D. Th condition taks th fom τ 0 ρ = K V z = C D V V (4.5) and w shall apply it at z = 0 instad of 10 m. Substituting (4.4) into (4.10) and tidying up th quation givs (1 i)a = ν 1 A (1 A) (4.6) wh ν = C D and = V g δ/k is a ynolds numb to th bounday lay. Stting 1 A = B iβ givs aft a littl algba an quation fo B and an xpssion fo β in tms of B (s Excis 4.1). Th quation fo B may b solvd using a Nwton-aphson algoithm. Pofils of u(z) andv(z) obtaind fo valus V g =10 ms 1, f =10 4 s 1, K =10m s and C D =0.00 a shown in Fig. 4.a also and th cosponding hodogaph is shown in Fig. 4.b. Fo ths valus, th bounday lay dpth scal, δ = 450 m and th sufac valus of v and u a about 0.7V g and 0.V g, spctivly.

6 CHAPTE 4. FICTIONAL EFFECTS, VOTEX SPIN-DOWN 86 (a) (b) Figu 4.: (a) Vtical pofils of tangntial and adial wind componnts fo th Ekman lay (d cuvs) and th modifid Ekman lay basd on a sufac dag fomulation (blu cuvs). (b)hodogaph of th two solutions showing th spial of th wind vcto. 4.4 Th modifid Ekman lay Fom th scal analysis in Tabl 4.1, th full u and v momntum quations in th bounday lay a: u t + u u + w u z + V v + f(v v) =K u z, (4.7)

7 CHAPTE 4. FICTIONAL EFFECTS, VOTEX SPIN-DOWN 87 v t + u v + w v z + uv + fu = K v z, (4.8) wh V () is th gadint wind spd abov th bounday lay. With th substitution v = V ()+v, ths quations bcom: u t + u u + w u ( ) z v V + f v = K u (4.9) z ( v t + u v + w v z + uv dv + d + V ) + f u = K v (4.10) z W cay out now a scaling of th tms in ths quations as shown in Tabl 4.. Lt U, V and V b scals fo u, V and v, spctivly. Examination of th u- and v-pofils fo th cas with a sufac dag cofficint in Fig. 4. suggsts that U V 0.V 0.3V. Accodingly w assum that U = V and dfin S = U/V, which may b as lag as 0.3. u-momntum u t +u u +w u z v ( V ) + f v = +K u (4.9) z U T U W U Z V V V S S S S fv K U Z (1a) 1 o 1 Z (1b) v-momntum v t +u v +w v z + uv ( V + + V ) + f u = +K v (4.10) z V T U V W V Z UV VU S S S S 1 1 VU fu KV 1 o Z 1 Z (a) (b) Tabl 4.: Scaling of th tms in Eqs. (4.9) to (4.10). H S = U/V = V /V, o = V /(f)and = VZ/K. With th backgound gadint wind balanc movd, th is lss of a scal spaation btwn th vaious tms in th bounday-lay quations. If ζ is gadd

8 CHAPTE 4. FICTIONAL EFFECTS, VOTEX SPIN-DOWN 88 as small compad with unity, w could linaiz Eqs. (4.9) and (4.9) to obtain ( ) V + f v = K u z, (4.11) ( dv d + V ) u = K v z. (4.1) It is intsting to xamin this appoximation vn though th nglct of tms of magnitud compad with unity is unlikly to b vy accuat. Fo on thing th quations (4.11) and (4.1) a lativly asy to solv and thy a a gnalization of th Ekman lay thoy divd in th pvious subsction. Th diffnc in th cofficints of v and u on th lft-hand-sid of th quations pcluds th mthod usd fo th Ekman quations (4.1) and (4.). Now w diffntiat on of th quations twic with spct to z and us th maining quation to liminat ith u o v. Thn ach of ths quantitis satisfis th quation 4 x z 4 + I x K = 0 (4.13) Th solutions hav th fom x = a αz wh α 4 =(I /K ) iπ+nπi, (n =0, 1,, 3), and a is a constant. Thus th fou oots a α = ±(1 ± i)/δ, whδ =(K/I) 1/ is th bounday-lay scal thicknss. Th solutions fo v andu which dcay as z can b wittn succinctly in matix fom ( v u ) = V z/δ ( a1 a b 1 b )( cos z/δ sin z/δ ) (4.14) wh a 1,a,b 1,b a constants. Two lations btwn ths constants may b dtmind by applying th bounday condition (4.5), mmbing that this applis to th total sufac wind fild (V +v,u). Two futh lationships may b obtaind by substituting fo v and u in Eq. (4.1). Th dtails fom th basis of Excis (4.). Givn a adial pofil of V () such as that shown in Fig. 4.3, it is possibl to calculat th full bounday-lay solution (u(, z), v(, z), w(, z)) on th basis of Eq. (4.14). Th vtical vlocity, w(, z) is obtaind by intgating th continuity quation (s Excis 4.3). Figu 4.4 shows th isotachs of u and v and th vtical vlocity at th top of th bounday lay fo th tangntial wind pofil shown in Fig It shows also th adial vaiation of th bounday lay dpth scal, δ. Not that δ dcass makdly with dcasing adius, whil th inflow incass. Th maximum inflow occus in this cas at a adius of 90 km, 50 km outsid th adius of maximum wind spd abov th bounday lay, m. In contast, th maximum vtical vlocity at lag hights paks just outsid m. Th is a gion of wak outflow abov th inflow lay coinciding oughly with th gion wh tangntial flow bcoms subgadint.

9 CHAPTE 4. FICTIONAL EFFECTS, VOTEX SPIN-DOWN 89 Figu 4.3: Tangntial wind pofil as a function of adius usd in th calculations fo Fig (a) (b) (c) (d) Figu 4.4: Isotachs of (a) adial wind, and (b) tangntial wind in th z plan obtaind by solving Eqs. (4.11) and (4.1) with th tangntial wind pofil shown in Fig (c) Vtical vlocity at th top of th bounday lay, and (d) adial vaiation of bounday-lay scal dpth, δ().

10 CHAPTE 4. FICTIONAL EFFECTS, VOTEX SPIN-DOWN 90 Excis 4.1 Show that th substitution A =1 B iβ in q (4.6) lads to th quation [ (νb +1) +1 ][ B { (νb +1) +1 } ] =0 and th β is givn by tan β = νb νb +1 [Hint: aft substitution, quat al and imaginay pats to obtain a pai of simultanous quations fo cos β and sin β.] Excis 4. Show that th substitution (4.14) into th bounday condition (4.5) givs th two lationships a a 1 = ν [ ] (1 + a 1 )+b 1 1 (1 + a 1 ), and b b 1 = ν [ (1 + a 1 )+b 1 ] 1 b 1 and that th substitution into Eq. (4.1) givs two mo lationships b 1 = K ξ a δ a and b = K ξ a δ a 1, wh ξ a = ζ + f. Show th points (a 1,a ) li on a cicl cntd at th point ( 1, 1 ) with adius 1 and may b xpssd as a, Hnc div an quation fo θ. V a 1 = 1 V cos θ 1 a = 1 V sin θ 1. abc