Wind Forced Motion. Equation of motion a=σf/ρ 12/09/11. What forces might cause a parcel of water to accelerate?

Transcription

1 Physical ocanography, MSCI 3001 Ocanographic Procsss, MSCI 5004 Dr. Alx Sn Gupta Wind Forcd Motion Equation of motion aσf/ρ What forcs might caus a parcl of watr to acclrat? Movmnt of th ocan (a) (1/ρ)x( wind - friction + rotation + tids gravity+ buoyancy + prssur diffrncs.) A sum of forcs. 1 ( Fg + FC + FP + Ff ρ +...) Nwton s Laws of Motion Vrtical dirction: PF/A F top F bottom Wight (mg) z P P+ΔP What ar th forcs acting in th up-down dirction? Th boxs wight is acting downwards (mg) Th prssur at th top of th box is also trying to forc th box downwards. But th prssur at th bottom of th box is trying to forc it upwards (th diffrnc in th prssur forcs is just th buoyancy forc discussd for Archimds) Acclration 1/ρ(sum of forcs) 1/ρ(wight + F top - F bottom ) dw g 1 ρ Bouyancy to diffrnc in prssurs dw 1 ρ + g In gnral ovr th ocan vrtical acclration is much smallr than g. This mans that in th vrtical quation g and 1 must b of similar magnitud ρ ρg This is calld th hydrostatic quation W can intgrat this quation sinc dnsity and g ar ssntially constant. p h 0 0 ρg Or simply p hρg Which you ar hopfully familiar with alra! 1

2 Nwton s Laws of Motion Horizontal dirction Hydrostatic quation tlls us that prssur wight of watr abov you (in this cas highr on th right) Acclration 1/ρ(sum of forcs) 1/ρ(F lft - F right ) Barotropic and Baroclinic Motion Rmmbr, p ρgz, and 1 ρ p x P P+ ΔP 1 ρ 0 Mixd situation F lft F right Or doing th sam in th y-dirction: x 1 ρ 0 ρ 1 < ρ 2 Motion to dnsity diffrncs Barotropic Ocan For a constant dnsity ocan, w can writ th prssur gradint in an asir way. 1 Rmmbr, p ρgz, and ρ d P 1 h 1 ρg h 1 d+η 1 η 1 η 2 Δx P 2 h 2 ρg h 1 d+η 1 p x p Δp p2 p1 x Δx Δx h2ρg h1 ρg Δx ( d + η2) ρg ( d + η1) ρg Δx ( η2 η1) ρg Δx p Δη ρg x Δx Coriolis Forc - Summary Th Acclration to th Coriolis forc is fv (x dirction) and -fu (y dirction) i. th Coriolis Forc x th vlocity It only acts if watr/air is moving (i. a scondary forc) Acts at right-angls to th dirction of motion causs watr/air to mov to th right in th northrn hmisphr causs watr/air to mov to th lft in th southrn hmisphr So w ar lft with η g x E.g. Foucault s Pnlum 2

3 Th Equations of Motion d! u 1 " #! F 1 " #F x 1 " #F y dw 1 " #F z Simplifying th Equations: Considr th situation whr thr ar no prssur forcs Acclration Prssur Gradint Forc + Coriolis Horizontal Equations: Acclration Prssur Gradint Forc + Coriolis " 1 # + fv " 1 # " fu Vrtical Equation: Or, for a Barotropic Ocan: Prssur Gradint forc Gravitational Forc d# "g + fv d# "g " fu ρg " 1 # + fv " 1 # " fu If prssur gradints ar small: fv " fu Inrtia currnts Acclration Prssur Gradint Forc + Coriolis Scaling argumnts: " 1 # + fv " 1 # " fu What forcs ar important in a bath tub? What kind of spds will th watr gt up to? What kind of acclrations? What surfac slops? Nd to look at th rlativ sizs of th trms in th quation. But first, choos which vrsion of th quations ar most appropriat. If prssur gradints ar small: fv " fu Inrtia currnts th watr flows around in a circl with frquncy f. T2π/f " 1 # + fv " 1 # " fu d# "g + fv d# "g " fu T(Sydny) 21 hours 27 minuts In a bathtub, dnsity is prtty much constant, so conditions will b BARATROPIC 3

4 Scaling argumnts: What forcs ar important in a bath tub? What kind of spds will th watr gt up to? What kind of acclrations? What surfac slops? d# "g + fv d# "g " fu Siz of th prssur forc: dη 0.1m /1m 0.1 dη g 10(0.1) 1ms Siz of th Coriolis forc: fu 7x x10 ms So Coriolis<<Prssur, so w can nglct rotation ffcts dη g f1.5x10-4 x sin(lat) 2 Acclration in th bathtub is drivn by prssur diffrncs ( to changs in surfac slops) But th ocan is not a bathtub. W will conct a scaling analysis on our quations of motion... to find furthr simplifications for motions with a priod gratr than ~10 days " 1 Scaling Analysis: # + fv T~10 days 8.64 x 10 5 s ~ 10 6 s " 1 # " fu u,v ~ U ~ 1cms -1-1ms -1 f~ 10-4 s -1 Acclration << Coriolis Gostrophic Balanc Acclration is much smallr than Coriolis and Prssur Gradint Forc (this is tru almost vrywhr in th ocan) Th ocan is in Gostrophic Balanc ( balanc btwn Prssur Gradint and Coriolis Forcs) Prssur forcd motion Gostrophic Transport Ingrdints: (1) Prssur forc acts from high prssur to low prssur (2) Coriolis always tris to push a moving objct to th lft (SH) or right (NH) Ocan is in Sta Stat (no acclration) / is ngligibl " 1 # + fv " 1 # " fu Or if conditions ar baratropic (constant dnsity) 1 " fv 1 " # fu dη fv g dη fu g L Imagin that somhow watr has bn pild up in som ara.g. by th ac7on of winds. What happns whn th wind stops? H 4

5 Prssur forcd motion Gostrophic Currnt Ingrdints: (1) Prssur forc acts from high prssur to low prssur (2) Coriolis always tris to push a moving objct to th lft (SH) or right (NH) SH Exampl (looking down on th ocan) L Extnds down to th ocan Prssur Forc Prssur Forc Coriolis Forc So, no nt forc on th watr So kps on going H Watr flow Coriolis Forc Gostrophic Currnts High prssur Low prssur horizontal prssur gradint forc 1 Gostrophic Ed! fv (Northrn Hmisphr) 18 Which dirction is th Gostrophic wind? (f <0 SH) y x Gostrophic Currnt Currnt that is procd whn Prssur Forc and Coriolis Forc balanc In th NH th final gostrophic currnt will flow at 90 dgrs to th right of th prssur forc In th SH th final gostrophic currnt will flow at 90 dgrs to th lft of th prssur forc PG CF V 1! fv 1!! fu dη fv g dη fu g Lik this th quations say: Prssur forc is balancing Coriolis Lik this th quations say: If w know th prssur forc (or surfac slop), w can calculat how fast th watr is moving v 1! f u! 1! f v g f u! g f d! d! 5

6 So far w hav nglctd friction. Effcts of Friction A simpl modl for th frictional forc at th sa floor in th x and y dirction is: ru rv and h h h is th th and r is th dissipation constant CORRECTION Thrmal Wind Balanc So far w hav assumd that dnsity ρ is constant (barotropic) Small horizontal changs in ρ can rsult in larg vrtical changs in currnt/ wind.g. nar fronts and ddis Baroclinic vlocity changs with th Rayligh frictional dissipation, r is a cofficint (r ~ 10-7 ms -1 ) Hnc th quations of motion bcom : d# "g + fv " ru h d# "g " fu " rv h 21 ρ 1 < ρ 2 Motion to dnsity diffrncs For gostrophic conditions: Thrmal Wind Balanc Th vrtical structur of u and v is rlatd to th horizontal dnsity gradints g "f # g "f d" d" Baroclinic vlocity changs with th NO CORIOLIS Thrmal Wind Balanc Baroclinic vlocity changs with th WITH CORIOLIS THERMAL WIND What dos this man If thr ar dnsity changd in th ast-wst dirction Thn th north-south watr vlocity will chang with th d! ρ 1 < ρ 2 Motion to dnsity diffrncs ρ 1 < ρ 2 6

7 Us thrmal wind to calculat th ast-wst watr spd at 45 S? Us thrmal wind to calculat th ast-wst watr spd at 45 S? 2000 m S Pol 1000km Dpth of no motion Equator g d!! f! g d!! f d! 1027! , 000-5x10-7 kg/m 4 Givn that u & v 0 at 2000m (th of no motion) Givn that u & v 0 at 2000m (th of no motion).! (!10!4 ) (!5"10!7 ) m/s/s Us thrmal wind to calculat th ast-wst watr spd at 45 S? 2000 m S Pol 1000km Dpth of no motion Equator! m/s/s This mans that vry mtr that you go downwards th astward watr vlocity (u) dcrass by m/s OR that vry mtr that you go upwards th astward watr vlocity (u) incrass by m/s Moving upwards 2000m th incras in u vlocity will b 2000x m/s But w know at 2000m u0 So th surfac vlocity will b m/s Watr at th surfac is moving at 10cm/s to th ast (th Antarctic Circumpolar Currnt) Thrmal Wind Balanc For gostrophic flow (i.. prssur is balancd by Coriolis): Gostrophic flow in th prsnc of horizontal dnsity gradints Horizontal dnsity gradints (T,S) can xplain vrtical vlocity changs To know absolut vlocity w nd xtra information (i.. w nd to know absolut vlocity at som th) g "f # g "f d" - Can figur out surfac vlocity from surfac hights. - Oftn w assum that at a crtain th (.g. 2000m) vlocitis ar zro this is calld th th of no motion. - Onc w know th vlocity at th surfac or th th of no motion w can calculat vlocity at all othr ths using th thrmal wind quation. d" 7

8 12/09/11 Summary: Ocan Dynamics Most of th motion in th ocan can b undrstood in trms of Nwton s Law that th acclration of a parcl of watr (how fast its vlocity changs with tim /) is rlatd to th sum of forcs acting on that parcl of watr. W can split th forcs, vlocitis and acclrations into south-north (y,v), wst-ast (x,u) and up-down (z,w) componnts. In th vrtical dirction th acclration is rlatd to th diffrnc btwn th watr wight and th bouyancy (or prssur) forc. Whn thr is a vrtical dnsity gradint this lads to oscillations (Brunt Väisälä frquncy N). Th dnsity gradint tris to inhibit vrtical motion (and mixing). Ths vrtical acclrations ar gnrally vry wak, so w gt th hydrostatic quation. If th hydrostatic quation is intgratd ovr th, it just says that th prssur at a point just quals th wight of watr abov that point. ρg Acclration in th horizontal can b drivn by a numbr of diffrnt forcs: (1) Th prssur gradint forc. This xists whnvr thr is a surfac slop and/ or a horizontal dnsity gradint. (2) Coriolis (bcaus w liv on a rotating plant). It is vry wak, so w only fl its ffct ovr long tims (> fw days) and larg distancs (> 10s of km). Coriolis only affcts moving fluids, dflcting to th right in th NH and to th lft in SH. (3) Friction. Also only acts on moving watr. Always acts to slow down motion. Important at th boundaris of th ocan. Summary: Ocan Dynamics Or for a constant dnsity (barotropic ocan): 1 " + fv " ru # 1 " " fu " rv # d# "g + fv " ru d# "g " fu " rv ρg! Ovr much of th ocan, th flow is sta (i.. //0) and friction is ngligibl, so w ar lft with th gostrophic balanc i.. prssur gradint forcs! at right angls to th prssur gradint. balanc coriolis. Th currnt movs P 1 1 fv, # fu " " C Whn thr is a horizontal dnsity gradint th vlocity changs with th. This can b calculatd using th thrmal wind quations! g d" "f, g d" # "f! Wind Drivn Motion Background wind forcd motion Track of th Fram as Frijof Nansn attmpts to drift to th North Pol in his ship ( ). Vagn Walfrid Ekman ( ), Swdish ocanographr, dvlopd thory of Ekman spiral from Nansns obsrvations. 8

9 Wind forcd motion EkmanTransport Ingrdints: (1) Wind forc acts in th dirction of th wind (2) Coriolis always tris to push a moving objct to th lft (SH) or right (NH) Wind forcd motion EkmanTransport Ingrdints: (1) Wind forc acts in th dirction of th wind (2) Coriolis always tris to push a moving objct to th lft (SH) or right (NH) SH Exampl (looking down on th ocan) Wind Forc Wind forcd motion EkmanTransport Ingrdints: (1) Wind forc acts in th dirction of th wind (2) Coriolis always tris to push a moving objct to th lft (SH) or right (NH) SH Exampl (looking down on th ocan) Wind Forc Wind Forc Coriolis Forc So, no nt forc on th watr So kps on going Coriolis Forc Equations of Motion!g d! + fv + " x #h! ru d!!g! fu + " y #h! rv i. th acclration of watr towards th ast nds on: any ast-wst prssur forc Th vlocity of th watr in th north-south dirction (which dtrmins th strngth of th ast wst Coriolis forc Th ast-wst wind forc Th vlocity of watr in th ast-wst dirction (which dtrmins th strngth of th friction) Only affcts watr in th top fw 10s of mtrs Watr flow 9

10 Equations of Motion!g d! + fv + " x #h! ru d!!g! fu + " y #h! rv fv g g d! fu g!g d! For larg scal, long timscal, throughout most of th ocan th trms that mattr ar Coriolis and Prssur forcs Nar th surfac in th prsnc of sustaind wind, w also nd to worry about th wind forc. fv! " x #h fu " y #h Equations of Motion Wind drivn Ekman transport is a balanc btwn Coriolis and wind forcs fv!! x "h fu! y "h If w know th wind forc (normally calld wind strss) w can calculat th Ekman transport v!! x f "h u! y f "h In words: Ekman transport is proportional to th strngth of th wind. Ekman flow is at 90 to th right of th dirction of th wind in th NH and 90 to th lft of th dirction of th wind in th southrn hmisphr. NB th Ekman transport in ths quations ar th th avragd currnt. In rality Ekman transport only occurs in th top fw 10s of mtrs (th Ekman Layr) E.G Wind blowing to th north in th southrn hmisphr Τ y >0 and f<0, so u <0 i. Ekman transport is to th wst (90 to th lft of th dirction of th wind) Without Coriolis With Coriolis Ekman Layr - wind blowing on an ocan procs a forc pr unit ara calld a wind strss τ. - ffcts of a wind strss on th ocan surfac ar transmittd down through th watr column by th action of turbulnt ddis that ar thmslvs gnratd by th wind braking wavs boundary shar strsss - th to which th ffcts of wind ar flt is calld th Ekman layr thicknss or Ekman th (H ), whr Dpth Nt flow of watr is 90 to dirction of wind H 2K f 10

11 Ekman Layr cont d - th to which th ffcts of wind ar flt is calld th Ekman layr thicknss or Ekman th (H ), whr H 2K f - Coriolis forc and wind balanc th-avragd Ekman vlocitis (u and v ) τ x fv ρh - - But u and v in ths quations ar th avragd vlocittis τ and w know that flow is actually y fu onlu in th surfac Ekman layr ρh th (H ) - whr K is th d diffusivity and - K WL whr W and L dnot a charactristic d vlocity and siz Typically K m 2 s 1, so that with f 10 4 s 1, H 20 m. Howvr in rgions of high wind-drivn turbulnc, K can b up to 0.5 m 2 s 1, so that H can rach 100 m. H h So nd to scal up to gt th Ekman latr vlocitis Ekman layr vlocitis (U and V ): h τ y U u H ρh f V h H v τ x ρh f Exrcis: Find th Ekman layr th H and th Ekman layr vlocitis U and V whn a wind of strngth 0.5 Pascals (0.5 N/m 2 ) blows towards th north in th mid-latituds of th NH. Tak: vrtical diffusivity K 0.1m 2 /s and f 10-4 s -1 and h 1000m H 2K f U h H h V H u τ y ρh f τ x v ρh f Schmatic of movmnt in column of watr in th Ekman layr. Which hmisphr is this xampl from? 11

12 Upwlling & Downwlling E.g. Northrn Hmisphr Schmatic diagrams of winddrivn upwlling in SH. TEMP Winds SST in California Currnt undr th influnc of a northrly alongshor wind. NITRATE Tmpratur and nitrat distributions off Pt. Sur, California, from satllit and shipboard oprations. From Traganza t al. (1983). Mann & Lazir (1996) 12

13 Full ffct of alongshor wind SH Wind Ekman currnt Upwlling Gostrophic currnt 1. Wind drivs offshor Ekman Currnt in first fw 10 s of mtrs. 2. In boundary layr, offshor flow is compnsatd by upwlling 3. Bcaus thr is a surfac slop, thr must also b a prssur forc (towards th coast) 4. This driv a gostrophic currnt that strtchs through th full th of th watr column 5. Excpt at vry bottom (whr friction is important) and thr is an onshor currnt Idalisd modl of a storm surg. Which hmisphr? Storm Surg Storm Surg A storm surg is an offshor ris of watr associatd with a low prssur wathr systm, typically tropical cyclons and strong xtratropical cyclons. Storm Surg Whn rotation ffcts ar not important For xampl, in sminclosd bays 5) Th ffct of wavs, whil dirctly powrd by th wind, is distinct from a storm's wind-powrd currnts. Powrful wind whips up larg, strong wavs in th dirction of its movmnt 6) Th rainfall ffct is xprincd prdominantly in stuaris. Hurricans may mp as much as 300 mm of rainfall in 24 hours ovr larg aras, and highr rainfall dnsitis in localizd aras 13

14 Exrcis: Q 4) A low-prssur systm sits off th NSW coast as shown blow. Th southrly winds xrt a strss τ y 1 N/m 2 ovr a priod of 5 days. Assuming an Ekman layr th H 100m ovr th shlf rgion, calculat th onshor Ekman layr vlocity U. Calculat how far and in which dirction th front at A will mov ring th 5 day priod. Commnt on th likly salvl chang and rsulting gostrophic adjustmnt. Exrcis: Q 5) Th wind strss τ and wind spd U ar rlatd by τ C D ρ a U 2 whr ρ a 1.23 kg/m 3 (air dnsity) and C D 10-3 ( U) is a drag cofficint for U btwn 6m/s and 22m/s. Givn U 8m/s calculat τ. Th units of τ ar kg/(ms 2 ) 1 N/m 2 1 Pa. Q 6a) A northrly wind spd of U 10m/s blows off th wst coast of Africa ring 9-13 March. Draw an idalisd sktch of th rsulting Ekman transport, salvl chang, and gostrophic adjustmnt to this wind fild. Discuss this diagram in rlation to th obsrvations of upwlling shown blow. u < 0 Dnsity Eastward vlocity (u) 14

15 Q 7) Discuss th dnsity sctions drawn blow. Th sctions corrspond to watrs off th NSW coast. Why might th dnsity fild slop upward in th initial conditions? What can you say about summrtim swimming conditions in Sydny? 15