Equations of motion - summary

Transcription

1 MATH 2240 Wk 8 Summary Equations of motion - summary Starting from Nwton s Laws w av sown ΣF a m du 1 dp ru + fv + dt ρo d ρ dv 1 dp y ru fu + dt ρo dy ρ dw 1 dp + g + coriolis + friction dt ρo dz du dv dw d dy dz ρ ρ( ST,, p)

2 Equations of motion - summary du dt 1 ρ o dp d 1 dp 0 g ρ dz + o P P ρ ΔV Δ Δy Δz P+ΔP ρ Δz ΔV Δy Δ mg P+ΔP p gρ z o z z1 z2 ρ 0 ξ ρ 1 Vg mg ρ 1 Vg Buoyancy Effcts ρ1 ρ2 ρ 2 Vg ρ1 mg ρ 1 Vg ρ3 ρ 3 Vg ρ1 mg ρ 1 Vg 2 d ξ g dρ a ξ ( ) 2 dt ρ dz 2 d ξ 2 N ξ 2 dt 2 g dρ wr N ρ dz If N 2 >0 t solution is simpl armonic motion ξasin(nt), priod T2π/N If N 2 <0 i.. unstabl conditions, t solution is ponntial, ξa (Nt), 0 0

3 Equations of motion - summary 1 dp 0 g ρ dz + o Prssur wigt of fluid abov du dt 1 ρ o dp d p P + ρ gz+ ρ gη A o dp η ρog d o du dt dη g d P1 > P2 Barotropic and Baroclinic Motion Rmmbr, p ρgz, Barotropic Vlocity is constant wit dpt and du dt 1 ρ p (no rotation) Baroclinic vlocity cangs wit dpt ρ ρ 1 0 < ρ 2 Most ocan flow will actually av barotropic and baroclinic portions. Sa surfac slop givs t surfac flow, and trmal wind (on rotating plant) can b usd to calculat cangs wit dpt

4 T Coriolis Paramtr T strngt of t Coriolis forc varis wit latitud It is proportional to t Coriolis paramtr f2ωsin(φ), wr Φ is latitud And Ω is t angular vlocity (in radians pr scond) It is maimum at t pols, zro at t quator and cangs sign from NH to SH T acclration du to t Coriolis forc is: fv in t -dirction (ast-wst), and -fu - in t y-dirction (nort-sout) Rviw of last wk: Ocan/Atmospr Dynamics Foucault s Pndulum - simpl ampl of Coriolis C W

5 To amin t ffcts of friction, considr t simpl balanc: du ru dt A solution is: u u o rt / Mor common paramtrisation of friction u u u μ y z So tat at t0, uu o Vlocity dcrass wit tim bcaus of friction (-folding tim scal of frictional spin down) U U o 1/3U o U o -1 /r Tim Wind strss U y ρaircd u v (, ) (, ) c D c D (u,v) Wind strss as units of forc pr m 2, But Nwton s quation nds forc/mass ρ

6 Continuity quation For an incomprssibl fluid dρ 0 dt t continuity quation just bcoms: ρ, u δz ρ+δρ,u+ δu u v w y z δ δy ρ ( uρ) ( vρ) ( wρ) t y z Volum of watr ntring any rgion amount of watr laving. Tr is no cang in t dnsity. Equations of motion - summary du 1 dp ru + fv + dt ρo d ρ dv 1 dp y ru fu + dt ρo dy ρ dw 1 dp + g + coriolis + friction dt ρo dz du dv dw d dy dz ρ ρ( ST,, p) But can simplify quation undr crtain circumstancs scaling analysis & logic!

7 Equations of motion ampl 1 du dη ru g + fv+ dt d ρ Wind blows on an ocan at rst. No initial surfac slops Away from boundaris Equations of motion ampl 1 du dη ru g + fv+ dt d ρ Wind blows on an ocan at rst. No initial surfac slops Away from boundaris d g d η 0 ru 0 fv 0

8 Equations of motion ampl 1 du dη ru g + fv+ dt d ρ Wind blows on an ocan at rst. No initial surfac slops d g d η 0 fv 0 Away from boundaris ru 0 du dt ρ Wat dos tis man? Equations of motion ampl 2 du dη ru g + fv+ dt d ρ Wind blows on ocan and sts it in motion, but tn wind stops No initial surfac slops Away from boundaris

9 Equations of motion ampl 2 du dη ru g + fv+ dt d ρ Wind blows on ocan and sts it in motion, but tn wind stops dη 0 No initial surfac slops g 0 d Away from boundaris ru 0 du dt fv Wat dos tis man? Equations of motion ampl 3 du dη ru g + fv+ dt d ρ Surfac slops ist and ocan as sttld for a numbr of days. Away from surfac and sid boundaris

10 Equations of motion ampl 3 du dη ru g + fv+ dt d ρ Surfac slops ist and ocan as sttld for a numbr of days. Away from surfac and sid boundaris 0 0 du dt U T dη g + d? 10 fv 4 U ru dη g d fv Wat dos tis man? du dt du dt dη g d dη g + d fv 3. g dη d fv Gostropic Balanc:

11 dη + ρ fv g d W split t vlocity fild into t sum of t gostropic and wind drivn componnts i.. u u + u v v g g + v Gostropic part (dp) dη fvg g d dη fug g dy Ekman part (sallow) fv fu ρ y ρ Ekman layr vlocitis (U and V ): U V H H u v ρh ρh y f f } H 2K f

12 Convrgnc and Divrgnc T Ekman transport is dirctd prpndicular to t applid strss Rigt (NH, lft SH) driving a convrgnt flow (lft) if t strss is anticyclonic and divrgnt flow (rigt) if t strss is cyclonic. NH Eampl Tis causs Currnts in t ocan Gostropic currnts EQ HIGH HIGH HIGH HIGH HIGH PG CF 24 GEOSTROPHIC FLOW

13 Storm Surg 1. Ekman transport by winds paralll to t coast transports watr toward t coast causing a ris in sa lvl. 2. Winds blowing toward t coast pus watr dirctly toward t coast. 3. Wav run-up and otr wav intractions transport watr toward t coast adding to t first two procsss. 4. Edg wavs gnratd by t wind travl along t coast. 5. T low prssur insid t storm raiss sa lvl by on cntimtr for ac millibar dcras in prssur troug t invrtdbaromtr ffct. 6. T storm surg adds to t tids, and ig tids can cang a rlativ wak surg into a muc mor dangrous on. 7. Larg amounts of rainfall can add to t problm particularly in stuarin rgions Coastal Upwlling Soutrn Hmispr ampl fu y ρ dη fv g d g Friction braks gostropic balanc H 2K f

14 (continud)

15 Q 3: f s -1 H 100 m y 1 N m -2 1 / [ (-10-4 ) ] m s -1 Movmnt of A during 5 days: distanc U tim m s -1 ( ) s m 42.3 km to wst

16 Ekman Pumping Prviously assumd tat t wind strss is constant. If t wind strss cangs in t dirction, tn U E will also cang A divrgnc in t surfac layr will driv a vrtical Ekman vlocity w E du w E H E E d w E d y + d ρf Eampls of tis: subtropical winds (astrlis in tropics/wstrlis in subtropics) Rotating winds of a cyclon d dy ρf U V H H fv fu u v ρ y ρ ρh ρh y NH f f

17 Ekman Pumping w E d y d + d ρ f dy ρ f 40N 20N Wat will Ekman pumping b on SH? Wat will Ekman pumping b? Wat will Ekman pumping b on SH? Vorticity Two typs of rotation 1. Plantary Vorticity (rotating bcaus t plant is rotating) f 2Ωsin( φ) 2. Rlativ Vorticity (spinning on its own ais) dv du ζ d dy 3. Total or Potntial Vorticity Q f +ζ 34

18 Cang in Rlativ Vorticity Column Strtcing Production of rlativ vorticity by cangs in t igt of a fluid column. As t vrtical fluid column movs from lft to rigt: vrtical strtcing rducs t momnt of inrtia of t column, causing it to spin fastr. W ar not considring t ffct of plantary vorticity r. (Stwart) 35 dv ζ d du dy f Vorticity 2Ωsin( φ) Q Q. An clockwis spinning ddy (diamtr 50km, outsid spd 1m/s), is in watr of dpt 1000m at 20N a) Wat would appn to its rlativ vorticity if it movs to 30N. b) Wat ar t two possibilitis for t ddy if it movs into watr of dpt 700m? f +ζ

19 dv ζ d du dy f Vorticity 2Ωsin( φ) Q Q. An clockwis spinning ddy (diamtr 50km, outsid spd 1m/s), is in watr of dpt 1000m at 20N a) Wat would appn to its rlativ vorticity if it movs to 30N. b) Wat ar t two possibilitis for t ddy if it movs into watr of dpt 700m? 1m/s f +ζ f1510-5, f , ξ ( (-1)-(1) ) ( (1)-(-1) ) 50,000 1m/s 1m/s 50,000m 1m/s dv du ζ d dy f 2Ωsin( φ) Vorticity Q f +ζ 1m/s 1m/s 1m/s f , f , ξ 1 ( (-1)-(1) ) ( (1)-(-1) ) a) 50,000 1m/s Potntial vorticity is consrvd 50,000m f1+ ξ1 f2 + ξ2 ξ2 f1+ ξ1 f2? X X Latitud dosn t cang f2 + ξ 2 f3 f2 ξ3 3 f2 b) f2 +ξ 2 f3+ξ Latitud dos cang 3 f2 + ξ 2 ξ3 ξ2 f3 3 ξ2 2

20 du dt dη fv g + d ρ ru Doldrums 10S Equator 10N 20N du dt dη fv g + d ρ ru du dt dη fv g + d ρ ru On t quator, f0 SEC du dt EUC dη fv g + d ρ ru

21 10S SEC SEC NECC NEC. EUC. 10N 20N Equator Coriolis strongly constrains EUC to t quator dρ dy > 0, f < 0, u z > o du dη fv g + dt d ρ ru Gostropic flow T abov quation will giv t surfac flow but tr ar orizontal dnsity gradints, so flow cangs wit dpt dv dz du dz g dρ ρo f d g dρ ρ f dy So, u bcoms mor positiv wit dpt, Flow magnitud bcoms smallr wit dpt and dis out blow t trmoclin o

22 Scmitz (1996)

23 Surfac gravity Wavs 2002 Brooks/Col, a division of Tomson Larning, Inc. 45 Disprsion Rlationsip For a 2D wav travlling in t dirction: 2 2 d η d η g 2 2 dt d 0 Assumptions: No Coriolis No friction Constant dnsity Sallow watr/ motion of watr particls -dir only η Acos(k ωt) Try a wav lik solution ω 2 ω gk 2 2 Disprsion rlation Mor inclusiv disprsion rlation gk tan( k) 46

24 2 ω gk Pas Vlocity >>λ c ω k 2 ω 2 gk <<λ Dp watr wavs: > λ /2: g gλ c k 2π Sallow watr wavs: < λ/20: c g Biggr wavlngt fastr wavs, tus DISPERSIVE Sallow watr Slowr wavs 47 Group Vlocity Group Vlocity is t vlocity at wic a group of wavs travls, & t vlocity tat wav nrgy propagats: Dp Watr Group Vlocity g c c g 2 ω 2 Sallow-watr Group Vlocity c g g c g dω dk Rmmbr t Pas Vlocity is givn by ω c k 48

25 Applications Wav Rfraction: C g > C j j i g i Wav Sorting: c gλ 2π Wav Amplitud: F ce F c E i i i j j j A 2 j 1/ 2 2 ( i / j ) Ai Wav braking: A λ/12, A (Dubury) 50

26 Tid gnrating forcs: Sum of cntrifugal and gravitational forcs Cntrifugal forc: Gravitational forc: Cntrifugal forcs ar ar t sam vrywr Gravitational forc is gratst closst to t sun/moon If you wr on a watr world wr t solar tidal priod was 10 ours (lngt of day 20 rs) and t lunar tidal priod was 11 ours, ow long would t spring nap cycl b? 52

27 10r 11r 10r 11r

28 10r 11r 10r 11r Constructiv intrfrnc Dstructiv intrfrnc Constructiv intrfrnc

29 Tids Also nd to considr: Lots of otr tidal; componnts (>100) Coriolis Friction Rsonanc 57 Etra stuff T Soutrn Ocan X. 50S. 30S

30 T Soutrn Ocan X. 50S. 30S T Soutrn Ocan X. 50S. 30S

31 T Soutrn Ocan X. 50S. 30S dv dz du dz g ρo f g ρ f o dρ d dρ dy T Soutrn Ocan X. 50S. 30S

32 T Soutrn Ocan X. 50S. Midlatitud wstrlis 30S ACC Subtropical gyr AAIW Dacon Cll NADW AABW Modl Data (www-pcmdi.llnl.gov/projcts/cmip/) 20+ coupld climat modls (ddy-prmitting to ~5º ocan) wit rang of paramtrisations 20t C climat basd on t last dcad of 20C3M primnts 21st C climat basd on last dcad of 21st cntury SRES A1B primnts Drift calculatd from 100 yrs of pr-industrial control (spanning 21st C) Multipl ralisations usd wr availabl

33 Cangs to t Ocan Forcing to Multi-modl man Individual modls ERA40 Multi-modl man for zonally avragd surfac filds. Rd (blu) indicats a positiv (ngativ) cang ovr 21 st cntury. Black or yllow lin sows obsrvd long-trm-man. Yllow ERA40 21C sifts soutwards and intnsifis Cangs to t Ocan Forcing to Larg biass in ydrological cycl 21C ydrological cycl intnsifis

34 SST / SSS Multi- Modl cang ( to ) SST & SSS cangs modulatd by cang in Ekman transport Spatial pattrn of SST cang modulatd by MLD SSS cang consistnt wit cang in ydrological cycl Incrasd stratification drivn by frsning surfac, sout of 60S warming - lswr Cangs in zonally avragd potntial dnsity Surfac intnsifid stratification, drivn by Salinity at ig latituds Tmpratur at mid-latituds Soutward sift in mid-latitud proprtis Mottld rgion indicats dnsity cang primarily du to salinity cangs 20 t C 21 st C

35 Mid Layr Dpt Envlop (long trm man and 21 st Cntury Cang) Cangs in t latral circulation and dpt intgratd at contnt Polward sift of wind strss curl > soutward sift of gyrs Polward intnsification of wind strss > soutward intnsification of ACC Incrasd dpt intgratd at contnt intnsifid at mid-latituds Multi-modl man

36 Ovrturning circulation X. 50S Midlatitud wstrlis 30S ACC Subtropical gyr AAIW Dacon Cll NADW AABW Sa-Ic HadISST

37 Summary