u x A j Stress in the Ocean

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Angela Harrell
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1 Strss in t Ocan T tratmnt of strss and strain in fluids is comlicatd and somwat bond t sco of tis class. Tos rall intrstd sould look into tis rtr in Batclor Introduction to luid Dnamics givn as a rfrnc at t bginning of t class. T scaling u of turbulnt strsss to ocanic scals is vn mor sotric. Hr w will mrl summari wat is known about t form of t bod forc for ocanic flows. It involvs a tnsor wit tr comonnts of strss for ac of t tr momntum quations. [ [ [ T notation is tat t surscrit rfrs to t strss dirction of t comonnt and t subscrit rfrs to t drivativ of t comonnt. T rlationsi of strss to strain fluid rsons) can b comlicatd. or fluids lik watr and for scals tical of tos w will b discussing in tis class it can tak on t following form: i j A j u i j wr A j is a constant known as t dd viscosit. or fluids tat ar sallow rlativ to tir widts ts tr assumd constants A A A ) ar not t sam: t vrtical comonnt is muc smallr tan t two oriontal comonnts wic ar usuall not alwas) assumd to b qual. If w insrt tis dfinition into t abov quation for t vctor bod forc w can gt t following rlationsi wr A A A ) [ A u A u A u [ A v A v A v [ A w A w A w

2 Our dnamical quations av alrad incororatd on of ts bod forcs into tm: it is t tird comonnt of ac of t abov sar strsss. Considr t vrtical intgral of on of ts tr forcs sa t first on). 0 d d[ d d H 0) H ) T aroimation is bcaus w ar taking H to b a constant dt lvl and assuming t surfac to b flat or it s variation to b small comard to t dt of t ocan). Considr now t last two trms in t abov. T last on is roortional to t strss on t bottom and t rcding on is du to t strss on t to or fr surfac. If w writ t bottom strss as u H ) A H ) ru w s tat w can now associat tis trm wit t siml bottom friction w av rtaind from our ucks on ic modl. It is tis trm tat las an imortant rol in t Stomml modl for t wstrn intnsification of t wind-drivn grs. As for t wind driving tis is t just t surfac comonnt of t strss 0). or t Svrdru balanc w obtaind t following quation βv ) or β dv d 0) wr w av again ignord satial variations of H st t strss at t bottom to ro and ignord t oriontal comonnts of t strss as wll. As ou will rcall tis was t assumtion inrnt in t Svrdru balanc tat w could ignor bottom strss or t frictional trm roortional to r. T abov statmnt of t Svrdru balanc is mor gnral tat t rvious on bcaus w av licitl includd t surfac wind strss and w av NOT mad t assumtion tat t vlocit v is constant wit dt. So t Svrdru balanc can b alid to t vrtical intgral of wind-drivn currnts including t baroclinic vlocit) and subjct to som rquirmnts on t lvlnss of t ocan dt is a usl dscrition of wind-drivn frictionlss ct for t surfac lar) flow. Not tat it dos

3 includ t wind driving and trfor t dnamics of t surfac Ekman lar. Gostroic and Ekman comonnts of t Svrdru Circulation T Svrdru balanc is a vrtical intgral of t simlifid otntial vorticit quation. Lt s st back now and look at wat vlocitis mak u tis balanc. W will b using t mor gnral quations for baroclinic flow r wic ou will rcall ar t following: du dt ru dv dt 0 g u v w 0 d dt rv u v t w W could now insrt t form of t bod forc drivd abov and w would av t ll turbulnt quations usd in larg-scal sical ocanogra. But r w ar intrstd in t Svrdru balanc wic is for a stad linarid ocan wit wind forcing but otrwis no licit friction. So t abov bcom

4 0 0 w v u g Suos w writ t vlocit comonnts as t sum of a gostroic art and an Ekman art wit surscrits dnoting ts. T first two quations bcom g g W av alrad lookd at t gostroic balanc. Now w look rtr at t Ekman balanc intgrating t Ekman balanc btwn t fr surfac 0) and a dt at t bas of t Ekman lar -Z ). f Z w v u d f dv f du 0) ) 0) 0) In t first two quations abov w s tat t intgratd Ekman transort is to t rigt of t alid wind strss. In t tird w s tat tr is a

5 oriontal divrgnc of t nt Ekman transort wic will roduc a vrtical vlocit at t bottom of t Ekman lar t vrtical vlocit at t surfac bing ro). Ts rssions ar not valid at t quator wr t Corilios aramtr vaniss! T vrtical vlocit is suc tat tr is sinking wit subtroical grs and uwlling witin subolar grs irrsctiv of misr. Now if w know t intgratd Ekman transort and w also know t vrticall intgratd Svrdru transort tn t vrticall intgratd gostroic transort must just b t diffrnc btwn t Svrdru and Ekman transorts sinc b dfinition v s v g v. Som invstigators av usd tis constraint to l dtrmin t unknown rfrnc lvl vlocit for gostroic motion but tis is onl as good as t assumtions bind it. In fact ou will b amining tis in on of our nt omwork roblms. T fact tat tr ar larg diffrnc btwn t actual and Svrdru volum transorts in t Gulf Stram 50 vs. 30 Sv. rsctivl) sould mak on aroac tis mtod wit caution! W will b sing tis diffrnc latr. Munk s gr vs. Stomml s gr Munk 950) roducd anotr modl of t N. Atlantic wind-drivn circulation and laind its wstward intnsification using a diffrnt friction modl tan Stomml. Instad of vrtical friction as containd in t frictional aramtr r oosing t motion Munk usd a oriontal dd viscosit. W will not go into tis modl as it also rquird a ositiv β to av t Gulf Stram strongr on t wstrn boundar. W will just writ down t linar vorticit balanc wit bot frictional trms. A ψ ψ ) r ψ ψ ) βψ [ 0) 0) Hr w av rlacd t bod forc wit t wind strss at t surfac and t stramnction is now t vrtical intgral of t rvious stramnction: dv ψ & du ψ

6 Munk solvd t abov nglcting t bottom friction trm roortional to r) but rtaining oriontal friction roortional to A ) Bcaus tr ar now four -drivitivs to t stramnction t wstrn boundar currnt is vn sarr and mor intns tan Stomml s wic ad 2 -drivativs. As w indicatd arlir t intrior Svrdru Balanc olds in t intrior of t basin but in t boundar nar t wstrn boundar a diffrnt dnamical balanc must b rsnt. W can dtrmin t widt of t wstrn boundar currnt in ac of t two modls vr asil witout actuall solving t roblm. In ac modl w av t following aroimat balanc nar t wstrn boundar wr tings cang raidl in t -dirction: rψ A ψ βψ βψ 0 s 0 r Stomml' s wbc widt β m A β / 3 Munk' s wbc widt Tim-dndnt & non-linar numrical modls of t ocan circulation using itr bottom or latral friction must b abl to rsolv t structur of t wbc. T abov satial scals for t two ts of boundar currnts bcom imortant minimum scals rquird for numrical ocan modls. Tis comlts t tra nots on t wind-drivn circulation. urtr discussion will cntr around t ttbook until w mov on to otr arts of t ocan.

Physics 43 HW #9 Chapter 40 Key

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