Ocean Modeling - EAS 8803 Equation of Motion

Ocean Modeling - EAS 8803 Equation of Motion

Conservation equations (mass, momentum, tracers)

Boussinesq Approximation

Statistically average !ow, Reynolds averaging

Scaling the equations

Primitive Equations for a geophysical !ow

Chapter 3-4 Chapter 3

Equations of Fluid Motion

(July 26, 2007) SUMMARY: The object of this chapter is to establish the equations govern- ing the movement of a stratified fluid in a rotating environment. These equations are then simplified somewhat by taking advantage of the so-called Boussinesq approximation. The cha3p.t7e.r BcoOnUcSluSdINesESbQy AinPtrPoRdOuXciInMgAfiTnIOitNe-volume discretizations and showing their r7e7lation to the budget calculations used to establish the mathematical equations of motion. In most geophysical systems, the fluid density varies, but not greatly, around a mean value. For example, the average temperature and salinity in the ocean are T = 4◦C and S = 34.7, to which corresponds a density ρ = 1028 kg/m3 at surface pressure. Variations in density within one ocean basin rarely exceed 3 kg/m3. Even in estuaries where fresh river 3.1waterMs (Sa=s0s) ubltuimdateglyetturn into salty seawaters (S = 34.7), the relative density difference is less than 3%. A neceBsysacroyntsratastt,etmheeanitr oinf thfleuaidtmmosepchhearenbicescoims etshgartamduaasllsy bmeorceornarseefirevdedw.ithTahltaituidse,, andy imbal- ancietsbdeetnwsietyenvacroiensvferorgmenacmeaaxnimdudmivaetrggreonucnedilnevtehletothnreeaerlyspzaetrioalatdgirreacttihoenigshmts,utshtucsrceoavt-e a local ering a 100% rangMasse of Budgetvariati o-n Continuitys. Most o fEquationthe density changes, however, can be attributed to comhpydrerossstiaotnicoprreesxsupraenesfifoecntso,fletahveinflguoindl.yMa matohdeemraatetivcarlilayb,itlhitey csatautseemd beyntothaekrefsacthtoersf.oFrmur-: thermore, weather patterns are confined to the lowest layer, the troposphere (approximately ∂ρ ∂ ∂ ∂ 10 km thick), within whic+h the de(nρsuit)y v+ariation(sρrve)sp+onsible (foρrwth)e =win0d,s are usually no (3.1) more than 5%. ∂t ∂x ∂y ∂z As it appears justifiable in most instances3 to assume that the fluid density, ρ, does not 3 whedreepaρrt imsutchhefrdoemnBoussinesqsaitmyeaonf rte hApproximationfereflnuciedva(ilune,kρg0/,mwe),taaknedth(euli,bver,tywto) warreitet:he three components of velocity (in m/s). All four variables "generally vary in the" three spatial directions, x and y in the horizontal, z in the vρer=ticaρl0, a+s wρe(lxl,ays,tizm, et)t. with |ρ | ! ρ0, (3.16) wThheirse ethqeuvaatriioanti,onofρt"ecnaucsaeldlebdy tthee ecxoisntitningusittryatiefiqcuataiotinoann,di/sorcfllausidsimcaoltiionnstrisadsmitiaollncaolmfl-uid me- chapnaircesd. toSttuhremre(fe2r0e0nc1e, vpaalgue ρ40). rAeprmoretds wthiatht LtheisonasasrudmopdtiaonV, iwnecipr(o1c4e5e2d–to15s1im9p)lhifaydthdeerived a simgpolvifierendinfgoerqmuaotifontsh.e statement of mass conservation for a stream with narrowing width. The continuity equation, (3.1), can be expanded as follows: However, the three-dimensional differential form provided here was most likely written much later and credit ought probably to go to Leonhard Euler (1707–1783). For a detailed deriva- ∂u ∂v ∂w " ∂u ∂v ∂w tion, the reρa0der is r+eferred+to Batche+lor ρ(1967), F+ox and+McDonald (1992), or Appendix A of the present te!xt∂.x ∂y ∂z " !∂x ∂y ∂z " ∂ρ" ∂ρ" ∂ρ" ∂ρ" Note that spherical geometry i+ntroduces +adudition+al vcurvat+urew terms,=wh0i.ch we neglect to be consistent with our previous restr!ic∂titon to l∂enxgth sca∂lyes subs∂tazn"tially shorter than the globGaelospchaylseic.al flows indicate that relative variations of density in time and space are not larger than – and usually much less than – the relative variations of the velocity field. This implies that the terms in the third group are on the same7o1rder as – if not much less than – those in the second. But, terms in this second group are always much less than those in the first because " |ρ | ! ρ0. Therefore, only that first group of terms needs to be retained, and we write ∂u ∂v ∂w + + = 0. (3.17) ∂x ∂y ∂z Physically, this statement means that conservation of mass has become conservation of vol- ume. The reduction is to be expected because volume is a good proxy for mass when mass per volume (= density) is nearly constant. A hidden implication of this simplification is the elimination of sound waves, which rely on compressibility for their propagation. The x– and y–momentum equations (3.2a) and (3.2b), being similar to each other, can be treated simultaneously. There, ρ occurs as a factor only in front of the left-hand side. So, " " wherever ρ occurs, ρ0 is there to dominate. It is thus safe to neglect ρ next to ρ0 in that pair of equations. Further, the assumption of a Newtonian fluid (viscous stresses proportional

3 The situation is obviously somewhat uncertain on other planets that are known to possess a fluid layer ( Jupiter and Neptune, for example), and on the sun. 3.7. BOUSSINESQ APPROXIMATION 77

In most geophysical systems, the fluid density varies, but not greatly, around a mean value. For example, the average temperature and salinity in the ocean are T = 4◦C and S = 34.7, to which corresponds a density ρ = 1028 kg/m3 at surface pressure. Variations in density within one ocean basin rarely exceed 3 kg/m3. Even in estuaries where fresh river 3.7. CBOUhSSaINpEStQeArPPR3OXIMATION waters (S = 0) ultim77ately turn into salty seawaters (S = 34.7), the relative density difference is less than 3%. In most geophysical systems, the fluid density varies, but not grBeaytlyc,oanrotruansdt,a tmheaanir of the atmosphere becomes gradually more rarefied with altitude, and value. For example, the average temperature and salinity in the ocean are T = 4◦C and S = 34.7, to which corresponds a density ρ = 1028 kg/m3 at surfiatcsedperenssuitrey. vVaarriiaetisonfsroinm a maximum at ground level to nearly zero at great heights, thus cov- densiEty wqithiun onae otceiaon bnasinsrareoly fexceFed l3 ukg/mi3d. EvMen in eoersituntagriieaos 1wn0h0er%e frreashngrieveor f variations. Most of the density changes, however, can be attributed to waters (S = 0) ultimately turn into salty seawaters (S = 34.7), thehryeldartiovestdaetniscitpyrdeifsfseurerneceeffects, leaving only a moderate variability caused by other factors. Fur- is less than 3%. thermore, weather patterns are confined to the lowest layer, the troposphere (approximately By contrast, the air of the atmosphere becomes gradually more rarefied with altitude, and its density varies from a maximum at ground level to nearly zero1a0t gkremat htheiigchkts),,thwusitchoivn- which the density variations responsible for the winds are usually no ering a 100% range of variations. Most of the density changes, however, can be attributed to (July 26, 2007) SUMMARY: The object of this chapter mis otoreesttahbalnish5t%he.equations govern- hydrostatic pressure effects, leaving only a moderate variability caused by other factors. Fur- 3 ing the movement of a stratified fluid in a rotating environmAenst.itTahpespeeeaqrusajtuiosntsifiaarebtlheenin most instances to assume that the fluid density, ρ, does not thermore, weather patterns are confined to the lowest layer, the troposphere (approximately 10 kmsimthpiclikfi)e, dwistohminewwhhiacthbthyetadkeinnsgityadvvaarinattaiognesorfestphoensiob-lceaflolderdethpBeaowrutisnmsdinsueacsrqhe aufpsrupoarmollxyiamnomatieoann. Trehfeerence value, ρ0, we take the liberty to write: morecthhaapnt5e%r c.onclude3.s7.bByOUinSStIrNoEdSQucAiPnPRgOfiXInMiAteTI-OvNolume discretizations and show77ing their relation to the budget calculations used to establis3h the mathematical equations of motion. " " As it appears justifiabInlemoisnt gmeopohsytsicianl ssytastnemcse, sthe tflouidadsesnusimty evartihesa, tbutthneot flgrueaitdly,daeronunsditay,meρa,n does not ◦ ρ = ρ0 + ρ (x, y, z, t) with |ρ | ! ρ0, (3.16) depart much from a mevaalune.reFofreerxeanmcplee, vthaelauveer,agρe0te,mwpeeratuarke eandthsaelinlitby einrtthye tooceawn raireteT: = 4 C and S = 34.7, to which corresponds a density ρ = 1028 kg/m3 at surface pressure. Variations in " density within one ocean basin rarely exceed 3 kg/m3. Even in estuaries where fresh river " " where the variation ρ caused by the existing stratification and/or fluid motions is small com- waters (S = 0) ultimately turn into salty seawaters (S = 34.7), the relative density difference ρ = ρ0 + ρ (x, y, z, t) with |ρ | ! ρ0, (3.16) is less than 3%. pared to the reference value ρ0. Armed with this assumption, we proceed to simplify the where3t.h1e variMatioansρs" cbauBusyecdodngtbrayset,tttheeaier xofitshteiantmgosspthrearteibfiecoamteisognradaunadlly/omrorfle ruariedfiemd woithioalntitsudies, asnmd all com- its density varies from a maximum at ground level to nearly zerogaot gvreaet rhenigihnts,gthues cqovu- ations. pared to the referenceervinaglau1e00ρ%0r.angAe orfmvaeridatiownsi.tMhostthoifstheasdesnusimty pchtainognes,, hwoweevperr, coacnebedattrtiboutesdimto plify the hydrostatic pressure effects, leaving only a moderate variability caused bTy ohtheer facctoorns. Ftuirn- uity equation, (3.1), can be expanded as follows: goverAninngeceeqsusaatrioynst.attheemrmoerne,tweiantheflr puaittdernms areecchonafinneidctso thise lotwheasttlamyera, tshse trbopeoscphoernes(aeprpvroexidm.atelTy hat is, any imbal- Tahnecceonbteitnwuieteyneqcuo1an0tvikomenrtgh,ie(ck3n).,c1we)it,hacinnawdnhidbcheivtheeexrdpgenaesnnitydcveeadrianatisotnhfsoerelsltpohownrseisbe:le sfopr athteiawlinddsiarree custuiaollny snomust create a local morMasse than 5Budget%. - Continuity Equation compression or expAas nit sapiopenarsojufsttifihaebleflinumidos.t Minstaantchese3mto aastsiucmaeltlhyat, tthhe eflusidtadetnesmity,eρn, dtoteas knoet s the form: depart much from a mean reference value, ρ0, we take the liberty to write: ∂u ∂v ∂w " ∂u ∂v ∂w ∂u ∂v ∂w " ∂u ∂v ∂w ρ0 + + + ρ " + + " ρ0 + + + ρ + + !∂x ∂y ∂∂ρz " ∂ρ = ρ0 !+ ∂ρ x(x, y,∂z,∂t)y with |ρ∂|z!∂"ρ0, !(3.1∂6) x ∂y ∂z " !∂x ∂y ∂z " + " (ρu) + (ρv) + (ρw) = 0, (3.1) where the variation ρ caused by the existing stratification and/or fluid motions is small com- ∂t ∂x ∂ρ" ∂y∂ρ" ∂ρ∂"z ∂ρ" " " " " pared to the reference value ρ0. Armed with this assumption, we proceed to simplify the ∂ρ ∂ρ ∂ρ ∂ρ governing equations.+ + u + v + w = 0. ! ∂t ∂3 x ∂y ∂z " + + u + v + w = 0. where ρ is the denBoussinesqTsheitcyontoinfuitty hApproximationeqeuafltioun,id(3.1(),incankbege/xmpand)e,d aasnfodllow(us:, v, w) are the three components of ! ∂t ∂x ∂y ∂z " Geopvheylsoicaitlyfl(oiwnsmin/sd)i.caAtellthfoaturevlaatriivaebvleasriagteionnesraolflydevnasriytyininthtiemteharened sspactiealardeirneocttiloarngse,rx and y in ∂u ∂v ∂w " ∂u ∂v ∂w the horizontal, z in thρe0 vert+ical, +as well+asρtime t+. + than – and usually much less !th∂axn –∂tyhe re∂zla"tive var!ia∂txions∂yof th∂ez G"veeloocpityhyfieslidc.aTlhflisoiwmpsliiensdicate that relative variations of density in time and space are not larger ∂ρ" ∂ρ" ∂ρ" ∂ρ" that the teTrmhissienqtuheattihoinrd, gorfoteunp acraelloendtthheesacmone+toinrdueitrya+seuq–uifatn+ioovtnm, uis+chcwllaesssitch=aal0n.i–n tthroasdeitiinonthael fluid me- ! ∂t ∂x ∂ythan∂z–"and usually much less than – the relative variations of the velocity field. This implies seconcdh.aBniucts,.terSmtusrimn t(h2i0s 0se1c,opnadggero4u)preapreoratlswtahyastmLuecohnlaersdsothdaan Vthionscei i(n1t4h5e2fi–r1s5t 1b9ec)ahuasde derived a " Geophysical flows indicate that relative variations of density in titmhe antd stphacee arteenrotmlargserin the third group are on the same order as – if not much less than – those in the |ρ | !simρ0p.liTfiheedreffoorrme, otohnafnContinuityly–thatnhedauststufiaalrtlye sEquationtmguecrhnoleutspsot hfoinafnm the–teathrse moceansreslcatnoiveenevsdaersiratvtioantsbiooef ntrheftvoaelironcieatydfis,etalrdne. daThmwis iemwpwliiterhsitenarrowing width. However, the thrtehaet-thdeitmermesninstihoe nthiardl gdroiufpfearreeonn thieaslamfoe rormderpasr–oivf niodtemsduechcleeosrsenthwdan.a–stBhomsueoitns,tthteleikremlyswirnitttehnims uscehcond group are always much less than those in the first because second. But, term∂suin this sec∂onvd group ∂arewalways much less than th"ose in the first because later and credit o|ρu"|g!hρt p. Trhoebrefaobrel,yon+ltyothgatofirstt+ogroLupeoofntehrm=asrnde0e.dEs utolbeerre(ta1in7ed0, 7an–d1w7e w8r3ite). For a(d3e.1ta7i)led deriva- 0 ∂x ∂y ∂z |ρ | ! ρ0. Therefore, only that first group of terms needs to be retained, and we write tion, the reader is referred to Batchelo∂ru (19∂6v 7),∂Fwox and McDonald (1992), or Appendix A of Physically, this statement means that conservati+on o+f mas=s h0.as become conserv(3a.t1i7o) n of vol- the present text. Conservation∂x ∂y of∂ zVolume ∂u ∂v ∂w ume. TheNroetdeucthtiaotnsipsPhtheoysribiccaelalyel, txhgipseesotcamtteemdeentbrt ymeecaiannusttsrheoatdvcouoncsleuervmsatieaondisodf aimtaigsosonhoaasdlbepccrouomrxevcyaotnfusoerrrveamtitoeanrsomsf vwsol,-hwenhmichasswe neglect + + = 0. (3.17) per volume (= density)umies. nTheearreldyuctcioonnissttoabnete.xpAecthedidbedcaeunse ivmolupmleicisaatigooond oprfoxtyhfiosr msaimss wphleinfimcastsion is the to be consistent pwer ivtohlumoeu(=r dpenrseitvy)iios nuesarlryecsontsrtiacntt.iAonhidtdoen limepnligcatthionsocf athlies simspulifibcsattioannitsitahelly shorter than the ∂x ∂y ∂z elimination of sound welaimviensat,iown ohfiscouhndrewlayveso, nwhcicohmrelpy roen scosmibprielsistiybilfitoy rfotrhtheeirrpprorpoagpataiogna. tion. global scale. The x– and y–momentum equations (3.2a) and (3.2b), beingPsihmyilasr tioceaaclhloyth,er,tchanis statement means that conservation of mass has become conservation of vol- The x– and y–mombeetrneattuedmsimeuqltaunaeotuioslyn.sTh(e3re.,2ρao)ccaurnsdas (a3fa.c2tobr)o,nlby einifnrogntsoifmtheillaefrt-htaondesiadce.hSoo, ther, can " " be treated simultaneouwshleyr.evTerhρeorcec,urρs, oρ0cics uthresreatos daomfiancatteo. rItoisnthlyus isnafeftroonnuegtmleocfteρt.hneeTxltehtofetρ-0hrianenthddatusicdtei.oSno,is to be expected because volume is a good proxy for mass when mass " pair of equations. Further, the assumption of a 7Ne1wtonian fluid (viscous stress"es proportional wherever ρ occurs, ρ0 is there to dominate. It is thus safe to neglect ρ next to ρ0 in that 3 The situation is obviously somewhat uncertain on other planets that are knowpn teo prossvessoa flluuid mlayere( Ju(pi=ter density) is nearly constant. A hidden implication of this simplification is the pair of equations. Furthaned Nr,epttuhne, faorsexsaumpmle)p, atnidoonntheosfuna. Newtonian fluid (viscous stresses proportional elimination of sound waves, which rely on compressibility for their propagation. 3 The situation is obviously somewhat uncertain on other planets that are known to possess a fluid layer ( Jupiter and Neptune, for example), and on the sun. The x– and y–momentum equations (3.2a) and (3.2b), being similar to each other, can be treated simultaneously. There, ρ occurs as a factor only in front of the left-hand side. So, " " wherever ρ occurs, ρ0 is there to dominate. It is thus safe to neglect ρ next to ρ0 in that pair of equations. Further, the assumption of a Newtonian fluid (viscous stresses proportional

3 The situation is obviously somewhat uncertain on other planets that are known to possess a fluid layer ( Jupiter and Neptune, for example), and on the sun. 3.3. EQUATION OF STATE 73 the pressure p and the density ρ. An equation for ρ is provided by the conservation of mass 3.3. EQUATION OF STATE 3.3. EQUATION(3O.1F)S, TaAndTEone additional equation is still r7e3quired. 73

3.3. EQUATION OF STATE 73 the pressure p and the density ρth. eAp3nr.e3es.qsuuEraeQtipoUnaAnfTdoIrtOhρeNidsOepnFrsoiStvyTidAρe.TdAEbnyetqhueatcioonnsfeorrvρatisonproofvmideadssby the conservation of mass 73 (3.1), and one a3d.3d.itEioQnUaAl eTqIOuaNtiO(o3nF.31Si.)s3T,.sAatnTiEldElQorUenAqeTuaiIdrOedNdit.iOonFaSlTeAqTuaEtion is still required. 73 73 the pressure p an3d.t3he denEsiqty uρ.aAtnioeqnuatoionf fsortρaitseprovided by the conservation of mass the pressure p and th(et3hd.te1he)pne,rsapeintsrydseuρsors.neuAepraneadnepdqditauitnaohtndeioadtlnheenfeqosuridtayeρtinoρisn.iptAiysronsρvteii.lqdlAuerdaentqbioueyniqrtehufodear.tciρoninssefprorvoravtρidoiensdopbfrymovtahisedsecdonbseyrvthateiocnoonfsmeravsastion of mass (3.1), and one additio(n3a(.l31e).1q, au),nadtainoondneiosansdetidlailtdiroednqiautilioreenqdau.lateioqnuaistisotinllirseqstuiilrlerde.quired. The description of the fluid system is not complete until we also provide a relation between 3.3 Equation of sta3te.3 Equatdieonnsitoyfansdtaprteessure. This relation is called the equation of state and tells us about the nature of the fluid. To go further, we need to distinguish between air and water. Th3e.d3escrEiptqiounaotfitohenfloufidsstyastteem is not complete until we also provide a relation between The description of the fluid system is not completeFuonrtialnwienaclosmo pprreosvsiidbeleaflreuliadtisouncbheatws epeunre water at ordinary pressures and temperatures, 3.3 Equatiodenn3so.i3tfy asntdEaptqereussautrieo. nThoisfresltataiotneis called the equation of state and tells us about the nature density and pressure. This relationThi3se.cd3aesllcerdipEtihoqenteuohqfeauthastetiiaflotuennimdofesoynstfstatectsmeatnaainsbtdneeottealcslossmiumpslpeatbleeouuantstitlρhwe=enacltosuonrseptraonvti.deIna rtehliastiocnasbee,twtheenpreceding set of equations of the fluid. To go further, we need to distinguish between air and water. of the fluid. ToTghoe fduerstchrieprt,iowneonf ethededTeflhntoeusidtdyeisssayctnisrdnitpegpmtiuroeiinssishsounfrbocetteo.htceTmwohflmepiuselpienrdleteatlsea.yitrsuiotaInennntmidilstwihwsceanalaoltoeeltscdrco.etohapmenrop,evlqehiudtoeaewtuiaonerntveileolawfrt,sieotwanatlaebstoeaetnprwdredoteevenlinldsseuitasyraebilsoautaitotnchoebemntawptuelirecenated function of pressure, FTohreadneisnccroipmtpiornesosifbtleheflufliudidsuscyhstaesmpuisrenwotacteormatploertdeinuanrtyilpwreessaulrseosparnodvitdeme paerrealtautrieosn, between For an incodmenpsriteyssainbdleprflesusiudres.oudTfcehthnhisseiatrflsyeulapaintduid.orneTporiwesgstcaesoatumelfrlureepr.dateThtrhehaoreit,sruewdrrqeieeunlanaaatetirnioeyoddnnptiososrafecdlssaiitsnlsalutteiiertndyega.tsnuhdieaDsnhteeedqbltuleastateiuwtlmisoeanpecbneoaorfnauaistrttbuatahertneedfsano,wnaudtaunttreederl.lsinusGaiblolu(t1th9e8n2at–urAe ppendix 3), but for most the sdtaetnesmiteynatncdanprbeessausrsei.mTphleisarseρlat=iocnonisstcaanltl.edInththeiseqcuaaseti,otnheopf rsetacetedianngdsteetllosfuesquaabtoiountsthe nature the statement coafnthbeeflausids. iTmopgloefausrothρfetFr=h,oewcrfleoaunnnidseit.enadTcnottom.agdpopIipnrsfeutlsiitrnshctighaibsuetliericos,hflawnsubseei,edn,twietstehuedecehnatponaardiesrbicspaeetunidnrdaegisnwuwsgiausathsmeterb.retedatowtftoeheerqadntiunaaitarhtriyaeonnpddsrewsnassuteirtrey.s aonfdsteeamwpaertaetruriess,independent of pressure ( is tchoeomfstptahlteetmefl.eunitIdnc.atTnhoebegooacsefausnirm,thpheloerw,awesveρenr=,ecweodanttseotranddtie.sntIisnnitgtyhuiisschabsece,otwtmhepelpincreaactierdainnfgudnswcettaiootenf re.oqfuaptrioesnssure, is complete. In thFeoroacneiannc,omhopwreessviebrl,FeowflruaaitdnersinudcceohInmnaspscirtopyemussriiespbrlwaeasflctseuoirbimdailtpsiutolyicrcd)haiantaenasdrdpyuflpiurnreneecwsastraiuoltryenersdaoaetfnpodeprdnrteeidnmseaspnruyetrreupa,trpueorsesnsu,breostahntdemtempeprearatuturrees(,warmer waters are lighter) the statement can bteemitashpsecesorsimmatatFpupteollremeerteeaaa.nsntρdIcinna=sncathcolbioemneniostpactysare.enasstniD.ms,ieIpbhntlolaeetwihlasfleissvucρecairadn,=sewsbc,uaoetcthnefhesrotapuadnrnseetnd.cpseIiuidntnyrientGhigswiisslalaecta(ce1sore9fm,8aeptt2qhliueoc–araptdtiAreoiednnpcaspefdrueyinncdgptiirxsoeens3ts)oou,ffrbepuqsrteuasafsnotuidrorenmt,seomstperatures, temperature and salinity. Details can be foundanind Gsailliln(it1y98(s2al–tieArpwpaetnedrisxa3re),dbeuntsefor)r, macocsotrding to: is complete. In thaeptpoiesclmtiehccpaaoenetmr,isoapthtnaluoestrwet,eeme.iavteneIcndrna,tsntwcahlaabeitneeoirtbcayedes.asenuanD,smeihtseyaoidimwlissetphvcalaaeetnrc,aotbwhsmeeaρptfledoi=rceuanndctsdeoindtinysnfistutoyGanfnicilststli.eoa(a1nIwc9noo8afmt2theppi–rrsleiiAcscsaspautipesnredeedn,,efdtuphinxeecn3tdpi)oe,rnnebctuoeotfdffpionrpregrsmessuoesrsutetr,oef(equations applications, it can be assumed that the density of seawater is independent of pressure ( temperature and saIlnincatipoetimpmys.lpipceDraoretaemisotsuanpiirbslles,ielaticietntay.dnc)asabInaneldibnfteolhiituneyan.sedsoauDrcimlneeyateaGddnieli,sltplhheca(ona1tnwd9te8hebn2evete–dufroep,AnuowsnpnidptaybetinoenodrtfhGixdstieel3lamn)w(,s1piab9tetuye8rtar2itfisuo–srraeAimn(pcdwopoesaepmtnremdnpidlexiercn3wat)t,aeotbdfeurptfsrufeaonsrrsceutmlriioegonhs(ttoerf) pressure, Incompressibility) and linearly dependent upon both temperature (warmer waters are lighter) applications, it cananbIdanepctsepaaosmllmisicnupaimrtetieyroesadns(tissubat,hrillietaititteyacrt)anhwandenabddsteeealirnlnasissneiastaiyurtreylmyo.defdedDesnpetsehaeetwnarat)dai,letsaehnrcetccuiadospneroρindnnbsidni=ebtegyopftteohooρnuf:t0densem[edn1aptwieon−rafatGtepurαrirelei(lsT(sw(ui1nra−d9erem8(pT2ee0rn–)wdeaA+ntetprposβfea(nrpSedreli−isxgshu3Stre)e0r,))(b],ut for most (3.4) and salinity (saIlnticeormwparetsesrisbialirtey)daennadIsnneldcirnos)ema,alaiprnclryiectsyodseri(dpbsaeilnlnittgdiyee)rtnoawtn:uadpteloirnsebaaroreltyhdedtenemspeeprne)d,raeatncuctrouerp(dowinagrbmototeh:r twematepresraatruerleig(whtaerrm) er waters are lighter) applications,wihtecraenTbies athsseutmemedpethraattutrhee(idnednesgitryeeosf CseealswiuasteorrisKienlvdienp)eannddenSt othfepsreaslisnuirtey ((defined in the and salinity (saltier waantedrssaalrienidtyen(ssearl)ti,earcwcoartdeρirnsg=arteoρ:d0en[1ser−), aαcc(oTrd−inTg 0to): + β(S − S0)], (3.4) Incompressibility) anρd=linρea[r1ly−deαp(eTnd−enTt u) p+onβb(oSth−tSem)]p,erature (warmer w(a3t.4e)rs are lighter) ρ = ρ0 [1 − α(T − pTa0s)t +aEquations βgr(aSm− sof0So Statef0)s]a, l(tLinearper kandil0o gNonlinearram of) (s3e.a04w)ater, i.e., in parts per thousand, denoted by ‰, whearendT siaslitnhietyte(msapletrieartuwrea(teinrsdaergeredeesnCseerls)i,uascocor rKdeinlvginto) :and S the salinity (defined in the ρ = ρ0 [1 a−ndα(mTo−rρeTr=0e)ce+ρn0 t[βl1y(S−b−yαtS(hT0e)]−s,oT-c0)al+ledβp(Sra−ctiSc0a)l](,3s.a4l)iLinearnity unit “psu”(3, .d4e) rived from measurements where T is the temperature (inpdaewsgthraeesreesgTrCaiemslssthiuoesfteosmarlptKepreaeltvruirknei)l(oiangnrddaemgSreotehfsesCesaealwlsiianutisetryo,r(iK.deee.,filvininne)dpaainrndtstShpeethrethsaoluinsiatnyd(,dedfienneodteidn bthye ‰, where T is the tempepwraahsttueraes(TginriasdmethsgeroeotfeefssmaCcpltoeelpnrsaeidturuusrkceoitlri(oviKgnirtedaylmevgianrone)fdeassnehCdaaewSvlsaiittnuhegser,osniar.eolKi.n,ueiitlnyviip(tndsa))era.tfisnndpTeedShrietnhcetohousensaalsintnadi,ntytds(ednρeofi0te,ndeTdb0yi,n‰athn,ed S0 are reference values past as grams of salt per kilogaranmd moofrseeraewceantetlry, bi.ye.t,hiensop-acratρsllep=derprρtah0cot[iu1csaa−lnsdaα,lin(dTietyn−outneTidt0)“bpy+su‰”β,,d(Seri−veSd 0fr)o],m measurements (3.4) past as grams of saltapnpadesrtmkaosilroeggrrareamcmesnootolffyfssbeadyaletwtnphasetiertsryok,,-iilc.oteaeg.l,mlreaidnpmepproraafrcttustiercpaeaew,lrasattahenlroidn,uiists.yaean.,ludinin,niitdtp“eyapn,rsotrutsee”dsp,pedbreeyrctihtv‰ioevud,eslafyrno,dm,wdmheeneraoestueadrsebmαyeni‰sts,the coefficient of thermal and more recently by the so-calolfedcopnrdacutcitcivailtysaalinndityhauvninitg“nposuu”n,idtse)r.ivTedhefrcoomnsmtaneatssuρr0e,mTe0n,tsand S0 are reference values 1 and more recently byoatfnhwdceohmsneodor-eucreacTltrlievcdiisteypntrethaalxynecpdtbitaceyhnamaltsvhsipieoanelnsgironaa-inttcnuoyadrlueuleβnn(diiittinsps“)rp.dcascaeuTtgli”lhcr,eaedleec,ssroabivlnCyiesnetdaialtnsynfirtauouslmnsoρig0otm,y“r,peTKast0huse,u”elra,vecndimodneer)eSinfvafi0tenscdadirfeerSnortmetohfemfersesanaaslcliueinrnevimatcyleuon(endtssteraficnteiodnin. tThyepical seawater of conductivity and having nooufondfietdsn)es.nitsyiTt,yh,tetmecmpopenresartatautunrretes,,aρan0nd,d Tssaa0ll,iinaitnyyd,, rrSeess0ppeeaccrteitvi3vereleylfy,e,wrwehnehcreeeCoefra◦esvaas!αlcientsuαiessitsh ofeth cthermaleoecfofiecfifi eexpansioncniteonft tohfe rtmhearlmal −4 −1 of conductivity and hoafvcinogndnuoctuivniittysv)a.anldTuhehesavacironengsρtnaon=tsun1ρi00ts,2).T80kT, gha/enmdcoS,n0sTtanre=tsr1eρf00e,reCTn0c=,ea2vn8adl3uSeK0s ,aSre 1r=efe3r5en,cαe =va1lu.e7s× 10 K , and β = 7.6 expapnassiot nasangdraβmisscoafllesda,ltbypea0rnakliologyg,rtahme coofefsfiecaiwe0natteorf,sia.lei.n,eicnopnatrratcstipo1en0r .thToyupiscaanlds,eadwenatoetred by ‰, of density, temopfedreantusirtey, taenmdpesraaltiuneorxieftp,yda,anennsrdiesoistsnpyaa,elinctnedtiimtvβyp,eielsryreca,satpuwl−elreceh4dt,ei,varbeenyladysa,snwaαallhioneigisrtyey,ta,htshreeαesccpiooesectftfhfiifivecceiliceyeon,netwtfofifhocsefiareandelntiahntsee osalinerαcfmotinhastelr tracontractionhmcetailcoone.ffiTcyipenictaol fsetahweramtearl × 10 . 3 ◦◦ − −4− −1 valuaens darme ρore=r1e0ce2n8tklyg/bmy3,thTe s=o-1c0alCled= p2r8a13cKtic,aSl 1sa=li3n5i,tyαu=n1it.7“p×su1”04,1 deKr1ived, afnrodmβ m= e7a.6surements expansion and eβxpisancsailolneda,ndbyβaisncaalvolelaxgelpduya,e,nsbthsyaieroaennc0ρaoa0lneod=fgfiyβ1c,0iths2een8catcklogleef/dmfsfi,acb,liyieTnna00etn=oacflo1osn0gatylriC,natech=teci2oc8non3terf.aKficTc,tyiSoep0nni0t=c.oaTf3ly5ssap,elαiiacnwa=elacs1toe.e7anrwt×ratc1et0rion .KTyp,iacnaldsβea=wa7t.e6r −4− 3 × 10of◦ c3.4onductiv◦ity Faonrdahi3ra,vwinhgicnho◦isuncoitms−).4preTs−hs−1eib4cleo,n−ts1hteanstistuρa0ti,oTn0i−,s4aqnud−it1eS0diaffreererenfte. rDenryceaivrailnuetshe atmosphere values are ρ0 =v1al0u2e8s akrge/ρm0 =, 1T0028=×kv1ga01l/um0eCs,a=.Tr0e2ρ=8031=0K1C,0S2=802k=8g33/mK5,,,αST00===131.570, ×αC=1=012.873×K1,0S0, =aKn3d5,β,αa=n=d71β..67=×7.160 K , and β = 7.6 −4 × 10−4. ×Fo1ofF0rod−are4iarn.,irsw,itwhyih,cithcbeheimsihspcaceovormemastppuarrrepeessp,ssriiabbonllxeedi,mtshhaeeltisesnilittyiuutayat,tisioroanennsispiisedqqcuetuiaitievltegdlaidyfsfi,fe,frwaeenrnhedten.rtsDe.oarDyswraαyeirawiiisnrrititthnheee:thactemoaeotfsmfipohcseieprenhteroef thermal × 10 . 1 For air, which ibsecbhoeaemhvxaeFpvpsroeaearsnspasaspipibrrop,loernwxo,haixtmhinicmedahsatβieitsteluylicaysoataimcsosanpaalnrlniesisdidqsd,ieuebbaailtlyee,gadtaahnissfe,af,ealsaorninetgdudnyats.s,toiotDohwnwreeyiescwawoqirreuirtifientfi:ect:hdieifnfaettrmoenfotss.paDhliernryeeaciroinntrthaectaiotmno.spThyepreical seawater For air, which is compressible, the situation is quite differ3ent. Dry a◦ir in the atmosphere p −4 −1 behaves approximatelbyevahasalvauenessidaaperapelrogρxa0ism,=aant1ed0lys2oa8swkaegnw/imdrieNonlineartae,l:Tg0as=, a1 n(empirical)0d sCo w=e2w8r3iteK: , S0 =ρ35=, α = 1.,7 × 10 K , and β = 7.6 (3.5) behaves approximately as an ideal gas, a−nd so we write: pp × 10 4. ρρ == , , RT (3.5()3.5) p RRpTT For air, wwphhieρcrhe=iRs ciosmaρ,pcro=ensssρitba(Tlnet,ρ,tSeh=q,eupsa)iltutoa,t2io8n7ims q2u/(ist2e Kd(3i)f.f5ae)treonrtd.inDarryyateirminp(3et.hr5ae)tuartmesoasnpdheprreessures. In the where Rρis=a constan,t, equRaTl to 287 m22/(s22K) aRt oTrdinary tem(p3e.r5a)tures and pressures. In the whebreehRaviseas acopnpRsprtoTraexncitm,eedaqitnuegalyletaqosu2aa8nt7ioimdne,a/T(lsgisaKst,h)aeantadobrssdooinlwuatereywtteermmitep:eerraattuurrees (atnedmppreersasutureres.iInndthegerees Celsius + 273.15). where R is a constpanreptw,creehqcdeeuridenailgRntgoeiqes2uq8aau7ctaiomtoinon2sn,/t(,aTsTn2ti,Ksise)tqthhaueteaoalarbtbdossioo2nl8aur7tyemttteee2mm/m(pspp2eeerrKaraatt)uutuarreterseo(art(edntmeidnmpapeprryreeasrttseauumtrureeprsieen.riaIdnnteudgtrhereegseraseneCdsepClsreiuslssiu+urse2s+7.3I2.n175t3h).e.15). where R is a constant, equal to 287 m2/(s2 K) at ordAinirariyn ttehmepaetrmatousrepsheanred mproestsuorfetse.nIncothnetains water vapor. For moist air, the preceding preceding equation, TpAirseiActrheiedrinianinbgtshteoehqleuuatateamttitmoeonmos,sppTphehreieasrrteutehrmem(aootbsessttmooplufettreeeannteuccmroeopnnietntaraiadnitenusgsrrweew(aetatseetmCrerepvlpeasvripauaoptsruo.+rre.2Fi7noF3rdo.1merg5omr)ei.seotsisaCtire,alistrih,uesthp+er2epc7er3de.1cin5egd).ing equation is geneeraqluizaetdiobny iisntgroednuecrianlgizaefdacbtoyr ithnρatrt=ovdaruiecsinwgi,tah tfhaectsoprectihfiacthvuamrideistywqi:th the specific(3h.u5m) idity q: preceding equationA,irTinistthheeaatmbseooqslpuuhateteiroAetenimrmisoipnsgetetrhonaefettureaarntelmizc(ooetesdnpmthbapieynreseirnawmttrauotrsdeetruiocvnfiantepdgnoearg.crfoeaFnectostaroiCnrmsetholwsaisiattutvesaari+Rrrv,ieaT2tsph7owe3r.ipt1rh5eFct)ohe.redmisnpogeisctifiacir,huthmeidpirteyceqd:ing equation is generalizeedqbuyatiinotnroisdugceinnegraalifzaecdtobrythinatrvoadruiecsinwgiathfathcetosrptehcaitfivcahriuems iwdityh qth: e specific humidity q: Air in the atmosphere most often contains water vapor. For moist2aipr,2 the preceding where R is a constant, equalρto=287 m /(sp K) at o. rdinary tempperatures and(p3r.6e)ssures. In the equation is generalized by introducing a factor that variespwith tρhe=spRecTifi(1c +hup0m.6i0d8itqρy) q=:. . (3.6) (3.6) precedingρeq=uation, T is the ρab.=solRutTe (te1m+p0e.r6a0tu8rqe).(temRpTe(r3a1.t6u+)re0i.n60d8eqg)rees Ce(3ls.6iu) s + 273.15). The spAeciirficinhutmhpeidRaittyTmq(o1is+pdhe0efi.r6ne0e8dmqa)os st oRftTe(n1 c+o0n.t6a0in8qs)water vapor. For moist air, the preceding Thρe sp=ecific humTidhiteysqpiescdifie.ficnheudmasidity q is defined as (3.6) The specific humidityTqheieqsusdpaeetRfiiconiTfinecd(i1hsaus+gme0ind.ei6try0al8qiqzis)eddebfiyneidntarsoducing a factor that varies with the specific humidity q: mass of water vapor mass of water vapor q = mass of water vapomr a=ss of water mvaapssorof water vapor mass.of wa(t3e.r7)vapor The specific humidity q is defimnaesds aosf waterqva=por masms oafsswoafqteamri=rvaasspoorf w=atemravsaspoofrdmryaassirpo+f mw=atsesrovfawpoarter vapor . (3.7) . (3.7) q = q = = mass of air =ρ m=ass of dry air +. mass(o3..f7w) ater vapo.r (3.7) (3.6) ma1 ss of air mmaassssooffdariyr air + mamsmsasoasfsswoRofafTtaed(irrr1yva+apiro0r+.6m0a8sqsm)oaf swsaotefrdvraypoarir + mass of water vapor mass of water vaporThe latter expressionmisaasms iosnfomwear,tesirncveasaplionrity increases density not by contraction of the water but by the q = 1 a1dde1d m=ass of dissolved salt. . (3.7) The lmattearsesxporfesasiiorn isTaThmheTiehlsaentsotlepamrtetemeercrx, iapesfiixrsnepsccsreseohissosfaunilo1imdnTisrithiiyysadeaimianltamiciyrtsritenseqa+norsomeiemssexerdpa,dre,rsneesisinsfiinstcocynieeofensnsodawatliliisbannyiatsteycmoriinnnivtcscrranareecopatamsiosoeensresrdo,defsentinshnisetciytewynasontaetolrbtinbybuiytctyobcnoyitnrntacthcrreateicoatnsioeonsf odthfeenthwseiattywerantboeurtt bbuyttcbhoyentthreaction of the water but by the added mass of dissolvedadsdaaletdd.dmedamssaossf odfisdsioslsvoaeldvdeddseadslat.mlt.ass of dissolved salt. 1The latter expression is a misnomer, since salinity increases mdeansistyonfowt baytecronvtraapctoiorn of the water butmbyasthseof water vapor q = = . (3.7) added mass of dissolved salt. mass of air mass of dry air + mass of water vapor

1The latter expression is a misnomer, since salinity increases density not by contraction of the water but by the added mass of dissolved salt. 80 CHAPTER 3. EQUATIONS usually a main contributor to the flow field. The only place where total pressure comes into play is then the equation of state.

80 CHAPTER 3. EQUATIONS 80 CHAPTER 3. EQUATIONS 3.8 Fluxusfuoalrlyma muailnactoinotrinbutoarntodthecfloownfiseeldr. vThaetoinvlyeplfacoerwmhere total pressure comes into playuissutahlelynathme aeiqnucaotinotnriobfutsotartteo. the flow field. The only place where total pressure comes into play is then the equation of state. The preceding equations form a complete set of equations and there is no need to invoke further physical laws. Nevertheless we can manipulate the equations to write them in another form, which, though mathematically equivalent, has some practical advantages. Consider 3.8 Flux formulation and conservative form for example the eq3u.8ationFflourxtefmopremrautulraet(i3o.2n3a),nwdhiccohnwsaesrdveadtuicveedfforormmthe energy equa- tion using the BThoeuspsreinceedsiqngaepqpuraotixoinms afotiromna. UconmdpelertethseetsoafmeequBatoiounsssiannedstqhearpepisronxoimneaetdioton,inmvoakses conservation was reduced to volume conservation (3.17) and we can write the temperature furthTEquationsehrephpyresciceadli nlforagwe sTracersq. uNaetivoenrst h(feTemperature,olermss wa ecocmanpmle taSalinityenispeutlaotfe etandhqeuaet qiothersounastioann)sdtothwerreiteisthneomnienedantoothinevroke equation by firsfotremfxu,prwtahhneidrcpihnh, gythstiohcuaeglhlmawmastae.thrNieaemvlaedtritechraeillvleyassteiqwvueeivc(aa3lne.3nm)t,anhiapsuslaotme ethpereaqcutiactaiol nasdtvoanwtarigtest.heCmoinnsiadneorther for efxoarmm,plwehtihceh,eqthuoautigohn fmoarttheemmpaetricaatullrye (e3q.u2i3v)a,lewnht,ichhaws assomdeedpurcaecdtifcraolmadthveanetnaegregsy. eCqounas-ider tionfuosrinexgatmheplBeothuesseinqeusaqtiaopnpfroorxtiemmaptieorna.tuUrend(3e.r2t3h)e, wsahmicehBwoausssdiendeusqceadppfrroomximthaetieonne,rmgyasesqua- ∂T ∂T ∂T ∂T 2 constieornvautisoingwt+ahse rBueoduscseidne+tsoqvavoplupmroexicm+oantsieowrnv.aUtionnde(=r3.t1h7eκ)sTaanm∇dewBTeo,ucsasninwesriqteapthperotxemimpae(tr3iao.tn2u,5rme)ass equactoionnseb∂rvytafitirosnt ewxapsa∂nrdexdinugcethdetomav∂toeylruiamledecroinvas∂etirvzveat(i3o.n3)(3.17) and we can write the temperature equation by first expanding the material derivative (3.3) and then using volumeadvection-diffusionconservati∂oTn (3.1 equation,7∂) Tto obt awherei∂nT T can∂T be any tracer + u + v + w = κ ∇2T, (3.25) ∂t ∂T ∂x ∂T ∂y ∂T ∂z ∂T T + u + v + w = κ ∇2T, (3.25) ∂t ∂x ∂y ∂z T ∂Tand tFluxhen∂ uFormulationsing volume c ∂ofon sTracerervatio Equationn (3.∂17) to obtain +and the(nuuTsin)g+volume(cvoTns)erv+ation (3(.w17T) t)o obtain ∂t ∂x ∂y ∂z advective !ux ∂T ∂ ∂ ∂ ∂ ∂T + (u∂T ) + ∂T(vT ) + ∂(wT ) ∂T − κT ∂t ∂T −∂x ∂ κT∂y ∂ − ∂z ∂ κT = 0. (3.26) ∂x ! ∂x " + ∂y(!uT ) +∂y "(vT ) +∂z !(wT∂) z " ∂∂t ∂Tx ∂ ∂y ∂T ∂z ∂ diffusive∂T !ux − κ − κ − κ = 0. (3.26) ∂x !∂ T ∂x "∂T ∂y !∂ T ∂y "∂T ∂z !∂ T ∂z "∂T The latter form is called a co−nservaκtive form−ulationκ, the reas−on for κwhich wi=ll b0.ecome cle(a3r.26) ∂x ! T ∂x " ∂y ! T ∂y " ∂z ! T ∂z " upon applying Tthhee dlaitvteerrfgoermncies tchaelloedreamco. nTsehrivsatthiveeoforermmu,latlisoon,ktnheowreansoans fGorauwshsi’cshTwhilel obreecomm,estcaletears that for any vecutpoornT(haqpexpl,alytqtiyenr,gfqtohzre)mdthivseecrvagloelenlducemathceoeionnrseteemrgv.arTtaihvlieosfftohiretmosurdelamivti,eoanrlgs,oethnkencroeewaisnsoaensqfGuoaraulwsthso’iscthThhweeioilnlretbemegc,orsamtalteeoscflear the flux over thtehaetunfpocorlonansayipnpvgleycsitnuogrft(haqecx,edq:ivy,erqgze)nthceetvhoelourmeme .inTtehgisrathl eoofrietsmd,iavlesrogkencoewins aesquGaalutsos’tsheThinetoergermal,osftates the flthuaxt ofovreranthyeveencctolors(iqnxg, squyr,fqazc)et:he volume integral of its divergence is equal to the integral of the flux over the enclosing surface:

∂qx ∂qy ∂qz + ∂+qx ∂qy dx ∂dqyz dz = (q n + q n + q n ) dS (3.27) + + dx dy dz =x x(qxnxy +yqyny z+ qzznz) dS (3.27) #V ! ∂x #V∂y! ∂x ∂q∂x z∂"y ∂qy ∂z ∂"qz #S #S + + dx dy dz = (qxnx + qyny + qznz) dS (3.27) #V ! ∂x ∂y ∂z " #S where the vectworhe(rnexth,envye,ctnorz)(nixs, tnhye, nozu)twisathrde ouuntwitavrdecutnoitrvneoctromr naolrtmoatlhtoe tshuersfuarcfeacSe Sddeelilmimiittingg the volume V (thFeigvwuohlreuerme3et-hV1e)v.(FeIicgntoutreeg(n3rxa-1,t)in.nygI,nntehzge)raicstionthngesteohruevtcwaotanirvsdeeurvfnaoitrivmecf(too3rr.m2n6o(3r)m.2oa6vl)etorvtaehrefiasxufierxfdeadcvevoSoluludmmeleeimiisisting thenthpearvtiocluulmarelyVsi(mFpigleuraen3d-l1e)a.dIsnttoegarnaetixnpgrethsseiocnonfoserrtvhaetievveoflourtimon(3o.f2t6h)eohveeart acofinxtendtvinoltuhme e is then particularlvyolsutimhmeenpalpseaaratfinucdunlclateriolayndssoimftotphleaenflaunexdxelpseareednstestroiionangnafeonxdrprtleehsaesviioennvgoftolhurethvioeonleuvmoofleu:tthioenhoefathtechoenattecnotnitnentthinethe volume as a functivoonluomfetahseaflfuunxcetisonenotfetrhienflguaxnesdelnetaervininggantdhelevavoilnugmthee:volume: d T dt + q ·n dS = 0. (3.28) d dt #d # T dt +V Tqd·tn+dSS =q ·n0.dS = 0. (3.2(83).28) dt #V #S The flux q of dtetm#pVerature is com#pSosed of an advective flux (uT, vT, wT ) due to flow across tThehesuflrufaxcqe aonfdteamdpiefrfautsuivree i(scocnodmupcotisveed) oflfuxan−aκdve(c∂tTiv/e∂flxu,x∂T(u/T∂,yv, ∂TT, w/∂Tz)).duIef ttoheflow The flux q of temperature is composed of an advective flTux (uT, vT, wT ) due to flow valuaecroofsesatchhe flsurxfaicsekannodwna odniffauscilvoese(dcosnudrufacctiev,et)hfleuexvo−luκtTio(n∂Tof/∂thxe, ∂avTe/ra∂gye, ∂teTm/p∂ezr)a.turIef the across the surfacevaalnued oaf edaicfhfuflsuivxeis(cknoonwdnucotnivaec)loflsuedx s−urκfaTce(,∂tThe/e∂vxol,u∂tiTon/o∂fyt,h∂e Tav/e∂razg)e.temIfptehraeture value of each flux is known on a closed surface, the evolution of the average temperature 72 CHAPTER 3. EQUATIONS

3.2 Momentum budget

For a fluid, Isaac Newton’s second law “mass times acceleration equals the sum of forces” is better stated per unit volume with density replacing mass and, in the absence of rotation (Ω = 0), the resulting equations are called the Navier-Stokes equations. For geophysical flows, rotation is important and acceleration terms must be augmented as done in (2.20):

du ∂p ∂τ xx ∂τ xy ∂τ xz x : ρ + f∗w − fv = − + + + (3.2a) ! dt " ∂x ∂x ∂y ∂z dv ∂p ∂τ xy ∂τ yy ∂τ yz y : ρ + fu = − + + + (3.2b) ! dt " ∂y ∂x ∂y ∂z dw ∂p ∂τ xz ∂τ yz ∂τ zz z : ρ − f∗u = − − ρg + + + , (3.2c) ! dt " ∂z ∂x ∂y ∂z 72 CHAPTER 3. EQUATIONS where the x–, y– and z–axes are directed eastward, northward and upward, respectively, f = 2Ω sin ϕ3i.s2theMCoomrioelinstpuamrambuetdegr,eft∗ = 2Ω cos ϕ the reciprocal Coriolis parameter, ρ density, p pressure, g the gravitational acceleration, and the τ terms represent the normal and For a fluid, Isaac Newton’s second law “mass times acceleration equals the sum of forces” shear stressesisdbueettetrosftartiecdtipoenr .unit volume with density replacing mass and, in the absence of rotation That the p(rΩes=su0Momentumr)e, fthoercreesuilsti neBudgetgqueqaulaat in-o ndNavier-Stokessoarpepcoasllietdetthoe EquationstNhaeviperr-eSstoskuerseegquraatdioinesn. tFaonrdgetohpahtytshicealviscous force involveflsotwhse, rodteatriiovnaitsivimepsorotafntaansdtraecscseletreantisoonrtersmhsomuludst bbeeaufgammeinltieadrastodotnheeins(t2u.2d0e):nt who has had an introductory course in fluid mechanics. Appendix A retraces the formulation of those du ∂p ∂τ xx ∂τ xy ∂τ xz terms for the student nxew: tρo fluid+mfe∗cwha−nifcvs. = − + + + (3.2a) The effective gravitation!aldftorce (sum of"true grav∂ixtation∂axl force∂aynd the∂czentrifugal force; dv ∂p ∂τ xy ∂τ yy ∂τ yz see Section 2.2) is ρgyp:erρunit v+olufmu e a=nd−is dir+ected v+ertically+downward. So(3,.2tbh)e corre- ! dt " ∂y ∂x ∂y ∂z sponding term occurs only in the third equation, for the vertical direction. dw ∂p ∂τ xz ∂τ yz ∂τ zz z : ρ − f∗u = − − ρg + + + , (3.2c) Because the acceleration! idnt a fluid"is not c∂ozunted as th∂ex rate o∂fy chang∂ez in velocity at a fixed location but as the change in velocity of a fluid particle as it moves along with the flow, the time derivwahtievre sthienxt–h,eya–cacnedlezr–aatxieosnPressurearceodmirepcot eForcendeenatsstw, darud,/ndotr,thdwva/Frictionalrddtanadnudpwd Stressesward/,drte,spceoctnivseilsyt, of both f = 2Ω sin ϕ is the Coriolis parameter, f∗ = 2Ω cos ϕ the reciprocal Coriolis parameter, ρ the local timedreansteityo, pfDepcr!heasnitionsnugree, ga Materialtnhde gtrhaevitsa oDerivativeti-ocnaalllaecdcelaedravtieocnt,iavned theermτ tse:rms represent the normal and shear stresses due to friction. That the pressdure force∂is equal and∂opposite to t∂he pressure g∂radient and that the viscous = + u + v + w . (3.3) force involves thdetderivati∂vets of a str∂esxs tensor sh∂oyuld be fam∂ilizar to the student who has had an introductory course in fluid mechanics. Appendix A retraces the formulation of those This derivativteermisscfoarlltehedsttuhdeenmt naetwertioaflludidemrievcahtainvices.. The effective gravitational force (sum of true gravitational force and the centrifugal force; The precesdeeinSgecetiqonua2.t2i)oinssρgaspseur muneit avolCumareteansdiains dsiyrescttedmveortficaclolyodrodwinnawtaersd.aSnod, tthheucsorrheo- ld only if the dimensisoponndoinfgthteermdoocmcuarsinonulynidnethrecthoinrdsiedqueartaiotino, fnoristhemvuerctihcasl hdiorerctteiornt.han the earth’s radius. On Earth, a lengBtehcasucsealtehenaoccteelexracteioendiinnagfl1u0id0i0s nkomt coiusntuedsuaasltlhye raacteceofpctahbanlgee. inTvheelocnietygaltecat of the curvature termfixseidslioncastioomn beutwaas ythseachnaanlgoeginouveslotocittyhoefdaiflsutoidrptiaortniclienatsroitdmuocveeds abloyngmwaipthptihnegflothwe, curved the time derivatives in the acceleration components, du/dt, dv/dt and dw/dt, consist of both earth’s surfacteheolnotcoalatimpelarnatee.of change and the so-called advective terms: Should the dimensions of the domain under consideration be comparable to the size of d ∂ ∂ ∂ ∂ the planet, the x–, y– and z–axes nee=d to b+e rueplac+edvby s+phwerica.l coordinates, an(d3.c3)urvature dt ∂t ∂x ∂y ∂z terms enter all equations. See Appendix A for those equations. For simplicity in the expo- This derivative is called the material derivative. sition of the basiTcheprpirnecceidpilnegseoqufagtioenospahssyusmiceaal Cflaurtiedsidanysnyasmtemicosf, cwooerdsihnaatlelsnaengd ltehcust thhorldouonglyhout this book the extraifnteheoudismceunsrivoantoufrtehetedrommsainanundduersceonCsaidretreastiioanniscmooucrhdisnhoartteerstehxanclthuesievaretlhy’s. radius. EquationsO(n3E.2arath), tahlreongutghhsca(l3e.2noct)exccaenedbineg 1v0i0e0wkemdisasusuthalrlyeeaceceqputaabtlieo. nTshepnroegvleidctinofgthtehe three curvature terms is in some ways analogous to the distortion introduced by mapping the curved velocity compeoarnthe’ns stsu,rfuac,evonatnoda wpla.nTe.hey implicate, however, two additional quantities, namely, Should the dimensions of the domain under consideration be comparable to the size of the planet, the x–, y– and z–axes need to be replaced by spherical coordinates, and curvature terms enter all equations. See Appendix A for those equations. For simplicity in the expo- sition of the basic principles of geophysical fluid dynamics, we shall neglect throughout this book the extraneous curvature terms and use Cartesian coordinates exclusively. Equations (3.2a) through (3.2c) can be viewed as three equations providing the three velocity components, u, v and w. They implicate, however, two additional quantities, namely, 72 CHAPTER 3. EQUATIONS

3.2 Momentum budget

For a fluid, Isaac Newton’s second law “mass times acceleration equals the sum of forces” is better stated per unit volume with density replacing mass and, in the absence of rotation (Ω = 0Momentum), the resulti nBudgetg equat i-o nNavier-Stokess are called the EquationsNavier-Stokes equations. For geophysical flows, rotation is important and acceleration terms must be augmented as done in (2.20):

du ∂p ∂τ xx ∂τ xy ∂τ xz x : ρ + f∗w − fv = − + + + (3.2a) ! dt " ∂x ∂x ∂y ∂z dv ∂p ∂τ xy ∂τ yy ∂τ yz y : ρ + fu = − + + + (3.2b) ! dt " ∂y ∂x ∂y ∂z dw ∂p ∂τ xz ∂τ yz ∂τ zz z : ρ − f∗u = − − ρg + + + , (3.2c) 78 ! dt " ∂z ∂x CHA∂PyTER 3. E∂QzUATIONS where thteo vxe–lo,cyit–y garnadiezn–tsa)x, ewsiPressuretahrteheduirseecot efForcedthearsetdwucaerd,connotirnthuwityaFrictionalredquaantidonu, p(3w. 1Stressesa7r)d,,perermspitescutisvteoly, f = 2Ωwsirniteϕthise cthoempCoonreinotlsisofptahreasmtreetsesrt,efns∗or=as2Ω cos ϕ the reciprocal Coriolis parameter, ρ density, p Viscouspressure ,stressg the gproportionalravitational a ctoce lvelocityeration, agradientsnd the τ terms represent the normal and shear stresses due to frict∂ioun. ∂u ∂u ∂v ∂u ∂w τ xx = µ + , τ xy = µ + , τ xz = µ + That the pressure fo!rc∂exis eq∂uxal"and opposite!to∂ythe pr∂exss"ure gradient a!nd∂zthat th∂exv"iscous force involves the deriva∂tvives o∂fva stress tensor s∂hvould ∂bwe familiar to the student who has τ yy = µ + , τ yz = µ + had an introductory cou!r∂sey in flu∂iyd"mechanics. A!p∂pzendix∂Ay "retraces the formulation of those terms for the student new∂wto fluid∂wmechanics. τ zz = µ + , (3.18) The effective gravit!at∂ioznal fo∂rcze"(sum of true gravitational force and the centrifugal force; see Section 2.2) is ρg per unit volume and is directed vertically downward. So, the corre- spondingwtheerrme µociscucarslleodntlhyeicnotehffiectiheinrtdoefqduyantaimonic, vfoisrcothsietyv.eArtiscuablsdeqirueecnttiodniv.ision by ρ0 and the Becaiunstreodthuectiaocncoeflethraetkioinneminataicflviusicdosiistynνo=t cµo/uρn0teydielads the rate of change in velocity at a fixed location but as the change in velocity of a fluid particle as it moves along with the flow, the time derivatives in the acceleration components, du/dt, dv/dt and dw/dt, consist of both du 1 ∂p 2 the local time rate of change and+thef∗swo-c−allfevd a=dvec−tive term+s: ν ∇ u (3.19) dt ρ0 ∂x dv 1 ∂p d ∂ + f∂u = −∂ + ∂ν ∇2v. (3.20) = d+t u + v ρ0 +∂yw . (3.3) dt ∂t ∂x ∂y ∂z The next candidate for simplification is the z–momentum equation, (3.2c). There, ρ ap- This derivative is called the material derivative. pears as a factor not only in front of the left-hand side, but also in a product with g on the The preceding equations assume a Carte"sian system of coordinates and thus hold only right. On the left, it is safe to neglect ρ in front of ρ0 for the same reason as previously, if the dimbuetnosniotnheorfigthhteitdiosmnoati.nInudnedeedr, tchoentesirdmerρagtiaocncoiusnmtsufocrhthsehowreteigrhtthoafnthteheflueiadr,twh’hsicrha,daisus. On Earthw,eaklneonwg,tchaussceasleanniontcreexacseeodfipnrges1su0r0e0wkitmh diesputhsu(oarl,lya daecccreepastaebolfep. reTshsuerenewgitlhechteiogfht,he curvaturedetepremndsinisg ionnswohmeethwerawyes athnianlkoogfotuhse toocethane dorisattomrtoisopnheinretr).oWduitchedthbeyρ0mpaaprtpoinfgthtehdeecnusritvyed earth’s sugrofeascaehoyndtroosataptilcanpree.ssure p0, which is a function of z only: Should the dimensions of the domain under consideration be comparable to the size of " the planet, the x–, y– apnd=zp–0a(xze)s+neepd(xto, yb,e zr,ept)lacewdithby spp0h(ze)ric=alPc0oo−rdiρn0agtze,s, and cu(r3v.a2t1u)re terms enter all equations. See Appendix A for those equations. For simplicity in the expo- so that dp0/dz = −ρ0g, and the vertical-momentum equation at this stage reduces to sition of the basic principles of geophysical fluid dynamics, we shall neglect throughout this " " book the extraneous curvatudrewterms and use Ca1rte∂spian coρorgdinates e2xclusively. − f∗ u = − − + ν ∇ w, (3.22) Equations (3.2a) throughdt(3.2c) can be viρe0we∂dz as thρr0ee equations providing the three velocity components, u, v and w. They implicate, however, two additional quantities, namely, after a division by ρ0 for convenience. No further simplification is possible because the " remaining ρ term no longer falls in the shadow of a neighboring term proportional to ρ0. Actually, as we will see later, the term ρ"g is the one responsible for the buoyancy forces that are such a crucial ingredient of geophysical fluid dynamics. Note that the hydrostatic pressure p0(z) can be subtracted from p in the reduced momen- tum equations, (3.19) and (3.20), because it has no derivatives with respect to x and y, and is dynamically inactive. For water, the treatment of the energy equation, (3.8), is straightforward. First, continuity of volume, (3.17), eliminates the middle term, leaving

dT ρC = k ∇2T. v dt T 78 CHAPTER 3. EQUATIONS

7to8velocity gradients), with the use of the reduced continuity eqCuHatAioPnT,E(3R.137.),EpQerUmAiTtsIOusNtSo write the components of the stress tensor as to velocity gradients), with the use of the reduced continuity equation, (3.17), permits us to 78 CHAPTER 3. EQUATIONS write the compone∂nuts of th∂eustress tensor as ∂u ∂v ∂u ∂w τ xx = µ + , τ xy = µ + , τ xz = µ + to velocity gradie!n∂tsx), with∂txh"e use of the red!uc∂eyd cont∂inxu"ity equation, (3!.17∂)z, perm∂itxs "us to write ythye componen∂tuvs of th∂∂evustress teynzsor as ∂∂vu ∂∂wv ∂u ∂w ττxx = µ ++ ,, ττ xy == µµ ++ , τ xz = µ + !∂xy ∂∂yx"" !!∂∂zy ∂∂yx" ! ∂z ∂x " ∂wv ∂∂vw ∂v ∂w τ yzyz = µ ∂u ++ ∂u , , τ yz = µ ∂u + ∂v ∂u ∂w (3.18) τ xx = µ !∂∂yz + ∂∂yz"", τ xy = µ !∂z + ∂y " , τ xz = µ + !∂x ∂x" ! ∂y ∂x" ! ∂z ∂x " ∂w ∂w wheτrzezµ i=s calµled∂thve co+e∂ffivcient ,of dynamic v∂isvcosity∂.wA subsequent division by ρ an(3d.1th8e) τ yy = µ ! ∂z + ∂z ", τ yz = µ + 0 introduction of !the∂ykinem∂atyic"viscosity ν = µ!/∂ρz0 yield∂y " 78 where µ is called∂thwe coef∂fiwcient of dynamic viscosity.CAHsAubPsTeEquRen3t.dEivQisUioAnTbIyOρNSand the 78 τ zz = µ + , CHAPTER 3. EQU0AT(3IO.1N8)S introduction of !the∂kzinema∂tizc "viscosity ν = µ/ρ0 yield du 1 ∂p 2 + f∗w − fv = − + ν ∇ u (3.19) to velocity gradients), with the usdet of the reduced continuityρ0eq∂uxation, (3.17), permits us to to vwehloecreityµgisracdailelendtst)h,ewciotheftfihceieunsteooffdtyhneamreidcuvciesdcocsoitnyt.inAuistuybesqequuaetinotnd,i(v3is.1io7n),bpyeρrm0 iatnsdutshteo write the components of the stress tensor davs 1 ∂p introduction of the kinemduatic viscosity ν = µ/ρ yiel1d ∂p 22 write the components of the st+resfs∗twens+−orffauvs = 0 − + νν ∇∇ vu. (3.1290) dt dt ρ0 ∂xy dv 1 ∂p xx ∂u ∂u xy ∂u ∂v xz 2∂u ∂w The next candidatedufor simplific+atiofnuis t=he z−–m1om∂epntu+m νeq∇u2atvio.n, (3.2c). There,(ρ3.2ap0-) τ = xxµ + ∂u , ∂τu+ =f∗dwtµxy− fv +=∂u− ρ,0∂τv∂y +=νxµ∇z u +∂u ∂w(3.19) τpears!a=∂s xa fµacto∂rxno"+t odntly in f,roτnt o!f=∂thyeµleft-∂haxn"+dρsid∂e,xbu,t aτlso !in=∂azpµrodu∂ctxw+"ith g on the !∂x ∂x" " ! ∂y 0∂x" ! ∂z ∂x " right. On the left, it is safe to neglect ρ in front of ρ0 for the same reason as previously, yy The∂nvext ca∂nvdidate foyrzsimpdlvifica∂tiovn is th∂ewz–m1om∂epntum equ2ation, (3.2c). There, ρ ap- τ =byuytµon the r+ight∂ivt is n,o∂t.τvInde=ed,µytzh+e tferum+ρ=g∂avc−coun∂tswfor+theνw∇eigvh.t of the fluid, wh(i3ch.2,0a)s τpears!a=∂s ya fµacto∂ryn"o+t only in ,frodτntt !of=∂thzeµleft-∂hyan+"dρs0id∂e,ybut also in a product with g on the we know, cau!se∂syan inc∂reya"se of pressure "w!it∂hzdepth ∂(oyr,"a decrease of pressure with height, right. O∂nwthe le∂ftw, it is safe to neglect ρ in front of ρ0 for the same reason as previously, τ zz =bdueptµTeohnnedtinhnegxroti+gnchawtn∂hdiwteiditshaetnero,wft∂o.erwIntshidmienepkdlio,fiftchtaehteitoeonrcmiesaρntghoearczac–tomuonmstspehfnoetruetm)h.eeWwqiuetahigtithohtneo,ρf(03th.p2eacr)flt.uoT(ifd3ht,.eh1wre8eh)d,ieρcnhas,piat-ys 78 τ zz !=∂z µ ∂z "+ , CHAPTER 3. EQU(3A.1T8I)ONS pgweoaeerksnaaoshwya,dfcraaocusttsoaertsincaopntroiensncsluryeraeinspef0ro,ofwnpthrioecfshstuhisreealwefufittn-hhctadinoednptsohifd(zeo,ro,bnaulytd:aelcsroeaisneaopf rpordeusscutrweiwthitgh ohneigthhet, ! ∂z ∂z " " rdigehpte.ndOingthoen lwefht,etihteirswsaefethtionknoefgltehcetoρceiannforronatmoofsρp0hefroer).thWe istahmtheerρe0aspoanrtaosfptrheevdioeunssliyty, where µ is called the coefpfic=ienpt o(fz)dy+napm"(ixc,vyis,czo,sity). Awistuhbsepqu(ze)nt=divPisio−n bρygρz0, and the (3.21) towhvebegruloeotecoµsintaiysthhgcyeardarlrildogeishdettantithtiscei)spc,rnoweoes0itfst.fiuhcIrneitehdpne0te,udow,sfethdhioyechnfteaitrsmhmeaicfρruevgndicasucticocoeosndiutonycf.toszAnfo0otinsrnultyuhb:iesteywqeueqieg0nuhtattdoiiofvnit0hs,ie(o3fln.u1bi7dy),,ρw0pheaircnmhd,ittahsseus to introductiowneofknthoewk, icnaeumseastiacnviinscroesaistye νof=preµs/sρur0eywieilthd depth (or, a decrease of pressure with height, winritrteodthuceticoMomentumonmopfothneenktisn eoBudgetmf athtiec sv ti-rs ecNavier-Stokessossitteynνso=r aµs/ρ 0Equationsyield so that dp0/dz = −ρ0g, and the vertical-momentum equation at this stage reduces to depending on whether we think o"f the ocean or atmosphere). With the ρ0 part of the density p = p0(z) + p (x, y, z, t) with p0(z) = P0 − ρ0gz, (3.21) goes a hydrostatic pressure p0, which is a function" of z on" ly: du dw 11 ∂∂p ρ g 2 2 sxoxthat dp0/dz ∂=+u−fρd∗0ugw,∂au−n−d fthfve∗ vue=xr=tyica−l-mome∂nutu1m+−e∂νqpu∂∇avtio+un aνt t∇h2xiszws,tage redu∂c(u3es.1to9) (3∂.w22) τ = µdt +dt+ f∗w," τ− fv= =ρµρ00 ∂−∂xz + ρ0 + ν, ∇τ u = µ + (3.19) !p∂x= pd0t(z∂)x+" p (x, y, z, t) !w∂iyt"ρh0 ∂p0x∂(" zx)"= P0 − ρ0gz,! ∂z (3∂.2x1)" ddvw 11 ∂∂pp ρ g 2 2 after a division by ρ0 fo+r −cofdnfvuv∗euni=e=nce−. No furth1+e−r ∂sνipm∇pli+vfi.caνti∇o2nwis, possible(3b.e2c0a)us(3e.2th2e) soyythat dp0/d"z =∂v−ρ0g,∂anvd the v+erytzifcaul-m=ome∂n−tvum equ∂awti+onνat∇thisv.stage reduces to (3.20) τremain=ing µρ term n+odldtotngerdf,tallτs in th=e sµρhρ0a0d∂o∂wyzρof+a∂nyρe0ighboring term proportional to ρ0. !∂y ∂y " " !∂z 0 ∂y " Aafcteturaallyd,ivasiswioenwbiyll ρseefloartecro, tnhveenteiremncρe.g Nisothfeuro"tnheerressi"pmopnlsifiibclaetifoonr tihsepbousosyibalnecybefocarcuesseththaet The next candidate for simp0dliwfication is the z–m1om∂epntum eρqguation, (32.2c). There, ρ ap- Tarheesuncehxtaccar"nudci∂adlwaitnegfroerd∂iseiw−mntpoflfi∗figuceaot=piohny−siiscatlhfleuzid–mdyo−nmamenitcusm.+ eνqu∇atiwon, , (3.2c). There(,3ρ.2a2p)- τrzezmain=ing µρ term no+longer fa,lls in the shadow of a neighboring term proportional to ρ0(.3.18) pears as a factoNronteotthoantltyheinhyfdrorodnstttaotficthperelsesfutr-ehapn"(dzρs)i0cdae∂n,zbbeutsuablstρro0acinteda fproromdupcitnwthiethregduocnedthmeomen- pearAscatsuaallyfa,catsowr!eno∂wtzilolnsleye iln∂atzefrr"o, tnht"eotfertmheρ0legfti-shtahnedonsiedree,spbount sailbsloe fionrathperboudouycatnwcyithfogrceosnththaet right. On thtuemleefqt,uaitioisnss,a(f3e.1t9o)naengdl(e3c.t2ρ0), ibnecfarou"nset iotfhaρs0nfoodr etrhiveastiavmesewrietahsroenspaesctptroevxioanudslyy, and is righaatf.rterOsuancdhthivaeicslrieoufnct,iabiltyiniρsg0rseafdfoeirentcotoonnfveeggnelioeecpnthcyρes.iciNnalofflruofuindrtthdoeyfrnρas0mimifcposlri.fithcaetisoanmies rpeoasssoibnleasbepcraeuvsioeutshley, but on the rdigyhntamit iicsanlloy" ti.nIancdtieve.d, the term ρg accounts for the weight of the fluid, which, as wbhuetrroeemnµatNhinieositnreciggathhlρlatetidtttehritemshhenynocdotor.oleoIsfnntfiadgcteeiicerednpf,arteltlohsssfeuidntreeytrnhpme0a(mρszhg)iaccadavconciwsobcueoonfstuisatbyfnto.reariAcgtthhesebduowfbrriseoneimggqhutpeterionmnfttthpdherieovrpieflsoduiruoitdcinoe,ndbwaymlhitρoocm0hρ,ea0nan.-sd the we know, causeFsoranwaintecrr,ethaseetroefatpmrensstuorfethweiethnedrge"yptehqu(oarti,oan,d(e3c.8re),aissesotrfaipgrhetsfsourwreawrdi.tFhirhset,igchotn,tinuity inwteroAktducnumtoucwateil,qolyucn,aautoissofewntshse,aew(n3ki.li1ln9sce)ermeaanalsadteitec(3or,v.f2tihsp0cer)e,otsebssriemutcyraeρuνswge=iiitsthµhthd/aeesρpo0ntnohyedi(reeoelrrsdi,pvaaotndivseiebcsrlewafisother trohefesppberuceotsystouanrxecyawnfiodthryc,ehsaenitghdahitts, depending oonf vwohluemthee,r(w3.e17t)h,ienlkimoifntahteesotcheamnioddr laetmteromsp, hleearvei)n.gWith the ρ part of the density depaedrnyednsiaunmcghicoaanlclwryuhicneiathcl teiinvrgewr.eedtiehnintkofogfetohpehoycseicaanl floruiadtmdyonsapmheicres.). With0 the ρ0 part of the density goes a hgyodersosatNahFtyoiocdtrerpwotrhsaetatsaetstrtui,hcrtehepehrpeyt0rsde,srawuotrsmhetaiepctnih0ct,oipwsfreahthsifsceuuhenrneicsetpria0og(ynfzue)onqfcuatzainotoinbonenoly,sfu:(z3b.to8rn)a,clyitse:dstfrraoigmhtpfoinrwtharedr.eFdiurcste,dcomnotimnuenit-y Only dynamic pressure is importantdT to motion2 toufmveoqluumateio, n(3s,.1(73).1, 9el)idmaunidna(t3e.s20th)e, bmeciρdaCdulseetietrhma=,slneaokvdin∇egr1ivTa.∂tivpes with res2pect to x and y, and is ∗ v T dynamically inactive. " + f" w − fvdt = − + ν ∇ u (3.19) p = p0p(z=) +p0(pdzt()x+, yp, z(x, ,t)y, zw,itth) pw0i(thz) ρ=p00(∂Pz)x0 =− Pρ0g−z, ρ0gz, (3.21) (3.21) For water, the treatment of the energydeTquation, (3.28), is straightforward. First, continuity dv ρCv = kT ∇1 T∂. p 2 so that dspo0to/hfdavztod=lpu0m−/edρ,z0(g3=.,1a7−n)d,ρe0tlghime, aivneadrttetihsceathlv-ememroti+mcdadellfn-emtutuedomrtmm=ee,qnleutuaavm−tiionegnqautatihoins +sattatgνheis∇restdavug.ceersetdouces to (3.20) dt ρ0 ∂y " "" " dw 1 ∂pdT ρ g 2 dw ρC 1 =∂p k ∇ ρTg. 2 2 The next candid−atef∗four−s=imf∗p−ulifi=cati−vondits−the zT–−m+oνm∇en+twumν, ∇eqwua,tion, (3.2(c3)..22T)h(e3r.2e,2)ρ ap- pears as a factodrtnot odntly in frontρo0f t∂hze ρle0ft∂-hρza0nd siρd0e, but also in a product with g on the " after ardaigifvtheitsr.ioaOndnibvytihsρieo0nleffbotyr, icρto0insvfoesrnaficeonntcovee.nneiNegnolececf.ut rρtNheoirnfsufirrmtohpneltrifioscifmaρtpi0olinffiocirsattiphooensssiiasbmlpeeobsrseeicbaalseuosnbeeatchsaeupsreevthioeusly, " " remainbinruegtmoρanintteihnremg rρingohtetlroimnt ginseornfolaotl.nlsgIenirndeftaehldels,stihhnaetdhtoewrsmhoafρdgaownaceoicgfohuabnontresiinfgoghrbteothrrmeinwgperteoigrpmhotrptoirofonptaholertofliounρia0dl.,two hρi0c.h, as ActuallwyA,ecatksunwaolelwyw,,aiclsalwuseseewslaialtnlers,ientehcleraetaesr,emthoρef"gtperiresmstshρue"rgoeinswetihrtehesopdnoeenpsrtihebsl(peoofrn,osraibtdhleecfborueroatyhseaenbocufyopfyroaernsccseyusrfteohrawcteitshthhaetight, are sucdhaearpecesrnuudccihinaagl iconrngurcweidahlieeitnhgterorefdwgieenottphohifnykgsiecooafplthflhyuesiidoccadelyaflnnuaimodridcaystn.maomsipchs.ere). With the ρ0 part of the density NotgeotehsaNtaothteyedthhryaodtsrtthoaestitchatypidcrreposrssetusasrtieucrpep0rp,e0sw(szuh)riecchpan0is(bzae)fscuuanbnctbrtaieocsnteudobftfrrzaocmotendplyfir:nomthepriendtuhceerdemduocmedenm-omen- tum equtautmionesq,u(a3t.i1o9n)s,a(n3d.1(93).2a0n)d, b(3e.c2a0u)s,ebietchaausseniot dhearsivnaotidveersivwatiitvhersewspiethctretospxecatntdoyx, aannddyis, and is dynamically inactive. " dynamically inactive. p = p0(z) + p (x, y, z, t) with p0(z) = P0 − ρ0gz, (3.21) For wateFro,rthweattreera, ttmheentrteoatfmtheenteonfertghye enqeuragtyioenq,u(3at.i8o)n, ,is(3s.t8ra),igishtsftorariwgahrtdfo.rFwirasrtd,.cFoinrtsitn, uciotnytinuity of volusmoofet,vh(oa3lt.u1dm7p)e0,,/e(dl3iz.m17=in),a−etelρism0thgine,aamtneisddtdthhleeemtveierdmrdti,lcelaetlae-vrmmino,gmleeavnitnugm equation at this stage reduces to

dT " " dw dT 12 ∂p 2 ρ g 2 −ρCfv∗ uρC==v −kT ∇= Tk.T ∇−T. + ν ∇ w, (3.22) dt dt dt ρ0 ∂z ρ0 after a division by ρ0 for convenience. No further simplification is possible because the " remaining ρ term no longer falls in the shadow of a neighboring term proportional to ρ0. Actually, as we will see later, the term ρ"g is the one responsible for the buoyancy forces that are such a crucial ingredient of geophysical fluid dynamics. Note that the hydrostatic pressure p0(z) can be subtracted from p in the reduced momen- tum equations, (3.19) and (3.20), because it has no derivatives with respect to x and y, and is dynamically inactive. For water, the treatment of the energy equation, (3.8), is straightforward. First, continuity of volume, (3.17), eliminates the middle term, leaving

dT ρC = k ∇2T. v dt T 78 CHAPTER 3. EQUATIONS

7to8velocity gradients), with the use of the reduced continuity eqCuHatAioPnT,E(3R.137.),EpQerUmAiTtsIOusNtSo write the components of the stress tensor as to velocity gradients), with the use of the reduced continuity equation, (3.17), permits us to 78 CHAPTER 3. EQUATIONS write the compone∂nuts of th∂eustress tensor as ∂u ∂v ∂u ∂w τ xx = µ + , τ xy = µ + , τ xz = µ + to velocity gradie!n∂tsx), with∂txh"e use of the red!uc∂eyd cont∂inxu"ity equation, (3!.17∂)z, perm∂itxs "us to write ythye componen∂tuvs of th∂∂evustress teynzsor as ∂∂vu ∂∂wv ∂u ∂w ττxx = µ ++ ,, ττ xy == µµ ++ , τ xz = µ + !∂xy ∂∂yx"" !!∂∂zy ∂∂yx" ! ∂z ∂x " ∂wv ∂∂vw ∂v ∂w τ yzyz = µ ∂u ++ ∂u , , τ yz = µ ∂u + ∂v ∂u ∂w (3.18) τ xx = µ !∂∂yz + ∂∂yz"", τ xy = µ !∂z + ∂y " , τ xz = µ + !∂x ∂x" ! ∂y ∂x" ! ∂z ∂x " ∂w ∂w wheτrzezµ i=s calµled∂thve co+e∂ffivcient ,of dynamic v∂isvcosity∂.wA subsequent division by ρ an(3d.1th8e) τ yy = µ ! ∂z + ∂z ", τ yz = µ + 0 introduction of !the∂ykinem∂atyic"viscosity ν = µ!/∂ρz0 yield∂y " 78 ∂w ∂w CHAPTER 3. EQUATIONS whτerzez µ =is caµlled the c+oefficient,of dynamic viscosity. A subsequent division by ρ0 a(n3d.1t8h)e introduction of !the∂kzinema∂tizc "viscosity ν = µ/ρ0 yield du 1 ∂p 2 + f∗w − fv = − + ν ∇ u (3.19) to velocity gradients), with the usdet of the reduced continuityρ0eq∂uxation, (3.17), permits us to where µ is called the coefficient of dynamic viscosity. A subsequent division by ρ0 and the write the components of the stress tensor davs 1 ∂p introduction of the kinemduatic viscosity ν = µ/ρ yiel1d ∂p 22 + f∗w +− ffuv = 0 − + νν ∇∇ vu. (3.1290) dt dt ρ0 ∂xy dv 1 ∂p xx The∂nuext ca∂nduidate forxysimplific+at∂iofunuis t=h∂evz−–momexnztu+m νeq∇ua2tv∂io.un, (3.2∂c)w. There,(ρ3.2ap0-) τ = µ + ,duτ = µ + 1, ∂τp = µ 2 + pears!a∂s xa facto∂rxno"t only +in ffr∗odwntt o−!f ∂tfhyve le=ft-∂hax−n"d ρsi0de∂, ybu+t aνlso∇!inu∂azprodu∂ctxw"ith g (o3n.1t9h)e dt " ρ0 ∂x right. On the left, it is safe to neglect ρ in front of ρ0 for the same reason as previously, yy The∂nvext ca∂nvdidate foyrzsimpdlvifica∂tiovn is th∂ewz–m1om∂epntum equation, (3.2c). There, ρ ap- τ =butµon the r+ight it is n,ot.τ Inde=ed,µth+e tferum+ρ=g ac−counts for+theνw∇ei2gvh.t of the fluid, wh(i3ch.2,0a)s pears!a∂s ya facto∂ryn"ot only in frodntt !of∂thze left-∂hyan"dρs0id∂e,ybut also in a product with g on the we know, causes an increase of pressure "with depth (or, a decrease of pressure with height, right. O∂nwthe le∂ftw, it is safe to neglect ρ in front of ρ0 for the same reason as previously, τ zz =bdueptµTeohnnedtinhnegxroti+gnchawtnhditeiditshaetnero,wfto.erIntshidmienepkdlio,fiftchtaehteitoeonrcmiesaρntghoearczac–tomuonmstspehfnoetruetm)h.eeWwqiuetahigtithohtneo,ρf(03th.p2eacr)flt.uoT(ifd3ht,.eh1wre8eh)d,ieρcnhas,piat-ys ! ∂z ∂z " pgweoaeerksnaaoshwya,dfcraaocusttsoaertsincaopntroiensncsluryeraeinspef0ro,ofwnpthrioecfshstuhisreealwefufittn-hhctadinoednptsohifd(zeo,ro,bnaulytd:aelcsroeaisneaopf rpordeusscutrweiwthitgh ohneigthhet, " rdigehpte.ndOingthoen lwefht,etihteirswsaefethtionknoefgltehcetoρceiannforronatmoofsρp0hefroer).thWe istahmtheerρe0aspoanrtaosfptrheevdioeunssliyty, where µ is called the coefpfic=ienpt o(fz)dy+napm"(ixc,vyis,czo,sity). Awistuhbsepqu(ze)nt=divPisio−n bρygρz0, and the (3.21) bguoteosn9a2thhyedrriogshttaitticisprneos0ts.uIrnedpe0e,dw, thhiechteirsma fρugncatcicoonuonftszfo0onrltyh:e weig0ChHt AofPtT0hEeRflu4i.d,EwQhUiAchT,IaOsNS introductiowneofknthoMomentumewk, icnaeumseastiacn vBudgetiinscroesaistye -νo Navier-Stokesf=preµs/sρur0eywieilthd dEquationsepth (or, a decrease of pressure with height, so that dp0/dz = −ρ0g, and the vertical-momentum equation at this stage reduces to depending on whether we think o"f the ocean or atmosphere). With the ρ0 part of the density p = p0(z) + p (x, y, z, t) with p0(z) = P0 − ρ0gz, (3.21) goes a hydrostatic pressure p , which is a function of z only: du dw 0 11 ∂∂p" ρ"g so that dp /dz = −ρ g, and the vertical-momentu! m=e0quat2ion at!th=i2s0stage r!ed! uces to 0 + f!∗0uwv"−−=ffv∗!u!u="=!v"−" + !!u" v "+− ν+∇!!+uv" uν "∇ w,+ !u v " (3.19) (3.22) dt dt " ρρ00 ∂∂xz# ρ0 # p = p (z) + p (x, y, z, t)! !with p (z) = P − ρ gz, (3.21) 0 = !u" !v" + !u v " " 0" 0 0 (4.2) ddvw 11 ∂∂pp ρ g 2 2 after a division by ρ0 fo+r −cofnfuv∗euni=e=nce−. No furth+e−r sνim∇pli+vfi.caνti∇onwis, possible(3b.e2c0a)us(3e.2th2e) so that dp0/d"z = −ρ0g, and the vertical-momentum equation at this stage reduces to remaining ρ term nodldtotnger falls in the sρhρ0a0d∂o∂wyz of a nρe0ighboring term proportional to ρ0. and similarly for !uu", !wu", !uρ" et"c. We recognize here that the average of a product is not Aafcteturaeaqlluyda,ilvatsioswitohenewbpiryloldρsuecetflooarftetcrho,etnhavveeentreiaregmnecsρe..TghNiissotihsfeuaro"dtnhoeeurrbelssei"pm-eopdnglsiefiibdclasewtifoonrdt:ihsOepbnouosonsyiebahlneacnybdef,ociartcugesesentehtrhaaettes The next candidate for simp0dliwfication is the z–m1om∂epntum eρqguation, (32.2c). There, ρ ap- are such a cr"ucial ingredie−nt off∗gueo=phy−sical fluid dy−namics.+ ν ∇ w, (3.22) remaminainthgemρ atiecraml cnoomlpolnicgaetirofnasllbsuitn, otnhethsehoatdhoewr hoafnda, niteaiglshobcorreiantgesteinrmterepsrtoinpgorstiitounaatilotnos.ρ0. pears as a factoNronteotthoantltyheinhyfdrorodnstttaotficthperelsesfutr-ehapn"(dzρs)i0cdae∂n,zbbeutsuablstρro0acinteda fproromdupcitnwthiethregduocnedthmeomen- Actually,Oausrwoebjwecitlilvseeeisltaoteers,ttahb"elitsehrmeqρu0agtiiosntshgeoovneernriensgpothnesimbleeafnoqruthaentbituieosy,a!nuc"y, !fovr"c, e!swt"h,a!tp" right. On thtuemleefqt,uaitioisnss,a(f3e.1t9o)naengdl(e3c.t2ρ0), ibnecfarounset iotfhaρs0nfoodr etrhiveastiavmesewrietahsroenspaesctptroevxioanudslyy, and is aafrtersuacndhdiva!ρics"ri.ouSnctiaabrlytinρg0rwefdiotihrencthtoeonfvavegenerioaepgnhecyeos.fictNhaleoflxuf−uidrmthdoeymrneasnmimtuicpmsli.efiqcuaatitoionni(s3.p1o9s)s,iwbleehbaevcea: use the but on the rdigyhntamiStatisticallyt iicsanlloy" ti.nIancdtiev eaverage.d, the te r"mowρ g- aReynoldsccounts f averagingor the weight of the fluid, which, as remaNinoitnegthρatttehremhyndorolosntagteicr pfarellsssuinrethpe(szh)acdaonwbeofsuabntreaicgthebdofrrionmg tperimn thperorpeodrutcioendaml toomρe0n.- 0" we know, cAatucumtsueFaesolqlrayunw,aataiisnotewcnrrs,ee,thaw(3seie.lt1lro9es)feaetapmnlradetnse(st3ru,o.2trfhe0te)hw,etebiertehmnceadrρugesgypeteiihstqhut(hoaersti,onanoe,dd(ere3ecr.s8ripve)ao,atnisisevsisebotsrlfaewipfgiorthhertsftrsoheureswrpbeeaucwrotdyit.toaFhnxicrhsyaetn,ifgodchroyctn,etsainntuhdiatitys depending oonf vwohlue∂mth!eeu,r"(w3.e1∂7t)h!,iuenulki"moifnta∂htee!svoutch"eamni∂od!drwlaeutmt"eromsp, hleearvei)n.gWith the ρ pa1rt o∂f!pth"e densi2ty adryensaumchicaalclryu+icniacl tiinvger.ed+ient of ge+ophysical+flufi∗d!dwy"na−mfic!sv." = 0− + ν ∇ !u" (4.3) ∂t ∂x ∂y ∂z ρ0 ∂x goes a hydrostNaFtoioctrepwtrhaeatsetsrtu,hrteheehpyt0rde,rawotsmhtaiectnihct oipsfreathsfseuuenrnecetpri0og(ynze)oqfcuazantoibonenly,su:(3b.t8r)a,citsedstfrraoigmhtpfoinrwtharedr.eFdiurcste,dcomnotimnuenit-y tum equations, (3.19) and (3.20), becausediTt has no der2ivatives with respect to x and y, and is of vowluhmiceh, (b3e.c1o7m),eesliminates the miρdCdlve term=, leakvTin∇g T. dynamically inactive. " dt p = p0(z) denotes+ p (x ,time/spacey, z, t) averagewith p0(z) = P0 − ρ0gz, (3.21) For water, the treatment of the energydeTquation, (3.28), is straightforward. First, continuity ρCv = kT ∇ T. so that dp0o/fdvzo=lum−eρ,0(g3.,1∂a7n!)du, e"tlhime i∂vne(a!rtutei"sc!atuhl-e"m)moimd∂del(ne!tutued"mrt!mve",q)leuaavti∂ion(gn!ua"t!twhi"s) stage reduces to + + + + f∗ !w" − f !v" = ∂t ∂x ∂y ∂z " " dw 1 ∂pdT ρ g !2 ! ! ! ! ! 1 ∂ !p" ρC 2 = k∂ !∇u uT". 2∂ !u v " ∂ !u w " − f∗ −u = − + νv∇dt!−u" − T + ν ∇− w, − (.3.22) (4.4) dt ρ0 ∂x ρ0 ∂z ρ0 ∂x ∂y ∂z after a division by ρ0 for convenience. No further simplification is possible because the " We note that this last equation for the mean field looks identical to the original equation, remaining ρ teremxcnepot lfoonrgtheer fparellssenincethoef tshhreaedonwewotferamnseaitghthbeoerinndgotferthme rpirgohpt-ohratniodnsaildeto. Tρh0.ese terms " Actually, as we wreipllresseeentlathteer,eftfheecttseromf thρegtuisrbtuhleenotnfleurcetusaptoionnssibolne fthoer mtheeabnuflooywa.ncCyomfobricneisngthtahtese terms are such a cruciawl iitnhgcroerdrieesnpot nodfigngeofprihctyiosincaallteflrumids dynamics. Note that the hydrostatic pressure p0(z) can be subtracted from p in the reduced momen- tum equations, (3.19) and (3.20), because it has no derivatives with respect to x and y, and is ∂ ∂ !u" ! ! ∂ ∂ !u" ! ! ∂ ∂ !u" ! ! dynamically inactive. ν − !u u " , ν − !u v " , ν − !u w " ∂x ∂x ∂y ∂y ∂z ∂z For water, the treatm!ent of the energy e"quation,!(3.8), is straightf"orward. F!irst, continuity " of volume, (3.17i)n,deicliamteisntahtaetstthheeamveirdadglees toefrmve,lolecaitvyinflguctuations add to the viscous stresses (for example, − !u!w!" adds to ν∂ !u" /∂z) and can therefore be considered as frictional stresses caused by turbulence. To give credit todTOsborne Rey2nolds who first decomposed the flow into mean and ρC = k ∇ T. ! ! ! ! ! ! fluctuating components, vthedetxpressioTns − !u u ", − !u v " and − !u w " are called Reynolds stresses. Since they do not have the same form as the viscous stresses (ν∂ !u" /∂x etc.), it can be said that the mean turbulent flow behaves as a fluid governed by a frictional law other than that of viscosity. In other words, turbulent flow behaves as a non-Newtonian fluid. Similar averages of the y- and z-momentum equations (3.20)–(3.22) over the turbulent fluctuations yield

∂ !v" ∂(!u" !v") ∂(!v" !v") ∂(!v" !w") 1 ∂ !p" + + + + f !u" − = ∂t ∂x ∂y ∂z ρ0 ∂y ∂ ∂ !v" ! ! ∂ ∂ !v" ! ! ∂ ∂ !v" ! ! ν − !u v " + ν − !v v " + ν − !v w " (4.5) ∂x ∂x ∂y ∂y ∂z ∂z ! " ! " ! " Chapter 4

Equations Governing Geophysical

78 CHAPTER 3. EQUATIONS

Flows 7to8velocity gradients), with the use of the reduced continuity eqCuHatAioPnT,E(3R.137.),EpQerUmAiTtsIOusNtSo write the components of the stress tensor as to velocity gradients), with the use of the reduced continuity equation, (3.17), permits us to 78 CHAPTER 3. EQUATIONS write the compone∂nuts of th∂eustress tensor as ∂u ∂v ∂u ∂w τ xx = µ + , τ xy = µ + , τ xz = µ + to velocity gradie!n∂tsx), with∂txh"e use of the red!uc∂eyd cont∂inxu"ity equation, (3!.17∂)z, perm∂itxs "us to Cha∂pv te∂vr 4 ∂v ∂w write ythye componen∂tus of th∂eustress teynzsor as ∂u ∂v ∂u ∂w ττxx = µ ++ ,, ττ xy == µµ ++ , τ xz = µ + !∂xy ∂∂yx"" !!∂∂zy ∂∂yx" ! ∂z ∂x " ∂wv ∂∂vw ∂v ∂w τ yzyz = µ ∂u ++ ∂u , , τ yz = µ ∂u + ∂v ∂u ∂w (3.18) τ xx = µ !∂∂yz + ∂∂yz"", τ xy = µ !∂z + ∂y " , τ xz = µ + !∂x ∂x" ! ∂y ∂x" ! ∂z ∂x " ∂w ∂w (July 26, 2007) SUMMwhAeτrzeRzµ iY=s ca:lµleTd∂thhve icso+e∂fficvchienat ,opf dtyenramcicov∂isvncotsiitny∂.wuA esusbsetqhuenet ddiviesiovnebylρopan(m3d.1th8ee) nt of the equations that τ yyE= qµ !u∂za+ t∂zi"o, τnyz =sµ G+overning0 Geophysical introduction of !the∂ykinem∂atyic"viscosity ν = µ!/∂ρz0 yield∂y " form the basis of d78ynamical mete∂woro∂lwogy and physiCcHaAlPToEcRe3.aEnQoUAgTrIOaNpShy. Averaging is performed whτerzez µ =is caµlled the c+oefficient,of dynamic viscosity. A subsequent division by ρ0 a(n3d.1t8h)e introduction of !the∂kzinema∂tizc "viscosity ν = µ/ρ0 yield Flowdsu 1 ∂p 2 over turbulent fluctuations and furthe+rf∗swim− fpvli=fic−ation+s νa∇reu justified(3b.19a) sed on a scale analysis. to velocity gradients), with the usdet of the reduced continuityρ0eq∂uxation, (3.17), permits us to where µ is called the coefficient of dynamic viscosity. A subsequent division by ρ0 and the write the components of the stress tensor davs 1 ∂p In the process, some imintrpodoucrtiotnaonf tthedkiniemmduateicnvisciosoitynνl=eµs/sρ nyieul1dm∂pbers 22are introduced. The need for an ap- + f∗w +− ffuv = 0 − + νν ∇∇ vu. (3.1290) dt dt ρ0 ∂xy propriate set of initial and boundary codnv ditions is1 a∂lpso explored from mathematical, physical xx The∂nuext ca∂nduidate forxysimplific+at∂iofunuis t=h∂evz−–momexnztu+m νeq∇ua2tv∂io.un, (3.2∂c)w. There,(ρ3.2ap0-) τ = µ + ,duτ = µ + 1, ∂τp = µ 2 + pears!a∂s xa facto∂rxno"t only +in ffr∗odwntt o−!f ∂tfhyve le=ft-∂hax−n"d ρsi0de∂, ybu+t aνlso∇!inu∂azprodu∂ctxw"ith g (o3n.1t9h)e and numerical points of view. dt " ρ0 ∂x right. On the left, it is safe to neglect ρ in front of ρ0 for the same reason as previously, yy Th(e∂Jnuvelxyt c2a∂n6vd,id2a0te0f7oyr)zsSimUpdlvMificMa∂tiovAn RisYth∂e:wzT–hm1iosm∂cephntaumpteeqruactoionnt,i(n3.u2ce)s. Tthhered, eρvaep-lopment of the equations that τ =butµon the r+ight it is n,ot.τ Inde=ed,µth+e tferum+ρ=g ac−counts for+theνw∇ei2gvh.t of the fluid, wh(i3ch.2,0a)s pears!a∂s ya facto∂ryn"ot only in frodntt !of∂thze left-∂hyan"dρs0id∂e,ybut also in a product with g on the we knfoowr, mcautsheseabnainscisreoasfedoyf pnraesmsuircea"wlimth edetpetohr(oorl,oagdyecarneadsepohfyprseiscsaurleowciethahneoigghrt,aphy. Averaging is performed right. O∂nwthe le∂ftw, it is safe to neglect ρ in front of ρ0 for the same reason as previously, τ zz =bdueptµTeohnnedotinhvnegexrroti+gncthauwtnrhdibteiditushaletneron,wfto.terIflntshidumiencepkdtliuo,fiftachtatehiteiotoeonrcmsiesaρantghnoeadrczac–ftomuurontmsthspehefnoertruestm)hi.emeWwqpiuetahligtiitfihohtneco,aρf(0t3tih.po2eacnr)flt.suoT(iafd3hrt,.eh1ewre8ehj)d,iueρcnshas,tpiiatfi-ys ed based on a scale analysis. ! ∂z ∂z " pgweoaeerksnaaoshwya,dfcraaocusttsoaertsincaopntroiensncsluryeraeinspef0ro,ofwnpthrioecfshstuhisreealwefufittn-hhctadinoednptsohifd(zeo,ro,bnaulytd:aelcsroeaisneaopf rpordeusscutrweiwthitgh ohneigthhet, In the process, some imp"ortant dimensionless numbers are introduced. The need for an ap- rdigehpte.ndOingthoen lwefht,etihteirswsaefethtionknoefgltehcetoρceiannforronatmoofsρp0hefroer).thWe istahmtheerρe0aspoanrtaosfptrheevdioeunssliyty, where µ is calledpthroe pcoreiafpfitec=iesneptto(ofzf)dyi+naipmti"(aixcl,vayisn,cdzo,sbitoy).uAnwdistuhabrsyepqcu(oze)nntd=diitviPoisnio−snibρsygaρzl0s, oanedxthpelo(3r.2e1d) from mathematical, physical bguoteosn9a2thhyedrriogshttaitticisprneos0ts.uIrnedpe0e,dw, thhiechteirsma fρugncatcicoonuonftszfo0onrltyh:e weig0ChHt AofPtT0hEeRflu4i.d,EwQhUiAchT,IaOsNS introductiowneofknthoaMomentumewnk, dicnaenumsueasmtiacen vrBudgetiiinsccaroeslaisptye o-νo iNavier-Stokesfn=ptrseµso/sρufr0evywiieeiwlthd .dEquationsepth (or, a decrease of pressure with height, so that dp0/dz = −ρ0g, and the vertical-momentum equation at this stage reduces to depending on whether we think o"f the ocean or atmosphere). With the ρ0 part of the density p = p0(z) + p (x, y, z, t) with p0(z) = P0 − ρ0gz, (3.21) 4.1 Reynolds-goaes va hyedrorstaatic pgresesurde p , wehiqch isua faunctioin oof znonlsy: du dw 0 11 ∂∂p" ρ"g so that dp /dz = −ρ g, and the vertical-momentu! m=e0quat2ion at!th=i2s0stage r!ed! uces to 0 + f!∗0uwv"−−=ffv∗!u!u="=!v"−" + !!u" v "+− ν+∇!!+uv" uν "∇ w,+ !u v " (3.19) (3.22) dt dt " ρρ00 ∂∂xz# ρ0 # p = p (z) + p (x, y, z, t)! !with p (z) = P − ρ gz, (3.21) 0 = !u" !v" + !u v " " 0" 0 0 (4.2) ddvw 11 ∂∂pp ρ g 2 2 after a division by ρ0 fo+r −cofnfuv∗euni=e=nce−. No furth+e−r sνim∇pli+vfi.caνti∇onwis, possible(3b.e2c0a)us(3e.2th2e) Geophysical flows are tsoythpatidcp0a/ld"zly= −inρ0ga, ansdtthaetveertiocalf-mtoumernbtuum leqeuantiocneat,thaisnstadge mreduocess to often we are only interested remaining ρ term nodldtotnger falls in the sρhρ0a0d∂o∂wyz of a nρe0ighboring term proportional to ρ0. an4d.s1imilarlyRfore!uyu"n, !owul"d, !suρ-"aet"vc. eWerraecoggneizde heereqthuat tahetaivoerangesof a product is not Aafcteturaeaqlluyda,ilvatsioswitohenewbpiryloldρsuecetflooarftetcrho,etnhavveeentreiaregmnecsρe..TghNiissotihsfeuaro"dtnhoeeurrbelssei"pm-eopdnglsiefiibdclasewtifoonrdt:ihsOepbnouosonsyiebahlneacnybdef,ociartcugesesentehtrhaaettes in the statistically aThveenerxat cganediddatfle forwsim,p0dlliwfiecativoniins tghe za–sm1iodm∂epntuamleρlqgutautiornb, (u32.2lce).nTtherfle,uρ capt-uations. To this effect and are such a cr"ucial ingredie−nt off∗gueo=phy−sical fluid dy−namics.+ ν ∇ w, (3.22) remaminainthgemρ atiecraml cnoomlpolnicgaetirofnasllbsuitn, otnhethsehoatdhoewr hoafnda, niteaiglshobcorreiantgesteinrmterepsrtoinpgorstiitounaatilotnos.ρ0. pears as a factoNroGnteoetthooapntlhtyhyeinshiyfcdroarodnlsttfltaootficwthpserealsesrfuetr-ehtyapnp"(dizcρs)i0acdlael∂n,yzbbeiuntsuaablstρsro0atcaintteeda foprofromtduuprcibtnuwthliethnregcdeuo,cneadtnhmedomeons- t often we are only interested Actually,Oausrwoebjwecitlilvseeeisltaoteers,ttahb"elitsehrmeqρu0agtiiosntshgeoovneernriensgpothnesimbleeafnoqruthaentbituieosy,a!nuc"y, !fovr"c, e!swt"h,a!tp" following Reynolridghst. (O1n t8htue9mle4efqt),uai,tioiwsnss,ae(f3e.1dt9o)enaencgdlo(e3cm.t2ρ0)p, ibnoecfarsouenset ioetfhaaρs0cnfohodr etvrhiveaastiravmiesaewbrietahlseroenspiaensctpttrooevxioaanudslmyy, anedaisn, denoted with a set of aafrtersuacnidhndiva!tρichs"ri.eouSnctisaabtrlyatintρig0srwtefidioctihreanctlhtoleoynfvavegaenevrioaeepgnrhecayeogs.ficetNhadleoflxfluf−uiodrmwthdoey,mrnleasenmimatuivcpmsili.nefiqgcuaataitoisonindi(se3.p1ao9ls)ls,iwbtuleerhbaevucela:eusnettflheuctuations. To this effect and but on the rdigyhntamiStatisticallyt iicsanlloy" ti.nIancdtiev eaverage.d, the te r"mowρ g- aReynoldsccounts f averagingor the weight of the fluid, which, as remaNinoitnegthρatttehremhyndorolosntagteicr pfarellsssuinrethpe(szh)acdaonwbeofsuabntreaicgthebdofrrionmg tperimn thperorpeodrutcioendaml toomρe0n.- brackets, and a fluctuationf,oldloewninogtReedynboyldsa(1p89r0" 4im), wee: decompose each variable into a mean, denoted with a set of we know, cAatucumtsueFaesolqlrayunw,aataiisnotewcnrrs,ee,thaw(3seie.lt1lro9es)feaetapmnlradetnse(st3ru,o.2trfhe0te)hw,etebiertehmnceadrρugesgypeteiihstqhut(hoaersti,onanoe,dd(ere3ecr.s8ripve)ao,atnisisevsisebotsrlfaewipfgiorthhertsftrsoheureswrpbeeaucwrotdyit.toaFhnxicrhsyaetn,ifgodchroyctn,etsainntuhdiatitys depending oonf vwohlube∂mtrh!aeeu,cr"(kw3.ee1t∂7ts)h!,iuenaulkin"modifnata∂htee!flsvouutch"cetamunai∂odt!driwolaeutnmt"e,romdspe, hlneeaorveti)en.dgWbityh tahepρrimpae1r:t o∂f!pth"e densi2ty adryensaumchicaalclryu+icniacl tiinvger.ed+ient of ge+ophysical+flufi∗d!dwy"na−mfic!sv." = 0− + ν ∇ !u" (4.3) ∂t ∂x ∂y ∂z ρ0 ∂x goes a hydrostNaFtoioctrepwtrhaeatsetsrtu,hrteheehpyt0rde,rawotsmhtaiectnihct oipsfreathsfseuuenrnecetpri0og(ynze)oqfcuazantoibonenly,su:(3b.t8r)a,citsedstfrraoigmhtpfoinrwtharedr.eFdiurcste,dcomnotimnuenit-y tum equations, (3.19) and (3.20), becausediTt has no der2ivatives with respect to x and y, and is of vowluhmiceh, (b3e.c1o7m),eesliminates the miρdCdlve term=, leakvTin∇g T. ! ! dynamicallyintroduce:inactive. " dt u = !u" + u , (4.1) p = p0(z) + p (x, y,uz, t)=with!up0"(z) += P0u− ,ρ0gz, (3.21) (4.1) For water, the treatment of the energydeTquation, (3.28), is straightforward. First, continuity ρCv = kT ∇ T. so that dp0o/fdvzo=lum−eρ,0(g3.,1∂a7n!)du, e"tlhime !i∂vne(a!rtutei"sc!atuhl-e"m)moimd∂del(ne!tutued"mrt!mve",q)leuaavti∂ion(gn!ua"t!twhi"s) stage reduces to suchwith:that !+u " = 0 by+ defiaveragenitio+n. perturbation+ f∗ !w" − f !v" = ∂t ∂x ∂y ∂z ! " " dw 1 ∂pdT ρ g !2 ! ! ! ! ! such that !u " = 0 by definitiTohner.e are s1ev∂e!pr"alρwCvay2s to= dk∂eT!fi∇unueT".th2e∂ a!uvevr"agi∂n!gu pwr"ocess, some more rigorous than others, − f∗ −u = − + ν ∇dt!−u" − + ν ∇− w, − (.3.22) (4.4) but wdte shall nρ0ot ∂bxeρc0o∂nzcerneρd0 h∂exre with∂ythose is∂szues, prefering to think of the mean as a

There are sevaefterraaldiwvisaionybstyemtρo0pfoodrracelofianvneenrieaengcteeh.oevNeorafruvartephierdrastigumrpiblniufilgceantiptonflruioscpctouesasstibioslen,bsse,coaunmseatehtiemme ionrteervraligloongroenuosugthhtaonobotatinhers, " We note that this last equation for the mean field looks identical to the original equation, remaining ρ teremxacnespottalfotoinrsgtheiecr afpalrellyssenisnciegthonef itsfihhrceaeadonnwetwmotfeeramannse,aitgyhthebetoersinnhdgootfretrthmeenrpiorgohuptg-ohhratniodtonsaildreeto.taTρih0n.esethteermsslower evolution of the flow but we shall not be concerned here w" ith those issues, prefering to think of the mean as a Actually, as we wreuiplnlresdseeeenrtlatchteoern,eftsfheiedctteseromaf tthρioegntui.srbOtuhleuenrotnhfleuyrceptusoapttoihonnessisboilsne ftihosertmthheeaatbnusflouoywcah.ncCayonmfobirnicnetisengrthmtahteesde iteartmestime interval exists. temporal averageareosvuceh ra cruacipawl iitnhdgcroetrdruieesnprotbnodfuigngeloefprinhctyitosincaflallteuflrumcidstduynaamtiicos.ns, on a time interval long enough to obtain Note that the hydroQstuataicdpraretsiscureexpp0(rze)sscaionnbes ssuubctrhactaesd ftrhoemppriondthuecrteduuvceod fmtowmoenv- elocity components have the fol- tum equations, (3.19) and (3.20), because it has no derivatives with respect to x and y, and is low∂ing p∂ro!up"erty:! ! ∂ ∂ !u" ! ! ∂ ∂ !u" ! ! a statistically sigdnyniafimcicaallny itnacmtivee. anν, ye−t !su hu "or,t enνough− !utov " re, tainν the− !sulwo"wer evolution of the flow ∂x ∂x ∂y ∂y ∂z ∂z under considerationFo.r wOateur,rthehtryeaptmo!entthofethseiesnerigsy e"tqhuaatiotn,!s(3u.8c),his staranighitfn"ortwearrdm. F!irestd, cionatitneuitytim" e interval exists. of volume, (3.17i)n,deicliamteisntahtaetstthheeamveirdadglees toefrmve,lolecaitvyinflguctuations add to the vis9c1ous stresses (for example, Quadratic expressions− !suu!w!c" ahddsatosν∂t!hu"e/∂zp) arndocdanuthecretforue bve coonsifdertewd asofricvtioenall ostrcessietsycauscedobmy ponents have the fol- turbulence. To give credit todTOsborne Rey2nolds who first decomposed the flow into mean and ρC = k ∇ T. ! ! ! ! ! ! fluctuating components, vthedetxpressioTns − !u u ", − !u v " and − !u w " are called Reynolds lowing property: stresses. Since they do not have the same form as the viscous stresses (ν∂ !u" /∂x etc.), it can be said that the mean turbulent flow behaves as a fluid governed by a frictional law other than that of viscosity. In other words, turbulent flow behaves as a non-Newtonian fluid. Similar averages of the y- and z-momentum equations (3.20)–(3.22) over the turbulent fluctuations yield 91

∂ !v" ∂(!u" !v") ∂(!v" !v") ∂(!v" !w") 1 ∂ !p" + + + + f !u" − = ∂t ∂x ∂y ∂z ρ0 ∂y ∂ ∂ !v" ! ! ∂ ∂ !v" ! ! ∂ ∂ !v" ! ! ν − !u v " + ν − !v v " + ν − !v w " (4.5) ∂x ∂x ∂y ∂y ∂z ∂z ! " ! " ! " 78 CHAPTER 3. EQUATIONS

7to8velocity gradients), with the use of the reduced continuity eqCuHatAioPnT,E(3R.137.),EpQerUmAiTtsIOusNtSo write the components of the stress tensor as to velocity gradients), with the use of the reduced continuity equation, (3.17), permits us to 78 CHAPTER 3. EQUATIONS write the compone∂nuts of th∂eustress tensor as ∂u ∂v ∂u ∂w τ xx = µ + , τ xy = µ + , τ xz = µ + to velocity gradie!n∂tsx), with∂txh"e use of the red!uc∂eyd cont∂inxu"ity equation, (3!.17∂)z, perm∂itxs "us to write ythye componen∂tuvs of th∂∂evustress teynzsor as ∂∂vu ∂∂wv ∂u ∂w ττxx = µ ++ ,, ττ xy == µµ ++ , τ xz = µ + !∂xy ∂∂yx"" !!∂∂zy ∂∂yx" ! ∂z ∂x " ∂wv ∂∂vw ∂v ∂w τ yzyz = µ ∂u ++ ∂u , , τ yz = µ ∂u + ∂v ∂u ∂w (3.18) τ xx = µ !∂∂yz + ∂∂yz"", τ xy = µ !∂z + ∂y " , τ xz = µ + !∂x ∂x" ! ∂y ∂x" ! ∂z ∂x " ∂w ∂w wheτrzezµ i=s calµled∂thve co+e∂ffivcient ,of dynamic v∂isvcosity∂.wA subsequent division by ρ an(3d.1th8e) τ yy = µ ! ∂z + ∂z ", τ yz = µ + 0 introduction of !the∂ykinem∂atyic"viscosity ν = µ!/∂ρz0 yield∂y " 78 ∂w ∂w CHAPTER 3. EQUATIONS whτerzez µ =is caµlled the c+oefficient,of dynamic viscosity. A subsequent division by ρ0 a(n3d.1t8h)e introduction of !the∂kzinema∂tizc "viscosity ν = µ/ρ0 yield du 1 ∂p 2 + f∗w − fv = − + ν ∇ u (3.19) to velocity gradients), with the usdet of the reduced continuityρ0eq∂uxation, (3.17), permits us to where µ is called the coefficient of dynamic viscosity. A subsequent division by ρ0 and the write the components of the stress tensor davs 1 ∂p introduction of the kinemduatic viscosity ν = µ/ρ yiel1d ∂p 22 + f∗w +− ffuv = 0 − + νν ∇∇ vu. (3.1290) dt dt ρ0 ∂xy dv 1 ∂p xx The∂nuext ca∂nduidate forxysimplific+at∂iofunuis t=h∂evz−–momexnztu+m νeq∇ua2tv∂io.un, (3.2∂c)w. There,(ρ3.2ap0-) τ = µ + ,duτ = µ + 1, ∂τp = µ 2 + 92pears!a∂s xa facto∂rxno"t only +in ffr∗odwntt o−!f ∂tfhyve le=ft-∂hax−n"d ρsi0de∂, ybu+t aνlso∇!inu∂CazHprAodPuT∂cEtxwR"it4h. gE(o3Qn.1Ut9hA)eTIONS dt " ρ0 ∂x right. On the left, it is safe to neglect ρ in front of ρ0 for the same reason as previously, yy The∂nvext ca∂nvdidate foyrzsimpdlvifica∂tiovn is th∂ewz–m1om∂epntum equation, (3.2c). There, ρ ap- τ =butµon the r+ight it is n,ot.τ Inde=ed,µth+e tferum+ρ=g ac−counts for+theνw∇ei2gvh.t of the fluid, wh(i3ch.2,0a)s pears!a∂s ya facto∂ryn"ot only in frodntt !of∂thze left-∂hyan"dρs0id∂e,ybut also in a product with g on the we know, causes an increase of pressure "with depth (or, a decrease of pressure with height, right. O∂nwthe le∂ftw, it is safe to neglect ρ in front of ρ0 for the same reason as previously, zz depending on whether we think of the ocean or a!tm=os0phere). With! th=e0ρ0 part!of! the density τ =butµTohnetnhexrti+gchatn!dituidivsa"tneo,fto=. rInsidm!e!epudli,"fitc!hvaet"ito"enr+mis!ρt!ghuea"czvc–om"uonmtsefn+otru!tmh!vee"wquueai"gtihotno, f(3+th.2e!cu)fl.uvT(id3h",.e1wr8eh),iρcha,pa-s goes!a h∂yzdrostat∂iczp"ressure p , which is a functi#on of z only: # pweaerksnaoswa, fcaacutsoersnaont oinnclryeainsef0roofnptroefsstuhreelwef!itt-h!hadnedptshid(eo,r,bautdaelcsroeaisneaopf rpordeusscutrweiwthitgh ohneigthhet, = !u" !v" + !"u v " (4.2) rdigehpte.ndOingthoen lwefht,etihteirswsaefethtionknoefgltehcetoρceiannforronatmoofsρp0hefroer).thWe istahmtheerρe0aspoanrtaosfptrheevdioeunssliyty, where µ is called the coefficient of dynam" ic viscosity. A subsequent division by ρ0 and the bguoteosnathhyedrriogshttapittici=sprnepos0ts.(uzIr)nedp+e0e,dpw, (thhxiec,htyei,rsmza,fρutg)ncatciwcoonituhonftszpfo0o(nrzlty)h:e=wePig0ht−ofρt0hgezfl, uid, whic(h3,.2a1s) introductiaonwndeosfkimnthoiMomentumewlak,riclnyaeumfsoearst!iacun vuBudgetiin"s,cro!ewsaistuye "-νo, Navier-Stokesf!=uprρeµ"s/seρutr0ce.ywWieiltehdr dEquationseecpotghn(iozre, ahedreecrtehaastetohfeparvesesruargeewoifthahperiogdhtu,ct is not so that dp0/dz = −ρ0g, and the vertical-momentum equation at this stage reduces to eqdueapletnoditnhgeopnrowdhuecthteorfwtheethaivnekroa"fgtehse. oTcheiasniosraadtmouosbplhee-ered)g. eWdisthwtohredρ:0Opnarot nofethhaendde,nistitgyenerates p = p0(z) + p (x, y, z, t) with p0(z) = P0 − ρ0gz, (3.21) mgaothesema haytidcraolstcaotimc prleicssautiroenps0,bwuth,icohnitshaefounthcetironh"aonfdz,oitn" layl:so creates interesting situations. du dw 11 ∂∂p ρ g 2 2 soOthuartodbpj0e/cdtziv=e+i−s fρto∗0gwe,sat−na−dbfltihfsve∗hvuee=qr=tuicaat−li-omnosmgeonvtuemr+−neinνqgu∇atthioe+unmaνtet∇ahniswqs,utaagnetirteideusc,(3e!s.u1t"o9,)!(3v."2,2!)w", !p" dt dt " ρρ00 ∂∂xz ρ0 and !ρ". Startinpg w=itph0t(hze) a+vepra(gxe, oyf, tzh,etx) −wmiot"hmepn0tu(" zm) e=quPa0tio−n (ρ30.g1z9,), we hav(3e.:21) ddvw 11 ∂∂pp ρ g 2 2 after a division by ρ0 fo+r −cofnfuv∗euni=e=nce−. No furth+e−r sνim∇pli+vfi.caνti∇onwis, possible(3b.e2c0a)us(3e.2th2e) so that dp0/d"z = −ρ0g, and the vertical-momentum equation at this stage reduces to remaining ρ term nodldtotnger falls in the sρhρ0a0d∂o∂wyz of a nρe0ighboring term proportional to ρ0. Actually, as we will see later, the term ρ"g is the one responsible for the buoyancy forces that ∂af!teur"a di∂vis!iuoun"by ρ∂ !fvoru"conve∂ni!ewncue". No fur"ther si"mplification is1pos∂si!bple" because2the The next candidate for simp0dliwfication is the z–m1om∗∂epntum eρqguation, (32.2c). There, ρ ap- are suc+h a cr"ucial +ingredie−nt of+f∗gueo=phy−sic+al flfui!dwd"y−n−amfic!sv.+" ν=∇ −w, + ν (∇3.22!u) " (4.3) re∂mtaining ρ∂xterm no l∂onyger falls ∂inzthe shadow of a neighboring terρm0 pr∂opxortional to ρ0. pears as a factoNronteotthoantltyheinhyfdrorodnstttaotficthperelsesfutr-ehapn"(dzρs)i0cdae∂n,zbbeutsuablstρro0acinteda fproromdupcitnwthiethregduocnedthmeomen- Actually, as we will see later, th"e term ρ0 g is the one responsible for the buoyancy forces that right. On thtuemleefqt,uaitioisnss,a(f3e.1t9o)naengdl(e3c.t2ρ0), ibnecfarounset iotfhaρs0nfoodr etrhiveastiavmesewrietahsroenspaesctptroevxioanudslyy, and is wahafirtcehrsuabcedhciovaimcsrieousnciablyinρg0refdoirenctoonfvegneioepnhcyes.icNaloflufuidrthdeyrnasmimicpsli.fication is possible because the but on the rdigyhntamiStatisticallyt iicsanlloy" ti.nIancdtiev eaverage.d, the te r"mowρ g- aReynoldsccounts f averagingor the weight of the fluid, which, as remaNinoitnegthρatttehremhyndorolosntagteicr pfarellsssuinrethpe(szh)acdaonwbeofsuabntreaicgthebdofrrionmg tperimn thperorpeodrutcioendaml toomρe0n.- 0" we know, cAatucumtsueFaesolqlrayunw,aataiisnotewcnrrs,ee,thaw(3seie.lt1lro9es)feaetapmnlradetnse(st3ru,o.2trfhe0te)hw,etebiertehmnceadrρugesgypeteiihstqhut(hoaersti,onanoe,dd(ere3ecr.s8ripve)ao,atnisisevsisebotsrlfaewipfgiorthhertsftrsoheureswrpbeeaucwrotdyit.toaFhnxicrhsyaetn,ifgodchroyctn,etsainntuhdiatitys depending aoodrnfyenvwsaouhmlcuehimtc∂haaeelc!,lrruy(uw3"ic.ne1ia7tcl)hti,∂iinveng(elki!r.meuodi"fine!tanhuttee"so)oftchgeeaom∂npi(hod!ydruslae"itcm!taevlro"flms)up,ihldeeadr∂vyei)n(n.!aguWm"iic!tshw. t"h)e ρ0 part of the density goes a hydrostatic pressur+e p , which is a+function of z +only: + f∗ !w" − f !v" = NFootrewth∂aattetrt,htehehyt0rderaot∂smtxaetnict opfrethsseuerneepr∂0g(yyze)qcuaantiboen,su(3b.t∂8r)az,citsedstfrraoigmhtpfoinrwtharedr.eFdiurcste,dcomnotimnuenit-y tum equations, (3.19) and (3.20), becausediTt has no der2ivatives with respect to x and y, and is of volume, (3.17), eliminates the miρdCdlve term=, leakvTin∇!g T! . ! ! ! ! dynamically inactive.1" ∂ !p" 2 dt ∂ !u u " ∂ !u v " ∂ !u w " p = p0(z) +− p (x, y, z+, tν) ∇wi!tuh" −p0(z) = P−0 − ρ0gz, − (3.21.) (4.4) For water, the treaρt0men∂t xof the energydeTquation,∂(3x.28), is straig∂hytforward. F∂irzst, continuity ρCv = kT ∇ T. so that dp0o/fdvzo=lum−eρ,0(g3.,1a7n)d, etlhime ivneartteiscathl-emmoimddelnettuedmrtme,qleuaavtiiongn at this stage reduces to We note that this last equation for the mturbulentean fiel d"uctuationslooks ide nofti ctheal tmeano the "oowriginal equation, d" T " except for thdewpresence of three1ρnCe∂wpterm=sρakgt t∇he2Te.nd2of the right-hand side. These terms − f∗ u = − v dt− T + ν ∇ w, (3.22) represent thedetffects of the turbuρl0en∂t zfluctuaρti0ons on the mean flow. Combining these terms with corresponding frictional terms after a division by ρ0 for convenience. No further simplification is possible because the " remaining ρ term no longer falls in the shadow of a neighboring term proportional to ρ0. Actually, as we will see later, the term ρ"g is the one responsible for the buoyancy forces that ∂ ∂ !u" ! ! ∂ ∂ !u" ! ! ∂ ∂ !u" ! ! are such a crucial ingνredient o−f g!euopuh"ysica,l fluid dyνnamics.− !u v " , ν − !u w " ∂x ∂x ∂y ∂y ∂z ∂z Note that the hy!drostatic pressure p"0(z) can b!e subtracted from p"in the red!uced momen- " tum equaitniodnisc,a(t3e.s1t9h)aatntdhe(3a.2v0e)r,abgeecsaoufsevietlhoacsitnyoflduecrtiuvativoensswaidthd rteosptheectvtoiscxoaunsdsytr,easnsdesis(for example, dynamica−lly!uin!wac!t"ivaed.ds to ν∂ !u" /∂z) and can therefore be considered as frictional stresses caused by For wtautrebr,utlheenctree.aTtmoegnitvoefctrheedeitnteorgOysebqouranteioRn,ey(3n.o8)ld, isswsthraoigfihrstftodrewcaormd.pFoirssetd, cthoentflinouwityinto mean and of volume, (3.17), eliminates the middle term, leaving ! ! ! ! ! ! fluctuating components, the expressions − !u u ", − !u v " and − !u w " are called Reynolds stresses. Since they do not hadvTe the same form as the viscous stresses (ν∂ !u" /∂x etc.), it ρC = k ∇2T. can be said that the mean turvbudltent flowT behaves as a fluid governed by a frictional law other than that of viscosity. In other words, turbulent flow behaves as a non-Newtonian fluid. Similar averages of the y- and z-momentum equations (3.20)–(3.22) over the turbulent fluctuations yield

∂ !v" ∂(!u" !v") ∂(!v" !v") ∂(!v" !w") 1 ∂ !p" + + + + f !u" − = ∂t ∂x ∂y ∂z ρ0 ∂y ∂ ∂ !v" ! ! ∂ ∂ !v" ! ! ∂ ∂ !v" ! ! ν − !u v " + ν − !v v " + ν − !v w " (4.5) ∂x ∂x ∂y ∂y ∂z ∂z ! " ! " ! " 92 CHAPTER 4. EQUATIONS

! =0 ! =0 ! ! !uv" = !!u" !v"" + !!u" v#" + !!v" u#" + !u v " ! ! = !u" !v" + !u v " (4.2) and similarly for !uu", !wu", !uρ" etc. We recognize here that the average of a product is not equal to the product of the averages. This is a double-edged sword: On one hand, it generates mathematical complications but, on the other hand, it also creates interesting situations. Our objective is to establish equations governing the mean quantities, !u", !v", !w", !p" and !ρ". Starting with the average of the x−momentum equation (3.19), we have:

∂ !u" ∂ !uu" ∂ !vu" ∂ !wu" 1 ∂ !p" 2 + + + + f∗ !w" − f !v" = − + ν ∇ !u" (4.3) ∂t ∂x ∂y ∂z ρ0 ∂x which becomes

92 ∂ !u" ∂(!u" !u") ∂(!u" !v") ∂(!u" !w") CHAPTER 4. EQUATIONS + + + + f∗ !w" − f !v" = ∂t ∂x ∂y ∂z 1 ∂ !p" ∂ !u!u!" ∂ !u!v!" ∂ !u!w!" − + ν ∇2 !u" − − − . (4.4) ρ0 ∂x ∂x ∂y ∂z ! =0 ! =0 ! ! !uv" = !!u" !v"" + !!u" v " + !!v" u " + !u v " We note that this last equation for the mean fi#eld looks ident#ical to the original equation, ! ! except for the presence=of t!hure"e!vne"w+te!rumvs a"t the end of the right-hand side. These terms (4.2) represent the effects of the turbulent fluctuations on the mean flow. Combining these terms andwsitihmciMomentumolarrelyspfoonrd!iun guBudgetf"r,ic!twioun "a-,l Navier-Stokes!teurρm"setc. We r Equationsecognize h +e rReynoldse that the averagingaverage of a product is not equal to the product of the averages. This is a double-edged sword: On one hand, it generates mathematical complications but, on the other hand, it also creates interesting situations. ∂ ∂ !u" ! ! ∂ ∂ !u" ! ! ∂ ∂ !u" ! ! ν − !u u " , ν − !u v " , ν − !u w " Our∂oxbjectiv∂exis to establish e∂qyuations∂gyoverning the me∂azn quan∂tzities, !u", !v", !w", !p" and !ρ". St!arting with the av"erage of!the x−momentum" equatio!n (3.19), we have": indicates that the Reynoldsaverages ostressesf velocity fluctuations add to the viscous stresses (for example, − !u!w!" adds to ν∂ !u" /∂z) and can therefore be considered as frictional stresses caused by ∂tu!rub"ulenc∂e.!Tuoug"ive c∂re!dvitut"o Osb∂o!rnweuR"eynolds who first decomposed1the∂fl!opw"into mea2n and + + + + f∗! !w! " − f! !v! " = − ! ! + ν ∇ !u" (4.3) fl∂utctuating∂coxmponent∂s,ythe expres∂szions − !u u ", − !u v " and − !u ρw0 " a∂rexcalled Reynolds stresses. Since they do not have the same form as the viscous stresses (ν∂ !u" /∂x etc.), it can be said that the mean turbulent flow behaves as a fluid governed by a frictional law other which bStatisticallyecomes average "ow - Reynolds averaging than that of viscosity. In other words, turbulent flow behaves as a non-Newtonian fluid. Similar averages of the y- and z-momentum equations (3.20)–(3.22) over the turbulent fluctuat∂io!nus"yield∂(!u" !u") ∂(!u" !v") ∂(!u" !w") + + + + f∗ !w" − f !v" = ∂t ∂x ∂y ∂z ! ! ! ! ! ! ∂ !v" 1∂(!∂u!"p!"v") ∂(2!v" !v") ∂ !∂u(!uv""!w")∂ !u v " ∂1 !∂u!wp"" −+ ++ν ∇ !u" − + − + f !u" −− =. (4.4) ∂t ρ0 ∂xx ∂y ∂x ∂z ∂y ρ0 ∂∂zy ∂ ∂ !v" ! ! ∂ ∂ !turbulentv" ! "! uctuations∂ of∂ the!v" mean! "!ow ν − !u v " + ν − !v v " + ν − !v w " (4.5) We no∂texthat t∂hixs last equation∂fyor the∂maddyea ntofi viscouseld loo stressesk∂szidentic∂azl to the original equation, except for!the presence of"three ne!w terms at the e"nd of t!he right-hand sid"e. These terms represent the effects of the turbulent fluctuations on the mean flow. Combining these terms with corresponding frictional terms

∂ ∂ !u" ! ! ∂ ∂ !u" ! ! ∂ ∂ !u" ! ! ν − !u u " , ν − !u v " , ν − !u w " ∂x ∂x ∂y ∂y ∂z ∂z ! " ! " ! " indicates that the averages of velocity fluctuations add to the viscous stresses (for example, − !u!w!" adds to ν∂ !u" /∂z) and can therefore be considered as frictional stresses caused by turbulence. To give credit to Osborne Reynolds who first decomposed the flow into mean and fluctuating components, the expressions − !u!u!", − !u!v!" and − !u!w!" are called Reynolds stresses. Since they do not have the same form as the viscous stresses (ν∂ !u" /∂x etc.), it can be said that the mean turbulent flow behaves as a fluid governed by a frictional law other than that of viscosity. In other words, turbulent flow behaves as a non-Newtonian fluid. Similar averages of the y- and z-momentum equations (3.20)–(3.22) over the turbulent fluctuations yield

∂ !v" ∂(!u" !v") ∂(!v" !v") ∂(!v" !w") 1 ∂ !p" + + + + f !u" − = ∂t ∂x ∂y ∂z ρ0 ∂y ∂ ∂ !v" ! ! ∂ ∂ !v" ! ! ∂ ∂ !v" ! ! ν − !u v " + ν − !v v " + ν − !v w " (4.5) ∂x ∂x ∂y ∂y ∂z ∂z ! " ! " ! " 92 CHAPTER 4. EQUATIONS

4.2. EDDY COEFFICIENTS 93 ! =0 ! =0 ! ! !uv" = !!u" !v"" + !!u" v#" + !!v" u#" + !u v " ! ! = !u" !v" + !u v " (4.2) ∂ !w" ∂(!u" !w") ∂(!v" !w") ∂(!w" !w") 1 ∂ !p" and similarly for !+uu", !wu", !uρ+" etc. We recog+nize here that th−efa∗ve!rua"g−e of a produc=t is−ngot!ρ" equal to the p∂rotduct of the∂xaverages. Thi∂s yis a double-ed∂gzed sword: On one hρa0nd,∂itzgenerates ∂ ∂ !w" " " ∂ ∂ !w" " " ∂ ∂ !w" " " mathematical cνomplicati−on!subwut,"on +the otherνhand, it−als!ovcwrea"tes+interestνing situat−ion!sw. w " . (4.6) ∂x ∂x ∂y ∂y ∂z ∂z Our objec!tive is to establish e"quations!governing the mean"quantitie!s, !u", !v", !w", !p"" and !ρ". Starting with the average of the x−momentum equation (3.19), we have: 94 CHAPTER 4. EQUATIONS

∂ !u" ∂ !uu" ∂ !vu" ∂ !wu" 1 ∂ !p" 2 + + + + f∗ !w" − f !v" = − + ν ∇ !u" (4.3) ∂t 4.2 ∂xEddy∂ycoeffic∂zients ρ0 ∂x which bCecoommpeuster models of geophysical fluid systems are limited in their spatial resolution. They are therefore incapable of resolving all but the largest tu∂rbwulent fluct∂uawtions, and∂awll motions∂w + u + v + w − f∗u = 94 of lengths shorter than the mesh size. In one wayCoHr aAnoPt∂ThetEr,Rwe4.muEs∂QtxsUtaAteTsIoOmNe∂thSying about ∂z 92 ∂ !u" ∂(!u" !u") ∂(!u" !v") ∂(!u" !w") CHAPTER 4. EQUATIONS these un+resolved turbu+lent and subg+rid scale moti+onfs∗in!wo"rd−erfto!vi"n=corporate their aggregate effe∂ctt on the l∂arxger, r1eso∂lvpe∂dyflow. Thgisρ∂przocess∂is called∂swubgrid-scal∂e param∂etwerization. ∂ ∂w − − − ! ! + ! ! A ! ! + A + νE , (4.7c) Here, we pre1sen∂t !tph"eρ0sim∂pzle2st of al∂l !sucρh0uem" es. ∂∂M!xuovre"sop∂h∂xi!suticwate"d pa∂raymeteriz∂atyions will ∂z ∂z − + ν ∇ !u" − − ! − " . !(4.4) " ! " follow in lateρr0sec∂tixons of the book, par∂tixcularly Ch∂aypter 14. ∂z ∂w ∂w !∂w=0 ∂w! =0 ! ! The!purvim" aryS=ienffce!ec!tuwo+"f!eflvuuw"i"di+ltlur!wb!+uuol"ernvkvc"eeaxncdl+u+ofsi!mwv!ovet"liyounws" a−itthsuf+ba∗gvur!eiudr=asvgca"eldes e(sqmuaalltieodndisesinantdhe rest of the book (unless We note that this last equation for the mean fi#eld looks ident#ical to the original equation, billows) is dissipat∂iotn. It is th∂exref!or!e tem∂pyting to rep∂rezsent both the Reynolds stress and the othe=rwis!ue"s!pve"c+ifi!eudv),"there is no longer any need to denote a(v4e.2r)aged quantities with brackets. 1 ∂p except feoffrgeρtchteopfruesne∂rnecsoelovfedt∂hmrweoetinoenws atesrms∂osmaet tfhoe∂rmewnodfosfuptheer rv∂iigshcto-hsiatny.d∂Tswihdies. isThdeosneetesrummsmarily by − re−pres−enretpthlaecei+nffgectChtseoomnfosthelAeeqctuulerabnruvtllieysnc,t+o"flsuiutcy#tuνhaaotisfontAbhseoeflnnutihrdeebpmyl+eaaacmnefludcohwb.ylaνCrugEoemar nbedidndiysnigv,mitshcielosaseri(tlt4yye.rt7mofcosb)re adellfinoetdher variables. ρ0 ∂z ρ0 ∂x ∂x ∂y ∂y ∂z ∂z andwsitihmciMomentumolarinrelytsepfromonrsd!iounf guBudgettfu"r,ribc!I!utwniloeuntn "ah-c,l eNavier-Stokes!teuaern"ρmnd"segerrtgicdy. pW(rodeper !enEquationserctsioietgsy.n)Tizhee"iqsh u+reaa rtReynoldsethietohrnac,truth!hdeee aaveragingatvppearnoadga"cehsoawfltaasmpfirrosdlteupccruot pilsaosrneoddtibfyfusion needs likewise to be equal to thBeopursosidnuecsqts.oufptehresaevdeerdagbesy. Tthheisdisisapdeorusbinleg-eedfgfeedcstwoofrdu:nOrensoonlevehdantdu,ribt guelennertamtesotions and subgrid-scale pro- Since wemwatihllemwaotrikcaelTxchocemluppasrliaivcmealetyitoerwnisziatbthuiotan,v,oehnoratwhgeevodetrh,eerqeruchoaagtninodizn,esist ioannlsetohecesrseeranettseitasloipnfrtoethpreersttbyino: gothkseit(auunaintsilooetnrsosp. y of the ∂ ∂ !u"cesse!s.! Using∂the s∂am!u"e hori!z!ontal e∂ddy v∂is!cuo"sity A! !for energy as for momentum is generally otherwise specifiOeudr),otflbhojewνcretfiivieseldnisaon−todlo!oeunfsgtuiaetsb"rlmaisnoh,dyenlqienuegadtgiνortinods.dgHeonovro−eiztre!onuniantvagle"arthanegd,emvdeerqatinucaaqlνnudtairnteitecitsioew−ns,si!t!uahurew"b,trr"!aevca"kte,ed!twsd.i"f,fe!rpe"ntly ∂x ∂x adequate, bec∂ayuse th∂ey larger turbule∂nzt mot∂iozns and subgrid processes act to disperse heat and Consequentlya,n"du!#ρ"h.aSsbty!abreatseinsniggrnweinpitglhatcwtheoeddabivsy"teirnuacgtaeendoddfy!sthvimiescxiol−asrimtliyeosfm,oAerniantlultm"hoetehqoeurriazvtoiao!nrntiaa(l3ba.nl1ed9sν).,Ewinethhaevve"e:rtical. Because turbulent msoatlitonassaendffmecetsihvseilzye caosvemr loomngerndtuismtan.ceIsninththee vhoerritziocnatla,l thhoanwienvtheer,vethrteicapl,ractice is usually to distin- In the energinydi(cdaetensstihtyat)theeqReynoldsuavaetriaogne,s h ostressesef avtelaoncidtysflaulcttumatoiolencsualdadrtoditfhfeuvsisocnounsesetrdesssleisk(efworiesxeamtoplbee, superseded by t−he!ud!wis!A"peacdrosdvisentrgos νaeg∂fmuf!euiusc"hht/∂oldazfrig)supeanrnedrsrepscsaiaononlnotvhfeoeudrfnerfteoeunsrreoeblbrvuegeldecyonmnftrsomoitdimoeorntesitdohaannassdtfrnoaiecnfetdidmosnsoatuolmbsbgtererenissditsgue-nssmiccfiacbulaesynetpdliyrnbotlyar-rogderutchianng a vertical eddy diffusivity ∂ !u" νE∂.!Fuuur"therm∂o!rve,ua"s they∂o!uwguh"t to depend in some elementary1wa∂y!opn"flow properties and grid cesses. Using thtuerbsualmeneceh.oTroizgoivnκetcEarledethidtadtotyOdvsiifbsfoceronrseiRtfyreoyAnmofldotshrweehnvoeefirrgrtsiytcdaeslcofeomdrpdmoysoevdmitshecenofltsuoiwmtyiinsνtoEgme.neaTe2nrhaailnsldydifference stems from the specific +dimension+s, each of w+hich may v+arfy∗f! r!ow! m" p−lafce! !tv!o"pla=ce,−ed!dy!viscositi+esνsh∇ould!ub"e e(x4p.e3c)ted adequate, becaufl∂suetcttuhaetinlagr∂cgoxemrptuotnruebnrubt∂lsue,yntlhetenmetxobptreieohs∂nsazisovnaison−rdo!suuf buega"r,ci−hd !psutraovtce"easvnsaders−iaa!bcutlρewto0a"dnai∂drsexpwcearillsleledbhReeeayfntuoarlndthds er discussed in Section 14.3. The stressest.oSeixnhciebtiht esyomdoe nspoatthiaalvevatrhieatsioamnse. fRoremturansinthgetovitshceoubsastircesmseasnn(νer∂b!yu"w/h∂ixchettch.e),mitomentum salt as effectiveclayn abse smbauioddmgtheeat tnwtthausemnemse.tearaIbgnnlyitsuth(hrebdedue,lvnewnesitrittflhiyocs)wtareleb,sqeshuhdoaaivwftefieseorvanesentratih,afletlushniaedmbgpeoorcnvagoecrmftnioecredceseb:isysoanufstrhiuceatirloilgnyhalt-olhaawdnidosttshiiendr-es, we are which bStatisticallyecomes average "ow - Reynolds averaging guish dispersionthaonftehnalteodrfgtovyirsfecrtooasimintyt.thhIenasetoteohdfedrmywcooomredfsefi,nctuiteurnbmtuslebinystifldinoewttrhobedefihuracsvtiendsgearsiavaavnteiovrnet-siNc, aaeswl fetoodlnldoiaywnsdfl:iufifdu.sivity κE that differs froSmimtihlaer vaveerrtaicgaesl oefdtdhye yv-isacnodszit-ymoνmEe.ntTumhiseqduiaftfioenresn(3c.e20s)t–e(m3.2s2f)rovmer tthhe tsuprbeucliefinct turbulent behavflioucrtuoaft∂ieo!anusc"hyiesltda∂te(!uva"r!iua"b)le a∂∂nud(!uw"i!llv∂"b)ue fu∂rt(h!∂euur" d!wis"c)u∂∂susρed in Se∂cρtion 14.∂3.ρ The ∂ρ + + + u ++ v + w + f∗++!wf∗u"w−−f !fv+v" ==v + w = energy (density) equati∂otn then be∂cxomes: ∂t ∂y ∂x ∂y∂z ∂zt ∂x ∂y ∂z ! ! ! ! ! ! ∂ !v" 1∂(1!∂u!"∂p!p"v") ∂∂(2!v" !∂v∂u") ∂ !∂u(!u∂v∂"ρ"!w")∂∂u!u v∂" ∂∂1 !∂∂uρ!wp∂"u" ∂ ∂ρ −+− +++ν ∇ !Au" − + + −A + f !u+" −− νE =. , (4(.44).7a) ρ0ρ0∂∂xx ∂x ∂x A∂x∂y ∂+y∂y ∂zA ∂z ∂z + κE , (4.8) ∂t ∂ρ∂x ∂ρ !∂y ∂∂x"ρ ∂∂xz∂!ρ "∂y ρ0!∂∂yy " ∂z ∂z ∂ ∂ !v" ! ! + u∂ +turbulent∂ !vv" ! +"!uctuations!w " ∂= of the∂ ! !meanv" "ow! "! ! " We note thaνt this l−ast!u∂eqvtu"ati+on f∂oxr tνhe me∂a−yn !fivevld" lo∂+ozks ideνntical to− t!hvewo"riginal(4e.q5u) ation, ∂x ∂x The linear c∂oyntinaddu∂i∂yt vytoe viscousqua∂tivo stressesn is∂n∂zov t=s eddyub∂jze∂ stressescvted to any such adaptation and remains unchanged: except f∂or!the p∂rρesence of∂"three n∂e!wρ terms+at∂uthe e"n+d∂voρf t!he+riwght-ha+ndfsuid"=e. These terms A + A ∂+t ∂xκ ∂y , ∂z (4.8) represe∂nxt the ef∂fexcts of the∂yturbule∂nyt fluctuati∂ozns onEth∂ezmean flow. Combining these terms with corres!ponding"frict1ion∂apl t!erm∂s " ∂v !∂ ∂∂"vu ∂∂v ∂v∂w − + A + A + ν+E , = 0. (4.7b) (4.9) ρ0 ∂y ∂x ∂x ∂y ∂y ∂z ∂z The linear continuity equation is not subjected t!o any s"uch adap!tatio∂nx"and re∂mya!ins un∂"czhanged:

∂ ∂ !u" F! o!r more ∂details∂ o!un"eddy! v!iscosity∂and d∂i!ffuu"sivity! a!nd some schemes to make those de- ν − !u∂uu " ∂, v ν∂w − !u v " , ν − !u w " ∂x ∂x pend on+flow∂py+ropert∂iey=s, t0h.e reader is∂rzeferred∂zto textb(o4o.9k)s on turbulence, such as Tennekes ! " ! " ! " and∂Lxumle∂yy(1972∂)zor Pope (2000). A widely used method to incorporate subgrid-scale pro- indicates that the averages of velocity fluctuations add to the viscous stresses (for example, For more deta! ils! on eddy vciesscsoessityinatnhdedhioffruizsiovnittyalaendddsyomviescsocshietymiesstthoamt parkoeptohsoesde bdye-Smagorinsky (1964): pend on flow−p!ruopwer"tiaedsd,sthtoeνre∂a!due"r/i∂szr)eafnedrrceadnttoheterexftobroeobkescoonstiuderbreudleanscferi,cstiuocnhalasstrTesesnenseckaeussed by turbulence. To give credit to Osborne Reynolds who first decomposed the flow into mean and and Lumley (1972) or Pope (2000). A widely used method to incorporat2e subgrid-scal2e pro- 2 fluctuating components, the expressions − !u!u!", − !u!v!"∂anud − !u!w!" ar∂evcalled Reyn1olds∂u ∂v cesses in the horizontal eddy viscosity is that pArop=ose∆d xby∆Symagorinsky (19+64): + + , (4.10) stresses. Since they do not have the same form as th#e visc∂oxus stresses (ν∂∂y!u" /∂x etc2.), it∂y ∂x can be said that the mean turbulent flow behaves as a flui!d gove"rned by a!fricti"onal law oth!er " 2 2 2 than that of viscositiyn. Iwn∂houitchher∆woxrdasn, dt∂uv∆rbuyleanrteflothw1ebleoh∂caauvlesgaris∂davdniomn-eNneswiotonnsi.anBfleuciadu. se such subgrid-scale parameteri- A = ∆x ∆y + + + , (4.10) Similar averagezsaotifo∂tnhxeisy-maenadnzt-tmo∂oyrmepernetusmenet2qpuhayti∂soiyncsa(l3p.∂2rxo0)c–e(s3s.2e2s), iotvoerugthhetttuorboubleenyt certain symmetry properties, #! " ! " ! " fluctuations yield notably invariance with respect to rotation of the coordinate system in the horizontal plane. in which ∆x and ∆y are the local grid dimensions. Because such subgrid-scale parameteri- zation is meant to represent phAysnicaapl proocpersiasetes,fiot romuguhlatttiooonbfeoyrcAertoauingshytmthmereetrfyorperotopebrteieesx, pressed solely in terms of the ro- ∂ !v" ta∂ti(o!un"al!vi"n)vari∂a(n!tvs"o!vf"t)he t∂en(!svo"r!wfo"r)med by the1ve∂lo!cpi"ty derivatives. One should note that the notably invariance with respe+ct to rotation+of the coord+inate system+inft!hue"h−orizontal p=lane. An appropriate formulati∂otn foprrAecoe∂udxginhgt tfhoerrmefuol∂raeytitoonbeunefxoprrte∂usznsaedtelsyoldeloyesinntoetrρm0sastio∂sfyfythtehrios-condition. Nonetheless it is often ∂ ∂ !v" use!d !in num∂erical∂m!vo"dels. ! ! ∂ ∂ !v" ! ! tational invariants of tνhe tenso−r !fuorvm"ed+by the vνelocity−d!evriva"tive+s. Oneνshould−no!tvewth"at the (4.5) ∂x ∂x ∂y ∂y ∂z ∂z preceding formulati!on unfortunately "does no!t satisfy this cond"ition. N!onetheless it is o"ften used in numerical models. 4.3. SCALES 95

Table 4.1 TYPICAL SCALES OF ATMOSPHERIC AND OCEANIC FLOWS 4.3. SCALES 95 Variable Scale Unit Atmospheric value Oceanic value x, y L m 100 km = 105 m 10 km = 104 m 3 2 z Table H4.1 TYmPICAL SCAL1ESkOmF A=T1M0OSPmHERIC AND 1O0C0EAmNI=C F1L0OWmS 1 4 4 t T s ≥ 2 day " 4 × 10 s ≥ 1 day " 9 × 10 s uV,avriable UScale mU/snit 1A0tmm/osspheric value O0.c1eamni/cs value x, y L m 100 km = 105 m 10 km = 104 m w W m/s 3 2 z H m 2 1 km = 10 m 100 m = 10 m p P kg/(m·s ) 1 v4ariable 4 ρ t ∆ρT kgs/m3 ≥ 2 day " 4 × 10 s ≥ 1 day " 9 × 10 s u, v U m/s 10 m/s 0.1 m/s w W m/s p P kg/(m·s2) variable 3 4.3 Scaleρs of m∆ρotikog/nm

Simp9l6ifications of the equations established in the preceding sCeHctAioPnTEarRe4p.osEsQibUlAeTbIeOyNonSd the Bou4s.s3inesqSacpaprloexsimoaftiomn oantdioavneraging over turbulent fluctuations. However, these require a preliminary discussion of orders of magnitude. Accordingly, let us introduce a scale for eveSryimvpalriifiacbaltei,oans owfethaelreeqaudaytidoindsienstaablilmishiteHedd!inwatLhy,eipnre1c.1ed0i.nBgysescctiaolne,awreepmosesaibnlea bdei(my4o.en1n3ds)itohenal conBsatoanundstswionefeesdxqipmaepcetpnWrsoioxtinomsbaietdivoeansttailcnyaddliaftvfoeerrtehangatitnforgofmotvhUeer.vtuarrbiaublelnetaflnudcthuaavtiionngs.aHnouwmeevreirc,athl evsaelureeqrueiprere- senataptirveTelihmoeficntoahnreytinvduaiisltucyueessqsuiooafntitoohnfaitnosridtasemrresedouvfcaemrdiaafboglrnemi.tu(4dT.ea9.b) lcAeocn4ct.ao1irndpsirnthogrvleyied, teleesrtmiulsls,uoisnftrtrraeotsidpveuecctesivcaealosecrsdaelferosfrotrhe 4.3. SCALES 95 vareiavobeflrmeysavgoanrfitaiunbdtle:,reasstwine aglereoapdhyydsiicdailnflauliidmfliteodww. aOybivni1o.u1s0l.y,Bsycsaclealev,awlueesmdeoanvaardyimweintshioenvaelry appcloicnasttiaonnt,oafnddimtheensviaolnuseisdleScalinginstiecdalinto theTtahba ltequationseo4f .t1hearvearoianblylesaungdghesatviivneg. aEnvuemn esroi,cathl evaclounecrleupsrieo-ns U U W draswentafrtiovme othf ethuesveaolufetshoefsethpaatrstiacmuelaTavrbavlr,eaia4l.ub1leesT.,YsTPtaIaCnbAdlLeSi.4nC.A1tLhEpeSrOovFvaiAsdTteMmsOiaSlPjloHuErsiRttrIyaCtAiovNfeDcsOacCsaEelAesN.sICfIofFrLdOtohWueSbt variables of interest in geophysical flLuid flLow. OHbviously, scale values do vary with every arises in a specific situation, the foVllaoriwabilneg Sscaallee aUnnaitlysis caAntmaolwspaheyrsicbvealrueedonOe.ceanic value We ought to examine three cases: W/H is much less than, on the orde5r of, or much greate4r application, and the values listed inx,Tyable 4L.1 arme only sug1g0e0sktimve=. 1E0vemn so, th1e0ckomnc=lu1s0iomns dtrhaawnnUf/roLm. Tthe tuhsierdocfastheemseusptabreticruzlleadr ovualtHu. eInsdsemteadn,difinWt/hHe1 kv"mas=tUm1/0aL3jo,mtrhiteyeoqfuactai1so0en0sr.medI=fu1cd0eo2sumbt in first approximation to ∂w/∂z = 0, which implies that w is co1nstant in the v4ertical; because 4 arises in a specific situation, the folltowing sTcale sanalysis ca≥n 2aldwayay"s 4be×r1e0dosne.≥ 1 day " 9 × 10 s IonftaheboctotonmstrsuocmtieownhoerfeT, athbaltefl4o.w1,muw,uevstwbeeresUucpaprleimefud/slbtyolsaatetirs1af0lycmoth/nsevecrrgietenrciea(osefeglea0ot.1eprhmsy/esscitcioanl fluid dyna4m.6i.c1s),oauntdliwneeddienduSceectthioatntshe1.t5eramnswd∂1u./6∂, xWand/mor/s∂v/∂y may not be both neglected at the same time. In sum, w must be muchpsmallerPthankign/i(tmia·lsl2y)thought. variable In the construction of Table 4.1, we were careful t3o satisfy the criteria of geophysical fluid In the first case, the leading balρance is∆twρo-dk1igm/mensional, ∂u/∂x + ∂v/∂y = 0, which diymnpamlieiscsthoauttclionnevderigneSneccetinonosn1e.h5oarnizdon1Tt.a6l, d!irection, must be compensated by divergence i(n4.11) Ω the other horizontal direction. This is very possible. The intermediate case, with W/H on 1 for ththeetoimrdeerscoaf lUe/aLn,dimplie4s.3a threSec-waaleysbTaolfanm!ceo, wtihoi,nch is also acceptable. In summary, (th4e.11) vertical-velocity scale must be constrained by Ω Scales of Geophysical Flows 96 Simplifications of the equations established iCnHthAe pPrTecEedRin4g.seEctQioUn aArTe IpOosNsibSle beyond the for the time scale and U H Boussinesq approWxim!at"ion aΩnUd, averaging over turbulent fluctuations.(4H.o1w4()e4v.e1r,2t)hese require a preliminary discLussion Lof orders of magnitude. Accordingly, let us introduce a scale for U for tahned,vbeylovciirttyueaonfd(4le.1n3g),thevsecryalveasr.iabIlte,iassgweenaelr"eaaldlyyΩdn,idotinreaqliumiirteed wtoaydinis1c.r1i0m. Binyastcealbee, wtwe(em4e.e1na2nt)haedimensional constant of dimensHLion!s ideLn,tical to that of the variable and having a num(4e.r1ic3a)l value repre- two horizontal directions, asnendtawtivee aosfstihgenvtahlueessaomf tehaltesnagmtehvsacraialbeleL. Ttoabbleo4th.1cporovriddiensaitlleusstarantidvethsecales for the saanmfdoewrvteehleoxcvpieteylcotsccWitayletaonUdbteloevnbagosvttahlhryisavdcbealileflosfeecsori.fetyinInttcteiforsremoWsgmtpeionnU!negre.eanolUltpysh..ynTsoichtaelresflqauumidirefle,odhwot.owOdebivsvecioru,imsclayin,nasnctoealtebbeveatwl(su4aee.ise1dn5do)otfhvteahrye with every The continuity equation in its reduced form (4.9) contains three terms, of respective orders vertwiIcnoaolhtdhoierrirezwcotonirtodanls.,dliGarreegcoet-ipsohcnayslas,epiacgpnaelidolcapflwthiooeywnsa,iscasanaildgrflentohtewyhvsepaailscruaeaemslslhleyiaslltcelodnwnginfit(hHTnasebcd!laelteo4L.L1d) aotrnomedboannoienltyahsrslctuyhogtaogwterodsat-iirndveeiam.tmeEesuvnaecsnihodsnwota,hiltdeheerconclusions thoafnsma(tWmahgeeny!viteaulrUodece)t.i:htyicskc,aalenUd tthoebdaroastwphnevcfertloormacitttihyoecHuosm/eLpofointsheesnsmetsap.laTlr.thiTceuhslaaermavteaml,uhoeosswpstheavnederiric,ncltahnyenevorattshbt aemtsadajoierdtietoyrfmotfhicneaesses. If doubt vertical direction. Geophayrissiecsailnflaoswpesciafircestiytupaitcioanl,lythceofonlfilonweidngtoscdaloemanaainlyssitshcaatnaarlewmayus cbhe rwedidoneer. our weaLtheet rusisnoownlcyonasbioduertt1h0e xk–mmothmiceknUt,uymeteqcUuyactlioonWeins iatsnBdoaunstsiicnyecsqloannedstusprbruelaedncoev-aevretrhaoguedsands of kthiflaoonrmmthe(et4ey.r7asar.)e.StIhitmsicvikala,riraolnyud,stotheceremaassnpscecuacltrerreasentiqtosu,eHanr/teiLaglleiy,snasesmraalllly. .Tcohnefiantmedostpohtehrieculapypeerrthhautndderteedrmmineetesrs L L H of tohuerwwaetaetrhceroliusmonnlybuabt oeuxtte1n0dkomvethritcekn,syoeft ckyilcolomneetserasndoramntiocryec,luonpetsostphreeawdiodvtherotfhothuesaoncdesan of kilometers. Similarly, oceInanthecucorrnesntrtusctaioren ogfeTnaebrlael4ly.1,cwonefiwneeredctaoretfhuletoupsaptiesrfyhtuhnedcrietedrima oeftegresophysical fluid bWaseino.uIgthfot ltloowexsatmhai2ntefothrrlea2ergcea-ssecsa:leWm/oHtioisnsmuch less than, on the order of, or much greater of the waUter coUlumn bUut edxytneanWmdicUosvoeurtltienneds ionfSkeicltoiomnset1e.5rPsaondr m1.6oA,rUe, up tAoUthe wiνdEthUof the ocean than U/L. Th,e third,case m, ust be r,uleΩdWou,t. IΩnUde,ed, if W,/H "2 ,U/L2, t,he equ2at.ion reduces T L L H ρ0L L L H in bfiarssitna.pIpt rfooxlliomwastitohnattofo∂r wla/rg∂ez-s=ca0le, wmhoitciohnismplies that w is consta1nt in the vertical; because T ! , (4.11) of aTbhoettpormevisooums erewmhaerrke,imthmatedflioatwelymsuhsotwbsetshuapt pthlieedfifbthy tleartmera(Ωl cWon)viΩesraglewnacyes(mseuechlastemraslleecrtion 4.6.t1h)a,natnhde wsixethde(ΩduUc)eatnhdatcfoathnr etbhetetsriammfeeslsy∂canule/ga∂lnexdctaendd1./or ∂v/∂y may not be both neglected at the same tiBmeec.auInsesuomf t,hwe fmunudsatmbeenmtaul cimh psmoratallnecrethofanthienirtoiatalltyionthtoeurmghst.in geophysical fluid dy- U nInamthices,fiwrsetccaansaen, ttihciepalteeatdhiantgthbeaplarenscseuries-gtwraod-iednimt ternmsio(tnhaeld, r∂ivu"in/g∂Ωxf,or+ce∂) wv/il∂l syca=le a0s, twhehich (4.12) impClioersiothliasttecromnsv,ei.reg.e, nce in one horizontal direction must beLcompensated by divergence in the other horizontal directfiornt.heTvheilsociistyvaenrdylpenogstshibscleal.esT. hIteisingteenremraelldyiantoet rceaqsueir,ewd tiothdiWscr/imHinaotne between the two hPorizontal directions, and we assign the same length scale L to both coordinates and the the order of U/L, implies a three-=waΩyUbala→nce, wPhi=chρi0sΩaLlsUo.acceptable. In summ(4a.1ry6,) the samρe0vLelocity scale U to both velocity components. The same, however, cannot be said of the vertical-velocity scale musvtebrteiccalodnisretrcatiionne.dGbeoyphysical flows are typically confined to domains that are much wider 1 Note, however, that near ththeaEnqtuhaetoyr,arwehtehreicfk,gaoneds tohezearsopwechtilreafti∗orHea/chLesisitsmmaalxl.imTuhme,atthme osismphpleifiriccatliaoyner that determines may be invalidated. If this is the case, a re-examination of the scales is warranted. The fifth term is likely to remain our weather is only aboutH10 km thick, yet cyclones and anticyclones spread over thousands much smaller than some other terms, such as theWpres!sure gradUient, but there may be instances when the f∗ te(r4m.14) must be retained. Because suchoaf skitiuloatmioenteisrse.xcSeipmtiiolnaarll,yw, oeLcweiallndcisuprernesnetswaitrhethgeenfe∗rtaelrlmy choernefi. ned to the upper hundred meters and, by virtue of (4.13), of the water column but extend over tens of kilometers or more, up to the width of the ocean basin. It follows that for large-scale motions W ! U. (4.15) In other words, large-scale geophysical flows are shallow (H ! L) and nearly two-dimensional (W ! U). Let us now consider the x–momentum equation in its Boussinesq and turbulence-averaged form (4.7a). Its various terms scale sequentially as

2 2 U U U W U P AU AU νEU , , , , ΩW , ΩU , , 2 , 2 , 2 . T L L H ρ0L L L H The previous remark immediately shows that the fifth term (ΩW ) is always much smaller than the sixth (ΩU) and can be safely neglected1. Because of the fundamental importance of the rotation terms in geophysical fluid dy- namics, we can anticipate that the pressure-gradient term (the driving force) will scale as the Coriolis terms, i.e.,

P = ΩU → P = ρ0ΩLU. (4.16) ρ0L

1Note, however, that near the Equator, where f goes to zero while f∗ reaches its maximum, the simplification may be invalidated. If this is the case, a re-examination of the scales is warranted. The fifth term is likely to remain much smaller than some other terms, such as the pressure gradient, but there may be instances when the f∗ term must be retained. Because such a situation is exceptional, we will dispense with the f∗ term here. 4.3. SCALES 97

For typical geophysical flows, this dynamic pressure is much smaller than the basic hydro- static pressure due to the weight of the fluid. Although horizontal and vertical dissipation due to turbulent and subgrid-scale processes is retained in the equation (its last three terms), it cannot dominate the Coriolis force in geophysical flows, which ought to remain among the dominant terms. This implies AU ν U and E ! ΩU. (4.17) L2 H2 Similar considerations apply to the y–momentum equation (4.7b). But, the vertical mo- mentum4.3e.qSuCatAioLnES(4.7c) may be subjected to additional simplifications. Its variou9s7 terms scale sequentially as For typical geophysical flows, this dynamic pressure is much smaller than the basic hydro- static pressure due to the weig2ht of the fluid. W AlUthWough hoUrizWontal anWd vertical dissipatioPn due togtu∆rbρulent aAndWsubgridA-sWcale proνcEesWses , , , , ΩU , , , 2 , 2 , 2 . Tis retainLed in theLequationH(its last three teρrm0Hs), it canρn0ot domiLnate the CLoriolis forHce in 4.3. 4S.C3.ASL4.CE3AS. LSECSA4L.3E. SSCAL4.E3S. SCgAeLopEhSysical flows, which ought to remain among the 9d7omi9n7ant 9te7rms. Th9i7s implies97 The first term (W/T ) cannot exceed ΩW , which is itself much less than ΩU, by virtue of 94 CHAPUTER 4. νEEQUUATIONS (F4o.r1t1y)piacnaldg(e4o.p1h5y)s.icaTlhfleownse,xtthitshdreyenatmericmpsreaasnrsdeuraelissommu!cuhchsΩmUsam.llearlltehranththaenbΩasUic,htyhdirso-tim(4e.1b7)ecause For tFypoirctayFlpogirceaotylppghieycFosaoiplcrhgaytleysoflpipcoihacwlyaslfls,igoctewhaolisspfl,hdotyhywsinssiac,dmatylhincflisaopmdwryeiscns,saptumhrreieisscsidsupyrmneasuismscuhimrcesupmicsrhaemlslsseumurcraLhtehl2lisaesmnrmtathhlulaeecnrhbttahshsmHeaicnab2lhatlhsyeierdcrtbohha-aysndicrtohh-eydbraos-ic hydro- ostfat(i4c.p1r2e)s,sAnu(r4 eexample.1d4ue) taontdh efor(4w. e1theig5h).t verticaloTfhthues,flu tcomponenthide. first fou roft emomentumrms may all be neglected compared to the staticstparteicssptuarreteiscdsupureestsdtosauuteihrceetopdwruteehesiesgtouwhrttehoidegfuhwtheteeotiofgflhtStuhhtieemdo.flfwilutaehiridegc.hflotunoisdfid.thereafltiuoinds. apply to the y–momentum equation (4.7b). But, the vertical mo- fifthA. ltBhouut,ghthhiosrfiizfotnhtatlearnmd viserittiscealfdiqsusiiptaetisomn adlule. tIotsturrabtuiloenttoanthdesufibrgsrtidte-srcmaleopnrothceessreigs ht-hand side AlthoAulgthhohAuogltrhhizohouongrtihaAzlohlatnonhtrdoaizluvoagennhrtdtahiclvoaaerlnrimztddioicesvnansetlitarpdutlaiimacstniasoildepnqdavuidtesiausrottieinipoctadnoatulito(edu4nirt.sbo7dsuciutp)lueearmntboituoaatlynnuendrbbtuesuaeulnsebtudongbtrstjiuaedrnbc-btdgseucrdsliaeudlnteb-ostgpcararainoddldecd-esipstcsurisaoobelncgesearpslidrssoe-iscsmceapslsleiefispcraotcioesnsse.s Its various terms is retiasinretdiasiinnreedthaieneeitsdqhuerieanteaiqtoihunneaet(dieiissotqsinunreal(atihaistotsienn9tle4eha(dqrs∂ietutsiweantthltiateorhserntemet(sehtie)tqr∂s,reumweialtasttis)ceot,arnnmtihtn(srioce)t∂as,tewndilntaoeoscrmttmaintdnshno)ar,omet∂eteiitwndtthcaoeeatrmenmCintnsho)ae,trtieodiCtlotiohcmsreaiifnonoCnlariosctetreifodotihlonriecmseCifnoionarrctieeoltihCsneHfoCArocPreTioiElniRs f4o.rcEeQinUATIONS geophyssiccaalleflsoew+qus,eunwthiailclhy+oausvght to r+emwain am−ongf∗thue d=ominant terms. This implies geopgheyosipchgaylesoflipcohawlysfls,iogwcweaholsipfl,chwohywhossiuicc,ghawhlothflutioocgwhhresto,mtuowgar∂hientitmcsmalltahomaiornoeunmaggmha∂tiohnxtnosmalleagdmretohommenaigdin∂notahymnasmallemtintdeoaornnmmgtisttne.h∂arTesmallmznhdtsiot.semTrimihnispas.lnismalliTemtshtepisrlmiiemss.pTlihesis implies 1 ∂p gρ ∂ ∂w ∂ ∂w ∂ ρ0ΩH∂Uw H AU 2 νEU ∼ , − − − + AU AAUWAνUEU+UνWEAUUνEAUUW νEW+Uand νE ! PΩ,U. (g4∆.7ρc) AW AW (4ν.1E7W) ρ0 ∂z ρ0 ∂x an∂dxand an∂!dy Ω!Ua.n∂ΩdyU!. 2ΩU. ∂!z ΩUP2. ∂z (4L.17()4.17()4.17) (4.17) 2 2 ,2 2 2,2 2 L, 2 , ΩHU , Scales, of terms, 2 , 2 , 2 . L !L T L"H LHL !H L "HH ∂!w ρ"0∂Hw ρ0∂w L ∂w L H whiScihm,ialacrccoorndsiindgeratotio(n4s.1a6pp)layntdo t(h4e.1y3–)miosmmenu+tcuhmuleqsusatth+ioannv(o4.n7eb.)+. Bwut, the−verfti∗cual =mo- SiSminilScaiermcwiolSeanriswmicdioilelnlarsrawitcdioSooerinnrkmasstieiadloxapenrcprslalcuytoaisponitvopnseilsdtylhyaeetprowapytlti–yhtomhentosoayvma–thepmerepnaolytgymu–emtemdonoetuhqmmeuueaayntet–itiqouomunmnaos(tme4iioqn.e7nunbta(ht)tu4.ieom.7Bnrbeue)(s∂qt.4t,ut.Bot7ahfbutei)tot.,hvnteBh(rueb4t∂ito.cv,7xoaetblhkr)tme.i(cuBvoane-lurltmet∂,iscyotsha-lemveor-tic∂azl mo- menFtuimnaellqyu,atthioenla(4s.t7tch)rmeeaytebremssubajreectsemd atoll.adWdihtieonnaWl simispsliufibcsattiotuntse. dItfsorvaUrioiuns(t4er.1m7s), we have moetnhtmeurmewnietsumqemuesanpteteiquocumnifia(tmeei4doqe.n)7un,catt()uth4imome.7nracee)y(qi4smub.7naeatcoyi)solunobmbTne(ajghe4ysce.u7trbbeficaedj)rensscmyttuotetnabedayejredemtcbddoteie(taWidosodundtd/boaeiTjtlneia)oocsdnittcdmeeaaidlntapinvsoltioioenfimrtacapleadglxtsdieicifioidmetcnieqaopsdut.nliiaoafiΩInlnctWtssai.itmvi,eoaIwspntrsilswhoi.fiviuiccatsIhartistobiiesourvnmasiatcsrsst.kieeoelruIftmtsssm.stveuarcmrhiosluesstethrmans ΩU, by virtue of scale seq(4u.e1n1t)iaalnl1yda(s∂4p.15). Thegnρext thr∂ee term∂sware also m∂uch sm∂waller than∂ΩU, thi∂swtime because scCaloensscseaeqlqeuusesecnenaqttliuleayel,slnye"tquiaasu#sclelahynlateaisassbleleqyeuanesnretipalalyceads by−u and sim−ilar−ly for al+l other vAariables. + A + νE , (4.7c) ρ0 ∂z ρA0 W ∂x ∂νxEW ∂y ∂y ∂z ∂z of (4.12), (4.14) and (4.15). Thus,!the first"four terms!may al"lsmallbe neglec!ted comp"ared to the In the energy (density) equation, heat and salt molecular2diffuasniodn neesmalld2s lik!ewiΩseWto b"e ΩU. small (4.18) fifth. But, this fifth term2isLitself quite smHall. Its ratio to the first term on the right-hand side superseded by the dispersing 2efWfec2t oSfUiun2WncerewsoelUvwe2WdilltuwroburWkleenxtcmluostiivoenlsy awnPidthsuabvegrrgai∆dg-eρsdcaelqeuAparWtoio-ns inAtWhe restνofWthe book (unless W WUWWUWUUWWWUWWUUWWWis, UWW ,P W P g,∆Pρg∆ρA,gWP∆ΩρAUW, AgAW∆WρAW, νAAEWWνEW, AνEWW , νEW , E . cesses.,Usin,g th,e, sam, e h,o,,rTizh,ounst,alto,,heΩtdeh,Udelyraw,ΩsvitiUs,se,tch,osΩrpseieUtecy,it,fiAe,erdfm,o)Ω,rsUteho,ne,n,erertghiy,se2ansroi,fgl,oo2hrntmg-,he2ora2man,ned,,ny2tsnuied,mee22d2isotog.f,,e2dtnheeenr.oa2elt22leqyu,a.vaetiroagn2e2adr.qeumanutict2iheslewsisththbraancktehtes.fifth T T L T L LLT L HLLTH HLL ρ0HHρL0H ρ0HHρ0 Lρρ0H0 L LLρ0HL HLLρ0 H LHL HL H adequate, because the largeterrtmurbouCnleontnthsmeeqoluteiefontn,tslywa,nh"duic#shuhbawgsrabidseepanrlorrceepasldsayecsefdaocbuρty0ntoΩuddHatinsoUpdebsreismevHihleaerrayltyasfnmodraallll.otIhnersvuamrimabalersy., only two terms The first term (W/T ) cannot exceed ΩW , which is its∼elf mu,ch less than ΩU, by virtue of ThsaeltfiTarhsset etfiTefrfrhsemtecttfii(evWrremslt/yTTt(ehWa)resmc/fimaT(rnoWs)ntmoc/trateTenrnemn)xtuoccamt(eaiWeenn.dxn,/IcoΩaTnetneW)Itednhxce,caΩthenwvWeeneehdoreiv,tcΩniewhceeWraxrhitlgcsi,ec,yihehawt(osdldiehw-sΩlemifceniWthvmosseiemitu,rlsy,fcwe)ihtmthnheseleqtiueculucspfhmhsarmtaitilshcobeutaisnacitsncs,hleathΩhlnefiaUscsamnPsteu,uΩatsbrhncueyUahdadnlv,ulseiyΩbarcsytlUestutLosevtm,hidtobraoitfynsuletteScalesivhΩcnioue-rUfltaus,riebmd oyofipffvfl uietermsrsthiuoeyndonrfoeesdtastliickebwailsaentcoebe (4g.1u1i(s)4ha.1dn1ids)(p4(a4e.n1r.1dsi5)o()a4n.n.(1oTd45fh.)(1ee.41n.n)1Tee5rhaxg)ne.tydn(tTf4he(rh4.rxo1ee.tm1etn5)htt)eearh.sxernuamteTdtptshteoh(eer4frarse.mrmen1eede5sotaxe)ema.ldtrsrmoteTbhnshyramteeluasetumrhonceteehembaxHydrostaticrdlysmtusimsotcishpnharmeatlerlsreoeuesmridcntaahuetlglhrsclsmaeoiemnr fsmBalancegftaΩehaluaclraUecetnvrh,oeaΩttfrlshhstmUaiuoicsnna,amlrtlΩtliehemuesUircdosehtd,lhtvbyitsamehemnddicesiaΩftufltbuuliUsermsebrci,euavttuhilhbtesayienesncttaΩmiumUsoet,ibothenicssaautinsmedesubbecgaruids-escale pro- of (4.12w),h(i4c.h1,4a)cacnodrd(i4n.g15to).(T4.h1u6s),athned fi(4r.s1t3f)ouisrmteurmchslmesasythaallnboenne.eglected compared to the ofκ(E4o.1tfh2a()4t,.o(d14fi2f.)(1f,4e4.(r)14s2a.f1n)ro,4dof)(m4((a4.4n1.t.1hd415e)2())4av.,n.e1T(dr45ht.i)(u1c.4s4a.T,1)lht5eahc)udne.esds,Tfiys(trFehh4svsuie.tn.i1ssfafi,5Ucolr)tolushs.ysirte,nTifttgfiehoyrruetmsνhsrtlE,eastfets.osmhtraumeTtamrhhyfisrteiermsaserhltmdalotfyeibrsofriefuazmmelornrlsaneetbyeangtearrclameleenlcelsetsdmbgetmdedlaeymalnclyv.etoseigamWsfdl1rclepohcocbamoestree∂nmietdntdypWphecatAegorleimsestfdphcopsetetruaecgorebdiρefintshdctceioerttmugotypethdaersfeofdorrtUomtihonem(4e.n1t7u)m, wise gheanverally fifttuhr.bfiBufltuhetn.,fitBthfbutiehsth,.fiatBhfvtiuihsotfi,rfitfettfohrhtmfih.seBtfiieasufrctmithht,sfitseithfsetltaifrhsimtt.qesfiueBifvilstautfaheditrqt,ietsauetmqehbrilutmilfaesealqtlsfiia.eusmfn,iItitabdthtesslelewtsrc.elamafriItlumtiaqlsoslubelrit.iaseottteIhiiftttoesuhsmeretrltoalahfafirtlteigqrlhorse.uetrtdiIotfittiesusrtrcr0hsrmbutameutstoia=fieslonelrerlmnsdt.totthIi−otmtetneshnrroeSrmitatghefiitohceoirotnsn-ritsohittgotanhehnetrt1hdmd-r4eh−iss.gaoifi3udnhn.rbedts-gtTthsrthiaieeddnerre.dmpigrshooidtcn-ehtsahsneedsrisagicdhtett-ohadnidspseirdsee heat an(d4.19) is ρ0 ∂z ρ0 iseneirsgy (disensity) iesquation then becsoalmt eass:effectively as momenAtuWm. In theνEveWrtical, however, the practice is usually to distin- and ! ΩW " ΩU. (4.18) In gthueishadbissepnecrseioonfofsternaetrigfiycafrtLoiom2nth(adteonfsmitHyom2peenrttuumrbbaytiionntroρdunciiln)g, athveerntiecxaltetdedrymdiifnfusliivnitey that ρ0ΩHρ0UΩHρU0HΩHUHρ0ΩHHU ρ0HΩHU H shouldκTbEheutshc,aottnhdesiifldaf∼estrrsethdf∼re,oaemste∼atrh,mepsovose,srntiibtc∼hlaeel rebidgad,hlyta-nhv∼acinsecdotsosiid,ttyehoeνfEpt.hreTshesqiusurdaetiifogfenrraeadrneiceemnsuttce(hm1l/seρsfsr0o)tmh(a∂ntphte/h∂espzfie)fcthiifiscf∗u. ∂ρ P ∂Pρ LP ∂Lρ PL ∂ρ LP L Howevtteuerrr,bmuu+lnoendutetrhbeshula+ecvfhti,ovbrwaohlfiacen+haccwehw,astthaetlerev=vaedarrytiiacfboalulenvadantrdoiawbtielolvnbeeroyffustmrhtheaellpr.rdeIinscssuuusrmseemdpaiwrny,oSuoencldtliyobntew1og4i.tv3e.remnTshbey the whicwh,haiccchwo,hradiccicnhog,radtocwinc(hog4ir.ctd1ohi6n,()4gaac.t1ncow6do)(hr(4dai4.cin1.nh1d6g,3)(a∂)t4aocti.ncs1(do43mr.)(d14uii6.scn1)hgm3∂al)txnueoisdcss(h(4mt4l.he1.ua1s6cns3)h∂)toayhlinesaesdnms.(ot4uhn.ca1ehn3.∂l)oezisnssem.thuacnholneess. than one. verticaelrneinmertageiygnr,(adatenindosntihtyoe)fveeρqr0tuifac∗tailuo-mnatonhmdeneintbstuescmcoamblaeelsab:neceρr0eΩduHceUs t.oSthinecseimthplies hisydmrousctahtilcebssaltahnacne the already FinalFlyin, athllFeyi,lnatashltelythl,ar∂tseFhteietnhtlaearlreslmyet∂,tsthρteharreFremeeilnsastaemsalrtrlmaeythl,∂lsr.tmehaWereaelthlleas.ermsnmW∂tatWρshhllaer.erniWeesWstshemuerbinmass∂lWtlssi.utauWbritessehtdsisetmunuf∂botaWeρsrldtlUi.tfiuWositnreshduU(e4bnf.soi1nWtri7t(Uu)4,it.sewi1nds7eu)(fh,4bowa.sr1tvUie7etu)h,itanewvd(e4fh.o1ar7vU)e, wine(h4a.1v7e), we have Aestabli+shed preAssure s+cale (4.1κ6E), it is, negligible, and w(4e.8c)onclude that the vertical variation ∂x ∂x ∂y ∂y A∂zW ∂z ν W 1 ∂p gρ ! AW"AW AWνE!WνEAWW"νEW ν!EW "0E = − − . (4.19) of p is very weak. In other woradnsd, p ∂isρnea!rlyρ∂Ω0zρW–∂izn"deΩ∂pρeUρn0.dent ∂inρ the absence(4o.f18s)tratification: 2 a2nd a2nd 2and!22 Ω!Wa2nΩd"WL!2Ω"ΩU2W. Ω"!U. ΩΩHWU2. +" uΩU. +(4v.18()4.+18()w4.18) =(4.18) The linear continuity equatiLon isLnot sLubHjecteHdLto aHny such aHdaptation and remains unchanged: In the absence of stratification∂t(density∂xperturb∂atyion ρ ni∂l)z, the next term in line that ThusT, hthues,Tlathhsutestlh,artsehteeTthtlhearuresmste,tsthtehorremneTlstathhseoeutrnmstrh,itsghrtehheoeetn-rthiletgaahrshnmettd-trshhisgaoriendhnedet-tthoseiaefrdnmtredhisgeoshofiedtntq-ehhuteahoanetfeidqotrhuinsgeaidhateiertoq-ehnoumafaantturidheocehnsmiedlauqeercsueshoamtftliheotuahsncnseha1tterhlhqeaesunm∂safitputfihhtocheahnnfilafetrhtsehesmtfihufatchnhtlheessfitfhthan the fifth should be considere∂d as a ∂poρssible0ba∂=lance−t∂oρthe pres.s∂ure grad∂ieρnt (1/ρ0)(∂p/∂z) is f∗u.(4.20) termtoernmthtoenrmlethftoe,nlwetfhetrie,cmwhleohfwtni,cahstwhtawhelriracmleeshaftadow,lynr∂waeusftahodaieucylnrhledefaofw∂tdu,oavynswdbfaheotliuorcvenhe∂abdrdewyytovassfemborayeuallnrlvsedA.meartdIayonylls.bsfmueoImuanvln+melsd.ruaytmIronysm,mbsoaueanrmlylAvl.y,meroItaynwrlysoys,mutotmaewn+lrmlmlo.yastIrtenwyr,mosoustnκmelErymmtaswryo, to,enrmlystwo terms (4.8) How+ever, un+der su∂chx=ba0la. n∂cxe, the ver∂tiycal va∂riyaρt0ion∂ozf (∂t4hze.9p)ressu∂rze p would be given by the remarienm, aanirnde,mtahaneidnv,teharretneidcmvaetalhr-itmenic,voaaemln-rrmtdeeincmotahtmuale-imemnvn,∂ebotaruxamtnmildcaeannbtlcht-auemelm∂arvoneyembdcraeuteilcrncaeantsdulc∂-umtemozcreboteshamdeltuaoesc!nnitecmthsueepmtrosleeibdtmhhau"eplycadlesnersicmohtesoyptrdalteerhtdoiechus!ystcbadieamtrsilocaptsonlbteacat"hethliaeycndsbcriamoelsaptanletciech!ybdarloasntace"tic balance So, thevheyrtdicraolsitnatetigcrabtiaolnanofcρe0(f4∗.u19an)dciotsnstcinalueebsetρo0ΩhoHldUi.nSitnhceeltihmisiitsρmu→ch0le.ss than the already For more details on eddy viscToehstieatyblilnaisnehaderddciopfnfruetisnsivuuiirtteyyseacqnaudleast(oi4om.n1e6is)s,nciohttei1msuneb∂esjgpetlcoitgemidbaltgekoρ,eaantnhydosswuecedhceao-dnacplutadteiothnaatntdherevmeratiincasluvnacrhiaatniogned: Since the1pr∂eps1su∂rep1gpρ∂ipsgρal01reg=a∂ρdpy−a sgmρall p−erturb. ation to a much larger (p4r.1e9s)sure, itself in pend on flow properties, the rea0de=orf0isp−r=iesfv0ee−rr=eydwt−eo−a0tke.x=It−nboo.−othke−sr.ownotrud.rsb,−uplρei0sncn∂ee.,zasrluychz–ρai0sn(dT4ee.p1ne9n(n)e4dk.e1en9st()4in.1t9h)e ab(s4e.1n9c)e of stratification: hydrostatic bρa0la∂nρzc0e,∂wzρ0ρe0c∂ozρn0cρlu0 dρ∂0eztha∂tugρ0eop∂hvysica∂l wflows tend to be fully hydrostatic even in and Lumley (1972) or Pope (2I0n00th).eAabwseidnecleyoufsesdtramtiefitchaotdiotno(idnecnosriptyorpa+teertsuurbgatr+iiodn-sρcanlei=lp),ro0th-.e next term in line that (4.9) ceIsnsetshIienattbhhseeIenhnaocbthresieezonoafcnbeIstsnaterolantfehtcidesefidtcraoyaabfttvisifioseistcnrcacaot(etisdiofiieotncnyfas(tisidisttoyertannhtpsai(ifietdtrycpetaunrptorsibeopitranoyttusi(orepdbndeearnbtρiusyoirnntbSyialmρ)tp,iaoentgnrhitoleu)ρr,rinbn∂tneahsxixtkeli)tyo,nnt(ete1hrxρ9met∂6nynt4eieln)r)x:m,tl1itnthiene∂er∂mztlpnhineaixentttlhtienartemthinatline that should be considered as a possible balance t0o t=he p−ressure gr.adient (1/ρ0)(∂p/∂z) is f∗u. (4.20) shoulsdhobuelsdchoboneuslicddoebnressehidcdooeaunrsledsdaidbapeesorecsadosinpbasosliesdasebibrpaeloledasnsbaciasbellaetnopbcotaehslaetsonibpctlreheeestbospaurltraheensesgcuperraetdosigseturnhartede(ip1egrn/reatρsd0(si1u)e(/rn∂eρtp0g(/)1r(∂a/∂dzρpiρ)0e/0)ni∂(st∂z∂f(p)z1∗//ius∂ρ.z0f)∗(ui∂s.pf/∗∂uz.) is f∗u. However, uFnodrermsourcehdbeatlaainlsceo,ntheeddveyrtvicisaclovsaitryiatainodn doifffthuesivpriteyssaunrde psowmoeusldchbeemgeivsetno bmyatkhee those de- HowHevoewr,eHuvnoedrw,eeurvnsedurHec,rhuosnwbudaecelvharensbrcu,aecul,ahnntdhbceaerl,vastenuhrcceteihcv,abetlhr2atveliaacvnraeiclarevtt,iiaoctrhanialeotvivfoaetrn2rhitaeoitcfipaotrlnhevesoasfpruirtraehetseispoupnwreoosfpustulhwdreeobpupe2rldegwsibovsueuernlgedibvpbyeewntghobieuvyledtnhbebeygtihveen by the pS∂eoun,dthoenhflyodwrosp∂travotipc∗ebrtaileasn,c1tehe(4r.e∂1a9ud)ecroin∂stvirneufesrrteodhtooldteixntbthoeoklismoitnρtu→rbu0l.ence, such as Tennekes verticvaelritnictvaeelgrirtnaictteiaoglnrianvotteifeorgρtnir0caofatf∗liouρinn0atvfoene∗gfdrurρtai0acttsfinao∗sldncuiniaotatlsfensgρdbcr0eaaiftltρs∗eio0usbnΩceaaoHnρlfed0UρbΩi0te.sHfSρsiUc0unaΩc.aleHnSdbtiUhneiict.sρseSi0stsΩichnmaiHcslueiUcsbthme.ilsSρueis0cnssΩhcmtHelheuatsUhcnshi.sthlSieaesisnnamsclttrhueheecaathnadillystreheiseassdamtyhlruaecnahdthlyeesasltrheaandythe already A = ∆x ∆y and SLiunmc+eletyhe(1p9r7es2s)uorer+Ppoipsea(l2re0a0d0y+).aAsmwaildle,plyerutusrebdamti(oe4nt.h1to0od)atomiuncchorlparogreartepsruesbsgurried,-sitcsaellef pinro- estabelissthabedleisspthareebdslisspuhrreesdstaucpbareliessssh(cu4ear.dlee1es6pst(a)cr#4,eba.sil1ties6suih)(s∂r,4eenx.id1tes6cpgisa)rl,ielngesieitsb(gui4lslre.ie1g,n6iaesb)∂cgn,laldeyil,tgewaiib(sne4ldne.c1e,ow6gane)lnc,i2dglciuitowbdnilesece∂l,ntcuhyaeodagnnetldcittglhwhui∂eabdextlveecteh,torheatnaincvtdlaetulhrwdtveieeacvratchielaoravttntiiaoctrlhnaiualedtviveoaerntrhitaaitctiaotlhnveavriearttiiocnal variation of p is!vcheeyrsyds"rewosseitanaktti.hcIenb!haolotahrnie"zcroew,nwtoareldecsd,odpny!cilsvunidseecaotrhlsyaittyzg–ies"inotdphehapytespnircdoaeplnofltsoienwdsthbteyenaSdbmsteoangbcoeriofnufslsklytyrah(t1yifi9dcr6oa4ts)ito:anti:c even in ofinp wiosfhvipecrhiyos∆fvwpexeriaysakwnv.deeIorna∆fykopw.ythiIesanearrvkoeew.ttrhIhoyenerrwdowlseot,haocpekrard.liswIg,nnropieordaditsrhsdl,enyimrpezawie–rsnoliynsrdiedzoase–n,rpilspney.ndizBsde–peneinecenntaddiurenlesypentehztnse–iudniacenbthndhsteseipunaebcbtneghsdereoeindfanc-bstestsircnoeaanfttlichesfietecpraoaattfbriiafisosemtcnrna:aectttiieeofirnoci:f-atsitorant:ification: zation is meant to represent physical processes, it ought to obey certain sy21mm∂eptry prop2erties, 2 1 ∂p1 ∂p1 ∂p 01 =∂p∂−u . ∂v 1 ∂u ∂v (4.20) notably invariance with respect to 0rot=at0io−n=o0f−th=eAc−o.=o0rd∆=i.nxat∆−e .ysystem in. tρh0e h+∂ozrizon(t4a.l2p0(l)4a+.n2e0.()4.20) (+4.20) , (4.10) ρ0 ∂ρz0 ∂zρ0 ∂z ρ0 ∂z∂x ∂y 2 ∂y ∂x An appropriate formulation for A ought therefore to be express#ed!sole"ly in ter!ms of"the ro- ! " So, thSeo,htyhSderooh,sytthdaertiochsSytbadoatr,lioacthnsbtecaaethlia(ycn4Sdbc.ro1aeo,9ls(a)t4thnac.ect1oie9chn)(ytb4idcna.or1luoan9enst)sticanctetouoi(cenh4stbo.i1tnalod9lua)heinnscocolttedohn(etih4innlo.i1utlmhd9ee)isitncltρioomthn→hietotinlρ0dium.→einisttt0ρoh.e→hloilm0d.iitnρth→e l0im. it ρ → 0. tational invariants of the tensor foirnmwedhibcyh t∆hexvaenldoc∆ity daererivthaetivloecsa. lOgnriedsdhiomueldnsniotnes.thBaet cthauese such subgrid-scale parameteri- SinceSitnhceeSptirhneecssepurtrheeseSspiupnricseesapslturhierseeaaSdpplyirrneeiacsasedasuymltrrheeaealplsdpmyiprseeaasralstllsurmperebaaradlpttlyuioripasbnearasttmloiuroeraanballadmttoypiuoeacnrthmsutmorluabacaralghtlmieoplraunerpcrgtrthouersrlaabspuramgrrteieuos,rscniuphttsroela,lsafrsigutimsenrereul,fpcirhitensselaslufrgrien,r iptsrelsfsuinre, itself in preceding formulation unfortunatzealytiodnoeis mnoetasnatttiosfryepthreissecnotnpdhityisoinc.alNporonceethsseeless,sitiot uisghotftteonobey certain symmetry properties, hydrohsytdartiochsytbadatrlioacnsbtcaaethli,aycnwdbceraeol,csaowtnancetciechlc,uyobwddnaeercloaltcunshotdcaanettci,gtclwheubaodeatpelcghatoeynhnoscacipetclh,guaywledsoefleipcotchawolyasntflscitgocelweuanolddspefltthoeotyhwnsbadsitecttgaofeleunboflldlepoythwfohuysylsblidteyceranofhludsyflltdlatoyortwiochbssyeteadtvfetreiunocnldsleytiavtnotheinycbdeeirnvofeusnltlayitnichyedvreonstiantic even in used in numerical models. notably invariance with respect to rotation of the coordinate system in the horizontal plane. An appropriate formulation for A ought therefore to be expressed solely in terms of the ro- tational invariants of the tensor formed by the velocity derivatives. One should note that the preceding formulation unfortunately does not satisfy this condition. Nonetheless it is often used in numerical models. 98 CHAPTER 4. EQUATIONS the presence of substantial motions2. Looking back, we note that the main reason behind this reduction is the strong geometric disparity of geophysical flows (H ! L). In rare instances when this disparity between horizontal and vertical scales does not exist, such as in convection plumes and short internal waves, the hydrostatic approximation ceases to hold and the vertical-momentum balance includes a three-way balance between vertical acceleration, pressure gradient and buoyancy.

4.4 Recapitulation of equations governing geophysical flows

The Boussinesq approximation performed in the previous chapter and the preceding devel- opments have greatly simplified the equations. We recapitulate them here. Primitive Equations describing motions of geophysical "ows

∂u ∂u ∂u ∂u x − momentum: + u + v + w − fv = ∂t ∂x ∂y ∂z 1 ∂p ∂ ∂u ∂ ∂u ∂ ∂u − + A + A + νE (4.21a) ρ0 ∂x ∂x ∂x ∂y ∂y ∂z ∂z ! " ! " ! " ∂v ∂v ∂v ∂v y − momentum: + u + v + w + fu = ∂t ∂x ∂y ∂z 1 ∂p ∂ ∂v ∂ ∂v ∂ ∂v − + A + A + νE (4.21b) ρ0 ∂y ∂x ∂x ∂y ∂y ∂z ∂z ! " ! " ! " ∂p z − momentum: 0 = − − ρg (4.21c) ∂z ∂u ∂v ∂w continuity: + + = 0 (4.21d) ∂x ∂y ∂z ∂ρ ∂ρ ∂ρ ∂ρ energy: + u + v + w = ∂t ∂x ∂y ∂z ∂ ∂ρ ∂ ∂ρ ∂ ∂ρ A + A + κ , (4.21e) ∂x ∂x ∂y ∂y ∂z E ∂z ! " ! " ! " where the reference density ρ0 and the gravitational acceleration g are constant coefficients, the Coriolis parameter f = 2Ω sin ϕ is dependent on latitude or taken as a constant, and the eddy viscosity and diffusivity coefficients A, νE and κE may taken as constants or functions of flow variables and grid parameters. These five equations for the five variables u, v, w, p and ρ form a closed set of equations, the cornerstone of geophysical fluid dynamics, sometimes called primitive equations. Using the continuity equation (4.21d), the horizontal-momentum and density equations

2According to Nebeker (1995, page 51), the scientist deserving credit for the hydrostatic balance in geophysical flows is Alexis Clairaut (1713–1765).