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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
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