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Turbulence Closure Models for Shallow Water Equations

This article explores the compromise between complexity and pragmatism in turbulence closure models for the shallow water equations, starting from the Navier-Stokes equations and Reynolds decomposition.

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Turbulence Closure Models for Shallow Water Equations

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  1. From the Navier-Stokes equations via the Reynolds decomposition to a working turbulence closure model for the shallow water equations: The compromise between complexity and pragmatism. Hans Burchard Leibniz Institute for Baltic Sea Research Warnemünde hans.burchard@io-warnemuende.de

  2. Milk foam: light, becauseoffoamandfat Coffee: relatively light, becausehot Milk: less light, becausecolderthancoffee Whyarewestirringourcupofcoffee? Whythespoon? …OK, andwhythecoocky?

  3. Fromstirringtomixing … littlestirring strong stirring littlemixing strong mixing

  4. Tea mixing (analyticalsolution) Put 50% of milk intotea. z Letm(z)bethe milk fractionwithm=1 atthebottom andm=0 atthesurface. With a constantmixingcoefficient, the m-equationisthis: 10cm Letustakethespoon andstirthe milk-tea mix n-times such thatwe get a sinosodial milk-tea variation in thevertical andthenseethe resultingmixing after 1 min: Conclusion: stirringleadstoincreasedmixing.

  5. Set ofequationsthatdescribes turbulent mixing Navier-Stokes equations(forvelocityvectoru1, u2, u3): Incompressibilitycondition: pressure gradient tendency advection Earth rotation gravitational force stress divergence  6equationsfor 6 unknowns (u1, u2, u3, p, ,  ) Temperatureequation: Equationofstate: stirring mixing

  6. ExampleforsolutionofNavier-Stokes equations (KH-instability) DirectNumericalSimulation (DNS) by William D. Smyth, Oregon State University

  7. Reynolds decomposition Toreproduce system-widemixing, thesmallest dissipative scales must beresolvedbynumericalmodels (DNS). This does not work in modelsfornaturalwatersdue to limited capacitiesofcomputers. Therefore, theeffectsofturbulenceneedstobepartially (= Large Eddy Simulation, LES) orfully (Reynolds-averagedNavier-Stokes, RANS) parametersised. Here, wegoforthe RANS method, whichmeansthatsmall-scalefluctuationsare „averagedaway“, i.e., itisonlytheexpectedvalueofthestate variables consideredand not theactualvalue.

  8. Reynolds decomposition (withsynthetictidalflowdata) Any turbulent flowcanbedecomposed intomeanandfluctuatingcomponents:

  9. Reynolds decomposition Therearemanywaystodefinethemeanflow, e.g. time averaging (upperpanel) orensembleaveraging (lowerpanel). Fortheensembleaveraging, a high number N ofmacroscopicallyidenticalexperimentsiscarried out andthenthemeanofthoseresultsistaken. The limitfor N  isthentheensembleaverage (whichisthephysicallycorrectone). Time averaging Ensemble averaging

  10. Reynolds decomposition Fortheensembleaverage 4 basicrulesapply: Linearity Differentiation Double averaging Productaveraging

  11. The Reynolds equations These rulescanbeappliedtoderive a balanceequation fortheensembleaveragedmomentum. This isdemonstratedherefor a simplified (one-dimensional) momentumequation: The Reynolds stress constitutes a newunknown whichneedstobeparameterised.

  12. The eddyviscosityassumption Reynolds stress and meanshearare assumedto be proportional toeachothers: eddyviscosity

  13. The eddyviscosityassumption The eddyviscosityistypicallyordersofmagnitude larger thanthemolecularviscosity. The eddyviscosityishowever unknownaswellandhighly variable in time andspace.

  14. Parameterisationoftheeddyviscosity Like in thetheoryof ideal gases, theeddyviscositycanbeassumed tobe proportional to acharacteristiclengthscale l and a velocityscale v: In simple cases, thelengthscale l couldbetakenfromgeometricarguments (such asbeing proportional tothedistancefromthewall). The velocityscale v canbetakenas proportional tothesquarerootofthe turbulent kineticenergy (TKE) whichisdefinedas: such that (cl = const)

  15. Dynamic equationforthe TKE A dynamicequationforthe turbulent kineticenergy (TKE) canbederived: with P: shearproduction B: buoyancyproduction e: viscousdissipation

  16. Dynamic equationforthelengthscale (here: eeq.) A dynamicequationforthedissipation rate ofthe TKE) isconstructed: withtheadjustableempiricalparametersc1, c2, c3, se. Withthis, itcanbecalculatedwith simple stabilityfunctionscmandcm‘. All parameterscanbecalibratedtocharacteristicpropertiesoftheflow. Example on nextslide: howtocalibratec3.

  17. Layerswithhomogeneousstratificationandshear Forstationary & homogeneousstratifiedshearflow, Osborn (1980) proposedthefollowingrelation: whichisequivalentto (N isthebuoyancyfrequency), a relationwhichisintensivelyusedtoderivethe eddydiffusivityfrommicro-structureobservations. Forstationaryhomogeneousshearlayers, thek-emodelreducesto whichcanbecombinedto . Thus, after havingcalibratedc1andc2, c3adjuststheeffectofstratification on mixing. Umlauf (2009), Burchard andHetland (2010)

  18. Mixing = micro-structurevariancedecay Example: temperaturemixing Temperatureequation: Temperaturevarianceequation: Mixing

  19. Second-moment closures in a nutshell Insteadofdirectlyimposingtheeddyviscosityassumption Withonecould also derive a transportequationfor andthe turbulent heatflux (secondmoments). These second-moment equationswouldcontainunknownthirdmoments, forwhich also equationscould bederived, etc. The second-moments areclosedbyassuminglocal equilibrium (stationarity, homogeneity) forthesecondmoments. Together withfurtheremipiricalclosureassumptions, a closed linear systemofequations will thenbefoundforthesecondmoments. Interestingly, theresultmaybe formulationsasfollows: , wherenowcmandcm‘are functionsofandwiththeshearsquared, M2.

  20. Such two-equationsecond moment-closuresarenowtheworkhorses in coastaloceanmodelling (andshouldbeit in lakemodels) andhavebeen consistentlyimplemented in theone-dimensional General OceanTurbulence Model (GOTM) whichhasbeenreleased in 1999 by Hans Burchard and Karsten Bolding underthe Gnu Public Licence. Sincethen, ithadbeensteadily developedandisnowcoupledtomanyoceanmodels.

  21. GOTM application: Kato-Phillips experiment Stress-induced entrainment intolinearly stratified fluid Dm(t) Empirical G<0.2 G>0.2 Empirical:

  22. GOTM application: Baltic Seasurfacedynamics unstable Reissmann et al., 2009

  23. Take home: Due tostirring, turbulenceleadsto an increaseofeffective mixinganddissipationbyseveralordersofmagnitude. Forsimulatingnaturalsystems, the Reynolds decompositioninto mean (=expected) andfluctuatingpartsisnecessary. Higher statisticalmomentsareparameterisedbymeansof turbulenceclosuremodels. Algebraicsecond-moment closuresprovide a goodcompromise betweenefficiencyandaccuracy. Therefore such modelsare ideal forlakesandcoastalwaters. Question: Will webeabletoconstruct a robostandmore accurateclosuremodelwhichresolvesthesecondmoments (inclusionofbudgetequationsformomentumandheatflux)?

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