An idealized model for dust-gas interaction in a rotating channel

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by  O. M. Umurhan

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An idealized model for dust-gas interaction in a rotating channel
⋆
O.M. Umurhan ,
DepartmentofAstronomy,CityCollegeofSanFrancisco,SanFrancisco,CA94112,USA
DepartmentofGeophysicsandPlanetarySciences,Tel–AvivUniversity,Israel
DepartmentofPhysics,Technion-IsraelInstituteofTechnology,32000Haifa,Israel
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5February2008
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a ABSTRACT
J
A2D model representing the dynamical interaction ofdustand gasin aplanetary channel is explored. Thetwo

1 componentsaretreatedasinterpenetrating fluidsinwhichthegasistreatedasaBoussinesqfluidwhilethedustis
1 treatedaspressureless.Theonlycouplingbetweenbothfluidstatesiskinematicdrag.Thechannelgasexperiencesa
temperaturegradientinthespanwisedirectionanditisadversetheconstantforceofgravity.Thelattereffectsonly

2 thegasandnotthedustcomponentwhichisconsideredtofreefloatinthefluid.Thechannelisalsoconsideredon
v anf-planesothatthebackgroundvorticitygradientcancauseanyemergingvortexstructuretodriftlikeaRossby
7 wave.Alineartheoryanalysis isexploredandanonlinearamplitude theoryisdeveloped fordisturbances ofthis
1 arrangement. Itisfoundthat the presence ofthe dustcanhelp generate andshape emerging convection patterns
2 anddynamicsinthegassolongasthestateofthegasexceedsasuitablydefinedRayleighnumberappropriatefor
1 describingdrageffects.Inthelinearstagethedustparticlescollectquicklyontositesinthegaswherethevorticityis
0 minimal,i.e.wherethedisturbancevorticityisanticylonicwhichisconsistentwithpreviousstudies.Thenonlinear
7 theoryshowsthat,inturn,thelocalenhancementofdustconcentrationinthegaseffectsthevigoroftheemerging
0 convectiverollbymodifyingtheeffectivelocalRayleighnumberofthefluid.Itisalsofoundthatwithoutthef-plane
/ approximationbuiltintothemodelthedynamicsthereisanalgebraicrunawaycausedbyunrestrainedgrowthinthe
h
dustconcentration.ThebackgroundvorticitygradientforcestheconvectiverolltodriftlikeaRossbywaveandthis
p
causesthedustconcentrationenhancementstonotrunaway.
-
o
1 INTRODUCTIONANDSUMMARYOFRESULTS removedsufficientlyoroneisinvestigatingthedynamicstakingplacenear
r
t thediscmidplane, where planetesimals arelikely tobestrongly concen-
s Therehasbeengrowinginterestinrecentyearsinthepossibilitythatpersis-
trated.
a tentvortexstructuresinprotoplanetarydiscsmaybethesiteswhereplan-
Asathoughtexperiment,supposethegasinamodelshearingenvi-
: etesimalsareformed(Barge&Sommeria,1995,Tangaetal.,1996,Bracco
v ronmentdevelops,throughsometypeofdynamicalinstability,aseriesof
etal,1999,Barranco&Marcus,2000,Barranco&Marcus,2006).These
i long-livedvortexstructures.Inadditiontoobviouslyeffectingthedusttra-
X investigators haveputforththescenarioinwhichadisc,ladenwithdust,
jectoryinthewayitisusuallyenvisioned,itseemsreasonabletosuppose
supportssometypeoflong-lastingvortex(drivenbysomeunspecifiedexci-
r thatthepresenceofdustwoulddynamicallyinfluence,becauseoftheirmu-
a tationmechanism,exceptforKlahr&Bodenheimer,2003andBarranco&
tualdynamiccoupling(e.g.kinematicdrag),themannerinwhichemerging
Marcus2006)thatmanagestoattractthedustthroughthecombinedaction
gasstructuresevolve.Thisallwouldbeplausibleunderconditionsinwhich
ofgas-particle dragandCoriolis forcing (forexample, Barranco &Mar-
thegasanddusthaveequaldynamicalinfluenceuponeachother.Youdin
cus,2000).Separately, itisalsowell-known that stronglyshearingflows
andGoodman(2005)haveconductedasimilarstudyconsideringthelinear
(Keplerian discsqualifyunderthisclassification) inrotatingframespref-
instability emergingfrominterpenetrating streamsofgasanddustwhose
erentially supportthepersistenceofanti-cyclonic structures whilealmost onlyinteractionisthroughkinematicdrag2.
entirely wiping outcyclonic vortices 1 (Bracco, etal.,1999). Inlight of
Presentedhereisasimpleandidealizedmodelforthewaythistypeof
this,oneoftheinterestingresultsoftheaforementioneddust-vortexinter-
interactiondevelopsinamodelconfinedenvironment.Thephysicalmodel
actionstudiesisthatanti-cyclonicvorticesalsohappentobethesitesonto
employedhereisaflowinarotatingchannel.Thematerialistreatedastwo
whichdiscdustcollectsmostrapidly(mostnotably,Braccoetal.,1999).
interpenetrating fluidsofuniformdensity,onerepresenting apressureless
Intheinvestigationsmentioned,thedynamicsofthesedust-discsce-
dust“fluid”andtheotheranincompressibleBoussinesqgas.Thegasther-
nariosaretreatedasone-way:thedustpassivelyrespondstothegasflow
modynamicsisgovernedbythermalconductionandwesupposethatthere
withoutanyback-reactionofthedustuponthegas.Giventhatthecurrent
isaconstantgradientofthegastemperaturefromonewalltotheother(in
hypothesis of protoplanetary discs is that their dust contributes no more
thespanwiseor“wall-normal” direction). Aspartoftheidealization that
thanafewpercentofthetotalmassdensityofthedisc(Hayashi,1983),it
goesintothemodelconsideredhere,thegascomponentaloneissubjectto
isreasonabletotreatthedustasacollection ofLagrangiantracerswhich
aconstantforceinthewall-normaldirectionandthattheBoussinesqbuoy-
respondstothevortexinducedgasflowandhavingnodynamicalinfluence
ancyeffects areassociated withthisforceintheusualway.Wesuppose,
onthegasitself.
further, that thegasanddustarebothsubject toanexternal forcewhich
However,itisinterestingandinstructivetoturnthequestionaround
givesrisetoalinearCouetteshearofbothfluidsinthedirectionparallelto
andexaminewhatwouldresultifthedustdoesdynamicallyeffectthegas.
thechannelwalls.Thecouplingbetweenthegasanddustisthroughasim-
Onecouldenvisionasituationinwhicheitherthegasinthedischasbeen
plekinematic (Darcy)dragprescription like,forexample,inTangaetal.
1 Acoherentvortexstructureissaidtobeanti-cyclonicifthesignatureof 2 Therehavebeenanumberofstudiesofthissortinwhichtheinterpene-
thebackgroundvortexstateinwhichitisembeddedisoppositethesignof tratingstreamsarecoupledtoeachotherthroughgravity.Foramorerecent
thevortexstructureitself. studyseeBertin&Cava(2006)andreferencestherein.
2 O.M. Umurhan
(1995).Thereisnoviscosity.Theingredientsofthishypotheticalscenario fu(y)xˆ
+ρ (2)
alreadypredisposesthegascomponentsusceptibletobuoyantinstabilityala (cid:26) Fk−fu(x)yˆ (cid:27)
RayleighBenard.Finally,becausethedustfluidistreatedasbeingpressure-
∂
less,noapriorirestrictionsuponthecompressibilityofthedustismade. ρCp +u T = K 2T. (3)
∂t ·∇ ∇
InSection2weshowthedevelopmentofRayleigh-Darcytypeofcon- (cid:16) (cid:17)
vection.Intheformulationoftheproblem,thedustdensitydecouplesfrom Inwhich u is the gas velocity and ud is the dust velocity, P is the gas
thelinearanalysisofthedevelopingconvectionroll.Asimplerelationship pressure, The dust density is ρd while the gas density is simply ρ. The
is demonstrated that further shows that a steady vortex roll at the corre- superscript(x)and(y)appearinginfrontofvelocityexpressionsarere-
spondingcriticalwavenumbercausesthedustdensitytogrowalgebraically spectivelytheirxˆandyˆcomponents.Thekinematicdragbetweenphases
whentherollisanti-cyclonicand,conversely,thedensityisdepletedwhere ismitigated bythe“stopping-time” ts andweassumeittobeconstant-
therolliscyclonic. seesimilartreatmentsbyCuzzi(1993)andTangaetal,(1995).Ifthechan-
InSection3anonlinearasymptoticanalysisisdevelopedinthelimit nelrotateswithrateΩindirectionˆzthentheCoriolistermf isgivenby
oflargeaspectratioandunderconditionsoffixedthermalfluxonthechan- 2Ω.BecausethegasphaseisassumedtobeBoussinesqitsthermodynam-
nelwallsandwherethebackgroundvorticityismodeledasanf-planewith icsismodeledbytheusualexpression ofafluidundergoingfluctuations
aweakgradient.Inadditiontothis,theimpositionofaweakexternalforce underconstantpressureconditionswhereCpisthespecificheatatconstant
willpromoteasteadyflowexhibitingaweakamountofshear(hereitwill pressure.
beCouette). Theasymptoticanalysis proceedsundertheassumptionthat Sincethisisatwo-dimensionalmodel,itwillbemoreconvenientfor
both(i)thatthethermaltimeismuchgreaterthantheduststoppingtime ustoconsiderthegasphasemomentumequationsintermsofasingleequa-
and,(ii)thatthetimescalederivedfromthegeometricmeanofthether- tionforthegasvorticity.Sincethegasphaseisdivergencefreeexceptwhen
malandrotationtimesisalsomuchgreaterthanthestoppingtimeofthe coupledtog,wemayconsiderthegasvelocityasresultingfromastream-
dust.Thetwoconditions translates toasituation inwhichtheduststop- functionψsuchthat
pingtimeisactuallymuchlongerthanthelocalrotationtime.Thisscaling
∂ψ ∂ψ
regimealsoimpliesthatthedustvelocitybehavesasifitwereirrotational u(x),u(y) = , . (4)
atleadingorder. (cid:2) (cid:3) h−∂y ∂xi
Themodelshowsalgebraicinstability(inthedustcomponent)unless It follows that by defining the vorticity as the curl of the fluid velocity,
someamountofbackgroundvorticitygradientispresentintheflow(aspro- ω= u,thenwehave
videdbytheweakf-planeprescription).Thisgradientstabilizesthegrowth ∇×
bypromotingaRossbywavedriftofthetheconvective roll(vortex)pat- ∂u(y) ∂u(x) ∂2ψ ∂2ψ
ω = + . (5)
tern.BecausetheRossbywavedriftspeedofthevortexpatternisdifferent ≡ ∂x − ∂y ∂x2 ∂y2
thanthebackgroundvelocityfieldtheeffecthereisforthefluidpatternto
Bytakingthecurlof(2)wefindthefollowing
driftpastthedustcomponent.Asgivenregionsofthedustareexposedto
overallreductionsofthetotalfluidvorticity(asmeasuredwithrespectto ∂ω ∂ρ 1
thebackgroundvorticityarisingfromthechannel’s rotation)thedustbe- ρh∂t +J(ω,ψ)i = −g∂x − tshρd(ω−ωd)
gHadioniwsntefiovnecirtoublmleeccbateuacssaeuetxshepeetphcateettldeorwfnrovimsonrttohicteistbtyaetphioaarnvtaiooryrftpthhreeedvdiocurtsettedxcabpnyanttotheterlnolicwnaeilalllyrptchaoselsolerocynt. + (u(x)−u(dx))∂∂ρyd −(u(y)−u(dy))∂∂ρxdi(,6)
throughagivenlocaldustregion.Inturn,thedustregionwillbeexposed therebyeliminatingdirectreferencetothegasphase’sdivergencefreena-
topatches offluidpassingbywithenhanced vorticity and,consequently, tureandreducingthenumberofequationsfromthreetotwo.TheJacobian
willexperienceareductioninthelocaldustconcentration.Itisfound,how- appearingin(6)isdefinedas
ever,thatsecondarygrowingoscillationsalsoappearinthisformulationand
∂f ∂g ∂f ∂g
thesearewipedoutwhenacertainamountofdiffusioninthedustconcen- J(f,g) .
≡ ∂y∂x − ∂x∂y
trationisaddedtothemodel.Itisalsofoundthataccordingtothemodel,
placeswherethedustconcentrationisenhanced(depleted)correspondsto Unlikeitsgascounterpart, thedustphaseisassumedtobecompressible,
zones where the convective roll is slightly weakened (strengthened). We andthisiswhywehaveretainedallspatialdependencesofρdin(6).Fur-
showthat this canbeexplained byaneffective modification ofthelocal thermore,thedustphasevorticity,ωd,isdefinedfromthedustphaseveloc-
Rayleighnumberduetothenonlinearrearrangementofthedustconcentra- ity,ud,intheusualway,viz.,
tion.
∂u(y) ∂u(x)
ωd≡ ∂dx − ∂dy (7)
Thedustphaseismodeledasapressureless,compressiblefluidunder
2 TWODIMENSIONALFLOWEQUATIONS
theinfluenceofparticledrag,theforce andtheCorioliseffect,
Fk
ConsideratwofluiddescriptionofadustladensimpleBoussinesqfluidin
arotatingchannel.Forthesakeofgeneralitywesupposethatonlythegas ∂∂ρtd +∇·(ρdud) = 0 (8)
phasehascertainpropertieswhichcauseittobedynamicallyinfluencedby
anexternalconstantforceofmagnitudegandwithdirectionnormaltothe ∂ fu(y)xˆ
channelwalls(viz.inyˆdirection).Inthistoymodelweadditionallyposit ρd(cid:16)∂t +ud·∇(cid:17)ud = ρd(cid:26) Fk−dfu(dx)yˆ (cid:27)−ρd(ud−u)/t(9s).
theretobeasecondforcerepresentedbytheterm whichalsoactsinthe
yˆdirectionanditsdependenceisonyonly:inothFekrwords,Fk =Fk(y). Sbointhcefluthidesu,nitdwerillylipnrgovmeabsesnedfiencisailtytostwatreitseadreepaprrteusruemsoedfthtoebdeusctodnesntsaintyt iinn
Udynnliakmeitchaelloythcoerupfolerdcetog,ththeedfuosrtcbeeFcakusaeffoefcstsimboptlhekgianseamnadtidcudsrt.agThmeeflasuuidreids termsofafractionaldensityparameter,δ,definedby,
bythedifferenceinvelocitiesbetweenthegasanddustphase.Finally,we
assumethatthethermodynamictransferpropertiesofthegasaremodeled ρd=ρ¯d(1+δ). (10)
bysimplethermalconduction. Thus,theequations ofmotionforthegas
phasearesummarizedtobe,
2.1 SteadyState
u = 0 (1)
∇·
∂ Athermal steady stateiswhenthegasconducts intheyˆ direction. This
ρ(cid:16)∂t +u·∇(cid:17)u = −∇P −ρgyˆ−ρd(u−ud)/ts meansthatthereislineartemperaturegradientinygivenby,T¯ = T∗−
An idealizedmodelfordust-gasinteractionin a rotatingchannel 3
βTy,inwhichT∗isthebackgroundtemperaturescaleandwherethecon- ThevariableC˜,definedby
svtiarnotnmβTenrt.epDreenseontitns,giun¯aansdenu¯sde,atshethsetesatedaydbyavceklgorcoiutyndprhoefialtesfl,uaxsoofluthtieonenis- C˜ ∂u˜(dx) + ∂u˜(dy), (21)
u¯=u¯d,alongwith, ≡ ∂ξ ∂Y
fu¯(x)=fu¯(dx)=Fk(y), u¯(dy)=u¯(y)=0, and ∂∂Py¯ =−ρ¯g.(11) vreeplorecsiteyntsscathleeddtioladta/tiτo.nTohfetRheaydluesigthphnausme.bTerh,eRte,rimsdu˜edfinreedprteosebnetsthedust
aAsnsudmasedprteovbioeucsolynsmtaenntt.ioWneedc,otnhseiddeernsstietiaedsyofcobnodthitiflounisdtsotahtaevse,ρ¯noanxˆddρ¯idre,cis- R= 1gαβTts Cpρ¯d2 ρ¯ , (22)
tiondependences.Thus,thetemperaturesteadystateconditionis ρ¯ (cid:18) K (cid:19)(cid:16)ρ¯d(cid:17)
∂2T¯ whileaneffectivePrandtlnumberPrisidentifiedwith
K =0, (12)
∂y2 Pr= τ = Cpρ¯d2. (23)
whichyieldsasteadyfluxstateofthegaswithatemperatureprofileT¯ = ts Kts
T∗−βy,whereβisaconstantrelatedtothebackgroundfluxstateofthe Finally,theCoriolisparameterisscaledtothestoppingtime:f˜ fts.
gasandwhereT∗isthebackgroundtemperaturescale. Wenotethatinthedustmomentumequation,outofwhich≡both(19)
and(20)areformed,allappearancesofthedustdensityanditsfluctuation,
2.2 LinearTheory ρ′d/ρ¯d,explicitlydropout,Inpracticethismeansthat(18)explicitlydecou-
plesinanormalmodeanalysisoftheperturbationequationset(15)-(20).
For the sake transparency, we consider linear disturbances of the steady Thisobservationhasaninterestingconsequenceforthenonlineartheoryin
arrangementprescribedabovewhentheforce iszero.Thismeansthat theupcomingsection.
Fk
perturbations ensue under static conditions: u¯ = u¯d = 0. Temperature Thelimited coupled set ofequations (15)-(17)and (19)-(20)admit
fluctuationsarerepresentedby steady(i.e.∂/∂t = 0)solutions.Theparticlephasevariables aresimply
relatedtothegasphasevariablesby,
T =T∗ βy+θ, (13)
− ω˜
whereθrepresentsthetemperaturedisturbanceawayfromthestaticstate. ω˜d = 1+f˜2 (24)
Except for buoyancy effects, the gas density is constant. Therefore the
Bousinessqapproximationsaysthat, C˜ = f˜ω˜ , (25)
′ 1+f˜2
ρ =ρ ρ¯= αθ, (14)
− −
Notethat (25)saysthat thedustdilatation is related tothe gasvorticity,
whereherethecoefficientofthermalexpansionatconstantpressureisrep-
whichwillhaveconsequenceshortly.Thegasphaseequationsbecome,
resentedbyα.Aprimeonanydensityquantityismeanttodesignate its
deviationfromthesteadyuniformstate,whichwillbedesignatedwithan ∂Θ
overbaroverthequantity. 0 = R∂ξ −(ω−ωd), (26)
Wesupposethatalldynamical disturbances occuronalengthscale
∂ψ˜ ∂2 ∂2
ofd.Consequentlyspatialscalesarenondimensionalizedondand,further- 0 = + + Θ, (27)
more,theyarewrittenas ∂ξ (cid:18)∂ξ2 ∂Y2(cid:19)
x=ξd , y=Yd alongwith(17)unalteredasitappears.Combining(17)with(24-27)yields
∂ 1 ∂ ∂ 1 ∂ asingleequationforψ˜,
,
∂x → d∂ξ ∂y → d∂Y ∂2 ∂ 2 1+f˜2∂2ψ˜
τ =Cpρd2/Kisathermaltimescaleand,assuch,wescalealltemporal (cid:18)∂Y2 + ∂ξ2(cid:19) ψ˜+R f˜2 ∂ξ2 =0 (28)
derivativesbyit.Temperaturedisturbancesarescaledbyβdandiswritten
innondimensionaltermsasθ=Θβd.Velocitiesscalewithd/τwhichsets (28)issupplementedwithboundaryconditions.Asforthethermalcondi-
thescalingforboththestreamfunctionandvorticity(forbothfluidphases), tions:weexploretwodifferenttypesofrequirements,namelythatthe(i)the
temperaturesarefixedattheboundariesor(ii)thethermalfluxesarefixed
d2 1
ψ= ψ˜, ω= ω˜, attheboundaries.Theformerconditionisthemorefamiliarofthetwoin
τ τ which,inpractice,itrequiresthatthetemperaturefluctuations arezeroat
inwhich ψ˜and ω˜ arethe nondimensional stream function andvorticity, theboundaries.Thesecondconditionis,effectively,thermalinsulationand
respectively.Inthisformalismthenondimensionallinearizedperturbation it,inpractice,amountstosetting∂Θ/∂Y =0attheboundaries.
equationsforthegasare, Assuming,ingeneralforallquantities,periodicsolutionsinthehori-
zontalξdirectionwithcorrespondinghorizontalwavenumberq,e.g.
ρ 1 ∂ω˜ ∂Θ
(cid:16)ρd(cid:17)Pr ∂t = R∂ξ −(ω˜−ω˜d) (15) ψ˜=ψ¯(Y)expiqξ+c.c.,
∂Θ = ∂ψ˜+ ∂2 + ∂2 Θ (16) wefindthatforfixed-temperature boundaryconditions thelowestnormal
∂t ∂ξ (cid:18)∂ξ2 ∂Y2(cid:19) modesolutioninY forψ˜is
ω˜ = ∂2 + ∂2 ψ˜. (17) ψ˜∼expiqξcosπY, (29)
(cid:18)∂ξ2 ∂Y2(cid:19)
wherethesystemeigenvalue,whichisaneffectiveRayleighnumberdefined
Whilstforthedustphase, by,
∂ ρ′d +C˜ = 0, (18) Reff 1+f˜2 R, (30)
∂t(cid:18)ρ¯d(cid:19) ≡(cid:18) f˜2 (cid:19)
P1r∂∂ωtd = −(ωd−ω)−f˜C˜, (19) mustsatisfy,
P1r∂∂Ct˜ = −C˜+f˜ωd. (20) Reff = (π2+q2q2)2. (31)
4 O.M. Umurhan
(a) Conducting Boundaries (b) Insulating Boundaries (a) s =0.21 (b) s =1.26 (c) s =1.8711
100 4 5
Rayleigh Number5678900000 Unstable to Rolls Rayleigh Number345000 Unstable to Rolls Temperature−000...0252 −2 0 2 −−02642 −2 0 2 −085 −2 0 2
FieffR: Effective gu1234r000000e1R.ecfCf =2r 4iqπt:2i sccaaqllc4e =dR πwaayvSeltn6eaubimlgeb hteor nR8oullmsbe10rasafeffuR: Effective n12c0000tiono2fqs: cscaalRle4eecdfdf =w ha1v2oeSrn6tiuazmbolben etrot a8Rlolwlsav10e- VorticityDust Concentration−−000...00555 −2 0 2 −−−−−13210000024042 −2 0 2 −−−−−264200000246042 −2 0 2
lengthq.(a)Conductingboundariesand(b)insulating (fixed-flux)bound- −2 spa0ce 2 −2 spa0ce 2 −2 spa0ce 2
aries. (d) Maximum Absolute Amplitude, Time Series
8 30
fTohresruecihsmaamrgininimalusmoluvtailounesotfoRexeifstf.Wateadcernitoictealthweamveinnuimmubmervqaclureeqoufirtehde Max |F|χ246 ∆Max ||1200
befofuenctdivareyRcaoynlediigtihonnsumqcbe=rasπRaecnfdf,aRnecdfwfe=find4πth2at≈for4fi0xe-d-wtehmicpheriastuthree 00 0.5 Tim1e: s 1.5 2 00 0.5 Tim1e: s 1.5 2
numberexpectedforRayleigh-Darcyconvection(Horton&Rogers,1945).
Thesolutionforperfectlyinsulatingboundariesresiststhestraightfor- Figure2.RepresentativeprofilesforModelA:β˜=˜b1=0andr2=5.30
wardformcharacterizingthefixedtemperatureresult.Itprovestobeeasier andφ˜ = 10. Plotted arethe lowest order temperature amplitude F,the
toseeknumericalsolutionshereinstead.Figure1(b)displaystheRayleigh lowest ordervorticity amplitude Fχ,followed bythedustconcentration,
numberasafunctionofhorizontalwavenumberforthesemarginalcondi- ∆.Panel(a),attimes∼ 0.21.Panel(b),times∼ 1.25.Panel(c),blow
tions.ItturnsoutthatRecff = 12anditoccursatq = 0.Theresultthat uptime s ∼ 1.87.Thescaled dustconcentration always seemstotrace
thecriticalRayleighnumberoccursatinfinitehorizontalscales,thoughcu- the gas vorticity Fχ even throughout the blow up time. Panel (d) is the
rious, is a well known result in standard Rayleigh-Benard problems and timeseriesfortheabsolutevaluesofvorticityscaleanddustconcentration,
itsimplicationshavebeenstudiedbymanyinvestigators(e.g.Chapman& respectively.
Proctor,1980).
Irrespectiveofthethermalboundaryconditionassumedthemarginal
rollstatecancausethedustdensitytogrowalgebraically.Toseethiscon-
This extreme ordering says physically that the dust stopping time, ts, is
sider(20)withthedustdilatation termre-expressedintermsoftherela- muchlongerthanthelocalrotationtime Ω−1.Thisscalinghoweveralso
tionship(25).Itisfoundthatthetimerateofchangeofthedustdensityis ∼
saysthattheduststoppingtimeismuchlessthanthegeometricmeanofthe
expressedby
thermalandrotationtimes.Inotherwords,
∂∂t(cid:18)ρρ¯d′d(cid:19)=−1+f˜ω˜f˜2. (32) Ωτ ≫t2s, and Ωts≫1.
ForthesakeofthisasymptoticcalculationwetakePr .
Inotherwords,thedustdensitychangeslinearlyintime(sincethereis →∞
Furthermore we will suppose that the horizontal lengths scales are
notimedependenceinω˜)and,additionally,thischangeisfastestatplaces
muchlargerthanthescalesindirectionofgravityandwewillintroducethe
wherethevortexamplitudeisextremal.Giventherelationshipin(32)itis
smallscalingparameterǫtorepresentthisextremestate.Correspondingto
apparentthatdustdensitygrowsfastestnearanti-cyclonicvortexperturba-
thissituationweintroducethescaledhorizontalvariableX=ǫξ,whereX
tions:thisisclearsincetheRHSoftheexpressionispositiveonlyifthe
product f˜ω˜ ispositiveandthiscanonlyhappenifthesignsoff˜andω˜ isanorderonequantity.Additionallyweintroduceanorder1scaledtime
− variable T suchthatT = ǫ4t.Thus,wereplace allderivative operators
areopposite.
with
Tobemoreaccurateinthisdescriptionofgrowingdustconcentration
it should be stated that if the sign of the growing vortex perturbation is ∂ ∂
negativeitonlymeansthatthispartofthefluidstatestartsappearingtohave ∂ξ →ǫ∂X, ∂t →ǫ4∂T (33)
lowervorticitythantheoriginalglobalstate.Theresultofthislineartheory,
Fromnowonwewillappeal totheshorthandnotationforderivative op-
takeninthislight,isnottoosurprisingbecauseearlierworkonthebehavior
erationsforcompactnessofnotationandclarityofderivation.Correspond-
ofLagrangiantracerparticlesinfullydevelopedisotropicturbulenceflow
ingly,wefindthatadistinguishedlimitexistswhenthestreamfunctionis
(SquiresandEaton,1991)showthatparticlesconcentrateinthosepartsof
(ǫ)thetemperaturefluctuation.Specifically,wesaythat
theflowwhichareregionsoflowervorticityandhighstrainrate. O
ψ˜=ǫΨ, ω˜ =ǫΩ,
whereΨandΩarenoworder1quantities.
3 ANASYMPTOTICNONLINEARMODEL Weintroduce the external force, , into the analysis in which its
Fk
nondimensionalizationisexpressedthroughtherelationship,
Inorder to understand how to handle the curious situation involving the
algebraic growthoftheparticle density perturbation eventhoughthegas d
phaseisneithergrowingnordecayingweasymptoticallyanalyzethegov- Fk(y)= τ2F˜k(Y),
erning equations under certain extreme conditions. Inparticular, wewill
explore the nonlinear development ofdisturbances forfixedthermal flux whereF˜kisnondimensional.Weassumethatthesteadyexternalforcehas
boundaryconditions.FurthermorewewillassumethatthePrandtlnumber theform
ismuchlargerthanthescaledCoriolisparameter,f˜whichwillalsobemuch 1
greaterthanone,viz., F˜k = ǫ4bφY, (34)
Pr f˜ 1. wherebandφareorder1constants.
≫ ≫
An idealizedmodelfordust-gasinteractionin a rotatingchannel 5
Inordertoachieveproperasymptoticbalanceweexploitthefreedom ThenewJacobiansymbol, isdefinedonthevariablesXandY.Inthe
J
wehaveinchoosingthe“largeness”oftheCoriolisparameterbyscalingit nextsectionwesetouttosolve,orderbyorder,thesetofequations(43-48)
(adhoc)as withfixedfluxthermalboundaryconditions.
1 β
f˜= φ 1+ ǫ6Y (35)
ǫ5 (cid:16) φ (cid:17) 3.1 Expansions
whereφis,asbefore,order1.Thebizarrechoicebehindthisasymptotic
orderingismotivated byintroducing thedustdisturbance termδ˜intothe Thesetofequations(43-48)aresolvedasaperturbationseriesexpansionin
subsequentanalysisattherightstage.TheβY term,whichtacitly repre- powersofǫ2.Thus,thefollowingexpansionsarepresumedforthispurpose,
sentsanassumptionofaweakf-planeeffect(seePedlosky),isintroduced Θ=Θ0+ǫ2Θ2+ǫ4Θ4+ (49)
aprioriinordertoensurestabilityofthenonlinearmodel.Thoughitisan ···
artificialdevice,ithastheeffectofcausingaRossbywavetypeofdriftofa Ψ=Ψ0+ǫ2Ψ2+··· (50)
vortexpattern. Ω=Ω0+ǫ2Ω2+ (51)
···
Withrespectto(11),theresultingsteadyx-directionflowisweakly δ=δ2+ǫ2δ4+ (52)
Couette.DenotingthesteadyflowbyUandUdthenwehave ···
(44)tolowestorderbecomessimply,
U =Ud=ǫbY +O(cid:0)ǫ7(cid:1). (36) D2Θ0=0. (53)
Fromhereonout,weconsiderdisturbancesofthedustandgasvelocities
ontopandoverthisbasestate. Thesolutionofthisequationsubjecttotheconditionthatthefluxremain
Withthesescalingsinplacewefindthatthedustquantitiesrequirethe fixedontheboundaryis,
followingscalingsinorderfortheretobenontrivialbalance:thefractional
dustdensityscalesasǫ2andfromhereonoutitiswrittenas Θ0=f(X,T), (54)
ρ′ meaningtosaythatthelowestordertemperatureperturbationisindepen-
d =ǫ2δ. (37) dentoftheverticalcoordinate.Therestoftheexpansionprocedureisstan-
ρ¯d dardandwerelegate itsfullexposition toAppendix A.Whatresults are
Wefindthatthedustdilatationisfirstnontrivialatorderǫ6anditiswritten twocoupledevolutionequationsforthetemperaturedisturbancef andthe
as, lowestorderparticledensityδ2in(A23)and(A32).Thesearereproduced
below,
C˜=ǫ6C6+ ǫ8 (38)
whereastheduOst(cid:0)vor(cid:1)ticity scalesevensmallersuchthatitisnontrivial at fT = −κfXXXX+µ(fX3)X−[(r2−δ2)fX]X
orderǫ11!Inotherwords, −b1S(fX2)X−βBfXXX, (55)
ω˜d=ǫ11Ωd. (39) δ2T = −φR0fX. (56)
Becauseofthislastobservationwewill,forallintentsandpurposes,treat Thepositiveconstantsκ,µ,B,SaredefinedinAppendixA.r2 isanef-
thedustvelocityasderivedfromapotentialwhichsimilarlyscalesonorder fectivedeparturefromthecriticalRayleighnumberR0=12givenby
ǫv6el.oIcnityotshcearlewsoarsds, ud = ǫ6∇Γ.As such, each component ofthe dust r2= R122 − 1b2210.
u˜(dy)=ǫ6v6=ǫ6∂∂YΓ, u˜d(x)=ǫ7w7=ǫ7∂∂XΓ. (40) Tfrhoempcarriatmiceatleitryrw2hciacnhbinecilnutdeerpsrethteedfaacstrethpartesaenctoinngstaannte(flfiencetaivr)esdheepaarrtiusrea
stabilizinginfluenceagainsttheonsetofrollsinthelinearregime.
Togetherwiththescalingrelationship(38)andgeneraldefinition(21)itis
clearthatattheseinitialordersthedustvelocitysatisfiesaPoissontypeof
equation,
3.2 CanonizationandIdentificationofEffects
2Γ=C6. (41)
∇ Itismoreconvenienttodefineanumberofrescalingsof(55)and(56)in
Withthesescalingsandassumptionsinplaceweconsiderthetemporal ordertotransparently displaythestructure ofthetwocoupledequations.
developmentofdisturbanceswhentheRayleighnumberdeviatesfromits Onemaydefinenewspaceandtimecoordinates,ξandsalongwithnew
criticalvaluebyanorderǫ2amountwrittenas, amplitudescalingsF and∆by,
R R0+ǫ2R2. (42) T sκ, X √κχ,
→ → →
Forthisanalysis wewillconsiderthedevelopment intermsoffixed-flux κ 1/2
boundaryconditionswhichmeansthatthecriticalRayleighnumberabout f(X,T)→(cid:16)µ(cid:17) F(χ,s), δ2(X,T)→∆(χ,s). (57)
which we will be expanding around will be 12. However, the following
Consequently,(55)and(56)rewrittenintermsofthedefinitions(57)is
calculationself-consistentlyselectsthepropervalueofR0.Thegoverning
equations(inthePr limit)withthesescalingsinsertednowreads
forthegasphase, → ∞ Fs = −Fχχχχ+(Fχ3)χ− (r2−∆)Fχ
(cid:16) (cid:17)χ
R0(cid:16)1+ǫ2RR02(cid:17)∂XΘ−(1+ǫ2δ2)Ω = ǫ2β∂XΨ (43) ∆s = φ˜−F˜bχ1,(F,χ2)χ−β˜Fχχχ, ((5598))
ǫ4∂TΘ+ǫ2J(Θ,Ψ)+ǫ2b1Y∂XΘ = ǫ2∂XΨ+ǫ2∂X2Θ+D(244Θ) where −
Ω = D2+ǫ2∂2 Ψ, (45)
X S B κ
andfortheparticlephasetheyare, (cid:0) (cid:1) ˜b1≡b1µ, β˜≡β√κ, φ˜≡φR0√µ.
0 = ǫΩ ǫφC6+ ǫ7 (46) Themodelset(58)and(59)arenowinamoretransparentformandallows
− O
0 = −ǫ6C6+ǫ6φΩd(cid:0)+O(cid:1) ǫ12 (47) uRsHtSodoifsc(5u8ss)athreewefefellctksntohweync(osenetaCinh.aTphmearnol&eoPfrothcetofir,rs1t9t8w0o)treersmpescotinvetlhye,
ǫ6∂Tδ+ǫ6C6 = 0+O ǫ8 (cid:0) (cid:1) (48) (i) todamp the temperature profile due to horizontal thermal dissipation
(cid:0) (cid:1)
6 O.M. Umurhan
(a) Time Series: no dissipation (b) Time Series: with dissipation Vorticity Peaks
4 4
Max. Amplitude123 VorticityDust Concentration Max. Amplitude123 DVourstti cCitoyncentration Time s 23..23455
1.5
00 5 10 15 20 00 5 10 15 20 1
Time: s Time: s 0.5
(c) s =0.84 (d) s =1.722 (e) s =20.958 0
−3 −2 −1 Spa0ce 1 2 3
mperature 0.05 −00..055 02 Figure4.Positionofthepeaksofthevorticity profileforModelB.The
Te−0.5 −1 −2 maximumpeakislabelledwithacircleandallsecondarypeaksaredenoted
−2 0 2 2 −2 0 2 −2 0 2
Vorticity −011 −011 −022 stwhyiesthtpemexa’ksg.oiAsestfrtaefrcroetmdheomutrtualwntisipitehlenatpdreeaaaskdhsjeutdostlmoinneelnyatnopdnheiat.sieAsfiestveriodvtehenrist(ttnhiemaateri,tstphr∼eoppa2agt)hattheoesf
Dust Concentration−00..022 −2 0 2 −00..055 −2 0 2 −022 −2 0 2 from0l.e5fttorighton(ath) isΘsp aTcemetpiemrateured, i sa =g1r8a.9m.
−2 0 2 −2 0 2 −2 0 2 3
space space space
2
1
Figure3.Representative profilesforModelB:˜b1 = 0andr2 = 5.30, Y 0 0
φ˜ = 10,β˜ = 10.(a), maximum amplitude ofthe dustconcentration −1
andthescaledvo−rticitywhenthereisdissipation( =0).Theamplitudes −2
oscillatewithgrowingsizeonalong-timescale.(Bb),maximumamplitude −0.5−3 −2 −1 χ0 1 2 3 −3
ofthedustconcentrationandthescaledvorticitywithdissipationpresent,
( = 1).Theoscillationsarequelledwhiletheoverallaveragemaximum
B (b) Ψ Stream Function, s =18.9
amplitudeofthestructuresarepreserved.Panels(c-e)representingprofiles 0.5 15
ofF,Fχ,∆atvariousstagesintheensuingevolution.Itshouldbeobserved 51 0
thatthedustconcentration,∆,eventuallylocksontothetemperatureprofile, 0
F,atlatetimes. Y 0 HVoigrhti c i t y Low Vorticity −−150
−15
−20
−25
−30
and(ii)tocontrolamplitudeduetononlinearadvectionoftemperature.The −0.5−3 −2 −1 χ0 1 2 3
r2 expression inthethird term onthe RHSof(58)constitutes theusual
sourceofamplitudegrowthbecauseofpositivedeparturesfromthecritical
Figure5.RepresentativecontoursforModelBwhenthegrowthhassettled
Rayleighnumber.The∆termexpressesthefactthatastheparticleconcen-
ontothesteadystate.Forthesecontoursǫ=1.(a)Temperaturefluctuation
trationgrowsthelocalRayleighnumberofthefluidgoesdownsomewhat.
contours,Θ.(b)Streamfunctioncontours,Ψ.
ThisisbestseenbyobservingthedefinitionoftheRayleighnumber(22):
ifthedustdensityρdgoesupthenRgoesdownandviceversa.Theother
termsontheRHSof(58)represent,respectively,(iii)thenonlineartwist- whosetotalamplitudedidnotexceed10−2.Thedustconcentrationalways
ing of the temperature profile due to shear and (iv) an effective Rossby
startedoutwithzeroinitialamplitude.
wavetypeofdriftofanemergingprofile.Asdiscussedintheprevioussec-
Weconsiderherefourselectedmodelstodiscuss;
tion,(59)representsthelocalenhancementofdustconcentrationduetothe
emergingconvectionroll’svorticity,whichis,tolowestorder,proportional ModelA.Nobackgroundvorticitygradientandnoshear,˜b2=β˜=0,
toFχasdemonstratedinAppendixA. wh•iler2=5.3andthedustcouplingissetφ˜=10,
knownNototeptrhoamtottheesnuobnclriintiecaarl tterarmnsit(iFonχ2s)χinaRssaoycleiaigtehd-Bwenitahrdthceonsvheecatrioins the•reMisondoeslhBe.arT˜bh2e=ba0ck,galroonugndwvitohrtri2cit=y g5r.a3daienndtφ˜is=ne1g0ataivned,,β˜ = −10,
(Gertsberg&Sivashinsky,1981)andwilldosohereinthismodelaswell ModelCSomeamountofshearwithaprogradesense,˜b1 = 10.0
butwewillnotexplorethisfeatureinthisstudy.Thistermappearsgeneri- alo•ngwiththesamevaluesofr2,φ˜andβ˜ofModelB. −
callysolongasthereissomesortofsymmetry-breakingeffectinthesystem
It should be noted that in all models the background vorticity is getting
(e.g.seeDepassier&Spiegel,1982).
smallerwithincreasingY.Additionally,theresultsreportedhererequired
theretobesomedissipationaddedtotheequationdescribingthedustcon-
centrationevolutionorotherwisethesolutionsblowup.Thisisdiscussed
3.3 SelectedSolutions inmoredetailforModelB.
Thereducedasymptoticequations(58-59)areevolvedaccordingtothenu-
mericalspectralschemeoutlinedinAppendixBandissimilartothetac-
3.3.1 ModelA
ticsusedinUmurhan&Regev(2004).Therobustnessoftheschemewas
checkedagainsttheanalyticsolutions ofasimilarevolutionequation de- Thisandothermodelslikeit,whereβ˜ = 0,areunstable.Theamplitude
rivedbyChapmanandProctor(1980).Thetestsdemonstratedagreement anddustconcentrationsteadilyrunawayoncethedustconcentrationlocks
between the numerical scheme and the analytical results to one part in intothestandingvorticityprofilethatemerges.Toseethismorereadilyre-
10−7.Theresultsreportedhereweregenerallycomputedon256Fourier fertoFigure2.Atearlytimesinthesimulationthelocaldustconcentration
modesandwhenconvergencewasdoubted,theresultswerecheckedagainst growsfastestwhereverG = Fχ,whichisproportionaltotheleadingor-
simulationsbasedon512and1024modes.Becausetherearefourthorder dervorticity, is greatest. Thisbehavior isinline with theconclusions of
derivativesinthelineartermsof(58)wenotethat,(a)noartificialviscos- thelineartheorysection.Forr2 = 5.3andφ˜ = 0,thefastestgrowing
itywasneededand,(b)smalltimestepsontheorderofdt = 10−4were wavenumberisn =2andthisreadilyshowninthelineartheoryanalysis
typicallytaken.AllsimulationsstartedoutwithrandomwhitenoiseforF of(58).Despitethelineartheoryexpectationshowever,thefinalrollprofile
An idealizedmodelfordust-gasinteractionin a rotatingchannel 7
willgofrombeingdoublehumpedtobeingsinglehumpedaswasshownby (a) Time Series: with dissipation
ChapmanandProctor(1980).Thisisbecausethenonlinearstabilityanal-
ysisnigslsehWrooiwtlhls.tφ˜hantothne-zseyrost,emtheprseafmerestsooretvoenftulaatlelytismetetlebeohnatvoioarpirsofiolbeswerivtheda. Max(|Amplitude|)246 46 1 1.5 2 2.5 VorticDituyst Concentration
Thedustconcentration alsoqualitatively resembles thevorticity distribu-
tion and it, similarly to the linear theory, appears to grow/deplete with- 00 2 4 6 8 10 12 14 16 18 20
Time: s
outbound.Thoseplaceswherethevorticityamplitudeisgreatestarethose (b) s =0.84 (c) s =1.722 (d) s =20.958
2 4
plRolaacycaleteisoignwhsheninruemtthhbeeerdgutahssetrcceoobnryrceescnpatuorasnitdniogntotihsgegrerroeaatllltylpyredonefihplaelnetcteheded.reCvaotolnusecesoqnuoteifnnutthleye,gtlrhooocwsae-l Temperature −011 02 −0242
ing, inwhat appears tobe, without bound. Naturally, the validity ofthis −2 0 2 −42 −2 0 2 5 −2 0 2
predictedblowupshouldbequestionedsinceunboundedgrowthmeansthe 2 2
asymptoticsolutionhasceasedbeingvalid. Vorticity −02 −−042 −05
Dust Concentration−00−..0242 −2 0 2 −00−−..05615 −2 0 2 −011 −2 0 2
3.3.2 ModelB:VorticityGradientwithNoShear −2 0 2 −2 0 2 −2 0 2
space space space
Introductionofβ˜stabilizestherunawaygrowthseeninModelA.Theam-
plitudeoftheemergingconvectionrollsettlesontothevicinityofagiven Figure6.Representative profiles forModel Cshowing aprograde shear
value though onoscillation setsin. Onaverylongtime scale this oscil- profile:r2 = 5.30andφ˜= 10,β˜= 10alongwith˜b1 = 10.These
− −
lation, whichshowsupinboththerollamplitude anddustconcentration resultsweregeneratedwith = 1.(a)Timeseriesoftheabsoluteampli-
B
amplitude,exhibitsagrowthinamplitudeanditeventuallyblowsupasis tudeofFχand∆.Theamplitudesreachasteadyaveragevaluebuteach
evidencedinFigure3a.Thishappensforallmodelsinwhichφ˜isnotzero profileexperiencesasmallamplitudeoscillationthatneithergrowsnorde-
andtheseculargrowthtimescaleisshorterasφ˜becomeslarger.Inorder cayswithtime.Insetisshownahigherresolutionprofileofthetransient
tostabilizethelongtimebehavioradissipationtermisintroducedintothe phaseforthevorticitynears 2.Theprofilegoesfromnominallytwo
∼
dustevolutionequationsothat,andfromhereonout,insteadof(59)the largescalehumpstoone.Panels(b-d)showprofilesofvariousquantitiesat
followingequationisevolved, threedifferenttimes:(b),initialgrowthphase,times 0.84,(c),transient
∼
adjustmentphase,times 1.72,(d),steadystatephase,times 20.0.In
∆s= φ˜Fχ+ ∆χχ, (60) (d)itisclearthatthevorti∼citytakesonsignificantandcomplicate∼dstructure
− B
anditisalsoapparentthatthoseregionswherethevorticityisnegativeare
is the scale of the dissipation and in the simulations it is taken to be stronglyconcentrated.
B
unity.Withthesimulationrunwith =1,theresultingmaximumabsolute
B
amplitudeisshowninFigure3b.Theoscillationsobservedfor = 0are
removedandtheoverallaverageamplitudeoftheunderlyingstruBcturesare 3.3.3 ModelC:ProgradeShear
preserved.
SheardoesnotchangethebasicfeaturesoftheresultsfoundforModelB.
Thetimeevolutionofthesesimulationsbeginwithaninitialgrowth
However,itdoesaltertheappearenceofthefinallydevelopedconvection
phase. Forr2 = 5.3, the linear theory predicts that the fastest growing rollbygivingitamuchmorecomplicatedstructure.Asbefore,theevolution
Fouriermodeisk = 2and,assuch,intheearlystages thetemperature
goesthroughaninitialgrowthstagefollowedbyatransientreadjustment
profileisdoublehumpedasinFigure3b.AsChapmanandProctor(1980)
phasebeforesettlingontoaroughlysteadyprofile(Figure6a).Thegreat
demonstrated,thistypeofprofileisnonlinearlyunstableandeventuallythe
difference betweentheseresultsandthatofModelBisthatthenegative
systemundergoes aphaseoftransient readjustment (arounds = 1.5)in
vorticity oftheroll is strongly concentrated into asmallregion inspace
whichthetemperature andvorticity gofrombeingatwo-humpedprofile
whereasthepositivevorticity,includingallthesubstructureslikeinFigure
down to a one-humped profile. Figure 3c shows the state of this profile
6d,arespreadoutoverlargerareasofspace(Figure7b).
readjustmentnearthetimes 1.7.
∼
Additionally, Figure 4 demonstrates how the profile drifts with in-
creasingχasafunctionoftime.Inthissteadydriftingstate,theprofilesare
singlehumpedand,asinFigure3d,afterthepostreadjustmentphasethe
dustconcentrationprofilelocksontothetempertureprofiletightly.Therea- ACKNOWLEDGMENTS
sonthatthedustconcentrationdoesnotgrowwithoutboundwhentheroll
patterndriftsissimple.InModelA,becauseβ˜ = 0,thetemperatureand IwouldliketothanktheDPWatBRC(2002)forprovidingandmaintaining
vorticityprofilethatdevelopsdoessoinplace.Becausethedustconcentra- thedesertfieldfacilitiesinwhichtheideasforthisworkfirstoriginated.
tionrespondstolocalvariationsofthevorticity,placesinthegrowingroll
profilewherethevorticityismostreducedarealsowheretherateofdust
collectionisgreatest.Withoutasuitablesaturationmechanismthegreatly
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enhanceddustconcentrationbeginstomodifythelocalRayleighnumberof
thefluid.This,inturn,causestherollamplitudetogrowwithferocity.With BargeP.&Sommeria,J.,1995,A&A,295,L1.
alargeramplitudeinthevorticitywillcomeaneverincreasedconcentra- BarrancoJ.A.&Marcus,P.S.,2000,SummerProgramProceedings,Center
tionofdust.Inthiswaythecyclefeedsonitselfanditrunsaway.Onthe ForTurbulenceResearch,NASAAmes/StanfordUniv.97.
otherhand,whentherollprofiledrifts,localenhancementsofdustdonot BarrancoJ.A.&Marcus,P.S.,2005,ApJ,623,1157.
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thedusttoconcentrate willeventually driftpast.Thelocaldustwillthen Bracco,A.,ChavanisP.H.,Provenzale,A.&SpiegelE.A.,1999,Phys.of
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thefull2Ddomain. Chiang,E.I.&Goldreich,P.,1997,ApJ,490,368.
8 O.M. Umurhan
0.5 (a) Θ Temperature, s =18.9 togethertheseequationsyieldthesolution
20
y2 1
10 Ψ0=R0P0fX, P0= 2 − 8, (A13)
Y 0 0 whereP0isdesignedsothatthereisnonormalflowoffluidatthevertical
−10 boundaries,orinotherwords,Ψ0=0aty= 1/2.Wenotethatbecause
±
oftherelationshipsbetweenthelowestordervorticity,streamfunctionand
−20
−0.5−3 −2 −1 χ0 1 2 3 thetemperaturein(A12)and(A13),thevorticityisproportionaltothegra-
dientofthetemperaturestructurefunction,orinotherwords,Ω0∼FX.
(b) Ψ Stream Function, s =18.9 Inorderforasolutiontoexistforthetemperatureatthenextorder,
0.5 1000
asolvability conditionmustbeenforcedupontheequations atthisorder.
800
600 Thismeansthatstartingwith,
400
Y 0 Lvoorwti c i t y 02 0 0 D2Θ2=−∂XΨ0−∂X2Θ0+b1y∂XΘ0+∂X(Ψ0)DΘ0−D(Ψ0)∂XΘ0,(A14)
−200 thesolvabilitycondition,inwhichthethermalfluxattheverticalboundaries
−400
−600 isfixed,amountstorequiring,
−0.5−3 −2 −1 χ0 1 2 3
1/2
Figure 7. Representative contours for Model c in the quasi-steady state Z−1/2D2Θ2dy=DΘ2|yy==1−/12/2=0. (A15)
regime,s 18.9.Forthesecontourcalculations,ǫ =1.(a)Temperature
fluctuation∼contours,Θ.(b)Streamfunctioncontours,Ψ. SinceΘ0isconstantwithrespecttoyandsinceΨ0isanevenfunctionof
y,wefindthatthecondition(A15)appliedto(A14)resultsinachoicefor
R0,
Cuzzi,J.N.Dobrovolskis,A.R.&Champney,J.M.,1993,106,102. 1/2
Depassier,M.C.&Spiegel,E.A.,1982,Geophys.Astrophys.FluidDynam- 1+R0 P0dy=0. (A16)
Z
ics,21,167. −1/2
Gertsberg,V.L.&Sivashinsky,G.I.,1981,Prog.Mod.Phys.,66,1219. From(A13),itfollowsthatR0=12.ThesolutionofΘ2becomes
Hayashi,C.,1981,Prog.Theor.Phys.Suppl.,70,35.
Horton,C.W.&RogersF.T.,1945,J.Appl.Phys.,16,367. Θ2=T2fXX+M2(fX)2+b1N2fX, (A17)
Klahr,H.HandBodenheimer,P.,2003,ApJ,582,869.
Pedlosky,J.,1987,GeophysicalFluidDynamics:SecondEdition,Springer- wheretheindividualfunctions,T2,M2,N2aregivenby
Verlag,NewYork. y2 y4
Squires,K.D.&EatonJ.K.,1991,Phys.FluidsA,3,1169. D2T2=−1−R0P0, T2= 4 − 2 , (A18)
Tanga,P.,Babiano, A,B.Dubrulle &Provenzale, A.,1996,Icarus, 121, 3y
158. D2M2=−R0DP0, M2= 2 −2y3, (A19)
Umurhan,O.M.&RegevO.,2004,A&A,427,855. y y3
Youdin,A.&Goodman,J.,2005,ApJ,620,459. D2N2=z, N2= + . (A20)
−8 6
these are also designed so their y-derivatives are zero at the boundaries
y = 1/2.Becauseu(x)andu(y)areorderǫ7andǫ6respectively,and
APPENDIXA: ASYMPTOTICEXPANSIONS furthe±rmore, because odf the relatdionship between the particle divergence
andthegasvorticityin(A4),thelowestorderexpressionof(A6)is
AtinfinitePrandtlnumbers,thefullgoverningequations(43-48)arerepro-
ducedhereinslightlydifferentformandwithalittlebitmoredetail, ∂Tδ2+φΩ0=0. (A21)
R0(cid:16)1+ǫ2RR02(cid:17)∂XΘ−(1+ǫ2δ)Ω = ǫ2β∂XΨ (A1) Butbecauseofthesolutions(A12)and(A13)wemaywrite
ǫ4∂TΘ+ǫ2J(Θ,Ψ)+ǫb1z∂XΘ = ǫ2∂XΨ+ǫ2∂X2Θ+D(A22Θ) δ2=δ2(X,T), (A22)
Ω = D2+ǫ2∂2 Ψ (A3) inotherwords,thestructurefunctionδfortheparticleconcentrationisin-
ǫ6C6 = ǫ(cid:0)6φΩ+···X(cid:1) (A4) rdeewperinttdeennttoorfeflyeactttthheissoolrudteior.nTΩhe0,governing evolution equation for δ2 is
1
Ωd = ǫ11φΩ (A5) ∂Tδ2=−φR0fX. (A23)
ǫ6∂Tδ+ǫ6(1+ǫ2δ)C6 = −ǫ3ud(x)∂Xδ−ǫ2u(dy)(AD6δ) functiWoneΨm2a.yTnhoewgopvroercneiendgteoqouradteiornǫ2beacnodmseosl,veforthenextorderstream
Thefollowingexpansionsarepresumed
Θ=Θ0+ǫ2Θ2+ǫ4Θ4+ (A7) D2Ψ2=R0∂XΘ2+R2∂XΘ0−∂X2Ψ0−δ2Ω0−β∂XΨ0 (A24)
Ψ=Ψ0+ǫ2Ψ2+······ (A8) wUrtiiltitzeinngasthesolutionsfromthepreviousorderswefindthattheΨ2maybe
Ω=Ω0+δǫ=2Ωδ22++··· (A(A190)) Ψ2 = S2fXXX+(R2−δ2R0)P0fX+Q2∂X(fX)2
andsimilarlyfortheotherquan·t·it·ies.Atthelowestorder(A2)is, + b1L(2odd)−βL(2even) fXX (A25)
(cid:16) (cid:17)
D2Θ0=0 (A11) wherethestructurefunctionsappearingabovearegivenby
y6 y4 3y2 27
whichadmitsthesolutionΘ0 = f(X,T).Nextsolvethenextordermo- D2S2=R0T2 R0P0, S2= + ,(A26)
mentumequation(A1)andstreamfunctionrelationship(A3) − − 5 − 4 4 − 160
6y5 27y
R0∂XΘ0−Ω0=0, Ω0=D2Ψ0, (A12) D2Q2=R0M2, Q2=− 5 +3y3− 40 , (A27)
An idealizedmodelfordust-gasinteractionin a rotatingchannel 9
D2L(2odd)=R0N2, L(2odd)= y105 − y43 + 196y0, (A28) δeqsuaastiothnesatirme,eincrement betweentimestepnandn+1thediscretized
D2L(2even)=R0P0, L(2even)= y24 − y42 + 352. (A29) Fkn+1−Fkn−1 = LkFkn+1+LkFkn−1 + n, (B7)
2δs 2 Nk
Tyh=estr1uc/t2u.reNfeuxntcwtieontusranretodtehseigonreddersǫo4thhaetattheeqyuaartieoznewrohaicththies,boundaries ∆nk+1−∆nk−1 = n. (B8)
± δs Mk
∂TΘ0+J(Ψ2,Θ0)+J(Ψ0,Θ2)+b1y∂XΘ2= Rearrangingthetermsandisolatingthen+1timestepreveals,
∂XΨ2+∂X2Θ2+D2Θ4. (A30) Fn+1 = 1+δsLkFn−1+ 2δs n, (B9)
Likeatthepreviousorderthesolvabilityconditionis, k 1−δsLk k 1−δsLkNk
1/2 ∆n+1 = ∆n−1+2δs n. (B10)
Z−1/2D2Θ4dy=DΘ4|yy==1−/12/2=0. (A31) InkthefirstdiscretkizationabovMe,kthecoefficienttermsinfrontofFkn−1and
n looklikePa´deapproximants toexponentials of .Thisisthenas-
Thesolvabilityconditionproducesanevolutionequationforf,namely, Nk Lk
sumedandthefollowingreplacements,
f+T[(+r2κ−fXδ2X)XfXX]X−+µ(bf1X3S)(XfX2)X+βBfXXX =0, (A32) 11+−δδssLLkk →exp[2δsLk], 1−1δsLk →exp[δsLk], (B11)
wheretheconstantsandparametersaredefinedby, aremadein(B9).Theresulting discretized equations that aresimultane-
ouslysolvedateachtimestepare(B10)alongwith,
1/2
κ = −Z (S2+T2)dy= 221, (A33) Fkn+1=exp[2δsLk]Fkn−1+2δsexp[δsLk]Nkn. (B12)
−1/2
Timecentereddifferencedschemeslikethesimpleoneusedhereareknown
1/2
6 tosometimesincuraperiod2δslong-timenumericalinstability.Inorderto
µ = R0 (P0DM2)dy= , (A34)
− Z 5 washthiseffectout,atevery50thtimestep(B12)and(B10)areadvanced
−1/2
throughforonetimestepusinganEulertypeevolverasinthefollowing,
r2 = −R2Z−11//22Pdy+b21Z−11//22zN2dy= RR02 − 1b2210, (A35) Fkn+1 = exp[δsLk]Fkn+δsexp[δsLk]Nkn, (B13)
∆n+1 = ∆n+δs n. (B14)
1/2 1 k k Mk
S = R0 P0DN2dy= , (A36) ThisEuleradvancerisalsousedastheveryfirsttimestepofallsimulations.
Z 10
−1/2
1/2
B = R0Z L(2even)dy= 110. (A37)
−1/2
APPENDIXB: NUMERICALMETHOD
Thegeneralformoftheequationsweseektosolveare,
∂F
= F+ (F,∆) (B1)
∂s L N
∂∆
= (F), (B2)
∂s M
where issomelinearoperatorandwhere isanonlinearoperatorin-
L N
volvingsomecombinationoftheargumentsand islinearinF.
M
Theseonedimensionalequations willbesolvedintermsofFourier
expansionsofF and∆asin,
F = Fk(s)expikχ+c.c. (B3)
X
∆ = ∆k(s)expikχ+c.c., (B4)
X
suchthateachFouriercomponentevolvesaccordingto
∂∂Fsk = LkFk+Nk(F,∆) (B5)
∂∂∆sk = Mk(F). (B6)
Thescalars arethelinearoperator actingontheappropriateFourier
Lk L
wavekandwhereNkandMkarethekthcomponentsofeachrespective
operator and .
TheNsymboMlFnrepresentsthenthtimeiterateofF.Thissamecon-
vention carries over to ∆, and . The evolution is carried out us-
M N
ingasecondordertimecentereddifferenceschemetogetherwithaCrank-
Nicholsonschemeforthespatialwavenumbersofthelinearoperatorterms.
All and expressionsareevaluatedatthecenteredtimepoint.With
Nk Mk