enhanced angular momentum transport in accretion disks

45 Pages·2007·0.46 MB·English

by  Steven A. Balbus

23Jul2003 20:32 AR AR194-AA41-14.tex AR194-AA41-14.sgm LaTeX2e(2002/01/18) P1:GCE
10.1146/annurev.astro.41.081401.155207
Annu.Rev.Astron.Astrophys.2003.41:555–97
doi:10.1146/annurev.astro.41.081401.155207
Copyright(cid:176)c 2003byAnnualReviews.Allrightsreserved
E A M T
NHANCED NGULAR OMENTUM RANSPORT
A D
IN CCRETION ISKS
Steven A. Balbus
g
or DepartmentofAstronomy,VITA,UniversityofVirginia, Charlottesville,Virginia22901;
ws. email:[email protected]
e
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7. Y o vationsoftheGalacticCentersupporttheexistenceoflowluminosityaccretion,which
59R mayultimatelyproveamenabletoglobalthree-dimensionalnumericalsimulation.
55-TO
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AE
o. ST
Astrby
v. 1.INTRODUCTION
e
R
u.
n Inrecentyears,accretiondisktransporttheoryhasdevelopedsorapidlythatany
n
A
review is destined to be significantly dated the moment it appears in print. This
willputthereaderatadisadvantage.However,itisanexhilaratingtimefordisk
theorists.
Thecurrenthappystateofaffairsinthiscomputationallydrivenfieldislargely
duetotheswiftevolutionofthree-dimensionalmagnetohydrodynamical(MHD)
codesandtheirsupportinghardware.Thesepowerfultoolsarrivewithprovident
timing,coincidingwithadeepeningtheoreticalunderstandingoftheroleofmag-
neticfieldsinaccretiondiskdynamics.Theresultisthataccretiondiskturbulence
theoryhasgrownfromamereviscositycoefficienttoafullyquantitativescience.
Inthisreview,Ifocusonwhatisnowknownoftherelationshipbetweenturbulence
0066-4146/03/0922-0555$14.00 555
23Jul2003 20:32 AR AR194-AA41-14.tex AR194-AA41-14.sgm LaTeX2e(2002/01/18) P1:GCE
556 BALBUS
andenhancedangularmomentuminaccretiondisks,andtheresultingimplications
forsomeselectedastrophysicalsystems.
Theclassicalproblemwithaccretiondisksisthattheydoofcourseaccrete.How
isitthatfluidelementsorbitinginacentralforcefieldlosetheirspecificangular
momentum and spiral inward? One may quickly rule out ordinary particulate
viscosity.Astrophysicaldisksaresimplytoobig.Tofixideas,notethatdisturbances
arepropagatedbyviscousdiffusionoveradistancelonatimescaleoforderl2=”,
where ” is the kinematic viscosity, or about 3£107 years forl » 1010 cm and
” D105cm2s¡1.Thisisordersofmagnitudetoolongforthetimevariabilityseen
g
or incompactobjectaccretiondisks.
s.
w Thewayaroundthisdifficultywasperceivedtobeturningthewoefullyinade-
e
vi quateviscositytoone’sadvantagebyappealingtotheassociatedlargeReynolds
e
ualry. number.This,itwasthought,wouldturndifferentialrotationintoshear-driventur-
m arjournals.annpersonal use onl botsotufrlabsebhenlecteareirlgo(flgeco.eagwrl.e,ldilCnabmreayawirnnfaioonnirsnldtiatno&beitalKuirtrirflbeaousfwltef1oni9rnc5esa6thaK,baSesihlpibalteikeereusinra.anTk&nhrooeStwafuatnnicoytsniatnhepcvareto1ttfi9hhl7eeer3eww).woaTrsekhrteehobefnrroReefaedokyerdnmeooonwldnons-t
oaded fro5/07. For immomSmheeednaitaruttemulrytbrvuailneeswnpceoedrtiasissanadneeeesdimreadbb.alTrerhatrissasimitsiebnneatctaflouostwheisswhsheceaerrne-dagrririoveea.ntlytuernbhualenncceedi(sacnhgaurlaacr)-
wnl1/2 terizedbyahighdegreeofcorrelationbetweentheradialandazimuthalvelocity
Don 0 fluctuations. This, as we shall see, has the direct effect of raising a disk’s angu-
597. RY o lar momentum flux orders of magnitude above what would be possible with an
55-TO ordinarycollisionalviscosity.
5A
1:V Notopicinfluiddynamicsismorecontentiousthantheonsetanddevelopment
4R
3.E ofturbulence,andaccretiondiskturbulencehasnotbeenanexception.Keplerian
ys. 200D OBS dfoisrckessddoranmotarteicsaelmlybsltealbaibliozreartootraytisohneaalrflfloowwos,nhloarwgeevsecralloecsaalnlydosmnealple,earfse.aCtuorreionloist
ophAR sharedbyclassicalplanarCouettefloworPoiseuilleflow.1YetCoriolisforcesdo
strW
AE noworkonthefluid;theyareinfactabsentfromtheenergyconservationequations.
Astro. by ST BorewcaiuthsoeuatCpootreinotlifsrfeoerceense,rdgeybsaoteurhcaesicnetnhteerfeodrmonowfhsehtehaerrrfleutaidinnsoanplirneesaernitcieeswaittha
v.
e high-enoughReynoldsnumberwouldstillfindawaytotapintothissourcewhen
R
u. lineardisturbancesfailtodoso.
n
n
A NopublishedlaboratoryexperimenthasshownthebreakdownofaKeplerian-
likeCouetteflowprofile,buttrulydefinitivestudieshaveyettobeperformed.The
onsetofnonlinearinstabilitiesthathavebeenreportedinCouetteflowexperiments
aregenerallyrelatedtoverysharprotationalvelocitygradientstypicalofKelvin-
Helmholtzinstabilities,notther¡1=2powerlawcharacteristicofaKeplerianprofile
(Triton1988).Theissueisreceivingrenewedattention,withgroupsinSaclay,Los
1Anotherkeyfeaturethatdisksdonotsharewiththeselaboratoryflowsisthepresenceof
aboundarylayerataretainingwall.
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ANGULARMOMENTUMTRANSPORTINDISKS 557
Alamos,andPrincetonlookingatReynoldsnumbersRe»105¡6,muchinexcess
ofthoseavailableintheclassicalexperiments(e.g.,Coles1965).
Theoretical developments have been more brisk. A theory for the onset of
turbulenceinPoiseuilleflowwaselucidatedinthe1970sand1980s(Bayly,Orszag
&Herbert1988).Thekeyistheexistenceofneutrallystable(orslowlydecaying)
finiteamplitudedisturbances.Thesesolutionsaretime-steadybutspatiallyperiodic
inthestreamwisedirection(Zahnetal.1974).Itistheseamendedflowprofilesthat
findthemselvessubjecttoarapid,short-wavelengththree-dimensionalinstability,
leadingtoabreakdownintoturbulence(Orszag&Patera1980,1981).Asimilar
g
or process appears to be at work in shear layers (Pierrehumbert & Widnall 1982,
s.
w Corcos & Lin 1984). It is now generally accepted that the triggering of a rapid,
e
evi linearthree-dimensionalinstabilityofanearlyneutrallystable,two-dimensional,
ualry. finite-amplitudedisturbanceisaverygenericmodeofthebreakdownoflaminar
m arjournals.annpersonal use onl fltwaonilwHtlppoionrwoitnopdtatoiguseartsbtheutahliteinesnvtcbheeeen.afrionorcnmomooupfrrleuinsnsediaberlresintaaexnritdsiiaynlmgwmoafevtearscicc(rLdeiitsgitohuntrhbdialilnsk1cse9?s7i8Tn)hawerimothtoasatitncighmaflpruaoicrd--
oaded fro5/07. For titenhraiatsstitisrcocfprrihetiyqcsuaielcnatcolydthpiserkotspr,aontrhtsieiotinfioannlittteooattuhmrebpluloiltceuandlceve,oinrnetiumctirtiayxlilnoygfstlhtaaeyberlroestaactxaiionsnnyomptrmofofiertlmrei.c.TOshtnaultsye,
wnl1/2 in a low-vorticity rotational profile, in which the oscillation frequency is much
7. DoY on 0 smallerthantheshearingrate,wouldweexpectabreakdownofflowlaminainto
59R turbulence,similartowhatisseeninamixinglayer.Thisisingoodaccordwith
55-TO three-dimensionalnumericalsimulations(Balbus,Hawley&Stone1996;Hawley,
5A
1:V Balbus&Winters1999),whichfindnononlinearlocalinstabilitiesinKeplerian
4R
3.E disksbutdoindeedfindthatshearlayersandlow-vorticitydisksshowanonlinear
strophys. 200WARD OBS bshyryemdarmkoddeoytrwnicanmsttoiacbtauillrirbtoyuu,lethenottwofleotvuwerr.b,TualhneednrcetehiesinnasKotoeiopfnlyeetrhtiaanntotdhaiensrkaelsy,mtbiaceyyporbnoedoaftnhooefthlhoeicgrahnleonsntolnirneaesxoair--
AE lutionavailableinsupercomputersimulations,retainssomeadvocates(Richard&
o. ST
Astrby Zahn1999).
v. Accretiondiskshaveonecriticallyimportantattributenotsharedwiththeclass-
Re icalhydrodynamicalfluids:Theyaregenerallymagnetized.Byengenderingnew
u. degreesoffreedomintheirhostfluid,evenveryweakmagneticfieldscompletely
n
n
A alterthestabilitybehaviorofastrophysicalgases,bothrotationallyandthermally
(Balbus2001).Freeenergysourcesintheformofangularvelocityandtemperature
gradientsbecomedirectlyavailabletodestabilizetheflow.
Thecounterintuitivepointhereisthataweakmagneticfieldcanhavesucha
potentinfluence.Thestabilitybehaviorofstronglymagnetizeddisksisrichand
astrophysicallyinteresting(e.g.,Papaloizou&Terquem1996;Varnie`re&Tagger
2002),buttheemphasisinthisreviewisdecidelyonsubthermalmagneticfields.
Ratherthandependingdirectlyonthestrengthoftheequilibriummagneticfields,
weakfieldinstabilitiesdependdirectlyuponhydrodynamicpropertiesoftheun-
perturbeddisk.Iftheangularvelocitydecreasesoutwardinaweaklymagnetized
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558 BALBUS
accretiondisk,whichisgenerallythecase,therotationprofileislinearlyunstable
(Balbus&Hawley1991).Thisinstabilityisknownasthemagnetorotationalin-
stability,orMRIforshort.Asweshallsee,thephysicsoftheMRIisverysimple.
Nevertheless, a full understanding of its mathematical generality and wide ap-
plicabilitywasmuchbelated,followingphenomenologicaldisktheory(Shakura
& Sunyaev 1973) by nearly two decades. Knowledge of the instability itself
significantly predated modern accretion disk theory (Velikhov 1959), albeit in
aratherformalglobalguise.
ThenumericalstudyoftheMRIdoesnotrequireunattainablegridresolutions,
g
or and it can be readily simulated. Both local (Hawley, Gammie & Balbus 1995;
s.
w Brandenburgetal.1995)andglobal(Armitage1998,Hawley2000)investigations
e
vi unambiguouslyshowabreakdownoflaminarKeplerianflowintowell-developed
e
ualry. turbulence.TheMRIistheonlyinstabilityshowntobecapableofproducingand
m arjournals.annpersonal use onl sioinnuusptnteraoroinntr-oeisngsetgileoflt-nlhgaserroaedvfniischtkaaasttnaionccneglydAdssUmitsrkiescssc.savAlnaeretsieal(dobGewleadmt(feCommrViape)ce1csryra9ets9utt6iero)emn,ststhao(enGpdleraovhmceigemlehoidfedMoe&nnRsvMiIit-aieiebnnsldoe,uuetci.1mge9d.e,9tsi8ucn)ra,btlheuoes-r
oaded fro5/07. For lwbeinhlicitlyeesctheaeenmucnshdcaeanrpglayebirnlaegpoKidfeltyph,leeefrruiualplntrpianrngogfieolreoefrveaemcncartieuntrsinoeinsnscgeonomtfifpalcleloyxmifitpyxlemedtae.nIlyni.fseAhsotlelrdtt,hitinhsenoaicntcusutrares-.
wnl1/2 Forallthesereasons,despitethedifficultiesofcopingwithMHDturbulence,the
Don 0 MRIisnowatthecenterofnumericalaccretiondiskstudies.
597. RY o Letus,however,postponeourdiscussionofmagneticmattersandturnourat-
55-TO tention first to the study of simple hydrodynamical waves in disks. These are of
5A
1:V greatpracticalinterestintheirownright,especiallyinprotoplanetarydisks.But
4R
3.E it is also the case that understanding the transport properties of waves deepens
ys. 200D OBS odnroed’synuanmdeicrs,taanndditnhgatoisfotuurrbpurliemnatrtyrarnesapsoornt,fobrortehvhieywdirnogdythneammihceraen.dWmeasghnaelltothheyn-
ophAR followwithadiscussionofhydrodynamicalinstability,withafocusonhowglobal
strW
AE instabilitycaninprincipleemergeinadifferentiallyrotatingdiskevenwhenthe
Astro. by ST Rthaeytloepigichscoofntdhietionnexitssseactitsiofine.dM.Magangenteicticstirnesstsaebsilairteytahnedmmoasgtnimetipcotrutarnbutlteranncsepaorret
v.
e mechanisminnon-self-gravitatingdisks,providedthegasisminimallyionizedto
R
u. coupletothefield.Thefinaltwosectionsareapresentationofrecentnumerical
n
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A studiesofMHDturbulence,andasummaryfollows.
2.PRELIMINARIES
2.1.FundamentalEquations
Foreaseoffuturereference,welistherethefundamentalequationsofmagneto-
hydrodynamics.
@‰
Cr¢(‰v)D0 (1)
@t
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ANGULARMOMENTUMTRANSPORTINDISKS 559
(cid:181) ¶ (cid:181) ¶
@v B2 B
‰ C(‰v¢r)v D¡r P C ¡‰r8C ¢r B
@t 8… 4…
(cid:181) ¶
1
C· r2v C r(r¢v) (2)
V
3
@B
Dr£(v £B¡· r£B) (3)
@t B
P dlnP‰¡(cid:176)
org (cid:176) ¡1 dt D QC¡Q¡ (4)
s.
w
vie Equation1ismassconservation;Equation2isthedynamicalequationofmotion;
e
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Downloaded fron 01/25/07. For isadirect@e(x‰p@Rrtevs`si)oCnorfa¢n•g‰ulRarv`mvo¡meRn4tB…u`mBcoCns(cid:181)erPvaCtio8Bn…:2¶e`‚D0; (5)
597. RY o wheree`isaunitvectorinthe`direction.Thedissipativetermshavebeendropped
55-TO because they appear only in the flux term, transporting a negligible amount of
1:5VA angularmomentum.
4R
3.E Itisalsousefultohaveathandanequationfortotalenergyconversation.This
ys. 200D OBS iisntseorpmreewtehda:tlengthytoderive(Balbus&Hawley1998),buttheresultisreadily
o. AstrophSTEWAR @@Et Cr¢FE D¡Q¡; (6)
Astrby wheretheenergydensityE is
v.
e
u. R E D 1‰v2C P C‰8C B2 (7)
nn 2 (cid:176) ¡1 8…
A
andtheenergyfluxis
(cid:181) ¶
1 (cid:176)P B
FE Dv 2‰v2C (cid:176) ¡1 C‰8 C 4…£(v £B): (8)
Theenergydensityconsistsofkinetic,thermal,gravitational,andmagneticcom-
ponents; the flux is similar with the magnetic component present as a Poynting
flux. The heating term QC, an entropy source, is assumed to arise from micro-
scopicdissipation,anditdoesnotexplicitlyappearinthetotalenergyequation—it
simplyconvertsoneformofenergytoanother.TheradiativeQ¡term,ontheother
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560 BALBUS
hand,representsgenuinesystemicenergylossesandappearsexplicitlyinthecon-
servationequation.
2.2.NonlinearFluctuations
Bothwavesandturbulenceinvolvetheconceptofwell-defineddeparturesofthe
flow from a smooth background. Velocity fluctuations are of particular interest
becauseitispossibletoformulateanexactenergyconservationlawforthefluctu-
ationsthemselves.This,inturn,explicitlyshowstheroleofdifferentialrotationas
g asourceoffreeenergyforthe(correlated)turbulentfluctuationsassociatedwith
or
s. outwardtransportofangularmomentum.
w
vie Letusdefinethevelocityfluctuationuby
e
Downloaded from arjournals.annualrn 01/25/07. For personal use only. ˜˜atehcq(ecuiRrsaien)tititniioeosnrpnanrfaflironleorcaewtis@[email protected]¢sioraiFtbyydti,lEoeaflbunputuopDct(urEtcoouDoq¡fxamutic(cid:181)ivmaoobt‰u¡iniaonrtsuinseoRaeRtn4l˜huot)hte`n(oteeRoe¡a:qm)onueobBau`tt4tRnia:iv…oidBnane`triao¶olnyfniddmenfx˜gRooartrcio¡ttoh,tnai`Qst(iEo¡danveq:fieuprnaratoigitfioieonlden2efi)onsrwe(t1rth(igh09tayhe))t
597. RY o Here,Eu isthefluctuationenergydensity
555-ATO 1 P B2
1:V E D ‰(u2C8 )C C ; (11)
3.4ER u 2 eff (cid:176) ¡1 8…
ophys. 200ARD OBS 8effisaneffectivepotentialfunction Z R
AstrEW 8eff D8¡ R˜2dR; (12)
o. ST
Astrby andFEuistheenergyfluxofthefluctuationsthemselves:
nu. Rev. FEu Du(cid:181)21‰u2C (cid:176)(cid:176)¡P1 C‰8eff¶C 4B… £(u£B): (13)
n
A
BecauseEquation10hasbeenaveragedover`,onlyRandZcomponentsappear
intheflux.
Thecombination
TR` D‰uRu` ¡ B4R…B` (14)
is an important quantity in both turbulent and wave transport theories of accre-
tiondisks.Ithasappearedoncebefore:withintheangularmomentumconserva-
tion equation (Equation 5), where it emerges as a component in the flux term.
Its constituents may be separately identified as Reynolds (‰uRu`) and Maxwell
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ANGULARMOMENTUMTRANSPORTINDISKS 561
(¡BRB`=4…)stresses.(Notethatbothwave-likeandturbulentdisturbancescan
create tight radial–azimuthal correlations in the velocity and magnetic fields.)
Thesecorrelationsevidentlyservetwoconceptuallyquitedifferentfunctions:They
directlytransportangularmomentum,andasshowninEquation10,theytapinto
thefreeenergysourceofdifferentialrotation.Thelatterroleisparticularlycrucial
forsustainingturbulence.Withoutexternaldriving,theonlyenergysourceforthe
fluctuationsisthiscouplingofthestresstothedifferentialrotation.Inastrophys-
ical accretion disks that make use of this free energy source, TR` must have the
samesignas¡d˜=dR,i.e.,itmustbepositive.
g
or
s.
w
vie 3.HYDRODYNAMICWAVESINDISKS
e
oaded from arjournals.annualr5/07. For personal use only. 3.1.TCpehnootelnhysaLtirldoipnepyriecfaaunernqcuWutniaomatnivoaHengnEo:efqtisuzHteaadtDtei,doiZPsnkDdi‰nPKwD‰h(cid:176)i(cid:176)c(cid:176),hwP¡th=he‰1erepDrKes(cid:176)issau¡ar2ec1oa;nnsdtadnetn.sWityemobaeyydaefisinm(e1pa5lne)
7. DownlY on 01/2 wdrhicearelcao2oirsditnhaeteasdi(aRb;at`ic;sZo)u.nTdhespgeaesdr.oIttaitsescoinnvtehneiegnratvtiotawtioornkalinfiestladnodfaardcceynltirna-l
59R mass.Theangularvelocity˜mustbeconstantoncylinders,˜D˜(R)(Tassoul
55-TO 1978).
5A
1:V Althoughitisastandardapproximation,theassumptionofabarotropicequa-
4R
ys. 2003.D OBSE ttihoenpoofssstiabtieliitsyoobfvaiobuusloyyaanntidreesaplioznastieonin.Athmeofnogrmothoefrinshteorrntcaolmgrianvgist,yitwparveecsludduees
ophAR toBrunt-Va¨isa¨la¨ oscillations(Ogilvie&Lubow1999).Instandarddiskmodels,
strW however,entropystratificationarisesbecauseofradiativeheatdiffusionfromtur-
AE
o. ST bulent heating. A linearized wave treatment of such an “equilibrium” is at best
Astrby adelicatematter.Ingeneral,theverticaltemperaturestructureofaccretiondisks
v. isnotwellunderstood,andthevirtueofadoptingabarotropicpressureisthatit
e
R allowsimportantdynamicalbehaviortoberevealed.
nu. Our goal is to study how linearized wave disturbances transport energy and
n
A angular momentum through a Keplerian disk. To this end, we introduce small
perturbationstotheequilibriumsolution,denotedas–‰; –v,etc.Theequilibrium
solutionisaxisymmetric,soaperturbedflowquantityXhastheform
–X D–X(R;Z)exp(im`¡i!t); (16)
where m is an integer and ! is the wave frequency. For the moment, the R;Z
dependenceoftheamplitudeisunrestricted.Thelinearizeddynamicalequations
ofmotionare
@–H
¡i!¯ –vR ¡2˜–v` D¡ @R (17)
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562 BALBUS
•2 m
¡i!¯ –v` C 2˜–vR D¡i R –H (18)
@–H
¡i!¯ –v D¡ : (19)
Z @Z
WehaveintroducedtheDoppler-shiftedwavefrequency,
!¯ D!¡m˜ (20)
andwhatisknownastheepicyclicfrequency•:
g
ws.or •2 D4˜2C d˜2 : (21)
vie dlnR
e
ualry. The epicyclic frequency is the rate at which a point mass in a circular motion,
oaded from arjournals.ann5/07. For personal use onl 0ad(BxiisiistTsnuyknhrmnbeeoyemrwde&emnitnraaTitcisrhnedteihmnipsegatluRaienrnqabeeuyaa1lonet9fciio8geitn7s¡hss)asio.ra!tr¯eAarbbeh–i‰itn‰ytl,hiedwtgeCyroaolctidu‰i1nrvyliedtrenaeaorrvmiis¢azoc(leini‰ucld.lea–almtloveyaf)asu•Dsbn2osc0tuoqatnubisiltteeser.avTgvaehetnieroearnrgeaeqellquryauirdaiemitmaiolpenllnoietcs•a(tt22iho2>an)t
wnl1/2
Don 0 andtheequationofstate(relatingdensityandenthalpyperturbations):
7. Y o –‰
55-59TOR –HDa2 ‰ (23)
5A
3.41:ERV Thethreedynamicalequationsmaybesolvedfor–v intermsof–H:
Astrophys. 200EWARD OBS –vR D 1Di ••!•¯2@@–@RH–H¡ 2˜mRm!¯ –H‚‚; (24)
Astro. by ST –v` D D 2˜ @R ¡ R –H ; (25)
v. i @–H
u. Re –vZ D¡!¯ @Z ; (26)
n
An where
D D•2¡!¯2: (27)
UsingEquations24–26inEquation22andsimplifyingtheresults,oneobtainsthe
linearwaveequationforthedisk:
• (cid:181) ¶ (cid:181) ¶
1 @ R‰ @ 1 @ @ m2‰
¡ ‰ ¡
R@R D @R !¯2@Z @Z R2D
(cid:181) ¶ ‚
1 @ 2˜m‰ ‰
C C –HD0: (28)
R!¯ @R D †2a2
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ANGULARMOMENTUMTRANSPORTINDISKS 563
We have inserted an artificial † factor in the sound speed term, which, although
formally equal to unity, will be used as an aid for sorting out asymptotic or-
dersinaWKBanalysis.(Thediskisherepresumedtobecoldinthesensethat
R˜(cid:192)a.)
ThelocationsatwhichD D0and!¯ D0aresingularitiesofthewaveequation,
althoughinthecaseoftheformerthesingularityisonlyapparent,notreal.They
areknownrespectivelyasLindbladandcorotationresonances,andtheirneighbor-
hoodsarezoneswherewavescouplestronglytothedisk.Theyareofimportance
inthestudyoftidallydrivenwavesandarecriticaltoanunderstandingofplanetary
g
or migration (Goldreich & Tremaine 1979, 1980; Ward 1997). In this section, our
s.
w emphasis will be on freely propagating WKB waves, and we shall assume that
e
vi neitherDnor!¯ issmall;i.e.,thatwearenotintheneighborhoodofresonance.
e
ualry.
m arjournals.annpersonal use onl 3.2.Tt3wi.o2.on1.s-DFoIfRimESTqeuOnaRtsiDiooEnRn2:a8DlIhWSaPvEKiRnSBgIOtWhNeaRfvEoLermAsTION A•ND GROUP‚VELOCITY We seek solu-
oaded fro5/07. For –HD A(R;Z) exp iS(R†;Z) : (29)
wnl1/2 Theideaisthatthephase S=† variesrapidly,andthe† factorensuresthisinthe
Don 0 formallimit† !0.Wewillsolvethewaveequationtoleadingandsecondorder
7. Y o ina1=† expansion.Notethattheabsolutephaseisnotrelevanthere,andwemay
59R
55-TO assume that A, the amplitude, is real. We shall also assume that the waves are
1:5VA tightlywound,i.e.,thatbothkR andkZ (cid:192)m=R.
3.4ER Inserting Equation 29 into Equation 28, we find that the leading order 1=†2
ys. 200D OBS termsgive
strophWAR !k¯2Z2 C !¯2k¡2R•2 D a12; (30)
AE
Astro. by ST where
Rev. (kR;kZ)D(@S=@R;@S=@Z): (31)
u.
n ThisisthedispersionrelationforWKBdiskwaves,andthegradientsofSare,in
n
A
essence,thewavenumbercomponents.
Figure1isaplotoftheconstantfrequencycurvesinthek a,k awavenumber
Z R
plane.For!¯ > •,theiso-!¯ curvesareellipses;for!¯ < •,theyarehyperbolae.
These two different conic sections define the two distinct wave branches, with
very different transport properties. Indeed, this diagram makes evident several
remarkablefeaturesofdiskwaves.
Theellipticaliso-!¯ surfacescorrespondtodensitywaves,whicharerotationally
modified sound waves (e.g., Goldreich & Tremaine 1979), and the hyperbolae
correspond to inertial waves (Vishniac & Diamond 1989), described below. It
is not difficult to show that the ellipses and hyperbolae always intersect at right
23Jul2003 20:32 AR AR194-AA41-14.tex AR194-AA41-14.sgm LaTeX2e(2002/01/18) P1:GCE
564 BALBUS
g
or
s.
w
e
vi
e
ualry.
m arjournals.annpersonal use onl
oaded fro5/07. For
wnl1/2
Don 0
7. Y o
55-59TOR Figure 1 Contours of constant !¯ for the dispersion relation (Equation 30) of a
1:5VA Kepleriandisk.Theseformasetofconformalellipses(densitywaves)andhyperbolae
3.4ER (inertialwaves).Thevalueof!¯ alongacurveisreadoffatthepointofintersection
ys. 200D OBS awriethsethpearkaRtedDb0yaoxnise;uthneitnfuomre!¯ric>al2scaanledi0s.i1nuunniittssfoofr˜1:1D<1.D!¯e<nsi2ty.wInaevretiaelllwipasvees
ophAR hyperbolaeareseparatedby.066units.
strW
AE
o. ST
Astrby angles,sothatthecurvesarelikeaconformalmapping.Thepracticalrelevanceof
ev. thisisthatbecausethewavegroupvelocityU isthewavenumbergradientof!¯,
R
u. (cid:181) ¶
nn @!¯ @!¯
A U D ; ; (32)
@k @k
R Z
thegroupvelocitydirectionofthedensitywavesliesalongtheinertialwaveiso-!¯
curves,andthegroupvelocitydirectionoftheinertialwavesliesalongthedensity
wave curves. Low frequency disturbances in disks have very different transport
propertiesfromhighfrequencydisturbances,apointtowhichweshallreturnmany
times.
Foragivenwavevector(k ;k ),thegradienttoaniso-!¯ curvecouldpointin
R Z
either direction because the dispersion relation (Equation 30) does not uniquely
determinethesignof!¯.Directcalculationrevealsthefollowing: