Enhanced Angular Momentum Transport in Accretion Disks

43 Pages·2003·0.33 MB·English

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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
DepartmentofAstronomy,VITA,UniversityofVirginia, Charlottesville,Virginia22901;
email:[email protected]
KeyWords accretion,instabilities,MHD,turbulence
n Abstract Thestatusofourcurrentunderstandingofangularmomentumtrans-
portinaccretiondisksisreviewed.Thelastdecadehasseenadramaticincreaseboth
intherecognitionofkeyphysicalprocessesandinourabilitytocarrythroughdirect
numerical simulations of turbulent flow. Magnetic fields have at once powerful and
subtleinfluencesonthebehaviorof(sufficiently)ionizedgas,renderingthemdirectly
unstabletofreeenergygradients.Outwardlydescreasingangularvelocityprofilesare
unstable.ThebreakdownofKeplerianrotationintoMHDturbulencemaybestudiedin
somenumericaldetail,andkeytransportcoefficientsmaybeevaluated.Chandraobser-
vationsoftheGalacticCentersupporttheexistenceoflowluminosityaccretion,which
mayultimatelyproveamenabletoglobalthree-dimensionalnumericalsimulation.
Ihatebeing“allowedfor,”asif
Iweresomeincalculablequantity
inanastronomicalequation.
—D.L.Sayers,TheDocumentsintheCase.
1.INTRODUCTION
Inrecentyears,accretiondisktransporttheoryhasdevelopedsorapidlythatany
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
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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
incompactobjectaccretiondisks.
Thewayaroundthisdifficultywasperceivedtobeturningthewoefullyinade-
quateviscositytoone’sadvantagebyappealingtotheassociatedlargeReynolds
number.This,itwasthought,wouldturndifferentialrotationintoshear-driventur-
bulence(e.g.,Crawford&Kraft1956,Shakura&Sunyaev1973).Thebreakdown
ofshearflowlaminaintoturbulencehasbeenknownsincetheworkofReynolds
tobetriggeredbynonlinearflowinstabilities.Thefactthattherewerenodemon-
strablelocallinearinstabilitiesforaKeplerianrotationprofilewasthereforenot
immediatelyviewedasanembarrassmenttothisscenario.
Shearturbulenceisadesirabletraitinaflowwheregreatlyenhanced(angular)
momentumtransportisneeded.Thisisbecauseshear-driventurbulenceischarac-
terizedbyahighdegreeofcorrelationbetweentheradialandazimuthalvelocity
fluctuations. This, as we shall see, has the direct effect of raising a disk’s angu-
lar momentum flux orders of magnitude above what would be possible with an
ordinarycollisionalviscosity.
Notopicinfluiddynamicsismorecontentiousthantheonsetanddevelopment
ofturbulence,andaccretiondiskturbulencehasnotbeenanexception.Keplerian
disksdonotresemblelaboratoryshearflows,howeverlocallyonepeers.Coriolis
forcesdramaticallystabilizerotationalflowonlargescalesandsmall,afeaturenot
sharedbyclassicalplanarCouettefloworPoiseuilleflow.1YetCoriolisforcesdo
noworkonthefluid;theyareinfactabsentfromtheenergyconservationequations.
Becauseapotentfreeenergysourceintheformofshearretainsapresencewith
orwithoutCoriolisforces,debatehascenteredonwhetherfluidnonlinearitiesata
high-enoughReynoldsnumberwouldstillfindawaytotapintothissourcewhen
lineardisturbancesfailtodoso.
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
process appears to be at work in shear layers (Pierrehumbert & Widnall 1982,
Corcos & Lin 1984). It is now generally accepted that the triggering of a rapid,
linearthree-dimensionalinstabilityofanearlyneutrallystable,two-dimensional,
finite-amplitudedisturbanceisaverygenericmodeofthebreakdownoflaminar
flowintoturbulence.
Howdoesthisbearonourunderstandingofaccretiondisks?Themostimpor-
tantpointisthatevenincompressibleaxisymmetricdisturbancesinarotatingfluid
willpropagateintheformoflinearinertialwaves(Lighthill1978)withacharac-
teristicfrequencyproportionaltothelocalvorticityoftherotationprofile.Thus,
in astrophysical disks, the finite amplitude, neutrally stable axisymmetric state
thatiscriticaltothetransitiontoturbulenceinmixinglayerscannotform.Only
in a low-vorticity rotational profile, in which the oscillation frequency is much
smallerthantheshearingrate,wouldweexpectabreakdownofflowlaminainto
turbulence,similartowhatisseeninamixinglayer.Thisisingoodaccordwith
three-dimensionalnumericalsimulations(Balbus,Hawley&Stone1996;Hawley,
Balbus&Winters1999),whichfindnononlinearlocalinstabilitiesinKeplerian
disksbutdoindeedfindthatshearlayersandlow-vorticitydisksshowanonlinear
breakdowntoturbulentflow.Thereisasofyetnoanalyticproofoflocalnonaxi-
symmetricstability,however,andthenotionthattheremaybeanothernonlinear
hydrodynamicalroutetoturbulenceinKepleriandisks,beyondthehighestreso-
lutionavailableinsupercomputersimulations,retainssomeadvocates(Richard&
Zahn1999).
Accretiondiskshaveonecriticallyimportantattributenotsharedwiththeclass-
icalhydrodynamicalfluids:Theyaregenerallymagnetized.Byengenderingnew
degreesoffreedomintheirhostfluid,evenveryweakmagneticfieldscompletely
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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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,
and it can be readily simulated. Both local (Hawley, Gammie & Balbus 1995;
Brandenburgetal.1995)andglobal(Armitage1998,Hawley2000)investigations
unambiguouslyshowabreakdownoflaminarKeplerianflowintowell-developed
turbulence.TheMRIistheonlyinstabilityshowntobecapableofproducingand
sustainingtheenhancedstressneededforaccretiontoproceedonviabletimescales
innon-self-gravitatingdisks.Atlowtemperaturesandhighdensities,e.g.,inthe
outerregionsofcataclysmicvariable(CV)systems(Gammie&Menou1998),or
inprotostellardisksonAUscales(Gammie1996),thelevelofMRI-inducedturbu-
lencecanchangerapidly,eruptingoreventurningoffcompletely.Allthisoccurs
whiletheunderlyingKeplerianprofileremainsessentiallyfixed.Inshort,theinsta-
bilityseemscapableofthefullrangeofaccretioncomplexitymanifestedinnature.
Forallthesereasons,despitethedifficultiesofcopingwithMHDturbulence,the
MRIisnowatthecenterofnumericalaccretiondiskstudies.
Letus,however,postponeourdiscussionofmagneticmattersandturnourat-
tention first to the study of simple hydrodynamical waves in disks. These are of
greatpracticalinterestintheirownright,especiallyinprotoplanetarydisks.But
it is also the case that understanding the transport properties of waves deepens
one’s understanding of turbulent transport, both hydrodynamic and magnetohy-
drodynamic,andthatisourprimaryreasonforreviewingthemhere.Weshallthen
followwithadiscussionofhydrodynamicalinstability,withafocusonhowglobal
instabilitycaninprincipleemergeinadifferentiallyrotatingdiskevenwhenthe
Rayleighconditionissatisfied.Magneticinstabilityandmagneticturbulenceare
thetopicsofthenextsection.Magneticstressesarethemostimportanttransport
mechanisminnon-self-gravitatingdisks,providedthegasisminimallyionizedto
coupletothefield.Thefinaltwosectionsareapresentationofrecentnumerical
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)
D QC¡Q¡ (4)
(cid:176) ¡1 dt
Equation1ismassconservation;Equation2isthedynamicalequationofmotion;
Equation3istheinductionequation;andEquation4istheentropyequation.Our
notationisstandard:‰isthemassdensity,vthefluidvelocity,Pthepressure(plus
radiationpressurewhenimportant),8thegravitationalpotential, Bthemagnetic
fieldvector,(cid:176) istheadiabaticindex, QC (Q¡)representheatgains(losses),·
V
themicroscopickinematicshearviscosity,and· themicroscopicresistivity.The
B
azimuthalcomponentoftheequationofmotiondeservesseparatemention,asit
isadirectexpressionofangularmomentumconservation:
• (cid:181) ¶ ‚
@(‰@Rtv`) Cr¢ ‰Rv`v ¡ R4B…`BC P C 8B…2 e` D0; (5)
wheree`isaunitvectorinthe`direction.Thedissipativetermshavebeendropped
because they appear only in the flux term, transporting a negligible amount of
angularmomentum.
Itisalsousefultohaveathandanequationfortotalenergyconversation.This
issomewhatlengthytoderive(Balbus&Hawley1998),buttheresultisreadily
interpreted:
@E
@t Cr¢FE D¡Q¡; (6)
wheretheenergydensityE is
1 P B2
E D ‰v2C C‰8C (7)
2 (cid:176) ¡1 8…
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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hand,representsgenuinesystemicenergylossesandappearsexplicitlyinthecon-
servationequation.
2.2.NonlinearFluctuations
Bothwavesandturbulenceinvolvetheconceptofwell-defineddeparturesofthe
flow from a smooth background. Velocity fluctuations are of particular interest
becauseitispossibletoformulateanexactenergyconservationlawforthefluctu-
ationsthemselves.This,inturn,explicitlyshowstheroleofdifferentialrotationas
asourceoffreeenergyforthe(correlated)turbulentfluctuationsassociatedwith
outwardtransportofangularmomentum.
Letusdefinethevelocityfluctuationuby
uDv ¡ R˜(R)e`: (9)
˜isinprinciplearbitrary,butofcoursethemotivationforthisdefinitionisthat
˜(R)isareasonablygoodapproximationtoanunderlyingrotationprofileforthe
accretionflow.Itispossibletocombinetheequationofmotion(Equation2)with
theinternalentropyequation(Equation4)toobtainanexact,` averagedenergy
equationfortheuvelocityfluctuationsalone:
(cid:181) ¶
@@Etu Cr¢FEu D¡ ‰uRu` ¡ B4R…B` dd˜R ¡Q¡: (10)
Here,E isthefluctuationenergydensity
u
1 P B2
E D ‰(u2C8 )C C ; (11)
u 2 eff (cid:176) ¡1 8…
8 isaneffectivepotentialfunction
eff
Z
R
8 D8¡ R˜2dR; (12)
eff
andFEuistheenergyfluxofthefluctuationsthemselves:
(cid:181) ¶
1 (cid:176)P B
FEu Du 2‰u2C (cid:176) ¡1 C‰8eff C 4… £(u£B): (13)
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.
3.HYDRODYNAMICWAVESINDISKS
3.1.TheLinearWaveEquation
Consideranunmagnetizeddiskinwhichthepressureanddensityobeyasimple
polytropicequationofstate, P D K‰(cid:176),whereKisaconstant.Wemaydefinean
enthalpyfunctionH:
Z
dP (cid:176)P=‰ a2
HD D D ; (15)
‰ (cid:176) ¡1 (cid:176) ¡1
wherea2 istheadiabaticsoundspeed.Itisconvenienttoworkinstandardcylin-
dricalcoordinates(R;`;Z).Thegasrotatesinthegravitationalfieldofacentral
mass.Theangularvelocity˜mustbeconstantoncylinders,˜D˜(R)(Tassoul
1978).
Althoughitisastandardapproximation,theassumptionofabarotropicequa-
tionofstateisobviouslyanidealization.Amongothershortcomings,itprecludes
the possibility of a buoyant response in the form of internal gravity waves due
toBrunt-Va¨isa¨la¨ oscillations(Ogilvie&Lubow1999).Instandarddiskmodels,
however,entropystratificationarisesbecauseofradiativeheatdiffusionfromtur-
bulent heating. A linearized wave treatment of such an “equilibrium” is at best
adelicatematter.Ingeneral,theverticaltemperaturestructureofaccretiondisks
isnotwellunderstood,andthevirtueofadoptingabarotropicpressureisthatit
allowsimportantdynamicalbehaviortoberevealed.
Our goal is to study how linearized wave disturbances transport energy and
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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•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•:
d˜2
•2 D4˜2C : (21)
dlnR
The epicyclic frequency is the rate at which a point mass in a circular motion,
disturbedintheplaneofitsorbit,wouldoscillateaboutitsaverageradiallocation
(Binney & Tremaine 1987). A negative value of •2 quite generally implies that
axisymmetricdisturbancesarehydrodynamicallyunstable.Therequirement•2 >
0isknownastheRayleighstabilitycriterion.
Theremainingequationsarethelinearizedmassconservationequation
–‰ 1
¡i!¯ C r¢(‰–v)D0 (22)
‰ ‰
andtheequationofstate(relatingdensityandenthalpyperturbations):
–‰
–HDa2 (23)
‰
Thethreedynamicalequationsmaybesolvedfor–v intermsof–H:
• ‚
i @–H 2˜m
–v D !¯ ¡ –H ; (24)
R D @R R
• ‚
1 •2 @–H m!¯
–v` D D 2˜ @R ¡ R –H ; (25)
i @–H
–v D¡ ; (26)
Z !¯ @Z
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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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
migration (Goldreich & Tremaine 1979, 1980; Ward 1997). In this section, our
emphasis will be on freely propagating WKB waves, and we shall assume that
neitherDnor!¯ issmall;i.e.,thatwearenotintheneighborhoodofresonance.
3.2.Two-DimensionalWKBWaves
3.2.1. FIRST ORDER: DISPERSION RELATION AND GROUP VELOCITY We seek solu-
tionsofEquation28havingtheform
• ‚
iS(R;Z)
–HD A(R;Z) exp : (29)
†
Theideaisthatthephase S=† variesrapidly,andthe† factorensuresthisinthe
formallimit† !0.Wewillsolvethewaveequationtoleadingandsecondorder
ina1=† expansion.Notethattheabsolutephaseisnotrelevanthere,andwemay
assume that A, the amplitude, is real. We shall also assume that the waves are
tightlywound,i.e.,thatbothk andk (cid:192)m=R.
R Z
Inserting Equation 29 into Equation 28, we find that the leading order 1=†2
termsgive
k2 k2 1
Z C R D ; (30)
!¯2 !¯2¡•2 a2
where
(k ;k )D(@S=@R;@S=@Z): (31)
R Z
ThisisthedispersionrelationforWKBdiskwaves,andthegradientsofSare,in
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
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Figure 1 Contours of constant !¯ for the dispersion relation (Equation 30) of a
Kepleriandisk.Theseformasetofconformalellipses(densitywaves)andhyperbolae
(inertialwaves).Thevalueof!¯ alongacurveisreadoffatthepointofintersection
withthek D0axis;thenumericalscaleisinunitsof˜D1.Densitywaveellipses
R
are separated by one unit for !¯ > 2 and 0.1 units for 1:1 < !¯ < 2. Inertial wave
hyperbolaeareseparatedby.066units.
angles,sothatthecurvesarelikeaconformalmapping.Thepracticalrelevanceof
thisisthatbecausethewavegroupvelocityU isthewavenumbergradientof!¯,
(cid:181) ¶
@!¯ @!¯
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: