Possible origin of viscosity in the Keplerian accretion disks due to secondary perturbation: Turbulent transport without magnetic field

0.2 MB·English

by  Banibrata Mukhopadhyay

ResearchinAstron.&Astrophys.Vol.0(200x)No.0,000–000
Researchin
(http://www.raa-journal.org)
Astronomyand
Astrophysics
Possible origin of viscosity in the Keplerian accretion disks due to
secondary perturbation: Turbulent transport without magnetic field
1
1
0 BanibrataMukhopadhyay andKanakSaha
2
AstronomyandAstrophysicsProgram,DepartmentofPhysics,IndianInstituteofScience,Bangalore
n
560012,India;[email protected]
a
J

4
Abstract The origin of hydrodynamicturbulencein rotating shear flow is a long standing
2
puzzle. Resolving it is especially importantin astrophysicswhen the flow angularmomen-

tumprofileisKeplerianwhichformsanaccretiondiskhavingnegligiblemolecularviscosity.
]
E Hence,anyviscosityinsuchsystemsmustbeduetoturbulence,arguablygovernedbymag-
H netorotationalinstabilityespecially whentemperatureT∼>105. However,suchdisks around
. quiescentcataclysmicvariables,protoplanetaryandstar-formingdisks, theouterregionsof
h
disksin activegalacticnucleiare practicallyneutralinchargebecauseoftheirlow temper-
p
ature,andthusexpectednottobecoupledwiththemagneticfieldappropriatelytogenerate
-
o any transportdue to the magnetorotationalinstability.This flow is similar to plane Couette
r flowincludingtheCoriolisforce,atleastlocally.Whatdrivestheirturbulenceandthentrans-
t
s port,whensuchflowsdonotexhibitanyunstablemodeunderlinear hydrodynamicpertur-
a
bation? We demonstrate that the threedimensional secondary disturbance to the primarily
[
perturbed flow triggering elliptical instability may generate significant turbulent viscosity

1 ranging0.0001∼<νt∼<0.1toexplaintransportinaccretionflows.
v
3
Keywords: accretion,accretiondisks—hydrodynamics—turbulence—instabilities
1
6
4
. 1 INTRODUCTION
1
0
Oneofthemainproblemsbehindtheoriginofhydrodynamicturbulenceinshearflowisthatthereisasig-
1
nificantmismatchbetweenthepredictionsoflineartheoryandexperimentaldata.Forexample,inthecase
1
ofplaneCouetteflow,laboratoryexperimentsandnumericalsimulationsshowthattheflowmaybeturbu-
:
v lentataReynoldsnumberaslowasRe∼350,whileaccordingtothelineartheorytheflowshouldbestable
i
X for all Re. Similar mismatch between theoretical results and observationsis found in astrophysical con-
texts,wheretheaccretionflowofneutralgaswithKeplerianangularmomentumprofile,whichessentially
r
a behaveslikerotatingshearflow,isacommonsubject.Examplesofsuchflowsystemsareaccretiondisks
around quiescent cataclysmic variables (Gammie&Menou1998), protoplanetary and star-forming disks
(Blaes&Balbus1994),andtheouterregionsofdisksinactivegalacticnuclei(Menou&Quataert2001).
A Keplerianaccretiondisk flow havinga verylow molecularviscosity mustgenerateturbulenceand
successively diffusive viscosity, which supportthe transfer of mass inwards and angular momentumout-
wards. However, theoretically this flow, in absence of magnetic field, never exhibits any unstable mode
which could trigger turbulence in the system. On the other hand, the laboratory experiments of Taylor-
Couette systems, which are similar to Keplerian disks, seem to indicate that although the Coriolis force
delays the onset of turbulence, the flow is ultimately unstable to turbulencefor Reynoldsnumberslarger
thanafewthousand(Richard&Zahn2001),evenforsubcriticalsystems.Indeed,Bech&Anderson(1997)
seeturbulencepersistinginnumericalsimulationsofsubcriticalrotatingflowsforlargeenoughReynolds
numbers.
How does shearing flow that is linearly stable to perturbationsswitch to a turbulentstate? Since last
decade, many authors including ourselves have come forward with a possible explanation of this fact
2 Mukhopadhyay&Saha
based on bypass transition (see, Butler&Farrell1992, Reddy&Henningson1993, Trefethenetal.1993,
Chagelishvilietal.2003, Umurhan&Regev2004, Mukhopadhyayetal.2005 and references therein)
where the decaying linear modes show an arbitrarily large transient energy growth at a suitably tuned
perturbation. In lieu of linear instabilities e.g. magnetorotationalinstability, the transient energy growth,
supplementedbyanon-linearfeedbackprocesstorepopulatethegrowingdisturbance,couldplausiblysus-
tainturbulenceforlargeenoughReynoldsnumbers.
The behavior of shear flows, however, in the presence of rotation is enormously different compared
to that in absence of rotation. The Coriolis effect is the main culprit behind this change in behavior
killing any growth of energyeven of transient kind in the presence of rotation. In the case of shear flow
with a varying angular velocity profile, e.g. Keplerian accretion flow, the above mentioned transient en-
ergy growth is insignificant for threedimensional perturbations. To overcome this limitation, it is nec-
essary to invoke additional effects. Various kinds of secondary instability, such as the elliptical insta-
bility, are widely discussed as a possible route to self-sustained turbulence in linearly perturbed shear
flows (see, e.g. Pierrehumbert1986, Bayly1986, Craik&Criminale1986, Landman&Saffman1987,
Hellberg&Orszag1988, Waleffe1989, Craik1989, LeDizes´etal.1996, Kerswell2002). These effects,
which generate threedimensional instabilities of a twodimensional flow with elliptical streamlines, have
been proposed as generic mechanism for the breakdown of many twodimensional high Reynolds num-
ber flows whose vortex structures can be locally seen as elliptical streamlines. Recently, one of the
present authors has studied the secondary perturbation and corresponding elliptical vortex effects in ac-
cretion disks and pinpointed that they can be the seed of threedimensional hydrodynamic instability
(Mukhopadhyay2006).Subsequently,bynumericalsimulation,thishasbeenshowntobeoneofthepos-
sible sources to generate turbulence to form large objects from the dusty gas surrounding a young star
(Cuzzi2007,Ormeletal.2008).Moreover,vortexgenerationintheunmagnetizedprotoplanetarydiskshas
beenfurnishedbyhydrodynamicturbulence(deVal-Borroetal.2007)whichleadstoplanetformation,and
angularmomentumtransportindisks.However,whethertheyleadtonon-linearfeedbackandthreedimen-
sionalturbulenceareyettobeshownexplicitly.
Herewe planto showin detailthatthreedimensionalsecondaryperturbationgeneratinglargegrowth
intheflowtimescalemaygeneratesignificantturbulentviscosityinrotatingshearflows,morepreciselyin
planeshearflowswiththeCoriolisforce.TheplaneshearflowwiththeCoriolisforceessentiallybehaves
asalocalpatchofarotatingshearflow.Possibilityofsignificantturbulenttransportinsuchflowsbythree-
dimensionalperturbationopensa new window to explainaccretion processin flows which are neutralin
charge.Inparticular,weaddresstheissueofderivingturbulentviscosityandtheShakura-Sunyaevviscosity
parameterα(Shakura&Syunyaev1973)fromapurehydrodynamicalperspective1.Thisisimportantfor
understandingaccretionflowsincoldchargeneutralmedium.
Itisimportanttonotethattransitiontoturbulenceisnotauniqueprocess,butitdependsontheinitial
condition/disturbance and the nature of the flow (Schmid&Henningson2001, Criminaleetal.2003). In
fact,itisknownthateveninthepresenceofsecondaryinstability,linearlyunstablebaseflowsmayreach
toanon-turbulentsaturatedstate.However,turbulencedefinitelybelongstothenonlinearregimeanditis
exhibitedonlyinthesituationswhenlargegrowthofperturbationswitchesthesystemoverthenon-linear
regime.Asourpresentgoalistounderstandthepossibleoriginofhydrodynamicturbulence,weconsider
thosesituationswhenlargeenergygrowthgovernsnon-linearity.
Thepaperisorganizedasfollows.Inthenextsection,wefirstrecalltheperturbationestablishedpre-
viously (Mukhopadhyay2006) due to secondary disturbance in the Keplerian flow and then discuss the
rangeofcorrespondingReynoldsnumberandthesolutions.Subsequently,weestimatethecorresponding
turbulentviscosityofhydrodynamicoriginin§3.Weendin§4bydiscussingimplicationsofourresults.
2 PERTURBATIONANDRANGEOFREYNOLDSNUMBER
Considering a twodimensional velocity perturbation w = (w (x,y,z,t),w (x,y,z,t),0), and pressure
x y
perturbationp (x,y,z,t)inasmallsectionoftheKeplerianshearflow/disk,thelinearizedNavier-Stokes
p
and continuity equations for the incompressible fluid with plane background shear in the presence of a
1 A preliminary calculation of such α has been appeared in a collected volume of Gravity Research Foundation
(Mukhopadhyay2008).
Originofviscosityinaccretiondisksduetosecondaryperturbation 3
Corioliscomponentcanbewrittenindimensionlessunitsas(seeMukhopadhyayetal.2005foradetailed
description)
dw ∂p 1
x =2Ωw − p + ∇2w , (1)
y x
dt ∂x Re
dw ∂p 1
y =Ω(q−2)w − p + ∇2w , (2)
x y
dt ∂y Re
∂w ∂w
x + y =0. (3)
∂x ∂y
We considerthe standardno-slip boundaryconditionsuch that w = w = 0 at x = ±1 and according
x y
tothechoiceofvariablesinthecoordinatesystemΩ = 1/q.Here(x,y,z)isalocalCartesiancoordinate
systemcenteredatapoint(r,φ)inthedisk(Mukhopadhyayetal.2005)suchthatdr =xandrdφ=y.
When the Reynolds number is very large, the solution of eqns. (1), (2) and (3) are given by
(Mukhopadhyayetal.2005)
k k
w =ζ y sin(k x+k y), w =−ζ x sin(k x+k y) (4)
x l2 x y y l2 x y
whereζ isthe amplitudeof vorticityperturbation,k andk are the componentsofprimaryperturbation
x y
wavevectorandl= k2+k2.Underthisprimaryperturbation,theflowvelocityandpressuremodifyto
x y
q
U = Up+w =(w ,−x+w ,0)=A.d, P¯ =p¯+p , (5)
x y p
whereUp,p¯arebackgroundvelocityandpressurerespectively,Aisatensorofrank2.Herek =k +k t,
x x0 y
whichbasicallyistheradialcomponentofprimaryperturbationwavevector,varyingfrom−∞toasmall
number,wherek isalargenegativenumber:|k |∼Re1/3 ∼t (Mukhopadhyayetal.2005).
x0 x0 max
Nowweconcentrateonafurthersmallpatchoftheprimarilyperturbedflowsuchthatthespatialscale
isverysmallcomparedtothewavelengthofprimaryperturbationsatisfyingsin(kxx+kyy)∼kxx=f∼<1.
Infact,f ∼1atclosetotheboundaryofthepatchwheny →0 and 2π/k ,andatanintermediatelocation
y
f ≪1.As|k |variesfromalargenumbertoclosetounity,thesizeoftheprimaryperturbationboxinthe
x
x-directionis 1/kx∼<1whenky ∼ 1,fixed.Hence,thisfurthersmallpatchmustbe confinedtoa region:
−a∼<x∼<a,whenf/|kx0|∼<a∼<f.Clearly,inthispatch,U ineqn.(5) describesa flowhavinggeneralized
ellipticalstreamlineswithǫ=(k /l)2,aparameterrelatedtothemeasureofeccentricity2,runningfrom0
x
to1astheperturbationevolves.Itwasalreadyshown(Mukhopadhyay2006)thatasecondaryperturbation
inthisbackgroundmaygrowexponentiallyleadingtheflowunstable.Weusethisunstableflowin§3,which
wasextensivelydiscussedearlier(Mukhopadhyay2006),toderiveν andα.
t
Aswefocusonthesecondaryperturbationatasmallpatchoftheprimarilyperturbedshearingbox,the
variationofprimaryperturbationappearsinsignificantinthepatchcomparedtothatofthesecondaryone.
Dependingonthe primaryperturbationwavevectorata particularinstant, thesize ofthe secondarypatch
isappropriatelyadjusted.Infactǫvariesveryveryslowlyandmarginallydeviatesfromunityin thetime
intervalwhen k varies from k (large negative)to, say, −10. Even when k tends to −3, ǫ changesto
x x0 x
∼0.9only.Therefore,ǫandthusApracticallyremainsconstant.
2.1 RangeofReynoldsnumber
Duetoconsecutivechoiceofsmallboxes/patches,theReynoldsnumberinthesecondaryflowisrestricted
withaparticularchoiceofthatintheprimaryflow.Hereintheinterestofclarity,weworkwiththeoriginal
dimensionedunits.TheReynoldsnumberattheprimaryboxisdefinedas
U L qΩ L2
Re = 0 = 0 , (6)
p
ν ν
2 Notethatǫisaparameterrelatedtothemeasureofeccentricitybutnottheeccentricityitself.
4 Mukhopadhyay&Saha
where 2L is the box size in the x-direction and 2U is the relative velocity of the fluid elements in the
0
boxbetweentwowallsalongthey-direction.Nowwerecallthesecondaryperturbationatasmallerpatch,
extendedfrom−L to+L ,suchthat|L |∼aL.Tomeetourrequirementsin(k x+k y)∼k x+k y,
s s s x y x y
weremindthatthesmallpatchsizeneedstobeadjusted.Therefore,theReynoldsnumberatthesecondary
boxisgivenby
qΩ L2 qΩ a2L2
Re = 0 s ∼ 0 . (7)
s
ν ν
Hence,
Re 1 k2
p ∼ ∼ x. (8)
Re a2 f2
s
At the beginningof the primaryperturbationk = k and thusǫ = 1. At this stage, the secondarybox
x x0
size Ls = Lf/kx0 and Rep∼>kx20Res. With time kx decreases in magnitude but ǫ deviates little from
unity until k ∼ −3 when ǫ = 0.9. Hence A can be considered constant approximately as described
x
above. At this stage Re ≥ 9Re , atleast an order of magnitude higher than Re . If the energy growth
p s s
due to primary perturbation is maximized for k = k = π (Mukhopadhyayetal.2005), then the
x x,min
range of Re for the secondary perturbationis given by Repf2/kx20∼<Res∼<Repf2/10. At kx = π, Res
is atleast an order of magnitudelower than Re . When k = 1, Re ∼ Re for f ∼ 1. In general
p x,mim p s
Repf2/kx20∼<Res∼<Repf2/kx2,min.
2.2 Solution
Followingpreviouswork (Mukhopadhyay2006), the generalsolution forthe evolutionof secondaryper-
turbationintheflowdiscussedabovecanbewrittenintermsofFloquetmodes
u (t)=exp(σt)f (φ)exp[i(k x+k y+k z)], (9)
i i 1 2 3
where φ = ̟t, f (φ) is a periodic function having time-period T = 2π/̟, σ is the Floquetexponent,
i
k ,k ,k arethecomponentsofwavevectorofthesecondaryperturbation.Notethatσisdifferentatdiffer-
1 2 3
entǫ.Clearly,ifσispositive,thenthesystemisunstable.Thedetailedsolutionswerediscussedelsewhere
(Mukhopadhyay2006)whatwewillnotrepeathere.
Inprinciple,k varieswithtimeandthusAdoesso.Thus,generalizingthesolution(9)fora(slowly)
x
varyingA,weobtain
u (t)=exp σ(t)dt f (φ)exp[i(k x+k y+k z)], (10)
i i 1 2 3
(cid:18)Z (cid:19)
where φ = ̟(t)dt. The eqns. (9) and (10) practically describe the solutions for the entire parameter
regimeexhibRitingellipticalvorticeswhichareveryfavorablefortheellipticalinstabilitytotrigger.
Forthepresentpurpose,thephysicallyinterestingquantityistheenergygrowthofperturbationwhich
isgivenby
|u (t)|2 f2(φ)
G= i =exp[2Σ(t)] i , (11)
|u (0)|2 f2(0)
i i
where Σ(t) = σ(t)dt and t = (k −k )/k . As k (t) varies from a large negativevalue, k , to 0,
x x0 y x x0
t increases fromR 0 to tmax = −kx0/ky. Thus, the energy growth is controlled by the quantity Σ(t), as
f2(φ)/f2(0) simplyappearsto be a phasefactor. Therefore,ouraim shouldbe to evaluateΣ forvarious
i i
possibleperturbations.
LetusspecificallyconcentrateontheKeplerianaccretionflows.Figure1ashowsthevariationofmax-
imum velocity growth rate, σ , as a function of eccentricity parameter, ǫ, for the various choices of
max
amplitudeof vorticity,ζ. By “maximum”we refer the quantity obtainedbymaximizingoverthe vertical
componentof the wavevector,k . At large ǫ (as well as large k ), when ζ is large, the backgroundflow
3 x
structure,A,isellipticalwithhigheccentricity.Thereforeaverticalperturbationtriggersthebestgrowing
Originofviscosityinaccretiondisksduetosecondaryperturbation 5
Fig.1 (a) Variation of maximum velocity growth rate as a function of eccentricity parameter.
Solid, dotted, dashed and long-dashed curves indicate the results for ζ = 0.01,0.05,0.1,0.2
respectively (Mukhopadhyay2006). (b) Variation of Σ as a functionof time for k = −105,
x0
whenvariouscurvesaresameasof(a).(c)Sameas(b)butfork = −104.Otherparameters
x0
arek =1,k =0,|k |=1,andq =3/2.
y0 10 0
6 Mukhopadhyay&Saha
modeintothesystem.However,withthedecreaseofζ,Aapproachestothatoftheplaneshearandthusthe
growthrate decreasessignificantly.Atthis stage,the correspondingbest perturbationisthreedimensional
butnottheverticalone.
At smallǫ (andthen small k ), when ζ is large the eccentricityof the backgroundelliptical flow de-
x
creases significantly, and thus the growth rate decreases. In this low eccentric flow, the best growth rate
arisesduetothetwodimensionalperturbation.Ontheotherhand,whenζ issmall,thebackgroundreduces
to that of the plane shear flow. Therefore, the growth rate increases according to the shearing effects, as
describedbyMukhopadhyayetal.2005.Aninterestingfacttonoteisthatexceptthecaseofsmallǫ(k )
x
withalargeζ,thegrowthratemaximizesforthethreedimensionalperturbation.Moreover,atalargeζ and
alargeǫ,thebestgrowthratearisesduetoavertical(oralmostvertical)perturbation.
Astheaccretiontimescaleisanimportantfactor,forthepresentpurpose,physicallyinterestingquantity
is Σ rather than σ itself. Figures 1b,c show the variation of Σ as a function of t at various ζ. As the
perturbationevolveswithtime,thecorrespondingΣincreases.ItisalsoclearthatΣandthencorresponding
growthincreaseswiththeincreaseof|k |(andthenRe),i.e.theincreaseofaccretiontimescale,inaddition
x0
to the increase of ζ. In Table 1, we enlist the approximatevalues of maximum growth factor, as follows
fromeqn.(11),correspondingtoΣ = tmaxσdt,forthecasesshowninFigs.1b,c.Whenk =−104,
max 0 x0
Rep ∼ 1012 (asRep ∼ t3max ∼ kx30)andRfromeqn.(8)Res(f = 1)∼>104,themaximumgrowthfactoris
significantforalargeamplitudeofvorticityperturbationi.e.ζ >0.1.However,thegrowthfactorincreases
withtheincreaseofRep andwhenRep ∼ 1015,andthenRes(f = 1)∼>105,itisquitesignificantforan
amplitudeofvorticityperturbationsassmallas0.05.Therefore,itappearsthatasuitablethreedimensional
secondary perturbation efficiently triggers elliptical instability and possible turbulence in rotating shear
flowsincludingaccretiondisks.
Table1
Maximumenergy growthcorresponding tocasesshowninFigs.1b,c
|k | ζ Σ G
x0 max max
105 0.2 6.1 2×105
105 0.1 5.2 3.3×104
105 0.05 4.43 7×103
105 0.01 1.97 52
104 0.2 3.65 1500
104 0.1 3 400
104 0.05 2.9 330
104 0.01 1.27 13
3 TURBULENTVISCOSITY
Hereweattempttoquantifytheturbulencebyparametrizingitintermsoftheviscosity.Thisisessentially
important, as explained in §1, in flows like astrophysical accretion disks, where molecular viscosity is
negligible,toexplainanytransporttherein.
Thetangentialstressatapoint(r,φ)ofarotatingflowexhibitingturbulenceis
dΩ
W =ν r =−ν qΩ, (12)
rφ t t
dr
where ν is the turbulent viscosity and Ω = Ω (r/r )−q. Note that q = 3/2 for the Keplerian angular
t 0 0
velocityprofile.Theperturbationdescribedaboveisexpectedtogovernthenonlinearityaftercertaintime,
say t . We also assume that the nonlinearity leads to turbulence attributing the fact that at the initiation
g
of turbulence the eddy velocity is same as the perturbation velocity. Therefore, we obtain the averaged
tangentialstressduetoperturbationatt=t
g
T (t )→T (t )=<u u >
rφ g xy g x y
k +Ls 2π/k2
= 2 u (t )u (t )dxdy, (13)
x g y g
4πLs Z−Ls Z0
Originofviscosityinaccretiondisksduetosecondaryperturbation 7
whereweremindthattheazimuthalflowisconsideredtobeperiodiciny =2π/k .
2
Nowcombiningeqns.(12),(13)andaftersomealgebraweobtain
T
ν¯ =− xy (14)
t qΩ h M
r
(cid:0) (cid:1)
whereT = W dxdy,M =Ωx/c andν¯ denotestheaveragedν inthesmallsection,computedhere
xy xy s t t
att=tg. R
WithoutanyproperknowledgeofturbulenceinKeplerianflowswhichariseinaccretiondisks,Shakura
&Sunyaev(Shakura&Syunyaev1973)parametrizeditbyaconstantαconsideringW tobeproportional
rφ
tothesoundspeed,c ,givenby
s
W =−αc2. (15)
rφ s
αiscalledtheShakura-Sunyaevviscosityparameter.Theyassumedthatthesmallsectionunderconsidera-
tiontobeisotropicsothatscaledthecharacteristiclengthl ofturbulenceintermsofthelargestmacroscopic
t
lengthscaleofthedisk,i.e.half-thicknessh,andtheeddyvelocityofturbulencev intermsofsoundspeed
t
c .Thustheydefinedtheturbulentviscosity
s
l v
ν = t t =αc h, (16)
t s
3
wherel =αh,v =α c ,α=αα /3.Obviouslyα ≤1.Iftheturbulentvelocitybecomessupersonic,
t l t v s l v l
thenshockformsandreducesthevelocitybelowthesoundvelocitywhichassuresαv ≤1.Therefore,α∼<1.
Fromeqns.(14)and(16)wewrite
T
α¯ =− xy , (17)
qΩ2 h 3Mr2
r
(cid:0) (cid:1)
whereα¯ denotestheaveragedαinthesmallsection.Therefore,ifweknowthestructureoftheflow,then
wecancomputetheturbulentviscosityduetovariousperturbations.Asweconsiderthesizeofthesection
tobeverysmall,α¯andν¯ areeffectivelyequivalenttoαandν ataparticularpositioninthedisk.Belowwe
t t
computeT forthevarioussecondaryperturbationsandthecorrespondingturbulentviscosities,atleastin
xy
certainapproximations.
3.1 Secondaryperturbationevolvesmuchrapidlythantheprimaryone
Fromeqn.(9)wecanwritethevelocityperturbationcomponents
u (x,y) = A eσtf (φ)sin(k x+k y+k z),
x x x 1 2 3
u (x,y) = A eσtf (φ)sin(k x+k y+k z), (18)
y y y 1 2 3
where A and A are the amplitudes of perturbation modes, k ,k are the radial and the azimuthal
x y 10 20
componentsrespectivelyofthesecondaryperturbationwavevectoratt = 0,A andA canbeevaluated
x y
bytheconditionthatthe velocitycomponentsofthesecondaryperturbationreducetothatoftheprimary
perturbationatt=0(atthebeginningoftheevolutionofsecondaryperturbation)givenby
k C k (ǫ) C
y x
A = ζ , A =−ζ ,
x l2(ǫ)f (0) y l2(ǫ) f (0)
x y
sin(k (ǫ)x+k y)
C = x y , (19)
sin(k x+k y+k z)
10 20 30
where k (ǫ) = ǫ/(1−ǫ)k , C is of the order of unity (for details see Mukhopadhyayetal.2005,
x y
Mukhopadhyay20p06).Therefore,fromeqn.(13)
k (ǫ)k
T (t ) ∼ −ζ2 x y e2σtgD,
xy g 2l4(ǫ)
f (φ)f (φ)
D = C2 x y . (20)
f (0)f (0)
x y
8 Mukhopadhyay&Saha
Nowbyconsideringatypicalcasewithk =0.71,ν andαcanbecomputedasfunctionsǫ(k ),whenwe
y t x
knowthetimeofevolutionofthesecondaryperturbationt .
g
Figure2describesν andαaccordingtoeqns.(14),(17)and(20)forvariousdiskparameters.Asthe
t
primary perturbationevolves, elliptical vortices form into the shearing flow which generate the turbulent
viscosity under a further perturbation. Figure 2a shows that the viscosity varies with the eccentricity of
vortices. At a veryearly stage when the primaryperturbationis effectivelya radialwave and ǫ → 1, the
maximumvelocitygrowthrateduetosecondaryperturbation,σ (showninFig.1a),andthecorrespond-
max
ing turbulentviscosity are verysmall, independentof the valueofζ. With time, the primaryperturbation
wavefrontsarestraightenedoutbytheshearuntilt = t ,whentheperturbationbecomeseffectivelyan
max
azimuthalwaveandǫ → 0.Atthisstage,σ andtheturbulentviscosityduetothesecondaryperturba-
max
tionbecomezeroagain.Thisfeatureisclearlyunderstoodfromeqn.(20).However,atanintermediatetime
whenk (ǫ)isfinite,ν maybe∼0.005eveninamoderatelyslimdiskwithh(r)/r =0.05,whenthetime
x t
of evolutionof secondaryperturbationt = 10. Thist is consideredto be the time at which turbulence
g g
istriggeredinthesystem.Figures2b-dshowthevariationofν andαwiththeeccentricityofvorticesat
t
variousζ whent =10,100.Itisinterestingtonote,particularlyfort =100,thatwiththeincreaseofζ,
g g
firstviscosityincreasesthendecreases.Thisisunderstoodfromtheunderlyingenergygrowthrateshown
inFig.1a,whenthereadersareremindedthatσ =σ(ζ,ǫ).Notethatthequalitativebehaviorofν issame
t
asthatofα.Ifwelookatatypicalcasewithζ = 0.05whereσ =σ atǫ= 0.86whichcorrespondsto
max
kx =−1.76,thenαandνtcomputedatt=tg areforRes∼<Rep ∼108.
3.2 Secondaryperturbationovertheslowlyvaryingprimaryperturbation
Inprinciple,theprimaryperturbationmayvarywithtimeduringtheevolutionofsecondaryperturbation.
Bynumericalsolutions,simultaneousevolutionoftheprimaryandthesecondaryperturbationalongwith
thecorrespondingenergygrowthhasalreadybeendiscussedearlier(Mukhopadhyay2006).Fortheconve-
nienceofanalyticalcomputationofviscosity,hereweconsidertheregimeofslowvariationoftheprimary
perturbationcomparedtothesecondaryone.Hencewerecalleqn.(10)andwritethevelocityperturbation
components
u → u (x,y)=B eΣ(t)f (φ)sin(k x+k y+k z),
x xΣ x x 1 2 3
u → u (x,y)=B eΣ(t)f (φ)sin(k x+k y+k z), (21)
y yΣ y y 1 2 3
with φ = ̟(t)dt. The amplitudes of perturbation modes B and B can be evaluated by the initial
x y
conditionofRsecondaryperturbation.Thesecondaryperturbationcouldtriggerellipticalinstabilityonlyafter
significantvortexformsintheflowduetotheevolutionofprimaryone.Atthebeginningoftheevolution
of primary perturbation k → −∞ (we choose the cases k = −105 and −104) which corresponds
x0 x0
to ǫ → 1 and thus effectively a plane shear background when ζ is small (see Mukhopadhyay2006). In
absenceofvortex,thiscannottriggerellipticalinstabilityunderasecondaryperturbation.Ask decreases
x0
in magnitude, ǫ deviates from unity giving rise to a background consisting of elliptical vortices. Above
certain ǫ = ǫ , the secondary perturbation does not have any effect to the primarily perturbed flow and
c
u andu reducetotheprimaryperturbation.Wehypothesizethatǫ = 0.9999.Hence,B andB are
xΣ yΣ c x y
computedinasimilarfashionasin§3.Agivenby
k C k (ǫ ) C
y x c
B = ζ , B =−ζ ,
x l2(ǫ )f (0) y l2(ǫ ) f (0)
c x c y
sin(k (ǫ )x+k y)
C = x c y . (22)
sin(k x+k y+k z)
10 20 30
Hence,fromeqn.(13)thestresstensor
k (ǫ )k
T (t ) ∼ −ζ2 x c y e2ΣmaxD,
xy max 2l4(ǫ )
c
f (φ)f (φ)
D = C2 x y (23)
f (0)f (0)
x y
Originofviscosityinaccretiondisksduetosecondaryperturbation 9
Fig.2 This is for the perturbation described in §3.A. (a) Variation of ν (dotted curve) and
t
α (solid curve) as functions of ǫ for ζ = 0.05 case described in Fig. 1a, when h(r)/r =
0.01,0.05,0.1 respectively for the top, middle, bottom curves of α; r = 30, k = 0.71,
y
t = 10. (b) Variation of ν as a function of ǫ for the cases described in Fig. 1a with
g t
h(r)/r = 0.05,t = 10,k = 0.71,whensolid,dotted,dashed,long-dashedcurvescorrespond
g y
toζ = 0.01,0.05,0.1,0.2respectivelywith|k | = 1.(c)Sameasin(b)exceptαisplottedin
0
placeofν .(d)Sameasin(c)exceptt =100.
t g
10 Mukhopadhyay&Saha
Fig.3 Thisisfortheperturbationdescribedin§3.B.Variationofν (dottedcurve)andα(solid
t
curve)asfunctionsofh(r)/rforcasesshowninFigs.1b,c,whenthecurvesfromtoptobottom
correspondtoζ = 0.2,0.1,0.05,0.01withr = 30for(a)k = −105,(b)k = −104.Other
x0 x0
parametersarek =1,ǫ =0.9999.
y c
wherek reducestozeroatt=t ,whichcorrespondstothebeginningofturbulencewhenΣ=Σ .
x max max
Itis foundfrom Fig. 3 thatin a thin disk with h(r)/r = 0.01,α at r = 30 may be as highas ∼>0.1
for k = −105 when ζ is very large. Although the viscosity decreases with the decrease of ζ, α still
x0
maybe∼ 0.001whenζ = 0.05.Theturbulentviscositydecreasesinaconsiderablythickerdisk,butstill