A note on transition, turbulent length scales and transport in differentially rotating flows

0.06 MB·English

by  Denis Richard

Astronomy&Astrophysicsmanuscriptno.0519 February2,2008
(DOI:willbeinsertedbyhandlater)
A note on transition, turbulent length scales and transport in
differentially rotating flows
DenisRichardandSanfordS.Davis
4 NASAAmesResearchCenter,MS245-3,MoffettField,California94035
0
0
Received;Accepted
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n Abstract. InthisnoteweaddresstheissueofhydrodynamicalinstabilitiesinAstrophysicalrotatingshearflowsinthelight
a ofrecentpublicationsfocusedonthepossibilityfordifferentialrotationtotriggerandsustainturbulenceintheabsenceofa
J
magneticfield.Wewishtopresentinasyntheticformthemajorargumentsinfavorofthisthesisalongwithasimpleschematic

0 scenarioofthetransitiontoandself-sustenanceofsuchturbulence.Wealsoproposethattheturbulentdiffusionlengthscale
2
scalesasthelocalRossbynumberofthemeanflow.Anewprescriptionfortheturbulentviscosityisintroduced.Thisviscosity

reducestotheso-calledβ-prescriptioninthecaseofvelocityprofileswithaconstantRossbynumber,whichincludesKeplerian
1
rotatingflows.
v
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0 Keywords.hydrodynamics–instabilities–turbulence
4
1
0 1. Introduction (2002), Balbus&Hawley (1991)), the most studied being
4
the Magneto-Rotational Instability (MRI) (Balbus&Hawley
0 If it is widely accepted that plane shear flows are subject
1991). This instability, linear in nature, is an excellent can-
/
h to non-linear instability, the situation concerning their rotat- didate as long as the disk is ionized enough, but is more un-
p ing counterpart is more controversial. Indeed, rotation alone
likelytobeefficientincolddiskssuchasproto-planetaryneb-
- is known to introduce constraints that could result in the sta-
o ulae (leading to the introduction of the ”dead zone” layered
r bilization of the flow. Differentially rotatingflows are ubiqui- accretiondiskmodelbyGammie(1996)).Theissueofdiffer-
t
s tous in astrophysics. From stellar interiors to accretion disks, entialrotationhasbeenrecentlyaddressedwithratherdifferent
a from galactic rotation to planetary disks, they are met every-
approaches in several publication such as Chagelishvilietal.
:
v where.Whileinmanycommunitiesitisgenerallyacceptedthat
(2003), Longaretti (2002), Longaretti (2003), Richard (2003)
i differential rotation could give rise to turbulence, the debate
X and Tevzadzeetal. (2003). It is our purpose to try to present
continuesin theaccretiondisk community.Since thedawnof
r thesedifferentresultsina syntheticmanner(section2)andto
a accretion disk modeling in the early 1970’s, and in the ab- proposeintheirlightapossiblebasiccoherentscenarioforthe
senceofquantifiedtransportpropertiessupportedbyanidenti-
developmentofinstabilitiesandthepropertiesoftheresulting
fied physical mechanism, turbulence transport has been mod-
turbulentstate(section3).
eled with ad hoc prescriptions for the anomalous viscosity
(the most popular being the so-called ”alpha prescription”).
Differential rotation has been proposed early as the possi- 2. Flowstability
ble source of this anomalous transport (Shakura&Sunyaev
1973). Recentnumericalsimulationsfail to show evidence of Inthissectionwebrieflycommentonvariouspublishedresults
such instability in accretion disks. However, whether or not concerningthestabilityofdifferentiallyrotatingflows.
these simulations are adequate or sufficient to give a defini-
tive answerto thisissue remainsopento question(Longaretti
2.1.Theeffectofrotation
2002; Richard 2001, 2003). The difficulty of studying turbu-
lence and instabilities has been reflected in the lack of a de- Ifplaneshearflowsareknowntobearnon-linearinstabilities,
tailedphysicalmechanismalongwithdynamicalandtransport thepictureisquitedifferentwheretheconstraintofrotationis
properties to apply in Astrophysical models. Several relevant added.Inawidelyreferred-toanalysis,Balbusetal.(1996)ar-
instabilities are currently being investigated in the accretion guedthathydrodynamicturbulencewouldbedifficulttosustain
disk community (Klahr&Bodenheimer (2003), Longaretti inKeplerianflowswithouttakingintoaccounthydro-magnetic
effects. Based on the equations of evolution of the turbulent
Send offprint requests to: D.Richard / e-mail: velocity fluctuations energy, they deduced that rotating flows
[email protected] couldnotexperiencetheturbulenceduetotheenergysinkin-
2 Richard&Davis:Anoteondifferentiallyrotatingflows
troduced by the Coriolis acceleration. Taking a closer look, 3.1.Rossbynumberandturbulentscaling
one should note that the Coriolis force embodying the rota-
Rotation has a known tendency to constrain larger scales in
tioninfluenceonthefluid motiondoesnothaveanyeffecton
a two-dimensional state while efficient turbulent diffusion is
thetotalenergyequation.Itseffectistoredistributeenergybe-
achieved by three-dimensional motions. Typical geophysical
tweentheradialandazimuthalcomponentsofthemotion,van-
rotating flows exhibit both two-dimensional (at large scales)
ishing when these equations are summed. This is the natural
andthree-dimensional(atsmallerscales)structures.Withinthis
consequence of the fictitious character of the Coriolis force.
picture, the relevant scale determining the turbulent diffusion
Therefore, the rotation merely stiffens the system, add con-
shouldbethelargerscaleforwhichtheturbulencecanachieve
straints, but does not introduce any energy sink. The rotation
to be three-dimensional.Baroudetal. (2003) performed labo-
andcurvatureoftheflowintroducesfundamentaldifferencesin
ratoryexperimentsonarotatingannulus,andhaveshownthat
comparisonwithplaneshearflow.Itdoesnotnecessarymean
lowRossbynumbersareassociatedwithtwo-dimensionaltur-
that the instability mechanisms present in plane flows disap-
bulencewhereashigherRossbynumbers(oforderunityintheir
pear, but more likely that the stability and the characteristics
experiments) are associated with three-dimensional turbulent
oftheresultingturbulencearemodified(Onecanreferforex-
structures. The Rossby number for the mean flow can be re-
ampletoLongaretti(2003)foracomparativediscussiononthe
writtenas
roleofrotation,curvatureandviscousdissipationinplaneand
rotatingshearflows).
1 r
Ro= · , (1)
2 (∂ lnΩ)−1
r
2.2.Finiteamplitudeperturbationsandtransitionto
whereit appearsasthe ratio oflocalradiusoverthe char-
turbulence
acteristic length scale of the shear. It can also be linked to a
Incontrastwiththeirlinearcounterpart,non-linearinstabilities quantity more often referred to in astrophysics, the epicyclic
requirethepresenceoffinite(non-infinitesimal)amplitudeper- frequencyω,throughtherelation
turbations to be triggered.Chagelishvilietal. (2003) (and se-
quelTevzadzeetal. (2003)) proposedthe mechanismof tran- ω2
=(1+Ro). (2)
sient growth and bypass mechanism (used for many years in 4Ω2
the aerodynamicscommunity) to provide the required ampli-
The Rossby number of a turbulent structure of character-
tudes. Even in a linearly stable flow, the linear operator can
istic length scale λ and velocity u (”turbulent Rossby num-
provide transient amplification of perturbations. The results
ber” hereafter), rotating with mean flow velocity, can be ap-
fromChagelishvilietal. (2003)showthattheinitialperturba-
proximated by Ro ∝ ru/2λ2Ω : The denominator of the
tionamplitudecanbegreatlyamplified.Thisamplificationcan λ
Rossby number is twice the rotation experienced by the tur-
introduceintotheflowperturbationsofamplitudessufficientso
bulent structure, i.e. 2(Ω±u/r) ≃ 2Ω ; The numeratoris the
thattheirnon-linearinteractionscannolongerbeneglected.
derivative with respect to the radius of the turnover time, i.e.
Richard(2003)recentlyproposedasimplemodeldescrib-
r∂ Ω+r∂ (u/λ)≃r∂ Ω+ru/λ2 ≃ru/λ2,wherewehaveused
ing the necessary conditions for self-sustained turbulence in r r r
therelation
differentially rotating flows, where it can be seen that the en-
ergyextractionis directly proportionalto the shear presentin
u/λ∝r∂ Ω, (3)
thebaseflow(aclassicalresultforshearflows).Italsoimplies r
thatthereis a criticalRossby number(Ro = r∂ Ω/2Ω, where
r from Richard (2003), and the condition λ << r. We can
r isthe localradiusandΩ isthe meanrotation.)abovewhich
thenwrite,fromEq.(3)that
the energy extraction is sufficient to compensate for the stiff-
ness introduced in the system by the mean rotation. It has to r
be pointed (as Chagelishvilietal. (2003) also suggested) that Roλ ∝ Ro. (4)
λ
the non-linearinteractions(also referredto as turbulentdiffu-
sion)donotparticipateintheenergyextraction,butonlyredis- Hence,theratioofthecharacteristicturbulentlengthscale
tributeitandcounteracttheeffectsofrotation.Inthisscenario, overthelocalradiuswritesas
oncethecriticalRossbynumberhasbeenreach(Richard2003)
λ Ro
andthecriticalamplitudeispresent,theflowcanthenundergo ∝ . (5)
a transition from its laminar state to a state where non-linear r Roλ
shearturbulenceisdeveloped.
From Eq. (4), it follows that (for a given Rossby number
associated with the mean flow) the turbulent Rossby number
increaseswhengoingtosmallerscalesalongtheturbulentcas-
3. Turbulentflowproperties
cade. Remembering that high (resp. low) Rossby number are
Richard(2003) focusedonthe stabilitypropertiesofdifferen- associated with three- (resp. two-) dimensional motions, this
tially rotating flows. Using a similar approach, we wish here result is coherentwith the classic picture of a turbulent spec-
toaddressthepropertiesoftheturbulentstateofthebifurcated trum showing two-dimensional structures at large scales and
flow,inparticulartheturbulenttransportandscaleproperties. three-dimensionalonesatsmallerscales.
Richard&Davis:Anoteondifferentiallyrotatingflows 3
3.2.Turbulentdiffusion reintroduced by Richard&Zahn (1999) (based on laboratory
fluidexperimentdata)andDuschletal.(2000),andappliedby
FromanAstrophysicalpointofview,themaininterestinstudy-
Hure´etal.(2001),Davis(2003),Wehrstedt&Gail(2003)and
ingturbulencemightwellbetocharacterizetheturbulenttrans-
Lachaumeetal.(2003).
port of angular momentum or mass, by deriving the relevant
model for the turbulent viscosity, which is classically written
as 4. Discussion
ν ∝u·λ , (6) Based on recent developments, we have proposed a simple
t ν
scheme for the transition and self-sustenanceof turbulencein
whereλ standforthelargestturbulentscale participating non-magneticdifferentiallyrotatingflows.Inthisscenario,ini-
ν
inthediffusionprocessanduitsassociatedvelocity.FromEqs. tialvelocityfluctuationsareamplifiedbytheirinteractionwith
(3)and(5),itderivesthat themeanshear(providedbythelinearoperatorintheequations
ofmotion).Whilethiscouplingbetweenperturbationsandav-
ν ∝ Ro 2·r3∂ Ω. (7) eraged flow can not by itself destabilize the flow (due to its
t r
Ro ! linearlystablenature)itcanprovidetransientgrowthintroduc-
λν
ingfiniteamplitudefluctuations.Thesefluctuationsinturnfeed
Two of the results from Richard&Zahn (1999), derived
thenon-linearinteractionsandtriggertheinstability,underthe
fromexperimentaldatafromWendt(1933)andTaylor(1936),
conditionthattheRossbynumberofthemeanflowhasreached
are:
a critical value sufficient for the non-linearenergy transfer to
overcome the stiffness introduced by the mean rotation. We
- The ratio λ /r is constant. From Eq.(5) it follows that
ν conjecture that once the flow has become unstable, two- and
(Ro/Ro )isaconstant.
λν three-dimensionalturbulencecoexistatdifferentscales.Thebi-
orthree-dimensionalcharacterofthemotionsisdictatedbythe
- The turbulent viscosity scales as ν = β · r3∂ Ω, where
t r value ofthe Rossby numberata givenscale. Finally we have
β is a numerical constant. This result combined with Eq.(7),
introducedanewformulationfortheturbulentviscosityofsuch
impliesthatβ∝(Ro/Ro )2.
λν turbulence, which reducesto the β prescription in the case of
constantRossbynumberflows,includingKeplerianrotation.
Experimentsfrom Taylor (1936) used by Richard&Zahn
(1999)toderivetheβformulationofturbulentviscosityinro-
Acknowledgements. TheauthorswishtothankDr.JeffreyCuzziand
tatingflows,allhadthesamemaximumRo.Thereasonisthat Dr. Robert Hogan for their comments and support. D.Richard is
theRossbynumberwithinalaminarCouette-Taylorflowwhen supported by a Research Associateship from the National Research
theinnercylinderisatrestisgivenbyRo = R/r,whereR is Council/NationalAcademyofSciences.Thisresearchhasmadeuse
i i
theradiusoftheinnercylinderandristhelocalradius.Itfol- ofNASA’sAstrophysicsDataSystem.
lowsthattheabsolutevalueoftheRossbynumberismaximum
attheinnerboundaryoftheflowanddecreasingoutward.Inhis
References
classicexperiments,Taylormodifiedtheaspectratioofhisap-
paratusbychangingtheradiusoftheoutercylinder,hencedid Balbus,S.,Hawley,J.,&Stone,J.1996,ApJ,467,76
notmodifythemaximumRossbynumbervalueintheflow.It Balbus,S.A.&Hawley,J.F.1991,ApJ,376,214
followsthatinEq.(7),Roistobetreatedasaconstantforthis Baroud,C. N.,Plapp,B. B., Swinney,H. L.,& She,Z. 2003,
set of experiments. We conclude that both the apparent scal- PhysicsofFluids,15,2091
ingofλν andthe constantvalueoftheβparameterderivedin Chagelishvili, G. D., Zahn, J.-P., Tevzadze, A. G., &
Richard&Zahn (1999), can be explainedif we postulate that Lominadze,J.G.2003,A&A,402,401
there exists a critical (transitional) Roλν in the turbulent cas- Davis,S.S.2003,ApJ,592,1193
cadeabovewhichtheturbulenceisthree-dimensionalandun- Duschl, W. J., Strittmatter, P. A., & Biermann, P. L. 2000,
derwhichitistwo-dimensional,andthatitsvalueisaconstant A&A,357,1123
thatdoesnotdependontheflowitself.Concludingontheform Gammie,C.F.1996,ApJ,457,355
of the turbulent viscosity, in the general case (from Eq.(7)) it Hure´,J.-M.,Richard,D.,&Zahn,J.-P.2001,A&A,367,1087
writes Klahr,H.H.&Bodenheimer,P.2003,ApJ,582,869
Lachaume, R., Malbet, F., & Monin, J.-L. 2003, A&A, 400,
1 (r2∂ Ω)3
ν ∝ · r . (8) 185
t Ro 4rΩ2
λν Longaretti,P.2002,ApJ,576,587
—.2003,http://www.ArXiv.org,physics/0305052
IntheparticularcaseofKeplerianflows,itreducesto
Lynden-Bell,D.&Pringle,J.E.1974,MNRAS,168,603
ν ∝Ω ·r2, (9) Richard,D.2001,The`sedeDoctorat(Universite´ParisVII,also
t K
availableathttp://tel.ccsd.cnrs.fr)
where Ω = GM/r3 is the Keplerian angular velocity. —.2003,A&A,408,409
K
This form is similpar to the β-viscosity prescription, first in- Richard,D.&Zahn,J.-P.1999,A&A,347,734
troduced by Lynden-Bell&Pringle (1974) and more recently Shakura,N.I.&Sunyaev,R.A.1973,A&A,24,337
4 Richard&Davis:Anoteondifferentiallyrotatingflows
Taylor,G.1936,Proc.Roy.Soc.LondonA,157,546
Tevzadze, A. G., Chagelishvili, G. D., Zahn, J.-P., &
Lominadze,J.G.2003,A&A,407,779
Wehrstedt,M.&Gail,H.-P.2003,A&A,410,917
Wendt,F.1933,Ing.Arch.,4,577