Viscous, Resistive Magnetorotational Modes

0.48 MB·

by  Martin E. Pessah

DRAFTVERSIONFEBRUARY3,2008
PreprinttypesetusingLATEXstyleemulateapjv.10/09/06
VISCOUS, RESISTIVE MAGNETOROTATIONALMODES
MARTINE.PESSAH
SchoolofNaturalSciences,InstituteforAdvancedStudy,Princeton,NJ,08540
AND
CHI-KWANCHAN
InstituteforTheoryandComputation,Harvard-SmithsonianCenterforAstrophysics,60GardenStreet,Cambridge,MA02138
8 DraftversionFebruary3,2008
0
0 ABSTRACT
2
Wecarryoutacomprehensiveanalysisofthebehaviorofthemagnetorotationalinstability(MRI)inviscous,

n resistiveplasmas.Wefindexact,non-linearsolutionsofthenon-idealmagnetohydrodynamic(MHD)equations
a describingthelocaldynamicsofan incompressible,differentiallyrotatingbackgroundthreadedbya vertical
J magneticfieldwhendisturbanceswithwavenumbersperpendiculartotheshearareconsidered. Weprovidea

9 geometricaldescriptionoftheseviscous,resistiveMRImodesandshowhowtheirphysicalstructureismodified
2 as a functionof the Reynoldsand magnetic Reynoldsnumbers. We demonstratethat when finite dissipative
effectsare considered, velocityand magneticfield disturbancesare no longerorthogonal(as it is the case in

] theidealMHDlimit)unlessthemagneticPrandtlnumberisunity.Wegeneralizepreviousresultsfoundinthe
h
ideallimitandshowthata seriesofkeypropertiesofthemeanReynoldsandMaxwellstressesalso holdfor
p
theviscous,resistiveMRI.Inparticular,weshowthattheReynoldsstressisalwayspositiveandtheMaxwell
-
o stress is alwaysnegative. Therefore,even in the presenceof viscosity and resistivity, the total mean angular
r momentumtransportisalwaysdirectedoutwards. Wealsofindthat,foranycombinationoftheReynoldsand
t
s magneticReynoldsnumbers,magneticdisturbancesdominateboththeenergeticsandthetransportofangular
a momentumandthatthetotalmeanenergydensityisanupperboundforthetotalmeanstressresponsiblefor
[
angularmomentumtransport. Theratiosbetweenthe Maxwelland Reynoldsstresses and betweenmagnetic

1 andkineticenergydensitiesincreasewithdecreasingReynoldsnumbersforanymagneticReynoldsnumber;
v thelowestlimitofbothratiosisreachedintheidealMHDregime.
0
Subjectheadings:blackholephysics—accretion,accretiondisks—MHD—instability—turbulence
7
5
4
. 1. INTRODUCTION across the entire disk (see, e.g., Balbus&Henri 2007). In
1 ordertounderstandthebehavioroftheMRIunderthesecon-
0 Themagnetorotationalinstability(MRI,Balbus&Hawley ditions it is necessary to relax the assumption of an inviscid
8 1991, 1998) has been widely studied in the inviscid and plasma.
0
perfectly conducting, magnetohydrodynamic (MHD) limit.
: A large fraction of the shearing box simulations address-
v The departures from this idealized situation are usually
ingthenon-linearregimeoftheMRIhavebeencarriedoutin
i parametrizedaccordingtotheReynoldsRe=vl/νandmag-
X the idealMHD limit, i.e., withoutincludingexplicitdissipa-
netic Reynolds Rm = vl/η numbers, where v and l stand
tioninthecodes(see,e.g.,Hawley,Gammie,&Balbus1995;
r forthe relevantcharacteristicvelocityand lengthscaleand ν
a Brandenburgetal.1995;Sanoetal.2004). However,evenin
and η stand for the kinematic viscosity and resistivity. The
the absence of explicit viscosity and resistivity, finite differ-
ideal MHD regime is then formally identified with the limit
ence discretization leads to numerical diffusion/dissipation.
Re , Rm . There are many situations of interest in
→ ∞ Therefore,eveninthistypeofsimulations,itisnecessaryto
whichtheeffectsofdissipationneedtobeconsidered.
understandtheimpactofthesenumericalartifactsthatleadto
From the astrophysical point of view, accretion disks departuresfromtheidealMHDregimeandhowsimilarthey
aroundyoungstellarobjectsconstituteoneofthemostcom- arewhencomparedwithphysical(resolved)dissipation.
pelling reasons for investigating the MRI beyond the ideal Ahandfulofnumericalstudieswithexplicitresistivitybut
limit. In particular, there is great interest in understanding
zerophysicalviscosity have beencarried outin order to un-
to what extent can MHD turbulence driven by the MRI en- derstandthe effectsof ohmic dissipation in the saturation of
able efficient angular momentum transport in cool, poorly
MRI-driventurbulence(see, e.g.,Sano,Inutsuka,&Miyama
conducting, protoplanetary disks (see, e.g. Blaes&Balbus 1998; Sano&Inutsuka 2001; Fleming,Stone,&Hawley
1994; Jin 1996; Gammie 1996; Sano&Miyama 1999; 2000;Sanoetal.2004;Turner,Sano,&Dziourkevitch2007).
Salmeron&Wardle 2005). Most of the studies addressing
In particular, Sano&Stone (2003) have shown that the sat-
the effects of dissipation in non-ideal MRI have usually fo- uration level of the stresses increases with increasing mag-
cusedininviscid,resistiveplasmas. However,accretiondisks
neticReynoldsnumberandseemtoconvergetoanasymptotic
are characterizedby a wide range of magnetic Prandtlnum- valueformagneticReynoldsnumberslargerthanunity.
bers,withPm=ν/ηvaryingbyseveralordersofmagnitude
Recent work has pointed out problems with conver-
Electronicaddress:[email protected],[email protected] gence in zero-net-flux numerical simulations of ideal
2 PESSAH AND CHAN
MHD driven by the MRI (Pessah,Chan,&Psaltis 2007; summarize our findings and discuss the implications of our
Fromang&Papaloizou2007)implyingthenecessityofincor- study.
poratingexplicitdissipationin the codes. Numericalstudies
withbothresistivityandviscosity,inthepresenceofamean
2. ASSUMPTIONS
vertical magnetic field (Lesur&Longaretti2007) and in the
caseofzeronetflux(Fromangetal.2007),havebeguntoun- Let us consider a cylindrical, incompressible background
coverhow the characteristicsof fullydevelopedMRI-driven characterized by an angular velocity profile Ω = Ω(r)zˇ,
turbulencedependson the Reynoldsand magnetic Reynolds threadedbyaverticalmagneticfieldB¯ = B¯ zˇ. Weworkin
z
numbers. EventhoughtherangesinReynoldsandmagnetic theshearingboxapproximation,whichconsistofafirstorder
Reynoldsnumbersthatcanbecurrentlyaddressedisstilllim- expansionin thevariabler r ofallthe quantitiescharac-
0
ited,theresultsobtainedfromthesimulationssuggestthatthe terizing the flow at the fidu−cial radius r . The goal of this
0
magneticandkineticenergiescontainedinturbulentmotions expansionis to retain the most relevantterms governingthe
inthesaturatedregimedependonthevaluesofthemicrophys- dynamicsoftheMHDfluidinalocally-Cartesiancoordinate
ical viscosity and resistivity. In particular, the mean angular systemco-orbitingandcorrotatingwiththebackgroundflow
momentumtransportin theturbulentstate increaseswith in- with local (Eulerian)velocityv = r Ω φˇ. (Fora more de-
0 0
creasingmagneticPrandtlnumber.
taileddiscussiononthisexpansionseeGoodman&Xu1994
From the experimental perspective, understanding the ef- andreferencestherein.)
fectsof non-vanishingresistivity and viscosity in the behav-
The equations governing the dynamics of an incompress-
ior of the MRI seems imperative, since the physical condi-
ibleMHDfluidwithconstantkinematicviscosityνandresis-
tions achievable in the laboratory depart significantly from
tivityηintheshearingboxlimitaregivenby
the ideal MHD regime (Ji,Goodman&Kageyama 2001;
Goodman&Ji 2002; Sisanetal. 2004; Liu,Goodman,&Ji ∂v
2006; Ru¨diger,Schultz,Shalybkov 2003). Liquid metals ∂t +(v·∇)v=−2Ω0×v + qΩ20∇(r−r0)2
(such as sodium, gallium, and mercury)are often character-
izedbyratherlowmagneticPrandtlnumbers(Pm 10−5– 1∇ P + B2 + (B·∇)B +ν∇2v,
10−7). AlthoughtheregimeofReynoldsnumbersin≃volvedis −ρ 8π 4πρ
(cid:18) (cid:19)
still ordersof magnitudesmaller than any astrophysicalsys- (1)
temwithsimilarmagneticPrandtlnumbers,MRIexperiments ∂B
offer one of the few prospects of studying anything close to +(v·∇)B=(B·∇)v+η∇2B, (2)
∂t
MHDastrophysicalprocessesinthelaboratory.
whereP isthepressure,ρisthe(constant)density,thefactor
A numberof analyses addressing some aspects of the im-
pact of viscosity and resistivity on the MRI in various dis- dlnΩ
sipative limits appear scattered throughout the literature on q , (3)
≡− dlnr
theoretical,numerical,andexperimentalMRI.Morerecently, (cid:12)r0
(cid:12)
Lesaffre&Balbus (2007) have found particular solutions of parametrizes the magnitude of the(cid:12)local shear, and we have
(cid:12)
theviscous,resistiveMHDequations(includingevenacool- definedthe(locally-Cartesian)differentialoperator
ing term) in the shearing box approximation. However, we
areunawareofanycomprehensive,systematicstudyaddress- ∂ φˇ ∂ ∂
∇ rˇ + +zˇ , (4)
inghow the MRI behavesin viscous, resistive, differentially ≡ ∂r r ∂φ ∂z
0
rotatingmagnetizedplasmasforarbitrarycombinationsofthe
Reynolds and magnetic Reynolds numbers. The aim of this whererˇ, φˇ, andzˇ are, coordinate-independent,orthonormal
workistocarryoutthisanalysisindetail. vectorscorrotatingwiththebackgroundflowatr . Thecon-
0
tinuityequationreducesto∇·v = 0andthereisnoneedfor
The rest of paper is organized as follows. In 2, we
§ anequationofstatesincethepressurecanbedeterminedfrom
state ourassumptions. In 3, we solvethe eigenvalueprob-
§ thiscondition.
lem defined by the MRI for arbitrary Reynolds and mag-
netic Reynolds numbers. We provide closed analytical ex- We focus our attention on the dynamics of perturbations
pressions for the eigenfrequenciesand the associated eigen- thatdependonlyontheverticalcoordinate.Underthecurrent
vectors. In 4,weaddresstheunexploredphysicalstructure set of assumptions, these types of perturbations are known
§
of MRI modes for finite Reynolds and magnetic Reynolds to exhibit the fastest growth rates in the ideal MHD case
numbers and derive simple analytical expressions that de- (Balbus&Hawley 1992, 1998; Pessah&Psaltis 2005). The
scribe these modes in various asymptotic regimes. In 6, equationsgoverningthedynamicsoftheseperturbationscan
§
we calculate the correlations between magnetic and veloc- be obtained by noting that the velocity and magnetic fields
ity MRI-drivenperturbationsthat are related to angular mo- givenby
mentum transport and energy densities. We find that some
keyresultspreviouslyshowntoholdintheidealMHDlimit v=δvr(z)rˇ+[ qΩ0(r r0)+δvφ(z)]φˇ +δvz(z)zˇ,(5)
− −
(Pessah,Chan,&Psaltis2006)arealsovalidinthenon-ideal B=δB (z)rˇ+δB (z)φˇ +[B¯ +δB (z)]zˇ, (6)
regime.Inparticular,weshowthateventhoughtheeffective- r φ z z
nesswithwhichtheMRIdisruptsthelaminarflowdependson where the time dependence is implicit, constitute a family
the Reynolds and magnetic Reynolds numbers, the instabil- of exact, non-linear,solutionsto the viscous, resistive MHD
ity always transportsangularmomentumoutwards. We also equations(1)-(2). As noted in Goodman&Xu (1994), even
findthatmagneticperturbationsdominateboththeenergetics in the dissipative case, the only non-linear terms, which are
andthetransportofangularmomentumforanycombination presentthroughtheperturbedmagneticenergydensity,areir-
ofthe ReynoldsandmagneticReynoldsnumbers. In 7we relevantinthecaseunderconsideration.
§
VISCOUS, RESISTIVE MAGNETOROTATIONALMODES 3
Wecanfurthersimplifyequations(1)and(2)byremoving 3. THEEIGENVALUEPROBLEMFORTHENON-IDEALMRI:
thebackgroundshearflow1v = qΩ (r r )φˇ andby AFORMALANALYTICALSOLUTION
shear 0 0
− −
realizingthatwecantakeδv (z)=δB (z)=0withoutloss
z z In this section we provide a complete analytical solution
ofgenerality.Wethenobtain
to the set of equations (12)–(15) as a function of the shear
∂ B¯ ∂ ∂2
δv =2Ω δv + z δB +ν δv , (7) parameterq,(or,equivalently,theepicyclicfrequency,κ)for
∂t r 0 φ 4πρ ∂z r ∂z2 r anysetofvalues(ν,η)definingtheviscosityandresistivity.
∂ δv = (2 q)Ω δv + B¯z ∂ δB +ν ∂2 δv ,(8) It is convenientto work in Fourier space, as this provides
∂t φ − − 0 r 4πρ ∂z φ ∂z2 φ theadvantageofobtainingexplicitlythebasisofmodesthatis
∂ ∂ ∂2 neededtoconstructthemostgeneralsolutionsatisfyingequa-
δB =B¯ δv +η δB , (9) tions(12)–(15). TakingtheFouriertransformofthissetwith
∂t r z∂z r ∂z2 r
respecttothez-coordinate,weobtainthematrixequation
∂ ∂ ∂2
∂tδBφ=−qΩ0δBr+B¯z∂zδvφ+η∂z2δBφ, (10) ∂tδˆ(kn,t)=Lδˆ(kn,t), (19)
wherethe first term onthe righthandside of equation(8) is
relatedtotheepicyclicfrequency wherethevectorδˆ(kn,t)standsfor
κ 2(2 q)Ω0, (11) δˆv (k ,t)
≡ − r n
antamwihcicdhistkh.eFfloorwKveaprliearbialpensroostactiilolantethienpaapraemrtuetrebredishqyd=ro3d/y2- δˆ(kn,t)=δδvˆˆbφ((kkn,,tt)) , (20)
andthustheepicyclicfrequencyisκ=Ω . r n
0  δˆb (k ,t) 
It is convenient to define the new variables δb  φ n 
i ≡  
δBi/√4πρ for i = r,φ, and introducedimensionless quan- andLrepresentsthematrix
titiesbyconsideringthecharacteristictime-andlength-scales
tsheetbdyim1e/nΩ∂st˜i0δov˜narnled=ssB2¯pδzev˜/rφ(tu√+rb4a∂πtz˜ρioδΩ˜nbrs0,)+δ.v˜νT˜i,∂hδz˜2e˜bδeiv˜,qrau,raetitohnesnsgaitvisefinebd(y1b2y) L=−−(2iνk0−knn2q) −iνk02kn2 −iηk0kqnn2 iηk00kn2  . (21)
∂t˜δv˜φ=−(2−q)δv˜r+∂z˜δ˜bφ+ν˜∂z˜2δv˜φ, (13) Thefunctionsdenotedby fˆ(kn,t) c−orresp−ondntothe Fourier
∂ δ˜b =∂ δv˜ +η˜∂2δ˜b , (14) n
t˜ r z˜ r z˜ r transformoftherealfunctions,f(z,t),andaredefinedvia
∂ δ˜b = qδ˜b +∂ δv˜ +η˜∂2δ˜b , (15)
t˜ φ − r z˜ φ z˜ φ 1 H
wheret˜andz˜denotethedimensionlesstimeandverticalco- fˆ(k ,t) f(z,t)e−iknzdz, (22)
n
ordinate,respectively. ≡ 2H Z−H
The dynamics of ideal MRI modes, with ν = η = wherewehaveassumedperiodicboundaryconditionsatz =
0, is completely determined by the dimensionless shear q H, with H being the (dimensionless) scale-height and k
n
(Pessah,Chan,&Psaltis 2006). The effects of viscous and t±hewavenumberinthez-coordinate,
resistive dissipation introduce two new dimensionless quan-
nπ
tities that alter the characteristics and evolution of the MRI. k , (23)
n
≡ H
Withourchoiceofcharacteristicscales,itisnaturaltodefine
theReynoldsandmagneticReynoldsnumberscharacterizing wheren is anintegernumber. In orderto simplifythe nota-
theMHDflowas2 tion,hereafterwedenotethesewavenumberssimplybyk.
v2 1
Re Az = , (16) Inordertosolvethematrixequation(19),itisconvenientto
≡νΩ0 ν˜ findtheeigenvectorbasis, ej withj =1,2,3,4,inwhichL
v2 1 isdiagonal.Thisbasisexis{tsfo}rallvaluesofthewavenumber
Rm Az = , (17)
k (i.e., the rank of the matrix L is equalto 4, the dimension
≡ηΩ η˜
0
ofthecomplexspace)withthe possibleexceptionofa finite
withassociatedmagneticPrandtlnumber
numberofvaluesofk. Inthisbasis,theactionofLoverthe
Rm ν˜ ν
Pm = = . (18) set e isequivalenttoascalarmultiplication,i.e.,
j
≡ Re η˜ η { }
L e =σ e for j =1,2,3,4, (24)
Inordertosimplifythenotation,wedrophereafterthetilde diag j j j
denotingthedimensionlessquantities.Intherestofthepaper,
where σ arecomplexscalars.
j
all the variables are to be regardedas dimensionless, unless { }
otherwisespecified.
3.1. Eigenvalues
1 Intheshearingboxapproximation, thedependence ofthebackground
flowontheradialcoordinateisstrictlylinearandthereforeviscousdissipa- Intheeigenvectorbasis,thematrixLhasadiagonalrepre-
tiondoesnotaffectitsdynamics. sentationL =diag(σ ,σ ,σ ,σ ). Theeigenvalues σ ,
diag 1 2 3 4 j
2 Inacompressible fluid, thesoundspeed, cs, provides another natural with j = 1,2,3,4, are the roots of the characteristic{poly}-
characteristicspeedtodefinetheReynoldsandmagneticReynoldsnumbers.
Thesedefinitions,e.g.,Re′andRm′,arerelatedtothoseprovidedinequa- nomialassociatedwithL,i.e.,thedispersionrelationassoci-
tions (16)and(17)viatheplasmabetaparameter β = (2/Γ)(cs/v¯Az)2, atedwiththenon-idealMRI,whichcanbewrittenincompact
wRiet′h=v¯Az(Γβ=/2B)¯Rz/e√an4dπρRmthe′ A=lfv(Γe´nβ/sp2e)eRdminfothreazpodliyrtercotpioicn,eqsiumatpiolynboyf formas
stateP =KρΓ,withKandΓconstants. (k2+σνση)2+κ2(k2+ση2)−4k2 =0, (25)
4 PESSAH AND CHAN
FIG. 1.—Growthratesγ+,eq.(44),asafunctionoftheverticalwavenumberkfordifferentcombinationsofReynoldsandmagneticReynoldsnumbersfor
Keplerianrotation. Inallthreepanels,thethicksolidlinecorrespondstotheidealMHDlimit,i.e.,Re,Rm . ForanycombinationoftheReynoldsand
magneticReynoldsnumbers,thegrowthratehasawelldefined,singlemaximumγmaxthatcorrespondstothe→mo∞stunstablemodekmax.Therangeofunstable
modes,0 < k < kc,isalwaysfinite,thecriticalwavenumberkcsatisfieseq.(25)whenσ 0,see 4.1. Left: Growthrateγ+ fordifferentvaluesofthe
magneticReynoldsnumberintheinviscidlimit,i.e.,Re . Thethinsolidlines,indecr≡easingord§er,correspondtoRm = 10,1,0.1. Middle: Growth
rateγ+ formagneticPrandtlnumberPm = Rm/Re =→1.∞Thethinsolidlines,indecreasingorder,correspondtoRe = Rm = 10,1,0.1. Inallofthe
casesshownintheleftandmiddlepanels,thecriticalwavenumber,kc,belowwhichunstablemodescanexistdecreaseswithincreasingresistivity, see 4.3
and 4.4foranalyticexpressionsofthesemarginallystablemodes. Right: Growthrateγ+fordifferentvaluesoftheReynoldsnumberintheidealcondu§ctor
limit§,i.e.,Rm .Thevariouscurves,indecreasingorder,correspondtoRe=10,1,...,10−3.Inthiscase,therangeofunstablemodesisinsensitiveto
theReynoldsnu→mb∞er,allthemodeswithwavenumbersshorterthankc =√2qareunstable,see 4.2. Itisevidentthatthegrowthratesandthecharacteristic
scales,bothkmaxandkc,aremoresensitivetochangesintheresistivitythantochangesinthev§iscosity. Thesimultaneousanalysisofallthreepanelsleads
totheconclusionthatviscous,resistivemodeswithmagneticPrandtlnumberequaltounityresemblemorecloselyinviscid,resistivemodesratherthanviscous,
conductiveones,see 4.4fortheexplanationofthisbehavior.
§
wherewehavedefinedthequantities andyisanyofthesolutionstothecubicequation
σν≡σ+νk2, (26) y3+ 5αy2+(2α2 λ)y+ α3 αλ β2 =0, (37)
σ σ+ηk2. (27) 2 − 2 − 2 − 8
η≡ (cid:18) (cid:19)
whichhasclosedanalyticsolutions
Thedispersionrelation(25)isafourthorderpolynomialwith
non-zero coefficients in σ and σ3. In order to find its roots 5 1P
itisconvenienttotakethispolynomialtoitsdepressedform. y = α+ U, (38)
−6 3U −
Thiscanbeachievedbydefiningthenewvariablesσ andµ
µ with
suchthat3
α2
1 P= λ, (39)
σµ (σν +ση), (28) −12 −
≡2
α3 αλ β2
µ 1(ν η)k2. (29) Q= + , (40)
−108 3 − 8
≡2 −
1/3
Theresultingpolynomialcanthenbewrittenas Q Q2 P3
U= + . (41)
σ4 +ασ2 +βσ +λ=0, (30) 2 ±r 4 27!
µ µ µ
NotethatthechoiceofeithersigninU isimmaterial.
wherethecoefficientsα,β,andλaregivenby
Itisnowtrivialto writethe solutionsofthe dispersionre-
α 2(k2 µ2)+κ2, (31) lation (25), σ, in terms of the variable σ . Using equations
≡ − µ
β 2µκ2, (32) (26)–(28)weobtain
≡−
λ≡(k2−µ2)2+κ2(k2+µ2)−4k2. (33) σ =σµ 1(ν+η)k2. (42)
− 2
Thesolutionstoequation(30)are Theeigenfrequenciesoftheviscous,resistiveMRImodesare
thengivenby
β
σ = ( Λ √∆)1/2 , (34)
µ ±a − ∓b ±b 4√∆ σ= ( Λ √∆)1/2
a b
± − ∓
wherethe subscriptsa andb in the “+”and “ ” signslabel ν κ2 η κ2
thefourpossiblecombinationsofsignsandw−ehavedefined −2k2 1±b 2√∆ − 2k2 1∓b 2√∆ . (43)
thequantities4 (cid:18) (cid:19) (cid:18) (cid:19)
3α y 3.2. Fourclassesofsolutions
Λ= + , (35)
4 2
All of the quantitiesΛ, ∆, and y, dependon the viscosity
∆=(y+α)2 λ, (36) ν and the resistivity η only through µ2 (ν η)2. This
− ∝ −
has a series of important implications, in particular, there is
3Aphysicalinterpretationofthevariableµisprovidedin 3.3. alwaysarangeofwavenumbersforwhichthediscriminantin
4DefiningthequantitiesΛand∆inthiswayallowsustos§howexplicitly
equation(43)ispositive,i.e.,√∆ Λ>0.Itcanalsobeseen
thatinthelimitν,η 0thesolutionstoequation(25)convergesmoothly
tothesolutionsfound→intheidealMHDcase(seeAppendixA). thatthe last two termsbetween p−arenthesesin equation(43)
VISCOUS, RESISTIVE MAGNETOROTATIONALMODES 5
FIG. 2.—Left: Geometricalrepresentationofthevelocityfield(black)andmagneticfield(gray)perturbationsforviscous,resistiveMRImodes. Notethat
thisisaprojectionofasinglemode,whichisinherentlythree-dimensional,ontothediskmid-plane(r,φ,z=0).Thevelocityandmagneticfieldcomponents
arealwaysoutofphaseintheverticaldirectionzbyπ/4,seeeq.(63). Theanglesθv andθb,definedineqs.(50)and(51),respectively, correspondtothe
physicalanglesdefiningtheplanes(perpendiculartothediskmidplane)containingtheMRI-drivenperturbations,seeeqs.(64)and(65).Therelativemagnitude
ofvelocityandmagneticfieldperturbationsisdeterminedbyeq.(54).Right:Evolutionofthegeometricalrepresentationofthefastest-growing,non-idealMRI
mode,withassociatedwavenumberkmax,withmagneticPrandtlnumberequaltounity,asafunctionoftheReynolds/magneticReynoldsnumber. Whenthe
Reynolds/magneticReynoldsnumbervariesaccordingtoRe=Rm: 0,theanglesevolveaccordingtoθv :π/4 0andθb :3π/4 π/2andthe
relativeamplitudeoftheperturbationsevolvesaccordingtob0/v0 :5/∞3 → . Notethatthevelocityandmagneticfield→perturbationsarealw→aysorthogonal
forPm=1,seeeq.(68). →∞
are always positive, i.e., √∆ κ2/2, and thus they always producedamping,i.e.,(1 κ2/2√∆)and(1+κ2/2√∆),in
≥ −
producedamping.Becauseofthis,wecanclassifythemodes theexponentialgrowthcharacterizedbyγ inequation(44).
+
infourtypes:two(damped)growinganddecaying“unstable” Iftheoscillatorymodes,ω± inequation(45),areconsidered
modeswitheigenvalues instead,therolesoftheplusandminussignsinthesetermsare
interchanged. Fromthisanalysiswecaninferthatthe“oscil-
γ±= (√∆ Λ)1/2 latory”modeisaffected(damped)morestronglybyviscosity
± −
ν κ2 η κ2 thanbyresistivity.
k2 1 k2 1+ , (44)
−2 − 2√∆ − 2 2√∆ ThesimultaneousanalysisofthevariouspanelsinFigure1
(cid:18) (cid:19) (cid:18) (cid:19)
leadstotheconclusionthatviscous,resistiveunstablemodes
andtwo(damped)“oscillatory”modeswitheigenvalues
with magneticPrandtlnumberequalto unity resemblemore
iω±= i(√∆+Λ)1/2 closelyinviscid,resistivemodesratherthanviscous,conduc-
±νk2 1+ κ2 ηk2 1 κ2 . (45) tthiviescoonnecs.luIsnio§n4. weprovideanalyticalexpressionstosupport
−2 2√∆ − 2 − 2√∆
(cid:18) (cid:19) (cid:18) (cid:19)
Wearbitrarilylabeltheseeigenvaluesas 3.3. NormalizedEigenvectors:GeometricalRepresentation
σ1 ≡γ+, σ2 ≡γ−, σ3 ≡iω+, σ4 ≡iω−. (46) Thesetofnormalizedeigenvectors,{eσj},associatedwith
theeigenvalues(46)aregivenby
Figure 1 shows the growth rate γ+ as a function of the e
vertical wavenumber k for different combinations of the e j for j =1,2,3,4, (47)
σj≡ e
Reynoldsand magnetic Reynoldsnumbersfor Keplerianro- j
k k
tation. These growth rates are more sensitive to changes in where
the resistivity than to changesin the viscosity. A qualitative σ
ηj
understanding of this behavior can be obtained by realizing (k2+σ σ )/2
e (k)= νj ηj ,
thatviscositytendstoquenchtheinstability,withoutaltering j  ik 
thelargescalemagneticfield. Thus,aslongastheresistivity ik[2σ +q(ν η)k2]/(k2+σ σ )
isnegligible,therangeofunstablelenghtscalesarethesame  − ηj − νj ηj (48)
inbothidealandviscous,perfectlyconductingfluids. Onthe σ =σ +νk2,σ =σ +ηk2,andthenormsaregivenby
νj j ηj j
other hand, resistivity tends to destroy the magnetic field at
small scales having a stronger impact on the stability of the 4 1/2
perturbationsatthesescales. kejk≡" eljelj∗# . (49)
Mathematically,theasymmetricresponseofthegrowthrate Xl=1
tochangesintheviscosityν ortheresistivityη originatesin Here, el is the l-th componentof the (unnormalized)eigen-
j
the different functional form of the terms that contribute to vectorassociatedwiththeeigenvalueσ .
j
6 PESSAH AND CHAN
FIG.3.—Criticalwavenumberkc,seeeq.(70),correspondingtothemarginallystableMRImodeforKeplerianrotationindifferentdissipativeregimes.The
horizontallinesatkc=√3representtheidealMHDlimit,eq.(76).Left:CriticalwavenumberkcasafunctionofthemagneticReynoldsnumberfordifferent
valuesoftheReynoldsnumber. Thethicksolidlinedenotestheinviscidlimit,i.e.,Re . Notethateq.(87)describesthiscurveexactly. Thethinsolid
lines,indecreasingorder,correspondtoRe = 1,0.1,.... ForsmallmagneticReynolds→nu∞mberkc Rm,seeeq.(87). ForfiniteReynoldsnumbers,such
thatReRm .1thereisatransitionbetweentheregimesRm 1andRm 1suchthatkc ∝Rm1/3,seeeq.(98). Middle: Criticalwavenumberkc
asafunctionoftheReynoldsnumberfordifferentmagneticPrand≪tlnumbers. Pm≫increases/decreas∝esbyanorderofmagnitudeforeachcurvetotheleft/right
ofthethicksolidlinedenotingthePm = 1case. Thedashedlineskc RePm = Rmarecalculatedaccordingtoeq.(87),whichgivesthecorrectresult
evenforPm > 1,providedthattheReynoldsnumberissufficientlysma∝ll. Thedottedlineskc (ReRm)1/3arecalculatedaccordingtoeq. (98). Right:
CriticalwavenumberkcasafunctionoftheReynoldsnumberfordifferentvaluesofthemagneticR∝eynoldsnumber.Thethicksolidlinecorrespondstotheideal
conductorlimit,i.e.,Rm .Thethinsolidlines,indecreasingorder,correspondtoRe=105,104,...,10−3.ForsmallReynoldsnumberskc Re1/3,
seeeq.(98),whileforRey→nol∞dsnumberslargerthanafew,thecriticalwavenumberisindependentofReregardlessofthevalueofRm. ∝
Thesetoffoureigenvectors e ,togetherwiththesetof where, for the sake of simplicity, we have omitted the sub-
{ σj}
complexscalars σ inequation(46),constitutethefullso- script j on the left hand side. The expressions for the nor-
j
lutiontotheeige{nva}lueproblemdefinedbytheMRIforany malizedeigenvectors{eσj},forj = 1,2,3,4,arethengiven
combination of the Reynolds and magnetic Reynolds num- by
bers. v0cosθvj
1 v sinθ
A geometrical representation of the eigenvectors (47) can e (k)= 0 vj . (55)
bebroughttolightbydefiningtheanglesθvj andθbj accord- σj v02+b20 iibb0csoinsθθbj 
ingto  0 bj 
p  
e2 k2+σ σ Itisinterestingtonotethatinthisgeometricrepresentation
tanθvj≡e1j = 2σνj ηj , (50) thedispersionrelation(25)canbeobtainedfromthetrigono-
j ηj metricidentity
e4 2σ +q(ν η)k2
tanθbj≡e3j =− ηkj2+σ −σ . (51) cos2θvj +sin2θvj =1, (56)
j νj ηj wheretheexpressionsfor
It is important to remark that each of the four eigenvectors
t1dy,ep2fie,ns3eo,,4fi.nmopWdrieenscladipbisleecl,utsfhoseeudranisngetlses3o.af2saasncocgcoilaertsdeid{nθgwvtjio,thθbtjh}e,dfoifrfejre=nt cosθvj=sq(k22σ+η2jση2j), (57)
θv1 ≡θvγ+, θv2 ≡θvγ−, § θv3 ≡θvω+, θv4 ≡θvω−, (52) sinθvj= k2+σνjσηj , (58)
withsimilardefinitionscorrespondingtoθ forj =1,2,3,4. 2q(k2+σ2 )
bj ηj
Note that these angles are defined in spectral space and de-
pend, in general, on the wavenumber k, the epicyclic fre- canbeobtainedfromthedqefinitionoftheangleθvj inequa-
quency,κ,theviscosityν,andtheresistivityη.Theanglesas- tion(50).
sociatedwiththemodeslabeledbyγ+andγ−arealwaysreal
while the onesassociated with the modesω+ and ω− are in 3.4. TemporalEvolution
generalcomplex. Forthesakeofbrevity,inwhatfollowswe
In physical space, the most general solution to the set of
willrefertothesetofanglesdescribingunstableMRImodes
θγ+,θγ+ simplyas θ ,θ . equations(12)–(15),i.e.,
{ v b } { v b}
A normalizedversion of the MRI eigenvectorscan be ob- δvr(z,t)
tained by multiplying the set of vectors in equation (48) by δ(z,t)= δvφ(z,t) , (59)
theamplitudes  δbr(z,t) 
δb (z,t)
 φ 
A 2 v0 . (53) evolvesintimeaccordingto 
j ≡sq(k2+ση2j) v02+b20 δ(z,t) δˆ(k,t)eikz, (60)
wherewehavedefined p ≡
k
1/2 X
2kv0 (ν η)[4σηj +q(ν η)k2]k2 where
b0 ≡ k2+σνjσηj 1+ − 2(k2+ση2j−) ! , δˆ(k,t)= 4 a (k,0)eσjte , (61)
j σj
(54)
j=1
X
VISCOUS, RESISTIVE MAGNETOROTATIONALMODES 7
FIG. 4.—Wavenumberkmax correspondingtothefastestgrowingnon-idealMRImodesforKeplerianrotationindifferentdissipativeregimes. Thedot-
dashedhorizontallinesatkmax =p15/16representtheidealMHDlimit,eq.(85).Left:FastestgrowingmodekmaxasafunctionofthemagneticReynolds
numberfordifferentvaluesoftheReynoldsnumber. Thethicksolidlinedenotestheinviscidlimit,i.e.,Re . Thethinsolidlines,indecreasingorder,
correspondtoRe=1,0.1,...,10−5.FormagneticReynoldsnumberslargerthanunity,thiswavenumberisi→nde∞pendentofRmregardlessofthevalueofRe.
Thedashedline,calculatedaccordingtoeq.(90),providesthecorrectasymptoticlimitkmax RmforsmallmagneticReynoldsnumbers. Middle: Fastest
growingmodekmax asafunctionoftheReynoldsnumberfordifferentvaluesofthemagneti∝cPrandtlnumber. Fromlefttoright,thecurvescorrespondto
Pm=103,102,...,1(thicksolidline),...,10−6.Thedashedlineskmax RePm=Rmarecalculatedaccordingtoeq.(90),whichleadstothecorrect
resultevenforPm&1providedthattheReynoldsnumberissufficientlysma∝ll.Thedottedlinekmax Re1/2resultsfromeq.(101).Right:Fastestgrowing
modekmaxasafunctionoftheReynoldsnumberfordifferentvaluesofthemagneticReynoldsnumber∝.Thethicksolidlinecorrespondstotheidealconductor
limit,i.e.,Rm . Thethinsolidlines,indecreasingorder,correspondtoRm= 1,0.1,0.01. ForReynoldsnumberslargerthanunity,thegrowthrateis
independentof→Re∞regardlessofthevalueofRm. Thedottedline,calculatedaccordingtoeq.(101),providesthecorrectasymptoticlimitkmax Re1/2for
smallReynoldsnumber. ∝
with σ and e , for j = 1,2,3,4, given by equations tionsinphysicalspace,seeFigure2,with
{ j} { σj}
(46) and (55). The initial conditions a(k,0) are related to
δv (z,t)
the initial spectrum of perturbations, δˆ(k,0), via a(k,0) = tanθvj= φ =const., (64)
δv (z,t)
Q−1δˆ(k,0). Here,Q−1 isthematrixforthechangeofcoor- r
δb (z,t)
dinatesfromthestandardbasistothenormalizedeigenvector tanθ = φ =const.. (65)
bj
basis5 and can be obtained by calculating the inverse of the δbr(z,t)
matrix
Finally,definingtheangleθ suchthat
Q=[e e e e ]. (62) bvj
σ1 σ2 σ3 σ4 π
θ =θ θ + , (66)
bvj bj vj
− 2
The temporalevolution of a single MRI-unstablemode in
(cid:16) (cid:17)
physicalspacecan beobtainedfroma linearcombinationof whichimpliesthattanθbjtan(θvj +θbvj) = 1,andusing
−
e (k)ande ( k)asdefinedinequation(55).Inparticular, thefactthat
σj σj −
sseutbtisntigtuati1n(gk,th0e)r=esual∗1t(i−nkeq,0u)at=ion−(i6/1√)w2einobeqtauiantion(60)and tan(θ1+θ2)= 1tantθa1n+θttaannθθ2 , (67)
1 2
−
itisnotdifficulttoshowthat
v cosθ sin(kz)
0 v
√2eγ+t v sinθ sin(kz) ν η
δ(z,t)= v2+b2 b00cosθbvcos(kz) . (63) tanθbvj =−µ=− −2 k2. (68)
0 0 b sinθ cos(kz) (cid:18) (cid:19)
p  0 b  This means that µ 6= 0 provides a measure of how non-
orthogonalvelocityandmagneticfieldperturbationsare.
These solutionsare of particular importancefor the linear
It is evident that when the magnetic Prandtl number
late-timeevolutionofMRI modes. Notethatanyreasonable
approaches unity viscous, resistive, MRI-driven magnetic
spectrumofinitialperturbationsofthetypeusedinnumerical
and velocity perturbations tend to be orthogonal, i.e.,
simulations of shearing boxes will have a non-zero compo-
nent along the unstable eigenvector e . If the value of the tanθvjtanθbj = 1,andtherefore
σ1 −
magneticfield is such thatthe MRI can be excitedfor given π
θ θ θ = for Pm=1, (69)
values of the viscosity and resistivity then the exponentially diff ≡ bj − vj 2
growingperturbationsinphysicalspacewillevolvetowardsa
for every wavenumber k. This is illustrated in Figure 2
modeoftheform(63)dominatedbythelengthscalek =k
max which shows the evolution of the angles θ and θ corre-
forwhichthegrowthratereachesitsmaximumvalueγ . b v
max sponding to the most unstable MRI mode as a function of
Notethatifaperturbationinphysicalspaceiscomposedby theReynolds/magneticReynoldsnumberwhenthemagnetic
asinglemodeofthetypedescribedin 3.2,nomatterwhich Prandtlnumberisequaltounity.
§
class, then the angles defined in equations(50) and (51) are
constant in time and are identical to the physical angles be- 4. PHYSICALSTRUCTUREOFMRIMODES
tween the planescontainingmagneticand velocityperturba-
Theevolutionofthephysicalstructureofasinglegrowing
for5jT=heje′i.gIefndveescitroerds,(a5n5o)rathreognoontailnbgaesnisercaalnobrethcoognosntraul,ctie.ed.,uesσinjg·tehσejG′r6=am0- MitsRgIromwotdherawteithγ+w,atvheenruemlabtievre0m<agkni<tudkecbisetcwheaerancttheerizaemdpbliy-
Schm6idtorthogonalizationprocedure(see,e.g.,Hoffman&Kunze1971). tudesofmagneticandvelocityfieldperturbations,b0/v0,and
8 PESSAH AND CHAN
FIG. 5.—MaximumgrowthrateγmaxforKeplerianrotationindifferentdissipativeregimes. Thedot-dashedhorizontallinesatγmax = 3/4representthe
idealMHDlimit,eq.(82). Left: MaximumgrowthrateasafunctionofthemagneticReynoldsnumberfordifferentvaluesoftheReynoldsnumber. Thethick
solidlinedenotestheinviscidlimit,i.e.,Re . Thethinsolidlines,indecreasingorder,correspondtoRe = 1,0.1,...,10−6. FormagneticReynolds
numberslargerthanunity,thegrowthrateisi→nde∞pendentofRmregardlessofthevalueofRe. Thedashedline,calculatedaccordingtoeq.(91),providesthe
correctasymptoticlimitγmax RmforsmallmagneticReynoldsnumbers. Middle: MaximumgrowthrateγmaxasafunctionoftheReynoldsnumberfor
differentvaluesofthemagnetic∝Prandtlnumber. Fromlefttoright,thecurvescorrespondtoPm=103,102,...,1(thicksolidline),...,10−6. Thedashed
linesγmax RePm = Rmarecalculatedaccordingtoeq.(91),whichleadstothecorrectresultevenforPm &1providedthattheReynoldsnumberis
sufficientlys∝mall.Thedottedlineγmax Re1/2resultsfromeq.(102).Right:MaximumgrowthrateγmaxasafunctionoftheReynoldsnumberfordifferent
valuesofthemagneticReynoldsnumber.∝Thethicksolidlinecorrespondstotheidealconductorlimit,i.e.,Rm .Thethinsolidlines,indecreasingorder,
correspondtoRm=1,0.1,0.01.ForReynoldsnumberslargerthanunity,thegrowthrateisindependentofRe→reg∞ardlessofthevalueofRm.Thedottedline,
calculatedaccordingtoeq.(102),providesthecorrectasymptoticlimitγmax Re1/2forsmallReynoldsnumber.
∝
thetwoanglesdefiningtheplanescontainingthem,θ andθ . itsderivativetoeliminatek betweenthesetwoandobtain
b v max
Foranyreasonablespectrumofinitialperturbationsthemode a polynomialin γ . The largest of the roots of this poly-
max
thatexhibitsthe fastest exponentialgrowth, γ , whichwe nomial is the desired maximum growth rate. It is possible
max
refertoask ,willdominatethedynamicsofthelatetime to find k following a similar methodology, but eliminat-
max max
evolutionoftheviscous,resistiveMRI.Itisthereforeofpar- ingbetweenthetwopolynomialsγ instead. However,for
max
ticular interest to characterize the physicalpropertiesof this arbitrary values of the viscosity and resistivity, both proce-
fastestgrowingmodeindifferentdissipativeregimes. dureslead to a seventhdegreepolynomialwhose rootsmust
befoundnumerically,defeatingaltogethertheattempttofind
analyticalexpressionsfork andγ .
4.1. MarginalandFastestGrowingMRI-modes max max
Usingas a guidethe resultsshownin Figures4 and5, we
Becausetheeigenvalueassociatedwiththeunstablegrow- follow an alternative procedure. The goal is to find simple
ing mode, γ+, is always real for any combination of the analyticalexpressionstodescribetheasymptoticbehaviorof
Reynolds and magnetic Reynolds numbers, it is possible to themostunstablemode,k ,andthemaximumgrowthrate,
max
find the marginally stable mode kc such that γ+(kc) ≡ 0. γmax,indifferentdissipativeregimes. ItisevidentfromFig-
Settingσ =0inequation(25),weobtainapolynomialinkc ure1thatkmax < 1andγmax < 1forallthenon-idealMRI
modes. Thisinformationcanbe usedto simplifythe disper-
k2(1+νηk2)2+κ2(1+η2k2) 4=0, (70)
c c c − sion relation and its derivative so as to decrease their order
validforanyvalueoftheviscosityandresistivity.Notethatk without loosing vital information. This makes it possible to
c
setstheminimumdomainheightfornumericalsimulationsof obtain manageable, but accurate, expressions for kmax and
viscous,resistiveMRI-driventurbulence. Figure3showsthe γmaxindifferentlimitingregimes.
solutions of equation (70) in various dissipative regimes for
Figure 6 showscontour plotsfor the critical wavenumber,
Keplerian rotation. The analytic solutions of equation (70)
k , the mostunstablewavenumber,k , andthe maximum
c max
are algebraically complicated but their asymptotic limits are
growthrate,γ ,asafunctionoftheReynoldsandmagnetic
max
rathersimple. Wefindexpressionsforthiscriticalwavenum-
ReynoldsnumbersforKeplerianrotation. Inallthreepanels,
berinseveralregimesofinterestbelow.
lightergrayareascorrespondtolargervaluesofk ,k ,and
c max
In the ideal MHD limit, it is straightforward to find sim- γmax, respectively. Note that in all the cases, the functional
pleanalyticalexpressionsforthemostunstablewavenumber, formofthecontoursnaturallydividestheplane(Re,Rm)in
k ,anditsassociatedgrowthrate,γ . However,thean- threedistinctiveregionsthatwedenoteaccordingtoI(ideal),
max max
alyticalexpressionsthatwe derivedforthe eigenfrequencies R(resistive),andV(viscous).Notethatwhenthemostunsta-
in the non-idealcase, equation(46), are notamenableto the blewavenumber,kmax,andthemaximumgrowthrate,γmax,
usualextremizationprocedure.Moreprecisely,itisverychal- are considered, these regions can be associated with the re-
lengingtofindthevaluesofk andγ thatsatisfy gionswhere Re, Rm 1, Re Rm, and Re Rm, re-
max max ≫ ≫ ≪
spectively. Theoverlapbetweentheseregionsisnotasclear
dγ+ =0. (71) whenthecriticalwavenumberkcisconsideredandsomecare
dk isneededwhenderivingapproximatedexpressionsforit.
(cid:12)kmax
(cid:12)
Figures4and5showthesol(cid:12)utionsofthisequationinvarious
(cid:12) 4.2. IdealMRIModes
dissipativeregimes.
Anotherpossiblepathtofindthevaluesofthewavenumber Let us first demonstrate briefly how the formalism pre-
k , and the associated growth rate, is to use the fact that sentedin 3reducestopreviouslyknownresultsintheideal
max
§
γ satisfiessimultaneouslythedispersionrelation(25)and MHD limit. In the absence of dissipation, the eigenvalues
max
VISCOUS, RESISTIVE MAGNETOROTATIONALMODES 9
FIG.6.—Contourplotsforthecriticalwavenumber,kc,themostunstablewavenumber,kmax,andthemaximumgrowthrate,γmax,forKeplerianrotation.In
allthreepanels,lightergrayareascorrespondtolargervaluesofkc,kmax,andγmax,respectively.Thesolidlineshighlightthecontoursforkc=1,...,10−7
andkmax =10−1,...,10−8. ThelabelsI(ideal),R(resistive),andV(viscous),denotethethreeregionsofthe(Re,Rm)planewhereequations(76),(85),
and(82);(87),(90)and(91);and(98),(101),and(102)arevalid,respectively.Thedashedlinesdividingthethreeregionsareobtainedbyequatingneighboring
approximationsforkc,kmax,andγmax.
σ ,withj = 1,2,3,4,aretherootsofthedispersionre- The temporal evolution of a single MRI-unstable mode in
0,j
{ }
lationassociatedwiththeidealMRI(Balbus&Hawley1991, physicalspacereducesto
1998),
v cosθ sin(kz)
0 v
(k2+σ02,j)2+κ2(k2+σ02,j)−4k2 =0, (72) δ(z,t)= √v22e+γ0bt2  bv00ssiinnθθvvcsoins((kkzz)) . (81)
andaregivenby(Pessah,Chan,&Psaltis2006) 0 0 b cosθ cos(kz)
− 0 v 
1/2 Theseareessenptiallythe(normalized)perturbationsfoundin
σ0,j = Λ0 ∆0 , (73) equation(4)inGoodman&Xu(1994)6.
± − ±
wherewehavedefinedth(cid:16)equantitpiesΛ(cid:17)and∆ suchthat Fromthedefinitionoftheangleθv,seeequation(50),itcan
0 0 beseenthat
κ2 The maximumgrowth rate can be obtainedby notingthat
Λ +k2, (74)
0≡ 2 γ0 = qsinθvcosθv andthereforethemaximumgrowthcor-
κ4 respondsto
∆0 +4k2. (75) q κ2
≡ 4 γ = =1 . (82)
max
2 − 4
The criticalwavenumberfor the onsetof the idealMRI is Itthenfollowsthat, in the absence ofdissipation, the planes
obtainedbysettingσ =0inthedispersionrelation(72),this containing the exponentiallygrowing velocity and magnetic
0
leadsto fieldperturbationsarecharacterizedbytheangles
k = 2q = 4 κ2. (76) π
c − θv= , (83)
4
For all the modes with pwavenumpbersk < k the difference
c 3π
√∆0 Λ0ispositiveandwecandefinethe“growthrate”γ0 θb= , (84)
− 4
andthe“oscillationfrequency”ω by
0
regardlessofthevalueoftheshearingparameter/epicyclicfre-
1/2 quency.
γ ∆ Λ , (77)
0 0 0
≡ − Finally, noting that the wavenumber for which the maxi-
(cid:16)p (cid:17)1/2 mumgrowthrateisrealizedis
ω ∆ +Λ , (78)
0 0 0
≡
κ4
bothofwhicharereala(cid:16)npdpositive((cid:17)forallpositivevaluesof kmax = 1 , (85)
− 16
theparameterq).Thisshowsthattwoofthesolutionsofequa- r
andusingequation(54)fortheratiobetweentheamplitudes
tion(72)arerealandtheothertwoareimaginary.Wecanthus
ofthemagneticandvelocityfieldsweobtain
writethefoureigenvaluesincompactnotationas
b 4+κ2
σ0,1 =γ0, σ0,2 =−γ0, σ0,3 =iω0, σ0,4 =−iω0. v0 = 4 κ2 . (86)
(79) 0 r −
In 6.2 we deriveequationsforthe MRI-drivenReynolds
§
In the ideal MHD limit, it is evident that the velocity and andMaxwellstresses,aswellasthekineticandmagneticen-
magneticfieldperturbationsareorthogonalforanymode,i.e., ergy densities associated with the perturbations. Equations
tanθ tanθ = 1,seeequation(68),andtherefore
v b
− 6NotethattheangleγinGoodman&Xu(1994),inournotationdefined
θb =θv+ π2 . (80) abnydtiatntaγke=sin−crδeBasrin/gδlByφp,oissitsivuechvatlhuaetsγin=the0cionutnhteerp-colsoitcikvweiaszeimdiurtehcatiloanx.is
10 PESSAH AND CHAN
FIG.7.—Openingangle,θdiff =θb θv,betweentheplanescontainingthefastestexponentiallygrowingmagneticandvelocityperturbationsforKeplerian
rotationinvariousdissipativeregimes.−IntheidealMHDlimittheopeningangleisθdiff =π/2,seeeqs.(83)and(84).Left:openingangleθdiffasafunction
ofthemagneticReynoldsnumberfordifferentvaluesoftheReynoldsnumber. Thethicksolidlinedenotestheinviscidlimit,i.e.,Re . Thethinsolid
lines,indecreasingorderaccordingtoθdiff forfixedRm,correspondtoRe=10,1,...,10−12.FormagneticReynoldsnumbersmuch→lar∞gerthanunity,the
openingangleisindependentofRmregardlessofthevalueofRe. Forsufficientlysmall/largeRm,θdiff π/2providedthatRe Rm. Forsufficiently
small/largeRm,θdiff π/2 arctan(κ/2)providedthatRm Re.Notethatthiscorrespondstoθdiff =→63◦26′foraKepleriand≫isk.Theonlyconditions
underwhichθdiff exce→edsπ/−2aresuchthatRe Rm 1. M≪iddle: openingangleθdiff asafunctionoftheReynoldsnumberfordifferentvaluesofthe
magneticPrandtlnumber.Thethicksolidlineatθd≥iff =π≃/2correspondstoPm=1,seeeq. (69)andFig.2.Thethinsolidlineswithpeaksatθdiff >π/2
correspond,fromlefttoright,toPm=10−1,10−2,....Thethinsolidlines,withθdiff <π/2,indecreasingorderaccordingtoθdiffforfixedRe,correspond
toPm = 10,102,.... NotethatforsmallRe,θdiff π/2forPm 1,whileθdiff π/2 arctan(κ/2)forPm 1. Right: openingangleθdiff
asafunctionoftheReynoldsnumberfordifferentvalu→esofthemagneti≃cReynoldsnumbe→r. Theth−icksolidlinecorresponds≫totheidealconductorlimit,i.e.,
Rm .Thethinsolidlines,fromrighttoleft,correspondtoRm=10,...,10−3.ForReynoldsnumberslargerthanunity,theopeningangleisindependent
ofRe→re∞gardlessofthevalueofRm.
(121), (122), (129), and(130), show why,in the idealMHD and
limit, equations(83),(84),and(86)arethereasonforwhich 14 κ2
the ratio between the Maxwell to the Reynolds stresses is γmax = η 4−κ2 . (91)
identical to the ratio between magnetic and kinetic energy
In this case, γ = ηk2 for any value of the epicyclic
densitiesforanyshearparameterandequalto5/3intheKe- max max
frequencyκ.
pleriancase(Pessah,Chan,&Psaltis2006).
Thedependenceofbothk andγ inthislimitingcase
max max
isshownwithdashedlinesintheleftpanelsofFigures4and
4.3. MRIModeswithRe Rm
≫ 5, respectively. The agreement between equations (90) and
Lets us consider the inviscid, poorly conducting limit de- (91) and the solutions to the full dispersion relation (25) in
scribed by ν = 0 and η 1. In this case, the marginally thecase ν = 0 andη 1 is excellent, onlybreakingdown
stablemodesatisfyingequ≫ation(70)isgivenby closetomagneticReyn≫oldsnumbersoforderunity.Notethat
even thoughthe equations(90) and (91) were derived under
4 κ2 theassumptionofaninviscidfluid,i.e.,ν = 0,theseexpres-
kc =s1+−η2κ2 . (87) sionscandescribetheasymptoticbehaviorofbothkmax and
γ forfiniteReynoldsnumbersprovidedthattheconditions
max
Thedependenceofthiscriticalwavenumberonthemagnetic Re RmandRm 1aresatisfied.
≫ ≪
ReynoldsnumberisshownontheleftpanelinFigure3,which
Substitutingtheasymptoticexpressionsfork andγ
max max
showsthatforsmallmagneticReynoldsnumbersk Rm.
c inequations(90)and(91)intoequation(54)weobtainthera-
∝
As discussed in 4.1, finding an analytic expression for tiobetweentheamplitudesofthemagneticandvelocityfield
the maximum grow§th rate and wavenumber associated with perturbations
it is not as straightforward. The left panel of Figure 5 sug- b ηκ3
0
= . (92)
gests that in the limit Re → ∞ and Rm ≪ 1, the maxi- v0 √4 κ2
mumgrowthrateislinearinthemagneticReynoldsnumber, −
γ Rm η−1. Thisinformationcanbeusedtoderive Therefore,inviscid, resistive MRI-unstablemodesare domi-
max ∝ ∝ natedbymagneticfieldperturbations. Notethattheratiobe-
asymptoticexpressionsforthedispersionrelation(25)andits
tweenamplitudesincreaseslinearlywithresistivity.
derivative.Theleadingordercontributionsaregivenby
The asymptotic behavior for the angles characterizing ve-
κ2γm2ax+2κ2ηkm2axγmax+κ2η2km4ax+(κ2−4)km2ax =0, locity and magnetic field perturbations, equations (50) and
(88) (51),aregivenby
1
and tanθ = , (93)
v 2κ2η
2κ2ηγ +2κ2η2k2 +κ2 4=0, (89)
max max − and
respectively. tanθ = ηκ2. (94)
b
−
Eliminating either γ or k between equations (88) InthelimitRe andRm 0,weobtain
max max
→∞ →
and(89)weobtain lim θ =0, (95)
v
η→∞
1 4 κ2 π
kmax = ηr 4−κ2 , (90) ηl→im∞θb= 2 . (96)