Table Of ContentDouble-diffusive magnetorotationalinstability ofsuper-rotation
under the influence ofazimuthal magneticfields
G. Ru¨diger, M. Gellert, M. Schultz
Leibniz-Institutfu¨rAstrophysik Potsdam, Ander Sternwarte16, D-14482 Potsdam, Germany
F. Stefani
Helmholtz-Zentrum Dresden-Rossendorf, PF 510119, 01314 Dresden, Germany
(Dated:January11,2016)
ItisdemonstratedthattheAzimuthalMagnetoRotationalInstability(AMRI)alsoworkswithsuper-rotation
6
contrarytothestandardMRIforaxialfieldswhichrequiresnegativeshear. Thestabilityagainstnonaxisym-
1
metricperturbationsofaconductingTaylor-Couetteflowwithpositiveshearundertheinfluenceofatoroidal
0
magneticfieldisconsideredwhichcanindeedbeunstableevenifthefieldbetweenthecylindersiscurrent-free.
2
Thisphenomenon isidentifiedasadouble-diffusive instabilitywhichonlyworksforsmallorlargemagnetic
n PrandtlnumberPm.ForallPmtheunstablecombinationsofrotationandfieldleadtomagneticMachnumbers
a
belowunity. ForsmallPmthecurvesofmarginalinstabilityintheHartmann/Reynoldsnumber planedonot
J
differfordifferingPm.
7 FormodelswithrestinginnercylinderitisdemonstratedthatexperimentstorealizethisformofAMRIwith
liquidsodium need axial electric currents of about 33 kAmp flowing inside the inner cylinder at rin = 0.8,
]
independent oftheglobalscalesofthecontainer. Moreover, theinstabilityofthesuper-rotationwithnottoo
n
y small Reynolds number is only weakly influenced by the radial profile of the field. The lines of marginal
d instabilityareverysimilarforthefieldswiththeprofilesbetweenBφ ∝1/R(‘current-free’)andBφ ∝R(‘z
- pinch’).
u
fl PACSnumbers:
.
s
c
i INTRODUCTION ricperturbationsareconsidered.Ithasbeendemonstratedthat
s
forfieldswithB 1/RallTaylor-Couetteflowsarestable
y φ
∝
h Inrecentyearsinstabilitiesinrotatingconductingfluidsun- against axisymmetric perturbations. The Azimuthal Magne-
p der the influence of magnetic fields became of high interest. toRotationalInstability(AMRI)isthusbasicallynonaxisym-
[ Especiallyinviewofastrophysicalapplicationstheconsidera- metric[10].
1 tionofdifferentialrotationisrelevant.Ithasbeenknownfora Taylor-Couetteflows with positiveshear are the prototype
v longtimethatdifferentialrotationwith negativeshear(‘sub- of hydrodynamic stability, [13, 14]. The more surprising is
4 rotation’) becomes unstable under the influence of an axial thefindingthatevenweakmagnetictoroidalfieldscaneasily
6
field[1]. Whenconsideredseparately,bothingredientsofthe destabilize such flows if also nonaxisymmetricperturbations
9
1 MagnetoRotationalInstability (MRI), i.e. the axial field and are taken into account, [18, 20]. We shall even demonstrate
0 alsotheKeplerrotation,arestable. Thefullsystemprovesto thattheradialprofileofthefields–ifnottoosteep–doesnot
. beunstableifandonlyifthefluidissub-rotating. playanimportantroletoallowinstability. Ontheotherhand,
1
0 Foridealfluidsthecriterionofstabilityagainstaxisymmet- the new instability is a double-diffusive phenomenon which
6 ricperturbationsofdifferentialrotationunderthepresenceof doesnotappearif the molecularviscosity andthe molecular
1 toroidalfieldsreads magneticresistivityareofthesameorder.
:
v 2
1 d R d B
i (R2Ω)2 φ >0, (1)
X R3dR − µ ρdR(cid:18) R (cid:19)
0 THEEQUATIONS
r
a where µ is the permeability and ρ the density [2]. In this
0
relation the existence of axisymmetric kinetic and magnetic Theequationsoftheproblemare
perturbationsdependssolelyontheradialprofileoftherota-
tionlaw.Thestabilityproblemoftoroidalfieldsplusdifferen- ∂U 1 1
+(U )U = P +ν∆U + curlB B
tialrotationincylindricgeometryhasbeenstudiedbyseveral ∂t ·∇ −ρ∇ µ ρ ×
0
authorsoften consideringthe perturbationsas axisymmetric, ∂B
[3–11]. = curl(U B)+η∆B (2)
∂t ×
After(1)solid-bodyrotationandallkindsofsuper-rotation
(“positive radial shear) are stabilized by the magnetic field withdivU =divB =0foranincompressiblefluid.U isthe
unless the field strongly increases outwards. The profiles velocity,Bthemagneticfield,P thepressure,νthekinematic
B 1/R(current-freebetweenthecylinders)andB R viscosity and η is the magnetic diffusivity. The basic state
φ φ
∝ ∝
(z-pinch, homogeneousaxial electric-current, see [12]) can- in thecylindricsystem with thecoordinatesR,φ,z is U =
R
notdestabilizeanyformofsuper-rotationifonlyaxisymmet- U = B = B = 0 forthe poloidalcomponentsandΩ =
z R z
2
aΩ +bΩ/R2fortherotationlawwiththeconstants conus where both branches have a positive slope, [15, 16].
AbsoluteminimaoftheHartmannnumberandthe Reynolds
aΩ = µ1Ω −rr2i2nΩin, bΩ = 11−µr2ΩΩinRi2n. (3) nthuemmbeinriemxaistt,hbeeilnoswtabwihliitcyhdtohmearointaitsioanlwlaawysislismtaibteled.bAybtowvoe
− in − in
criticalvaluesoftheHartmannnumberortheReynoldsnum-
Here r = R /R with R and R as the radii and
in in out in out ber. For a given (supercritical) Reynolds number there is a
Ωin astheangularvelocityoftheinnercylinder. WithµΩ = minimummagneticfieldfortheinstabilityandthereisamax-
Ωout/Ωin super-rotationisrepresentedbyµΩ >1. imum magnetic field suppressing the instability. Above the
Forthemagneticfieldthestationarysolutionis
weak-fieldbranchtherotationalshearistoostrongtosupport
b nonaxisymmetricperturbations.
B
B =a R+ . (4)
φ B
R
TheradialmagneticprofileB Risduetoanappliedho-
φ
∝
mogeneousaxialelectriccurrent(‘z-pinch’)whileB 1/R
φ
∝
is current-freein the fluid. We define µ = B /B . Al-
B out in
mostalwaysr =0.9ischosen.Thecurrent-freefieldisthen
in
givenbyµ =0.9andthefieldinthez-pinchisµ =1.1.
B B
The dimensionless physical parameters of the system are
the magnetic Prandtl number Pm, the Hartmann number Ha
andtheReynoldsnumberRe,i.e.
ν B D Ω D2
in out
Pm= , Ha= , Re= . (5)
η √µ0ρνη ν
FIG.1:Theinstabilityconesforthetwosuper-rotationprofileswith
ThesethreeparameterscombinetothemagneticMachnum- µΩ = 1.9(cross-hatched) andµΩ = 2.3(hatched). Pm = 10−5.
ber Mm = √PmRe/Ha indicating whether the rotation en- rin=0.9,µB =0.9,m=±1,perfect-conductingcylinders.
ergydominatesthemagneticenergyornot.
A linear code is used to solve the set of ordinary differ- The magnetic Mach number for both curves is always
ential equations for the radial functions of flow, field and smallerthanunityhenceforinstabilitythemagneticfieldmust
pressure which results after a Fourier mode transformation bestrong.
F =F(R)exp i(kz+mφ+ωt) ofthefluctuations.Theso-
lutionsareoptim(cid:0) izedwithrespec(cid:1)ttotheReynoldsnumberfor
given Hartmann numberby varying the wave number. Only
solutions for m = 1 are discussed in this Letter. The hy-
| |
drodynamicboundaryconditionsatthecylinderwallsareas-
sumedasu = u = u = 0. Thecylindersaremadefrom
R φ z
perfect-conductingor – in one case – from insulating mate-
rial. For the conducting walls the fluctuations b must fulfill
db /dR+b /R = b = 0atbothR andR sothatten
φ φ R in out
boundaryconditionsexistforthesetoftendifferentialequa-
FIG.2: ThewavenumberskD (leftpanel) andthedriftfrequency
tions (see [15]). For Bz = 0 one can easily show that the ωdr/Ωout (right panel) along the lines of marginal instability for
systemisdegeneratedunderthetransformationm → −mso Pm = 10−5 (µΩ = 5.0,2.3,1.9, fromlefttoright)andPm = 10
thatalleigenvalues(forspecificReandHa)arevalidforeach (µΩ = 5.0). rin = 0.9, µB = 0.9, m = ±1, perfect-conducting
pairm= 1. cylinders.
±
To become unstable the flat rotation law with µΩ = 1.9
AZIMUTHALMAGNETOROTATIONALINSTABILITY requiresmuchhigherHartmannnumbersandReynoldsnum-
bersthan the steeper rotation law with µΩ = 2.3. The rota-
Westartwithrotationlawswheretheoutercylinderrotates tion law with µΩ <1.9 nowhere within the gap reaches the
with higher frequencythan the inner cylinder. For the small minimumRossbyn∼umberRo=(1/2)dlogΩ/dlogR=4.8
magneticPrandtlnumberof Pm = 10−5 Fig. 1 presentsthe which is found by local approximations of the solution of
linesofmarginalinstability(i.e.forvanishinggrowthrate)for (2) as a lower limit which should be exceeded for instabil-
thetwo rotationprofileswithµΩ = 1.9andµΩ = 2.3. The ity [17, 18]. The reason might be that, as the plots also
magneticfield between the cylindersis fixed to the vacuum- show, this Rossby numberis by far notuniformover the ra-
type,B 1/R. Theformoftheresultinglinescorresponds dius for Taylor-Couette flows. They also demonstrate that a
φ
∝
to the general form of the lines for the AMRI (with sub- lower limit does not exist for perfect-conductingboundaries
rotation)wherethe line ofmarginalinstability formsa tilted at the mentioned value. It is clear, however, that for more
3
andmoreflatrotationlawthesupercriticalReynoldsnumbers
andHartmannnumbersaregrowing.Thequestioniswhether
a minimum value µΩ > 1 indeed exists for which the crit-
ical Reynolds number and Hartmann number reach infinity.
WhethersuchalimitexistsbelowµΩ =1.9isnotyetfinally
clearfornumericalreasons.
Wavenumbersandazimuthalpatternmigration
TheleftpanelofFig. 2givestheoptimizedwavenumbers
ofthe instabilitynormalizedwith thegapwidthD alongthe
FIG. 3: The minimum values of Ha and their corresponding
twobranchesofinstabilityforsmallandlargePm. Theaxial
Reynoldsnumbersfromthecurvesofmarginalinstabilityshownin
sizeofcellisδz/D≃π/k. Cellswithk <πarethusprolate Fig. 1forµΩ =2.3vs. Pm. ForPm>2·10−3nosolutionexists.
whilecellswith k > π areoblatewith respectto therotaion Forrin = 0.8theHartmannnumbers(crosses)andReynoldsnum-
axis. Thelimitk = π fornearlysphericalcellsismarkedby bers(squares)forµΩ =4(up)andµΩ =128(down)aregivenbut
a horizontal dotted line. The plot demonstrates that the cell onlyforPm=10−5.
geometrystrongly dependson the magnetic Prandtlnumber.
ForsmallPmalsothewavenumbersaresmallhencethecells
Moreimportantis thatthe curvesstop atPm 2 10−3 in-
areprolateorspherical.Alongthestrong-field(lower)branch
≃ ·
dicating that for higher values of Pm no solutions unstable
of the instability cone the wave numbersexceedthose at the
againstperturbationswithm=1exist.ForPm>1,however,
low-field(upper)branchwherethecellsareratherlonginax-
the instability resurrects with strikingly low Reynolds num-
ial direction. For Pm 1, however, the wave numbers at
≫ bers (Fig. 4, right). The low Reynolds numbersensure that
bothbranchesoftheconusarelargerthanπ so thatthecells
also for large magnetic Prandtl numbers the magnetic Mach
arealwaysveryflat.
number remains smaller than unity. Instabilities which only
Also the drift rates possess a strong Pm-dependence. The
existforν = η belongto the class of double-diffusiveinsta-
drift values as given in the rightpanel of Fig. 2 are the real
6
bilities[21]. TheydonotappearforPm 1.
parts ω of the frequency ω of the Fourier mode of the in-
dr ≈
To findouthow strongthe influenceofthe boundarycon-
stability,normalizedwiththerotationrateoftheoutercylin-
der. Because of φ˙ = ω /m the azimuthal migration has dition is, Fig. 4 (left) gives the instability map for µΩ = 5
dr
− for the two cases of perfect-conductingwalls and insulating
the opposite sign of ω . For AMRI with negativeshear we
dr
boundaryconditions. The application of the rather steep ro-
alwaysfoundthatthepatternmigratesforallPminpositiveφ-
direction([19]). ForTaylerinstability undertheinfluenceof tation law with µΩ = 5 has only numerical reasons, the
super-rotationthesituationismorecomplicatedasthepattern magneticPrandtlnumberofPm = 10−5 characterizesliquid
sodium (for laboratory experiments). Note that for vacuum
counterrotates for small Pm while it corotates for Pm > 1,
boundariesthesuper-rotationlawsaremuchmorestablethan
[20]. The plot in Fig. 2 shows similar results. For small
for perfect-conductingconditions. For AMRI with negative
Pmpositiveω occurandtheperturbationpatternindeedro-
dr
shearthecriticalHartmannnumbersforbothkindsofbound-
tatesbackwards.LargePmprovidenegativedriftvalueshence
the pattern migrates with the rotation. For ω /Ω = 1 aryconditionsdoonlyslightlydiffer(butalsowithhigherval-
dr out
− ues for vacuum conditions). The influence of the boundary
thepatternrotatesjustastheoutercylinder(onlypossiblefor
largePm). conditions is also large if the instability of the z-pinch un-
derthepresenceofdifferentialrotationisconsidered. Often,
however,vacuumboundaryconditionslead to a mucheasier
InfluencesofthemagneticPrandtlnumberandboundary excitationoftheinstabilitythantheconductingconditionsdo,
conditions see[20].
OnefindsfromtheFigs. 1–4theexpectedtrendwithµΩ
Figure 3 gives the coordinates of the locations with mini- for the minimum Reynolds number. For perfect-conducting
mumHartmannnumbershowninFig. 1forµΩ =2.3butfor boundariesandPm=10−5itdecreasesfrom3500fortheflat
variousPm. One finds that for Pm 0 both the Hartmann rotation law (µΩ = 1.9) to 250 for the steeper rotation law
numberandtheReynoldsnumbercon→verge(toHa 400and (µΩ =5).
≃
Re 2000). Thecurvesscale,therefore,withHaandRefor
≃
small Pm, so that the numerical values of Ha and Re given
in Fig. 1 for Pm = 10−5 (sodium) can also be used for the RESTINGINNERCYLINDER
smallervalueofgallium.
On the other hand, for increasing Pm the minimum Hart- ATaylor-Couetteflowwithrestinginnercylinderisclearly
mannnumberandtheReynoldsnumberare(slightly)reduced. the most simple model for super-rotating fluids. As it also
4
1000
R)
E
T
U 800 TI RI
O M
R ( A
BE 600
M
U
N
S 400
D
L
O
N 200
Y
FIG. 4: Instability maps for µΩ = 5. Left panel: Pm = 10−5 RE
0
forperfect-conductingandinsulatingcylinders. Rightpanel: Pm = 0 500 1000 1500
10, perfect-conductingcylinders. Allcurvesofmarginalinstability HARTMANN NUMBER
fulfilltheconditionMm<1.rin=0.9.
FIG.5: Instabilitymapfor Taylor-Couetteflowswithrestinginner
cylinder. Theradialprofilesoftheazimuthalbackgroundfieldsvary
forms the steepest radial profile (µΩ = ∞) we expect the lfirnoems).µBBlu=eco1l.o1r:(zP-mpin=ch0,.t0h2ic.kRleidnecso)lotor:µPBm==100.9−5(.ArMinR=I, 0th.i9n,
minimal Hartmann numberswhich are suitable for a labora-
perfect-conductingboundaries.
tory experiment (with cylinders from conducting material).
The thin red line in Fig. 5 valid for the current-free field
B 1/R and for Pm = 10−5 provides a minimum Hart-
φ
∝ in MHD Taylor-Couette flows does not strongly depend on
mannnumberof about145. In orderto transformthis value
theradialprofileofthefield. Italreadyworksforcurrent-free
tothenecessaryelectric-currentalongtheaxis(i.e. withinthe
fields but axial electric-currents do not basically change the
innercylinder)thewell-knownrelationI = 5R B can
axis in in
system, which is even true for the z-pinch with its uniform
berewrittenas
currentdensity.
R
in
Iaxis =5 Ha√µ0ρνη =5Hain√µ0ρνη, (6)
D
withHa astheHartmannnumberafter(5)formedwithR
in in
insteadwiththegapwidthD.WefindHain =1300orIaxis [1] E.P.Velikhov,Sov.Phys.JETP36,995(1959)
≃
53.5 kAmp for liquid sodium as the conducting fluid (with [2] D.Michael,Mathematica1,45(1954)
√µ0ρνη 8.2 in c.g.s.). This is the minimum current for [3] S.Chandrasekhar,HydrodynamicandHydromagneticStability
≃ (ClarendonPress,Oxford,1961)
instabilitywith the gapwidthD = 0.1. Anothercalculation
[4] L.N.Howard&A.S.Gupta,JFM14,463(1962)
for D = 0.2 and µΩ = 128 presented in Fig. 3 leads to
[5] G.Chanmugam,MNRAS187,769(1979)
Ha =800withonly33kAmpasthecriticalelectric-current.
in [6] D.Fearn,GAFD30,227(1984)
Thoughthesizeofthecontainerdoesnotinfluencethecritical [7] E.Knobloch,MNRAS255,25(1992)
valueoftheelectric-currentbutthegapwidthdoes.Thewider [8] B. Dubrulle & E. Knobloch, Astron. Astrophys. 274, 667
the gap the weaker the necessary currentwhich is necessary (1993)
toexcitetheAMRIforsuper-rotation. [9] M.E.Pessah&D.Psaltis,ApJ628,879(2005)
[10] I.Herron&F.Soliman,Appl.Math.Lett.19,1113(2006)
Sofar we did not vary the radial profile 1/R of the mag-
[11] D.Shalybkov,Phys.Rev.E73,016302(2006)
netic background field. One of the new questions is, how-
[12] H.Goedbloed&S.Poedts,PrinciplesofMagnetohydrodynam-
ever,howsensitivethedescribedinstabilityofsuper-rotating
ics(Cambridge,UniversityPress2004)
fluidsreacts on the presenceof axialelectric-currentswithin [13] F.Wendt,Ingenieur-Archiv4,577(1933)
the fluid. In addition to the current-free field, in Fig. 5 the [14] F.Schultz-Grunow,ZAMM39,101(1959)
lines of marginal instability for the field B R of the z- [15] G.Ru¨diger,L.L.Kitchatinov,&R.Hollerbach,MagneticPro-
φ
pinch with uniform axial current are given fo∝r two different cessesinAstrophysics(Wiley-VCH,2013)
[16] R.Hollerbach,V.Teeluck,&G.Ru¨diger,Phys.Rev.Lett.104,
magneticPrandtlnumbers.Asthenewexamples(thicklines)
044502(2010)
containaxialelectric-current,theTaylerinstabilityforRe=0
[17] W.Liu,J.Goodman,&I.Herron,Phys.Rev.E.74,6302(2006)
appearsatcertainvaluesHa oftheHartmannnumber[22–
Tay [18] F.Stefani&O.Kirillov,Phys.Rev.E92,051001(R)(2015)
25], which themselves do not change for changingPm [26]. [19] G. Ru¨diger, M. Gellert, M. Schultz, et al., MNRAS 438, 271
(2014)
For larger Reynolds numbers – if the rotation becomes [20] G.Ru¨diger,M.Schultz,M.Gellert,&F.Stefani,Phys.Fluids,
important – one finds only little differences of the lines of acc..(2016)
[21] D.J.Acheson,Phil.Trans.R.Soc.A289,459(1978)
marginal instability for the two field profiles with or with-
[22] R.J.Tayler,Proc.Phys.Soc.70,31(1957)
out axial electric backgroundcurrent in the fluid. The min-
[23] R.J.Tayler,MNRAS161,365(1973)
imum Hartmann number for instability are slightly reduced
[24] Y.V.Vandakurov,Sov.Astron.16,265(1972)
by the currents but the differences are small. The main re- [25] M.Seilmayer,F.Stefani,Th.Gundrum,etal.,Phys.Rev.Lett.
sult is thatthe magnetic-inducedinstability of super-rotation 108,244501(2010)
5
[26] G.Ru¨diger&M.Schultz,Astron.Nachr.331,121(2010)
