Stratorotational instability in MHD Taylor-Couette flows

a r X i v :0808.0577v 1 [a s t r o -p h ] 5 A u g 2008

Astronomy &Astrophysics manuscript no.srimag c

ESO 2008August 5,2008

Stratorotational instability in MHD Taylor-Couette ?ows

G.R¨u diger 1and D.A.Shalybkov 2

1Astrophysikalisches Institut Potsdam,An der Sternwarte 16,D-14482Potsdam,Germany

A.F.Io ?e Institute for Physics and Technology,194021,St.Petersburg,Russia

August 5,2008;accepted

ABSTRACT

Aims.The stability of dissipative Taylor-Couette ?ows with an axial stable density strati?cation and a prescribed azimuthal magnetic ?eld is considered.

Methods.Global nonaxisymmetric solutions of the linearized MHD equations with toroidal magnetic ?eld,axial density strati?cation and di ?erential rotation are found for both insulating and conducting cylinder walls.

Results.Flat rotation laws such as the quasi-Kepler law are unstable against the nonaxisymmetric stratorotational instability (SRI).The in?uence of a current-free toroidal magnetic ?eld depends on the magnetic Prandtl number Pm:SRI is supported by Pm >1and it is suppressed by Pm <～1.For too ?at rotation laws a smooth transition exists to the instability which the toroidal magnetic ?eld produces in combination with the di ?erential rotation.This nonaxisymmetric azimuthal magnetorotational instability (AMRI)has been computed under the presence of an axial density gradient.

If the magnetic ?eld between the cylinders is not current-free then also the Tayler instability occurs and the transition from the hydrodynamic SRI to the magnetic Tayler instability proves to be rather complex.Most spectacular is the ‘ballooning’of the stability domain by the density strati?cation:already a rather small rotation stabilizes magnetic ?elds against the Tayler instability.

An azimuthal component of the resulting electromotive force only exists for density-strati?ed ?ows.The related alpha-e ?ect for magnetic SRI of Kepler rotation appears to be positive for negative d ρ/dz <0.Key words.methods:numerical –magnetic ?elds –magnetohydrodynamics (MHD)

1.Motivation

This work is motivated by theoretical and experimen-tal progresses in studies of the stratorotational instability (SRI)and the magnetorotational instability (MRI)in MHD Taylor-Couette experiments.It has been shown theoretically (Molemaker,McWilliams &Yavneh 2001;Yavneh,McWilliams &Molemaker 2001;Dubrulle et al.2005;Shalybkov &R¨u diger 2005;Umurhan 2006)and in the laboratory (Le Bars &Le Gal 2007)that a combination of a centrifugal-stable nonuniform ro-tation law and a stable axial density strati?cation leads to the so-called stratorotational instability (SRI)in the Taylor-Couette ?ow.This instability exists only for nonaxisymmetric distur-bances.On the other hand,there are also nonaxisymmetric insta-bilities for a combination of Rayleigh-stable rotation laws and azimuthal magnetic ?elds (Pitts &Tayler 1985).The question is whether the combination of density strati?cation,di ?erential rotation and toroidal ?elds acts stabilizing or destabilizing or whether even new instabilities arise.

Such a combination of axial density strati?cation,stable ro-tation law and strong toroidal magnetic ?eld is the typical con-stellation in accretion disks (Ogilvie &Pringle 1996;Curry &Pudritz 1996;Papaloizou &Terquem 1997).There the rota-tion is Keplerian with ?∝R ?3/2and the toroidal ?eld is gen-erated from weak large-scale poloidal ?elds by induction due to the di ?erential rotation.The standard case is that the result-ing toroidal ?eld strongly exceeds the amplitude of the original poloidal ?eld if the magnetic Reynolds number of the di ?eren-tial rotation Rm =?R 2/ηis much larger than unity.If this is true then the standard MRI,i.e.the in?uence of the poloidal ?eld on the stability of the di ?erential rotation,would be of minor im-

portance 1.The question is whether the toroidal magnetic ?eld can reach such high amplitudes or whether it becomes unstable already for much smaller values.It is known that toroidal ?elds with strong electric currents become unstable (Tayler instability,‘TI’)but it is also known that in combination with di ?erential ro-tation also toroidal ?elds become unstable which are current-free in the considered domain (azimuthal MRI,‘AMRI’,see R¨u diger

et al.2007a).In the latter case,with B φ∝R ?1

,the questions are whether the density strati?cation destabilizes AMRI and /or whether the toroidal ?eld stabilizes the SRI too strongly so that its real existence becomes basically suppressed.

We shall consider the interaction of the di ?erential rota-tion with both an axial density strati?cation and a toroidal mag-netic ?eld in the simpli?ed Taylor-Couette geometry.The den-sity strati?cation is always supposed to be stable but the mag-netic ?eld and the rotation law between the cylinders can be both stable or unstable.If,in particular,the toroidal ?eld is Tayler un-stable then the interaction of di ?erential rotation,density strati?-cation and magnetic ?eld becomes highly complex.We ?nd sta-bilization and destabilization in strong dependence on the mag-netic Prandtl number.Again,in experiments,for small magnetic Prandtl number,the ?ows are predicted to be stabilized.For galaxies and protoneutron stars (PNS)with their high magnetic Prandtl numbers we ?nd the opposite:the magnetic in?uence supports the SRI leading to even smaller Reynolds numbers than in hydrodynamics.

2G.R¨u diger&D.A.Shalybkov:Stratorotational instability in MHD Taylor-Couette?ow

out

Fig.1.The geometry of the problem–two concentric cylinders

with radii R in and R out rotating with?in and?out.Bφis the az-

imuthal magnetic?eld which,generally,is produced by both

an axial current inside the inner cylinder and an axial current

through the?uid.

2.The Taylor-Couette geometry

A Taylor-Couette container is considered which con?nes a

toroidal magnetic?eld with given amplitudes at the cylinder

walls which rotate with di?erent rotation rates?(see Fig.1).

The?uid between the cylinders is assumed to be incompress-

ible and dissipative with the kinematic viscosityνand the mag-

netic di?usivityη.Derived from the conservation of angular mo-

mentum the rotation law?(R)in the?uid is

?=a+

1??η2?in,b=

1?μ?

R out ,μ?=

?out

,(4)

where the?rst term corresponds to a uniform axial electric cur-rent at radius R and the second term is current free for R>0.In analogy withμ?it is useful to de?ne the quantity

μB=B out

AR in+BR?1in

,(5)

measuring the variation in Bφacross the gap between the cylin-ders.The coe?cients A and B are

A=B in

1??η2

,B=B in R in

1?μB?η

+(u?)u=?

μ0

curl B×B,

+(u?)ρ=0,

and

div u=0,div B=0,(7)

where u is the velocity,B is the magnetic?eld,P is the pressure,

g is the gravitational acceleration(supposed as vertical and con-

stant),νis the kinematic viscosity,ηis the magnetic di?usivity

andμ0the magnetic constant.

In the presence of a vertical density gradient(ρ=ρ(z))it has

been shown for B=0,that the system(7)allows the angular

velocity pro?le(1)only in the limit of slow rotation and small

strati?cation

R?2d log(z)

?1.(8)

It easy to show that the same is true for B 0.So,we are inter-

ested in the stability of the basic state

U=(0,R?(R),0),B=(0,Bφ(R),0),

P=P0(R)+P1(R,z),ρ=ρ0+ρ1(z),(9)

where?is given by(1),Bφby(4),ρ0is the uniform reference

density,P is the total pressure including the magnetic part with

|P1/P0|?1and|ρ1/ρ0|?1.

The linear stability problem is considered for the perturbed

state of U+u,B+b,ρ0+ρ1+ρ′,P0+P1+P′.Using the conditions

(8)the linearized system(7)takes the Boussinesq form with the

coe?cients depending only on the radial coordinate,and a nor-

mal mode expansion of the solution F=F(R)exp(i(mφ+kz+ωt))

can be used,where F represents any of the disturbed quantities.

Finally after a normalization we?nd

d2u R

d u R

R2?

k2+m2R uφ?

?i Re(ω+m?)u R+2Re?uφ?d P′

Ha2Bφb R?2Ha2Bφ

d R2

d R?

uφ

R2 uφ+2i m

P′?Re d R(R2?)u R

Ha2

d R BφR

b R+i m

d R2

d R?

k2+m2

Ha2Bφb z=0,

i(ω+m?)ρ′?N2u z=0,

d u R

d R2

d R?

b R

R2 b R?2i m

G.R¨u diger &D.A.Shalybkov:Stratorotational instability in MHD Taylor-Couette ?ow 3

?i Pm Re(ω+m ?)b R +i

d R 2+1

d R ?b φR 2

b φ+2i m d R

b R ?

u R +i

d R 2

+1d R

k 2

m 2R

B φu z =0,

(10)

where the same symbols are used for the normalized quanti-ties except P ′which denotes P ′/ρ0and rede?ning ρ′as PmRe g ρ′/ρ0.The dimensionless numbers of the problem are the magnetic Prandtl number Pm,the Hartmann number Ha,the Reynolds number Re Pm =

√

,(11)

and the buoyancy frequency N after N 2=?

g d z .

(12)

We used R 0=(R in (R out ?R in ))1/2as the unit of length,η/R 0as the unit of the perturbation velocity,B in as the magnetic ?eld unit,?in as the unit of ω,N ,and ?,R 0?2as the unit of g ,ρ0as

the density unit and νη/R 20as the unit of the rede?ned P ′

.It is convenient to describe the in?uences of the density strati?cation and the basic rotation by the Froude number Fr =

?in

4G.R¨u diger &D.A.Shalybkov:Stratorotational instability in MHD Taylor-Couette ?ow

in the following whether such results are in?uenced by the exis-tence of a toroidal magnetic

?eld.

Fig.3.The marginal stability lines for small-gap (top,?η=0.78),

modest-gap (?η=0.5,middle)and wide-gap (?η=0.3,bottom).The vertical dotted lines denote the Rayleigh limit,the pseudo-Kepler rotation and the pseudo-galactic rotation (μ?=?η).The curves represent di ?erent density strati?cations with Fr =2.2(dot-dashed),Fr =1(dashed),Fr =0.7(dotted),Fr =0.5(solid,maximum strati?cation).

4.SRI with azimuthal magnetic ?eld

Without density strati?cation the ideal Taylor-Couette ?ow with

imposed azimuthal magnetic ?eld is stable against axisymmetric disturbances if

1d R (R 2?)2?R

d R

B φ?

η≡?μ0.

(16)

Therefore,all magnetic pro?les with positive μB or with μB <2(for ?η=0.5)are stable against axisymmetric perturbations.As the dissipative e ?ects are

stabilizing the ?ow a magnetic ?eld with μB beyond the interval (16)becomes unstable against ax-isymmetric disturbances in case that the magnetic ?eld ampli-tude (or its Hartmann number)exceeds a critical value.

The stability condition for the azimuthal magnetic ?eld against nonaxisymmetric disturbances after Tayler (1961)is d

3?2?η2??η4

≡?μ1

(18)

for the m =1(kink)mode which is the most unstable mode.Hence,all azimuthal magnetic ?elds with positive μB smaller than 0.62(for ?η=0.5)are stable against nonaxisymmetric dis-turbances.Current-free ?elds with B φ∝1/R ,i.e.μB =0.5for ?η=0.5are thus always stable against m =0and m =1.

As always ?μ1

In the present paper the in?uence of stable vertical density strati?cations on the Taylor-Couette ?ow stability with imposed azimuthal magnetic ?elds is considered for three di ?erent cases:1)both the rotation law and the magnetic ?eld are individually stable and both together are unstable (i.e.μB =0.5,no criti-cal Hartmann number)

2)the magnetic ?eld is stable for m =0but unstable for m =1(i.e.μB =1,one critical Hartmann number)

3)the magnetic ?eld is so steep that it is unstable for m =0and for m =1(i.e.μB =3,two critical Hartmann numbers).4.1.AMRI with density strati?cation

The azimuthal magnetic ?eld with B φ∝1/R (i.e.μB =0.5for ?η=0.5)is considered which is current-free between the cylinders as the simplest example of an azimuthal magnetic ?eld which is basically stable without rotation for both axisymmetric and asymmetric disturbances.

G.R¨u diger &D.A.Shalybkov:Stratorotational instability in MHD Taylor-Couette ?ow 5

The typical stability diagram is presented by the

Figs.4and 5

for containers with conducting /isolating cylinders,medium gap width (?η=0.5)and rather ?at rotation law (pseudo-Kepler ro-tation with μ?=0.35and a slightly steeper rotation law with ?μ=0.3)which are stable without density strati?cation and with-out magnetic ?eld.We ?nd that the results depend only slightly on the conducting properties of the cylinder material.

Fig.4.The marginal stability lines for the ?ow with conducting cylinders,?η=0.5,current-free (μB =0.5)azimuthal magnetic ?eld,pseudo-Kepler rotation (μ?=0.35),Fr =0.5for distur-bances with m =1(solid)and m =2(dotted).The curves are labeled by their magnetic Prandtl numbers.For comparison the dashed line demonstrates AMRI (m =1)for homogeneous ?uids (N =0).

Fig.5.The same as Fig.4but for insulating cylinders with μ?=0.3(as in Fig.2)but without m =2mode.

The solid line in Figs.4and 5is the marginal stability line of a homogeneous ?uid with Pm =1.There are no solutions at both the vertical axis and the horizontal axis.The MHD ?ow is only unstable as a combination of di ?erential rotation and the azimuthal magnetic ?eld.For any supercritical Hartmann num-ber there are two critical Reynolds numbers between which the ?uid is unstable.Similarly,for any supercritical Reynolds num-ber there are two critical Hartmann numbers between which the ?uid is unstable.We have called this phenomenon as the Azimuthal MagnetoRotational Instability (AMRI)which scales with the magnetic Reynolds number in the same way as it is known from the standard MRI in Taylor-Couette experiments.However,magnetic Reynolds numbers of order 100are too high to be realized in the MHD laboratory.

With a density strati?cation (Fr =0.5)the situation changes.The following results can be interpreted as either AMRI for density-strati?ed ?uids or as the in?uence of an azimuthal mag-netic ?eld on SRI.There is now an instability (SRI)without any magnetic ?eld for Re >296at m =1and for Re >366at m =2(for conducting cylinders,pseudo-Kepler rotation law,Fig.4)and for Re >264at m =1(insulating cylinders,μ?=0.3,Fig.5).For this latter case the calculations are restricted to the kink (m =1)instability.

For conducting boundaries (see Fig.4)and for a special case (Ha =100,m =1)the radial eigenfunctions are given for Pm =1(Fig.6)and Pm =10(Fig.7)which demonstrate that the SRI modes do not form boundary phenomena.The pro?les are normalized since in the linear theory the amplitudes have no own physical meaning.For the linear growth time in units of the rotation period of the inner cylinder for the solution with Pm =1one obtains τgrowth

Re ?341

(19)

Hence,for Re =400the e-folding time is about four rota-tions.The vertical wave number at the critical Reynolds number Re =341is 8.63what means that the vertical cell size in units of the gap width is about 0.36.Without magnetic ?eld the wave number is https://www.360docs.net/doc/d62317140.html,mon action of magnetic ?eld and density strati?cation leads to rather ?at cell con?gurations.

Fig.6.The eigenfunctions for the radial components of ?ow (left)and ?eld (right)for Pm =1.Solid:real part,dashed:imag-inary part;the (arbitrary)amplitude is normalized.Fr =0.5,Ha =100,m =1,see Fig.4.

Note that the in?uence of the azimuthal magnetic ?eld on the SRI strongly depends on the magnetic Prandtl number Pm.For Pm ≤1the SRI is stabilized by the magnetic ?eld but it can be supported for Pm >1(see Figs.4,5).For Pm =10

6G.R¨u diger &D.A.Shalybkov:Stratorotational instability in MHD Taylor-Couette

?ow

Fig.7.The same as in

Fig.6but for Pm =10.The eigenfunc-tions do not re?ect a boundary phenomenon.

the critical Reynolds number is reduced (by a factor of two)for

Ha ?200.In this case AMRI supports the SRI.This behavior is similar to that of the standard MRI with axial magnetic ?eld and hydrodynamically unstable rotation where also for Pm >1the magnetic ?eld supports the centrifugal instability (R¨u diger &Shalybkov 2002).

Note also that the magnetic suppression of the SRI for small Pm is rather weak.Even for Pm =10?5a (large)Hartmann num-ber of order of 100only leads to a small critical Reynolds num-ber of about 500.The lines for Pm =10?5in Figs.4and 5are identical with the lines for all smaller Pm,i.e.they also hold for gallium (10?6)and also mercury (10?7).Here all the curves scale with Re rather than Rm so that the corresponding mag-netic Reynolds numbers are quite small compared with those for AMRI.As also the corresponding Hartmann numbers with Ha <～100are not too high,laboratory experiments with ?uid metals should be possible provided a su ?ciently density strat-i?cation can be produced.

It is also shown in Fig.4that for a given Pm the critical Reynolds numbers for m =2exceed those for m =1.The di ?er-ences,however,are not too big so that for slightly supercritical Re several modes should be excited.This is insofar surprising as the known smoothing action of di ?erential rotation upon modes with high m here seems to be very weak.

The rather small Reynolds numbers and Hartmann numbers lead to the impression that the instability should be observable into the laboratory.For hydrodynamically unstable situations we have only a suppression of the instability by the magnetic ?eld for small Pm.For hydrodynamically stable situations (solid lines in Figs.4and 5)it was shown for uniform ?uids that the appro-priate numbers of the problem are the magnetic Reynolds num-ber Rm and the Lundquist number S Rm =Pm ·Re ,

S =Pm 1/2·Ha ,

(20)

rather than Re and Ha.Figure 8demonstrates that the same is true for density-strati?ed ?uids with so ?at rotation laws that hy-dromagnetic SRI does not operate (see Fig.3).Both the critical magnetic Reynolds number and Lundquist number do not vary remarkably when the magnetic Prandtl number varies over 3or-ders of magnitude.

Strati?ed ?ows without magnetic ?eld become stable for μ?-values slightly greater than μ?=0.5.According to Fig.8the ?ow is stable for μ?=0.6without a magnetic ?eld but it be-comes unstable with magnetic ?elds.The critical Rm are not monotonous with Pm.The values of Rm decrease with Pm for small Pm and v.v.The result can be seen in Fig.8which di ?ers in this respect to the AMRI without density strati?cation (see R¨u diger et al.2007a).

Fig.8.AMRI for strati?ed ?uids with Fr =0.5between insu-lating cylinders (?η=0.5,μB =0.5).The rotation law is so ?at (μ?=0.6)that nonmagnetic SRI does not exist (see Fig.3,mid-dle).Lines are labeled by their Pm numbers.

The solid line in Fig.9demonstrates that the AMRI (like the standard MRI)exists for all rotation laws with decreasing angular velocity as a function of radius (i.e.for μ?<1).

Fig.9.The minimum Reynolds numbers for insulating cylin-ders,?η=0.5,Pm =1and for the kink (m =1)mode.The verti-cal lines are the same as in Fig.3.The numbers on the curves are

Hartmann numbers which correspond to the minimum Reynolds numbers.The solid line is AMRI without a strati?cation,the dot-dashed line is AMRI with strati?cation (Fr =0.5,see Fig.8)and the dashed line is SRI without magnetic ?eld.

According to the Figs.4and 5the marginal stability lines have always a minimum for some Hartmann number including

G.R¨u diger &D.A.Shalybkov:Stratorotational instability in MHD Taylor-Couette ?ow 7

Ha =0.These minimum Reynolds numbers are plotted in

Fig.9

as functions of μ?.The dashed line is SRI without magnetic ?eld which disappears for too ?at rotation laws (μ?>～0.52).There is a smooth transition,however,to the instability AMRI (current-free toroidal ?eld plus di ?erential rotation)which has been com-puted here for the ?rst time with a density gradient (dot-dashed).The comparison with standard AMRI without d ρ/d z (solid line)yields the expected suppression by density strati?cation.Note,however,that for quasi-galactic rotation (μ?=0.5in Fig.9)the SRI with magnetic ?eld can easier be excited than without mag-netic ?eld.As the ?gure shows the necessary magnetic ?eld has a Hartmann number of about 137.

As AMRI scales with Rm rather than Re.the ‘smooth’tran-sition interval from the hydrodynamic solutions to the magneto-hydrodynamic scales with Pm ?1.Hence,the dot-dashed line in Fig.9becomes more and more steep for decreasing Pm.4.2.Density strati?cation and Tayler instability (m =1)

The in?uence of the vertical density strati?cation on the non-axisymmetric Tayler instability (i.e.?μ1<μB

For m =1the critical Hartmann numbers above which the ?ow is unstable for Re =0are 150for conducting cylin-ders and 109for insulating cylinders.These critical numbers are not in?uenced by the density strati?cation.For m =0no critical Hartmann numbers here exist;the magnetic ?eld stabi-lizes this Rayleigh-mode for both homogeneous (dotted line)and density-strati?ed (dot-dot-dot-dashed lines)?uids.For homoge-neous ?uids the (dotted)marginal stability line for axisymmetric disturbances does not depend on the magnetic Prandtl number Pm (Shalybkov 2006)!

The critical Reynolds numbers for the Rayleigh instability for Ha =0do not depend on the conducting properties of the cylinders.They are 68(m =0)and 75(m =1)without strati?-cation.The density-strati?cation stabilizes the ?ow;the critical Reynolds numbers are 294for m =0and 226for m =1(for Fr =0.5).With density-strati?cation the kink mode (m =1)proves to be the most unstable mode.

Moreover,for given Pm the kink mode is the most unstable one for all Ha for Fr =0.5.For small Pm this mode is stabi-lized by weak magnetic ?elds before a dramatic destabilization happens for larger magnetic ?elds.For increasing Pm the ?ow becomes unstable for decreasing Reynolds numbers at a ?xed Hartmann number.The opposite is also true:the smaller the Pm the stronger is the magnetic ?eld in?uence and the ?ow becomes unstable for smaller Hartmann numbers at a ?xed Reynolds number.

Note that the competition between magnetic and centrifu-gal instabilities can lead to a rather complex behavior as illus-trated by the marginal stability line with Pm =10.Generally,however,the stability region is much larger for density-strati?ed ?uids so that we ?nd that a stable strati?cation increases the ?ow stability.Of particular interest is the phenomenon that for density-strati?ed ?uids generally the critical Hartmann numbers are larger than without rotation and the same is true for small Pm with respect to the critical Reynolds number (‘ballooning’).For

Fig.10.The marginal stability lines for a ?ow with insulating (top)and conducting (bottom)cylinders.It is ?η=0.5,μ?=0(resting outer cylinder)and μB =1.The curves are labeled by their magnetic Prandtl Pm.The m =0(dotted)mode (no dependence on Pm)and m =1(solid)mode for homogeneous ?uid and m =0(dot-dot-dot-dashed)and m =1(dot-dashed)modes for density-strati?ed ?uids with Fr =0.5.

large magnetic Prandtl number,however,the magnetic ?eld ba-sically destabilizes the Rayleigh instability.Even for weak ?elds the critical Reynolds number is smaller than the Reynolds num-ber for the nonmagnetic case.

4.3.Density strati?cation and Tayler instability (m =0,1)

The most complex situation appears when both the modes with m =0and m =1are Tayler-unstable.Only conducting cylin-ders with medium-sized gap (?η=0.5),outer cylinder at rest and a toroidal magnetic ?eld with strong currents (μB =3)are analyzed.The di ?erential rotation is Rayleigh unstable without magnetic ?eld and density strati?cation and the magnetic ?eld is

8G.R¨u diger &D.A.Shalybkov:Stratorotational instability in MHD Taylor-Couette

?ow

Fig.11.The marginal stability lines for conducting cylinders with ?η=0.5,μ?=0,μB =3,Pm =10?5(top)and Pm =1(bot-tom).The lines connect the Rayleigh instability without mag-netic ?eld with the Tayler instability without rotation.The dotted lines are for m =0;the solid lines for m =1.The inner curves are for homogeneous ?uids (N =0)and the outer curves for strati?ed ?uids with Fr =1.The density strati?cation stabilizes in both directions.

Tayler-unstable without rotation (for m =0and m =1).The crit-ical Hartmann number for Re =0is nearly equal for all cases:m =0gives Ha 0=21and m =1gives Ha 1=20.6.For other pa-rameters it can be more di ?erent.Either the m =0or the m =1mode can be the most unstable mode.Often also the m =0mode is the most unstable for small Ha numbers and fast di ?erential rotation (see R¨u diger et al.2007b).

The critical Reynolds numbers above which the rotation be-comes unstable without a magnetic ?eld (Ha =0)are called Re 0and Re 1.Re 0is smaller than Re 1for ?ows with 0<μ?Ha 0.Note that ‘ballooning’of the stability region is produced by the density strati?cation.The density strati?cation in combina-tion with the basic rotation stabilizes the ?ow for larger mag-netic ?elds.Under the in?uence of di ?erential rotation and den-sity strati?cation stronger magnetic ?eld amplitudes prove to be stable than they were without rotation.The e ?ect already exists for rather slow rotation rates (see Fig.11,Pm =1).This phe-nomenon is a consequence of the explicit inclusion of the sta-

ble density strati?cation into the MHD equations.One ?nds that

both the Rayleigh instability and the Tayler instability are stabi-lized by the density strati?cation.Without rotation,however,the density in?uence of the Tayler instability would remain negligi-bly small.

5.Electromotive force for AMRI

Stratorotational instability under the in?uence of a toroidal ?eld is tempting to apply the concept of the mean-?eld electrodynam-ics in turbulent ?elds.The nonaxisymmetric components of ?ow and ?eld can be used as ?uctuations while the axisymmetric components are considered as the mean quantities.Simply the averaging procedure is the integration over the azimuth φ.

It is standard to express the turbulence-induced electromo-tive force as

E = u ′×B ′ =α? B ?ηT curl B

(21)

with the alpha-tensor αand the (scalar)eddy di ?usivity ηT .In cylindric geometry the mean magnetic ?eld B has only a φ-component and the mean current curl B only has a z -component.Hence,E φ=αφφ B φ and E z =?ηT curl z B =0.The latter relation only holds for AMRI where the mean mag-netic ?eld is current-free.

In the present paper we only consider the alpha-e ?ect in the frame of a linear theory where all functions are free to one and the same (complex)arbitrary parameter.This is only possible if the second-order quantities such as E φare normalized with a second-order quantity.In the considered case it makes sense to form the correlation function

f α

u ′×B ′ φ

u ′2 B ′2 )

,(22)

where all the correlations along the radius R are normalized with one and the same parameter.Hence,this function does no longer contain the arbitrary factor of the eigenfunctions and by de?ni-tion it is smaller than unity.The main questions are the sign of this quantity and the in?uence of the density strati?cation.Note that the B φhas been given as positive in the model setup where rotation axis and density gradient are antiparallel.The latter is quite characteristic for the situation at the poles of rotating stars or disks so that the old question whether an alpha-e ?ect at the poles does exist or not is here concerned.

One can easily show that second-order correlations of quan-tities running with exp(i(kz +m φ))after integration over φdo not depend on the coordinate z .The term kz only ?xes the phase where the integration starts.

The correlation function f αcan be estimated for fast-rotating magnetoconvection in the high conductivity limit α?

G.R¨u diger &D.A.Shalybkov:Stratorotational instability in MHD Taylor-Couette ?ow

Fig.12.The correlation function

(22)for AMRI with conduct-ing cylinders and with ?η=0.5,μ?=0.35(pseudo-Kepler),

μB =0.5,Pm =1.Density strati?cation is zero (top)and ?nite (bottom,Fr =0.5).The Reynolds numbers Re =141(top)and Re =341(bottom)were taken from Fig.4.

correlations for N =0are much more antisymmetric than the values for Fr =0.5.Averaged over the radius the f αvanishes for N =0but not for the presence of a density strati?cation.

This cannot be a boundary e ?ect.A boundary e ?ect is im-plausible as the preferred radial direction is perpendicular to the rotation axis for cylinders which mimics the equator in rotating spheres and there the α-e ?ect must vanish.Indeed,the antisym-metry is reduced if a density strati?cation is allowed (Fig.12,bottom).Then the density gradient is now the preferred direc-tion in the system parallel to the rotation allowing the formation of large-scale helicity (as in rotating spheres at the poles).We are thus tempted to assume that the presented calculations for the presence of a density gradient indeed have demonstrated for the existence of an α-e ?ect for the magnetic SRI.

Figure 13presents similar results but for the small magnetic Prandtl number Pm =0.01.The correlation functions do pro-vide a dominance of the positive contributions but for higher and higher magnetic ?elds the di ?erence becomes smaller.It seems that the numerical results do here re?ect a magnetic quenching of the α=-e ?ect.

Fig.13.The correlation function (22)for ?η=0.5,μ?=0.35

(pseudo-Kepler),μB =0.5,Pm =0.01.Fr =0.5).The Reynolds numbers are Re =531for Ha =100and Re =999for Ha =200and Re =2137for Ha =400.Note that the alpha -e ?ect is quenched for stronger magnetic ?elds.

6.Conclusions

The stability of the dissipative Taylor-Couette ?ow under the joint in?uence of a stable vertical density strati?cation and an azimuthal magnetic ?eld is considered.The problem is of inter-est for future laboratory experiments but also within the frame of accretion disk physics.The Kepler rotation generates strong toroidal magnetic ?elds dominating the poloidal components.The standard MRI which works with only axial ?elds may be of minor relevance for the stability of the Kepler rotation law compared with the azimuthal MRI in connection with the den-sity strati?cation and Tayler instability.Mainly nonaxisymmet-ric ‘kink’modes (m =1)are considered but there are also exam-ples where the axisymmetric modes are the most unstable ones.

We started with a discussion of the SRI without magnetic ?elds.For a ?at rotation law a strati?cation value Fr exists for which the critical Reynolds number of the rotation has a min-imum (see Fig.2).The SRI is basically stabilized by both too weak or too strong strati?cation.The instability only appears if the characteristic buoyancy time approaches the rotation period.Also if the rotation is too ?at the instability disappears.

The limiting ratio μ?strongly depends on the gap width (see Fig.3).For small gaps rotation laws with μ?>?ηeven prove to be unstable while for wide gaps the condition μ?

for exciting SRI.Our previous ?nding that μ?<～?η

limits the SRI is reproduced for medium-sized gaps.In all cases,however,the quasi-Kepler rotation proves to be unstable.New experiments with di ?erent gap sizes and larger Reynolds numbers could ver-ify these results.

Figures 4and 5yield the basic results for SRI subject to toroidal ?elds.The magnetic ?eld is assumed as current-free in the ?uid between the cylinders,i.e.B φ∝1/R excluding Tayler instability.Without density strati?cation no Rayleigh instabil-ity exists for quasi-Kepler rotation but nonaxisymmetric modes with m =1are unstable as a result of the interaction of di ?eren-tial rotation and magnetic ?eld (‘AMRI’).

The magnetic in?uence strongly depends on the magnetic Prandtl number.The instability needs higher Reynolds number

for Pm <～1and it needs lower Reynolds number for Pm >～1.

10G.R¨u diger&D.A.Shalybkov:Stratorotational instability in MHD Taylor-Couette?ow After our experiences with MHD instabilities this is not a sur-

prise.It is a surprise,however,that always the magnetic in?u-

ence is only weak.Up to Hartmann numbers of Ha?100only

a magnetic-induced factor of two plays a role2.Hence,our con-

clusion is that the SRI survives for rather high magnetic?elds.

For large Pm it is even supported by the toroidal magnetic?eld.

The combination of rotation,density strati?cation and mag-

netic?eld leads to complex results.However,a basic observation

is that the critical Reynolds numbers,if existing,above which the

?ow becomes unstable without a magnetic?eld,are increased by

the stable density strati?cation.In contrast,the critical Hartmann

numbers,if existing by the Tayler instability,above which the

?eld becomes unstable without a rotation,do not depend on the

strati?cation.Di?erent transition are possible between the two

limits.

Generally speaking,the density strati?cation‘balloons’the

stability region and in this sense it stabilizes the?ow.For slow

rotation the maximal stable magnetic?eld exceeds the critical

magnetic?eld without rotation while for faster rotation the max-

imal stable magnetic?eld is smaller than this critical value.Even

rather slow values of the Reynolds numbers lead to a stabiliza-

tion of those?elds which are unstable for Re=0.The e?ect is

strong for Pm=1but it becomes smaller for decreasing mag-

netic Prandtl numbers.

For steep radial pro?les of the magnetic?eld(i.e.strong ax-

ial currents)and magnetic Prandtl numbers Pm>～1one also?nds

the magnetic?eld destabilizing the Rayleigh instability,i.e.the

critical Reynolds numbers with magnetic?eld are lower than the

Reynolds numbers without magnetic?eld.This magnetic desta-

bilization only exists for not too small magnetic Prandtl num-

bers.It exists for both uniform and density-strati?ed?uids(Fig.

11).

Finally,theφ-component of the electromotive force repre-

senting theα-e?ect of the mean-?eld dynamo theory has been

computed.The computations require an extreme degree of ac-

curacy.The results demonstrate the importance of the density

strati?cation for the existence of theα-e?ect.Without density

strati?cation the correlations are vanishing in the radial average.

With an axial density strati?cation the calculations model the

polar region of a rotating sphere or disk.This interpretation ac-

cepted we found theαφφat the northern pole as positive(Fig.12).

Again,the basic ingredient of thisα-e?ect model is the density

strati?cation.

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