Destabilization of super-rotating Taylor-Couette flows by current-free helical magnetic fields
aa r X i v : . [ phy s i c s . f l u - dyn ] A ug Under consideration for publication in J. Plasma Phys. Destabilization of super-rotating Taylor-Couetteflows by current-free helical magnetic fields
G. R ¨udiger † , M. Schultz and R. Hollerbach University of Potsdam, Institute of Physics and Astronomy, Karl-Liebknecht-Str. 24-25, 14476 Potsdam,Germany Leibniz-Institut f¨ur Astrophysik Potsdam, An der Sternwarte 16, D-14482 Potsdam, Germany Department of Applied Mathematics, University of Leeds, Leeds, LS2 9JT, United Kingdom(Received xx; revised xx; accepted xx)
It is known that the combination of azimuthal magnetic fields and super-rotation in Taylor-Couette flows of conducting fluids can be unstable against non-axisymmetric perturbations ifthe magnetic Prandtl number of the fluid is Pm = 1 . We show here that the addition of a weakaxial magnetic field component allows axisymmetric perturbation patterns for Pm of order unity.The unstable domain in the stability map becomes increasingly narrow if the axial magnetic fieldbecomes too strong in comparison with the applied azimuthal field. The axisymmetric modes,however, only occur for azimuthal magnetic Mach numbers of order unity, while for highervalues only non-axisymmetric modes exist. The characteristic time scale of the axial migrationof the axisymmetric mode is long compared with the rotation time, but short compared withthe magnetic diffusion time. The modes travel in the positive or negative z -direction along therotation axis in the same sense as the sign of B φ B z . We also demonstrate that the azimuthalcomponents of flow and field perturbations travel in phase if | B φ | ≫ | B z | , independent ofthe form of the rotation law. It is finally shown that for ideal fluids the considered helicalmagnetorotational instability only exists for rotation laws with negative shear.
1. Introduction
Cylindrical Taylor-Couette containers filled with a conducting fluid and subject to externallyapplied large-scale magnetic fields can be used as a ‘virtual’ laboratory to study magneticinstabilities. The simplest geometry of the external magnetic field is a homogeneous axial field,reproducing the standard magnetorotational instability (MRI) if the outer cylinder rotates at aslower frequency than the inner one. Both axisymmetric and non-axisymmetric perturbationpatterns are unstable, with axisymmetric modes excited first, that is, at slower rotation rates.Due to diffusion, this instability requires a minimum magnetic field for excitation, with a criticalLundquist number S ≃ (see below for the exact definitions of parameters such as S ).Once the axisymmetric mode (“channel flow”) is excited, any further increase of the Reynoldsnumber Re does not restabilize the flow. This, however, is not true for the non-axisymmetricmodes, which can always be restabilized by faster rotation. The minimum rotation rates of thelines of neutral stability scale with PmRe ≃ const for small Pm , and with √ Pm Re ≃ const forlarge Pm , where the magnetic Prandtl number Pm = νη (1.1)is the ratio of kinematic viscosity ν and magnetic diffusivity η . Hence, the lowest critical rotationrates Ω for Pm ≪ run as Ω ∝ η and as Ω ∝ √ νη for Pm ≫ . They are obviously minimal † Email address for correspondence: [email protected] for Pm of order unity. This well-known standard type of MRI does not exist for rotation profileswith positive shear (super-rotation).Another type of MRI appears if the applied magnetic field is azimuthal and curl-free in thegap between the cylinders. This configuration exhibits only non-axisymmetric unstable modes,but independent of the sign of the shear of the rotation. There is again a minimum Reynoldsnumber for excitation, but unlike the standard MRI, the azimuthal magnetorotational instability(AMRI) is suppressed again if the rotation is too rapid. The Hartmann number Ha exhibits thesame behaviour, with a minimum value required, but the AMRI also suppressed again if theapplied field is too strong. The lines of neutral stability of these modes thus form typical obliquecones in the ( Ha / Re ) plane, where the slopes dRe / dHa of the two branches are positive, andthe Hartmann number Ha min at the point where dRe / dHa = ∞ defining the overall weakestmagnetic field amplitude for instability.A very special situation holds for AMRI flows with super-rotation, when the outer cylinderrotates with a higher frequency than the inner one. For small magnetic Prandtl numbers the linesof neutral stability coincide in the ( Ha / Re ) plane, whereas for large Pm they coincide in the( Ha / Rm ) plane, where Rm = PmRe is the magnetic Reynolds number. One might not expectproblems in the limit Pm → but they do exist. Approaching Pm = 1 , the critical values forboth Ha and Re go to infinity, for both Pm < and Pm > . The magnetized flow for Pm = 1 isstable, but is unstable for Pm = 1 ; that is, this is a so-called double diffusive instability. We havenumerically demonstrated this behaviour of the critical values for a container with an almoststationary inner cylinder (R¨udiger et al. b ). The existence of solutions for Pm = 1 is ofparticular relevance if turbulent fluids are considered, as the effective magnetic Prandtl numberin turbulent media basically approaches unity.The present paper focusses on the problem of how the characteristics of this double-diffusiveinstability for azimuthal field B φ and super-rotation are modified if the azimuthal field is com-plemented by a small axial component B z . The resulting field then possesses a helical structure,as we considered before but with sub-rotation (Hollerbach & R¨udiger 2005; Stefani et al. et al. a ). One finds the similar situation for the nonaxisymmetric AMRI: it existsfor ideal fluids only for sub-rotation rather than for super-rotation (see Appendix).The basic parameter in the present study is the ratio of the azimuthal to the axial fieldcomponent, β = B φ ( R in ) B z , (1.2)where R in is the radius of the inner cylinder. We are interested in the limit where β is large, andcan be positive or negative. For comparison, in the solar convection zone the equivalent β is oforder , and is negative (positive) in the northern (southern) hemispheres.It has been suggested that the migration toward the equator of the latitude of maximal solaractivity over 11 years (the solar cycle) might be understood as a drifting axisymmetric mode of amagnetic instability driven by the super-rotation which exists beneath the equator at the bottom ofthe convection zone (Mamatsashvili et al. et al. agnetic Taylor-Couette flows · − .The paper is structured as follows. The basic differential equations and boundary conditionsare formulated in the following Section. In Section III the lines of marginal stability of thelinearized system for various container sizes and for fixed Pm = 1 and | β | = 25 are discussed.One finds axisymmetric and non-axisymmetric modes to be unstable, where the latter requiresstronger fields and faster rotation for excitation. For the axisymmetric mode the inclination angle β and the magnetic Prandtl number Pm are varied in Sections IV and V, where the axial driftrelations in dependence on the sign of β are also demonstrated. In the final sections the phaserelations of the azimuthal components of flow and field for super-rotation and sub-rotation willbe discussed. The results are reviewed in the last Section, where their possible connection to thecyclic activity of the Sun will also be discussed.
2. The Equations
The equations of the problem are ∂ U ∂t + ( U · ∇ ) U = − ρ ∇ P + ν∆ U + 1 µ ρ curl B × B ,∂ B ∂t = curl ( U × B ) + η∆ B (2.1)with div U = div B = 0 for an incompressible fluid. U is the velocity, B the magnetic field, P the pressure and ρ the density. The basic state in the cylindrical system with coordinates ( R, φ, z ) is U R = U z = B R = 0 for the poloidal components and Ω = a Ω + b Ω /R for the rotation law,with the constants a Ω = µ − r − r Ω in , b Ω = 1 − µ − r Ω in R , (2.2)where r in = R in /R out is the ratio of the two cylinders’ radii, and Ω in and Ω out are their angularvelocities. If we define the ratio µ = Ω out /Ω in , then super-rotation is represented by µ > .The current-free azimuthal field is given by B φ = R in BR . (2.3)Together with a uniform axial component B z , the externally imposed field is therefore B =(0 , R in B/R, B z ) .In addition to the magnetic Prandtl number Pm , which is a material property of the fluid, theother dimensionless parameters of the system are the Hartmann number Ha and the Reynoldsnumber Re , Ha = B z R √ µ ρνη , Re = Ω out R ν , (2.4)which measure the strength of the imposed axial field and the outer cylinder’s rotation rate,respectively. Alternative measures are the Lundquist number S = √ Pm Ha and the magneticReynolds number
Rm = Pm Re . Different choices of Ha versus S , and Re versus Rm , areappropriate in different limiting parameter regimes. The parameter R = p ( R out − R in ) R in isa suitably scaled measure of length.Recalling the ratio β = B/B z (1.2), it is useful to also define an azimuthal Hartmann number Ha φ = β Ha , which measures the strength of the azimuthal field B φ rather than the axial field B z . These quantities may be combined to yield the magnetic Mach number of the azimuthal field, Mm = √ Pm Re β Ha , (2.5)measuring whether the rotation energy dominates the magnetic energy or not. The magneticMach number of cosmical objects almost always exceeds unity. Adopting solar values,( U φ ≃ B φ ≃ kG, one finds Mm ≃ p r in (1 − r in ) , which already exceedsunity for the very small gap width of km. For the solar tachocline with its thickness of 50,000km, the magnetic Mach number is Mm ≃ , or Mm ≃ for the stronger azimuthal field B φ ≃ kG.The equations are linearized, and instability modes of the form f = f ( R )exp (cid:0) i( kz + mφ + ωt ) (cid:1) are sought. The result is a linear, one-dimensional eigenvalue problem, with only the radialstructures f ( R ) still to be solved for, and with ω being the eigenvalue. This eigenvalue system issolved by finite-differencing in R , as in Shalybkov et al. (2002). For a given Hartmann number,solutions are optimised with respect to the Reynolds number by varying the axial wave number k .The azimuthal wave number m is either for axisymmetric modes, or ± for non-axisymmetricmodes. Higher non-axisymmetric modes can also be excited, but typically at higher Hartmannand/or Reynolds numbers than m = ± , so we focus on m = 0 and ± here. There are alsovarious symmetries that apply to positive versus negative m . For purely azimuthal fields ( B z =0) , m → − m are directly equivalent, whereas for general helical fields m → − m are equivalentif additionally one takes either of k → − k or β → − β . One can therefore restrict attention toeither positive m or positive β , for example, as long as the other one is allowed to take on bothsigns.The associated boundary conditions are no-slip for the velocity perturbations, u = 0 . Forthe boundary conditions on b one can take the cylinders to be either perfectly conducting orinsulating. Conducting boundary conditions are d b φ / d R + b φ /R = b R = 0 at both R in and R out . Insulating boundary conditions are slightly more complicated, and different at R in and R out , i.e. for R = R in b R + i b z I m ( kR ) (cid:16) mkR I m ( kR ) + I m +1 ( kR ) (cid:17) = 0 , (2.6)and for R = R out b R + i b z K m ( kR ) (cid:16) mkR K m ( kR ) − K m +1 ( kR ) (cid:17) = 0 , (2.7)where I m and K m are the modified Bessel functions. (Note that these satisfy I − m = I m and K − m = K m .) Additionally, the toroidal field at both boundaries must satisfy kR b φ = m b z . Amore detailed derivation of the boundary conditions, including the option of finitely conductingboundaries, is given in R¨udiger et al. (2018 c ). Note also that in all cases the total number ofboundary conditions (ten) correctly matches the number of equations in the eigenvalue problem.The linear code works with length scales normalized with R , and frequencies normalizedwith Ω out . Positive values of the drift frequencies denote negative axial phase velocities, so thatthe instability pattern migrates in the negative z -direction, anti-parallel to the rotation axis. Fornegative drift frequencies it is vice versa. The drift frequency can also be normalized with themagnetic diffusion frequency ω diff = Ω out / Rm . (2.8)Note finally that in any linear eigenvalue problem the overall solution amplitude is undetermined,so that only ratios of variables have clearly defined physical meanings.We mainly deal with a flow with almost stationary inner cylinder, in narrow-gap configura- agnetic Taylor-Couette flows m = ± (R¨udiger et al. a ). It is a double-diffusive instability which requires Pm = 1 for its existence. It also exists in the inductionlessapproximation, Pm → , which automatically means that the relevant parameters for smallmagnetic Prandtl number are Re and Ha . This is relevant for possible experiments with liquidmetals with very small Pm as for Pm → the Reynolds number does not grow to infinity as isthe case for instabilities where the relevant parameter is Rm rather than Re . For Pm ≫ therotational parameter scales with Rm .If an axial component is added to the imposed field, the first mode to go unstable becomes theaxisymmetric m = 0 mode. This is even true if the axial field is much smaller than the azimuthalfield, i.e. for β ≫ . For much smaller wave numbers the existence of another mode (‘type 2’)is reported which does not exist for Pm = 1 or for
Pm = 0 (Mamatsashvili et al.
Pm = 1 and
Pm = 0 will scale with Re for small Pm , which favourspossible experiments with liquid metals.
3. Axisymmetric and non-axisymmetric solutions for Pm = In this Section we consider the instability which exists for magnetic Prandtl number unity,thus excluding all types of double-diffusive instabilities. We shall demonstrate that even in thiscase axisymmetric as well as non-axisymmetric perturbation modes can be unstable for rotationlaws with positive shear. We are mainly interested in magnetic background fields where theazimuthal component dominates; the general choice here is β = 25 . The gap width between thetwo cylinders is a free parameter, and we are interested in narrow gaps. The parameters whichallow instability are the Reynolds number and the Hartmann number. They define an unstabledomain which is limited by a lower and an upper Reynolds number, and similarly a lower and anupper Hartmann number. That is, the system is stable both for too slow and too fast rotation, andsimilarly for too weak and too strong fields, as seen in Fig. 1. The absolute minimal Hartmannnumber for marginal instability is called Ha min .For cylinders with r in = 0 . , . and . , Fig. 1 presents the lines of marginal stability ofthe axisymmetric perturbation modes in the (Ha / Re) plane. The influence of the gap widths onthe neutral stability lines is weak. One finds a minimal Hartmann number of order 100, with aweak dependence on the gap width. The instability only exists for magnetic Mach numbers (2.5)beween 0.06 and 0.4. These values, defined with the azimuthal field amplitude, are strikinglysmall. Related to the Alfv´en frequency of the magnetic field the rotation rate must be rather low to destabilize the flow. Figure 1 also demonstrates that the instablity only occurs in a rather smallpart of the ( Ha / Re ) plane. The opening of the instability cone depends on the precise value of β .For β → and β → ∞ the axisymmetric instability will disappear, so that one must expect thatthe opening of the cone becomes smaller and smaller for deceasing and increasing β (see below).For larger and larger Hartmann numbers the lines of marginal instability in Fig. 1 remain linearlines of constant first derivative as we have probed for the blue line up to Ha = 10 . We did notfind any indication of (island) instability domains limited in the Hartmann/Reynolds numbers.The axial wave numbers (normalized with R ) and the drift frequencies (normalized with therotation frequency Ω out ) are given by the two panels of Fig. 2. By definition the cell size δz along the rotation axis normalized with the gap width D is δz/D ≃ π/kR . Hence, a magneticpattern which is nearly spherical in the gap between the cylinders should have a normalized wavenumber kR = 2 π . The wave number values in the left panel of Fig. 2 are much smaller, so thatthe cells are instead rather long in the vertical direction z . For large Hartmann numbers the wave F IGURE
1. Stability maps for the axisymmetric mode of the super-rotating flow. Green line: r in = 0 . ;blue line: r in = 0 . ; red line: r in = 0 . . m = 0 , µ = 128 , Pm = 1 , β = 25 . Insulating cylinders.F IGURE
2. Similar to Fig. 1 but for normalized wave number (left panel) and drift frequency (3.1) (rightpanel). numbers decrease. The cells, therefore, become increasingly elongated for stronger axial fields,in agreement with the magnetic Proudman theorem.The characteristic values of the drift ω dr = ω R Ω out , (3.1)where ω R is the real part of ω , are negative and of order 0.05, which is faster than the diffusiondrift ω diff < ∼ . of the magnetic pattern by one order of magnitude. More details of the driftphenomenon are also presented below in Section 5.Non-axisymmetric modes are also unstable. The boundary conditions (2.6) and (2.7) forinsulating cylinders also allow calculations for non-zero azimuthal wave numbers ± m . Oneexpects higher eigenvalues for the excitation of non-axisymmetric modes if axisymmetric modesexist. The two solutions for m = 1 and m = − form spirals of opposite chirality. The questionis whether the two modes due to a background field with a fixed value of β have differenteigenvalues or not.Fig. 3(left panel) gives the stability map for the non-axisymmetric modes with m = ± , for agnetic Taylor-Couette flows F IGURE
3. Left: Stability map of the non-axisymmetric modes for r in = 0 . (blue) and r in = 0 . (green)of a strongly super-rotating flow. Solid line: m = 1 , β = 25 . Dotted line: m = − , β = 25 . Dashed line: m = 1 , β = − . Middle panel: normalized wave numbers, right panel: drift rates. Note that for β = 25 the sign of ω dr differs for m = 1 (solid) and m = − (dotted). µ = 128 , Pm = 1 . Insulating cylinders. two choices of r in . The values of Ha min exceed those of the axisymmetric mode, and now Ha min also depends strongly on r in . Ha min increases from 350 for the wider gap (blue line) to about1000 for the narrower gap (green line). A general result is that the flow in the wide gap is moreunstable than the flow in the narrow gap. Recall also that the Hartmann number (2.4) is definedwith the weak axial field, so that the Hartmann number of the toroidal field is higher by the(large) factor β .The critical values for Ha and Re differ slightly for m = ± , as do the wave numbers(middle panel). The minimum Hartmann number for excitation of the mode with m = 1 issomewhat smaller than that for m = − . The drift rates, however, are significantly different,so that the phase velocities of the axial drifts also differ. The mode with m = 1 travelsupwards (in the direction of positive z ) while the mode with m = − travels downwards(in the direction of negative z ). The wave numbers k of the spirals are larger than the wavenumbers of the axisymmetric mode, but still they are rather small so that the cells are oblong.The two non-axisymmetric modes form two different spirals. The general phase relationship is d z/ d φ = − m/k , so that the mode with positive m forms a left-hand spiral while the mode withnegative m forms a right-hand spiral. The two spirals have slightly different excitation conditionsbut they travel in opposite directions.The question is what happens with the eigensolutions for β → − β . Then obviously thechirality of the background field is changed. The perturbations should react with m → − m .Indeed, Fig. 3 verifies that the transformation β → − β simply replaces the transformation m → − m . One only finds the two possible spirals which we already know for m = ± . Thecurve for m = 1 and β = − in the (Ha / Re) plane agrees with the curve for m = − and β = 25 . The same is true for the wave numbers, but is not true for the drift speeds, which changesign. Here only the azimuthal wave number m determines the ω dr value. Its values for m = 1 and β = ± are identical (right panel of Fig. 3).
4. The axisymmetric modes in their dependence on the inclination angle beta
For β → the system would turn into that of the standard magnetorotational instability which,however, does not exist for super-rotation. On the other hand, for β → ∞ the system approachesthat of the super-AMRI which also does not exist for axisymmetry. Hence, there should be anoptimal β at which the instability is most easily excited, depending only on Pm for any fixed r in . Figure 4 shows the stability lines for Pm = 0 . . The horizontal axes in the two plots arethe axial Hartmann number Ha (left panel) and by the azimuthal Hartmann number Ha φ (rightpanel), where we recall that the azimuthal Hartmann numbers are defined by Ha φ = β Ha formedwith the azimuthal field B φ rather than B z . One finds the minimum values of Re growing forboth large β (green lines) and small β (black lines). The red line for β = 62 represents the F IGURE
4. Stability map of axisymmetric modes for various inclination angles β . For the horizontal axisthe axial Hartmann number Ha is used in the left panel, and the azimuthal Hartmann number Ha φ = β Ha in the right panel. Black lines: β < (down to 24), red line: β = 62 , green lines: β > (up to 200). Thelines are marked with their values of β . r in = 0 . , µ = 128 , Pm = 0 . , m = 0 . Insulating cylinders.F IGURE
5. Similar to Fig. 4 but for normalized wave numbers (left) and drift rates (right). instability with the absolutely lowest Reynolds number; all other lines are located above thisline. With this low Reynolds number the azimuthal magnetic Mach number (2.5) takes on thelow value Mm ≃ . . It remains always constant for higher β (green lines). For Pm of orderunity magnetic Mach numbers exceeding 0.1 are necessary for instability, but they must not behigher than 0.3 (for β ≃ ). The axisymmetric modes are thus not unstable for magnetic Machnumbers exceeding unity. The minimum Hartmann numbers do not depend on β for large β (seeleft panel). For low β the minimum azimuthal Hartmann numbers do not depend on β (see rightpanel, if β is not too small).In view of its right panel Fig. 4 also demonstrates that the opening of the instability cone islargest for the optimal β ≃ . The cone becomes increasingly narrow for smaller β , i.e. forgreater B z (so tending toward the standard MRI limit). A similar behaviour can be observed formuch greater β . One finds the most extensive instability domain for an optimal value β ≃ . Forsmaller as well as larger values the axisymmetric instability is suppressed as the constellationswith β = 0 and β = ∞ are stable.We note that the wave numbers and also the drift rates (normalized with the outer rotation rate)are small (Fig. 5). The drift rates only depend slightly on β and the Hartmann number. For small β , i.e. β < , the wave numbers become increasingly small. In these cases the phase speed(normalized with R Ω out ) is of order 0.1, while for the larger values of β it is smaller, of order0.01. agnetic Taylor-Couette flows T ABLE
1. Eigenvalues of the axisymmetric solutions for super-rotating flows with µ = 128 , r in = 0 . and β = 62 . ω diff after Eq. (2.8). Minimal Hartmann numbers, insulating boundary conditions. Pm Re Rm Ha min
S Mm kR ω dr ω R /ω diff IGURE
6. Left panel: Stability map of the axisymmetric perturbation modes of super-rotating flows with µ = 128 . Middle panel: wave numbers. Right panel: the drift frequencies. The curves are marked with theirmagnetic Prandtl numbers. β = 62 , m = 0 , r in = 0 . . Insulating cylinders.
5. The axisymmetric modes in their dependence on the magnetic Prandtl number
The axisymmetric modes are the solutions with the lowest critical parameter values. They shallnow be considered for magnetic Prandtl number larger or smaller than unity. For Pm > . ,Table 1 gives the critical values of marginal instability of a flow with µ = 128 penetrated bya helical magnetic field with the optimal value β = 62 . These numbers may serve to modelthe interaction of a strongly super-rotating flow and a magnetic field with a moderate axialcomponent. The main result is that we find the flow is unstable also for Pm = 1 . For the modelsof Table 1 the critical magnetic Reynolds numbers hardly vary. The same is true for the criticalLundquist numbers. The lines of neutral stability for Pm > ∼ appear to scale with Rm and S rather than with Re and Ha .The negative sign of the drift rates (3.1) for positive β is another important result. As weshall demonstrate below, this sign is opposite to that for flows with sub-rotation. The instabilitypattern thus migrates “pole-wards” along the rotation axis for super-rotation and “equator-wards”for sub-rotation for all positive β . As noted above the axial migration is slower than the rotationtime scale but faster than the magnetic diffusion time scale.The stability maps for Pm . are shown in Fig. 6. They all have the typical conicalstructure of the stability line in Fig. 1. For supercritical Hartmann numbers there are always twoReynolds numbers between which the flow is unstable. The slopes dRe / dHa of both branchesare again positive and very similar. There is always a minimum Hartmann number Ha min at dRe / dHa = ∞ below which the flow is stable. We note that this ‘oblique-cone’ geometry ofthe instability domain previously only appeared for the non-axisymmetric modes of MRI andAMRI. The “helical” magnetorotational instability (HMRI) with super-rotation (“super-HMRI”)0 F IGURE
7. Stability map (left), wave numbers (middle) and drift rates (right) for background fields withpositive or negative helicity. µ = r in = 0 . , Pm = 0 . and β = 2 (red), β = − (green). m = 0 , perfectlyconducting cylinders. is the only magnetic instability where rapid rotation stabilizes the axisymmetric mode. Rotationexcites the instability, but it can also be too fast for its existence. This is quite opposite to theexcitation conditions of the axisymmetric (or channel) modes of standard MRI (Gellert et al. et al. et al. a ) which do notpossess upper limits of the Reynolds number. The HMRI with super-rotation (“super-HMRI”) isthus much more stable than HMRI with sub-rotation.The two branches of the curves in the left panel of Fig. 6 limit the magnetic Mach number(2.5) of the azimuthal field to the small value of O(0.1) for the unstable modes. Flows withhigher magnetic Mach numbers, i.e. with faster rotation or weaker field, are stable. Instabilityoccurs for azimuthal Mach numbers only between 0.05 and 0.1.On the other hand, a steep dependence of Ha min – and also the corresponding Reynoldsnumber – on the magnetic Prandtl number Pm appears. The critical values grow by a factorof two for the small variation Pm = 0 . to Pm = 0 . . One also finds very small wavenumbers k for the unstable modes with Pm < . , so that the vortices are extremely long in theaxial direction (Fig. 6, middle). As an immediate consequence, the radial components of flow andfield become smaller and smaller. Such small wave numbers also make the problem increasinglydifficult numerically. The right panel of Fig. 6 again shows the eigensolutions possessing verysmall values of the drift rates if normalized with the outer rotation rate. We note, however, thatwith ω R /ω diff ≃ , the mode migrates faster in the axial direction than the diffusion time scale.The phase velocity along the z -coordinate of the travelling axisymmetric mode is d z d t = − ω dr k R Ω max , (5.1)where ω dr and k are the normalized frequencies and wave numbers. Hence, negative ω dr valuesas given in Table 1 for positive β describe a wave pattern drifting in the “pole-ward” direction.The axial phase velocity of the models of Table 1 is . . . . . in units of R Ω out . The driftfrequency is thus much lower than the rotation rate. On the other hand, as the last row in Table 1shows, it is faster than diffusion by one order of magnitude.Figure 7 gives the critical values of the axisymmetric modes for two models with differentsigns of β for a rotation law with negative shear corresponding to the first model in Table 2. Thetop panel displays the dependence of the critical Reynolds number as a function of the Hartmannnumber. Here the Reynolds number takes its minimum value at Ha ≃ . The wave number hasa maximum there (middle panel). The curves are identical for β = ± . The drift rates, however,are different, always satisfying βω dr > , that is, ω dr and β always have the same sign. The flowand field patterns migrate downwards (to − z ) for positive β and upward for negative β (R¨udiger et al. b ). The correctness of this statement has been proven by the experiment PROMISE(Seilmayer et al. agnetic Taylor-Couette flows β → − β and ω dr → − ω dr . If the wave-likesolution with a certain β travels (say) parallel to the rotation axis then another solution existsfor − β travelling in the opposite direction. Models in Table 1 with negative β , therefore aretravelling in negative z -direction, as also the models in Table 2 do with positive β . In this Tablefor a demonstration also the numbers are given for β = ± where indedd only the sign of ω dr ischanged while the Hartmann/Reynolds numbers remain unaltered.
6. Phase relations
One may ask how the flow and field patterns that migrate along the z -axis relate to one another.Is there a shift between the maxima of flow and field and, if yes, what is its dependence onthe background field or the rotation law? For several well-defined models we shall present thephase relations between the azimuthal flow perturbations and the azimuthal field perturbations for m = 0 . For super-rotating Taylor-Couette flows four cases with . Pm are considered,whose critical values are given in Table 1. For comparison, we made similar calculations for a setof sub-rotating flows with various values of β and a fixed magnetic Prandtl number Pm = 0 . (Table 2).From b φ = ( b R + i b I )e i ψ , u φ = ( u R + i u I )e i ψ , (6.1)where the superscripts R and I denote the real and imaginary parts of the quantities, one obtainsfor the vertical waves of the azimuthal perturbations b φ and u φ b φ = b R cos ψ − b I sin ψ = b sin( ψ − δ b ) ,u φ = u R cos ψ − u I sin ψ = u sin( ψ − δ u ) (6.2)with ψ as the actual phase and the δ ’s as their phase shifts. Then δ b = arctan b R b I , δ u = arctan u R u I . (6.3)We are only interested in the phase difference δ = δ b − δ u . In order to exclude the boundarieswe consider this quantity only in a certain region between outer and inner radius. The two wavesare in phase if δ ≃ there. If the phase differences are given in degrees, then for δ ≃ ◦ thewaves are out of phase.Figures 8 and 9 show the radial profiles of b φ and u φ for the super-rotating flow with µ = 128 .Because the solutions contain a free arbitrary factor, only ratios of the components have anyphysical meaning. The magnetic Prandtl numbers vary between Pm = 1 and
Pm = 3 forfixed β = 62 (Fig. 8), and β is varied for fixed Pm = 0 . (Fig. 9). For all examples one finds δ ≃ ± ◦ , hence the waves of b φ and u φ are travelling nearly in phase for all Pm and large β .For smaller β the phase difference grows (left panel of Fig. 10).Also rotation laws are considered where the outer cylinder rotates slower than the inner one.For a fixed Pm = 0 . some eigenvalues of models with growing magnetic inclination angle β are summarized in Table 2. The radial profiles of b φ and u φ were calculated for all these flows.The profiles are used for the calculation of the phase differences between the maximum of b φ and the maximum of u φ . The right panel of Fig. 10 shows the phase shift δ which also prove tobe small for large β . These waves, therefore, travel in phase along the rotation axis. The result δ ≃ ◦ for small β ( | β | = 2 ) demonstrates that for sub-rotating Taylor-Couette flows, i.e. withnegative shear, the waves travel in phase, but only for β ≫ .2 F IGURE
8. The radial profiles of the azimuthal components b φ (top) and u φ (bottom) for Taylor-Couetteflows with µ = 128 (super-rotation) for various Pm . Solid lines: real parts, dashed lines: imaginary parts.From left to right: Pm = 1 , Pm = 2 , Pm = 3 . β = 62 . The eigenvalues of the models are given in Table1. m = 0 , r in = 0 . . Insulating cylinders.F IGURE
9. Similar to Fig. 8 but for
Pm = 0 . fixed. From left to right: β = 62 , β = 32 , β = 25 .
7. Summary
The stability problem for axisymmetric and non-axisymmetric perturbations of a magnetizedTaylor-Couette flow is analysed where the outer cylinder spins much faster than the inner one(“super-rotation”). The flow is penetrated by a current-free magnetic field of a helical structurewith non-vanishing azimuthal and axial components. The ratio β of the toroidal and the axialmagnetic field components plays an important role in determining the stability characteristics ofthe system. We already know that for both extrema β → and β → ∞ the flow is always stableagainst axisymmetric perturbations.For magnetic Prandtl numbers of order unity the flow is most unstable for large but not toolarge β , i.e. β ≃ . Non-axisymmetric modes also exist, but their excitation requires fasterrotation and stronger fields. The domains of instability always possess the characteristic geometryof oblique cones in the ( Ha / Re ) plane: for a given supercritical Hartmann number there are a agnetic Taylor-Couette flows F IGURE
10. Phase shifts δ in degrees after Eq. (6.3) between the azimuthal field and flow components.Left panel: the super-rotating flows of Figs. 8 and 9. The blue line corresponds to the model with highest β and smallest Pm (Fig. 9, top). The red line represents the model with β = 2 (Fig. 9, bottom). Right panel:sub-rotating flows defined in Table 2. The red line denotes β = 2 , and the blue line β = 128 .T ABLE
2. Eigenvalues for axisymmetric solutions of sub-rotating models with µ = r in = 0 . and Pm = 0 . . m = 0 . Minimal Reynolds numbers, perfectly conducting boundaries. β Re min Ha kR ω dr ω R /ω diff ±
532 23.8 1.65 ± . ± . lower and an upper Reynolds number between which the flow is unstable, and similarly for agiven supercritical Reynolds number there are a lower and an upper Hartmann number betweenwhich the flow is unstable. For axisymmetric patterns the rotation and azimuthal magneticfield must form a magnetic Mach number of order unity; systems with higher magnetic Machnumbers are stable against axisymmetric perturbations, but they may be unstable against non-axisymmetric perturbations.The instability pattern always migrates in the axial direction, where the sign of β determinesthe sign of the drift rate. The latter lies between the rotation rate and the diffusion frequency,hence the super-HMRI is basically slower than the standard MRI but faster than any diffusionwave, e.g. drifts and waves in dynamo theory.It has been suggested that the axial drift of the axisymmetric instability pattern mimics theequator-ward drift of the activity belts during the solar 11-year cycle (Mamatsashvili et al. β , hence ω dr ∝ − β d Ω/ d R . For super-rotation, therefore, negative β lead to ω dr < corresponding to a drift anti-parallel to the rotation axis (“equator-ward”).Positive β lead to negative drift frequencies ω dr which implies ˙ z > , i.e. pole-ward migrationin the northern hemisphere. Equator-ward migration, therefore, complies with negative β .For the solar convection zone the quantity R Ω is about 800 m/s, so that from Table 1 therelated phase velocity would become 16 m/s. The axial drift of the super-HMRI therefore exceedsthe drift of the solar butterfly diagram ( ≃ out ofphase toward the equator, i.e. the location of u φ = 0 matches the maxima of b φ . The axial wavesof both sorts of HMRI (with sub-rotation and with super-rotation), however, migrate in phase forlarge β . Only for small β do u φ and b φ migrate out of phase, in both cases.As mentioned in the Introduction, due to the positive latitudinal shear of the solar rotation law, cos θ d Ω/ d θ > , the field geometry parameter β should be negative in the northern hemisphereand positive in the southern hemisphere. By the induction of the differential rotation within thesolar convection zone we indeed expect β < (of order ) in the northern hemisphere.For sub-rotating Taylor-Couette flows equator-ward migration requires positive β . These con-clusions are not very strict, however, as the β values derived for the Sun are due to the action of anaxial gradient of the angular velocity, which does not exist at all in the considered Taylor-Couettesystem.D.A. Shalybkov (St. Petersburg) and F. Stefani (Dresden-Rossendorf) are acknowledged forcritical readings of the manuscript. Appendix A
Using a short-wave approximation we derived a dipersion relation γ + a γ + i b γ + a = 0 (A 1)(with γ = i ω/Ω ) from the linearized equation system (2.1) the solutions of which provide forthe stability/instability characteristics of axisymmetric perturbations (R¨udiger & Schultz 2008).This result may be used to probe the stability of helical magnetic fields under the presence ofdifferential rotation for ideal flows, i.e. ν = η = 0 . The coefficients in (A 1) were a = α (4 − q ) + (4 α + 2 β ) f Ω A2 ,b = − α β f Ω A2 , a = f Ω A4 β − q α β f Ω A2 . (A 2)Here α = k z / | k | is the axial wave number (normalized with the total wave number k ). Theshear is defined by the radial rotation law Ω ∝ R − q . For q = 2 (potential flow) and q = 1 agnetic Taylor-Couette flows F IGURE
11. The normalized growth rates γ/Ω of HMRI for m = 0 . The horizontal coordinate is f Ω A = Ω φ A /Ω . Left panel: various values of β (marked), α = 0 . . Right panel: various wave numbers α (marked) for β = 1 . The rotation profile is for q = 1 , i.e. Ω ∝ R − . (quasi-uniform flow) rotation profiles with negative shear are defined while negative q describesuper-rotation. We shall numerically determine the run of the growth rate as the real part ofthe complex γ with the magnetic system parameter f Ω A = Ω φ A /Ω which is the inverse of theazimuthal magnetic Mach number Mm . Ω φ A = B φ / √ µ ρR and Ω z A = k z B z / √ µ ρ are theAlfv´en frequencies of the azimuthal and the axial magnetic fields. Following Kirillov et al. (2014) the ratio β of the azimuthal field and the axial field may be written as β = Ω φ A /Ω z A (corresponding to but not not identical with (1.2)).Here we shall only work for q = ± . The ratio α must be considered as a free parameterwhich varies as < α < . Figure 11 demonstrates the existence of positive growth rates forthe rotation law Ω ∝ R − (i.e. for uniform linear rotation) up to a certain upper limit of Ω φ A /Ω corresponding to a magnetic Mach number of Mm ≃ . . The flow becomes unstable if its Machnumber exceeds this value – or, with other words, if its rotation is fast enough. The instabilitycondition for ideal flows obviously represents only the lower branch of the complete line ofmarginal stability of real fluids. The numerical value indeed approaches the Mach number for Pm = 1 of the lower branch of the instability cone for quasi-uniform linear rotation. Obviously,the upper branch of the instability cone which stabilizes the flow for higher Reynolds numbers isbasically due to finite diffusivities. We note that within the short-wave approximation the upperbranch with the maximally possible Reynolds number will basically not be provided (R¨udiger et al. q < does not provide solutionswith positive real part of γ . The axisymmetric super-HMRI, therefore, is a diffusion-originatedinstability which does not exist for ν = η = 0 . Both branches of its instability cone are thus dueto diffusion processes.The left panel of Fig. 11 demonstrates that the dimensional maximal growth rate γ ∝ Ω φ A β = Ω z A (A 3)is only determined by the axial magnetic field. On the other hand, the system is unstable forall Ω φ A /Ω < ∼ . , i.e. for all magnetic Mach numbers exceeding 1.4. One also finds a very weakinfluence of β on the critical ratio Ω φ A /Ω for marginal excitation ( γ = 0 ). The lower branches ofthe instability cones of real flows should thus be almost identical for all β (see Fig. 4).The right panel of Fig. 11 demonstrates a rather strong monotonous influence of the normal-ized axial wave number onto the growth rate profiles. As also the horizontal coordinate contains6 k z , the plot reflects mainly the influence of the radial wave number, i.e. the width of the cylindergap. R E F E R E N C E SG
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