Turbulent magnetic Prandtl numbers obtained with MHD Taylor-Couette flow experiments
aa r X i v : . [ a s t r o - ph ] S e p Turbulent magnetic Prandtl numbers obtained with MHD Taylor-Couette flowexperiments
Marcus Gellert & G¨unther R¨udiger
Astrophysikalisches Institut Potsdam, An der Sternwarte 16, D-14482 Potsdam, Germany ∗ (Dated: November 2, 2018)The stability problem of MHD Taylor-Couette flows with toroidal magnetic fields is considered independence on the magnetic Prandtl number. Only the most uniform (but not current-free) fieldwith B in = B out has been considered. For high enough Hartmann numbers the toroidal field isalways unstable. Rigid rotation, however, stabilizes the magnetic (kink-)instability.The axial current which drives the instability is reduced by the electromotive force induced bythe instability itself. Numerical simulations are presented to probe this effect as a possibility tomeasure the turbulent conductivity in a laboratory. It is shown numerically that in a sodiumexperiment (without rotation) an eddy diffusivity 4 times the molecular diffusivity appears resultingin a potential difference of ∼
34 mV/m. If the cylinders are rotating then also the eddy viscosity canbe measured. Nonlinear simulations of the instability lead to a turbulent magnetic Prandtl numberof 2.1 for a molecular magnetic Prandtl number of 0 .
01. The trend goes to higher values for smallerPm.
PACS numbers: 47.20.Ft, 47.65.+a
INTRODUCTION
Strong enough toroidal fields that are not current-free become unstable due to the Tayler instability (TI,[1, 2, 3]). Because the source of the energy is the electriccurrent, these (mainly nonaxisymmetric) instabilities canexist even without any rotation. On the other hand, itis becoming increasingly clear that the stability of dif-ferential rotation under the presence of magnetic fieldsis one of the key problems in MHD astrophysics. There,however, is no laboratory experiment so far and even therelated numerical simulations of TI are very rare [4, 5].We shall demonstrate here how the TI interacts with dif-ferential rotation, and how it is possible to verify themain results in laboratory experiments. In particular,the theoretical results are used to propose experimentsfor measuring the turbulent diffusivity via a TI-inducedreduction of the electromotive force. Such experimentswill be of high relevance as the knowledge of the mag-netic turbulent diffusivity is basic for many applicationsin fluid dynamics. Very often we have only limited infor-mations about the magnetic diffusivity. In the laboratoryonly a very small number of experiments have been done(see [6, 7]). The same is true for the eddy viscosity whichcan be measured with the same experimental device sothat finally the turbulent magnetic Prandtl number be-comes known for one and the same instability.Consider a Taylor-Couette (TC) flow with U as the ve-locity, B the magnetic field, ν the microscopic kinematicviscosity and η the microscopic magnetic diffusivity. Thebasic state in cylindrical geometry is U R = U z = B R = B z = 0 and U φ = R Ω = a Ω R + b Ω R , B φ = a B R + b B R . (1) Let ˆ η = R in R out , µ Ω = Ω out Ω in , µ B = B out B in . (2) R in and R out are the radii of the inner and outer cylin-ders, Ω in and Ω out their rotation rates and B in and B out are the azimuthal magnetic fields at the inner and outercylinders. In particular, a field of the form b B /R is gen-erated by an axial current only through the inner re-gion R < R in , whereas a field of the form a B R is gener-ated by a uniform axial current through the entire region R < R out including the fluid.The magnetic Prandtl number Pm, the Reynolds num-ber Re and the Hartmann number Ha,Pm = νη ,
Re = Ω in R ν , Ha = B in R √ µ ρνη , (3)are the basic parameters of the problem where R = √ DR in is the unit of length with D = R out − R in . Forthe velocity the boundary conditions are assumed as al-ways no-slip ( u = 0). For conducting walls the radialcomponent of the field and the tangential components ofthe current must vanish so that d b φ / d R + b φ /R = b R = 0at both R in and R out . Here u and b are the fluctuatingcomponents of flow and field.While the linear stability code works well with smallPm, the minimum microscopic Pm which can be handledwith our nonlinear code is (only) 10 − . The numericscannot deal with the very small magnetic Prandtl num-bers of liquid metals used in the laboratory (Pm < ∼ − ).Some of our results can only be obtained by extrapolationmethods. THE INSTABILITY MAP
The map for the marginal instability is the result ofa linear theory for liquid sodium with Pm = 10 − . Thelinearized equations of the MHD system in a TC flowunder the presence of a toroidal field are given elsewhere,[8].Figure 1 shows the results for various values of µ B .For µ B = 0 . >
68. From the given pro-files closest to being current-free is µ B = 0 and we do notfind that for Ha ≤
200 there is any sign of destabilizinginfluence of the magnetic field, for neither axisymmetricnor nonaxisymmetric perturbations.For certain µ B the m = 1 mode should be unstablewhile the m = 0 mode should be stable [9]. The values µ B = 1 and µ B = 2 in Fig. 1 are examples of thissituation. There is always a crossover point at whichthe most unstable mode changes from m = 0 to m = 1.Note also that for µ B = 1 the critical Reynolds numberfor the m = 0 mode steadily increases, while the m =1 mode is suddenly decreasing for a sufficiently strongmagnetic field. Hence, weak fields initially can stabilizethe flow, and stronger fields eventually destabilize via anonaxisymmetric mode. Beyond Ha = 150 the flow isunstable even for Re = 0.Except for the almost current-free profile µ B = 0 allother values share the feature that there is a critical Hart-mann number beyond which the basic state is unstableeven for Re = 0. Let Ha (0) and Ha (1) denote these criticalHartmann numbers for m = 0 and m = 1, resp. Flat rotation laws
The rotation law with resting outer cylinder destabi-lizes the magnetic field. The question arises what hap-pens for those flat rotation laws which are stable in thenonmagnetic regime. In Fig. 2 the marginal stabilitycurves are also given for µ Ω = 0 .
25 (Rayleigh limit), µ Ω = 0 . µ Ω = 0 .
45 and µ Ω = 1 (rigid rotation). Onefinds the instabilities more and more stabilized by therotation.Note the massive quenching of the TI by rigid rotation.Even a rather slow rotation prevents the TI to destabilizethe system. Rigidly rotating containers can keep muchstronger fields as stable than without rotation. This ro-tational stabilization is modified for nonuniform rotation.At the Rayleigh limit, where Ω ∝ R − , even a slow ro-tation destabilizes the system while it is stabilized forfast rotation. Generally, fast rotation stabilizes, slow ro-tation destabilizes. At a Hartmann number of (say) 50and at the Rayleigh line one finds for increasing rotationrate the regimes: stable, unstable, stable. The critical Reynolds numbers of the sequence are ∼
300 and ∼ µ Ω = 0 .
35 while for rotation laws with µ Ω > ∼ .
45 onlythe rotational stabilization can be observed.
Electric currents
For experiments the electric currents must not be toostrong. As an upper limit currents with 10–15 kA shallbe considered. In order to translate the obtained criti-cal Hartman numbers into amplitudes of electrical cur-rents we apply our results to liquid sodium with a den-sity of 0.92 g/cm , a microscopic magnetic diffusivity of810 cm /s and a magnetic Prandtl number of 10 − . Forgallium-indium-tin the necessary currents are stronger bya factor of 3.15 (see [10]).Let I axis be the axial current inside the inner cylinderand I fluid the axial current through the fluid (i.e. be-tween inner and outer cylinder). Then the toroidal fieldamplitudes at the inner and outer cylinders are B in = I axis R in , B out = ( I axis + I fluid )5 R out , (4)measured in cm, Gauss and Ampere. Expressing I axis and I fluid in terms of our dimensionless parameters onefinds I axis = 5Ha s ˆ ηµ ρνη − ˆ η , I fluid = µ B − ˆ η ˆ η I axis . (5)Table I gives the electric currents needed to reach the less of Ha (0) and Ha (1) for ˆ η = 0 .
5, and µ B ranging from − | µ B | the current I fluid approaches a constant value. TABLE I: Characteristic Hartmann numbers and electric cur-rents for a sodium-container (ˆ η = 0 .
5) with conducting walls.The experiment with the almost uniform field µ B = 1 is indi-cated in bold. µ B Ha (0) Ha (1) I axis [kA] I fluid [kA]-2 19.8 24.8 0.807 -4.04-1 59.3 63.7 2.42 -7.25 ∞
151 6.16 6.16 ∞ The most interesting experiment is that with the al-most uniform field µ B = 1. For a container with a gap ofˆ η = 0 .
5, parallel currents of 6.16 kA are necessary alongthe axis and through the fluid. The experiment does notpossess the weakest electric currents but both the cur-rents are parallel and have the same amplitudes. Figure1 (middle) shows that in this case a crossing point Mexists where the axisymmetric mode has the same char-acteristic Reynolds number and Hartmann number as thenonaxisymmetric mode with m = 1. FIG. 1: The marginal stability curves for m = 0 (dashed) and m = 1 (solid). Pm= 10 − , ˆ η = 0 . µ Ω = 0, and µ B as indicated.Conducting walls. The hatched domain is stable.FIG. 2: µ B = 1. The same magnetic constellation as in Fig.1 (middle) but for µ Ω as indicated. µ Ω = 0 (resting outercylinder), µ Ω = 0 .
25 (Rayleigh limit), µ Ω = 0 . µ Ω = 0 . µ Ω = 1 (rigid rotation). The dotted line gives the eigenvaluesfor m = 0 which for those parameters only exists for µ Ω = 0. THE EDDY DIFFUSIVITY
We now turn to the mean-field concept turbulent fluidsof electrically conducting material. It is known that theexistence of turbulence in the fluid reduces the electricconductivity or – with other words – the fluctuations en-hance the magnetic diffusivity called the turbulent mag-netic diffusivity. In MHD the turbulent diffusivity is amuch more simple quantity than the corresponding eddyviscosity. While the latter is also formed by the existingmagnetic fluctuations this is not the case for the turbu-lent diffusivity. In a simplified (‘SOCA’) approximationfor a turbulence field with a correlation time τ corr results ν T ≃ ( 215 h u i + 13 h b i µ ρ ) τ corr (6)for the eddy viscosity but only η T ≃ τ corr h u i (7) for the eddy diffusivity, [11] . The magnetic fluctuationsdo not contribute to the magnetic diffusivity. This basicdifference between both the diffusion coefficients is notyet proven by an experiment. The results (6) and (7)suggest that in turbulent magnetic fluids the effectivemagnetic Prandtl number exceeds the value 0.4 whichwas confirmed by numerical simulations for driven MHDturbulence with Pm of order unity, [12]. The knowledgeof the turbulent magnetic Prandtl number is of extraor-dinary meaning in fluid mechanics and geo/astrophysics.For its calculation one has to measure both quantitiessimultaneously in one and the same experiment.Simplifying, the nonaxisymmetric components of flowand field may be used in the following as the ‘fluctuations’while the axisymmetric components are considered as themean quantities. Then the averaging procedure is simplythe integration over the azimuth φ . It is standard to ex-press the turbulence-induced electromotive force (EMF)as E = h u × b i = − η T curl B (8)with the (scalar) eddy diffusivity η T which must be pos-itive. In cylindric geometry the mean current curl B hasonly a z -component. Hence, E z = − η T curl z B . Take from Table I that for µ B = 1 the current throughthe fluid is positive then for negative E z the η T results aspositive. This is indeed the case. We have shown that TIindeed provides reasonable expressions for the turbulentdiffusivity in rotating containers, [13]. Nonlinear simulations
The absolute values for E z can only be computed withnonlinear simulations. The minimum possible magneticPrandtl number for the code yielding robust results is10 − . Here the results without and with rotation butonly for Ha = 200 are reported. The used MHD Fourierspectral element code has been described earlier in moredetail, [14], [15].Either M = 8 or M = 16 Fourier modes are used,two or three elements in radius and twelve or eighteenelements in axial direction, resp. The polynomial orderis varied between N = 8 and N = 16.Figure 3 shows the negative TI-induced EMF for mag-netic Prandtl numbers varied between 0.01 to 1. Themain results are that i) the EMF is always negative ( η T positive!) and ii) it runs with E/ Pm with the factor E ≃ − ηEB /D . Hence, η T η ≃ . E ≃ . . (9)For the voltage difference δU due to this EMF one finds δU = ηEB H/D with H as the container height and δU = ηE Γ √ µ ρνη Ha D (10)with Γ = H/D the aspect ratio of the container. For D = 10 cm and H = 100 cm we find for sodium( √ µ ρνη ≃ .
15) the maximum value of 34 mV as the po-tential difference from endplate to endplate. This valuecan only be considered as an estimate basing on the scal-ing with 1 / Pm suggested by Fig. 3. But even in the casethat the slope of the curve decreases for smaller Pm theeffect should be observable in the laboratory. A Hart-mann number of 200 requires 163 G at the inner cylinder( R in = 10 cm) which can be produced with an axial cur-rent of 8.15 kA for R out = 2 R in . N E G . E L E C T R O M O T I VE F O RC E Re= 0Re=500
FIG. 3: Maximum values of the normalized axial EMF vsPm for Ha = 200 and µ B = 1. The lower curve is withoutrotation, the upper curve is with rotation and shear (Re =500 , µ Ω = 0 . In order to study the rotational influence also the Pm-dependence of the EMF under the presence of a differen-tial rotation is given in Fig. 3. It is µ Ω = 0 .
35 (quasikep-lerian) and the Reynolds number is Re = 500. Note thatthe influence of the rotation for the TI-induced EMF issurprisingly weak; with rotation the values are slightlyhigher than without rotation. The magnetic Reynolds number, Rm = MAX( u ) D/η ,of the fluctuations is considered next. With Fig. 4 arather weak magnetic Prandtl number dependence of Rmis found. Extrapolating the results to Pm = 10 − givesin both cases a value of Rm ≃ .
6. The associated ve-locity fluctuations for sodium are about 15 m/s in a gapof 1 cm and 1.5 m/s in a gap of 10 cm. The values arerather similar to those of the Riga ’ α -yashchik’ experi-ment, [16]. Even with resting cylinders it is possible toproduce rather high (azimuthal) velocities in TI experi-ments. M A G N E T I C R EY N O L D S NU M BE R Re= 0Re=500
FIG. 4: The same as in Fig. 3 but for the magnetic Reynoldsnumber of the fluctuations.
THE EDDY VISCOSITY
Experiments with Tayler instability under the presenceof differential rotation can also provide eddy viscositymeasurements due to the angular momentum transportby both Reynolds stress and Maxwell stress. Within thediffusion approximation it is T = h u ′ R u ′ φ i − h B ′ R B ′ φ i µ ρ = − ν T R dΩd R (11)for the torque in the fluid. The fluctuations of flow andfield can be calculated with the code. The patterns forthe instability-induced diffusivity values for Pm = 0 . µ Ω = 0 .
35, Ha = 200 and Re = 500. One findsthe turbulence-originated increase of the eddy viscosity ν T /ν much larger than for the turbulent diffusivity. Theturbulent magnetic Prandtl number Pm T = ν T /η T be-comes about 2.05. Similar calculations for Pm = 0 . T = 0 .
71 while for Pm = 1 the smaller value0.65 results. The results only weakly depend on the av-eraging procedure. The given numbers follow after aver-aging over the whole cylinder. The turbulent magneticPrandtl number slightly increases with decreasing micro-scopic Pm; and for small Pm it reaches values larger thanunity. Note the differing results of simulations with Pmmuch smaller than unity and those with Pm < ∼
1, [12].
FIG. 5: Ha = 200, Re = 500. Simulations for a flow with Pm = 0 . µ Ω = 0 .
35 and µ B = 1. Left: η T /η , right: ν T /ν . Theresulting magnetic Prandtl number ν T /η T is about 2.1. Small Pm are shown to produce large turbulent valuesPm T . We cannot provide results for Pm smaller than0 .