High Magnetic Shear Gain in a Liquid Sodium Stable Couette Flow ExperimentA Prelude to an α− Ω Dynamo

Stirling A. Colgate1,2, Howard Beckley2, (deceased), Jiahe Si2, Joe Martinic2, David Westpfahl2, James

Slutz2, Cebastian Westrom2, Brianna Klein2, Paul Schendel2, Cletus Scharle2, Travis McKinney2, Rocky

Ginanni2, Ian Bentley2, Timothy Mickey2, Regnar Ferrel2, and Hui Li1 Vladimir Pariev1, John Finn3

1T-2, MS B-227, Los Alamos National Laboratory, Los Alamos, NM 87545; colgate@lanl.gov2Department of Physics, New Mexico Institute of Mining and Technology, Socorro, NM 87801 and

3T-5, Los Alamos National Laboratory, Los Alamos, NM 87545;

The Ω-phase of the liquid sodium α-Ω dynamo experiment at NMIMT in cooperation with LANLhas been successfully demonstrated to produce a high toroidal field Bφ that is ' 8×Br, where Br isthe radial component of an applied poloidal magnetic field. This enhanced toroidal field is producedby the rotational shear in stable Couette flow within liquid sodium at magnetic Reynolds numberRm ' 120. The small turbulence in stable Taylor-Couette flow is caused by Ekman flow with anestimated turbulence energy fraction of (δv/v)2 ∼ 10−3. This high Ω-gain in low turbulence flowcontrasts with a smaller Ω-gain in higher turbulence shear flows. This result supports the ansatzthat large scale astrophysical magnetic fields can be created by semi-coherent large scale motions inwhich turbulence plays only a smaller diffusive role that enables magnetic flux linkage.

PACS numbers: 52.72.+v, 52.30.cv, 95.30.QdKeywords: Dynamos — MHD —turbulence

Major efforts are spent world-wide on astrophysicalphenomena that depend upon magnetic fields, e.g., plan-etary, solar, and stellar magnetic fields, X-rays, cosmicrays, TeV gamma rays, jets and radio lobes powered byactive galactic nuclei (AGNs), and in the interpretationof Faraday rotation maps of galaxy clusters. Yet, so far,there is no universally accepted explanation for the in-ferred larage-scale magnetic field. A process, called theα − Ω dynamo [10, 16, 23], has been proposed whichinvolves two orthogonal conducting fluid motions, shearand helicity. When the two motions are comparable, itis often described as a stretch, twist, and fold or ”fast”dynamo, and supposedly can be produced by turbulencealone [18], [33]. The problem is that a turbulent dynamomust create these two orthogonal motions turbulent mo-tions. Fluid turbulence maximizes entropy and the diffu-sion of the flux of momentum [31], so that a field twistedone way by turbulent eddies at one moment of time maybe partially twisted the opposite way with only a netbias, and hence partially diffusive.

One possible way to achieve dynamo is to use naturallyoccurring large scale near-stable rotational shear flows(as in AGN accretion disks and in stars) in combinationwith transverse, transient, rotationally coherent plumes.Such plumes may be driven either by star–disk collisionsin AGNs [21] or large scale (density scale height) convec-tive elements in the base of the convective zone in stars[3, 15, 32]. These flows can stretch and wind-up an em-bedded, transverse magnetic flux through a large numberof turns (the Ω effect). In both types of shear flows theturbulence is expected to be relatively small because ofthe stability imposed by either an angular momentumgradient or an entropy gradient. (Positive work must bedone to turn over an eddy in such a gradient.) The ad-vantage of a large Ω-gain is that the corresponding α-gain(the helicity deformation necessary to convert a fraction

of the toroidal flux back into poloidal flux) can be corre-spondingly smaller. Parker, the originator of the α − Ωdynamo concept for astrophysical dynamos, invoked cy-clonic motions to produce the helicity, [24]. Moffatt [16]countered that a typical atmospheric cyclone makes verymany turns before dissipating, therefore making the signof the resulting poloidal flux incoherent. Moffat soughtmotions where the rotation angle was finite, ∼ π/4. Wil-lette [32] and the first author suggested that buoyantplumes were a unique solution to the problem of coher-ent dynamo helicity. Beckley et al. [1] demonstratedthis property experimentally of finite, ∼ π/4, coherentrotation of a driven plume before it dissipated throughturbulent diffusion in the background fluid.

Two liquid sodium dynamo experiments have pro-duced positive exponential gain, but the flows were con-strained by rigid walls unlike astrophysical flows [8, 30].The rigid walls restrict turbulent eddy size by the dis-tance from the wall, log-law of the walls [12], therebyproducing low turbulence. The three recent experimentsuse the Dudley-James [6] or Von Karman flow (counterrotating flow converging at the mid-plane and driven bytwo counter rotating propellers or two vaned turbine im-pellers, respectively, [7, 13, 19, 20, 25, 26, 29]). Turbu-lence is induced by the Helmholtz instability at the shear-ing mid-plane. This combination of coherent motions andunconstrained shear-driven (relatively high level) turbu-lence resulted in a maximum Ω-gain of only ∼ ×2. Rec-ognizing the enhanced resistivity of the mid-plane tur-bulence, the team at Wisconsin added a mid-plane baffleto reduce the turbulence, following which the Ω-gain in-creased to ' ×4 [27]. The von Karman Sodium 2 (VKS2)experiment in the same geometry produced exponentialgain [17]; However, the dynamo action was explained notby turbulence but primarily by the production of helic-ity by the large coherent vortices produced by the radial

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The New Mexico α − Ω dynamo (NMD) experimentis a collaboration between the New Mexico Institute ofMining and Technology and Los Alamos National Lab-oratory [4, 5]. It is designed to explore the possibilityin the laboratory using coherent fluid motions in low-turbulence high shear Couette flow in the annular vol-ume between two coaxial cylinders, R2/R1 = 2 rotatingat different angular velocities (see Figure 1), closely anal-ogous to natural fluid motions that occur in astrophysicalbodies [2, 21, 22]. The results of the first phase (Ω-phase)of this two-phase experiment are presented here.

Fig. 1 also indicates the Ekman flow, a thin fluid layerflowing along the boundaries of the annular volume andincluding a small radial return flow from inner to outercylinder through the fluid volume. The Ekman flow pro-duces both a torque and a small but finite level of turbu-lence [9]. This turbulence adds a small turbulent resis-tivity to the resistivity of metallic sodium. Five pressuresensors are distributed radially at one end. They mea-sure the pressure distributions, giving information of theactual rotational flow angular velocity distribution.

FIG. 1. A schematic drawing of the fluid flow of the exper-iment shows the inner cylinder of R1 = 15.2 cm rotating atΩ1/2π = 68Hz relative to the outer or confining cylinder ofR2 = 30.5 cm at Ω2 = Ω1/4. Liquid sodium in the annularspace undergoes Couette rotational flow. In addition, a thinlayer in contact with the cylinder walls and end walls under-goes Ekman flow, circulating orthogonally around and thenthrough the Couette flow. Pressure sensors in the end walland the magnetic probe housing are also shown schematically.

For the magnetic measurements, a 50 kW AC induc-tion motor (20kW is used for stable Couette flow) drivesa gear train with clutches and power take-off that ro-tates the two cylinders at the fixed ratio of Ω1 = 4Ω2.The outer cylinder can also be disengaged from the geartrain by a clutch, and a DC motor is used to accelerate orbrake the outer cylinder independently from the driveninner cylinder. This allows different Ω1/Ω2 ratios to beexplored. The DC motor housing (stator) is mountedon bearings. A torque arm with two force sensors (+−)

connects the motor stator to ground, so that the torqueon the DC motor can be measured separately from thedrive of the inner cylinder. This arrangement allows usto measure the torque between the two cylinders due tothe Ekman flow along the surface of the end plates whichrotate more slowly at Ω2.

In particular, when the inner cylinder is driven athigher speed by the AC motor, the Ekman fluid torquetries to spin up or accelerate the outer cylinder. Twotorques counteract this acceleration: 1) the friction inthe bearings that support the rotation of the outer cylin-der and 2) the torque on the DC motor when used asa generator, or brake. (The generated power is dissi-pated in a resistor.) In Fig. 2 (left) the crosses are themeasurements of the braking force exerted by the DCmotor torque arm; the dashed line is the calibrated bear-ing torque (measured by disengaging the inner cylinderdrive and rotating the outer cylinder with the DC mo-tor alone). The sum of these two torques is equal to theEkman fluid torque spinning up the outer cylinder.

FIG. 2. (left) The lower crosses are the measured DC genera-tor torques; the dashed line is the measured bearings torque,and the sum of the two is the top solid curve, the torquedue to the Ekman flow. (right) Shown is the measured andtheoretical pressure distribution. The pressure measurementstarts at R1 and extends to R2. The finite pressure at R1,P1 ' 100 psi is due to the centrifugal pressure of rotatingfluid in the narrow gap between the inner cylinder rotatingat Ω1 and the end wall rotating at Ω2 (dot-dash curve). Thisthin end-gap is filled with rotating fluid, which extends fromthe shaft seal (where P = 1 atm) at radius, R3 ' R1/3 toR1 at which radius the Couette flow in the annular volume isinitiated. From there, the pressure in the annular volume isextrapolated either as Ω ∝ R−2 or R−3/2. The measurementsfall along the R−3/2 pressure distribution curve indicating lessshear than the ideal Couette flow.

The experiment was originally designed upon the sup-position that the primary torque between the two cylin-ders would be due to the radial Ekman flow at the endwalls and would be small, ∼ 1/10, compared to pipe flowwall friction stress, τ = ρCDv

2. In Fig. 2 (left) the mini-mal Ekman flow torque occurs at Ω1/Ω2 ∼< 4, somewhatless than the limit of stable Couette flow. The torque

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value is 2 × 108 dyne-cm, close to the approximationthat each Ekman layer thickness [9] is ∆ ' R1/Re

1/2 '5 × 10−3 cm and that the average radial flow velocitywithin the layer is vr ' (Ω1R1)/2. In Fig 2 (right),the measured pressures are compared to the calculatedpressures corresponding to two different angular velocitypower laws. The upper (dashed) curve corresponds toideal, maximum shear, stable Couette flow, or Ω ∝ r−2,but the experimental points follow the lower solid curvecorresponding to Ω ∝ r−3/2. The Ekman flow torquehas distorted the angular velocity profile and reduced theshear relative to the ideal Couette flow.

Turbulent flow at such high Reynolds number Re '107 is still well beyond current simulation capability. TheEkman flow is a flux of fluid of reduced angular momen-tum deposited at the inner cylinder. The torque reducingthis angular momentum is friction with the end walls,which in turn reduces the angular velocity of the Cou-ette flow in the annular volume. Unstable flow, turbulentfriction with the inner cylinder surface, counteracts thistorque by speeding up this flow. The torque in the Cou-ette volume is a constant, independent of radius and axialposition. Then this shear stress is maintained constantin two laminar sub layers [28] with Re ' 100 and in twoturbulent boundary layers in a log-law-of-the-walls solu-tion [12]. When the distance from the walls correspondsto an eddy scale of the radial gradient, a transition takesplace to a scale independent eddy size turbulent torqueconnecting the two regions at inner and outer boundaries.The mean velocity distribution in the above analysis cangive a reduced shear, Ω/Ω1 ∝ (R1/R)1.64, rather thanstable Couette flow.

FIG. 3. A schematic drawing of the poloidal magnetic fluxproduced by two coils (left). Superimposed (right) are the fluxlines from the Maxwell calculation [14], with the iron shieldand steel shaft included. This radial flux crosses the highshear of the Couette flow producing the enhanced toroidalfield. The Br field from the Maxwell calculations agrees withthe calibrated probe measurements to 10%.

Fig. 3 shows the magnetic field configuration. Themagnetic field Hall detector probe, internal to the liq-uid sodium at the mid-plane, is the primary diagnostic

of the experiment. It consists of 6 multiple, 3-axis, mag-netic field Hall effect detectors at the mid-plane in theannular space between the two cylinders and containedin an aero-dynamically shaped housing. (The fluid fric-tion drag produced by this housing, primarily Ekmanflow, is estimated to be ∼ 0.1 of the end-wall Ekmantorque.) The Ω-gain was then measured using an ap-plied calibrated Br magnetic field as a function of thecoil currents. Because of the high gain in Bφ and thelack of perfect orthogonality of the Hall detectors, themeasured Br would be expected to be contaminated bya small fraction of the much larger Bφ field.

FIG. 4. (left) The lower curve shows the applied radial fieldBr ' 12 G and the upper curves show the measured toroidalmagnetic field for four over-laid experiments at Ω1 = 68 Hz.The right panel shows the ratio Bφ/Br.

FIG. 5. The top trace shows the toroidal field Bφ ∼ 750Gvs. time from an applied radial field of ∼ 250 G. The Ω-gainis now reduced to ×3. The bottom traces are the AC motortorque and current respectively, each showing a 50% increasedue to back reaction. The delay of several seconds in theback reaction torque corresponds to the spin-down time forthe Couette flow to reach a new, modified velocity profile.

Fig 4 confirms that the measurements are repeatableby showing four experiments over-laid where the Ω-gainratio of ×8 is repeated all with a low bias field of Br ' 12G. The Bφ produced by the Couette flow shear is about' 8 × Br. Note that the ideal Ω-gain could be as highas ∼ Rm/4π ' ×20 [22], but the reduced shear in thesodium (i.e., not exactly Couette profile) partially ac-counts for the reduction in the Ω-gain. We then mea-

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sured whether an increase in the velocity shear in theCouette volume might increase the Ω-gain ratio, as onemight naively expect. The velocity shear is increased byslowing the outer cylinder by using the DC motor as agenerator or brake. Therefore the ratio Ω1/Ω2 was in-creased by 20%, but the Ω-gain remains constant. Thissuggests the gain expected due to shear is balanced bythe greater flux diffusion due to the increased turbulenceof the now unstable Couette flow (see Fig 2). Roughly100 electronic samples are recorded and averaged dur-ing the several seconds recording cycle of each of fourmagnetic experiments. The time variation between eachrun reflects slight changes in the Couette flow relaxationtime and hence, the angular velocity distribution. Therepeatability among these four runs as well as several ear-lier runs in the previous six months gives us confidencethat the conclusion of high Ω-gain is valid.

The expected back-reaction should occur at a largervalue of Br where the dissipation and torque of the en-hanced Bφ affects the flow field and the added torqueshows up as additional current and torque in the AC

drive motor. This effect is shown in Fig 5, where, in ad-dition, the expected reduction in the Ω-gain ratio, from×8 to ∼ ×3 is observed when the applied poloidal fieldis Br ' 250G. The back reaction stress will modify theCouette flow profile such as to reduce the Ω-gain andhence reduce the back reaction stress until a new steadystate is achieved at reduced Bφ. Fig 5 shows the mea-sured difference in power, ' 10kW, due to the back reac-tion. We estimate that the specific magnetic field energydensity, B2

φ/8π is dissipated at < Ω > ×(V ol) where

the volume of high Bφ is estimated as 1/3 the length,V ol ' (L/3)(π/2)(R2

2 − R21). This results in a power of

∼ 8 kW, roughly the measured 10 kW.

In conclusion, a large Ω-gain in low turbulent shear ina conducting fluid has been demonstrated. This is likelyto be the mechanism of the Ω-gain of a coherent α − Ωastrophysical dynamo. Driven plumes are planned to beused in the α phase of the NMD experiment.

This experiment has been funded by NSF, Univ. ofCalif. & LANL, IGPP LANL, and NMIMT.

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