Lorentz force effects in magneto-turbulence

Physics of the Earth and Planetary Interiors 135 (2003) 137–159

Lorentz force effects in magneto-turbulence

Daniel R. Sisan, Woodrow L. Shew, Daniel P. Lathrop∗Department of Physics, IREAP and IPST, University of Maryland, College Park, MD 20742, USA

Received 6 December 2001; received in revised form 12 July 2002; accepted 25 September 2002

Abstract

We experimentally characterize magnetic field fluctuations in a strongly turbulent flow of liquid sodium in the presenceof a large externally applied field. We reach high interaction parameter (up toN = 17) for moderate magnetic Reynoldsnumber (up toRem = 18), a previously unexplored parameter range for liquid metal flows. As the interaction parameter (i.e.the ratio of Lorentz to inertial forces) is increased, the system passes through distinct regimes, which we classify. We findthat for certain ranges of the applied magnetic field, particularly at high values, the induced magnetic field exhibits large,coherent oscillations. Spatial structure in these induced field oscillations suggests the formation of non-axisymmetric vorticesthat precess at a fraction of the impeller rotation rate. We also investigate the effect of rough versus smooth boundaries andrelate these results to topographic core–mantle coupling in the Earth.© 2003 Elsevier Science B.V. All rights reserved.

PACS:47.27.Jv; 47.65.+a; 98.38.Am; 91.25.Cw

Keywords:Dynamo; Geodynamo; MHD turbulence; Interaction parameter; Magnetic Reynolds number

1. Introduction

There is extensive research indicating that plane-tary magnetic fields arise from flowing electrically-conducting fluids. Since the 1960s, many researchershave found self-generating dynamo solutions of thegoverning equations for the magnetic field and fluidmotion(Steenbeck et al., 1966; Moffatt, 1978; Dudleyand James, 1989). More recently, numerical researchhas shown dynamo action, including field reversalsand more complicated dynamics, in approximately re-alistic simulations of the Earth’s core(Glatzmaier andRoberts, 1995; Kuang and Bloxham, 1997). Dormyet al. (2000), and Roberts and Glatzmaier (2000),present excellent reviews. Recently, groups in Riga,

∗ Corresponding author.E-mail address:[email protected] (D.P. Lathrop).

Latvia (Gailitis et al., 2000, 2001)and Karlsruhe,Germany(Stieglitz and Müller, 2001)have producedliquid metal laboratory dynamos, in geometries thatforce helical flow using baffling and duct-work. Alogical next step is producing a laboratory dynamo ina geometry where Lorentz forces can greatly modifythe velocity field. A number of groups including ourown are attempting this(Odier et al., 1998; Beckleyet al., 1998; O’Connell et al., 2001; Peffley et al.,2000).

Experimentally studying the interaction of magneticfields and electrically-conducting flows has the poten-tial to answer a number of open questions. First isthe effect of turbulence, which experiments are wellsuited to study. In the Earth’s core, turbulence mostlikely has a significant effect on the Earth’s field, asbest evidenced by the broad magnetic energy spec-tra at the core–mantle boundary. Satellite data imply

0031-9201/03/$ – see front matter © 2003 Elsevier Science B.V. All rights reserved.doi:10.1016/S0031-9201(02)00212-1

138 D.R. Sisan et al. / Physics of the Earth and Planetary Interiors 135 (2003) 137–159

that the Earth’s field at the core–mantle boundary hassignificant contributions at least up to degreel = 12(Hulot et al., 2002). As the Earth’s mantle filters thefield increasingly with higherl, the Earth’s field maythus be even more broadband than we are able to inferfrom the surface. Evidence that turbulence is signifi-cant is also found in experiments(Odier et al., 1998;Peffley et al., 2000).

Other approaches to the dynamo problem may missimportant effects of turbulence. In theoretical studies,turbulence is either treated using approximations andparameterizations, e.g. the- and-effects, or onlystationary flows are considered. In numerical work,hyperdiffusivities or an artificially large magneticPrandtl number (e.g. setting the ratio of the kinematicviscosity to the magnetic diffusivity,Prm ≡ ν/η, to or-der unity—whereasPrm ∼ 10−6 for the Earth’s core(Dormy et al., 2000)) purposefully limit turbulence. Inthe Riga and Karlsruhe experiments, turbulence doesnot play a direct role in the magnetic field dynam-ics. For instance, the Karlsruhe dynamo is two-scale:while the velocity field is small-scale and turbulent,the magnetic field is large scale and apparently onlyslightly affected by the fluid turbulence.

Furthermore, in the Riga and Karlsruhe experi-ments the magnetic field reaches equilibrium throughmechanisms that are different than those operating inplanetary core dynamos. In the Karlsruhe experiment,the equilibrium magnetic energy corresponds simplyto the pressure head in excess of the critical value fordynamo action. In a less constrained dynamo, on theother hand, the equilibrium magnetic energy would bemuch harder to predict as Lorentz forces would alterthe flow in a more complicated way. A recent paperby Gailitis et al. (2001)showed evidence of saturationin the Riga dynamo experiment that appears morecomplicated than the saturation in the Karlsruhe ex-periment. However, the saturated field value was small(8 G) for moderate magnetic Reynolds number, corre-sponding to an interaction parameter much less thanone, far from the regime thought present in the Earth.Although we have not produced a self-generatingdynamo like the experiments in Karlsruhe and Riga,we are nonetheless closer to the parameter rangewhere the Earth’s dynamo is thought to operate. For amore extensive discussion of the relation between dy-namo theory and experiments see a review byBusse(2000).

Previous liquid metal experiments attempting toproduce dynamos in unconstrained flows used smallexternal applied fields(Odier et al., 1998; Peffleyet al., 2000). The applied fields there were used toprobe the system in two ways. In one, induced fieldswere measured, which result from twisting and shear-ing of a constant field by the conducting fluid. In theother, the decay of a pulsed applied field probed howclose the system was to self-generation. Small fieldswere desirable for these purposes so that Lorentzforces did not alter the flow.

In this study, the applied fields are purposefullylarge enough to alter the flow. In other words, the in-teraction parameter, the ratio of the Lorentz to iner-tial forces, is made greater than unity. The interactionparameter isN = B2L/ρµ0ηU, whereB is the mag-netic field,U and L the characteristic velocity andlength scales,ρ the density, andµ0 the magnetic per-meability. Near unity interaction parameter should bean interesting regime, as a competition exists betweentwo dominant forces. Although the interaction param-eter in the Earth is much greater than one, there is alength scale within the outer core whereN is compa-rable to our experiments. One place where structuresat these length scales might be especially importantis in the outer core boundary layers. The boundarylayers may play a crucial role in the global state ofthe outer core, and specifically in the magnetic fieldsaturation.

In addition to a force balance, turbulence, or itssuppression by the magnetic field, could affect sat-uration of the magnetic field. For instance, in theEarth’s core small-scale currents (due to small-scaleturbulent velocity fluctuations) could contribute toJoule heat production, even if they do not contributedirectly to the observable field. Joule heat production,which obviously cannot exceed the power sourcedriving the dynamo, places limitations on the inten-sity of magnetic fields in the core.Brito et al. (1996)studied Joule heating in turbulent gallium flows for arange of applied fields and found that the fields had asignificant effect. As the applied field was increasedfor a fixed rotation rate, Joule heating increased andthen decreased, similar to some of our observations.However, their magnetic Reynolds number and inter-action parameter (as defined here) were less than one.Although we do not directly measure Joule heatingin our experiments, the sudden change in dynamics

D.R. Sisan et al. / Physics of the Earth and Planetary Interiors 135 (2003) 137–159 139

we observe at large interaction parameter could makeextrapolating from Brito’s study to higher interactionparameter problematic. It should be noted, however,that our applied field direction is parallel to the rota-tion axis, whereas Brito’s was transverse.

Many other experimental studies of magnetic fieldson electrically-conducting flows (Rayleigh–Benardconvection, duct flows, magnetically-driven flows,flow around obstacles, etc.) have been performed(Moreau, 1990; Brito et al., 1995). However, to ourknowledge, no experiments have been performed forhighN, Rem = UL/η > 1.

Lastly, our experiment relates to a paper byHollerbach and Skinner (2001). They studied, numer-ically, spherical Couette flow with large applied fieldsparallel to the axis of rotation. The applied fieldsinduced instabilities in their system that resemble inmany ways the dynamics in our experiment. Theycan only reach Reynolds numbersRe= Rem/Prm ∼O(103), however, and the system changes significantlypast the critical value for the instabilities. Thus ourexperiment, with its range of much larger ReynoldsnumbersRe∼ O(106) and similar geometry, signifi-cantly extends their result.

In this paper, we present magnetic field and torquemeasurements taken over a range of interaction pa-rameter. These data exhibit phenomena that dependmainly on the interaction parameter (and weakly onRem)—suppression of turbulence, regular oscillationsin the induced magnetic field, and spatial correlationswith complex structure. We also find that angularmomentum transfer between the fluid and the spher-ical vessel depends in rather unexpected ways on theapplied field and topography of the vessel. We demon-strate that the interaction parameter is the relevant pa-rameter describing these phenomena, and we classifythe major regimes. Finally, we make an interpretationof the flow dynamics and discuss implications for thedynamics in the Earth’s core.

2. The experiment

The liquid sodium is contained in a 31.2 cm hollowsphere of 304 (nonmagnetic) stainless steel. The flowis driven by two 12.7 cm diameter titanium impellerson 2.5 cm diameter shafts entering the sphere fromeither pole (seeFig. 1). Each shaft is belt-driven by

Table 1Important dimensional parameters for the experiment and sodiumat 120C

Symbol Description Value

b Impeller radius (cm) 6.35a Sphere radius (cm) 15.6η Sodium magnetic diffusivity (cm2/s) 830ν Sodium kinematic viscosity (cm2/s) 7.39× 10−3

ρ Sodium density (g/cm3) 0.927

a 7.5 kW electric motor. The results we report in thispaper are for co-rotating impellers.

Two coils in near-Helmholtz configuration supplythe external field concentric to the axis of impellerrotation. The applied field varies approximately 10%from the center of the sphere to the outer edge. Theapplied field decreases away from the center along theaxis of rotation, and increases in the other two perpen-dicular directions. Magnetic fields are monitored withHall probes near the sphere. Three Hall arrays wereused: a 15 cm diameter ring (six probes) encircling onepole azimuthally, a pole-to-pole arc (eight probes), andan equatorial arc (seven probes) all of which measurefields perpendicular to the applied field (seeFig. 1).The equatorial array is augmented by two single Hallprobes, one of which is offset slightly from the equa-tor. A Hall probe just outside the sphere next to oneshaft measured the field in the direction parallel tothe applied field.Table 1shows relevant experimentalparameters for our system. More detail regarding theexperimental apparatus can be found inPeffley et al.(2000).

We perform experiments both with baffles and withsmooth walls. The baffles are thin stainless steel platesthat run from pole to equator in each hemisphere.They extend 5% of the sphere diameter. These baf-fles increase the ratio of poloidal to toroidal flow,as originally motivated by Dudley and James’s study(1989) of simple velocity fields in a spherical geometryand their ability to self-generate. We found inPeffleyet al. (2000)that a trend toward self-generation us-ing pulse decay measurement was only possible withthese baffles in place. In this experiment, the baffles’role is to create topographic variation at the vesselwall, analogous to topographic variation at the Earth’score–mantle boundary.

Experimental runs were performed using the fol-lowing procedure. The impellers were spun at a fixed


Fig. 1. Schematic of experimental apparatus showing the spherical vessel and impellers, pole-to-pole baffles, coils supplying the externalfield, Hall probes, and Hall arrays (a pole-to-pole arc, a 15 cm diameter ring encircling one pole, and an equatorial arc). (b) shows aphotograph of the impellers. The two impellers have the same shape. The single Hall probe shown in the center of the figure is slightlyoffset from the equatorial array outside the sphere. The array encircling one pole nearly touches the sphere (approximately 5 cm from thesodium). All Hall probes measure the field perpendicular to the applied field, except for the probe at the pole (opposite the array encirclingone pole). Runs were performed with and without the baffles.

rotation rate, and the field was incrementally increasedthrough its full range. The maximum field obtainablewas 2×103 G. However, the magnets heated through-out the day and thus lowered the maximum obtainablefield value. For each field value, data were taken forseveral seconds (typically 4 or 16 s).

3. Equations of motion

The dimensionless equations of motion relevant toour system are the Navier–Stokes equation with the

added Lorentz force:

∂u∂t

+ (u · ∇)u = −∇P + Re−1∇2u+N Re−1

m ( ∇ × B) × B (1)

and the induction equation:

∂ B∂t

= ∇ × u × B + Re−1m ∇2B (2)

derived from Maxwell’s equations and Ohm’s law fora moving conductor, whereu is the velocity,P thepressure, andB the magnetic field. The dimensionless


Table 2Relevant dimensionless parameters

Dimensionless parameter

Re Rem N

Definition UL/ν UL/η B2L/Uηρµ0

Range for experiment 5× 105 to4 × 106

3.8–18 0–17

Estimate for Earth ∼108 ∼100 ∼105–107

Estimate for Sun ∼102–1011 ∼102–104 ∼1–104

Parameter estimates for the Earth use the westward drift rate offield patterns (U ≈ 10−4 m/s) for the velocity andB = 5 G (radialcomponent at core-mantle boundary) toB = 500 G (estimatedtoroidal component) for the magnetic field. Density, electricalconductivity, kinematic viscosity, and length scale of the fluid coreare taken fromRoberts (1988). Estimates for the Sun are discussedin Section 5.2.

parameters; the Reynolds numberRe = UL/ν, themagnetic Reynolds numberRem, and the interactionparameterN are described inTable 2 and the nextsection. In the equations above, the velocityu hasbeen scaled by the impeller tip speedΩb, time bythe rotational time scaleΩ−1, length by the sphereradiusa, and the magnetic fieldB by the imposed fieldstrengthB0 (Table 2) .

In addition, sodium is effectively incompressible inthese conditions:

∇ · u = 0. (3)

No-slip boundary conditions for the velocity apply atall surfaces. The magnetic field boundary condition isapproximately continuous with an exterior vacuum so-lution, as stainless steel is a poor electrical conductor.

3.1. Dimensionless parameters

The state of the fluid flow, and thus the induced field,depends on a competition between the inertial andLorentz forces (the second and fifth terms inEq. (1)).The interaction parameter quantifies the competition.An expression for the interaction parameter can befound using dimensional analysis. Using Ohm’s lawJ = σ( E + u × B) (whereσ is the electrical con-ductivity, and E the electric field) and a scaling fromFaraday’s law (Eind ∼ uBind), the dimensional formof the Lorentz force per unit mass (F = J × B/ρ)scales as:

FLor ∼ σ

ρuBextBind + σ

ρuB2

ext (4)

whereρ is the density. The induced magnetic field isabout 1000 times smaller than the external magneticfield in our experiments, so the second term abovedominates.

Since the advective term in the Navier–Stokes equa-tion scales asU2/L, whereL is a characteristic length,the interaction parameter can be expressed asN =B2

extL/ηρµ0u, whereη ≡ 1/σµ0 is the magnetic dif-fusivity, composed of the electrical conductivity andmagnetic permeability. Using the sphere radiusa, andthe impeller tip speedΩb, gives

N = B2exta

ηρµ0Ωb. (5)

The only other independent, adjustable dimension-less parameter is the magnetic Reynolds number,Rem ≡ UL/η. We defineRem using the sphere radiusand impeller tip speedRem ≡ Ωba/η. The mag-netic Reynolds number quantifies the ratio of fieldadvection (twisting, stretching, etc.) to resistive dif-fusion. Thus, this number must be greater than unityto produce a dynamo. The (hydrodynamic) Reynoldsnumber, Re ≡ Ωba/ν, is related to the magneticReynolds number by the ratio of kinematic viscos-ity to magnetic diffusivity. This ratio, known as themagnetic Prandtl numberPrm ≡ ν/η, is a property ofthe fluid. For sodium at 120C, this number is small(Prm = 8.3 × 10−6), meaning flows withRem > 1will be highly turbulent (Re> 105).

4. Results

4.1. Time series

The induced field dynamics change markedly withthe interaction parameter. The five time series, shownin Fig. 2, characterize the magnetic field fluctuationsobserved over the range of interaction parameter in ourexperiment. Except forFig. 2b, the time series for thesmooth wall case are qualitatively similar to those inthe case with baffles. The interaction parameter rangesfor the different regimes are shown inTable 3.

Fig. 2shows magnetic field time series for increas-ing applied fields forΩ/2π = 10 Hz. For small fields(Fig. 2a) the induced field fluctuates erratically, sim-ilar to the fluid velocity fluctuations of the underly-ing turbulence; the magnetic field is simply a passive


Table 3The five regimes

Regime Range(smooth walls)

Range(baffles)

Beginning boundary Character

I 0 < N < 0.3 0< N < 0.3 B passive vectorI∗ 0.3 < N < 0.6 Increasing slope ofσ (seeFig. 9a) m = 1 oscillationsII 0.6 < N < 2 0.3 < N < 3 First peak ofσ (seeFig. 9) Turbulence suppressionIII 2 < N < 9 3< N < 7 Start of induced field frequency peak

(seeFig. 3)m = 2 oscillations,f depends onBext

IV N > 9 N > 7 Start of frequency locking (seeFig. 3) m = 2 oscillations,f independent ofBext

vector. Increasing the applied field strength, large os-cillations emerge in regime I∗ (Fig. 2b), though onlywith smooth walls. In regime II (Fig. 2c), the fluctu-ations become smaller despite a larger applied field,indicating the suppression of turbulence. This is con-sistent with other studies of MHD turbulence, whereturbulence suppression by applied fields is well estab-lished(Cowling, 1957; Moreau, 1990). In the smoothwall case, regime II also shows a suppression of theregime I∗ oscillations. In regime III (Fig. 2d), anotherinstability forms as large oscillations in the time seriesemerge. The frequency of these oscillations increaseswith the applied field (detailed inSection 4.2). Finally,in regime IV (Fig. 2e), the oscillations acquire mod-ulations, and lock onto subharmonics of the impellerrotation rate.

4.2. Composite Fourier spectra

Now we examine the power spectra to show how thesystem evolves through the different regimes.Fig. 3shows power spectra of magnetic field as a function ofapplied field for a fixed rotation rate. Each horizontalline of the images is formed from the power spectrumat one value of the applied field. In the figure we haveindicated the regimes that are exemplified inFig. 2. Inregime I, the power spectra is broadband and increases

Fig. 2. Time series of the induced magnetic field measured outside the sphere at the equator for each of the regimes (for smooth walls).In each, the time-averaged magnetic field has been subtracted. The magnetic Reynolds number isRem = 7.5, corresponding to an impellerrotation rateΩ/2π = 10 Hz. The interaction parameter is shown for each time series. As the interaction parameter increases: (a) regimeI, the induced magnetic field exhibits turbulent fluctuations, as the magnetic field acts as a passive vector; (b) regime I∗, an instabilitywith large oscillations at approximately 30% the impeller rotation rate occurs (with smooth walls only); (c) regime II, the turbulence andoscillations are suppressed by the Lorentz forces; (d) regime III, large coherent oscillations occur again where the frequency depends onthe applied magnetic field; and (e) regime IV, the oscillations decrease in magnitude, acquire a modulation, and the frequency locks (forthe case with baffles) to half the impeller rotation rate.

in intensity as the applied field is increased. In regimeI∗ (Fig. 3aonly), we see the emergence of a frequencypeak. In regime II, where turbulence is suppressed,the intensity decreases with increasing applied field.Regime III begins with the emergence of an intensesingle frequency peak whose frequency increases withapplied field (nearly linearly for the case with baffles,Fig. 3b).

A standing Alfvèn wave resonance frequency wouldalso increase linearly with applied field; however, thereare at least four problems with an Alfvèn wave originfor these oscillations. First, the frequency range is toolow. The Alfvén velocity,vA = B/

√µ0ρ, at 1000 G

for our experiment is 2.8 m/s. The resulting resonancefor a box of length 2b, the sphere diameter, is thus9.9 Hz—several times too large for the observed oscil-lations at 1000 G for theRem explored in our experi-ment. Second, the induced field frequencies in regimeIII increase with rotation rate, as evidenced by thecollapses inFig. 4. No obvious fluid motion wouldmodify an Alfvén wave resonance in this way. Third,the (extrapolated) zero field induced field frequencyin our experiment is non-zero. Lastly, inFig. 3a, thefrequency increase in regime III deviates more signif-icantly from a linear trend, and there is an even lowerfrequency peak (∼0.1 Ω) that changes slowly (andnonlinearly) with applied field.


Fig. 3. Images composed of induced field power spectra. Applied field increases up the page, where each horizontal line is a powerspectrum of the induced field for smooth walls (a) and with baffles (b). The applied field increases in steps of≈22 G (a) and≈18 G (b).The deviation from a linearly-increasing field in this figure is estimated to be less than 1% and the nonlinearities are limited to the highestfield values due to coil heating. Color indicates logarithmic intensity, from highest to lowest: red–yellow–green–blue. The impeller rotationrate is 10 Hz (Rem = 7.5). The regimes are indicated on the right. Both spectra were obtained from a probe at the equator, though thequalitative features are similar at any point outside the sphere.


Fig. 4. The most intense induced field frequency in regimes I∗, III and IV for impeller rotation rates: 5 Hz,Rem = 3.8 (); 7.5 Hz,Rem = 5.6 (); 10 Hz,Rem = 7.5 (); 12.5 Hz,Rem = 9.4 (); 15 Hz,Rem = 11.3 (×); and 22.5 Hz,Rem = 18.75 (+). Data is shown in(a) with smooth walls and (b) with baffles. The frequency and applied field have been made dimensionless using the rotation rate and theinteraction parameter.N1/2 is used instead ofN here (and elsewhere) because the former is linear inB. In regime III, we have followedthe 0.2 < 2πf/Ω < 0.5 peak and not the peak 2πf/Ω < 0.1.


Regime IV begins where the single frequency ofregime III reaches 2πf = Ω/2, at which point thesingle frequency splits into three frequencies and locksonto the rotation rate. The central peak at 2πf = Ω/2is nearly independent of the applied field strength; twosideband modulations near 2πf = Ω/4 and 3Ω/4change slowly with the applied field.

We have extracted the dominant frequency peak inregimes I∗, III, and IV as it changes with applied fieldfor multiple rotation rates, as shown inFig. 4. Thefrequency and magnetic field strength were made di-mensionless using the rotation rate and the interactionparameter. Were the interaction parameter the onlyrelevant parameter, the curves would collapse. How-ever, the collapse is imperfect, especially in regimeIII—a given oscillation frequency occurs at a some-what higher interaction parameter for higher impellerrotation rates. Also, there are secondary features—anovershooting of the 2πf = Ω/2 frequency, as well asthe frequency remaining unchanged for small rangesof interaction parameter—that depend on magneticReynolds number. These are likelyRem effects, be-cause the magnetic Reynolds number is the only otherrelevant, independent non-dimensional parameter.

4.3. Induced field structure

The induced field oscillations in regimes I∗, III, andIV are present at every point near the sphere’s surface.To examine the spatial structure of these oscillations,we calculate correlations in azimuthal and poloidalHall probe arrays.

Fig. 5 shows the spatial correlations, along withschematic versions of the induced field derived fromthese correlations, for regimes I∗ and IV. Regime III(not represented in this figure), like regime IV, has an

Fig. 5. Correlations of induced field. The cross-correlation as a function of poloidal angle for regime I∗ (a) and regime IV (b), and as afunction of azimuthal angle at the equator for regime I∗ (c) and regime IV (d), and as a function of azimuthal angle near one pole inregime IV (e). The cross-correlation is defined as:C(x) = 〈A(t)B(t)〉/

√〈A(t)2〉〈B(t)2〉 whereA(t) andB(t) are magnetic field time series

with time averages subtracted, from Hall probes separated by an anglex; brackets indicate time averages. In each array, cross-correlationswere computed relative to one probe, giving correlation as a function of angular separation. In the poloidal array, the probe nearest onepole was chosen as a reference, since that permitted the largest angular separation. In the array encircling a pole, the reference probe waschosen arbitrarily (the results are independent of reference). These measurements were taken atRem = 7.5 (Ω/2π = 10 Hz) with smoothwalls. In (c), the probe at 103 is offset from the equator, which explains the deviation from them = 1 trend. In the lower left corner, aschematic shows the idealized induced field pattern consistent with correlation data for regimes I∗ and IV. The field component is in thecylindrical-radial direction. White and black indicate positive and negative induced field. The patterns precess in the prograde direction.

m = 2 correlation at the equator and is equatoriallyanti-symmetric, but it does not have a well-definedpoloidal wave numberl, as correlations near the polesare weak (seeFig. 8d). Also, in regime IV there is anm = 0 component mixed in with them = 2 correlationnear the pole (Fig. 5e). In Fig. 6, cross-correlationsbetween judiciously chosen probes as a function ofinteraction parameter show how the modes, describedabove, evolve with interaction parameter.

From temporal cross-correlations on the equatorialarray, we can determine if the induced field patternsprecess, and if so in which direction. We first computethe cross-correlation function for pairs of probes:

C(τ, φ) = 〈A(t)B(t + τ)〉√〈A(t)2〉〈B(t + τ)2〉

.

Here, A(t) and B(t) are magnetic field time serieswith time averages subtracted, from Hall probes sep-arated azimuthally by an angleφ; brackets indicatetime averages. The time lag that maximizes the cor-relation,τmax, determines the time for the pattern totravel the angular distanceφ. We find thatτmax in-creases linearly with angular separation, indicatingazimuthally-traveling waves or precessing vortices inthe fluid motion. From the relative signs ofτmax, φ,and the impeller rotation direction, we found that thismotion was prograde (i.e. in the direction of impellerrotation).

Fig. 7 shows the precession speed deduced withthis technique as a function of interaction parame-ter. Shown also is the peak induced-field-spectral-frequency scaled bym, for the sameRem. The twocurves collapse for most values of interaction param-eter, suggesting that the induced field oscillations arethe result of precessing spatial patterns. This collapseprovides new insight into the dynamics of regime


Fig. 6. Correlations vs. interaction parameter. The left two panels show cross-correlation between two probes separated by 170 on theequatorial array with smooth walls (a) and with baffles (b). The right two panels show cross-correlations between two probes, each 45from the equator on opposite sides with smooth walls (c) and with baffles (d). The cross-correlations in (c) and (d) were computed from4 s time series and in (a) and (b) from 16 s time series. This measurement time difference is reflected inFig. 6c and dwhere the correlationhas more scatter at low interaction parameter.

III: the frequency peak increasing with applied fieldis caused by an increasing precession velocity of anm = 2 pattern in the flow. However, forN1/2 > 3.0,the deduced precession velocity is somewhat greaterthan the peak frequency scaled bym. This discrepancyis likely associated with the observed modulations inregime IV (seeFigs. 2e and 3).

Since coherent structures (with relatively longcorrelation times) are precessing azimuthally, thepole-to-pole array should provide a nearly completepicture of the induced field dynamics.Fig. 8 showsinduced field space–time diagrams for this array. Sixdiagrams are shown—one each for regimes I, I∗, II,and IV, and two for regime III. In regime I∗, we see


Fig. 7. Dimensionless precession frequency from cross-correlations (), and frequency peak from induced field power spectra scaled bym (), vs. interaction parameter. The precession frequency was obtained by first maximizing the cross-correlation functionC(τ, φ) forprobes separated byφ on the equatorial array. The precession angular velocity is a linear fit to angular separationφ vs. optimum time lag,τmax(φ). The error bars for precession speed indicate the standard deviation of the fit. The error bars on the induced field frequency peakis the bin size in the Fourier transform algorithm. Both frequencies have been scaled by the rotation rateΩ/2π. These measurements weretaken atRem = 7.5 (Ω/2π = 10 Hz) with smooth walls.

the oscillations appear to be caused by precessing(perhaps columnar) structures tilted relative to theaxis of rotation (Fig. 8b). Precessing structures alsoappear to cause the oscillations in regime III (Fig. 8dand e). As one looks from pole to pole in that regime,it appears the structures are divided into zones. Thedynamics between the zones is not strongly correlated,reflecting the lack of a distinct poloidal wave numberin regime III. Two things happen inFig. 8d and eas the interaction parameter increases: the coherencebetween the zones increases, and the zonal boundaryshifts toward the equator. Both effects are also evidentin Fig. 6b and d, where the anti-correlation increasessteadily pastN1/2 = 2.

In regime IV, the coherence between zones increasesfurther still. Here, the structures seem more wavy than

divided discretely into zones. Additional boundarieshave formed in between the first and second probes andalso between the last two probes directly opposite theequator (not shown, seeFig. 8 caption). These wavystructures give rise to thel = 3 correlation shown inFig. 5b.

It should be noted that inFig. 8b, d–fthe strongestoscillations (i.e. where the peaks are brightest andmost sharply defined) occur at large angles. Thisasymmetry most likely results from asymmetry inthe impellers: the impellers have the same helicityso that faced opposite and co-rotating—as in ourexperiment—one impeller “scoops” while the otherdoes not (seeFig. 1b). The scooping impeller is atlargeθ in Fig. 8; thus, the greater entertainment of thescooping impeller apparently enhances the instability.


Fig. 8. Six different space–time diagrams of induced field in the poloidal array. Each horizontal line shows the magnetic field intensity(indicated by color) vs. poloidal angle; time increases up the page. The time averages of each probe have been subtracted. Red indicatesthe highest field values and blue the lowest. Six diagrams, (a)–(f), are shown with the regime indicated above each. The correspondinginteraction parameters are: (a)N1/2 = 0.28, (b) N1/2 = 0.75, (c) N1/2 = 1.1, (d) N1/2 = 2.1, (e) N1/2 = 3.0, and (f)N1/2 = 4.0. Alldiagrams are for the smooth wall case andRem = 7.5 (Ω/2π = 10 Hz). In each diagram, eight probes are used. Note that each diagramis not centered about the equator—the probe nearest the right pole failed during the run and was not included.

4.4. Induced field fluctuation magnitudes

Another way to observe interaction parameter ef-fects is through the magnitudes of the induced fieldfluctuations.Fig. 9 shows the standard deviation ofthe induced fieldσ as it changes with the externalfield for a single rotation rate. In regime I, the mag-netic field is a passive vector in the turbulent flow, and

the standard deviation increases linearly with appliedfield. In regime I∗ (Fig. 9aonly), as them = 1 in-stability forms, the fluctuations increase more sharply(though still linearly) with applied field. Then, inregime II, the fluctuations diminish with increasingapplied field—a sign that turbulence in the velocityfield is suppressed. The fluctuations increase again inregime III due to them = 2 instability. This rise in


Fig. 9. Standard deviation of the induced field vs. applied field (a) with smooth walls and (b) with baffles. Both (a) and (b) are forRem = 7.5. The field fluctuations were measured in the radial direction at the equator. The regimes are indicated.

σ occurs despite a further suppression of turbulencein regime III, evident from the decrease of broad-band content in the power spectra. The oscillationmagnitude then decreases in the latter half of regimeIII. Finally, in regime IV the oscillations decreasefurther and then change only slowly with the appliedfield.

The fall of the standard deviation in regimes IIIand IV is due partly to the shifting zones seen inFig. 8. Indeed, the standard deviation curves forprobes at different polar angles look somewhat dif-ferent. For instance, the peak in regime III occurs atdifferent interaction parameter, and for some probesthe standard deviation increases with applied fieldin regime IV. This difference can be seen quali-

tatively in Fig. 8, as described above inSection4.3. The fall of the standard deviation in regimeII, however, seems to be a general suppression ofturbulence.

Fig. 10 shows the induced field standard devia-tion for several magnetic Reynolds numbers. A col-lapse is attempted for these data by scaling the appliedfield using the interaction parameter, and by scalingthe magnitudes by a characteristic field value,Bc ≡√ρµ0ab3Ω3/η.This characteristic field value is arrived at from the

equations of motion. Assuming large Reynolds num-ber and moderate interaction parameter, the advectiveforce and the Lorentz force are both dominant andcomparable in magnitude. Thus, we obtain from the

Fig. 10. Dimensionless standard deviation of the induced field as a function of the applied field (a) with smooth walls and (b) with bafflesfor the magnetic Reynolds numbers: 3.8 (), 7.5 (), 15 (), and 18.75 (). The data are made dimensionless using the interactionparameter and a characteristic magnetic field,Bc ≡

√ρµ0ab3Ω3/η.


dimensional form ofEq. (1),

(u · ∇)u ∼ 1

ρµ0( ∇ × Bind) × Bext

which implies

ρµ0U2 ∼ BextBind. (6)

From the induction equation (where the time derivativeterm has been dropped, since it should be somewhatsmaller than the diffusive term):

∇ × u × Bext ∼ η∇2Bind.

This implies that

Bind ∼ BextUL/η. (7)

The proportionality inEq. (7)should not be taken asan equality due to ambiguities in the choice of velocityand length scales from the induction equation.

Fig. 11. Dimensionless torque as a function of interaction parameter for a rotation rate of 15 Hz (Rem = 11.3), with smooth walls () andwith baffles (). The torque shown is the average of both motors. Because the data sets were (necessarily) taken during different runs,there is an uncertainty in the relative offset of the two curves due to unknown differences in frictional losses in the seals. This offset isestimated to be less than 5% of the smooth wall zero field value.

EliminatingBext from (6) and (7) yields

Bind ∼ Bc ≡√ρµ0ab3Ω3/η

where the impeller tip speedΩb has been substitutedfor U and the sphere radiusa for L. Thus, we mightexpect the standard deviation of the induced field toscale asBc. From Fig. 10, we see that making thestandard deviation ofBind dimensionless asσ/Bc andBext dimensionless asN1/2 causes the differentRemcases to collapse.

Taken together,Figs. 4 and 10demonstrate that theinteraction parameter controls much of the qualitativechanges in the system for the full range ofRem valuesand applied fields explored in this experiment.

4.5. Motor torque

We also observe significant changes in the torque asthe interaction parameter is increased.Fig. 11 shows


the motor torque required to co-rotate the impellers ata fixed rotation rate (Rem = 11.25) versusN1/2. Thetop curve is with baffles and the bottom with smoothwalls. The salient and somewhat surprising feature ofthis figure is that with increasing magnetic field thetorqueincreaseswith smooth walls anddecreaseswithbaffles.

Two other differences in the two torque curves areapparent as well. First, at zero field the torque is lowerwith the smooth wall case. This is not surprising. Thetorque required to spin the impellers is a measure of themomentum transfer from the impellers to the spher-ical container. The baffles, which entrain fluid in theazimuthal direction—the dominant component of themean flow—will thus aid this momentum transfer. Fi-nally, the two curves converge (to within uncertainty)at high interaction parameter. This convergence, oc-curring also for other rotation rates, appears robust.

5. Discussion

In this section, we will interpret our measurements,and discuss their connection with prior scientific workand their relation to the Earth and the Sun.

Applied magnetic fields are known to increase therigidity of conducting fluids in the direction of thefield. As the applied field increases in our experiments,a cylinder of sodium between the impellers wouldbecome increasingly coupled to the impeller rotationrate. Correspondingly, the fluid outside the cylinderwould be increasingly tied to the boundary. As thishappens, we might expect momentum transport fromthe impellers to the wall to decrease, reducing drag andthus torque. Indeed, with baffles the required torquedecreases with imposed field over a large range ofN.

For small applied fields the drag is greater withrough walls than with smooth. One might thus ex-pect the torque decrease to be weaker without baffles.However, with smooth walls the torqueincreaseswithapplied field. A possible mechanism for this increaseinvolves how the applied field affects the boundarylayers. As the applied field increases, the Hartmannlayer at the sphere wall becomes thinner, increasingthe viscous drag across it. This would increase thecoupling between the fluid and the outer wall, caus-ing the torque to increase with applied field. The casewith baffles does not show the effect of the thinning

boundary layers however (i.e. the torque does not in-crease with applied field). This discrepancy is likelydue to the way angular momentum is transferred tothe wall in the two cases. With smooth walls, viscousboundary layers alone control the momentum trans-port. In the other case, pressure drops across the baf-fles apparently cause most of the transport.

The torque with smooth walls plateaus and then de-creases slightly aroundN1/2 = 1.5. A similar plateauoccurs with baffles aroundN1/2 = 2.5 (Fig. 11).There is no obvious connection between this appar-ent saturation and changes in the induced field char-acter. The best connection, which is rather tenuous,is found inFig. 6b and d: the trend toward equatorialanti-symmetry begins roughly at the same interactionparameter where the torque deviates from its lineartrend in Fig. 11. The torque curves may be best ex-plained by changes in the velocity field that are notreflected in the magnetic field data. In the future, weplan to measure the velocity directly using ultrasoundvelocimetry.

The picture involving the tendency for magneticfields to tie fluid along field lines also leads to an ex-planation for the observed oscillations in regimes I∗,III, and IV. In the large field limit, the fluid forms acylinder between the impellers undergoing solid bodyrotation while the surrounding fluid is stationary. As arotating cylinder of sodium forms, so would a stronglysheared boundary. Traveling waves or precessing vor-tices would likely form at this boundary, giving riseto the observed oscillations in the induced field.

Experiments byLehnert (1955)first showed, in avery different geometry, how magnetic rigidity canlead to vortex formation in liquid metals. In the ab-sence of applied fields Lehnert’s experiment was lam-inar, whereas ours is highly turbulent. Experimentssimilar to Lehnert’s have been performed byMoreau(1990), Brito et al. (1995).

In a system similar to ours,Hollerbach andSkinner (2001)find (numerically) vortices forming ina magnetically-induced shear layer. Like ours, theirflow was in a spherical container with strong mag-netic fields imposed parallel to the axis of rotation;however, their driving was a rotating inner sphere,not co-rotating impellers. At high fields they find asolidly-rotating cylinder tangent to the inner sphere;thus, we would expect the two systems’ dynamics toconverge at largeN. They found vortices precessing


at a fraction of the inner sphere rotation rate in theirsystem. The number of vortices depended mainly onthe applied field and the boundary conditions (con-ducting or non-conducting), though Reynolds numbereffects were also observed.

Though we are unable to measure the sodium ve-locity in our experiment directly, our magnetic fieldmeasurements are consistent with the precessing vor-tex patterns described by Hollerbach and Skinner. Inparticular, we infer velocity patterns in our experimentsimilar to them = 1 and 2 instabilities described byHollerbach and Skinner. However, extrapolating fromHollerbach and Skinner’s study to our parameter rangewe would expect much higherm modes to be mostunstable. For example, in our experiments for a mag-netic field of 1000 G (expressed in their paper usingthe Hartmann number, the ratio of the magnetic toviscous time scales), Hollerbach and Skinner predictm = 5 (m = 12) for insulating (conducting) boundaryconditions to be the first unstable mode.

The critical Reynolds numbers for both boundaryconditions areRe≈ 1000, which are much lower thanthe lowest obtainable Reynolds number in our experi-ment (extremely low rotation rates are difficult to con-trol accurately). However, for two values of appliedfield (for each boundary condition) they increased theReynolds number past the critical values to observehow the system equilibrates in the supercritical regime.They found that the torque needed to spin the innersphere increased at the onset of the first instability. Inour runs without baffles we also observe the torqueincreasing at the instability onset. In our experimentswe increase the magnetic field to induce the instabil-ity, since we can not lower the rotation rate below thecritical Reynolds number for even the highest field ob-tainable. Notice inFig. 11, in the smooth wall case, thetorque is nearly constant untilN1/2 = 0.4; in Fig. 4a,them = 1 instability also occurs atN1/2 = 0.4. Thetorque for that case then steadily increases and thendecreases with applied field, seemingly independentof further instabilities.

As the Reynolds number is increased past its crit-ical value, Hollerbach and Skinner observe that theazimuthal wave numberm is reduced, through a sub-critical bifurcation. For the range of Reynolds numberthat they study (up to approximately twice the criti-cal value) them value is reduced at most by one. TheReynolds number in our experiment, by contrast, is

hundreds of times their critical value. That we onlyobserve lowm instabilities in our experiments (in-stead of the higherm instabilities that Hollerbach andSkinner predict to be most unstable) is evidence foradditionalm reductions occurring at higher supercrit-icality. It is not clear whym = 2 is favored at highapplied field in our experiment, nor whym = 1 isfavored at intermediate applied field. In fact, Holler-bach and Skinner never find anm = 1 instability. Itis possible thatm = 1 modes are only seen at highReynolds number or are a feature associated with geo-metric differences. In our experiment, the progressionfrom m = 1 to 2 as the applied field increases is sim-ilar to the favoring of higherm modes as the appliedfield increases that Hollerbach and Skinner observe. Itappears that at high supercriticality some features ofthe regime near criticality remain (e.g. preferring highm modes at high applied fields) while other featureschange (e.g. the appearance and then suppression of apreviously unseenm = 1 mode).

Another feature that is due to either high supercrit-icality or geometric differences is the equatorial sym-metry of the instabilities. Hollerbach and Skinner findonly equatorially symmetric instabilities. Our exper-iments show both symmetric and anti-symmetric in-stabilities. More specifically, them = 1 instabilitycorresponds to anl = 2 (i.e. equatorially symmetric)pole-to-pole induced field correlation (Fig. 5a), andthe m = 2 to an l = 3 (anti-symmetric) correlation(Fig. 7b). Note that these correlations are not the sim-plest symmetric and anti-symmetric states possible.As seen inFig. 8 and as discussed above, the tran-sition to equatorial anti-symmetry is due to apparentundulations in columnar vortices.

Interestingly,Moreau (1990)speculates that suchundulating columnar vortices parallel with the appliedfield would form forN 1. This prediction camefrom a theoretical extension to much larger appliedfields for the experiment described inAlemany et al.(1979). The flow there was generated in a circular col-umn of mercury by driving a grid at constant velocity.They only reachedN ≈ 1 and did not observe thesevortices. For the largeN limit Moreau concluded thatthe columnar vortices would undulate in the directionof the applied field with the characteristic lengthl‖ ≈l⊥N1/2, wherel⊥ is a characteristic length transverseto the applied field. This contrasts with our findingthat as the interaction parameter is increased,l‖ de-


creases (as new zones are added) (Fig. 8). There is lit-tle change inl⊥ as that is slaved to the azimuthal wavenumberm. Indeed, in regimes III and IV, wherel‖ isdecreasing, the wave number (m = 2) remains con-stant. The resolution of our pole-to-pole array is toolow to more precisely determine howl‖ scales withN.

Although we have not directly measured the ve-locity field, it seems most likely that the oscillatoryinstabilities result from precessing, wavy, columnarvortices in the flow aligned with the applied field. Itis not clear, however, why the precession rate changesas it does with applied field (Fig. 7). Hollerbach andSkinner report precession rates for different appliedfields and Reynolds number, but for a given Reynoldsnumber they never report the precession rate for morethan one applied field value. Hollerbach and Skin-ner also find that, for a given applied field, higherazimuthal wave numbers precess faster. For exam-ple, for a field equivalent to 350 G and a Reynoldsnumber ofRe = 2000 they find that them = 3 so-lution precesses at 0.179Ω and them = 2 solutionat 0.15Ω. In contrast, we find in our parameter rangethat lowerm modes precess faster (seeFig. 7). Themagnitude of Hollerbach and Skinner’s precessionrates is comparable to ours.

Labbé et al. (1996)observe a similar phenomenonin a different system that may provide a clue to thischanging precession speed mechanism. They exper-imentally studied co-axial, co-rotating von Kármanflow in air and found precessing vortices near theedge of their impellers. The precession period wasfound to increase with separation distance of theirrotating disks. In other words, decreasing the lengthscale (set by the impeller separation) increases thevortex precession rate. In regimes III and IV, wefound a shortening length for the poloidal zones withincreasing interaction parameter. So it is possiblethis length-scale shortening leads to the increasedprecession rate. It would be interesting to vary theimpeller separation in our experiment and repeat themeasurements.

There is, however, a significant discrepancy in thedynamics of our system and Labbé et al.’s. In theirexperiment, the vortex precession is retrograde; in ourexperiment the precession, as measured at the equator(Section 4.3) and the array encircling one pole, is pro-grade. Their study, conducted using air, was of coursenon-magnetic, further suggesting that magnetic fields

play a crucial, non-trivial role in the dynamics of oursystem.

5.1. Effect of the baffles

With or without baffles the overall dynamics inthe system are quite similar. Note, for example, thesimilarity in the two cases inFigs. 3, 6, 9 and 10.However, there are four main effects produced by thebaffles. Most significant is how the torque changeswith applied field for a fixed rotation rate. With bafflesthe torque decreases with increasing applied field, andwithout baffles the torque increases. The magnitude ofthe torque decrease with baffles is larger than the in-crease without. Second, the baffles suppress them =1 oscillations that emerge in the intermediate range ofinteraction parameter. It appears this state is sensitiveto the boundaries. Thism = 1 instability begins atthe same applied field as the turbulence-suppressingregime II with baffles (seeFig. 9). Third, the rangeof N values for regime III is increased, beginning atsmallerN and ending at largerN (seeTable 3). Thedynamics in regime III, as described inSection 4.3,are postulated to result from precessing columnar vor-tices. Thus the baffles apparently suppress the forma-tion (or precession) of the vortices, and better lock thevortex precession rate to half the impeller rotation rate.The latter effect is evident inFig. 4, where the inducedfield frequency peak is one half for large interactionparameter with baffles but only approaches one halfwithout.

Finally, the baffles increase the magnitude of in-duced field oscillations in regime I. As seen inFigs. 9and 10, the slope ofσ versusBext is steeper withbaffles. The baffles either make a more turbulent flow(assumingBrms ∼ urms), or makes the flow moreefficiently generate induced fields (i.e.Rem is effec-tively made larger for a given rotation rate). Thiseffect of the baffles disappears at higher interactionparameter.

5.2. Application to the Earth and Sun

Core–mantle coupling—exchange of angular mo-mentum between the core and the mantle—is thoughtis to be responsible for variations in the Length of Day,and in the orientation of Earth’s axis of rotation. Thedegree of coupling is also an important parameter in


many dynamo models(Roberts, 1988). Coupling canoccur in a number of ways, including topographicallyor electromagnetically. These two processes are oftenconsidered separately (e.g.Jault et al., 1996; Aldridgeet al., 1990). However, our results show that Lorentzforces and topography need to be considered together,at least in our parameter range. Indeed, the magneticfield can either enhance or weaken angular momentumtransfer (≈50% over the range of interaction parameterin our experiment) to the outer case depending on thetopography of the vessel walls (seeFig. 11). In our ex-periment the angular momentum transfer, with bafflesand without, seems to converge at higher interactionparameter. This suggests that magnetic effects come todominate and “wash out” pure topographic couplingwhen the interaction parameter is greater than aboutN = 5. The estimated values of interaction parameterin the Earth are much larger than in our experiment;thus, the Earth operates either in regime IV or in somehigher regime of Lorentz force domination.

Maps of the radial component of the magneticfield for 1715–1980 at the surface of the Earth indi-cate structure having similarities to our data. In thatinterval, the radial component of the Earth’s fieldshows a fixed pattern of four flux concentrations(lobes) placed anti-symmetrically about the equator(Gubbins and Bloxham, 1987). Similar to regimesIII and IV in our experiment, these lobes are placedanti-symmetrically about the equator. Paleomagneticevidence, which samples the Earth’s field on longertime intervals, suggests that these lobes do drift,though more slowly than the characteristic westwarddrift velocity (Constable et al., 2000). In our experi-ment, magnetic structures also drift slowly comparedto the characteristic velocity (i.e. the impeller rotationrate). The relatively slowly moving flux pattern in theEarth is commonly thought to be an effect of heatflux inhomogeneities at the core–mantle boundary(Bloxham and Gubbins, 1987). However, our experi-ment, having no such inhomogeneities, suggests thismotion could be a more robust dynamical feature.Indeed, one might guess that global rotation or the dif-ferent forcing of the flow in the Earth would frustratethe instabilities seen in our experiment. Future exper-iments will determine how robust these instabilitiesare.

There is a length scale for which the flow withinthe core has the same interaction parameter as our

experiments. This is approximately 10−5 the radiusof the core, or about 30 m. One obvious locationwhere the dynamics at this scale could affect geomag-netic observables are the core–mantle and inner coreboundary layers. The boundary layers (thermal andEkman) act as valves for both heat and momentumflux through the outer core. Our observations of theeffects of Lorentz forces and roughness (baffles) onmomentum transport may apply to these boundarylayers. In addition to the effects on angular mo-mentum transport discussed above, there should beLorentz force effects on heat flux and core cooling.It has been argued that the turbulent momentum andheat flux are slaved together (by use of the Reynoldsanalogy, i.e. a unity turbulent Prandtl number)(Kaysand Crawford, 1980). Using this concept and our ob-servations of torque dependence one might expect thatthe heat flux from the rough core–mantle boundary isreduced when the dynamo first turns on. This mightact as one saturation mechanism for the geodynamo.Future experiments are needed to directly probe theeffects of Lorentz force on heat flux.

On this note, we recognize that our experiment isnot an ideal geophysical model. One obvious short-coming, as previously mentioned, is that our sphereis non-rotating—the fluid only rotates due to theco-rotating impellers. We are thus not in a low Rossbynumber regime as the Earth and most astrophysicalobjects are. Furthermore, a system driven by impellersis simpler than a system driven by convection, suchas the Earth. The experimental compromises in thisexperiment were necessary, however, due to the diffi-culty and safety concerns in performing these types ofexperiments. Rotating our sphere would cause safetyproblems associated with decoupling the system froman overflow reservoir. Also, without impellers wewould be unable to reach the magnetic Reynoldsnumbers reached in this experiment (without amuchlarger device), as convection causes weaker fluidvelocities. Experiments with a new 60 cm diameterrotating convection apparatus are underway that mayilluminate the effects of these simplifications.

The parameter ranges for the Sun come closer tooverlapping with our experiment, though there aresignificant differences (e.g. the sun is plasma, and hasa stratified structure). Unfortunately, uncertainties indiffusivity, field strength, and the relevant length scaleyield a large range for estimates of the interaction


parameter for the Sun 1< N < 104 (Fisher et al.,2000; Gough and McIntyre, 1998; Charbonneau andMacGregor, 1997; Moffatt, 1978; Christensen-Dalsgaard et al., 1996; Gilman, 2000). The Rossbynumber of the Sun isRo ∼ 10−1, four-orders ofmagnitude larger than in the Earth. It is possible thatthe relatively stronger effect of advection relative torotation in the Sun partially explains why the Sun hasmore periodic dynamics than the Earth.

6. Conclusion

We have observed the effect of Lorentz forces onturbulent flow by applying large external fields to aflow in liquid sodium. An interaction parameter ofN = 17 has been reached for magnetic Reynolds num-ber up toRem = 18.

Our observations indicate five regimes, which re-flect changes in the dynamics as Lorentz forces cometo dominate. As the applied field increases, the turbu-lence is suppressed and the system undergoes first anm = 1 instability (with smooth walls only) and thenanm = 2 instability. From magnetic data and compar-ison toHollerbach and Skinner (2001), we infer thatthe instabilities consist of precessing columnar vor-tices aligned with the field (we have no direct velocityor vorticity data). The induced field patterns are di-vided into zones at different polar angles. The numberof zones grows with increasing interaction parameter.The precession speed of these patterns increases withimposed field in regime III, and then remains approxi-mately constant with imposed field in regime IV. Thisincreasing precession speed may result from the short-ening of the poloidal wavelength of the inferred vor-tices, possibly related to the mechanism described inLabbé et al. (1996). When this wavelength is compa-rable to the sphere radius, equatorially anti-symmetricstates result.

The torque needed to co-rotate the impellerschanges with interaction parameter, and dependsnon-trivially on the topography of the spherical ves-sel. These results might be useful in future studies ofcore–mantle coupling. Our results also suggest thatin the absence of global rotation, at large interactionparameter, core–mantle coupling is insensitive to themantle topography.

Acknowledgements

This work was funded by the NSF Grants EAR-0116129, EAR-0207789, and DMR-9896037. Wegratefully acknowledge assistance from AlfredCawthorne, James Drake, Khurrum Gillani, DonaldMartin, Edward Ott, Nicholas Peffley, John Rodgers,James Stone, Santiago Triana, and Daniel Zimmer-man. We benefited greatly from helpful comments bytwo anonymous referees. D.P. Lathrop is a CottrellScholar of the Research Corporation.

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Lorentz force effects in magneto-turbulence

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