Bi-stability in turbulent, rotating spherical Couette flowDaniel S. Zimmerman,1, a) Santiago Andres Triana,1, b) and D. P. Lathrop2, c)

1)Department of Physics,Institute for Research in Electronics and Applied Physics, University of Maryland, College Park,MD 207422)Department of Physics, Department of GeologyInstitute for Research in Electronics and Applied Physics, University of Maryland, College Park,MD 20742

(Dated: 6 November 2013)

Flow between concentric spheres of radius ratio η = ri/ro = 0.35 is studied in a 3 m outer diameter experiment.We have measured the torques required to maintain constant boundary speeds as well as localized wallshear stress, velocity, and pressure. At low Ekman number E = 2.1 × 10−7 and modest Rossby number0.07 < Ro < 3.4, the resulting flow is highly turbulent, with a Reynolds number (Re = Ro/E) exceedingfifteen million. Several turbulent flow regimes are evident as Ro is varied for fixed E. We focus our attentionon one flow transition in particular, between Ro = 1.8 and Ro = 2.6, where the flow shows bistable behavior.For Ro within this range, the flow undergoes intermittent transitions between the states observed aloneat adjacent Ro outside the switching range. The two states are clearly distinguished in all measured flowquantities, including a striking reduction in torque demanded from the inner sphere by the state lying athigher Ro. The reduced angular momentum transport appears to be associated with the development of afast zonal circulation near the experiment core. The lower torque state exhibits waves, one of which is similarto an inertial mode known for a full sphere, and another which appears to be a strongly advected Rossby-typewave. These results represent a new laboratory example of the overlapping existence of distinct flow states inhigh Reynolds number flow. Turbulent multiple stability and the resilience of transport barriers associatedwith zonal flows are important topics in geophysical and astrophysical contexts.

PACS numbers: 47.27.-i,47.27T,47.32Ef

I. INTRODUCTION

An understanding of rapidly rotating turbulent flow iskey to many problems of geophysical and astrophysicalfluid dynamics. The large scale fluid motions of starsas well as planetary atmospheres, oceans, and cores arestrongly influenced by the Coriolis accelerations arisingfrom the rapid overall rotation in those systems. Theseflows can be described by the Navier-Stokes equationwritten for a frame rotating with constant angular ve-locity Ω = Ωz, which in dimensionless form is

∂u

∂t+Ro u · ∇u + 2z × u = −∇p+ E ∇2u. (1)

Using U as the characteristic velocity scale, ` as thecharacteristic length scale, and the kinematic viscosityν, we define the relevant dimensionless parameters. TheRossby number,

Ro =U

Ω`, (2)

a)email address: danzimmerman@gmail.comb)email address: triana@umd.educ)email address: lathrop@umd.edu

expresses the strength of the nonlinear term relative tothe Coriolis term 2z × u, and the Ekman number,

E =ν

(Ω`2), (3)

characterizes the importance of viscous drag relative toCoriolis acceleration. Throughout this paper we use, ` =ro − ri, Ω = Ωo, and U = (Ωi − Ωo)`, so that

Ro =∆Ω

Ωo(4)

and

E =ν

Ωo`2. (5)

The Taylor-Proudman constraint on rotating flows of-ten holds approximately: overall rotation about the z-axis leads to a tendency toward z-independence of theflow for large scales or slow motions. This is useful fornearly steady flow where (u ·∇)u and ∂u/∂t in Eq. 1 aresmall with respect to 2z × u. Due to the relatively largesize of this apparatus and practical limitations, all of ourflows are turbulent. Thus the Taylor-Proudman theoremis unlikely to hold, though it is expected that significantanisotropy will remain. When accelerations are signifi-cant, retaining time dependence but neglecting the non-linear term in Eq. 1 yields dynamics where flow distur-bances can propagate via linear Coriolis-restored inertialwaves1. In containers, wave modes arise. Modes for the

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interior of a sphere are treated by Greenspan1. Zhanget al.2 provide complete explicit analytical solutions forthe full sphere. The modes in a spherical annulus are notknown analytically but have been investigated numeri-cally3–5. Inertial waves and related Rossby waves playa role in the inverse energy cascade to large scales ob-served in rotating turbulence6. Experiments have shownthat inertial modes are important in turbulent sphericalCouette flow7–10, and in more general turbulent flowslike rotating grid turbulence11. We expect that iner-tial modes have substantial influence on the characterof the flow in rapidly rotating bounded systems for mo-tions with frequencies less than twice the rotation rate.Above that frequency, no inertial waves or modes exist.The question of completeness of inertial modes is still anopen one12, so it may not be possible to express arbitrarymotions with frequencies below ω = 2 Ωo in terms of in-ertial mode Fourier components. However,the nonlinearinteractions of a sea of modes with ω ≤ 2 Ωo may have astrong signature on a wide range of spatial and temporalscales.

Turbulent flow in a rapidly rotating spherical annu-lus with a radius ratio η = ri/ro = 0.35 has poten-tial geophysical relevance due to geometrical similarityto Earth’s liquid outer core. Differential rotation im-posed by the boundaries is at first glance considerablydifferent from the convection, precession, and tidal forc-ing that may drive flows in planetary systems. However,convection in rotating systems tends to set up differentialrotation13–18, as does the nonlinear interaction of inertialmode shear layers in spherical shells driven by precessionor tidal deformation19,20.

The study of spherical Couette flow with rapid outersphere rotation has been fairly limited. Experiments andtheoretical studies have focused more on the case withthe outer sphere stationary21–29. In experiments, thisis possibly due to of the difficulty of conducting mea-surements in the rotating frame. Furthermore, there isa complicated dependence of the observed laminar flowon Reynolds number and gap width even when the outersphere is fixed21,22. Nevertheless, some work in spheri-cal Couette with overall rotation has been carried out.Hollerbach, Egbers, Futterer, and More30 teamed ex-perimental observations with numerical simulations tostudy Stewartson layer instabilities in the case of smallto modest differential rotation. They found good agree-ment between simulation and observed visual patternsin experiments for parameters for E > 8 × 10−4 and0 < Ro < 0.6. Experiments by Gertsenshtein, Zhilenko,and Krivonosova31 investigated the transition to turbu-lence at some values of Ro, especially when Ro < 0.Schaeffer and Cardin32 studied the onset of Stewartsonlayer instabilities at E similar to our experiment at verylow Ro where the first instability happens.

The strongly turbulent behavior of spherical Couetteflow has been studied mostly in hydromagnetic apparatuswith liquid sodium as a working fluid. Sisan et al.24 stud-ied instabilities of a turbulent flow of sodium that arise

with a sufficiently strong axial magnetic field, with somehydrodynamic measurements to characterize the initialunmagnetized flow. Kelley et al.7,8 inferred the flow ina 60 cm rapidly rotating sodium apparatus with a mag-netic field sensor array and a dynamically passive appliedfield. These experiments demonstrated over-reflectionalexcitation of inertial modes at comparable Ekman num-ber to ours and −2 < Ro < 0. Rapidly rotating sphericalCouette flow of sodium strongly magnetized by a dipolepermanent magnet inner sphere has been studied in theDTS experiment10,33,34 in Grenoble. Measurements ofvelocity, magnetic field, and electric potential measure-ments exposed a number of interesting hydromagneticstates in the turbulent regime10,33,34.

At present, numerical work on spherical Couette flowhas not investigated the portion of the Ro, E parameterplane where we find turbulent bi-stability. However, nu-merical and theoretical work has demonstrated interest-ing phenomena with outer sphere rotation. Stewartson35,studied the case at infinitesimal Rossby number, deriv-ing the form of the free cylindrical shear layer tangentto the inner sphere equator that still bears his name. Asystematic three-dimensional numerical study of instabil-ities of the Stewartson was undertaken by Hollerbach36.Schaeffer and Cardin9,32 used depth-averaged equationsof motion coupled to realistic Ekman pumping to con-duct quasi-geostrophic simulations at low Ekman numberfor a split-sphere geometry similar to spherical Couetteflow. Schaeffer and Cardin32 showed good agreement be-tween experiment and simulation, predicting Stewartsonlayer instabilities correctly for very low Rossby number.Quasi-geostrophic simulations9 found Rossby wave tur-bulence at higher Ro.

Our experiments can not generally achieve Ro < 0.05at experimentally accessible Ekman number because oflimitations on motor minimum speeds. The lowest possi-ble Rossby number increases if we rotate the system moreslowly to raise E. It is impractical to reduce the Rossbynumber or increase the Ekman number enough to matchthe parameters of known numerical simulations.

Guervilly and Cardin37 performed simulations ofspherical Couette flow with outer rotation in an inves-tigation of magnetic dynamo action. They performedfully three dimensional numerical simulations for Ekmannumber higher than E = 10−4 while achieving Rossbynumber matching some presented here. The definitionof the Rossby number used by Guervilly and Cardin37

is different, RoGC = η∆Ω/Ωo = ηRo. They report anm = 2 Rossby type wave for Ro ∼ 2.9 (RoGC = 1),above our first bistable range. The existence and az-imuthal wavenumber of this wave is partially consistentwith our observations at that Rossby number. However,we observe strong turbulence, additional waves, and thebi-stable behavior that is the focus of this paper. Thesehave not been previously reported in spherical Couetteflow.

Turbulent flow transitions and multiple stability invery high Reynolds number flows have been reported

3

in a number of systems. Several examples exist in geo-physics and astrophysics. The dynamo generated mag-netic fields of the Earth and Sun reverse polarity. Oceancurrents, namely the Kuroshio current in the North Pa-cific near Japan and the Gulf Stream, both exhibit bi-stability in meander patterns.38 Polar vortices in Earth’sstratosphere are bounded by resilient transport barriersmuch of the time, mixing to higher latitudes only inter-mittently.39–41 This process is important to ozone deple-tion in polar regions and may have some similarities tothe dynamics we observe.

A number of laboratory flows are known to exhibitmulti-stability and hysteresis. The mean circulation inturbulent thermal convection cells has been observed toswitch direction abruptly.42 Hysteresis in the large scaleflow states has been seen in surface waves excited byturbulent swirling flows in a Taylor Couette geometrywith a free surface.43 Von Karman flow in a cylinder be-tween two independently rotating impellers has exhibitedmulti-stability and hysteresis of the mean flow despiteextremely high fluctuation levels.44–46 Magnetohydrody-namic experiments in the von Karman geometry havesucceeded in producing dynamos that show reversals ofthe generated magnetic field.47,48 The L-H transition inturbulent tokamak plasma confinement devices involvesthe formation of a wave-driven zonal flow transport bar-rier that greatly enhances confinement of the plasma.49,50

However, this barrier eventually breaks down in a burstthat can damage the confinement device.51 Understand-ing this particular form of turbulent multiple stabilityand its control is important issue in sustained confine-ment of fusion plasmas.

Spherical Couette flow is a dynamically rich systemin both laminar and turbulent regimes. Our appara-tus, initially designed, constructed, and eventually des-tined for magnetohydrodynamic experiments in moltensodium metal, presents a unique opportunity to measurethe properties of hydrodynamic turbulence in this geom-etry, including transition phenomena between multipleturbulent states. We operate in a novel region of pa-rameter space, simultaneously achieving moderately highRossby number and low Ekman number. This regime cancurrently only be directly accessed by experiments andnaturally occurring flows, and has not been the focus ofprevious purely hydrodynamic studies. Furthermore, weare able to make quantitative measurements in the rotat-ing frame, something that can be quite difficult in smallerapparatus.

II. APPARATUS

The three meter apparatus allows independent rota-tion of the inner and outer spheres. Instrumentationin the rotating frame allows measurements of velocity,wall shear and pressure, as well as the torques requiredto maintain the boundary speeds. Fig. 1 is a schematicsketch of the apparatus. The stainless steel outer vessel

1m

FIG. 1. A schematic of the apparatus showing the inner andouter sphere and locations of measurement ports in the vesseltop lid at 60 cm cylindrical radius (1.18 ri). Data from sensorsin the ports are acquired by instrumentation, including anacquisition computer, bolted to the rotating lid and wirelesslytransferred to the lab frame. Also shown is the wireless torquesensor on the inner shaft.

has an inner diameter of 2.92 ± 0.005 m and is 2.54 cmthick. It is mounted on a pair of spherical roller bearingsheld by a frame. The vessel top lid is installed in a 1.5 mdiameter cylindrical flanged opening, and the inside lidsurface is curved to complete the outer spherical bound-ary. The lid has four 13 cm diameter instrumentationports centered at 60 cm cylindrical radius. Due to designconstraints aimed at safe operation with liquid sodiummetal as the working fluid, these four ports are the onlypenetrations through the outer boundary, and so are theonly location from which we may make direct flow mea-surements. Port inserts hold measurement probes nearlyflush with the inner surface of the outer sphere.

The inner sphere has a diameter of 1.02±0.005 m and issupported on a 16.8 cm diameter shaft held coaxial withthe outer shell by bearings at the bottom of the outersphere and in the top lid. The inner sphere is driven froma 250 kW electrical motor through a calibrated Futekmodel TFF600 torsional load cell. The measured torqueincludes the torque from a pair of lip seals which addsome confounding error. The outer sphere is driven by a250 kW induction motor mounted to the support frame.A timing belt reduction drive with a 25:3 ratio couplesthe outer sphere motor to a toothed pulley on the lid rim.

Motor speeds are controlled to better than 0.2% byvariable frequency drives, and optical encoders monitorthe inner and outer sphere speeds. The drives estimate

4

the motor torque from electrical current measurementsand the torque estimate supplied by the inner motor driveagrees well with the calibrated strain-gauge torque sen-sor at motor speeds above about 2 Hz. This supportsthe use of the outer motor drive’s reported torque as areasonable estimate of the total torque required to drivethe outer sphere, provided that the outer sphere angu-lar speed is above about 0.24 Hz, as it is for the datapresented here. As the outer sphere speed is lowered be-low 0.24 Hz, the outer drive’s reported torque becomesincreasingly dominated by motor magnetizing current.The torque exerted by aerodynamic and bearing dragon the outer sphere is typically larger than the workingfluid’s contribution. However, this drag is repeatable andcan be subtracted off by measuring the torque demandedwith no differential rotation.

A rotating computer acquires data from sensors inthe instrumentation ports at a sampling rate of 512 Hz,recording data on a lab frame computer using a wire-less ethernet connection. Sensors include a Dantec model55R46 flush mount shear stress sensor driven by a TSImodel 1750 constant temperature anemometer and threeKistler model 211B5 pressure transducers. The threepressure transducers are installed in three ports 90 aparton the 60 cm radius port circle. A thermocouple is usedto monitor fluid temperature.

A Met-Flow UVP-DUO pulsed Doppler ultrasound ve-locimeter is mounted in the rotating frame and pairedwith Signal Processing transducers, also communicatingwith the lab frame using wireless ethernet. Some velocitydata in this paper was acquired with a Met-Flow UVP-X1-PS ultrasound unit in the lab frame using a resonanttransformer arrangement to couple the signal into the ro-tating frame. To scatter ultrasound, the flow is seededwith 150 µm− 250 µm polystyrene particles with nomi-nal density of 1.05 g/cm3. Limitations on the product ofmaximum measurable velocity and measurement depthconstrain the velocity measurements in this paper to bevery close to the wall, typically no more than 10 cm fromthe transducer face, which is either flush mounted withthe wall or intruding no more than 10 cm into the flow.Intrusive transducers do not seem to measurably changethe torque dynamics. When possible, we measure up-stream to avoid directly measuring the turbulent wakeshed from an intrusive mounting scheme.

III. TORQUE MEASUREMENTS

The torques required to maintain constant speeds ofthe boundaries of in a rotating flow provide a global pic-ture of angular momentum transport and power dissipa-tion. Experimental torque measurements on wall-drivenflows have been largely confined to the flow between con-centric cylinders52–57. The torque required to drive theflow between two spheres has received less attention, withprevious results in the hydrodynamic turbulent regimeseemingly limited to outer-sphere stationary measure-

ments of liquid sodium23. Following Lathrop et al.58, we

106 107

1011

1012

G A Re + B Re2

0 0.5 1 1.5 2 2.5 3 3.5

G

Ro0

14 x1010

10 x1010

6x1010

2x1010

H

L

LL

G

1x

1x (a)

(b)

Re

FIG. 2. (a) The dimensionless torque G vs. the Reynoldsnumber Re for the case Ωo = 0, stationary outer sphere, witha fit to G = ARe+BRe2 with A = 4.49× 104 and B = 0.05.(b) Mean value of the dimensionless torque G vs. Ro at E =2.1× 10−7. Ro is varied by increasing the inner sphere speedin steps of Ro = 0.067, waiting 450 rotations per step. Threecurves are differentiated by circles, triangles, and squares. Inthe ranges of Ro where the symbols overlap, flow exhibitsbistable behavior. H and L denote the torque curves of the“high torque” and “low torque” states. There is a secondbistable regime starting around Ro = 2.75, with LL labelingthe lower torque state.

define the dimensionless torque G on the inner sphere,

G =T

ρν2ri, (6)

where ρ is the fluid density, ν is the kinematic viscosity,ri is the inner sphere radius and T is the dimensionaltorque. The torque G as a function of Reynolds numberRe, here defined as

Re =(Ωi − Ωo)`2

ν, (7)

is bounded above by G ∝ Re2 from dimensional argu-ments. We fit the measured torque with the outer sphere

5

stationary to G ∝ Re2 with a correction for linearly in-creasing seal drag. Data and fit with the outer spherestationary are shown shown in Fig. 2(a) for the range ofRe accessible in this experiment. At the lowest rotationrates, the measurement is significantly confounded by thebearing and seal torque.

Figure 2(a) shows that G monotonically increases asRe is increased with the outer sphere stationary. This isnot necessarily the case when we impart global rotation,provided that we do not hold Ro constant. When wehold the outer sphere speed constant (constant Ekmannumber) and super-rotate the inner sphere (Ro > 0) asin Fig. 2(b), we find that the mean torque on the innersphere is non-monotonic as the dimensionless differentialspeed, Ro, and the Reynolds number, Re = Ro/E areconcurrently increased. As Ro is increased, the flow un-dergoes several transitions to significantly different tur-bulent flow states. In ranges of Ro in Fig. 2(b) wherethe torque data are presented in overlapping brancheswith different symbols, the flow exhibits bistable behav-ior with intermittent transitions between adjacent flowstates. In these ranges of Ro, we plot the mean torqueof each state, conditioned on state.

1000 2000 3000 4000 5000 6000

ot/2

G

0101

x

0101

x

0101

x

0101

x

0101

x

0101

x

0101

x6

7

8

9

10

11

12H H H H H H HL L L L L L L

FIG. 3. Time series of G at fixed Ro = 2.13 and E =2.1×10−7, with time made dimensionless by the outer sphererotation period. The raw torque signal has been numericallylow pass filtered (fc = 0.05 Hz, 15 rotations of the outersphere.).

The state switching means that the transitions arenot hysteretic, and long time mean torque through theswitching range decreases with increasing Ro. We chooseto plot the conditioned torque branches here instead ofthe long-time mean to emphasize our observation thatthese states seem the same as those present alone athigher and lower Ro. We will focus on the first bistableregime in this paper, between Ro = 1.8 and Ro = 2.75,and not the second bistable regime with lower state ”LL”that begins near Ro = 2.75. This second bistable regimehas several similarities to the first, but we will not discuss

it in detail in this paper.A representative time series of the torque in the first

bistable regime is shown in Fig. 3, with E = 2.1 × 10−7

and Ro = 2.13. The onset of the high torque state isabrupt, taking on the order of 10 rotations of the outersphere. The torque overshoots the high state mean valueat high torque onset by 10-15%.

The end of the high torque state exhibits a slow decayof the torque to the low torque value, approximately ex-ponential with a time constant of 40 rotations of the outersphere. In addition to the full transitions between the twotorque levels, there are “excursions” where the torque de-cays toward the low mean value or rises toward the highmean value without fully reaching the other state.

The qualitative bi-stability in Fig. 3 can be expressedquantitatively in the bimodal probability distribution ofthe torque shown in Fig. 4. The division of data into hightorque and low torque states was done manually so as toexclude the transition regions. The resulting individualdistributions of G for the high and low states based onthis conditioning are shown in Fig. 4, as well as the fulldistribution. There is a small region of overlap in theH and L state individual distributions due to the differ-ence drawn between “transitions” and “excursions.” Thesame manual division in states is used throughout thepaper to condition other data on state. In Fig. 4, thehigh state mean torque is 1.4 times that in the low state.The torque fluctuations in the high torque state are con-siderably higher than in the low; the standard deviationof the high state torque is 1.8 times that in the low state.The low frequency fluid fluctuations responsible for thisare also observed in the velocity and wall shear.

In both Fig. 3 and Fig. 4 the torque data has been nu-merically low pass filtered. In Fig. 3 the cutoff frequencyis 0.05 Hz. In Fig. 4, the filter cutoff frequency was cho-sen to be 0.5 Hz, where the fluid torque power spectrumappears to cross the mechanical vibration noise floor. Inthis way we retain the fastest measurable hydrodynami-cally relevant fluctuations.

TABLE I. Statistics of the interval between high torque on-sets. ∆t′H is the time interval between two subsequent hightorque onsets made dimensionless by Ωo/2π, so the time in-terval is measured in outer sphere rotations. Ro = 2.13,E = 2.1× 10−7.

〈∆t′H〉 σ∆t′HMax(∆t′H) Min(∆t′H)

717 313 1917 390

The interval between transitions is somewhat irregu-lar. Over 45 transitions at the parameters in Fig. 3, weobserve the statistics shown in Table I. We also observethat the probability that the system is in one state or theother depends on Ro. Above a threshold value of Ro, webegin to observe state transitions to the low state, andthe high torque state becomes less likely as Ro increases.Fig. 5 shows the probability that the system is in thehigh or low torque state for 4000 rotations of the outer

6

10-13

10-12

10-11

10-10

G

Pr(G)

6 8 10 121010

x1010

x 1010

x 1010

x

H

L

FIG. 4. Probability density of the dimensionless torque atRo = 2.13, E = 2.1 × 10−7. The full, unconditioned dis-tribution is denoted by small points. Solid circles denoteconditioning on low torque state, and open circles on thehigh, with Gaussian solid and dashed curves for the lowand the high respectively. The mean and standard devi-ation of the low torque state data are 〈G〉 = 6.82 × 1010

and σG = 2.74 × 109. In the high torque state they are〈G〉 = 9.53 × 1010 and σG = 5.02 × 109. The data were lowpass filtered at fc = 0.5 Hz to remove high frequency noisecaused by mechanical vibration.

sphere across the first bistable range of Ro. At valuesof Ro where transitions were not observed for more than4000 rotations of the outer sphere, a probability of oneor zero was assigned. The probability that the systemwas in the low state was fit to

Pr(L) =

0 : Ro < Roc

1− exp(−γ(Ro−RocRoc)) : Ro > Roc

(8)

with γ = 8.25 and Roc = 1.80. The high torque prob-ability is given by Pr(H) = 1-Pr(L). The physical im-plication inherent in the exponential form is that thereis no upper threshold where the high torque state be-comes impossible. Instead, it only becomes less likely asRo is increased. However, the lower threshold for statetransitions is well defined at Roc.

IV. FLUID ANGULAR MOMENTUM

The torque measurements presented so far only con-sidered the torque on the inner sphere. To see the com-plete picture of the angular momentum transport in thesystem, we examine the torque on both boundaries. Thetorque on the outer sphere is reported by the motor drive,though with less precision than the inner torque sensormeasurement.

The fluid filling the gap cannot undergo angular ac-celeration or deceleration for arbitrarily long times withthe boundaries rotating at constant speeds. Therefore,

1.6 1.8 2 2.2 2.4 2.6

0

0.2

0.4

0.6

0.8

1

Ro

Pr(H,L)

L

H

FIG. 5. Probability that the flow was in the high torque (opencircles) or low torque (closed circles) state as a function ofRo over the first bistable range with fixed E = 2.1 × 10−7.The dashed and solid lines are fits to the exponential form ofEq. (8).

the time averaged net torque on the fluid must be zero,and the boundary torques must be equal and opposite,provided averaging is done over sufficiently long times.In the bistable regime, however, the net torque, shown inFig. 6, reveals long periods of angular acceleration anddeceleration interspersed with plateaus where the angu-lar momentum of the fluid remains nearly constant. Thetorques on the inner and outer boundaries only balanceon averaging over many state transitions. We define thenet torque on the fluid as

Tnet = T + To, (9)

where T and To are the torques about the axis that thefluid exerts on the inner and outer spheres respectively.To is always negative when Ro > 0, as the fluid wouldtend to speed up the outer sphere rather than resist itsmotion. The total torque on the outer sphere as reportedby the outer sphere motor drive, To,meas, is largely dueto drag from the outer sphere bearings and air aroundthe outer sphere. However, it is reproducible, and wedefine the torque that the interior fluid exerts on theouter sphere To in Eq. 9, as

To = To,meas − Tdrag. (10)

The drag, Tdrag is the outer drive torque required to keepthe experiment in solid body rotation (with inner lockedto outer) at the same outer sphere angular speed. In theRo > 0 flow states, some of the total torque required tomaintain the speed of the outer shell against bearing andaerodynamic drag is supplied by the inner sphere via thefluid, and this is easily detected.

The net torque is shown in Fig. 6 along with the sepa-rate inner and outer torques with the steady outer sphere

7

bearing and aerodynamic drag subtracted. The dimen-sionless torques are defined as in Eq. 6, for exampleGnet = Tnet/ρν

2ri. Up to a constant of integration, wecan calculate the fluid angular momentum L(t) about therotation axis from the net torque,

L(t) =

∫ t

0

Tnet(t′)dt′. (11)

Due to measurement limitations, we cannot integrate thetorques from motor startup to fix the constant of inte-gration. Instead, we set the initial value of L to zeroat an arbitrary time and make the resulting quantity di-mensionless by dividing it by the angular momentum thefluid would have if it were in solid body rotation with theouter sphere,

L′ =L

IfluidΩo. (12)

The moment of inertia of the fluid filling the gap is

Ifluid =8πρ

15(r5

o − r5i ), (13)

with a value of (1.14± 0.02)× 104 kg m2.

At the onset of the high torque state, as shown inFig. 6(a), there is some prompt response of the outersphere torque, indicating a certain amount of increasedangular momentum transport. However, the increase inthe torque on the outer sphere is insufficient to fully op-pose the increased inner sphere torque. At this point,the net torque becomes steadily positive, and the fluidaccelerates. As this happens, the fluid torque Go onthe outer sphere tends to slowly decrease in magnitude,though with large fluctuations. Eventually the innersphere torque G starts the slower transition to the lowtorque state. At a point during the high to low transition,as shown in Fig. 6(b), the net torque becomes negativeand the total angular momentum starts to decrease. Thetorque on the outer sphere continues a slow decay towarda value opposite and equal to that on the inner sphere,occasionally reaching a net torque fluctuating about zeroas in times after Fig. 6(c).

When both spheres rotate the angular momentum fluc-tuations in the range of Ro considered here may indi-cate a slow “store and release” process where long last-ing imbalances in the torques on the inner and outerspheres lead to fluid spin up and spin down. We notethat the average magnitude of torque is similar betweenthe outer stationary and the outer rotating cases for thesame Reynolds number in Fig. 2(a) and Fig. 2(b). Inboth cases the mean torque on the inner sphere is be-tween G = 1010 and G = 1011. Although the dynamicsof the angular momentum transport seem quite different,the magnitude of the transport has not changed drasti-cally.

L

-0.9

0

0.9

1.8

x10-2

x10-2

x10-2

ot/2

6000 6500 7000 7500 8000

0

10 x1010

-10 x1010

G

Gnet

Go

a b c

8500

FIG. 6. The angular momentum, inner torque, outer torque,and net torque. Ro = 2.13, E = 2.1 × 10−7. The upper plotshows the dimensionless angular momentum L′ as defined inEq. (12). The lower plot shows the inner torque G, the outertorque Go, and their sum Gnet. The bearing and aerodynamicdrag on the outer sphere have been subtracted off, and thetorques have been low pass filtered as in Fig. 3.

V. MEAN FLOW MEASUREMENTS

In the bistable regime, all measured flow quantities un-dergo transition along with the inner torque. Fig. 7 showsthat the time averaged dimensionless wall shear τ ′w andmeasured dimensionless azimuthal velocity u′ decreasesharply the onset of the high torque state.

In Fig. 7, the velocity was measured in a shallow rangenear the surface of the outer sphere at 23.5 colatitude(60 cm cylindrical radius). This location is about 9 cmoutside a vertical cylinder tangent to the inner sphereequator (called simply the “tangent cylinder” from hereon). In Fig. 7, the transducer beam was in a plane normalto the cylindrical radius at the port and was inclined23.5 from pointing straight down (23.5 inclined fromnegative z). Since the transducer responds only to thevelocity component along the beam axis, this orientationmade it sensitive to the cylindrical radial, azimuthal, andvertical velocity components, us, uφ, and uz.

However, by using a remotely controlled rotatablemount, we determined that the time-averaged measuredvelocity is dominated by the azimuthal component, uφ.When the transducer is rotated 90 from the usual orien-tation, making it least sensitive to uφ, the mean velocitywas about 10% of that seen when it maximally respondsto the azimuthal flow. This indicates a meridional circu-

8

lation of some importance. But it also means that themeridional circulation provides a small contribution tothe mean measured velocity when the transducer is ori-ented to be most sensitive to azimuthal flow. This is thecase in Fig. 7, and so we treat the measured mean ve-locity as entirely azimuthal and correct for the verticalinclination, so that:

〈uφ〉t =〈umeas〉tsin 23.5

. (14)

We then divide by the outer sphere equatorial tangentialvelocity to make 〈uφ〉t dimensionless:

〈u′〉t =〈uφ〉tΩoro

. (15)

This non-dimensionalization means that u′ can be inter-preted as a local Rossby number.

The wall shear stress sensor was calibrated against themeasured torque T on the inner sphere with the outersphere stationary. We assumed that the mean wall shearτw on the outer sphere was

τw =T

4πr3o cos 23.5

. (16)

We fit the bridge voltage V and mean wall shear calcu-lated from the measured torque by Eq. (16) to

V 2 = Aτ2/3w +Bτ1/3

w + C, (17)

as was done by Lathrop et al.52 We then used the cali-bration coefficients A,B and C to calculate the wall shearstress τw from the measured bridge voltage for all subse-quent data with the outer sphere rotating.

In the torque switching regimes, the slow fluctuationsin wall shear and the mean azimuthal velocity shown inFig. 7 are both strongly anti-correlated with the innersphere torque. The simultaneous wall shear and torquedata suggest that the two different states have differentlatitudinal distribution of the shear stress on the outersphere. Fig. 6 shows that the torque the fluid exerts onthe whole outer sphere is indeed somewhat greater in thehigh torque state. However, the mean shear stress at themeasurement location in the high torque state is 65% ofthat measured in the low torque state (see also Fig. 8(b)),implying a shear stress concentration at high latitudes inthe low torque state relative to the distribution of shearstress in the high torque state.

These shear and velocity measurements suggest a fastcentral zonal flow in the low torque state that ceases sud-denly at the high torque onset. When the fluid around,above, and below the inner sphere is circulating fasterin the low torque state, there is less drag on the in-ner sphere. The unusual aspect of this is that it takesless torque and less power input to maintain the fastercirculation. This could indicate a transport barrier toenergy and angular momentum in the low torque state.This change in transport is also important in interpret-ing the observation that the total angular momentum

2000 3000 4000 5000

G

u′

−0.1

0

0.1

0.2

ot/2

0.03

0.06

0.09

w

y

11 0101

x

9 0101

x

7 0101

x

10

14

18

0101

x

101

x 0

101

x 0

< >t

FIG. 7. Simultaneous time series of velocity, wall shear stress,and torque. A space-time diagram of the low pass filteredvelocity u′ (defined in Eq. 15) is shown at the top. This mea-surement is dominated by the azimuthal velocity uφ. Thevelocity u′ is made dimensionless by the outer sphere tan-gential velocity, and so can be interpreted as a locally mea-sured Rossby number. The dimensionless wall shear stressτ ′w = 4πr3

oτw/(ρν2ri) is shown in the middle, and the di-

mensionless torque G is shown at the bottom. The wall shearstress and torque have been low pass filtered with fc = 0.05Hzas before, which is comparable to the time averaging of thevelocimetry.

is often decreasing in the low torque state (See Fig. 6,line (b)), while the measured circulation is increasing orsteady. This is only possible with a change in shape ofthe angular momentum profile as a function of cylindricalradius. An angular momentum transport barrier couldexplain this, possibly one associated with a fast zonalflow17,39–41,59–62. The anti-correlated torque and flowmeasurements will be discussed more in Sec. VIII.

VI. TURBULENT FLOW FLUCTUATIONS

The turbulent fluctuations are significantly different inthe two flow states. The measured velocity fluctuationsin the high torque state are larger than those in the lowtorque state, despite the lower mean velocity. Fig. 8(a)shows the probability density of the dimensionless veloc-ity conditioned on the low and high torque states, withGaussian curves for comparison. The mean azimuthalvelocity measured in the low torque state is 2.45 timesthat seen in the high torque state (in agreement withFig. 7), while the fluctuations in the high torque stateare 1.5 times the low torque state fluctuations. Based onthis measurement, the high torque state has a consider-ably higher turbulence intensity σu′/〈u′〉, of 57%. Thelow torque state turbulence intensity is 16%.

The wall shear stress fluctuations are similar in magni-tude between the two states, as shown in Fig. 8(b). Thelow torque state mean wall shear is about 1.5 times that

9

-0.1 0 0.1 0.2 0.3 0.4

10-2

100

u

LH

5 10 15 20 25

10 6

10 4

10 2

Pr(w

)

w

x1010

x1010

x1010

x1010

x1010

LH

Pr( )u

-

-

-

(a)

(b)

FIG. 8. (a) The probability density of the dimensionless ve-locity conditioned on the torque state, with solid circles de-noting the low torque state and open circles denoting the high.Dashed and solid lines are Gaussian. Standard deviations areσu′ = 0.043 for the high state, σu′ = 0.029 for the low state.The high state mean is 〈u′〉 = 0.076, and the low state meanis 〈u′〉 = 0.186. The means are dominated by azimuthal ve-locity, though the transducers are equally sensitive to uz anduφ in this measurement. (b) The probability density of thedimensionless wall shear conditioned on torque. Solid circlesagain denote the low torque state, and open circles the high,with solid and dashed Gaussian curves. The mean and stan-dard deviation in the low torque state are 〈τ ′w〉 = 1.63× 1011

and στ ′w = 1.81 × 1010. In the high torque state they are

〈τ ′w〉 = 1.06× 1011 and στ ′w = 1.71× 1010.

in the high torque state, and the standard deviationsin the two states agree to within 5%. The high torquestate wall shear distribution is significantly skewed, witha skewness of about 0.6.

VII. WAVES

Some of the most energetic fluctuations in the lowtorque state are system-scale wave motions with well de-fined frequencies. The high torque state has more broad-band, low frequency fluctuations. The conditional wall

shear stress power spectra of Fig. 9 show this clearly.Two wave motions feature prominently in the low torquestate but peaks are nearly absent in the high torque state.The lower peak in the low state spectrum of Fig. 9 hasa frequency of 0.18 Ωo. Measurements with the multi-ple pressure sensors show that this wave has azimuthalwavenumber m = 1. The higher peak has a frequency of0.71 Ωo at this Rossby number, and several higher har-monics are also visible. This higher frequency wave ap-pears to have m = 2.

The frequency and azimuthal wavenumber of the lowerfrequency wave are consistent with the full sphere iner-tial mode2 (3,1,-0.1766) in the (l,m,ω/Ωo) notation ofGreenspan1 and Kelley et al.7. The inertial modes arenot known analytically for the spherical shell3–5, butthere is good experimental agreement in frequencies andspatial patterns between inertial modes observed in tur-bulent spherical Couette flow7 for Ro < 0 and modesof the full sphere2. Like the inertial modes observedpreviously7, the dependence of this mode’s frequency onthe Rossby number is weak. The spectrogram of Fig. 10shows the variation of pressure power spectra from a sin-gle pressure sensor as Ro is varied and E held constant.Line (a) in Fig. 10 is at 0.16 Ωo. The strong lower fre-quency peak starts higher, 0.18 Ωo, and varies down to0.14 Ωo and back up to 0.16 Ωo as Ro is increased. Thispeak is featured in flow power spectra over a wide range,not disappearing until about Ro = 12. It is worth men-tioning that (3, 1,−0.1766) inertial mode of the sphere isone of a special class of slow, geostrophic inertial modesthat are equivalent to Rossby waves propagating on asolid body background63.

10-1

100

101

102

10-8

10-6

10-4

10-2

PSD

L

H

o

FIG. 9. Power spectra of wall shear stress at Ro = 2.13 andE = 2.1 × 10−7, conditioned on state. Angular frequencyhas been made dimensionless using the outer sphere angularspeed. The black curve is the spectrum from the low torquestate, and the gray curve is that of the high torque state. Thelow torque spectrum has prominent peaks at ω/Ωo = 0.18,0.71, and harmonics. In the high torque state there are broadpeaks at ω/Ωo = 0.40 and 0.53.

10

5.00

5.0

1

5.1

2

oR

o

1.0 2.01.5 2.5 3.5 4.03.0

7-6-5-4-3-2-1-01log(PSD)

(b)

H

L

LL

(a)

FIG. 10. A spectrogram of wall pressure at 23.5 colatitudeshows evidence of several flow transitions as Ro is varied.E = 1.3 × 10−7 and 0.1 ≤Ro≤ 4.4, waiting 430 rotationsper step with steps of Ro = 0.1. Instead of averaging thepower spectra over an entire step of Ro, there are 30 spectraper step in Ro, so some temporal evolution is visible at agiven Ro. In the L state, there are two strong waves, thelower, at (a), varies only slightly in frequency with Ro. Thehigher frequency wave at (b) varies more strongly with Ro,suggesting that advection by the mean flow is important insetting its frequency.

The frequency of the stronger, higher frequency m = 2wave varies more strongly with the Rossby number. Theequation for Fig. 10 line (b) is

ω

Ωo= 0.25Ro+ 0.16. (18)

This variation with Ro suggests a Rossby wave that isdoppler shifted by advection. Rossby waves propagatewhere there is a gradient of potential vorticity, whichin an isothermal, incompressible fluid is the quotient offluid absolute vorticity and fluid column height. In theβ-plane approximation common for Rossby waves64–68,the dimensionless frequency of the waves is given by

ω

Ωo= k

U

Ωo− β

kΩo, (19)

where k is the wavenumber and U is the velocity of amean flow advecting the waves. The parameter β is re-lated to the background potential vorticity gradient onwhich the waves propagate. In our system, this gradientcould result from the topographical effect of the slop-ing boundaries9,32 or from a gradient of relative vorticitycaused by the mean azimuthal flow profile. We observethat the zonal flow velocity U varies linearly with theRossby number Ro in the low torque state. Outside thetangent cylinder with the inner sphere super-rotating,both the topographical contribution to β and the prob-able contribution from the mean relative vorticity would

make it negative, consistent with the positive intercept ofEq. 18. Therefore Eq. 19 and Eq. 18 are indeed similar,and suggest an advected Rossby-type wave.

At E = 10−3 and Ro = 2.9, in the low torque staterange, Guervilly and Cardin observe an m = 2 Rossby-type wave37. They do not report an additional m = 1wave. However, if the physical mechanism that gives riseto the m = 2 mode there is robust to lowering the Ek-man number from 10−3 to 10−7, it could be that them = 1 wave is excited by the m = 2 advected wavewhen the inertial mode damping is low enough. Reynoldsstresses from nonlinear waves in rotating systems cantransport angular momentum by driving strong zonalflows18–20,69,70, and the strength of these zonal flows com-pared to the wave amplitude grows rapidly as the Ekmannumber is decreased19. Therefore, the waves could playan important role in the angular momentum dynamicswe observe.

VIII. DISCUSSION

One possibility for the state transitions discussed hereis the formation and destruction of a fast zonal circula-tion at small cylindrical radius, with a resilient barrier totransport in the low torque state. Such barriers are com-mon features in rotating turbulent flows17,39–41,59–61,71.The fast mean flow and strong wall shear stress at highlatitude in the low torque state, along with the fallingtotal angular momentum of Fig. 6 mean that the twostates must have a different shape of the profile of an-gular momentum with cylindrical radius (s2Ω(s)). Thelow torque state flow must favor, on average, fast circula-tion at the center of the experiment and slower at largerradius. A simple possibility would be a central zonal cir-culation bounded by a sharp shear layer modulated bythe advected waves discussed in Sec. VII. This is shownschematically in Fig. 11(a). A collapse of this shear layercould cause the abrupt low state to high state transition,suddenly surrounding the inner sphere with low angu-lar velocity fluid and increasing the torque. The veryshort timescale of the L to H transition, 10 rotations ofthe outer sphere, suggests that some sort of rapid mix-ing is responsible. When the fluid is angular momentumis closer to well mixed, we expect a relaxation to a lesssteeply varying angular velocity profile, as in Fig. 11(b).

In the low torque state schematic, Fig. 11(a), the az-imuthal flow outside the shear layer is slow, but notlocked to the outer sphere. The angular momentum mea-surements of Fig. 6 suggest that the Ekman suction onthe outer boundary drains angular momentum from theregion outside the shear layer faster than the flux acrossthe zonal flow boundary can replenish it until an equilib-rium is reached or a L to H transition occurs.

At the transition from the high torque to low torquestate, the slow decay of the inner torque and increase ofthe observed fluid velocity and wall shear suggest that

11

(a) L (b) H

FIG. 11. A sketch of two possible mean flow states. Thelow torque state is labeled L and the high labeled H. Thelow torque state at (a) is characterized by fast zonal circu-lation near the core of the experiment and large amplitudewaves. In the high torque state, the velocity profile variesmore gradually. The zonal circulation has been destroyed bymixing across the transport barrier, so the fluid near the innersphere is slower and the torque higher.

the transport barrier bounding the central zonal flow hasre-formed, but the inner sphere must still provide an-gular momentum and energy to spin up the fast centralflow. At this point, since the exterior fluid is now weaklycoupled to the fast inner sphere, it starts to spin down.

We mention finally that changes in mean profiles ofangular momentum by vigorous mixing have been stud-ied in rotating thermal convection72–74. Mixing by con-vectively driven fluctuations tends to flatten the angularmomentum profile associated with solid body rotation,and this results in faster velocities in the rotating frameat high latitudes72. Increasing vigor of convective mixingof the conserved angular momentum results in a switchfrom a prograde equatorial zonal jet to a retrograde onein the models of Aurnou et al.72. Advectively flattenedangular momentum in the bulk of turbulent swirling flowsis well established in Taylor-Couette flow52,53,75 when theouter vessel is stationary, though the profile is not so flatin outer-stationary spherical Couette24.

Since the total angular momentum of the system isfluctuating, we must be careful not to use argumentsbased on strict angular momentum conservation. An-gular momentum is freely exchanged with the rotat-ing walls. An alternative explanation based on angularmomentum mixing from vigorous turbulent fluctuationscould involve a profile closer to the quadratic solid bodyprofile in the high torque state and an increasingly flat-tened angular momentum profile in the low torque state,and would not involve the formation and destruction ofa mixing barrier. It does, however, require more thana simple redistribution of angular momentum to explainthe results of Fig. 6.

IX. DYNAMICAL BEHAVIOR

0

0.02

0.04

0.06

0.08

0

0.02

0.04

0.06

0.08

0 0.05 0.1

0

0.02

0.04

0.06

0.08

u (t- t)/Ro

u (t)/Ro

u (t)/Ro

u (t)/Ro

(a) Ro = 1.80

(b) Ro = 2.13

(c) Ro = 2.40

L

H

H

L

FIG. 12. Time delay embeddings of the slow velocity fluctua-tions at y′ = 0.02 for three values of Ro below, in, and abovethe bistable range. Arrows in (b) show the direction takenby the transitions. The time delay ∆t is 4.5 rotations of theouter sphere in all three cases. The dimensionless velocity asdefined previously has been scaled by Ro, roughly collapsingthe mean velocity and fluctuation levels of each state. Thesedata have been low pass filtered with fc = 0.1 Hz, or a periodof 7.5 rotations of the outer sphere.

We also investigated the evolution in phase space ofH↔L transitions using low dimensional time delay em-bedding of low pass filtered velocity data. Figure 12 isa 2D embedding of a low pass filtered velocity time se-ries. The filter is a 4th order Butterworth with a cutofffrequency fc = 0.1 Hz. This corresponds to a period of7.5 rotations of the outer sphere. This velocity signalwas plotted against the same signal 4.5 rotations prior.

12

Figure 12 (a) is Ro = 1.8, the lower threshold of the firstbistable range. At (b), we have Ro = 2.13, the same as inFig. 3, where the bistable switching is evident, and in (c),Ro = 2.4, above the range where spontaneous transitionsare observed.

The velocity is rescaled here by the outer sphere tan-gential speed to make it dimensionless and by Ro, the ex-pected dimensionless velocity scale relative to the outersphere. Figure 12(b) could be interpreted as a hetero-clinic connection between two turbulent attractors, withconnections between the high torque flow state in (a) andthe low torque flow state in (c). The arrows in Fig 12(b)show the direction of flow in phase space as the systemundergoes several transitions. Below the critical Rossbynumber in Eq. 8, we expect H↔L connection to be bro-ken. Above the bistable range, we recall the probabilityof state as a function of Ro of Eq. 8 and Fig. 5 and expectthat connections between attractors weaken but do notbreak.

X. CONCLUSIONS

The experimental results presented here reveal novelturbulent multiple stability and turbulent flow transi-tions in a previously unexplored parameter range of veryhigh Reynolds number, rapidly rotating spherical Cou-ette flow. The transitions appear to involve the formationand destruction of zonal flow transport barriers and ex-hibit strong waves and unusual angular momentum trans-port. These results suggest spherical Couette as anotherstraightforward laboratory testbed for studying turbu-lent multiple stability and could help lead to a betterunderstanding of similar phenomena in natural systems.

We are grateful to the NSF Geophysics program forfunding and to the two anonymous reviewers for manyhelpful suggestions.

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