^{1,a)}, Santiago Andrés Triana

^{1,b)}and D. P. Lathrop

^{1,c)}

### Abstract

Flow between concentric spheres of radius ratio is studied in a 3 m diameter experiment. We have measured the torques required to maintain constant boundary speeds as well as localized wall shear stress, velocity, and pressure. At low Ekman number and modest Rossby number , the resulting flow is highly turbulent with a maximum Reynolds number (*Re* = *Ro*/*E*) exceeding 15 million. Several turbulent flow regimes are evident as *Ro* is varied for fixed *E*. We focus our attention on 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 alone at adjacent *Ro* outside the switching range. The two states are clearly distinguished in all measuredflow quantities, including a striking reduction in torque demanded from the inner sphere by the state lying at higher *Ro*. The reduced angular momentum transport appears to be associated with the development of a fast zonal circulation near the experiment core. The lower torque state exhibits waves, one of which is similar to an inertial mode known for a full sphere and another which appears to be a strongly advected Rossby-type wave. These results represent a new laboratory example of the overlapping existence of distinct flow states in high Reynolds numberflow.Turbulent multiple stability and the resilience of transport barriers associated with zonal flows are important topics in geophysical and astrophysical contexts.

We are grateful to the NSF Geophysics program for funding and to the two anonymous reviewers for many helpful suggestions.

I. INTRODUCTION

II. APPARATUS

III. TORQUE MEASUREMENTS

IV. FLUID ANGULAR MOMENTUM

V. MEAN FLOWMEASUREMENTS

VI. TURBULENT FLOWFLUCTUATIONS

VII. WAVES

VIII. DISCUSSION

IX. DYNAMICAL BEHAVIOR

X. CONCLUSIONS

### Key Topics

- Torque
- 122.0
- Turbulent flows
- 55.0
- Angular momentum
- 34.0
- Torque measurement
- 26.0
- Couette flows
- 21.0

## Figures

A schematic of the apparatus showing the inner and outer sphere and locations of measurement ports in the vessel top lid at 60 cm cylindrical radius (1.18 *r* _{i}). Data from sensors in the ports are acquired by instrumentation, including an acquisition computer, bolted to the rotating lid and wirelessly transferred to the lab frame. Also shown is the wireless torque sensor on the inner shaft.

A schematic of the apparatus showing the inner and outer sphere and locations of measurement ports in the vessel top lid at 60 cm cylindrical radius (1.18 *r* _{i}). Data from sensors in the ports are acquired by instrumentation, including an acquisition computer, bolted to the rotating lid and wirelessly transferred to the lab frame. Also shown is the wireless torque sensor on the inner shaft.

(a) The dimensionless torque *G* vs. the Reynolds number *Re* for the case , stationary outer sphere, with a fit to *G* = *A Re* + *B Re* ^{2} with and *B* = 0.05. (b) Mean value of the dimensionless torque *G* vs. *Ro* at . *Ro* is varied by increasing the inner sphere speed in steps of *Ro* = 0.067, waiting 450 rotations per step. Three curves are differentiated by circles, triangles, and squares. In the ranges of *Ro* where the symbols overlap, the flow exhibits bistable behavior. H and L denote the torque curves of the “high torque” and “low torque” states. There is a second bi-stable regime starting around *Ro* = 2.75, with LL labeling the lower torque state.

(a) The dimensionless torque *G* vs. the Reynolds number *Re* for the case , stationary outer sphere, with a fit to *G* = *A Re* + *B Re* ^{2} with and *B* = 0.05. (b) Mean value of the dimensionless torque *G* vs. *Ro* at . *Ro* is varied by increasing the inner sphere speed in steps of *Ro* = 0.067, waiting 450 rotations per step. Three curves are differentiated by circles, triangles, and squares. In the ranges of *Ro* where the symbols overlap, the flow exhibits bistable behavior. H and L denote the torque curves of the “high torque” and “low torque” states. There is a second bi-stable regime starting around *Ro* = 2.75, with LL labeling the lower torque state.

Time series of *G* at fixed *Ro* = 2.13 and , with time made dimensionless by the outer sphere rotation period. The raw torque signal has been numerically low pass filtered (*f _{c} * = 0.05 Hz, 15 rotations of the outer sphere).

Time series of *G* at fixed *Ro* = 2.13 and , with time made dimensionless by the outer sphere rotation period. The raw torque signal has been numerically low pass filtered (*f _{c} * = 0.05 Hz, 15 rotations of the outer sphere).

Probability density of the dimensionless torque at *Ro* = 2.13 and . The full, unconditioned distribution is denoted by small points. Solid circles denote conditioning on low torque state, and open circles on the high, with Gaussian solid and dashed curves for the low and the high, respectively. The mean and standard deviation of the low torque state data are and . In the high torque state, they are and . The data were low pass filtered at *f _{c} * = 0.5 Hz to remove the high frequency noise caused by mechanical vibration.

Probability density of the dimensionless torque at *Ro* = 2.13 and . The full, unconditioned distribution is denoted by small points. Solid circles denote conditioning on low torque state, and open circles on the high, with Gaussian solid and dashed curves for the low and the high, respectively. The mean and standard deviation of the low torque state data are and . In the high torque state, they are and . The data were low pass filtered at *f _{c} * = 0.5 Hz to remove the high frequency noise caused by mechanical vibration.

Probability that the flow was in the high torque (open circles) or low torque (closed circles) state as a function of *Ro* over the first bistable range with fixed . The dashed and solid lines are fits to the exponential form of Eq. (8).

Probability that the flow was in the high torque (open circles) or low torque (closed circles) state as a function of *Ro* over the first bistable range with fixed . The dashed and solid lines are fits to the exponential form of Eq. (8).

The angular momentum, inner torque, outer torque, and net torque. *Ro* = 2.13, . The upper plot shows the dimensionless angular momentum as defined in Eq. (12). The lower plot shows the inner torque *G*, the outer torque *G* _{o}, and their sum *G* _{net}. The bearing and aerodynamic drag on the outer sphere have been subtracted off, and the torques have been low pass filtered as in Fig. 3.

The angular momentum, inner torque, outer torque, and net torque. *Ro* = 2.13, . The upper plot shows the dimensionless angular momentum as defined in Eq. (12). The lower plot shows the inner torque *G*, the outer torque *G* _{o}, and their sum *G* _{net}. The bearing and aerodynamic drag on the outer sphere have been subtracted off, and the torques have been low pass filtered as in Fig. 3.

(Color) Simultaneous time series of velocity, wall shear stress, and torque. A space-time diagram of the low pass filtered velocity (defined in Eq. (15)) is shown at the top. This measurement is dominated by the azimuthal velocity . The velocity is made dimensionless by the outer sphere tangential velocity, and so can be interpreted as a locally measured Rossby number. The dimensionless wall shear stress is shown in the middle, and the dimensionless torque *G* is shown at the bottom. The wall shear stress and torque have been low pass filtered with *f _{c} * = 0.05 Hz as before, which is comparable to the time averaging of the velocimetry.

(Color) Simultaneous time series of velocity, wall shear stress, and torque. A space-time diagram of the low pass filtered velocity (defined in Eq. (15)) is shown at the top. This measurement is dominated by the azimuthal velocity . The velocity is made dimensionless by the outer sphere tangential velocity, and so can be interpreted as a locally measured Rossby number. The dimensionless wall shear stress is shown in the middle, and the dimensionless torque *G* is shown at the bottom. The wall shear stress and torque have been low pass filtered with *f _{c} * = 0.05 Hz as before, which is comparable to the time averaging of the velocimetry.

(a) The probability density of the dimensionless velocity conditioned on the torque state, with solid circles denoting the low torque state and open circles denoting the high. Dashed and solid lines are Gaussian. Standard deviations are for the high state and for the low state. The high state mean is , and the low state mean is . The means are dominated by azimuthal velocity, though the transducers are equally sensitive to *u* _{z} and in this measurement. (b) The probability density of the dimensionless wall shear conditioned on torque. Solid circles again denote the low torque state, and open circles the high, with solid and dashed Gaussian curves. The mean and standard deviation in the low torque state are and . In the high torque state, they are and .

(a) The probability density of the dimensionless velocity conditioned on the torque state, with solid circles denoting the low torque state and open circles denoting the high. Dashed and solid lines are Gaussian. Standard deviations are for the high state and for the low state. The high state mean is , and the low state mean is . The means are dominated by azimuthal velocity, though the transducers are equally sensitive to *u* _{z} and in this measurement. (b) The probability density of the dimensionless wall shear conditioned on torque. Solid circles again denote the low torque state, and open circles the high, with solid and dashed Gaussian curves. The mean and standard deviation in the low torque state are and . In the high torque state, they are and .

Power spectra of wall shear stress at *Ro* = 2.13 and , conditioned on state. Angular frequency has been made dimensionless using the outer sphere angular speed. The black curve is the spectrum from the low torque state, and the gray curve is that of the high torque state. The low torque spectrum has prominent peaks at , 0.71, and harmonics. In the high torque state, there are broad peaks at and 0.53.

Power spectra of wall shear stress at *Ro* = 2.13 and , conditioned on state. Angular frequency has been made dimensionless using the outer sphere angular speed. The black curve is the spectrum from the low torque state, and the gray curve is that of the high torque state. The low torque spectrum has prominent peaks at , 0.71, and harmonics. In the high torque state, there are broad peaks at and 0.53.

(Color) A spectrogram of wall pressure at 23.5° colatitude shows the evidence of several flow transitions as *Ro* is varied. and Ro , waiting 430 rotations per step with steps of *Ro* = 0.1. Instead of averaging the power spectra over an entire step of *Ro*, there are 30 spectra per step in *Ro*, so some temporal evolution is visible at a given *Ro*. In the *L* state, there are two strong waves, the lower at (*a*) varies only slightly in frequency with *Ro*. The higher frequency wave at (*b*) varies more strongly with *Ro*, suggesting that advection by the mean flow is important in setting its frequency.

(Color) A spectrogram of wall pressure at 23.5° colatitude shows the evidence of several flow transitions as *Ro* is varied. and Ro , waiting 430 rotations per step with steps of *Ro* = 0.1. Instead of averaging the power spectra over an entire step of *Ro*, there are 30 spectra per step in *Ro*, so some temporal evolution is visible at a given *Ro*. In the *L* state, there are two strong waves, the lower at (*a*) varies only slightly in frequency with *Ro*. The higher frequency wave at (*b*) varies more strongly with *Ro*, suggesting that advection by the mean flow is important in setting its frequency.

A sketch of two possible mean flow states. The low torque state is labeled *L* and the high labeled *H*. The low torque state at (a) is characterized by fast zonal circulation near the core of the experiment and large amplitude waves. In the high torque state, the velocity profile varies more gradually. The zonal circulation has been destroyed by mixing across the transport barrier, so the fluid near the inner sphere is slower and the torque higher.

A sketch of two possible mean flow states. The low torque state is labeled *L* and the high labeled *H*. The low torque state at (a) is characterized by fast zonal circulation near the core of the experiment and large amplitude waves. In the high torque state, the velocity profile varies more gradually. The zonal circulation has been destroyed by mixing across the transport barrier, so the fluid near the inner sphere is slower and the torque higher.

Time delay embeddings of the slow velocity fluctuations at for three values of *Ro* below, in, and above the bistable range. Arrows in (b) show the direction taken by the transitions. The time delay is 4.5 rotations of the outer sphere in all three cases. The dimensionless velocity as defined previously has been scaled by *Ro*, roughly collapsing the mean velocity and fluctuation levels of each state. These data have been low pass filtered with *f _{c} * = 0.1 Hz or a period of 7.5 rotations of the outer sphere.

Time delay embeddings of the slow velocity fluctuations at for three values of *Ro* below, in, and above the bistable range. Arrows in (b) show the direction taken by the transitions. The time delay is 4.5 rotations of the outer sphere in all three cases. The dimensionless velocity as defined previously has been scaled by *Ro*, roughly collapsing the mean velocity and fluctuation levels of each state. These data have been low pass filtered with *f _{c} * = 0.1 Hz or a period of 7.5 rotations of the outer sphere.

## Tables

Statistics of the interval between high torque onsets. is the time interval between two subsequent high torque onsets made dimensionless by , so the time interval is measured in outer sphere rotations. *Ro* = 2.13, .

Statistics of the interval between high torque onsets. is the time interval between two subsequent high torque onsets made dimensionless by , so the time interval is measured in outer sphere rotations. *Ro* = 2.13, .

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