Abstract
The standard magnetorotational instability (SMRI) is a promising mechanism for turbulence and rapid accretion in astrophysical disks. It is a magnetohydrodynamic (MHD) instability that destabilizes otherwise hydrodynamically stable disk flow. Due to its microscopic nature at astronomical distances and stringent requirements in laboratory experiments, SMRI has remained unconfirmed since its proposal, despite its astrophysical importance. Here we report a nonaxisymmetric MHD instability in a modified TaylorCouette experiment. To search for SMRI, a uniform magnetic field is imposed along the rotation axis of a swirling liquidmetal flow. The instability initially grows exponentially, becoming prominent only for sufficient flow shear and moderate magnetic field. These conditions for instability are qualitatively consistent with SMRI, but at magnetic Reynolds numbers below the predictions of linear analyses with periodic axial boundaries. Threedimensional numerical simulations, however, reproduce the observed instability, indicating that it grows linearly from the primary axisymmetric flow modified by the applied magnetic field.
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Introduction
Astronomical accretion disks consist of gas or plasma orbiting a compact massive object such as a black hole or protostar, and slowly spiraling inward (accreting) by surrendering orbital angular momentum to other material in the disk or in an outflow^{1}. Driven by gravity, the angular velocity profile in a Keplerian flow has a decaying powerlaw dependence on the cylindrical radius, Ω(r) ∝ r^{−q}, with q = 3/2. According to Rayleigh’s criterion^{2}, purely hydrodynamic rotation profiles with 0 < q < 2 (“quasiKeplerian”) are linearly stable to axisymmetric perturbations, and apparently linearly and nonlinearly stable nonaxisymmetrically as well^{3,4}, at least without complications such as thermal effects or interactions with dust^{5}. Therefore, hydrodynamic modes cannot excite the turbulence required to explain rapid accretion^{6,7}. The standard magnetorotational instability (SMRI)—a unique magnetohydrodynamic (MHD) instability in a conducting Keplerian flow in the presence of an axial magnetic field—is thus regarded as one of the most promising mechanisms for unraveling the origin of turbulence in accretion disks^{3,8,9}, apart from possible rapid accretion due to laminar magnetized winds^{10}. Unlike the SMRI that requires only the magnetic field parallel to the rotation axis, other versions of MRI involving azimuthal fields have been found and experimentally demonstrated: helical MRI (HMRI) and azimuthal MRI (AMRI)^{11,12}. While their existence is intriguing, these instabilities are inductionless, incapable of generating and sustaining the needed magnetic field. They also require steeperthanKeplerian rotation profiles (q > 3/2), hence are unlikely to be relevant to most astrophysical disks^{13}.
Unlike other fundamental plasma processes such as Alfvén waves^{14,15,16} and magnetic reconnection^{17,18,19} which have been detected and studied in space and in the laboratory, SMRI remains unconfirmed long after its proposal^{8,20,21} other than its analogs^{22,23,24,25}, despite its widespread applications in modeling including recent black hole imaging^{26}. Due to its microscopic nature and the limitations of telescope resolution, SMRI cannot be captured by current astronomical observations. SMRI is also proposed^{27,28} to be realized in a terrestrial Taylor–Couette experiment, which consists of two independently rotating coaxial cylinders that viscously drive the liquid metal between them to a quasiKeplerian flow with a magnetic Reynolds number larger than unity. However, as the axial boundaries (endcaps) of a conventional Taylor–Couette cell are bound either to the inner cylinder or to the outer cylinder, their motions do not match the viscously driven flow profile in the bulk. Ekman circulation is thus excited, entailing ∂Ω/∂z ≠ 0 along the axial zdirection and some turbulence that prevent the detection of SMRI^{29,30}.
Here, we report a laboratory experiment searching for SMRI using a specially designed Taylor–Couette cell. The cell’s coppermade endcaps can rotate independently^{31}, which provide a quiescent quasiKeplerian flow in the bulk region despite that the shear Reynolds number exceeds a million^{3,4,32}, as well as their inductive coupling with the fluid^{33,34,35,36}. Through magnetic field measurements, we identify a global MHD instability occupying the entire bulk region, which exists only at sufficiently large rotation rates and intermediate magnetic field strengths, in line with typical requirements for SMRI from linear theories. The instability is nonaxisymmetric with a dominant m = 1 mode in the azimuthal direction, which spontaneously breaks the rotational symmetry possessed by the system. Our numerical simulations reproduce the experimentally observed instability and further reveal that it develops from an axisymmetric base flow modified by the applied magnetic field.
Results
We denote the radius of the inner and outer cylinders as r_{1} = 7.06 cm and r_{2} = 20.3 cm, and their height as H = 28 cm. The aspect ratio of the cell is thus Γ ≡ H/(r_{2} − r_{1}) ≃ 2.1, which is deliberately designed to be large to ensure the magnetic diffusion time is longer than the rotation period and the Alfvén crossing time, and thus help excite SMRI^{27,28}. The upper and lower sealing endcaps are split into two rings at r_{3} ≃ (r_{1} + r_{2})/2. The angular velocities of the inner cylinder, inner rings, outer rings and outer cylinder are, respectively, Ω_{1}, Ω_{3}, Ω_{2} and Ω_{2}. The corresponding frequencies and their differences are denoted as f_{i} ≡ Ω_{i}/2π and f_{ij} ≡ (Ω_{i} − Ω_{j})/2π, respectively. For all results described here, Ω_{1}: Ω_{2}: Ω_{3} = 1: 0.19: 0.58, generating a shear flow similar to what was studied previously^{36}. A set of six copper coils provides a uniform axial magnetic field B_{z}. The local radial magnetic field B_{r}(t) is measured by Hall probes installed on the surface of the inner cylinder at various azimuths and heights. Dimensionless measures of rotation and field strength are the magnetic Reynolds number \({{{{{{{\rm{Rm}}}}}}}}={r}_{1}^{2}{{{\Omega }}}_{1}/\eta\) and the Lehnert number \({B}_{0}={B}_{z}/({r}_{1}{{{\Omega }}}_{1}\sqrt{{\mu }_{0}\rho })\), which are varied in the ranges 0.5 ≲ R_{m} ≲ 4.5 and 0.05 ≲ B_{0} ≲ 1.2, respectively. The magnetic Prandtl number is P_{m} = ν/η = 1.2 × 10^{−6} for the working fluid GaInSn eutectic alloy (galinstan). Here, μ_{0} is the vacuum permeability and ν, η, and ρ are, respectively, the kinematic viscosity, magnetic diffusivity and density of galinstan. The device spins for 2 min (several Ekman times) to ensure a relaxed hydrodynamic flow before the introduction of B_{z}.
A representative time series of the imposed magnetic field B_{z}(t) is shown in Fig. 1a, increasing from zero to 2100 G in less than one second. The corresponding B_{r}(t) measured in the midplane is shown in Fig. 1b, which first increases synchronously with B_{z}(t), then saturates to a statistically stationary state. An intriguing finding is a strong oscillation in B_{r}(t), which emerges at t ≃ 125.6 s and saturates after t ≃ 127.0 s. Such an instability appears to be global as it is well correlated at different heights. Figure 1c shows the spectrogram of B_{r}(t) using a onesecond moving window, where all frequencies are shifted by − f_{1} relative to the lab frame. The transient power at low frequency (f ≲ 5 Hz) is believed to be due to a modification of the base flow caused by the imposed magnetic field. As indicated by the horizontal lines, energy of the instability appears between the machineinduced frequencies f_{12} and f_{13}, and has a maximum value at f ≃ 17.5 Hz. This is a general feature of the instability at different Rm and B_{0}, namely, its frequency is between the rotation frequencies of the inner rings and outer cylinder. We then define the normalized strength of the instability as
where P_{B}(f) and 〈B_{z}(t)〉 are, respectively, the power spectrum of B_{r}(t) and the mean value of B_{z}(t) sampled in a moving onesecond window. As shown by the vertical line in Fig. 1d, the measured χ(t) starts to grow about 0.3 s after the imposition of B_{z}, and then saturates. Such a growthsaturation process agrees well with the evolution of B_{r}(t) shown in Fig. 1b. The initial growth of χ(t) is well described by an exponential, indicating linear instability^{27,28}.
We also conduct threedimensional (3D) numerical simulations using the opensource SFEMaNS code, which solves coupled Maxwell and Navier–Stokes equations using spectral and finiteelement methods^{37}. It contains 32 Fourier modes in the azimuthal direction. Similar to the experiment, the entire simulation process consists of two stages. In the first hydrodynamic stage, simulations are run to reach a relaxed hydrodynamic state without the external magnetic field, starting from an initial piecewise solidbody condition that follows the angular speed of the endcaps and two cylinders. In the second MHD stage, the external magnetic field is imposed and lasts until a saturated MHD state is achieved. The main difference between simulation and experiment is the viscous Reynolds number \({{{{{{{\rm{Re}}}}}}}}={r}_{1}^{2}{{{\Omega }}}_{1}/\nu\), which is on the order of 10^{6} in the experiment but only 1000 in our simulations. The relatively large viscosity (low \({{{{{{{\rm{Re}}}}}}}}\)) in simulation gives rise to thick residual Ekman layers at the endcaps, which drive nonaxisymmetric hydrodynamic modes in the bulk (see “Flow characterization in 3D simulation” in “Methods”). It was found numerically that these modes’ amplitudes decrease with increasing \({{{{{{{\rm{Re}}}}}}}}\)^{38,39}, so they are undetected in the experiment. In the simulated radial magnetic field, we also observe instability similar to that in the experiment. The instability occupies the whole radial extent and rotates as a solid body (see Supplementary Movie 1), which is distinct from the Stewartson–Shercliff layer (SSL) instability, as the latter concentrates near r ≃ r_{3} and has a spiral structure^{35,40,41,42}. Due to the residual hydrodynamic modes, our 3D simulation cannot always reproduce the exponential growth of the instability seen in the experiment. Nonetheless, when only a single m = 1 mode is allowed in an otherwise twodimensional (2D) simulation, an exponential growth is observed with a growth rate behaving similar to that in the experiment (see below).
The normalized amplitude of the mth azimuthal Fourier mode 〈∣a_{m}∣〉/B_{z} averaged in the saturated MHD state are shown in Fig. 2a. For Rm = 1, all the 〈∣a_{m}∣〉/B_{z} from the experiment are negligibly small and thus the instability is absent. The 〈∣a_{2}∣〉/B_{z} at Rm = 1 from simulation is believed to be due to the residual hydrodynamic modes at low \({{{{{{{\rm{Re}}}}}}}}\)^{38,39}. For Rm ≳ 3, the experimental 〈∣a_{1}∣〉/B_{z} dominates, indicating that the instability is nonaxisymmetric and mainly at m = 1. Similar results are also obtained from simulations, namely, the nonaxisymmetric amplitudes increase for Rm ≳ 3, and m = 1 dominates for Rm ≳ 4. At the onset of the instability (Rm = 3) in simulation, the m = 2 mode is the strongest, which could result from the interaction between the instability and the residual hydrodynamic modes.
Figure 2b shows a comparison between the phase θ_{1}(t) of the m = 1 mode from experiment and simulation in the saturated MHD state. All θ_{1}(t) are well described by linear functions, θ_{1}(t)/(2π) = f_{0}t (colorcoded lines), where f_{0} is the characteristic frequency of the m = 1 mode. At Rm = 1, f_{0} = 5.4 Hz in the experiment is different from f_{0} = 6.1 Hz in the simulation. This discrepancy is caused by the \({{{{{{{\rm{Re}}}}}}}}\) difference between experiment and simulation, which gives rise to different hydrodynamic modes that are the main contributor to the m = 1 mode at low Rm. For Rm ≳ 3, on the other hand, f_{0} from experiment agrees well with that from simulation, suggesting that the frequency of the instability is insensitive to \({{{{{{{\rm{Re}}}}}}}}\). The frequency f_{0} = 17.5 Hz for Rm = 3 from experiment is also consistent with the characteristic frequency of the instability seen in Fig. 1c.
A “bubble plot” of the instability strength in the Ω_{1}B_{z} plane is shown in Fig. 3. The diameter of the experimental bubbles (black) denotes χ from Eq. (1) with a typical time window of 10 s in the saturated MHD state while their numerical counterpart (orange) follows the time average of the standard deviation of B_{r} among azimuths. Overall agreement between experiment and simulation is excellent, namely, the instability becomes particularly pronounced only for Ω_{1} ≳ 1500 rpm (Rm ≳ 3) and 1800 ≲ B_{z} ≲ 2800 G. The red curve shows the boundary for SSL instability in the midplane, which is stable on its left and unstable on its right. The SSL instability is nonaxisymmetric and develops from a free SSL spanning the entire vertical extent around r ≃ r_{3}^{34,40,41} (see “Characterization of the StewartsonShercliff layer instability” in “Methods”). In particular, the SSL instability is inductionless and thus can be excited in the limit of small Rm. For example, for Ω_{1} ≲ 1000 rpm (Rm ≲ 2), larger bubbles appear on the right of the red curve, indicating that the SSL instability is excited. On the other hand, the large bubbles in the upper middle area are far to the left of the red curve and require Ω_{1} ≳ 1500 rpm (Rm ≳ 3), implying that the identified instability is distinct from the SSL instability. In the presence of a weak magnetic field (B_{z} ≲ 1200 G), the bubbles from simulation are larger than their experimental counterpart. We believe that this is due to the residual hydrodynamic modes in these simulations, for which a weak field acts as a passive tracer.
Discussion
The above measurements reveal a global MHD instability in a modified liquidmetal Taylor–Couette experiment. The instability grows exponentially and becomes particularly pronounced once the radial shear rate is sufficiently large and the imposed axial magnetic field is moderate, consistent with typical requirements for SMRI in this system^{27,28,36,43}. On the other hand, it is nonaxisymmetric with a dominant m = 1 azimuthal structure, which contradicts the prediction of linear theories for SMRI with an axial magnetic field in an ideal Couette flow between infinitely long cylinders: the SMRI should be axisymmetric at onset^{27,28}. The instability also has minimum requirements for rotation (Ω_{1} ≳ 1500 rpm) and magnetic field (B_{z} ≳ 1800 G) smaller than predictions for SMRI based on local Wentzel–Kramers–Brillouin (WKB) analysis or global linear calculation (see “Linear theory predictions of SMRI” in “Methods”), implying that either it is not SMRI or the linear theories are not capable enough to describe our system without including its closed geometry and complex boundary conditions. Experiment and simulation generally agree as to the characteristic frequency, azimuthal structure, and distribution of amplitudes in the Ω_{1} − B_{z} plane. The instability is distinct from the hydrodynamic Rayleigh instability (see “Characterization of hydrodynamic Rayleigh instability” in “Methods”) and the SSL instability, which are often present and sometimes mistaken for SMRI in previous experiments^{44,45}. Even with conductive endcaps, our 3D simulation nonetheless shows that the induced azimuthal magnetic field B_{ϕ} is still less than 15% of the applied axial magnetic field (see “Azimuthal magnetic field” in “Methods”). Furthermore, it is found that most B_{ϕ} is mainly concentrated in the region near the endcaps rather than the bulk region where the observed instability is located. As a result, the instability is also unlikely to be the nonaxisymmetric AMRI or HMRI, which require a pure or predominant azimuthal magnetic field^{46}.
While we are currently unable to pinpoint the fundamental cause for the observed instability, our "2mode” simulations nonetheless provide a phenomenological description of its generation mechanism: it develops from an axisymmetric base flow modified by the applied magnetic field. In these simulations, only m = 0 and m = 1 modes are kept, and all the m ≥2mode amplitudes are set to zero. In the first hydrodynamic stage, a 2D flow is relaxed to a steady state from an initial piecewise solidbody rotation, where the m = 0 mode can evolve freely and the amplitude of the m = 1 mode is set to a negligibly small value (~ 10^{−20}) at each time step. When an axial magnetic field is applied in the second MHD stage, both m = 0 and m = 1 modes can evolve freely.
Figure 4a shows the time series of the dimensionless amplitude A_{0} of the m = 0 mode in the velocity field from the 2mode simulations. As separated by the vertical line, A_{0} (black curve) saturates at the end of the first stage at tΩ_{1} = 400, indicating that a hydrodynamic steady state is reached. When the axial field is turned on (B_{0} > 0), there is a transient variation in A_{0}, followed by a new MHD steady state. The red curve shows a continuation of the hydrodynamic evolution (B_{0} = 0) for comparison. Because the amplitude of the m = 1 mode is very small (see Fig. 4b) in the simulation time span, there is no coupling between the two modes in the second stage. The initial rapid variation of A_{0} after the imposition of magnetic field is caused by the magnetization of the residual hydrodynamic modes^{36}. It is found that under the influence of the imposed magnetic field, the A_{0} value in the final steady state of the second stage is different from that in the relaxed hydrodynamic state of the first stage, indicating that the base flow is modified by the applied magnetic field^{36}. Figure 4b shows the corresponding time evolution of the amplitude A_{1} of the m = 1 mode. As shown by the black curve, A_{1} in the first stage is negligibly small, as expected. Once A_{1} is allowed to evolve in the second MHD stage, it first rapidly increases to ~ 3 × 10^{−17} at tΩ_{1} ≃ 410 in all cases. Such an increase is a numerical selfadjustment to adapt to the sudden change of mode number, therefore is irrelevant to the physical flow dynamics. Similar to the amplitude of the instability identified in the experiment shown in Fig. 1d, A_{1} grows exponentially in the final steady state of the second stage, \({A}_{1} \sim {e}^{{\gamma }_{0}t{{{\Omega }}}_{1}}\), with γ_{0} being the dimensionless growth rate. The amplitude of the m = 1 mode in the magnetic field also has an exponential growth with a growth rate the same as that in the velocity field.
As shown by black squares in Fig. 5a, the hydrodynamic case (B_{0} = 0) in our 2mode simulation has a growth rate of γ_{0} = 0.0175 > 0, indicating that the m = 1 mode is unstable to the relaxed 2D hydrodynamic state in the first stage. This is as expected because nonaxisymmetric modes are found in the relaxed hydrodynamic state of the 3D simulation. The value of γ_{0} at B_{0} = 0 thus can be taken as a benchmark, and deviations from it in other B_{0} > 0 cases are caused by MHD effects. It is found that γ_{0} from simulation is not monotonic in B_{0}, but rather is maximized at an intermediate field strength, as is characteristic of the instability observed in our experiment. For comparison, we also plot the corresponding γ_{0} from experiment, which is determined by the growth of χ(t) at the beginning stage (t ≲ 127 s) after the external magnetic field is imposed (see Fig. 1d). Because nonaxisymmetric hydrodynamic modes are squeezed to regions adjacent to the endcaps and thereby are absent in the bulk region of the experiment, γ_{0} increases from zero with B_{0}. Except for this difference, the experiment agrees quite well with the simulation, including that γ_{0} has a nonmonotonic dependence on B_{0} and becomes significantly large for 0.15 ≲ B_{0} ≲ 0.3. Similarly, Fig. 5b shows that at fixed B_{0} = 0.2, γ_{0} from both 2mode simulations (black squares) and experiments (red circles) is largely enhanced for Rm ≳ 3, consistent with the amplitude of the instability identified at saturation (see Fig. 3). Again, the positive γ_{0} at Rm ≲ 2 in the simulation is mainly caused by the residual hydrodynamic modes, which are not present in the experiment and thus γ_{0} ≃ 0. All these findings indicate that in the presence of sufficiently large shear and a moderate axial magnetic field, the mean axisymmetric flow is driven into a new state at which nonaxisymmetric modes become linearly unstable. This mechanism exists only in bounded systems like our experiments and may not apply to a real accretion disk, where the Kepler flow is hardly affected by instabilities or turbulence in it.
Further investigations are needed to fully understand the reported instability, including its fundamental cause from first principles, saturation mechanism, transition to turbulence, and responses to different radial boundary conditions and geometric confinements. Different components of the velocity and magnetic fields inside the liquidmetal flow will be measured using Ultrasonic Doppler Velocimetry and Hall probe arrays, which help to better resolve its spatial structures and relationship to local angular momentum transport, and thus further determine its identity. It is also worth examining its possible connection to the nonaxisymmetric SMRI in a narrowgap annular cylinder observed recently by an incompressible linear theory^{47}. The possibility of nonaxisymmetric global instabilities due to unstable Alfvén continuum will also be investigated^{48}. Numerical simulations with a higher \({{{{{{{\rm{Re}}}}}}}}\) closer to the experimental setup should be explored, perhaps with an entropyviscosity method in SFEMaNS^{49}.
Methods
Taylor–Couette cell
Details about the device used in this experiment have been described elsewhere^{34}, and here we only mention some key points. As shown in Fig. 6, the inner cylinder is composed of five Delrin rings (green) and two stainless steel caps (cyan) that compress them axially. The upper (lower) stainless steel cap has a 1 cm protruding rim on its top (bottom) side, respectively, which help to further reduce the Ekman circulation^{4}. The outer cylinder is made of stainless steel. The endcaps between the two cylinders are made of 1inchthick silverplated copper and split into two rings at r_{3} = 13.5 cm. The upper and lower inner rings are bound together, while the upper and lower outer rings are bound to the outer cylinder. Driven by three independent motors, the inner cylinder, inner rings, and outerringbound outer cylinder can rotate independently.
This Taylor–Couette cell has three unique features for the experiment reported here. First, the gap between the inner and outer cylinders is purposely made wide, which corresponds to a small aspect ratio Γ that helps to excite the SMRI according to theoretical predictions^{28}. Second, the independently rotatable endcap rings reduce Ekman circulation that could destabilize the desired quasiKeplerian profile^{3,31}. For a conventional Taylor–Couette cell, the endcaps are either bound to the inner cylinder or to the outer cylinder, which leads to a boundary condition different from the flow profile in the bulk. As a result, the Ekman circulation is inevitably excited and highly disturbs the bulk flow ^{3,29}, which likely overwhelms signals from the SMRI. In our cell, the inner rings rotate at an angular speed between that of the inner and outer cylinders, thereby significantly reducing the velocity discontinuity at the endcap. Consequently, the Ekman circulation in our cell is significantly reduced, allowing the flow in the bulk to approach a quasiKeplerian profile^{3,4,32,50}. Finally, the conducting copper endcaps significantly enlarge the magnetic stress within the boundary layer attached to them, which reinforces the differential rotation in the bulk flow^{33,34,35,36}.
Measurement of local magnetic field
The local magnetic field is measured by highprecision Hall probes (Allegro MicroSystems, A1308 series) with an accuracy of 0.5 G. A Hall probe is a device whose output voltage is directly proportional to the magnetic field through it. As shown by arrows in Fig. 7, the Hall probes are mounted at the surface of the three Delrin rings in the middle and orientated outwards to measure the radial magnetic field B_{r}. Six Hall probes (red arrows) are placed in the upper 1/4 plane (z = 7 cm). Six Hall probes (blue arrows) are placed in the midplane (z = 0 cm) with a same azimuthal distribution. One probe (green arrow) is placed in the lower 1/4 plane (z = −7 cm). An Arduinobased system containing analogtodigital converter (ADC) chips and a micro SD card is used to measure and record the voltage signals from Hall probes with a sampling rate ~ 175 Hz. The Arduino system is placed in a container along with a 9 V battery that powers it and the Hall probes. This container is bound on top of the inner cylinder and thus corotates with it.
Characterization of nonaxisymmetric modes
We fit the time series of the radial magnetic field at different azimuths B_{r}(θ, t) to the azimuthal Fourier series,
where a_{0}(t), a_{m}(t), and θ_{m}(t) are fitting parameters. In Eq. (2), the absolute value ∣a_{0}(t)∣ describes the amplitude of the axisymmetric mode, while ∣a_{m}(t)∣ and θ_{m}(t) describe the amplitude and phase of the mth azimuthal mode. Since there are only six sensors in the midplane, we set N = 2 for data from the experiment, which leaves residuals an order of magnitude smaller than the smallest mode amplitude. For the simulations, we adopt N = 10, and the amplitude of the residuals relative to B_{z} is less than 10^{−5}. Before fitting, a bandpass filter with a passband 1.05f_{13} ≤ f ≤ 0.95f_{12} is applied to the experimental data in order to remove the mechanical signals (Fig. 1c). Supplementary Fig. 1 shows an example of the measured radial magnetic field variations as a function of azimuth angle θ from the experiment (Supplementary Fig. 1a) and simulation (Supplementary Fig. 1b). As shown by the solid lines, the data points are well described by Eq. (2) with N = 2 for experiment and N = 10 for simulation.
Linear theory predictions of SMRI
The axisymmetric (m = 0) SMRI in our system has been extensively studied by local Wentzel–Kramers–Brillouin (WKB) analysis^{27} and global linear analysis^{28}. Both methods assume no change in base flow in the axial direction, therefore do not include the conducting and noslip boundary conditions in real experiments and nonlinear simulations. The global linear analysis assumes that the base flow has a Couette profile. The base flow shear in the WKB method is the geometric mean of the Couette flow shears of the inner and outer cylinders. As shown in Fig. 8, predictions of m = 0 SMRI from WKB method (blue curve) and global linear analysis (green curve) are very similar, both requiring B_{z} ≳ 5000 G and Ω_{1} ≳ 4500 rpm (Rm ≳ 9). This is higher than the parameter space explored in current experiments and simulations, as shown by the bubbles, concentrated in the region of B_{z} ≲ 5000 G and Ω_{1} ≲ 2500 rpm (Rm ≲ 5). In particular, as shown by the magenta curve, the prediction for the m = 1 SMRI from global linear analysis requires even higher flow shear (Ω_{1} ≳ 12,000 rpm or equivalently Rm ≳ 24) and magnetic field strength (B_{z} ≳ 18,000 G).
Characterization of the Stewartson–Shercliff layer instability
As shown in Fig. 9, the Stewartson–Shercliff layer (SSL) is a local free shear layer with q > 2 that originates from the junction of the inner and outer rings, where substantial shear occurs due to velocity discontinuities. The flow in SSL is Rayleigh and Kelvin–Helmholtz unstable, entailing nonaxisymmetric modes. For a fixed shear (Rm), the vertical extent of the upper and lower SSLs monotonically increases with the applied magnetic field strength, until they merge in the midplane. It has been shown that for a “split” configuration with Ω_{3} = Ω_{1} and insulating endcaps, the two SSLs reach the midplane once the Elsasser number
is greater than unity, causes nonaxisymmetric fluctuations there^{40,41}. Because the SSL is inductionless, Eq. (3) is valid in the small Rm limit^{40}. Despite the use of conductive copper endcaps and different rotation speed ratios in our experiments, we find Eq. (3) works remarkably well for the onset of SSL instability in the midplane with Rm ≲ 2 (see Fig. 3). At the same time, our simulation also shows that the two SSLs reach the midplane only for Λ ≳ 1.6 for all Rm values studied here, consistent with experimental results. For example, as shown in Fig. 9, at Rm = 4 the two SSLs merge in the midplane only for B_{0} ≳ 0.4, which corresponds to Λ ≳ 1.64.
Characterization of hydrodynamic Rayleigh instability
To excite the hydrodynamic Rayleigh instability, an angular velocity ratio Ω_{1} : Ω_{3} : Ω_{2} = 1 : 0.507 : 0.05 called Rayleigh unstable configuration is adopted, as Rayleigh’s criteria demands \({{{\Omega }}}_{2}/{{{\Omega }}}_{1}\,\le \,{r}_{1}^{2}/{r}_{2}^{2}\simeq 0.12\) for the Rayleigh instability^{2}. The value of Ω_{3} is chosen such that the Ekman circulation is still suppressed. In the experiment, a weak B_{z} acting as passive tracers to the hydrodynamic flow is imposed, and we use the measured B_{r}(t) to characterize the hydrodynamic instability. Similar to the procedure discussed in the main text, a relaxed hydrodynamic flow is achieved before the imposition of B_{z}.
Supplementary Fig. 2a shows the measured normalized power spectrum \({P}_{B}(\,\,f)/{B}_{z}^{2}\) of B_{r}(t) at the angular velocity ratio used in the main text. The data are obtained in the frame corotating with the inner cylinder. The vertical lines indicated the machineinduced frequency, at which the power is irrelevant to the flow dynamics. Because the imposed magnetic field is weak, MHD effects are minute and thereby the measured B_{r}(t) represents hydrodynamic properties of the flow. As expected, the power over the whole frequency range is small, indicating that the flow is hydrodynamically stable. Supplementary Fig. 2b shows the measured power spectrum at the Rayleigh unstable configuration. Compared with Supplementary Fig. 2a, the power spectrum in Supplementary Fig. 2b has significant power at f/f_{1} ≲ 0.2. Such a lowfrequency power is believed to be from the hydrodynamic Rayleigh instability, which has a frequency spectrum distinct from the instability (0.42 ≲ f/f_{1} ≲ 0.81) discussed in the main text.
Using Eq. (2), mode decomposition of hydrodynamic Rayleigh instability in the azimuthal direction is also performed. Supplementary Fig. 3 shows the measured normalized mode amplitude 〈∣a_{m}∣〉/B_{z} as a function of mode number m. Before fitting, we applied a lowpass filter with a cutoff frequency 0.2f_{1} to the measured B_{r}(t) at different azimuths. The discretization of ADC chips gives rise to an accuracy of 0.5 G for the measured B_{r}(t), which is about 0.1% of the imposed B_{z} in experiments with Rayleigh unstable configuration. Segments with a variation less than 0.5 G in the filtered B_{r}(t) are thus inaccurate and discarded. Compared with modes of instability shown in Fig. 2a, there are three main differences for the azimuthal modes of the hydrodynamic Rayleigh instability. First, the overall mode amplitudes of the hydrodynamic instability are much smaller, indicating that at least its timevarying part is quite weak. Second, unlike the instability reported in the main text that only appears for Rm ≳ 3, modes of hydrodynamic Rayleigh instability at different Rm have a similar amplitude. Finally, the dominant mode of the hydrodynamic Rayleigh instability is m = 0, in contrast to the instability reported in the main text that has a dominant m = 1 mode. This is consistent with the typical features of hydrodynamic Rayleigh instability in a Taylor–Couette flow, in which it has an evolution from axisymmetric to nonaxisymmetric^{51,52}. All these differences further confirm that the instability reported in the main text is unlikely to be the hydrodynamic Rayleigh instability.
Numerical methods and simulation setup
The numerical code used in the simulation has been described in detail elsewhere^{33,35,36}, and only some key points are mentioned here. As in the experiment, a cylindrical coordinate system is adopted in our threedimensional (3D) simulation. The origin is set at the geometric center of the Taylor–Couette cell. Unit vectors in the radial, azimuthal and vertical directions are denoted by e_{r}, e_{θ}, and e_{z}. In the simulation, we set r_{2}/r_{1} = 3, r_{3}/r_{1} = 2, H/r_{1} = 4 and the radius of the inner cylinder rim \({r}_{{{{{{{{\rm{rim}}}}}}}}}/{r}_{1}=1.15\). The length, time, velocity, magnetic field and electrical conductivity are normalized, respectively, by r_{1}, \({{{\Omega }}}_{1}^{1}\), Ω_{1}r_{1}, \({{{\Omega }}}_{1}{r}_{1}\sqrt{{\mu }_{0}\rho }\) and σ_{Ga}, where σ_{Ga} is the electrical conductivity of galinstan. In order to mimic the experiment, the whole volume is divided into three coupled domains, including a fluid domain for galinstan, a solid domain for endcaps and a spherical vacuum domain surrounding them.
Supplementary Fig. 4a shows the mesh allocation in a quarter of the meridional plane, in which meshes with different colors belong to different domains. The fluid domain has a mesh resolution of 100 × 200 triangular cells. The governing dimensionless equations are the resistive MHD equation for an incompressible fluid, with
where \(\tilde{{{{{{{{\bf{u}}}}}}}}}\), \(\tilde{p}\), \(\tilde{{{{{{{{\bf{B}}}}}}}}}\) and \(\tilde{\sigma }=1\) (Rm = \({{{\Omega }}}_{1}{r}_{1}^{2}{\sigma }_{{{\mbox{Ga}}}}{\mu }_{0}\)) are the dimensionless velocity, pressure, magnetic field, and electrical conductivity in the fluid domain, respectively.
As shown by Supplementary Fig. 4b which is an enlarged portion of Supplementary Fig. 4a, the solid domain is further divided into two subdomains with one for the stainless steel rim of the inner cylinder (yellow) and the other for the copper endcap (blue). In the solid domain, only the induction equation in Eq. (4) for magnetic field is evolved, where \(\tilde{\sigma }=19.4\) for copper and \(\tilde{\sigma }=0.468\) for stainless steel, respectively. The dimensionless linear velocity at the endcap boundaries is \(\tilde{{{{{{{{\bf{u}}}}}}}}}=\tilde{r}{{{{{{{{\bf{e}}}}}}}}}_{\theta }\) for \(1\,\le \,\tilde{r}\,\le \,1.15\), \(\tilde{{{{{{{{\bf{u}}}}}}}}}={{{\Omega }}}_{3}\tilde{r}/{{{\Omega }}}_{1}{{{{{{{{\bf{e}}}}}}}}}_{\theta }\) for \(1.15\, < \,\tilde{r}\,\le \,2\) and \(\tilde{{{{{{{{\bf{u}}}}}}}}}={{{\Omega }}}_{2}\tilde{r}/{{{\Omega }}}_{1}{{{{{{{{\bf{e}}}}}}}}}_{\theta }\) for \(2\, < \,\tilde{r}\,\le \,3\). Here \(\tilde{r}\equiv r/{r}_{1}\) is the dimensionless radial position.
The vacuum domain has a radius of 20r_{1}. By introducing a scalar potential for the magnetic field with \(\tilde{{{{{{{{\bf{B}}}}}}}}}\equiv \tilde{\nabla }\phi\), the governing equation in the vacuum domain is
Equation (5) is a Laplace equation so its solution is uniquely determined by ϕ at the boundary of the domain. The boundary conditions for Eqs. (4)–(5) are the following: Noslip boundary conditions are applied at the fluidsolid interface; \(\tilde{{{{{{{{\bf{u}}}}}}}}}=1{{{{{{{{\bf{e}}}}}}}}}_{\theta }\) at \(\tilde{r}=1\) and \(\tilde{{{{{{{{\bf{u}}}}}}}}}={{{\Omega }}}_{2}/{{{\Omega }}}_{1}{{{{{{{{\bf{e}}}}}}}}}_{\theta }\) at \(\tilde{r}=3\) are adopted in the fluid domain with insulating boundary conditions; The external magnetic field is introduced by setting the scalar potential \(\phi={B}_{0}\tilde{z}\) at the outer boundary of the vacuum domain. In simulation, the hydrodynamic Reynolds number is fixed at \({{{{{{{\rm{Re}}}}}}}}\equiv {{{\Omega }}}_{1}{r}_{1}^{2}/\nu=1000\), while the Rm and B_{0} are varied over their experimentally accessible ranges. The angular speed ratio in simulation is fixed at Ω_{1} : Ω_{3} : Ω_{2} = 1 : 0.58: 0.19, the same as the experiment.
The 3D simulations presented here are more faithful to the current experiment than our past numerical efforts. Some of the latter were 2D and axisymmetric^{33,36}, and therefore could not exhibit the experimentally observed nonaxisymmetric modes. In one of our previous 3D simulations^{42}, pseudovacuum boundary conditions were adopted for the vertical boundaries, obviating the strong coupling between the liquidmetal flow and conducting endcaps in the experiment (at that time, the experiment had insulating endcaps). In the 3D simulations of ref. 35, the external magnetic field was imposed at the beginning of the first hydrodynamic stage, and the simulation only lasted for a short period of time: long enough for the axisymmetric mode to saturate but not for nonaxisymmetric modes to grow. We have continued to run that simulation for a longer period of time and found that eventually nonaxisymmetric modes became significant. Compared with all our previous simulation, the simulation provided in this work has the best consistency with our experiment. This includes the 3D domains, the conducting boundary conditions and the way to introduce the external magnetic field.
Flow characterization in 3D simulation
Figure 10 shows a representative radial velocity profile v_{r}/(Ω_{1}r_{1}) in the relaxed hydrodynamic state of 3D simulation. It is found that there are nonaxisymmetric modes dominated by m = 2 in v_{r}/(Ω_{1}r_{1}), which we believe are the main cause of the positive growth rate of the m = 1 mode at B_{0} or low Rm in our 2mode simulation (see Fig. 5). Similar nonaxisymmetric structures are also found in the corresponding vertical velocity profile. Such structures only exist in simulation with low \({{{{{{{\rm{Re}}}}}}}}\) (\({{{{{{{\rm{Re}}}}}}}}=1000\)), but are completely suppressed in our high \({{{{{{{\rm{Re}}}}}}}}\) (\({{{{{{{\rm{Re}}}}}}}} \sim 1{0}^{6}\)) experiments^{39}. In addition, the frequencies of the nonaxisymmetric hydrodynamic modes are also different from those of the instability reported in the main text, indicating that the corresponding mechanisms are different.
Supplementary Fig. 5a shows the azimuthally and vertically averaged angular speed profile Ω(r) in the bulk region at Rm = 6 and B_{0} = 0.2. The origin of time (t = 0) in these plots coincides with the moment when the magnetic field is imposed in the second MHD stage. Three representative epochs are examined: tΩ_{1} = 0 corresponds to the relaxed hydrodynamic state, tΩ_{1} = 30 corresponds to the moment with growing nonaxisymmetric MHD modes, and tΩ_{1} = 100 corresponds to the saturated MHD state. Although the difference is small, the angular speed profile is indeed modified by the imposed magnetic field. Supplementary Fig. 5b shows the corresponding time evolution of q profile, which also changes after applying the magnetic field, thus confirming the modification of the base flow.
Azimuthal magnetic field
The rotating conductive endcaps drag the imposed magnetic field lines that are static in the lab frame, inducing an azimuthal magnetic field B_{ϕ} in the liquidmetal flow. As shown in Supplementary Fig. 6, the induced normalized azimuthal magnetic field B_{ϕ}/B_{z} in our 3D simulation is mainly concentrated in the region close to the endcaps. This is as expected since the endcaps have the most influence on the flow in contact with them. Due to the system’s reflection symmetry about the midplane, B_{ϕ} in the upper and lower halfplanes changes sign. Overall, B_{ϕ}/B_{z} is small for all Rm and B_{0} studied here, less than 0.15 in the region close to the endcap and less than 0.05 in the bulk region. Similar results were obtained for all Rm values studied here.
Data availability
Source data of plots in the main text are deposited in the DataSpace at Princeton University that is available to the public via https://dataspace.princeton.edu/handle/88435/dsp01x920g025r. All other data that support the plots within this paper and other findings of this study are available from the author upon reasonable request.
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Acknowledgements
This research was supported by U.S. DoE (Contract No. DEAC0209CH11466), NASA (Grant No. NNH15AB25I), NSF (Grant No. AST2108871), and the MaxPlanckPrinceton Center for Plasma Physics (MPPC). This publication was supported by the Princeton University Library Open Access Fund. E.G. and H.J. acknowledge support by S. Prager and Princeton University. H.J. acknowledges the contribution by E. Schartman of Nova Photonics, Inc. funded by NSF.
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H.J. and J.G. initiated the research. Y.W. and E.G. performed the experiments. K.C. participated in the early experiments. F.E. updated the simulation setup with the latest version of the SFEMaNS code. Y.W. performed the simulations with the help of F.E. and H.W. Y.W. analyzed the data and prepared the figures with the help of E.G., F.E., J.G., and H.J. Y.W. drafted the manuscript. Y.W., E.G., F.E., J.G., and H.J. discussed the results and revised the manuscript together.
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Wang, Y., Gilson, E.P., Ebrahimi, F. et al. Identification of a nonaxisymmetric mode in laboratory experiments searching for standard magnetorotational instability. Nat Commun 13, 4679 (2022). https://doi.org/10.1038/s41467022322780
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DOI: https://doi.org/10.1038/s41467022322780
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