Abstract
The magnetorotational instability (MRI) has long been considered a plausibly ubiquitous mechanism to destabilize otherwise stable Keplerian flows to support radially outward transport of angular momentum. Such an efficient transport process would allow fast accretion in astrophysical objects such as stars and black holes to release copious kinetic energy that powers many of the most luminous sources in the universe. But the standard MRI under a purely vertical magnetic field has heretofore never been directly measured despite numerous efforts over more than a decade. Here we report an unambiguous laboratory demonstration of the springmass analogue to the standard MRI by comparing motion of a springtethered ball within different rotating flows. The experiment corroborates the theory: efficient outward angular momentum transport manifests only for cases with a weak spring in quasiKeperian flow. Our experimental method accomplishes this in a new way, thereby connecting solid and fluid mechanics to plasma astrophysics.
Introduction
Understanding angular momentum transport in astrophysical disks comprises a longstanding enterprise, spanning planetary, stellar, black hole, galactic, and laboratory astrophysics. The challenge originated 250 years ago^{1,2,3} with enduring questions about how the angular momentum distribution within the solar system evolved from its original nebular gas^{4,5,6}. In addition, luminous and jetted sources in the universe, including quasars, Xray binaries^{7,8,9}, preplanetary nebulae^{10,11}, and gammaray bursts^{12} are likely powered by the conversion of gravitational potential energy into kinetic energy and radiation, as matter accretes onto central engines^{13}. Since accreting plasma typically originates far from the core of the potential well, conserving even a modest initial angular momentum during infall would prevent matter from reaching the engines. Angular momentum must be extracted much faster than microphysical diffusivities alone allow.
Enhanced transport is typically parameterized by a “turbulent viscosity”, allowing practical accretion disk models to be compared with observations^{14}. What mechanisms supply enhanced transport and how to model it are longstanding physics problems of astrophysics^{15,16}. A ubiquitous source of turbulence is thought to be the magnetorotational instability (MRI)^{17,18} as applied to accretion discs^{19,20,21,22}: while purely hydrodynamic discs require a decreasing angular momentum gradient for linear instability, the MRI in a magnetohydrodynamic (MHD) disk requires only a radially decreasing angular velocity, so magnetized Keplerian disks of astrophysics should be unstable. Growth and saturation of the MRI are widely studied^{23,24,25,26,27,28,29,30}.
The scientific method establishes scientific fact by corroborating theory with experiment, no matter how widely assumed the veracity of a theoretically calculated mechanism may otherwise be. As such, there are substantial efforts to demonstrate the MRI in the laboratory using differentially rotating liquid metals^{31,32,33} and plasma^{34}, and even polymer fluids^{35,36} or an elastic beam^{37}. Purely hydrodynamic flow experiments confirm the Rayleigh criterion for stability^{38,39}. Measurements of the MRI in the standard setup with a purely vertical field in liquid metals are challenging, although recent evidence of related helical and azimuthal field MRI has been reported^{40,41}. The result of ref. ^{32}, for example, is now understood to result from boundary effects^{42}. There is further optimism as boundary control improves^{43,44}, but so far, none of these experiments have yet demonstrated the vertical MRI.
Here we take a different approach. We appeal to the known result that the dispersion relation of the MRI for an initially vertical magnetic field also characterizes the motion of two masses tethered by a weak spring^{16,22}. The spring represents the magnetic field and the mass represents a parcel of MHD fluid. It has been speculated^{16} that this analogue might be experimentally testable in the laboratory, distinct from multitethered configurations that have been previously theoretically explored^{45,46,47}.
Below we discuss the design and results from a new tethered ball experiment using the Princeton Taylor–Couette apparatus with water or Hydrodynamic Turbulence Experiment (HTX)^{48}. We compare the radial motion of the ball for cases when the ball is untethered, weakly tethered, and strongly tethered. As predicted by the MRI mechanism, angular momentum is transported efficiently outward only in the cases with a weak spring in quasiKeplerian flows. The experiment demonstrates a new way to use solid and fluid mechanics to study astrophysical processes in the lab.
Results
Theoretical model and predictions
A Keplerian flow with a weak vertical magnetic field, B_{z}, subjected to perturbations within the horizontal plane (r, θ) exhibits the MRI. The minimalist MHD version of the equations depends only on the displacement of the field lines in the plane perpendicular to the initial magnetic field. Two masses tethered by a weak spring orbiting in a central potential^{15} then provide an analogue of this local instability, although the minimalist MHD MRI equations most directly correspond to the motion of a single mass tethered to a fixed point in a corotating frame^{16} (Fig. 1a vs. Fig. 1b).
Physically, the linear phase of the instability interpreted in the context of Fig. 1b is expected to occur as follows: a light test mass is released from a post that is fixed to orbit with the flow at angular speed Ω_{3}. The mass is tethered to a weak spring. If the spring is weak enough such that oscillation time is significantly longer than an orbit time but still strong enough to couple the post and mass over this time scale, the post will transmit angular momentum to the test mass moving the latter outward. If the spring is too strong, outward motion is limited by the spring tension, effectively retaining the ball as part of the post.
Mathematical correspondence between the minimalist MRI unstable MHD equations and those of tethered mass motion is simplest in local Cartesian coordinates x, y, z in a rotating frame with radius r = r_{0} + x and r_{0}(θ − θ_{0}) = y, with fixed point at x = y = 0. This point moves in the lab frame with angular velocity Ω_{3} ≡ Ω(x = 0) and the shear flow away from the fixed point in the rotating frame is given by \(r(\Omega  \Omega _3) \simeq xr\partial _r\Omega _{r = r_0} =  xq\Omega _3\), with q ≡ −dlnΩ/dlnr. For the MHD case, when the centrifugal force is balanced by gravity and total pressure gradients are ignored, the local 2D MHD momentum equations are
Dots indicate time derivatives; \(T = 2q\Omega _3^2\) is the coefficient of the tidal force per unit mass; the second terms on the left sides come from the Coriolis force; K_{A} = (kv_{A})^{2}, arises from magnetic tension where v_{A} is the Alfvén speed associated with the vertical field.
Equations (1) and (2) also approximate motion of a mass tethered to a fixed point x,y = 0 by a spring with spring constant per unit mass K_{A}, as in Fig. 1b. [For Fig. 1a this requires Ω = Ω_{3} and K_{A} →2 K_{A}^{16}.] The Coriolis and tidal force terms arise whether supplied by gravity without pressure gradients, or by pressure gradients when the mass is embedded in a laboratory quasiKeplerian (qK) flow without gravity. For initial displacements [x(t)e^{ikz}, y(t)e^{ikz}] and q < 0, the system is stable. But for q > 0, when K_{A} < T, the MRI instability ensues. For K_{A} = 0 (no spring), the right side of Eq. (2) vanishes and \(\mathop {x}\limits^{...} = \dot x(T  4\Omega _3^2)\). The behavior then depends on q: the coefficient of \(\dot x\) changes sign at q = 2, and instability occurs only for q > 2—the Rayleigh unstable regime.
Although the Cartesian approximation captures the MRI mechanism, modeling the MRI mechanism with our our TaylorCouette experiment requires inclusion of the nonlinear curvature and damping terms. In cylindrical coordinates, the vector labframe equation of motion for a tethered mass in the rotating background flow is
where t is time; \({\mathbf{r}} = r\widehat {\mathbf{e}}_{\mathrm{r}}\) and \({\mathbf{r}}_{\mathrm{p}} = r_0\widehat {\mathbf{e}}_{{\mathrm{r}}_0}\) are the timedependent position vectors of the ball and its launch locus (the post) respectively; \(\bar K\) is the spring constant divided by the mass of the ball; \({\mathrm{\Omega }}(r) \simeq \Omega _0(r/r_0)^{  q}\), where q is a constant; and \({\mathbf{f}}_{\mathrm{c}} =  r\Omega ^2(r)\widehat {\mathbf{e}}_r\) is the centripetal force per unit mass on the ball, supplied by the background fluid pressure gradient transmitted from the outer wall. It is equal and opposite in magnitude to the centrifugal force per unit mass of the flow of the local rotating frame when the background flow is in equilibrium. Quantities D_{1} and D_{2} are the Stokes and Reynolds drag coefficients^{49} given by \(D_1 = 6\pi \rho _{\mathrm {H}_{\mathrm {2}}{\mathrm {O}}}\nu _{\mathrm {H}_{\mathrm {2}}{\mathrm {O}}}R/M\) and \(D_2 = C_D\pi \rho _{\mathrm {H}_{\mathrm {2}}{\mathrm {O}}}R^2/2M\), for water density \(\rho _{\mathrm {H}_{\mathrm {2}}{\mathrm {O}}}\), kinematic viscosity \(\nu _{\mathrm {H}_{\mathrm {2}}{\mathrm {O}}}\), test mass radius R, test mass M, and drag coefficient C_{D}. Using R = 1.27 cm and neutrally buoyant test mass, D_{1} = 0.0284 s^{−1} and D_{2} = 15.0 m^{−1} in our experiments.
Since \({\mathrm {d}}\widehat {\mathbf{e}}_r/{\mathrm {d}}t = \dot \theta \widehat {\mathbf{e}}_\theta\), Eq. (3) contains both the azimuthal and radial components of the force equation. For initial values \(r(0) = r_0;\theta (0) = \theta _{\mathrm{p}}(0) = \theta _0;\dot r_{\mathrm{p}} = 0,\dot \theta _{\mathrm{p}} = \Omega _3\), (where θ_{p} is the angular coordinate of the post), the coupled equations for r(t) and θ(t) are given by
where we have used \(\widehat {\mathbf{e}}_r \cdot \widehat {\mathbf{e}}_{r_p} = {\mathrm{cos}}(\theta  \theta _0  \Omega _3t)\) and \(\widehat {\mathbf{e}}_\theta \cdot \widehat {\mathbf{e}}_{r_p} = {\mathrm{sin}}(\theta  \theta _0  \Omega _3t).\) Eqs. (4) and (5) reduce to Eqs. (1) and (2) in the linear limit.
For realistic parameters, the D_{1} term is small. In the linear regime, the D_{2} term also does not contribute and Eqs. (4) and (5) then predict runaway displacement in the usual MRI unstable regimes, namely \(0 < q < 2\) and \(\bar K > 0\), but not \(0 < q < 2\) and \(\bar K = 0\) (Table 1). By choosing springs with proper strengths, the MRI mechanism can be directly tested using a tethered ball in qK flows.
We emphasize that even when D_{1} and D_{2} are small, the ball is still strongly coupled to the flow by the background fluid pressure forces. In the vertical direction the upward pressure force balances gravity to maintain neutral buoyancy which keeps the primary ball motion confined to 2D. The radial pressure force transmitted from the outer wall balances the outward radial force associated with rotation as we have discussed in defining f_{c} above.
Experimental measurements
For solidbody (q = 0) and qK (0 < q < 2) flows, we compare the motion of an untethered ball to that of a ball tethered to a post anchored at a local rotating frame (Ω_{3} = 80 rpm, clockwise) by a weak or strong spring. These cases are listed in Table 1.
Figure 2 shows polar coordinate and timedependent ball trajectories in the lab frame. Each solid line of a given color corresponds to a separate experimental run with the same initial conditions. The left and right column panels correspond to qK and solidbody flow cases respectively. For each run in the qK case, the ball is initially held to the post rotating at Ω_{3} which rotates slightly faster (and has more angular momentum) than the background flow at its radius, to minimize secondary Ekman flow, as in the cases with both caps^{48}. The ball therefore drifts to larger radii, regardless of whether it is tethered or untethered. However, the ball lags behind less in azimuth in the rotating frame for the tethered cases and thus advances ahead to more negative angles in the lab frame (Fig. 2a). The radial and azimuthal drift speeds are also different for tethered versus untethered cases. The radial velocity is lower for the tethered than untethered cases (Fig. 2c). The tethered cases exhibit faster angular speeds, as evidenced by their steeper slopes in Fig. 2e.
The dashed lines show the corresponding solutions to Eqs. (4) and (5). Amplitudes of oscillation modes across all presented cases are negligible compared to experimental noise. The very early time linear growth rate, within the noise, is consistent with the standard MRI growth rate with negligible Stokes drag D_{1}. At late times, saturation from nonlinear damping by the D_{2} term is most consistent with the data.
Most telling are the specific angular momentum evolution plots of Fig. 3. Figure 3a shows that for the qK flows, the angular momentum of the ball remains constant for the untethered case (solid black lines) as expected from angular momentum conservation. In contrast, the weak spring tethered ball gains angular momentum (solid red lines) as expected from the MRI. Figure 3c correspondingly shows that the tethered ball gains angular momentum as it moves outward.
For solid body flow, Fig. 2d shows that the ball hardly moves in radius from its initial position for either the weak spring case (red) or the untethered case (black). Correspondingly, Fig. 3b, d show little difference in the red and black lines for solidbody flow runs. The blue lines in the plots of Figs. 2 and 3 show the case of a strong spring where the MRI mechanism is predicted to be ineffective. All of these blue trajectories are consistent with theoretical expectation that outward motion is halted once the strong spring is taut and angular momentum transfer is abated. The initial radial drift and associated angular momentum gain in the strong spring case is due to a limitation of the experimental setup, namely that the spring anchor point is offset from the center of mass of the ball. This does not affect the physics conclusions.
Discussion
While many astrophysical processes are difficult to test and validate in the lab, theory should be experimentally validated when possible and this is one of the core pillars of the discipline of laboratory astrophysics. In this context, neither the standard MRI instability, nor its mechanical analogue have been previously demonstrated in the laboratory, despite their widespread use in theoretical astrophysics. Measurements from our new apparatus now experimentally confirm the mechanism of angular momentum transport by the MRI and thus support its validity.
The measurements are all consistent with the theoretical implications of Eqs. (4) and (5). Specifically, (i) only for the weak spring case with a qK (0 < q < 2) flow, does the MRIlike instability manifest, and sustain angular momentum transport from post to ball; (ii) measured trajectories of the ball agree with nonlinear model equations for weakspring tethered, strongspring tethered, and untethered cases for qK and solidbody flows; (iii) Reynolds drag eventually balances the spring force to saturate the instability in the tethered case. Larger experiments could better distinguish linear from nonlinear regimes and detailed investigations could further delineate the “weak” and “strong” spring transition.
Our springball apparatus highlights use of a novel combination of solid and fluid mechanics to test MHD principles in the lab. The apparatus requires careful choices of the experimental parameters to ensure that the MHD analogue is captured: the dominant forces governing the motion of the ball must directly correspond to the dominant forces governing the motion of a parcel of MHD fluid for the chosen experimental design.
Methods
Apparatus
The experiments were carried out in a modified Taylor–Couette device (Fig. 4) using water and an open top cap. Two coaxial cylinders with height h = 39.7 cm, and radii r_{1} = 6.9 cm and r_{2} = 20.3 cm, were driven by motors at two independent angular rotation rates Ω_{1} and Ω_{2}. qK flows in which Ω_{1} > Ω_{2} while \(\Omega _1r_1^2 < \Omega _2r_2^2\) can be established. To minimize secondary Ekman flow, axial boundaries are divided into three annuli. The innermost annulus with r < 8 cm corotates with the inner cylinder while the outermost annulus with r > 14 cm corotates with the outer cylinder. The intermediate annulus where 8 cm < r < 14 cm is driven by a third motor at a rotation rate Ω_{3}. The secondary flow can be minimized by a suitable choice of Ω_{3}, resulting in an extremely quiescent qK flow^{48}. Our experiments used only the bottom boundary, allowing top access to the interior. To avoid significant fluid height variation that occurs on a rotating free surface, the rotation rates were limited to Ω_{1} = 190 rpm, Ω_{3} = 80 rpm, and Ω_{2} = 22 rpm. Measurements of the azimuthal velocity at the midheight of the fluid using laser Doppler velocimetry confirmed that the flow had nearly the ideal Couette profile with negligible Ekman effect (as using both axial boundaries^{48}) with q ≤ 2 with little dependence on r and z. Practical limitations on rotation rates and spring constants led us to use 1inch diameter waterfilled plastic spheres, of total mass 8.43 g. With any tethering spring, they were nearly neutrally buoyant. The finite size of the spherical test masses, as compared with r_{1} and r_{2} is included in the analysis as discussed above. The test mass was held in place by a clamp attached to a vertical post mounted at r_{0} = 10.8 cm on the annular ring rotating at Ω_{3}. This radius was originally selected so that Ω_{3} = Ω_{TC}(r_{0}) where Ω_{TC}(r) is the ideal Couette profile with a Ω_{1}:Ω_{3}:Ω_{2} = 190:80:22. The height l = 12.7 cm of the vertical post was chosen so that the test mass would sit away from the lower boundary and the top surface at z = 31.1 cm. The clamp release was triggered by hand using a metal arm fixed in the laboratory frame. The test mass was either untethered to the vertical post, or tethered with either a weak or strong spring. The springs had measured spring constants of k_{weak} = 0.636 N m^{−1} and k_{strong} = 51.5 N m^{−1}. We estimate the effective Reynolds number of the flow around the ball using \({\mathrm{Re}} = 2{\mathrm{R}}[\dot{r}^{2} + {r}^{2}[\dot{\theta}  \Omega (r)]^2]^{1/2}/\nu _{{\mathrm {H}}_{2}{\mathrm {O}}}\), and find maximum values Re ≈ 5000–20,000 for qK runs and Re ≈ 1 for solid body. The former values are consistent with the importance of the D_{2} term in Eqs. (4) and (5).
Diagnostics
We mounted a compact batterypowered, waterproof, video camera in the rotating frame of the vertical post with rotation rate Ω_{3} so that the test mass appeared stationary until release at t = 0. The camera captured 120 frames per second and the lens was slightly immersed in the water to minimize further optical distortions due to the fluid free surface. After each run, the recorded video was transferred to a computer. The camera uses a “fisheye” lens for a wide fieldofview, but this distortion was readily removed using commonly available software. The location of the center of the test mass in each frame was determined automatically by object identification and tracking software. Cartesian image data were converted into polar coordinates. From the position data, velocities, acceleration, and the vertical component of the angular momentum were calculated. The accuracy of the position data is limited by factors such as motion blur, tracking errors, the abilities to correct for lens distortion and refraction.
Data availability
The digital data for this paper can be found at http://arks.princeton.edu/ark:/88435/dsp01x920g025r.
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Acknowledgements
D.H., K.C., E.G., and H.J. acknowledge support from NASA (NNH15AB25I), NSF (AST1312463), and DoE (DEAC0209CH11466). E.B. acknowledges support from the Simons Foundation and the Institute for Advanced Study (Princeton) while on sabbatical, and grants NSFAST15156489 and HSTAR13916, the Kavli Institute for Theoretical Physics (KITP) USCB with associated support from grant NSF PHY1125915. Authors acknowledge technical support by Peter Sloboda.
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E.B. and H.J. initiated the research. D.H. modified the apparatus and conducted the experiments with the help of K.C. and E.G., guided by H.J. and E.B. D.H. analyzed the data, performed theoretical calculations and generated result figures with the guidance of all other authors. H.J. generated analogue diagrams and E.G. generated apparatus figure. E.B., D.H., E.G., and H.J. drafted and revised the manuscript. All authors discussed the results and interpretations.
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Hung, D.M.H., Blackman, E.G., Caspary, K.J. et al. Experimental confirmation of the standard magnetorotational instability mechanism with a springmass analogue. Commun Phys 2, 7 (2019). https://doi.org/10.1038/s4200501801037
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