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From helical to standard magnetorotational instability: Predictions for upcoming liquid sodium experiments

Ashish Mishra1,*, George Mamatsashvili1,2, and Frank Stefani1

  • 1Helmholtz-Zentrum Dresden-Rossendorf, Bautzner Landstrasse 400, D-01328 Dresden, Germany
  • 2Abastumani Astrophysical Observatory, Abastumani 0301, Georgia and Institute of Geophysics, Tbilisi State University, Tbilisi 0193, Georgia

  • *a.mishra@hzdr.de
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Phys. Rev. Fluids 7, 064802 – Published 21 June, 2022

DOI: https://doi.org/10.1103/PhysRevFluids.7.064802

Abstract

We conduct a linear analysis of axisymmetric magnetorotational instability (MRI) in a magnetized cylindrical Taylor-Couette (TC) flow for its standard version (SMRI) with a purely axial background magnetic field and two additional types—helically modified SMRI (H-SMRI) and helical MRI (HMRI)—in the presence of combined axial and azimuthal magnetic fields. This study is intended as preparatory for upcoming new cutting-edge large-scale liquid sodium MRI experiments planned within the DRESDYN project at Helmholtz-Zentrum Dresden-Rossendorf, so we explore these instability types for typical values of the main parameters: the magnetic Reynolds number, the Lundquist number, and the ratio of the angular velocities of the cylinders, which are attainable in these experiments. In contrast to previous attempts at detecting MRI in the laboratory, our results demonstrate that SMRI and its helically modified version can in principle be detected in the DRESDYN-TC device for the range of the above parameters, including the astrophysically most important Keplerian rotation, despite the extremely small magnetic Prandtl number of liquid sodium. Since in the experiments we plan to approach (H-)SMRI from the previously studied HMRI regime, we characterize the continuous and monotonous transition between these two regimes. We show that H-SMRI, like HMRI, represents an overstability (traveling wave) with nonzero frequency linearly increasing with azimuthal field. Because of its relevance to finite-size flow systems in experiments, we also analyze the absolute form of H-SMRI and compare its growth rate and onset criterion with the convective one.

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References (56)

  1. E. P. Velikhov, Stability of an ideally conducting liquid flowing between rotating cylinders in a magnetic field, J. Exptl. Theoret. Phys. (U.S.S.R.) 36, 1398 (1959).
  2. S. A. Balbus and J. F. Hawley, A powerful local shear instability in weakly magnetized disks. I. Linear analysis, Astrophys. J. 376, 214 (1991).
  3. G. Rüdiger, M. Gellert, R. Hollerbach, M. Schultz, and F. Stefani, Stability and instability of hydromagnetic Taylor–Couette flows, Phys. Rep. 741, 1 (2018).
  4. G. R. J. Lesur, Magnetohydrodynamics of protoplanetary discs, J. Plasma Phys. 87, 205870101 (2021).
  5. D. R. Sisan, N. Mujica, W. A. Tillotson, Y.-M. Huang, W. Dorland, A. B. Hassam, T. M. Antonsen, and D. P. Lathrop, Experimental Observation and Characterization of the Magnetorotational Instability, Phys. Rev. Lett. 93, 114502 (2004).
  6. M. D. Nornberg, H. Ji, E. Schartman, A. Roach, and J. Goodman, Observation of Magnetocoriolis Waves in a Liquid Metal Taylor-Couette Experiment, Phys. Rev. Lett. 104, 074501 (2010).
  7. A. H. Roach, E. J. Spence, C. Gissinger, E. M. Edlund, P. Sloboda, J. Goodman, and H. Ji, Observation of a Free-Shercliff-Layer Instability in Cylindrical Geometry, Phys. Rev. Lett. 108, 154502 (2012).
  8. D. M. H. Hung, E. G. Blackman, K. J. Caspary, E. P. Gilson, and H. Ji, Experimental confirmation of the standard magnetorotational instability mechanism with a spring-mass analogue, Commun. Phys. 2, 7 (2019).
  9. R. Hollerbach and G. Rüdiger, New Type of Magnetorotational Instability in Cylindrical Taylor-Couette Flow, Phys. Rev. Lett. 95, 124501 (2005).
  10. W. Liu, J. Goodman, I. Herron, and H. Ji, Helical magnetorotational instability in magnetized Taylor-Couette flow, Phys. Rev. E 74, 056302 (2006).
  11. O. N. Kirillov and F. Stefani, On the relation of standard and helical magnetorotational instability, Astrophys. J. 712, 52 (2010).
  12. O. N. Kirillov, F. Stefani, and Y. Fukumoto, Local instabilities in magnetized rotational flows: a short-wavelength approach, J. Fluid Mech. 760, 591 (2014).
  13. O. N. Kirillov and F. Stefani, Extending the Range of the Inductionless Magnetorotational Instability, Phys. Rev. Lett. 111, 061103 (2013).
  14. G. Mamatsashvili and F. Stefani, Linking dissipation-induced instabilities with nonmodal growth: The case of helical magnetorotational instability, Phys. Rev. E 94, 051203(R) (2016).
  15. R. Hollerbach, V. Teeluck, and G. Rüdiger, Nonaxisymmetric Magnetorotational Instabilities in Cylindrical Taylor-Couette Flow, Phys. Rev. Lett. 104, 044502 (2010).
  16. M. Seilmayer, V. Galindo, G. Gerbeth, T. Gundrum, F. Stefani, M. Gellert, G. Rüdiger, M. Schultz, and R. Hollerbach, Experimental Evidence for Nonaxisymmetric Magnetorotational Instability in a Rotating Liquid Metal Exposed to an Azimuthal Magnetic Field, Phys. Rev. Lett. 113, 024505 (2014).
  17. F. Stefani, T. Gundrum, G. Gerbeth, G. Rüdiger, M. Schultz, J. Szklarski, and R. Hollerbach, Experimental Evidence for Magnetorotational Instability in a Taylor-Couette Flow under the Influence of a Helical Magnetic Field, Phys. Rev. Lett. 97, 184502 (2006).
  18. F. Stefani, G. Gerbeth, T. Gundrum, R. Hollerbach, J. Priede, G. Rüdiger, and J. Szklarski, Helical magnetorotational instability in a Taylor-Couette flow with strongly reduced Ekman pumping, Phys. Rev. E 80, 066303 (2009).
  19. A. Mishra, G. Mamatsashvili, V. Galindo, and F. Stefani, Convective, absolute and global azimuthal magnetorotational instabilities, J. Fluid Mech. 922, R4 (2021).
  20. M. Seilmayer, F. Stefani, T. Gundrum, T. Weier, G. Gerbeth, M. Gellert, and G. Rüdiger, Experimental Evidence for a Transient Tayler Instability in a Cylindrical Liquid-Metal Column, Phys. Rev. Lett. 108, 244501 (2012).
  21. G. Mamatsashvili, F. Stefani, R. Hollerbach, and G. Rüdiger, Two types of axisymmetric helical magnetorotational instability in rotating flows with positive shear, Phys. Rev. Fluids 4, 103905 (2019).
  22. J. Goodman and H. Ji, Magnetorotational instability of dissipative couette flow, J. Fluid Mech. 462, 365â382 (2002).
  23. G. Rüdiger, M. Schultz, and D. Shalybkov, Linear magnetohydrodynamic Taylor-Couette instability for liquid sodium, Phys. Rev. E 67, 046312 (2003).
  24. R. Hollerbach and A. Fournier, End-effects in rapidly rotating cylindrical Taylor-Couette flow, in MHD Couette Flows: Experiments and Models, AIP Conf. Series 733, edited by R. Rosner, G. Rüdiger, and A. Bonanno (AIP, New York, 2004), pp. 114–121
  25. J. Szklarski, Reduction of boundary effects in the spiral MRI experiment PROMISE, Astron. Nachr. 328, 499 (2007).
  26. C. Gissinger, J. Goodman, and H. Ji, The role of boundaries in the magnetorotational instability, Phys. Fluids 24, 074109 (2012).
  27. J. Priede and G. Gerbeth, Absolute versus convective helical magnetorotational instability in a Taylor-Couette flow, Phys. Rev. E 79, 046310 (2009).
  28. J. Priede, I. Grants, and G. Gerbeth, Inductionless magnetorotational instability in a Taylor-Couette flow with a helical magnetic field, Phys. Rev. E 75, 047303 (2007).
  29. G. Mamatsashvili, F. Stefani, A. Guseva, and M. Avila, Quasi-two-dimensional nonlinear evolution of helical magnetorotational instability in a magnetized Taylor-Couette flow, New J. Phys. 20, 013012 (2018).
  30. G. Rüdiger and R. Hollerbach, Comment on “Helical magnetorotational instability in magnetized Taylor-Couette flow,” Phys. Rev. E 76, 068301 (2007).
  31. J. Priede, Inviscid helical magnetorotational instability in cylindrical Taylor-Couette flow, Phys. Rev. E 84, 066314 (2011).
  32. P. Huerre and P. Monkewitz, Local and global instabilities in spatially developing flows, Annu. Rev. Fluid Mech. 22, 473 (1990).
  33. J.-M. Chomaz, Global instabilities in spatially developing flows: Non-normality and nonlinearity, Annu. Rev. Fluid Mech. 37, 357 (2005).
  34. A. Gailitis, Design of a liquid sodium mhd dynamo experiment, Magnetohydrodynamics 32, 58 (1996).
  35. F. Stefani, G. Gerbeth, and A. Gailitis, Velocity profile optimization for the riga dynamo experiment, in Transfer Phenomena in Magnetohydrodynamic and Electroconducting Flows, Fluid Mechanics and Its Applications, edited by A. Alemany, P. Marty, and J. P. Thibault (Kluwer Academic, Dordrecht, 1999), Vol. 51, pp. 31–44.
  36. A. Gailitis and Ya. Freibergs, Nature of the instability of a turbulent dynamo, Magnetohydrodynamics 16, 116 (1980).
  37. X. Wei, H. Ji, J. Goodman, F. Ebrahimi, E. Gilson, F. Jenko, and K. Lackner, Numerical simulations of the Princeton magnetorotational instability experiment with conducting axial boundaries, Phys. Rev. E 94, 063107 (2016).
  38. H. W. Winarto, H. Ji, J. Goodman, F. Ebrahimi, E. P. Gilson, and Y. Wang, Parameter space mapping of the Princeton magnetorotational instability experiment, Phys. Rev. E 102, 023113 (2020).
  39. S. C. Reddy, P. J. Schmid, and D. S. Henningson, Pseudospectra of the Orr–Sommerfeld operator, SIAM J. Appl. Math. 53, 15 (1993).
  40. L. N. Trefethen, A. E. Trefethen, S. C. Reddy, and T. A. Driscoll, Hydrodynamic stability without eigenvalues, Science 261, 578 (1993).
  41. P. J. Schmid and D. S. Henningson, Stability and Transition in Shear Flows (Springer, New York, NY, 2001).
  42. L. N. Trefethen and M. Embree, Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators (Princeton University Press, Princeton, NJ, 2005).
  43. P. J. Schmid, Nonmodal stability theory, Annu. Rev. Fluid Mech. 39, 129 (2007).
  44. S. Grossmann, The onset of shear flow turbulence, Rev. Mod. Phys. 72, 603 (2000).
  45. B. Eckhardt, T. M. Schneider, B. Hof, and J. Westerweel, Turbulence transition in pipe flow, Annu. Rev. Fluid Mech. 39, 447 (2007).
  46. G. Mamatsashvili, G. Khujadze, G. Chagelishvili, S. Dong, J. Jiménez, and H. Foysi, Dynamics of homogeneous shear turbulence: A key role of the nonlinear transverse cascade in the bypass concept, Phys. Rev. E 94, 023111 (2016).
  47. R. R. Kerswell, Nonlinear nonmodal stability theory, Annu. Rev. Fluid Mech. 50, 319 (2018).
  48. Á. Meseguer, Energy transient growth in the Taylor-Couette problem, Phys. Fluids 14, 1655 (2002).
  49. S. Maretzke, B. Hof, and M. Avila, Transient growth in linearly stable Taylor-Couette flows, J. Fluid Mech. 742, 254 (2014).
  50. G. R. Mamatsashvili, G. D. Chagelishvili, G. Bodo, and P. Rossi, Revisiting linear dynamics of non-axisymmetric perturbations in weakly magnetized accretion discs, Mon. Not. R. Astron. Soc. 435, 2552 (2013).
  51. J. Squire and A. Bhattacharjee, Magnetorotational instability: Nonmodal growth and the relationship of global modes to the shearing box, Astrophys. J. 797, 67 (2014).
  52. J. Squire and A. Bhattacharjee, Nonmodal Growth of the Magnetorotational Instability, Phys. Rev. Lett. 113, 025006 (2014).
  53. D. Gogichaishvili, G. Mamatsashvili, W. Horton, G. Chagelishvili, and G. Bodo, Nonlinear transverse cascade and sustenance of MRI turbulence in keplerian disks with an azimuthal magnetic field, Astrophys. J. 845, 70 (2017).
  54. D. Gogichaishvili, G. Mamatsashvili, W. Horton, and G. Chagelishvili, Active modes and dynamical balances in MRI turbulence of keplerian disks with a net vertical magnetic field, Astrophys. J. 866, 134 (2018).
  55. G. Mamatsashvili and F. Stefani, Nonmodal analysis of helical and azimuthal magnetorotational instabilities, Magnetohydrodynamics 53, 107 (2017).
  56. D. G. Meduri, F. Lignières, and L. Jouve, Nonaxisymmetric magnetorotational instability in spherical Couette flow, Phys. Rev. E 100, 013110 (2019).

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Phys. Rev. Fluids 7, 064802– Published 21 June, 2022
  • Received 2 December 2021
  • Accepted 1 June 2022
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