Dissipation-Induced Instabilities in Magnetized Flows

Abstract

We study local instabilities of a differentially rotating viscous flow of electrically conducting incompressible fluid subject to an external azimuthal magnetic field. A hydrodynamically stable flow can be destabilized by the magnetic field both in an ideal and a viscous and resistive system giving rise to the azimuthal magnetorotational instability. A special solution to the equations of ideal magnetohydrodynamics characterized by the constant total pressure, the fluid velocity parallel to the direction of the magnetic field, and by the magnetic and kinetic energies that are finite and equal—the Chandrasekhar equipartition solution—is marginally stable in the absence of viscosity and resistivity. Performing a local stability analysis, we find the conditions under which the azimuthal magnetorotational instability can be interpreted as a dissipation-induced instability of the Chandrasekhar equipartition solution.

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References

  1. 1.

    V. I. Arnold, “On matrices depending on parameters,” Russ. Math. Surv., 26, 29–43 (1971).

    MathSciNet  Article  Google Scholar 

  2. 2.

    S. A. Balbus and J. F. Hawley, “A powerful local shear instability in weakly magnetized disks 1. Linear analysis,” Astrophys. J., 376, 214–222 (1991).

    Article  Google Scholar 

  3. 3.

    S. A. Balbus and J. F. Hawley, “A powerful local shear instability in weakly magnetized disks 4. Nonaxisymmetric perturbations,” Astrophys. J., 400, 610–621 (1992).

    Article  Google Scholar 

  4. 4.

    V. V. Beletsky and E. M. Levin, “Stability of a ring of connected satellites,” Acta Astron., 12, 765–769 (1985).

    Article  Google Scholar 

  5. 5.

    H. Bilharz, “Bemerkung zu einem Satze von Hurwitz,” Z. Angew. Math. Mech., 24, 77–82 (1944).

    MathSciNet  Article  Google Scholar 

  6. 6.

    A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden, and T. S. Ratiu, “Dissipation-induced instabilities,” Ann. Inst. H. Poincaré Anal. Non Linéaire, 11, 37–90 (1994).

    MathSciNet  Article  Google Scholar 

  7. 7.

    O. I. Bogoyavlenskij, “Unsteady equipartition MHD solutions,” J. Math. Phys., 45, 381–390 (2004).

    MathSciNet  Article  Google Scholar 

  8. 8.

    S. Boldyrev, D. Huynh, and V. Pariev, “Analog of astrophysical magnetorotational instability in a Couette–Taylor flow of polymer fluids,” Phys. Rev. E, 80, 066310 (2009).

    Article  Google Scholar 

  9. 9.

    V. V. Bolotin, Nonconservative Problems of the Theory of Elastic Stability, Pergamon Press, Oxford–London–New York–Paris (1963).

  10. 10.

    O. Bottema, “The Routh–Hurwitz condition for the biquadratic equation,” Indag. Math., 18, 403–406 (1956).

    MathSciNet  Article  Google Scholar 

  11. 11.

    T. J. Bridges and F. Dias, “Enhancement of the Benjamin–Feir instability with dissipation,” Phys. Fluids, 19, 104104 (2007).

    Article  Google Scholar 

  12. 12.

    S. Chandrasekhar, “On the stability of the simplest solution of the equations of hydromagnetics,” Proc. Natl. Acad. Sci. USA, 42, 273–276 (1956).

    MathSciNet  Article  Google Scholar 

  13. 13.

    S. Chandrasekhar, “The stability of nondissipative Couette flow in hydromagnetics,” Proc. Natl. Acad. Sci. USA, 46, 253–257 (1960).

    Article  Google Scholar 

  14. 14.

    S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Oxford University Press, Oxford (1961).

    Google Scholar 

  15. 15.

    S. Chandrasekhar, A Scientific Autobiography: S. Chandrasekhar, World Scientific, Singapore (2010).

  16. 16.

    S. Dobrokhotov and A. Shafarevich, “Parametrix and the asymptotics of localized solutions of the Navier–Stokes equations in R 3, linearized on a smooth flow,” Math. Notes, 51, 47–54 (1992).

    MathSciNet  Article  Google Scholar 

  17. 17.

    F. Ebrahimi, B. Lefebvre, C. B. Forest, and A. Bhattacharjee, “Global Hall-MHD simulations of magnetorotational instability in the plasma Couette flow experiment,” Phys. Plasmas, 18, 062904 (2011).

    Article  Google Scholar 

  18. 18.

    B. Eckhardt and D. Yao, “Local stability analysis along Lagrangian paths,” Chaos Solitons Fractals, 5 (11), 2073–2088 (1995).

    MathSciNet  Article  Google Scholar 

  19. 19.

    K. S. Eckhoff, “On stability for symmetric hyperbolic systems, I,” J. Differ. Equ., 40, 94–115 (1981).

    MathSciNet  Article  Google Scholar 

  20. 20.

    K. S. Eckhoff, “Linear waves and stability in ideal magnetohydrodynamics,” Phys. Fluids, 30, 3673–3685 (1987).

    Article  Google Scholar 

  21. 21.

    S. Friedlander and M. M. Vishik, “On stability and instability criteria for magnetohydrodynamics,” Chaos, 5, 416–423 (1995).

    MathSciNet  Article  Google Scholar 

  22. 22.

    S. V. Golovin and M. K. Krutikov, “Complete classification of stationary flows with constant total pressure of ideal incompressible infinitely conducting fluid,” J. Phys. A, 45, 235501 (2012).

    MathSciNet  Article  Google Scholar 

  23. 23.

    H. Ji and S. Balbus, “Angular momentum transport in astrophysics and in the lab,” Phys. Today, August 2013, 27–33 (2013).

    Article  Google Scholar 

  24. 24.

    P. L. Kapitsa, “Stability and passage through the critical speed of the fast spinning rotors in the presence of damping,” Z. Tech. Phys., 9, 124–147 (1939).

    Google Scholar 

  25. 25.

    O. N. Kirillov, “Campbell diagrams of weakly anisotropic flexible rotors,” Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 465, 2703–2723 (2009).

    MathSciNet  Article  Google Scholar 

  26. 26.

    O. N. Kirillov, “Stabilizing and destabilizing perturbations of PT-symmetric indefinitely damped systems,” Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 371, 20120051 (2013).

    MathSciNet  Article  Google Scholar 

  27. 27.

    O. N. Kirillov, Nonconservative Stability Problems of Modern Physics, De Gruyter, Berlin–Boston (2013).

    Google Scholar 

  28. 28.

    O. N. Kirillov and A. P. Seyranian, “Metamorphoses of characteristic curves in circulatory systems,” J. Appl. Math. Mech., 66 (3), 371–385 (2002).

    MathSciNet  Article  Google Scholar 

  29. 29.

    O. N. Kirillov and F. Stefani, “On the relation of standard and helical magnetorotational instability,” Astrophys. J., 712, 52–68 (2010).

    Article  Google Scholar 

  30. 30.

    O. N. Kirillov and F. Stefani, “Standard and helical magnetorotational instability: How singularities create paradoxal phenomena in MHD,” Acta Appl. Math., 120, 177–198 (2012).

    MathSciNet  Article  Google Scholar 

  31. 31.

    O. N. Kirillov and F. Stefani, “Extending the range of the inductionless magnetorotational instability,” Phys. Rev. Lett., 111, 061103 (2013).

    Article  Google Scholar 

  32. 32.

    O. N. Kirillov, F. Stefani, and Y. Fukumoto, “A unifying picture of helical and azimuthal MRI, and the universal significance of the Liu limit,” Astrophys. J., 756(83) (2012).

  33. 33.

    O. N. Kirillov, F. Stefani, and Y. Fukumoto, “Instabilities of rotational flows in azimuthal magnetic fields of arbitrary radial dependence,” Fluid Dyn. Res., 46, 031403 (2014).

    MathSciNet  Article  Google Scholar 

  34. 34.

    O. N. Kirillov, F. Stefani, and Y. Fukumoto, “Local instabilities in magnetized rotational flows: A short-wavelength approach,” J. Fluid Mech., 760, 591–633 (2014).

    MathSciNet  Article  Google Scholar 

  35. 35.

    O. N. Kirillov and F. Verhulst, “Paradoxes of dissipation-induced destabilization or who opened Whitney’s umbrella?” Z. Angew. Math. Mech., 90 (6), 462–488 (2010).

    MathSciNet  Article  Google Scholar 

  36. 36.

    R. Krechetnikov and J. E. Marsden, “Dissipation-induced instabilities in finite dimensions,” Rev. Mod. Phys., 79, 519–553 (2007).

    MathSciNet  Article  Google Scholar 

  37. 37.

    E. R. Krueger, A. Gross, and R. C. Di Prima, “On relative importance of Taylor-vortex and nonaxisymmetric modes in flow between rotating cylinders,” J. Fluid Mech., 24 (3), 521–538 (1966).

    Article  Google Scholar 

  38. 38.

    V. V. Kucherenko and A. Kryvko, “Interaction of Alfvén waves in the linearized system of magnetohydrodynamics for an incompressible ideal fluid,” Russ. J. Math. Phys., 20 (1), 56–67 (2013).

    MathSciNet  Article  Google Scholar 

  39. 39.

    M. J. Landman and P. G. Saffman, “The three-dimensional instability of strained vortices in a viscous fluid,” Phys. Fluids, 30, 2339–2342 (1987).

    Article  Google Scholar 

  40. 40.

    W. F. Langford, “Hopf meets Hamilton under Whitney’s umbrella,” Solid Mech. Appl., 110, 157–165 (2003).

    MathSciNet  MATH  Google Scholar 

  41. 41.

    H. N. Latter, H. Rein, and G. I. Ogilvie, “The gravitational instability of a stream of coorbital particles,” Mon. Not. R. Astron. Soc., 423, 1267–1276 (2012).

    Article  Google Scholar 

  42. 42.

    W. Liu, J. Goodman, I. Herron, and H. Ji, “Helical magnetorotational instability in magnetized Taylor–Couette flow,” Phys. Rev. E, 74 (5), 056302 (2006).

    MathSciNet  Article  Google Scholar 

  43. 43.

    R. S. MacKay, “Movement of eigenvalues of Hamiltonian equilibria under non-Hamiltonian perturbation,” Phys. Lett. A, 155, 266–268 (1991).

    MathSciNet  Article  Google Scholar 

  44. 44.

    J. H. Maddocks and M. L. Overton, “Stability theory for dissipatively perturbed Hamiltonian systems,” Commun. Pure Appl. Math., 48, 583–610 (1995).

    MathSciNet  Article  Google Scholar 

  45. 45.

    D. H. Michael, “The stability of an incompressible electrically conducting fluid rotating about an axis when current flows parallel to the axis,” Mathematika, 1, 45–50 (1954).

    MathSciNet  Article  Google Scholar 

  46. 46.

    D. Montgomery, “Hartmann, Lundquist, and Reynolds: The role of dimensionless numbers in nonlinear magnetofluid behavior,” Plasma Phys. Control. Fusion, 35, B105–B113 (1993).

    Article  Google Scholar 

  47. 47.

    G. I. Ogilvie and J. E. Pringle, “The nonaxisymmetric instability of a cylindrical shear flow containing an azimuthal magnetic field,” Mon. Not. R. Astron. Soc., 279, 152–164 (1996).

    Article  Google Scholar 

  48. 48.

    G. I. Ogilvie and A. T. Potter, “Magnetorotational-type instability in Couette–Taylor flow of a viscoelastic polymer liquid,” Phys. Rev. Lett., 100, 074503 (2008).

    Article  Google Scholar 

  49. 49.

    G. I. Ogilvie and M. R. E. Proctor, “On the relation between viscoelastic and magnetohydrodynamic flows and their instabilities,” J. Fluid Mech., 476, 389–409 (2003).

    MathSciNet  Article  Google Scholar 

  50. 50.

    J. W. S. Rayleigh, “On the dynamics of revolving fluids,” Proc. R. Soc. Lond. A, 93, 148–154 (1917).

    Article  Google Scholar 

  51. 51.

    G. Rüdiger, M. Gellert, M. Schultz, and R. Hollerbach, “Dissipative Taylor–Couette flows under the influence of helical magnetic fields,” Phys. Rev. E, 82, 016319 (2010).

    Article  Google Scholar 

  52. 52.

    G. Rüdiger, M. Gellert, M. Schultz, R. Hollerbach, and F. Stefani, “The azimuthal magnetorotational instability (AMRI),” Mon. Not. R. Astron. Soc., 438, 271–277 (2014).

    Article  Google Scholar 

  53. 53.

    G. Rüdiger, L. Kitchatinov, and R. Hollerbach, Magnetic Processes in Astrophysics, Wiley-VCH (2013).

  54. 54.

    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 an azimuthal magnetic field,” Phys. Rev. Lett., 113, 024505 (2014).

    Article  Google Scholar 

  55. 55.

    D. M. Smith, “The motion of a rotor carried by a flexible shaft in flexible bearings,” Proc. R. Soc. Lond. A, 142, 92–118 (1933).

    Article  Google Scholar 

  56. 56.

    J. Squire and A. Bhattacharjee, “Nonmodal growth of the magnetorotational instability,” Phys. Rev. Lett., 113, 025006 (2014).

    Article  Google Scholar 

  57. 57.

    F. Stefani, A. Gailitis, and G. Gerbeth, “Magnetohydrodynamic experiments on cosmic magnetic fields,” Z. Angew. Math. Mech., 88, 930–954 (2008).

    MathSciNet  Article  Google Scholar 

  58. 58.

    G. E. Swaters, “Modal interpretation for the Ekman destabilization of inviscidly stable baroclinic flow in the Phillips model,” J. Phys. Oceanogr., 40, 830–839 (2010).

    Article  Google Scholar 

  59. 59.

    S. A. Thorpe,W. D. Smyth, and L. Li, “The efect of small viscosity and diffusivity on the marginal stability of stably stratified shear flows,” J. Fluid Mech., 731, 461–476 (2013).

    MathSciNet  Article  Google Scholar 

  60. 60.

    E. P. Velikhov, “Stability of an ideally conducting liquid flowing between cylinders rotating in a magnetic field,” Sov. Phys. JETP-USSR, 9, 995–998 (1959).

    MathSciNet  Google Scholar 

  61. 61.

    M. Vishik and S. Friedlander, “Asymptotic methods for magnetohydrodynamic instability,” Quart. Appl. Math., 56, 377–398 (1998).

    MathSciNet  Article  Google Scholar 

  62. 62.

    V. A. Yakubovich and V. M. Starzhinskii, Linear Differential Equations with Periodic Coefficients. V. 1 and 2, Wiley, New York (1975).

    Google Scholar 

  63. 63.

    H. Ziegler, “Die Stabilitätskriterien der Elastomechanik,” Arch. Appl. Mech., 20, 49–56 (1952).

    MATH  Google Scholar 

  64. 64.

    R. Zou and Y. Fukumoto, “Local stability analysis of the azimuthal magnetorotational instability of ideal MHD flows,” Prog. Theor. Exp. Phys., 113J01 (2014).

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Correspondence to O. N. Kirillov.

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Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 60, Proceedings of the Seventh International Conference on Differential and Functional Differential Equations and InternationalWorkshop “Spatio-Temporal Dynamical Systems” (Moscow, Russia, 22–29 August, 2014). Part 3, 2016.

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Kirillov, O.N. Dissipation-Induced Instabilities in Magnetized Flows. J Math Sci 235, 455–472 (2018). https://doi.org/10.1007/s10958-018-4081-9

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