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Generalised quasilinear approximation of the helical magnetorotational instability

  • Adam Child (a1), Rainer Hollerbach (a1), Brad Marston (a2) and Steven Tobias (a1)


Motivated by recent advances in direct statistical simulation (DSS) of astrophysical phenomena such as out-of-equilibrium jets, we perform a direct numerical simulation (DNS) of the helical magnetorotational instability (HMRI) under the generalised quasilinear approximation (GQL). This approximation generalises the quasilinear approximation (QL) to include the self-consistent interaction of large-scale modes, interpolating between fully nonlinear DNS and QL DNS whilst still remaining formally linear in the small scales. In this paper we address whether GQL can more accurately describe low-order statistics of axisymmetric HMRI when compared with QL by performing DNS under various degrees of GQL approximation. We utilise various diagnostics, such as energy spectra in addition to first and second cumulants, for calculations performed for a range of Reynolds and Hartmann numbers (describing rotation and imposed magnetic field strength respectively). We find that GQL performs significantly better than QL in describing the statistics of the HMRI even when relatively few large-scale modes are kept in the formalism. We conclude that DSS based on GQL (GCE2) will be significantly more accurate than that based on QL (CE2).


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Avila, M. 2012 Stability and angular-momentum transport of fluid flows between corotating cylinders. Phys. Rev. Lett. 108, 124501.
Bai, X. N. & Stone, J. M. 2014 Magnetic flux concentration and zonal flows in magnetorotational instability turbulence. Astrophys. J. 796, 31.
Balbus, S. A. & Hawley, J. F. 1991 A powerful local shear instability in weakly magnetized disks. I – Linear analysis. II – Nonlinear evolution. Astrophys. J. 376, 214233.
Bouchet, F., Nardini, C. & Tangarife, T. 2013 Kinetic theory of jet dynamics in the stochastic barotropic and 2D Navier–Stokes equations. J. Stat. Phys. 153, 572625.
Constantinou, N. C., Farrell, B. F. & Ioannou, P. J. 2013 Emergence and equilibration of jets in beta-plane turbulence: applications of Stochastic Structural Stability Theory. J. Atmos. Sci. 72, 16891712.
Farrell, B. F. & Ioannou, P. J. 2007 Structure and spacing of jets in barotropic turbulence. J. Atmos. Sci. 64, 36523665.
Farrell, B. F. & Ioannou, P. J. 2008 Formation of jets by baroclinic turbulence. J. Atmos. Sci. 65, 3353.
Flanagan, K., Clark, M., Collins, C., Cooper, C. M., Khalzov, I. V., Wallace, J. & Forest, C. B. 2015 Prospects for observing the magnetorotational instability in the plasma Couette experiment. J. Plasma Phys. 81, 345810401.
Gissinger, C., Goodman, J. & Ji, H. 2012 The role of boundaries in the magnetorotational instability. Phys. Fluids 24, 074109.
Gressel, O. & Pessah, M. E. 2015 Characterizing the mean-field dynamo in turbulent accretion disks. Astrophys. J. 810, 59.
Guseva, A., Willis, A. P., Hollerbach, R. & Avila, M. 2015 Transition to magnetorotational turbulence in Taylor–Couette flow with imposed azimuthal magnetic field. New J. Phys. 17, 093018.
Hollerbach, R. 2008 Spectral solutions of the MHD equations in cylindrical geometry. Intl J. Pure Appl. Maths 42 (4), 575.
Hollerbach, R. & Fournier, A. 2004 End-effects in rapidly rotating cylindrical Taylor–Couette flow. In American Institute of Physics Conference Series, vol. 733, pp. 114121. AIP Publishing, arXiv:astro-ph/0506081.
Hollerbach, R. & Rüdiger, G. 2005 New type of magnetorotational instability in cylindrical Taylor–Couette flow. Phys. Rev. Lett. 95 (12), 124501.
Hollerbach, R., Teeluck, V. & Rüdiger, G. 2010 Nonaxisymmetric magnetorotational instabilities in cylindrical Taylor–Couette flow. Phys. Rev. Lett. 104, 044502.
Ji, H., Goodman, J. & Kageyama, A. 2001 Magnetorotational instability in a rotating liquid metal annulus. Mon. Not. R. Astron. Soc. 325, L1L5.
Julien, K. & Knobloch, E. 2010 Magnetorotational instability: recent developments. Phil. Trans. R. Soc. Lond. A 368 (1916), 16071633.
Kirillov, O. N. & Stefani, F. 2010 On the relation of standard and helical magnetorotational instability. Astrophys. J. 712 (1), 52.
Kirillov, O. N., Stefani, F. & Fukumoto, Y. 2014 Local instabilities in magnetized rotational flows: a short-wavelength approach. J. Fluid Mech. 760, 591633.
Knobloch, E. 1996 Symmetry and instability in rotating hydrodynamic and magnetohydrodynamic flows. Phys. Fluids 8, 14461454.
Liu, W., Goodman, J., Herron, I. & Ji, H. 2006 Helical magnetorotational instability in magnetized Taylor–Couette flow. Phys. Rev. E 74, 056302.
Marston, J. B., Chini, G. P. & Tobias, S. M.2016 The generalised quasilinear approximation: application to zonal jets, arXiv:1601.06720.
Marston, J. B., Qi, W. & Tobias, S. M.2014 Direct statistical simulation of a jet, arXiv:1412.0381.
Meheut, H., Fromang, S., Lesur, G., Joos, M. & Longaretti, P. Y. 2015 Angular momentum transport and large eddy simulations in magnetorotational turbulence: the small Pm limit. Astron. Astrophys. 579, A117.
Nornberg, M. D., Ji, H., Schartman, E., Roach, A. & Goodman, J. 2010 Observation of magnetocoriolis waves in a liquid metal Taylor–Couette experiment. Phys. Rev. Lett. 104 (7), 074501.
Priede, J. 2011 Inviscid helical magnetorotational instability in cylindrical Taylor–Couette flow. Phys. Rev. E 84, 066314.
Priede, J., Grants, I. & Gerbeth, G. 2007 Inductionless magnetorotational instability in a Taylor–Couette flow with a helical magnetic field. Phys. Rev. E 75, 047303.
Roach, A. H., Spence, E. J., Gissinger, C., Edlund, E. M., Sloboda, P., Goodman, J. & Ji, H. 2012 Observation of a free-Shercliff-layer instability in cylindrical geometry. Phys. Rev. Lett. 108, 154502.
Rüdiger, G., Hollerbach, R., Stefani, F., Gundrum, T., Gerbeth, G. & Rosner, R. 2006 The traveling-wave MRI in cylindrical Taylor–Couette flow: comparing wavelengths and speeds in theory and experiment. Astrophys. J. Lett. 649 (2), L145.
Rüdiger, G. & Zhang, Y. 2001 MHD instability in differentially-rotating cylindric flows. Astron Astrophys. 378 (1), 302308.
Schartman, E., Ji, H. & Burin, M. J. 2009 Development of a Couette–Taylor flow device with active minimization of secondary circulation. Rev. Sci. Instrum. 80, 024501.
Seilmayer, M., Galindo, V., Gerbeth, G., Gundrum, T., Stefani, F., Gellert, M., Rüdiger, G., Schultz, M. & Hollerbach, R. 2014 Experimental evidence for nonaxisymmetric magnetorotational instability in a rotating liquid metal exposed to an azimuthal magnetic field. Phys. Rev. Lett. 113, 024505.
Squire, J. & Bhattacharjee, A. 2015 Statistical simulation of the magnetorotational dynamo. Phys. Rev. Lett. 114 (8), 085002.
Srinivasan, K. & Young, W. R. 2012 Zonostrophic instability. J. Atmos. Sci. 69 (5), 16331656.
Stefani, F., Gerbeth, G., Gundrum, T., Hollerbach, R., Priede, J., Rüdiger, G. & Szklarski, J. 2009 Helical magnetorotational instability in a Taylor–Couette flow with strongly reduced Ekman pumping. Phys. Rev. E 80 (6), 066303.
Suzuki, T. K. & Inutsuka, S. I. 2014 Magnetohydrodynamic simulations of global accretion disks with vertical magnetic fields. Astrophys. J. 784, 121.
Tobias, S. M., Dagon, K. & Marston, J. B. 2011 Astrophysical fluid dynamics via direct statistical simulation. Astrophys. J. 727 (2), 127.
Tobias, S. M. & Marston, J. B. 2013 Direct statistical simulation of out-of-equilibrium jets. Phys. Rev. Lett. 110 (10), 104502.
Velikhov, E. P. 1959 Stability of an ideally conducting liquid flowing between rotating cylinders in a magnetic field. Sov. Phys. JETP 36, 995.
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Generalised quasilinear approximation of the helical magnetorotational instability

  • Adam Child (a1), Rainer Hollerbach (a1), Brad Marston (a2) and Steven Tobias (a1)


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