Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-17T14:55:59.102Z Has data issue: false hasContentIssue false
  • English
  • Français

Non-ideal instabilities in sinusoidal shear flows with a streamwise magnetic field

Published online by Cambridge University Press:  06 October 2022

A.E. Fraser Open the ORCID record for A.E. Fraser [Opens in a new window] ,
I.G. Cresswell Open the ORCID record for I.G. Cresswell [Opens in a new window]  and
P. Garaud Open the ORCID record for P. Garaud [Opens in a new window]
A.E. Fraser*
Affiliation:
Department of Applied Mathematics, Baskin School of Engineering, University of California Santa Cruz, Santa Cruz, CA 95064, USA
I.G. Cresswell
Affiliation:
Department of Astrophysical and Planetary Sciences & LASP, University of Colorado, Boulder, CO 80309, USA
P. Garaud
Affiliation:
Department of Applied Mathematics, Baskin School of Engineering, University of California Santa Cruz, Santa Cruz, CA 95064, USA
*
†Email address for correspondence: adr.fraser@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate the linear stability of a sinusoidal shear flow with an initially uniform streamwise magnetic field in the framework of incompressible magnetohydrodynamics (MHD) with finite resistivity and viscosity. This flow is known to be unstable to the Kelvin–Helmholtz instability in the hydrodynamic case. The same is true in ideal MHD, where dissipation is neglected, provided the magnetic field strength does not exceed a critical threshold beyond which magnetic tension stabilizes the flow. Here, we demonstrate that including viscosity and resistivity introduces two new modes of instability. One of these modes, which we refer to as an Alfvénic Dubrulle–Frisch instability, exists for any non-zero magnetic field strength as long as the magnetic Prandtl number ${{{Pm}}} < 1$. We present a reduced model for this instability that reveals its excitation mechanism to be the negative eddy viscosity of periodic shear flows described by Dubrulle & Frisch (Phys. Rev. A, vol. 43, 1991, pp. 5355–5364). Finally, we demonstrate numerically that this mode saturates in a quasi-stationary state dominated by counter-propagating solitons.

JFM classification

Instability: Shear-flow instability MHD and Electrohydrodynamics: Plasmas
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 949 , 25 October 2022 , A43
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Ascher, U.M., Ruuth, S.J. & Spiteri, R.J. 1997 Implicit-explicit Runge–Kutta methods for time-dependent partial differential equations. Appl. Numer. Maths 25 (2), 151–167.CrossRefGoogle Scholar
Baines, P.G. & Gill, A.E. 1969 On thermohaline convection with linear gradients. J. Fluid Mech. 37 (2), 289–306.CrossRefGoogle Scholar
Balmforth, N.J. & Young, Y.-N. 2002 Stratified Kolmogorov flow. J. Fluid Mech. 450, 131–167.CrossRefGoogle Scholar
Barker, A.J., Jones, C.A. & Tobias, S.M. 2019 Angular momentum transport by the GSF instability: non-linear simulations at the equator. Mon. Not. R. Astron. Soc. 487 (2), 1777–1794.CrossRefGoogle Scholar
Brown, J.M., Garaud, P. & Stellmach, S. 2013 Chemical transport and spontaneous layer formation in fingering convection in astrophysics. Astrophys. J. 768 (1), 34.CrossRefGoogle Scholar
Burns, K.J., Vasil, G.M., Oishi, J.S., Lecoanet, D. & Brown, B.P. 2020 Dedalus: a flexible framework for numerical simulations with spectral methods. Phys. Rev. Res. 2 (2), 23068.CrossRefGoogle Scholar
Calzavarini, E., Doering, C.R., Gibbon, J.D., Lohse, D., Tanabe, A. & Toschi, F. 2006 Exponentially growing solutions in homogeneous Rayleigh–Bénard convection. Phys. Rev. E 73 (3), 035301.CrossRefGoogle ScholarPubMed
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.Google Scholar
Drazin, P.G. & Reid, W.H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Dubrulle, B. & Frisch, U. 1991 Eddy viscosity of parity-invariant flow. Phys. Rev. A 43 (10), 5355–5364.CrossRefGoogle ScholarPubMed
Faganello, M. & Califano, F. 2017 Magnetized Kelvin–Helmholtz instability: theory and simulations in the earth's magnetosphere context. J. Plasma Phys. 83 (6), 535830601.CrossRefGoogle Scholar
Fraser, A.E., Pueschel, M.J., Terry, P.W. & Zweibel, E.G. 2018 Role of stable modes in driven shear-flow turbulence. Phys. Plasmas 25 (12), 122303.CrossRefGoogle Scholar
Fricke, K. 1968 Instabilität stationärer rotation in sternen. Z. Astrophys. 68, 317.Google Scholar
Garaud, P., Gallet, B. & Bischoff, T. 2015 The stability of stratified spatially periodic shear flows at low Péclet number. Phys. Fluids 27 (8), 084104.CrossRefGoogle Scholar
Garaud, P., Ogilvie, G.I., Miller, N. & Stellmach, S. 2010 A model of the entropy flux and Reynolds stress in turbulent convection. Mon. Not. R. Astron. Soc. 407 (4), 2451–2467.CrossRefGoogle Scholar
Goldreich, P. & Schubert, G. 1967 Differential rotation in stars. Astrophys. J. 150, 571.CrossRefGoogle Scholar
Goodman, J. & Xu, G. 1994 Parasitic instabilities in magnetized. Differentially rotating disks. Astrophys. J. 432, 213.CrossRefGoogle Scholar
Gotoh, K., Yamada, M. & Mizushima, J. 1983 The theory of stability of spatially periodic parallel flows. J. Fluid Mech. 127, 45–58.CrossRefGoogle Scholar
Harrington, P.Z. & Garaud, P. 2019 Enhanced mixing in magnetized fingering convection, and implications for red giant branch stars. Astrophys. J. 870 (1), L5.CrossRefGoogle Scholar
von Helmholtz, H. 1896 Zwei hydrodynamische Abhandlungen: I. Ueber Wirbelbewegungen (1858) II. Ueber discontinuirliche Flüssigkeitsbewegungen (1868). W. Engelmann.Google Scholar
Henri, P., et al. 2013 Nonlinear evolution of the magnetized Kelvin–Helmholtz instability: from fluid to kinetic modeling. Phys. Plasmas 20 (10), 102118.CrossRefGoogle Scholar
Howard, L.N. 1961 Note on a paper of John W. Miles. J. Fluid Mech. 10 (4), 509–512.CrossRefGoogle Scholar
Hughes, D.W. & Tobias, S.M. 2001 On the instability of magnetohydrodynamic shear flows. Proc. R. Soc. Lond. Ser. A: Math. Phys. Engng Sci. 457 (2010), 1365–1384.CrossRefGoogle Scholar
Hunt, J.C.R. 1966 On the stability of parallel flows with parallel magnetic fields. Proc. R. Soc. Lond. Ser. A 293, 342–358.Google Scholar
Ivanov, P.G., Schekochihin, A.A., Dorland, W., Field, A.R. & Parra, F.I. 2020 Zonally dominated dynamics and dimits threshold in curvature-driven ITG turbulence. J. Plasma Phys. 86 (5), 855860502.CrossRefGoogle Scholar
Karimabadi, H., et al. 2013 Coherent structures, intermittent turbulence, and dissipation in high-temperature plasmas. Phys. Plasmas 20 (1), 012303.CrossRefGoogle Scholar
Latter, H.N., Lesaffre, P. & Balbus, S.A. 2009 MRI channel flows and their parasites. Mon. Not. R. Astron. Soc. 394 (2), 715–729.CrossRefGoogle Scholar
Lecoanet, D., McCourt, M., Quataert, E., Burns, K.J., Vasil, G.M., Oishi, J.S., Brown, B.P., Stone, J.M. & O'Leary, R.M. 2016 A validated non-linear Kelvin–Helmholtz benchmark for numerical hydrodynamics. Mon. Not. R. Astron. Soc. 455 (4), 4274–4288.CrossRefGoogle Scholar
Lecoanet, D., Zweibel, E.G., Townsend, R.H.D. & Huang, Y.-M. 2010 Violation of Richardson's criterion via introduction of a magnetic field. Astrophys. J. 712 (2), 1116–1128. arXiv:1002.3335.CrossRefGoogle Scholar
Longaretti, P.Y. & Lesur, G. 2010 MRI-driven turbulent transport: the role of dissipation, channel modes and their parasites. Astron. Astrophys. 516, A51. arXiv:1004.1384.CrossRefGoogle Scholar
Lucas, D. & Kerswell, R. 2014 Spatiotemporal dynamics in two-dimensional Kolmogorov flow over large domains. J. Fluid Mech. 750, 518–554.CrossRefGoogle Scholar
Mattingly, G.E. & Criminale, W.O. 1972 The stability of an incompressible two-dimensional wake. J. Fluid Mech. 51 (2), 233–272.CrossRefGoogle Scholar
Meshalkin, L.D. & Sinai, I.G. 1961 Investigation of the stability of a stationary solution of a system of equations for the plane movement of an incompressible viscous liquid. Z. Angew. Math. Mech. 25 (6), 1700–1705.CrossRefGoogle Scholar
Mikhaylov, K. & Wu, X. 2020 Nonlinear evolution of interacting sinuous and varicose modes in plane wakes and jets: quasi-periodic structures. Phys. Fluids 32 (6), 064104.CrossRefGoogle Scholar
Miles, J.W. 1961 On the stability of heterogeneous shear flows. J. Fluid Mech. 10, 496–508.CrossRefGoogle Scholar
Palotti, M.L., Heitsch, F., Zweibel, E.G. & Huang, Y.-M. 2008 Evolution of unmagnetized and magnetized shear layers. Astrophys. J. 678 (1), 234.CrossRefGoogle Scholar
Pessah, M.E. 2010 Angular momentum transport in protoplanetary and black hole accretion disks: the role of parasitic modes in the saturation of MHD turbulence. Astrophys. J. 716 (2), 1012–1027.CrossRefGoogle Scholar
Pessah, M.E. & Goodman, J. 2009 On the saturation of the magnetorotational instability via parasitic modes. Astrophys. J. 698 (1), L72–L76.CrossRefGoogle Scholar
Radko, T. & Smith, D.P. 2012 Equilibrium transport in double-diffusive convection. J. Fluid Mech. 692, 5–27.CrossRefGoogle Scholar
Rayleigh, Lord 1879 On the stability, or instability, of certain fluid motions. Proc. Lond. Math. Soc. s1-11 (1), 57–72.CrossRefGoogle Scholar
Reynolds, O. 1883 XXIX. An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels. Phil. Trans. R. Soc. Lond. 174, 935–982.Google Scholar
Rincon, F. 2019 Dynamo theories. J. Plasma Phys. 85 (4), 205850401.CrossRefGoogle Scholar
Rogers, B.N. & Dorland, W. 2005 Noncurvature-driven modes in a transport barrier. Phys. Plasmas 12 (6), 1–12.CrossRefGoogle Scholar
Tatsuno, T. & Dorland, W. 2006 Magneto-flow instability in symmetric field profiles. Phys. Plasmas 13 (9), 092107.CrossRefGoogle Scholar
Vogman, G.V., Hammer, J.H., Shumlak, U. & Farmer, W.A. 2020 Two-fluid and kinetic transport physics of Kelvin–Helmholtz instabilities in nonuniform low-beta plasmas. Phys. Plasmas 27 (10), 102109.CrossRefGoogle Scholar
Vorobev, A. & Zikanov, O. 2007 Instability and transition to turbulence in a free shear layer affected by a parallel magnetic field. J. Fluid Mech. 574, 131–154.CrossRefGoogle Scholar
Waleffe, F. 1995 Hydrodynamic stability and turbulence: beyond transients to a self-sustaining process. Stud. Appl. Maths 95 (3), 319–343. Available at: https://onlinelibrary.wiley.com/doi/pdf/10.1002/sapm1995953319.CrossRefGoogle Scholar
Waleffe, F. 1997 On a self-sustaining process in shear flows. Phys. Fluids 9 (4), 883–900.CrossRefGoogle Scholar
Zhu, H., Zhou, Y. & Dodin, I.Y. 2018 On the Rayleigh–Kuo criterion for the tertiary instability of zonal flows. Phys. Plasmas 25 (8), 082121.CrossRefGoogle Scholar