Hostname: page-component-76fb5796d-r6qrq Total loading time: 0 Render date: 2024-04-26T00:45:32.829Z Has data issue: false hasContentIssue false
  • English
  • Français

Equilibrium magnetohydrodynamic flows of liquid metals in magnetorotational instability experiments

Published online by Cambridge University Press:  27 January 2010

I. V. KHALZOV ,
A. I. SMOLYAKOV  and
V. I. ILGISONIS
I. V. KHALZOV*
Affiliation:
Center for Magnetic Self-Organization, University of Wisconsin, 1150 University Avenue, Madison, WI 53706, USA
A. I. SMOLYAKOV
Affiliation:
University of Saskatchewan, 116 Science Place, Saskatoon, Saskatchewan, S7N5E2, Canada
V. I. ILGISONIS
Affiliation:
Russian Research Center ‘Kurchatov Institute’, 1 Kurchatov Square, Moscow, 123182, Russia
*
†Email address for correspondence: khalzov@wisc.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

A theoretical analysis of equilibrium magnetohydrodynamic flows in annular channels is performed from the perspective of establishing required conditions for liquid metal magnetorotational instability (MRI) experiments. Two different types of fluid rotation are considered: electrically driven flow in an annular channel and Taylor–Couette flow between rotating cylinders. The structure of these flows is studied within a unified approach as a function of the Hartmann and Reynolds numbers. The parameters appropriate for realization of MRI experiments are determined.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 644 , 10 February 2010 , pp. 257 - 280
Copyright
Copyright © Cambridge University Press 2010

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

Acheson, D. J. & Hide, R. 1973 Hydromagnetics of rotating fluids. Rep. Prog. Phys. 36, 159–221.CrossRefGoogle Scholar
Balbus, S. A. 2003 Enhanced angular momentum transport in accretion disks. Annu. Rev. Astron. Astrophys. 41, 555–597.CrossRefGoogle Scholar
Balbus, S. A. & Hawley, J. F. 1991 A powerful local shear instability in weakly magnetized disks. I. Linear analysis. Astrophys. J. 376, 214–222.CrossRefGoogle Scholar
Balbus, S. A. & Hawley, J. F. 1998 Instability, turbulence, and enhanced transport in accretion disks. Rev. Mod. Phys. 70, 1–53.CrossRefGoogle Scholar
Baylis, J. A. & Hunt, J. C. R. 1971 MHD flow in an annular channel: theory and experiment. J. Fluid Mech. 48, 423–428.Google Scholar
Davidson, P. A. & Pothérat, A. 2002 A note on Bödewadt Hartmann layers. Eur. J. Mech. B. Fluids 21, 545–559.CrossRefGoogle Scholar
Ferraro, V. C. A. 1937 The non-uniform rotation of the Sun and its magnetic field. Mon. Not. R. Astron. Soc. 97, 458–472.CrossRefGoogle Scholar
Heiser, W. H. & Shercliff, J. A. 1965 A simple demonstration of the Hartmann layer. J. Fluid Mech. 22, 701–707.Google Scholar
Hollerbach, R. & Fournier, A. 2004 End-effects in rapidly rotating cylindrical Taylor–Couette flow. AIP Conf. Proc. 733, 114–121.Google Scholar
Hollerbach, R. & Rüdiger, G. 2005 New type of magnetorotational instability in cylindrical Taylor–Couette flow. Phys. Rev. Lett. 95, 124501.CrossRefGoogle ScholarPubMed
Ilgisonis, V. I. & Khalzov, I. V. 2007 Magnetorotational instability in a non-uniform magnetic field. JETP Lett. 86, 705–708.CrossRefGoogle Scholar
Ilgisonis, V. I., Khalzov, I. V. & Smolyakov, A. I. 2009 Negative energy waves and MHD stability of rotating plasmas. Nucl. Fusion 49, 035008.CrossRefGoogle Scholar
Ji, H., Burin, M., Schartman, E. & Goodman, J. 2006 Hydrodynamic turbulence cannot transport angular momentum effectively in astrophysical disks. Nature 444, 343–346.Google Scholar
Ji, H., Goodman, J., Kageyama, A., Burin, M., Schartman, E. & Liu, W. 2004 Magnetorotational instability in a short Couette flow of liquid gallium. AIP Conf. Proc. 733, 21–34.CrossRefGoogle Scholar
Kageyama, A., Ji, H., Goodman, J., Chen, F. & Shoshan, E. 2004 Numerical and experimental investigation of circulation in short cylinders. J. Phys. Soc. Japan 73, 2424–2437.CrossRefGoogle Scholar
Kaneda, M., Tagawa, T., Noir, J. & Aurnou, J. M. 2005 Variations in driving torque in Couette-Taylor flow subject to a vertical magnetic field. J. Phys.: Conf. Ser. 14, 42–47.Google Scholar
Khalzov, I. V., Ilgisonis, V. I., Smolyakov, A. I. & Velikhov, E. P. 2006 Magnetorotational instability in electrically driven flow of liquid metal: spectral analysis of global modes. Phys. Fluids 18, 124107.CrossRefGoogle Scholar
Khalzov, I. V. & Smolyakov, A. I. 2006 On the calculation of steady-state magnetohydrodynamic flows of liquid metals in circular ducts of a rectangular cross-section. Tech. Phys. 51, 26–33.CrossRefGoogle Scholar
Khalzov, I. V., Smolyakov, A. I. & Ilgisonis, V. I. 2008 Energy of eigenmodes in magnetohydrodynamic flows of ideal fluids. Phys. Plasmas 15, 054501.CrossRefGoogle Scholar
Lakhin, V. P. & Velikhov, E. P. 2007 Instabilities of highly-resistive rotating liquids in helical magnetic fields. Phys. Lett. A 369, 98–106.CrossRefGoogle Scholar
Liu, W. 2008 a Magnetized Ekman layer and Stewartson layer in a magnetized Taylor–Couette flow. Phys. Rev. E 77, 056314.Google Scholar
Liu, W. 2008 b Numerical study of the magnetorotational instability in Princeton MRI experiment. Astrophys. J. 684, 515–524.CrossRefGoogle Scholar
Liu, W., Goodman, J., Herron, I. & Ji, H. 2006 Helical magnetorotational instability in magnetized Taylor–Couette flow. Phys. Lett. E 74, 056302.Google ScholarPubMed
Liu, W., Goodman, J. & Ji, H. 2007 Traveling waves in a magnetized Taylor–Couette flow. Phys. Lett. E 76, 016310.Google Scholar
Moresco, P. & Alboussiére, T. 2003 Weakly nonlinear stability of Hartmann boundary layers. Eur. J. Mech. B. Fluids 22, 345–353.CrossRefGoogle Scholar
Moresco, P. & Alboussiére, T. 2004 Experimental study of the instability of the Hartmann layer. J. Fluid Mech. 504, 167–181.Google Scholar
Müller, U. & Bühler, L. 2001 Magnetofluiddynamics in Channels and Containers. Springer.CrossRefGoogle Scholar
Noguchi, K. & Pariev, V. 2003 Magnetorotational instability in a Couette flow of plasma. AIP Conf. Proc. 692, 285–292.Google Scholar
Noguchi, K., Pariev, V. I., Colgate, S. A., Beckley, H. F. & Nordhaus, J. 2002 Magnetorotational instability in liquid metal Couette flow. Astrophys. J. 575, 1151–1162.CrossRefGoogle Scholar
Pothérat, A., Sommeria, J. & Moreau, R. 2000 An effective two-dimensional model for MHD flows with transverse magnetic field. J. Fluid Mech. 424, 75–100.CrossRefGoogle Scholar
Press, W. H., Flannery, B. P., Teukolsky, S. A. & Vettering, W. 1992 Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. Cambridge University Press.Google Scholar
Rüdiger, G. & Hollerbach, R. 2007 Comment on ‘Helical magnetorotational instability in magnetized Taylor–Couette flow’. Phys. Lett. E 76, 068301.Google ScholarPubMed
Rüdiger, G., Schultz, M. & Shalybkov, D. 2003 Linear magnetohydrodynamic Taylor–Couette instability for liquid sodium. Phys. Rev. E 67, 046312.Google Scholar
Silk, J. & Langer, M. 2006 On the first generation of stars. Mon. Not. R. Astron. Soc. 371, 444–450.CrossRefGoogle Scholar
Sisan, D. R., Mujica, N., Tillotson, W. A., Huang, Y.-M., Dorland, W., Hassam, A. B., Antonsen, T. M. & Lathrop, D. P. 2004 Experimental observation and characterization of the magnetorotational instability. Phys. Rev. Lett. 93, 114502.CrossRefGoogle ScholarPubMed
Stefani, F., Gailitis, A. & Gerbeth, G. 2008 Magnetohydrodynamic experiments on cosmic magnetic fields. Z. Angew. Math. Mech. 88, 930–954.CrossRefGoogle Scholar
Stefani, F. & Gerbeth, G. 2004 MRI in Taylor–Dean flows. AIP Conf. Proc. 733, 100–113.CrossRefGoogle Scholar
Stefani, F., Gundrum, T., Gerbeth, G., Rüdiger, G., Schultz, M., Szklarski, J. & Hollerbach, R. 2006 Experimental evidence for magnetorotational instability in a Taylor–Couette flow under the influence of a helical magnetic field. Phys. Rev. Lett. 97, 184502.Google Scholar
Szklarski, J. 2007 Reduction of boundary effects in the spiral MRI experiment PROMISE. Astron. Nachr. 328, 499–506.Google Scholar
Szklarski, J. & Rüdiger, G. 2007 Ekman–Hartmann layer in a magnetohydrodynamic Taylor–Couette flow. Phys. Rev. E 76, 066308.CrossRefGoogle Scholar
Tagawa, T. & Kaneda, M. 2005 Numerical analyses of a Couette–Taylor flow in the presence of a magnetic field. J. Phys.: Conf. Ser. 14, 48–54.Google Scholar
Thess, A., Krasnov, D., Boeck, T., Zienicke, E., Zikanov, O., Moresco, P. & Alboussiére, T. 2007 Transition to turbulence in the Hartmann boundary layer. GAMM-Mitt. 30, 125–132.Google Scholar
Velikhov, E. P. 1959 Stability of an ideally conducting liquid flowing between cylinders rotating in a magnetic field. Sov. Phys. JETP 36, 995–998.Google Scholar
Velikhov, E. P., Ivanov, A. A., Lakhin, V. P. & Serebrennikov, K. S. 2006 a Magneto-rotational instability in differentially rotating liquid metals. Phys. Lett. A 356, 357–365.CrossRefGoogle Scholar
Velikhov, E. P., Ivanov, A. A., Zakharov, S. V., Zakharov, V. S., Livadny, A. O. & Serebrennikov, K. S. 2006 b Equilibrium of current driven rotating liquid metal. Phys. Lett. A 358, 216–221.CrossRefGoogle Scholar
Wendl, M. C. 1999 General solution for the Couette flow profile. Phys. Rev. E 60, 6192–6194.CrossRefGoogle ScholarPubMed
Youd, A. J. & Barenghi, C. F. 2006 Hydromagnetic Taylor–Couette flow at very small aspect ratio. J. Fluid Mech. 550, 27–42.CrossRefGoogle Scholar