Letter | Published:

Hydrodynamic turbulence cannot transport angular momentum effectively in astrophysical disks

Naturevolume 444pages343346 (2006) | Download Citation



The most efficient energy sources known in the Universe are accretion disks. Those around black holes convert 5–40 per cent of rest-mass energy to radiation. Like water circling a drain, inflowing mass must lose angular momentum, presumably by vigorous turbulence in disks, which are essentially inviscid1. The origin of the turbulence is unclear. Hot disks of electrically conducting plasma can become turbulent by way of the linear magnetorotational instability2. Cool disks, such as the planet-forming disks of protostars, may be too poorly ionized for the magnetorotational instability to occur, and therefore essentially unmagnetized and linearly stable. Nonlinear hydrodynamic instability often occurs in linearly stable flows (for example, pipe flows) at sufficiently large Reynolds numbers. Although planet-forming disks have extreme Reynolds numbers, keplerian rotation enhances their linear hydrodynamic stability, so the question of whether they can be turbulent and thereby transport angular momentum effectively is controversial3,4,5,6,7,8,9,10,11,12,13,14,15. Here we report a laboratory experiment, demonstrating that non-magnetic quasi-keplerian flows at Reynolds numbers up to millions are essentially steady. Scaled to accretion disks, rates of angular momentum transport lie far below astrophysical requirements. By ruling out purely hydrodynamic turbulence, our results indirectly support the magnetorotational instability as the likely cause of turbulence, even in cool disks.

Access optionsAccess options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.


  1. 1

    Shakura, N. I. & Sunyaev, R. A. Black holes in binary systems. Observational appearance. Astron. Astrophys. 24, 337–355 (1973)

  2. 2

    Balbus, S. A. & Hawley, J. F. Instability, turbulence, and enhanced transport in accretion disks. Rev. Mod. Phys. 70, 1–53 (1998)

  3. 3

    Zeldovich, Y. B. On the friction of fluids between rotating cylinders. Proc. R. Soc. Lond. A 374, 299–312 (1981)

  4. 4

    Dubrulle, B. Differential rotation as a source of angular momentum transfer in the solar nebula. Icarus 106, 59–76 (1993)

  5. 5

    Balbus, S. A., Hawley, J. F. & Stone, J. M. Nonlinear stability, hydrodynamical turbulence, and transport in disks. Astrophys. J. 467, 76–86 (1996)

  6. 6

    Richard, D. & Zahn, J-P. Turbulence in differentially rotating flows: What can be learned from the Couette-Taylor experiment. Astron. Astrophys. 347, 734–738 (1999)

  7. 7

    Richard, D. Instabilités Hydrodynamiques dans les Ecoulements en Rotation Différentielle. Ph.D. thesis, Univ. Paris 7. (2001)

  8. 8

    Longaretti, P. On the phenomenology of hydrodynamic shear turbulence. Astrophys. J. 576, 587–598 (2002)

  9. 9

    Chagelishvili, G. D., Zahn, J-P., Tevzadze, A. G. & Lominadze, J. G. On hydrodynamic shear turbulence in Keplerian disks: Via transient growth to bypass transition. Astron. Astrophys. 402, 401–407 (2003)

  10. 10

    Yecko, P. A. Accretion disk instability revisited. Transient dynamics of rotating shear flow. Astron. Astrophys. 425, 385–393 (2004)

  11. 11

    Umurhan, O. M. & Regev, O. Hydrodynamic stability of rotationally supported flows: Linear and nonlinear 2D shearing box results. Astron. Astrophys. 427, 855–872 (2004)

  12. 12

    Garaud, P. & Ogilvie, G. I. A model for the nonlinear dynamics of turbulent shear flows. J. Fluid Mech. 530, 145–176 (2005)

  13. 13

    Mukhopadhyay, B., Afshordi, N. & Narayan, R. Bypass to turbulence in hydrodynamic accretion disks: An eigenvalue approach. Astrophys. J. 629, 383–396 (2005)

  14. 14

    Dubrulle, B. et al. Stability and turbulent transport in Taylor-Couette flow from analysis of experimental data. Phys. Fluids 17 095103 doi: 10.1063/1.2008999 (2005)

  15. 15

    Lesur, G. & Longaretti, P-Y. On the relevance of subcritical hydrodynamic turbulence to accretion disk transport. Astron. Astrophys. 444, 25–44 (2005)

  16. 16

    Taylor, G. I. Stability of a viscous liquid contained between two rotating cylinders. Phil. Trans. R. Soc. Lond. A 223, 289–343 (1923)

  17. 17

    Wendt, F. Turbulente Strömungen zwischen zwei rotierenden konaxialen Zylindern. Ing. Arch. 4, 577–595 (1933)

  18. 18

    Taylor, G. I. Fluid friction between rotating cylinders. i. torque measurements. Proc. R. Soc. Lond. A 157, 546–578 (1936)

  19. 19

    Schultz-Grunow, F. Zur Stabilität der Couette-Strömung. Z. Angew. Math. Mech. 39, 101–117 (1959)

  20. 20

    Kageyama, A., Ji, H., Goodman, J., Chen, F. & Shoshan, E. Numerical and experimental investigation of circulation in short cylinders. J. Phys. Soc. Jpn 73, 2424–2437 (2004)

  21. 21

    Burin, M. J. et al. Reduction of Ekman circulation within a short circular couette flow. Exp. Fluids 40 962–966 doi: 10.1007/s00348-006-0132-y (2006)

  22. 22

    Hueso, R. & Guillot, T. Evolution of protoplanetary disks: constraints from DM Tauri and GM Aurigae. Astron. Astrophys. 442, 703–725 (2005)

  23. 23

    Beckley, H. Measurements of Annular Couette Flow Stability at the Fluid Reynolds Number Re = 4.4 × 106: The Fluid Dynamic Precursor to a Liquid Sodium αω Dynamo. PhD thesis, New Mexico Inst. Mining Technol. (2002)

  24. 24

    Colebrook, C. F. Turbulent flow in pipes with particular reference to the transitional region between smooth and rough pipes. J. Inst. Civil Eng. 11, 133–156 (1938)

  25. 25

    Hartmann, L., Calvet, N., Gullbring, E. & D'Alessio, P. Accretion and the evolution of T Tauri disks. Astrophys. J. 495, 385–400 (1998)

  26. 26

    Klahr, H. H. & Bodenheimer, P. Turbulence in accretion disks: Vorticity generation and angular momentum transport via the global baroclinic instability. Astrophys. J. 582, 869–892 (2003)

  27. 27

    Dubrulle, B. et al. An hydrodynamic shear instability in stratified disks. Astron. Astrophys. 429, 1–13 (2005)

  28. 28

    Johnson, B. M. & Gammie, C. F. Nonlinear stability of thin, radially-stratified disks. Astrophys. J. 636, 63–74 (2006)

  29. 29

    Goodman, J. & Balbus, S. A. Stratified disks are locally stable. Preprint at 〈http://arxiv.org/astro-ph/0110229〉 (2001)

  30. 30

    Rayleigh On the dynamics of rotating fluid. Proc. R. Soc. Lond. A 93, 148–154 (1916)

  31. 31

    Lathrop, D. P., Fineberg, J. & Swinney, H. L. Turbulent flow between concentric rotating cylinders at large Reynolds number. Phys. Rev. Lett. 68, 1515–1518 (1992)

  32. 32

    Lewis, G. S. & Swinney, H. L. Velocity structure functions, scaling, and transitions in high-Reynolds-number Couette-Taylor flow. Phys. Rev. E 59, 5457–5467 (1999)

Download references


We thank S. Balbus for discussions, R. Cutler for technical assistance with the apparatus, P. Heitzenroeder, C. Jun, L. Morris and S. Raftopolous for engineering assistance, as well as Dantec Dynamics for the contracted use of an LDV measurement system. This research was supported by the US Department of Energy, Office of Science – Fusion Energy Sciences Program; the US National Aeronautics and Space Administration, Astronomy and Physics Research and Analysis and Astrophysics Theory Programs; and the US National Science Foundation, Physics and Astronomical Sciences Divisions. Author Contributions H.J., M.B. and E.S. planned and executed the experiments, and analysed data; E.S. and M.B. prepared apparatus and diagnostics; H.J. drafted the paper; and J.G. suggested this subject and assisted in the interpretation of the results and in revising the paper.

Author information

Author notes

    • Michael Burin

    Present address: Department of Physics and Astronomy, Pomona College, Claremont, California, 91711, USA


  1. Center for Magnetic Self-organization in Laboratory and Astrophysical Plasmas, Plasma Physics Laboratory and Department of Astrophysical Sciences, Princeton University, Princeton, New Jersey, 08543, USA

    • Hantao Ji
    • , Michael Burin
    • , Ethan Schartman
    •  & Jeremy Goodman


  1. Search for Hantao Ji in:

  2. Search for Michael Burin in:

  3. Search for Ethan Schartman in:

  4. Search for Jeremy Goodman in:

Competing interests

Reprints and permissions information is available at www.nature.com/reprints. The authors declare no competing financial interests.

Corresponding author

Correspondence to Hantao Ji.

About this article

Publication history



Issue Date



Further reading


By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.