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Hydrodynamic turbulence cannot transport angular momentum effectively in astrophysical disks

Abstract

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.

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Figure 1: Experimental set-up.
Figure 2: Experimentally studied Taylor-Couette flows.
Figure 3: Experimentally measured Reynolds stress versus height in a quasi-keplerian profile.
Figure 4: Dimensionless Reynolds stress at Reynolds numbers up to 2 × 106.

References

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

    ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  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)

    ADS  Google Scholar 

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

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

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  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)

    Article  CAS  MATH  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  Google Scholar 

  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)

    Article  CAS  ADS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

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

    Article  CAS  ADS  Google Scholar 

  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. 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)

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  CAS  ADS  Google Scholar 

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

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

    Article  ADS  Google Scholar 

  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)

    Article  CAS  ADS  PubMed  Google Scholar 

  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)

    Article  CAS  ADS  Google Scholar 

Download references

Acknowledgements

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.

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Correspondence to Hantao Ji.

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Ji, H., Burin, M., Schartman, E. et al. Hydrodynamic turbulence cannot transport angular momentum effectively in astrophysical disks. Nature 444, 343–346 (2006). https://doi.org/10.1038/nature05323

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