- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review Fluids

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Threshold behavior of local gradient Richardson number in strongly stratified nonequilibrium turbulence

Qi Zhou*

  • Department of Civil Engineering, University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4

  • *qi.zhou1@ucalgary.ca
PDF

Phys. Rev. Fluids 7, 104802 – Published 20 October, 2022

DOI: https://doi.org/10.1103/PhysRevFluids.7.104802

Abstract

In this paper we examine the possible self-organization of strongly stratified turbulence around a local critical state by analyzing a dataset of a numerically simulated stratified turbulent wake. To facilitate the analysis, the turbulent flow field is decomposed into a “large-scale” flow of horizontal scales greater than the Ozmidov scale, ℓO, and a “small-scale” flow of scales below ℓO. A local gradient Richardson number, Ri, characterizing the large-scale flow is calculated and then utilized to produce conditional sampling of various turbulence statistics relevant to the local dynamics. While the bulk turbulence is observed to decay by approximately one order of magnitude in terms of the dissipation rate, the median Ri has remained nearly constant due to the self-organization of flow structures under strong stratification; the subsampled Ri distribution peaks around 1/4 for regions in the upper quartile of local dissipation. Regions of small Ri are found to be associated with large dissipation and large net transfer of energy to the small scales. Regions of “back-scatter” of kinetic energy to large scales, where the local eddy viscosity, νe, takes a negative value, are also observed. Occurrence of a large magnitude of both positive and negative νe appears to be most frequent around the critical value of Ri∼1/4, indicating an intense two-way exchange of kinetic energy between the large and small scales around the local critical state. The threshold behavior of Ri underscores the dynamical significance of the critical Ri of 1/4 for locally self-sustained turbulence in a strongly stratified configuration and bears some resemblance to the celebrated self-organized criticality dynamics [Bak et al., Phys. Rev. Lett. 59, 381 (1987)].

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (63)

  1. P. Bak, C. Tang, and K. Wiesenfeld, Self-Organized Criticality: An Explanation of the 1/f Noise, Phys. Rev. Lett. 59, 381 (1987).
  2. G. Pruessner, Self-organised Criticality: Theory, Models and Characterisation (Cambridge University Press, Cambridge, 2012).
  3. H. Salehipour, W. R. Peltier, and C. P. Caulfield, Self-organized criticality of turbulence in strongly stratified mixing layers, J. Fluid Mech. 856, 228 (2018).
  4. W. D. Smyth, J. D. Nash, and J. N. Moum, Self-organized criticality in geophysical turbulence, Sci. Rep. 9, 1 (2019).
  5. G. P. Chini, G. Michel, K. Julien, C. B. Rocha, and C. P. Caulfield, Exploiting self-organized criticality in strongly stratified turbulence, J. Fluid Mech. 933, A22 (2022).
  6. J. W. Miles, On the stability of heterogeneous shear flows, J. Fluid Mech. 10, 496 (1961).
  7. L. N. Howard, Note on a paper of John W. Miles, J. Fluid Mech. 10, 509 (1961).
  8. J. S. Turner, Buoyancy Effects in Fluids (Cambridge University Press, Cambridge, 1973).
  9. Q. Zhou, J. R. Taylor, and C. P. Caulfield, Self-similar mixing in stratified plane Couette flow for varying Prandtl number, J. Fluid Mech. 820, 86 (2017).
  10. S. A. Thorpe and Z. Liu, Marginal instability? J. Phys. Oceanogr. 39, 2373 (2009).
  11. W. D. Smyth and J. N. Moum, Marginal instability and deep cycle turbulence in the eastern equatorial Pacific Ocean, Geophys. Res. Lett. 40, 6181 (2013).
  12. W. D. Smyth and J. R. Carpenter, Instability in Geophysical Flows (Cambridge University Press, Cambridge, 2019).
  13. W. D. Smyth, H. T. Pham, J. N. Moum, and S. Sarkar, Pulsating turbulence in a marginally unstable stratified shear flow, J. Fluid Mech. 822, 327 (2017).
  14. M. J. Aschwanden, A macroscopic description of a generalized self-organized criticality system: Astrophysical applications, Astrophys. J. 782, 54 (2014).
  15. D. Hebert, D. R. Caldwell, C. A. Paulson, and J. N. Moum, Turbulence and internal waves at the Equator. Part II: Details of a single event, J. Phys. Oceanogr. 22, 1346 (1992).
  16. J. N. Moum, J. D. Nash, and W. D. Smyth, Narrowband oscillations in the upper equatorial ocean. Part I: Interpretation as shear instabilities, J. Phys. Oceanogr. 41, 397 (2011).
  17. H. Van Haren, L. Gostiaux, E. Morozov, and R. Tarakanov, Extremely long Kelvin–Helmholtz billow trains in the Romanche fracture zone, Geophys. Res. Lett. 41, 8445 (2014).
  18. J. Tu, D. Fan, Q. Lian, Z. Liu, W. Liu, A. Kaminski, and W. D. Smyth, Acoustic observations of Kelvin–Helmholtz billows on an estuarine lutocline, J. Geophys. Res. Oceans 125, e2019JC015383 (2020).
  19. W. D. Smyth, Marginal instability and the efficiency of ocean mixing, J. Phys. Oceanogr. 50, 2141 (2020).
  20. A. Mashayek, L. E. Baker, B. B. Cael, and C. P. Caulfield, A marginal stability paradigm for shear-induced diapycnal turbulent mixing in the ocean, Geophys. Res. Lett. 49, e2021GL095715 (2022).
  21. G. R. Spedding, The evolution of initially turbulent bluff-body wakes at high internal Froude number, J. Fluid Mech. 337, 283 (1997).
  22. P. J. Diamessis, G. R. Spedding, and J. A. Domaradzki, Similarity scaling and vorticity structure in high-Reynolds-number stably stratified turbulent wakes, J. Fluid Mech. 671, 52 (2011).
  23. Q. Zhou, Far-field evolution of turbulence-emitted internal waves and Reynolds number effects on a localized stratified turbulent flow, Ph.D. thesis, Cornell University (2015).
  24. Q. Zhou and P. J. Diamessis, Large-scale characteristics of stratified wake turbulence at varying Reynolds number, Phys. Rev. Fluids 4, 084802 (2019).
  25. B. Halawa, S. Merhi, C. Tang, and Q. Zhou, Three-dimensional visualization of stratified turbulent wakes at varying Reynolds number, J. Vis. 23, 437 (2020).
  26. P. J. Diamessis, J. A. Domaradzki, and J. S. Hesthaven, A spectral multidomain penalty method model for the simulation of high Reynolds number localized incompressible stratified turbulence, J. Comput. Phys. 202, 298 (2005).
  27. T. Watanabe, J. J. Riley, S. M. de Bruyn Kops, P. J. Diamessis, and Q. Zhou, Turbulent/non-turbulent interfaces in wakes in stably stratified fluids, J. Fluid Mech. 797, R1 (2016).
  28. Q. Zhou and P. J. Diamessis, Surface manifestation of internal waves emitted by submerged localized stratified turbulence, J. Fluid Mech. 798, 505 (2016).
  29. K. L. Rowe, P. J. Diamessis, and Q. Zhou, Internal gravity wave radiation from a stratified turbulent wake, J. Fluid Mech. 888, A25 (2020).
  30. W. D. Smyth and J. N. Moum, Length scales of turbulence in stably stratified mixing layers, Phys. Fluids 12, 1327 (2000).
  31. H. Salehipour, W. R. Peltier, and A. Mashayek, Turbulent diapycnal mixing in stratified shear flows: the influence of Prandtl number on mixing efficiency and transition at high Reynolds number, J. Fluid Mech. 773, 178 (2015).
  32. J. A. Redford, T. S. Lund, and G. N. Coleman, A numerical study of a weakly stratified turbulent wake, J. Fluid Mech. 776, 568 (2015).
  33. J. P. Boyd, Chebyshev and Fourier Spectral Methods (Dover Publications, Mineola, New York, 2001).
  34. J. R. Taylor, Numerical simulations of the stratified oceanic bottom boundary layer, Ph.D. thesis, University of California, San Diego (2008).
  35. Q. Zhou, J. R. Taylor, C. P. Caulfield, and P. F. Linden, Diapycnal mixing in layered stratified plane Couette flow quantified in a tracer-based coordinate, J. Fluid Mech. 823, 198 (2017).
  36. S. A. Orszag and Y.-H. Pao, Numerical computation of turbulent shear flows, Adv. Geophys. 18, 225 (1975).
  37. J. J. Riley and E. Lindborg, Recent progress in stratified turbulence, in Ten Chapters in Turbulence, edited by P. A. Davidson, Y. Kaneda, and K. R. Sreenivasan (Cambridge University Press, Cambridge, 2012), pp. 269–317.
  38. M. L. Waite, Stratified turbulence at the buoyancy scale, Phys. Fluids 23, 066602 (2011).
  39. J. J. Riley, R. W. Metcalfe, and M. A. Weissman, Direct numerical simulations of homogeneous turbulence in density-stratified fluids, in AIP Conf. Proc. on Nonlinear Properties of Internal Waves, edited by J. B. West (American Institute of Physics, College Park, Maryland, 1981), pp. 79–112.
  40. P. Billant and J.-M. Chomaz, Self-similarity of strongly stratified inviscid flows, Phys. Fluids 13, 1645 (2001).
  41. G. Brethouwer, P. Billant, E. Lindborg, and J.-M. Chomaz, Scaling analysis and simulation of strongly stratified turbulent flows, J. Fluid Mech. 585, 343 (2007).
  42. M. Falder, N. J. White, and C. P. Caulfield, Seismic imaging of rapid onset of stratified turbulence in the South Atlantic Ocean, J. Phys. Oceanogr. 46, 1023 (2016).
  43. B. R. Sutherland, U. Achatz, C.-c. P. Caulfield, and J. M. Klymak, Recent progress in modeling imbalance in the atmosphere and ocean, Phys. Rev. Fluids 4, 010501 (2019).
  44. E. Lindborg, The energy cascade in a strongly stratified fluid, J. Fluid Mech. 550, 207 (2006).
  45. J. J. Riley and E. Lindborg, Stratified turbulence: A possible interpretation of some geophysical turbulence measurements, J. Atmos. Sci. 65, 2416 (2008).
  46. L. G. Shapiro and G. C. Stockman, Computer Vision (Prentice Hall, New Jersey, 2001).
  47. S. B. Pope, Turbulent Flows (Cambridge University Press, Cambridge, 2000).
  48. D. K. Lilly, Stratified turbulence and the mesoscale variability of the atmosphere, J. Atoms. Sci. 40, 749 (1983).
  49. G. R. Spedding, F. K. Browand, and A. M. Fincham, The long-time evolution of the initially turbulent wake of a sphere in a stable stratification, Dyn. Atmos. Oceans 23, 171 (1996).
  50. O. Praud, A. M. Fincham, and J. Sommeria, Decaying grid turbulence in a strongly stratified fluid, J. Fluid Mech. 522, 1 (2005).
  51. A. M. Fincham, T. Maxworthy, and G. R. Spedding, Energy dissipation and vortex structure in freely decaying, stratified grid turbulence, Dyn. Atmos. Oceans 23, 155 (1996).
  52. J. J. Riley and S. M. de Bruyn Kops, Dynamics of turbulence strongly influenced by buoyancy, Phys. Fluids 15, 2047 (2003).
  53. D. A. Hebert and S. M. de Bruyn Kops, Relationship between vertical shear rate and kinetic energy dissipation rate in stably stratified flows, Geophys. Res. Lett. 33, L06602 (2006).
  54. S. M. de Bruyn Kops and J. J. Riley, The effects of stable stratification on the decay of initially isotropic homogeneous turbulence, J. Fluid Mech. 860, 787 (2019).
  55. H. Peters, M. C. Gregg, and J. M. Toole, On the parameterization of equatorial turbulence, J. Geophys. Res. 93, 1199 (1988).
  56. J. N. Moum, D. R. Caldwell, and C. A. Paulson, Mixing in the equatorial surface layer and thermocline, J. Geophys. Res. 94, 2005 (1989).
  57. W. D. Smyth, J. N. Moum, and D. R. Caldwell, The efficiency of mixing in turbulent patches: Inferences from direct simulations and microstructure observations, J. Phys. Oceanogr. 31, 1969 (2001).
  58. C. J. Howland, J. R. Taylor, and C. P. Caulfield, Shear-induced breaking of internal gravity waves, J. Fluid Mech. 921, A24 (2021).
  59. C. P. Caulfield, Layering, instabilities, and mixing in turbulent stratified flows, Ann. Rev. Fluid Mech. 53, 113 (2021).
  60. H. Salehipour, W. R. Peltier, C. B. Whalen, and J. A. MacKinnon, A new characterization of the turbulent diapycnal diffusivities of mass and momentum in the ocean, Geophys. Res. Lett. 43, 3370 (2016).
  61. J. R. Taylor and Q. Zhou, A multi-parameter criterion for layer formation in a stratified shear flow using sorted buoyancy coordinates, J. Fluid Mech. 823, R5 (2017).
  62. O. M. Phillips, Turbulence in a strongly stratified fluid—Is it unstable? Deep-Sea Res. 19, 79 (1972).
  63. E. S. Posmentier, The generation of salinity finestructure by vertical diffusion, J. Phys. Oceanogr. 7, 298 (1977).

Outline

Information

Phys. Rev. Fluids 7, 104802– Published 20 October, 2022
  • Received 17 July 2022
  • Accepted 5 October 2022
Crossmark - Check for updates button

©2022 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation