Skip to main content Accessibility help

Self-organized criticality of turbulence in strongly stratified mixing layers

  • Hesam Salehipour (a1) (a2), W. R. Peltier (a1) and C. P. Caulfield (a3) (a4)


Motivated by the importance of stratified shear flows in geophysical and environmental circumstances, we characterize their energetics, mixing and spectral behaviour through a series of direct numerical simulations of turbulence generated by Holmboe wave instability (HWI) under various initial conditions. We focus on circumstances where the stratification is sufficiently ‘strong’ so that HWI is the dominant primary instability of the flow. Our numerical findings demonstrate the emergence of self-organized criticality (SOC) that is manifest as an adjustment of an appropriately defined gradient Richardson number, $Ri_{g}$ , associated with the horizontally averaged mean flow, in such a way that it is continuously attracted towards a critical value of $Ri_{g}\sim 1/4$ . This self-organization occurs through a continuously reinforced localization of the ‘scouring’ motions (i.e. ‘avalanches’) that are characteristic of the turbulence induced by the breakdown of Holmboe wave instabilities and are developed on the upper and lower flanks of the sharply localized density interface, embedded within a much more diffuse shear layer. These localized ‘avalanches’ are also found to exhibit the expected scale-invariant characteristics. From an energetics perspective, the emergence of SOC is expressed in the form of a long-lived turbulent flow that remains in a ‘quasi-equilibrium’ state for an extended period of time. Most importantly, the irreversible mixing that results from such self-organized behaviour appears to be characterized generically by a universal cumulative turbulent flux coefficient of $\unicode[STIX]{x1D6E4}_{c}\sim 0.2$ only for turbulent flows engendered by Holmboe wave instability. The existence of this self-organized critical state corroborates the original physical arguments associated with self-regulation of stratified turbulent flows as involving a ‘kind of equilibrium’ as described by Turner (1973, Buoyancy Effects in Fluids, Cambridge University Press).


Corresponding author

Email address for correspondence:


Hide All
Aschwanden, M. J. et al. 2016 25 years of self-organized criticality: solar and astrophysics. Space Sci. Rev. 198 (1–4), 47166.
Baines, P. G. & Mitsudera, H. 1994 On the mechanism of shear flow instabilities. J. Fluid Mech. 276, 327342.
Bak, P., Tang, C. & Wiesenfeld, K. 1987 Self-organized criticality: an explanation of the 1/f noise. Phys. Rev. Lett. 59 (4), 381384.
Butler, S. & Peltier, W. R. 1997 Internal thermal boundary layer stability in phase transition modulated convection. J. Geophys. Res. 102 (B2), 27312749.
Caulfield, C. P. 1994 Multiple linear instability of a layered stratified shear flow. J. Fluid Mech. 258, 155285.
Caulfield, C. P. & Peltier, W. R. 2000 The anatomy of the mixing transition in homogeneous and stratified free shear layers. J. Fluid Mech. 413, 147.
Ellison, T. H. 1957 Turbulent transport of heat and momentum from an infinite rough plane. J. Fluid Mech. 2 (05), 456466.
Fischer, P. F. 1997 An overlapping Schwarz method for spectral element solution of the incompressible Navier–Stokes equations. J. Comput. Phys. 133 (1), 84101.
Gregg, M. C., D’Asaro, E. A., Riley, J. J. & Kunze, E. 2018 Mixing efficiency in the ocean. Annu. Rev. Marine Sci. 10, 443473.
Guha, A. & Lawrence, G. A. 2014 A wave interaction approach to studying non-modal homogeneous and stratified shear instabilities. J. Fluid Mech. 755, 336364.
Hogg, A. McC. & Ivey, G. N. 2003 The Kelvin–Helmholtz to Holmboe instability transition in stratified exchange flows. J. Fluid Mech. 477, 339362.
Holleman, R. C., Geyer, W. R. & Ralston, D. K. 2016 Stratified turbulence and mixing efficiency in a salt wedge estuary. J. Phys. Oceanogr. 46 (6), 17691783.
Holmboe, J. 1962 On the behaviour of symmetric waves in stratified shear layers. Geofys. Publ. Oslo 24, 67113.
Holt, S. E., Koseff, J. R. & Ferziger, J. H. 1992 A numerical study of the evolution and structure of homogeneous stably stratified sheared turbulence. J. Fluid Mech. 237, 499539.
Howard, L. N. 1961 Note on a paper of John W. Miles. J. Fluid Mech. 10, 509512.
Ivey, G. N., Winters, K. B. & Koseff, J. R. 2008 Density stratification, turbulence, but how much mixing? Annu. Rev. Fluid Mech. 40 (1), 169184.
Lawrence, G., Pieters, R., Zaremba, L., Tedford, T., Gu, L., Greco, S. & Hamblin, P. 2004 Summer exchange between Hamilton Harbour and Lake Ontario. Deep-Sea Res. II 51 (4–5), 475487.
Lawrence, G. A., Browand, F. K. & Redekopp, L. G. 1991 The stability of a sheared density interface. Phys. Fluids A 3 (10), 23602370.
Lefauve, A., Partridge, J. L., Zhou, Q., Dalziel, S. B., Caulfield, C. P. & Linden, P. F. 2018 The structure and origin of confined Holmboe waves. J. Fluid Mech. 848, 508544.
Lilly, D. K. 1983 Stratified turbulence and the mesoscale variability of the atmosphere. J. Atmos. Sci. 40 (3), 749761.
Lindborg, E. 2006 The energy cascade in a strongly stratified fluid. J. Fluid Mech. 550, 207242.
Linden, P. F. 1979 Mixing in stratified fluids. Geophys. Astrophys. Fluid Dyn. 13, 323.
Macagno, E. O. & Rouse, H. 1961 Interfacial mixing in stratified flow. J. Engng Mechanics Division. Proc. Am. Soc. Civil Engineers 87 (EM5), 5581.
Mashayek, A., Caulfield, C. P. & Peltier, W. R. 2013 Time-dependent, non-monotonic mixing in stratified turbulent shear flows: implications for oceanographic estimates of buoyancy flux. J. Fluid Mech. 736, 570593.
Mashayek, A. & Peltier, W. R. 2013 Shear induced mixing in geophysical flows: does the route to turbulence matter to its efficiency? J. Fluid Mech. 725, 216261.
Miles, J. W. 1961 On the stability of heterogeneous shear flows. J. Fluid Mech. 10, 496508.
Osborn, T. R. 1980 Estimates of the local rate of vertical diffusion from dissipation measurements. J. Phys. Oceanogr. 10, 8389.
Peltier, W. R. & Caulfield, C. P. 2003 Mixing efficiency in stratified shear flows. Annu. Rev. Fluid Mech. 35, 135167.
Pruessner, G. 2012 Self-Organised Criticality: Theory, Models and Characterisation. Cambridge University Press.
Rohr, J. J., Itsweire, E. C., Helland, K. N. & Van Atta, C. W. 1988 Growth and decay of turbulence in a stably stratified shear flow. J. Fluid Mech. 195, 77111.
Salehipour, H., Caulfield, C. P. & Peltier, W. R. 2016 Turbulent mixing due to the Holmboe wave instability at high Reynolds number. J. Fluid Mech. 803, 591621.
Salehipour, H. & Peltier, W. R. 2015 Diapycnal diffusivity, turbulent Prandtl number and mixing efficiency in Boussinesq stratified turbulence. J. Fluid Mech. 775, 464500.
Salehipour, H., Peltier, W. R. & Mashayek, A. 2015 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, 178223.
Shih, L. H., Koseff, J. R., Ferziger, J. H. & Rehmann, C. R. 2000 Scaling and parameterization of stratified homogeneous turbulent shear flow. J. Fluid Mech. 412, 120.
Smyth, W. D., Carpenter, J. R. & Lawrence, G. A. 2007 Mixing in symmetric Holmboe waves. J. Phys. Oceanogr. 37, 15661583.
Smyth, W. D. & Moum, J. N. 2013 Marginal instability and deep cycle turbulence in the eastern equatorial Pacific Ocean. Geophys. Res. Lett. 40 (23), 61816185.
Smyth, W. D., Moum, J. N., Li, L. & Thorpe, S. A. 2013 Diurnal shear instability, the descent of the surface shear layer, and the deep cycle of equatorial turbulence. J. Phys. Oceanogr. 43 (11), 24322455.
Smyth, W. D. & Peltier, W. R. 1989 The transition between Kelvin–Helmholtz and Holmboe instability: an investigation of the overreflection hypothesis. J. Atmos. Sci. 46 (24), 36983720.
Smyth, W. D. & Winters, K. B. 2003 Turbulence and mixing in Holmboe waves. J. Phys. Oceanogr. 33, 694711.
Smyth, W. D., Klaassen, G. P. & Peltier, W. R. 1988 Finite amplitude Holmboe waves. Geophys. Astrophys. Fluid Dyn. 43 (2), 181222.
Solheim, L. P. & Peltier, W. R. 1994 Avalanche effects in phase transition modulated thermal convection: a model of earth’s mantle. J. Geophys. Res. 99 (B4), 69977018.
Strang, E. J. & Fernando, H. J. S. 2001 Entrainment and mixing in stratified shear flows. J. Fluid Mech. 428, 349386.
Taylor, G. I. 1915 Eddy motion in the atmosphere. Phil. Trans. R. Soc. A 215, 126.
Thorpe, S. A. & Liu, Z. 2009 Marginal instability? J. Phys. Oceanogr. 39 (9), 23732381.
Thorpe, S. A. 1968 A method of producing a shear flow in a stratified fluid. J. Fluid Mech. 32, 693704.
Turner, J. S. 1973 Buoyancy Effects in Fluids. Cambridge University Press.
Winters, K. B., Lombard, P. N., Riley, J. J. & D’Asaro, E. A. 1995 Available potential energy and mixing in density-stratified fluids. J. Fluid Mech. 289, 115128.
Zhou, Q., Taylor, J. R. & Caulfield, C. P. 2017 Self-similar mixing in stratified plane Couette flow for varying Prandtl number. J. Fluid Mech. 820, 86120.
Zhu, D. Z. & Lawrence, G. A. 2001 Holmboe’s instability in exchange flows. J. Fluid Mech. 429, 391409.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification

Related content

Powered by UNSILO

Self-organized criticality of turbulence in strongly stratified mixing layers

  • Hesam Salehipour (a1) (a2), W. R. Peltier (a1) and C. P. Caulfield (a3) (a4)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.