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Diapycnal mixing in layered stratified plane Couette flow quantified in a tracer-based coordinate

  • Qi Zhou (a1), J. R. Taylor (a1), C. P. Caulfield (a1) (a2) and P. F. Linden (a1)

Abstract

The mixing properties of statically stable density interfaces subject to imposed vertical shear are studied using direct numerical simulations of stratified plane Couette flow. The simulations are designed to investigate possible self-maintaining mechanisms of sharp density interfaces motivated by Phillips’ argument (Deep-Sea Res., vol. 19, 1972, pp. 79–81) by which layers and interfaces can spontaneously form due to vertical variations of diapycnal flux. At the start of each simulation, a sharp density interface with the same initial thickness is introduced at the midplane between two flat, horizontal walls counter-moving at velocities $\pm U_{w}$ . Particular attention is paid to the effects of varying Prandtl number $\mathit{Pr}\equiv \unicode[STIX]{x1D708}/\unicode[STIX]{x1D705}$ , where $\unicode[STIX]{x1D708}$ and $\unicode[STIX]{x1D705}$ are the molecular kinematic viscosity and diffusivity respectively, over two orders of magnitude from 0.7, 7 and 70. Varying $\mathit{Pr}$ enables the system to access a considerable range of characteristic turbulent Péclet numbers $\mathit{Pe}_{\ast }\equiv {\mathcal{U}}_{\ast }{\mathcal{L}}_{\ast }/\unicode[STIX]{x1D705}$ , where ${\mathcal{U}}_{\ast }$ and ${\mathcal{L}}_{\ast }$ are characteristic velocity and length scales, respectively, of the motion which acts to ‘scour’ the density interface. The dynamics of the interface varies with the stability of the interface which is characterised by a bulk Richardson number $\mathit{Ri}\,\equiv \,b_{0}h/U_{w}^{2}$ , where $b_{0}$ is half the initial buoyancy difference across the interface and $h$ is the half-height of the channel. Shear-induced turbulence occurs at small $\mathit{Ri}$ , whereas internal waves propagating on the interface dominate at large $\mathit{Ri}$ . For a highly stable (i.e. large $\mathit{Ri}$ ) interface at sufficiently large $\mathit{Pe}_{\ast }$ , the complex interfacial dynamics allows the interface to remain sharp. This ‘self-sharpening’ is due to the combined effects of the ‘scouring’ induced by the turbulence external to the interface and comparatively weak molecular diffusion across the core region of the interface. The effective diapycnal diffusivity and irreversible buoyancy flux are quantified in the tracer-based reference coordinate proposed by Winters & D’Asaro (J. Fluid Mech., vol. 317, 1996, pp. 179–193) and Nakamura (J. Atmos. Sci., vol. 53, 1996, pp. 1524–1537), which enables a detailed investigation of the self-sharpening process by analysing the local budget of buoyancy gradient in the reference coordinate. We further discuss the dependence of the effective diffusivity and overall mixing efficiency on the characteristic parameters of the flow, such as the buoyancy Reynolds number and the local gradient Richardson number, and highlight the possible role of the molecular properties of fluids on diapycnal mixing.

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Corresponding author

Email address for correspondence: q.zhou@damtp.cam.ac.uk

References

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Baines, P. G. & Mitsudera, H. 1994 On the mechanism of shear flow instabilities. J. Fluid Mech. 276, 327342.
Balmforth, N. J., Llewellyn-Smith, S. G. & Young, W. R. 1998 Dynamics of interfaces and layers in a stratified turbulent fluid. J. Fluid Mech. 355, 329358.
Bouffard, D. & Boegman, L. 2013 A diapycnal diffusivity model for stratified environmental flows. Dyn. Atmos. Oceans 61–62, 1434.
Brethouwer, G., Billant, P., Lindborg, E. & Chomaz, J.-M. 2007 Scaling analysis and simulation of strongly stratified turbulent flows. J. Fluid Mech. 585, 343368.
Carpenter, J. R., Balmforth, N. J. & Lawrence, G. A. 2010 Identifying unstable modes in stratified shear layers. Phys. Fluids 22, 054104.
Carpenter, J. R., Tedford, E. W., Heifetz, E. & Lawrence, G. A. 2011 Instability in stratified shear flow: review of a physical interpretation based on interacting waves. Appl. Mech. Rev. 64, 060801.
Caulfield, C.-C. P. 1994 Multiple instability of layered stratified shear flow. J. Fluid Mech. 258, 255285.
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.
Crapper, P. F. & Linden, P. F. 1974 The structure of turbulent density interfaces. J. Fluid Mech. 65, 4563.
Deusebio, E., Caulfield, C. P. & Taylor, J. R. 2015 The intermittency boundary in stratified plane Couette flow. J. Fluid Mech. 781, 298329.
Eaves, T. S. & Caulfield, C. P. 2017 Multiple instability of layered stratified plane Couette flow. J. Fluid Mech. 813, 250278.
Eliassen, A., Hailand, E. & Riis, E. 1953 Two-dimensional Perturbation of a Flow with Constant Shear of a Stratified Fluid. Norwegian Academy of Sciences and Letters.
Fernando, H. J. S. 1991 Turbulent mixing in stratified fluids. Annu. Rev. Fluid Mech. 23, 455493.
Gregg, M. C. 1980 Microstructure patches in the thermocline. J. Phys. Oceanogr. 10, 915943.
Linden, P. F. 1979 Mixing in stratified fluids. Geophys. Astrophys. Fluid Dyn. 13, 323.
Maffioli, A., Brethouwer, G. & Lindborg, E. 2016 Mixing efficiency in stratified turbulence. J. Fluid Mech. 794, R3.
Marshall, J., Shuckburgh, E., Jones, H. & Hill, C. 2006 Estimates and implications of surface eddy diffusivity in the southern ocean derived from tracer transport. J. Phys. Oceanogr. 36, 18061821.
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.
Mellor, G. L. & Yamada, T. 1982 Development of a turbulence closure model for geophysical fluid problems. Rev. Geophys. 20, 851875.
Nakamura, N. 1996 Two-dimensional mixing, edge formation, and permeability diagnosed in an area coordinate. J. Atmos. Sci. 53, 15241537.
Oglethorpe, R. L. F., Caulfield, C. P. & Woods, A. W. 2013 Spontaneous layering in stratified turbulent Taylor–Couette flow. J. Fluid Mech. 721, R3.
Peltier, W. R. & Caulfield, C. P. 2003 Mixing efficiency in stratified shear flows. Annu. Rev. Fluid Mech. 35, 135167.
Phillips, O. M. 1972 Turbulence in a strongly stratified fluid – is it unstable? Deep-Sea Res. 19, 7981.
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.
Posmentier, E. S. 1977 The generation of salinity fine structure by vertical diffusion. J. Phys. Oceanogr. 7, 298300.
Ruddick, B. R., McDougall, T. J. & Turner, J. S. 1989 The formation of layers in a uniformly stirred density gradient. Deep-Sea Res. 36, 597609.
Salehipour, H., Caulfield, C. P. & Peltier, W. R. 2016a 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., Whalen, C. B. & MacKinnon, J. A. 2016b A new characterization of the turbulent diapycnal diffusivities of mass and momentum in the ocean. Geophys. Res. Lett. 43, 33703379.
Shih, L. H., Koseff, J. R., Ivey, G. N. & Ferziger, J. H. 2005 Parameterization of turbulent fluxes and scales using homogeneous sheared stably stratified turbulence simulations. J. Fluid Mech. 525, 193214.
Shuckburgh, E. & Haynes, P. 2003 Diagnosing transport and mixing using a tracer-based coordinate system. Phys. Fluids 15, 33423357.
Smyth, W. D., Klaassen, G. P. & Peltier, W. R. 1988 Finite amplitude Holmboe waves. Geophys. Astrophys. Fluid Dyn. 43, 181222.
Smyth, W. D., Moum, J. & Caldwell, D. 2001 The efficiency of mixing in turbulent patches: inferences from direct simulations and microstructure observations. J. Phys. Oceanogr. 31, 19691992.
Smyth, W. D., Moum, J. N. & Nash, J. D. 2011 Narrowband oscillations in the upper equatorial ocean. Part II: properties of shear instabilities. J. Phys. Oceanogr. 41, 412428.
Strang, E. J. & Fernando, H. J. S. 2001 Entrainment and mixing in stratified shear flows. J. Fluid Mech. 428, 349386.
Taylor, J. R.2008 Numerical simulations of the stratified oceanic bottom boundary layer. PhD thesis, University of California, San Diego.
Taylor, J. R. & Zhou, Q.2017 A multi-parameter criterion for layer formation in a stratified shear flow using buoyancy coordinates. J. Fluid Mech. (in press).
Thorpe, S. A. 2005 The Turbulent Ocean. Cambridge University Press.
Thorpe, S. A. 2016 Layers and internal waves in uniformly stratified fluids stirred by vertical grids. J. Fluid Mech. 793, 380413.
Tseng, Y. & Ferziger, J. H. 2001 Mixing and available potential energy in stratified flows. Phys. Fluids 13, 12811293.
Turner, J. S. 1973 Buoyancy Effects in Fluids. Cambridge University Press.
Venaille, A., Gostiaux, L. & Sommeria, J. 2017 A statistical mechanics approach of mixing in stratified fluids. J. Fluid Mech. 810, 554583.
Venayagamoorthy, S. K. & Koseff, J. R. 2016 On the flux Richardson number in stably stratified turbulence. J. Fluid Mech. 798, R1.
Winters, K. B. & D’Asaro, E. A. 1996 Diascalar flux and the rate of fluid mixing. J. Fluid Mech. 317, 179193.
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.
Woods, A. W., Caulfield, C. P., Landel, J. R. & Kuesters, A. 2010 Non-invasive turbulent mixing across a density interface in a turbulent Taylor–Couette flow. J. Fluid Mech. 663, 347357.
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.
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