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Asymmetry of vertical buoyancy gradient in stratified turbulence

  • Andrea Maffioli (a1)


We consider the asymmetry of the buoyancy field in the vertical direction in stratified turbulence. While this asymmetry is known, its causes are not well understood, and it has not been systematically quantified previously. Using theoretical arguments, it is shown that both stratified turbulence and isotropic turbulence in the presence of a mean scalar gradient will become positively skewed, as a direct consequence of the presence of stratification and mean scalar gradient, respectively. Assuming a rapid adjustment of isotropic turbulence to a stable stratification on a time scale $\unicode[STIX]{x1D70F}\sim N^{-1}$ , where $N$ is the Brunt–Väisälä frequency, a scaling for the skewness of the vertical buoyancy gradient is obtained. Direct numerical simulations of stratified turbulence with forcing are performed and the positive skewness of $\unicode[STIX]{x2202}b/\unicode[STIX]{x2202}z$ is confirmed ( $b$ is the buoyancy). Both the volume-averaged dimensional skewness, $\langle (\unicode[STIX]{x2202}b/\unicode[STIX]{x2202}z)^{3}\rangle$ , and the non-dimensional skewness, $S$ , are computed and compared against the theoretical predictions. There is a good agreement for $\langle (\unicode[STIX]{x2202}b/\unicode[STIX]{x2202}z)^{3}\rangle$ , while there is a discrepancy in the behaviour of $S$ . The theory predicts $S\sim 1$ and a constant skewness, while the direct numerical simulations confirm that the skewness is $O(1)$ but with a remaining dependence on the Froude number. The results are interpreted as being due to the concurrent action of linear and nonlinear processes in stratified turbulence leading to $S>0$ and to the formation of layers and interfaces in vertical profiles of buoyancy.


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Bartello, P., Métais, O. & Lesieur, M. 1994 Coherent structures in rotating three-dimensional turbulence. J. Fluid Mech. 273, 129.
Billant, P. & Chomaz, J.-M. 2000 Experimental evidence for a new instability of a vertical columnar vortex pair in a strongly stratified fluid. J. Fluid Mech. 418, 167188.10.1017/S0022112000001154
Billant, P. & Chomaz, J.-M. 2001 Self-similarity of strongly stratified inviscid flows. Phys. Fluids 13 (6), 16451651.
Bos, W. J. T. 2014 On the anisotropy of the turbulent passive scalar in the presence of a mean scalar gradient. J. Fluid Mech. 744, 3864.
Brethouwer, G., Billant, P., Lindborg, E. & Chomaz, J.-M. 2007 Scaling analysis and simulation of strongly stratified turbulent flows. J. Fluid Mech. 585, 343368.10.1017/S0022112007006854
Browand, F. K., Guyomar, D. & Yoon, S.-C. 1987 The behavior of a turbulent front in a stratified fluid: experiments with an oscillating grid. J. Geophys. Res. 92 (C5), 53295341.
de Bruyn Kops, S. M. 2015 Classical scaling and intermittency in strongly stratified Boussinesq turbulence. J. Fluid Mech. 775, 436463.
Davidson, P. A. 2013 Turbulence in Rotating, Stratified and Electrically Conducting Fluids. Cambridge University Press.
Desaubies, Y. & Gregg, M. C. 1981 Reversible and irreversible fine structure. J. Phys. Oceanogr. 11, 541556.
Deusebio, E., Boffetta, G., Lindborg, E. & Musacchio, S. 2014 Dimensional transition in rotating turbulence. Phys. Rev. E 90, 023005.
Donzis, D. A., Sreenivasan, K. R. & Yeung, P. K. 2005 Scalar dissipation rate and dissipative anomaly in isotropic turbulence. J. Fluid Mech. 532, 199216.
Eswaran, V. & Pope, S. B. 1988 An examination of forcing in direct numerical simulations of turbulence. Comput. Fluids 16 (3), 257278.
Gan, L., Baqui, Y. B. & Maffioli, A. 2016 An experimental investigation of forced steady rotating turbulence. Eur. J. Mech. (B/Fluids) 58, 5969.10.1016/j.euromechflu.2016.03.005
Gence, J.-N. & Frick, C. 2001 Naissance des corrélations triples de vorticité dans une turbulence statistiquement homogène soumise à une rotation. C. R. Acad. Sci. llb Mec. 329 (5), 351356.
Gregg, M. C., D’Asaro, E. A., Riley, J. J. & Kunze, E. 2018 Mixing efficiency in the ocean. Annu. Rev. Mar. Sci. 10 (1), 443473.
Holford, J. M. & Linden, P. F. 1999 Turbulent mixing in a stratified fluid. Dyn. Atmos. Oceans 30, 173198.10.1016/S0377-0265(99)00025-1
Hopfinger, E. J., Browand, F. K. & Gagne, Y. 1982 Turbulence and waves in a rotating tank. J. Fluid Mech. 125, 505534.
Kimura, Y., Sullivan, P. & Herring, J. 2016 Temperature front formation in stably stratified turbulence. In International Symposium on Stratified Flows, vol. 1. Available at:
Lilly, D. K. 1983 Stratified turbulence and the mesoscale variability of the atmosphere. J. Atmos. Sci. 40, 749761.10.1175/1520-0469(1983)040<0749:STATMV>2.0.CO;2
Lindborg, E. 2006 The energy cascade in a strongly stratified fluid. J. Fluid Mech. 550, 207242.
Maffioli, A. 2017 Vertical spectra of stratified turbulence at large horizontal scales. Phys. Rev. Fluids 2, 104802.10.1103/PhysRevFluids.2.104802
Maffioli, A., Brethouwer, G. & Lindborg, E. 2016 Mixing efficiency in stratified turbulence. J. Fluid Mech. 794, R3.
Maffioli, A., Davidson, P. A., Dalziel, S. B. & Swaminathan, N. 2014 The evolution of a stratified turbulent cloud. J. Fluid Mech. 739, 229253.
Mydlarski, L. & Warhaft, Z. 1998 Passive scalar statistics in high-Peclet-number grid turbulence. J. Fluid Mech. 358, 135175.
Park, Y.-G., Whitehead, J. A. & Gnanadeskian, A. 1994 Turbulent mixing in stratified fluids: layer formation and energetics. J. Fluid Mech. 279, 279311.
Phillips, O. M. 1972 Turbulence in a strongly stratified fluid is it unstable? Deep-Sea Res. Oceanogr. Abstr. 19 (1), 7981.
Pinkel, R., Sherman, J., Smith, J. & Anderson, S. 1991 Strain: observations of the vertical gradient of isopycnal vertical displacement. J. Phys. Oceanogr. 21, 527540.
Posmentier, Eric S. 1977 The generation of salinity finestructure by vertical diffusion. J. Phys. Oceanogr. 7 (2), 298300.
Riley, J. J. & de Bruyn Kops, S. M. 2003 Dynamics of turbulence strongly influenced by buoyancy. Phys. Fluids 15 (7), 20472059.10.1063/1.1578077
Riley, J. J. & Lelong, M.-P. 2000 Fluid motions in the presence of strong stable stratification. Annu. Rev. Fluid Mech. 32, 613657.10.1146/annurev.fluid.32.1.613
Rogallo, R. S.1981 Numerical experiments in homogeneous turbulence. Tech. Mem. 81835. NASA.
Schumacher, J. & Sreenivasan, K. R. 2003 Geometric features of the mixing of passive scalars at high Schmidt numbers. Phys. Rev. Lett. 91 (17).
Staplehurst, P. J., Davidson, P. A. & Dalziel, S. B. 2008 Structure formation in homogeneous freely decaying rotating turbulence. J. Fluid Mech. 598, 81105.10.1017/S0022112007000067
Taylor, J. R. & Zhou, Q. 2017 A multi-parameter criterion for layer formation in a stratified shear flow using sorted buoyancy coordinates. J. Fluid Mech. 823, R5.10.1017/jfm.2017.375
Thorpe, S. A. 2016 Layers and internal waves in uniformly stratified fluids stirred by vertical grids. J. Fluid Mech. 793, 380413.
Warhaft, Z. 2000 Passive scalars in turbulent flows. Annu. Rev. Fluid Mech. 32, 203240.
Watanabe, T. & Gotoh, T. 2004 Statistics of a passive scalar in homogeneous turbulence. New J. Phys. 6 (40).
Yaglom, A. M. 1949 On the local structure of the temperature field in a turbulent flow. Dokl. Akad. Nauk SSSR 69, 743746.
Yeung, P. K., Xu, S. & Sreenivasan, K. R. 2002 Schmidt number effects on turbulent transport with uniform mean scalar gradient. Phys. Fluids 14, 4178.
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