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Nonlinear evolution of linear optimal perturbations of strongly stratified shear layers

Published online by Cambridge University Press:  20 July 2017

A. K. Kaminski Open the ORCID record for A. K. Kaminski [Opens in a new window] ,
C. P. Caulfield Open the ORCID record for C. P. Caulfield [Opens in a new window]  and
J. R. Taylor Open the ORCID record for J. R. Taylor [Opens in a new window]
A. K. Kaminski*
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK College of Earth, Ocean and Atmospheric Sciences, Oregon State University, 104 CEOAS Administration Building, Corvallis, OR 97331, USA
C. P. Caulfield
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK BP Institute, University of Cambridge, Madingley Rise, Madingley Road, Cambridge CB3 0EZ, UK
J. R. Taylor
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
*
†Email address for correspondence: [email protected]
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Abstract

The Miles–Howard theorem states that a necessary condition for normal-mode instability in parallel, inviscid, steady stratified shear flows is that the minimum gradient Richardson number, $Ri_{g,min}$, is less than $1/4$ somewhere in the flow. However, the non-normality of the Navier–Stokes and buoyancy equations may allow for substantial perturbation energy growth at finite times. We calculate numerically the linear optimal perturbations which maximize the perturbation energy gain for a stably stratified shear layer consisting of a hyperbolic tangent velocity distribution with characteristic velocity $U_{0}^{\ast }$ and a uniform stratification with constant buoyancy frequency $N_{0}^{\ast }$. We vary the bulk Richardson number $Ri_{b}=N_{0}^{\ast 2}h^{\ast 2}/U_{0}^{\ast 2}$ (corresponding to $Ri_{g,min}$) between 0.20 and 0.50 and the Reynolds numbers $\mathit{Re}=U_{0}^{\ast }h^{\ast }/\unicode[STIX]{x1D708}^{\ast }$ between 1000 and 8000, with the Prandtl number held fixed at $\mathit{Pr}=1$. We find the transient growth of non-normal perturbations may be sufficient to trigger strongly nonlinear effects and breakdown into small-scale structures, thereby leading to enhanced dissipation and non-trivial modification of the background flow even in flows where $Ri_{g,min}>1/4$. We show that the effects of nonlinearity are more significant for flows with higher $\mathit{Re}$, lower $Ri_{b}$ and higher initial perturbation amplitude $E_{0}$. Enhanced kinetic energy dissipation is observed for higher-$Re$ and lower-$Ri_{b}$ flows, and the mixing efficiency, quantified here by $\unicode[STIX]{x1D700}_{p}/(\unicode[STIX]{x1D700}_{p}+\unicode[STIX]{x1D700}_{k})$ where $\unicode[STIX]{x1D700}_{p}$ is the dissipation rate of density variance and $\unicode[STIX]{x1D700}_{k}$ is the dissipation rate of kinetic energy, is found to be approximately 0.35 for the most strongly nonlinear cases.

JFM classification

Instability: Instability Geophysical and Geological Flows: Stratified flows Instability: Transition to turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 825 , 25 August 2017 , pp. 213 - 244
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Copyright
© 2017 Cambridge University Press 

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References

Andersson, P., Berggren, M. & Henningson, D. S. 1999 Optimal disturbances and bypass transition in boundary layers. Phys. Fluids 11 (1), 134–150.Google Scholar
Arratia, C., Caulfield, C. P. & Chomaz, J.-M. 2013 Transient perturbation growth in time-dependent mixing layers. J. Fluid Mech. 717, 90–133.CrossRefGoogle Scholar
Augier, P., Billant, P., Negretti, M. E. & Chomaz, J.-M. 2014 Experimental study of stratified turbulence forced with columnar dipoles. Phys. Fluids 26 (4), 046603.Google Scholar
Brucker, K. A. & Sarkar, S. 2007 Evolution of an initially turbulent stratified shear layer. Phys. Fluids 19, 105105.Google Scholar
Caulfield, C. P. & Peltier, W. R. 2000 The anatomy of the mixing transition in homogeneous and stratified free shear layers. J. Fluid Mech. 413, 1–47.CrossRefGoogle Scholar
Chapman, S. J. 2002 Subcritical transition in channel flows. J. Fluid Mech. 451, 35–97.Google Scholar
Cherubini, S., De Palma, P. & Robinet, J.-C. 2015 Nonlinear optimals in the asymptotic suction boundary layer: transition thresholds and symmetry breaking. Phys. Fluids 27, 034108.Google Scholar
Duguet, Y., Monokrousos, A., Brandt, L. & Henningson, D. S. 2013 Minimal transition thresholds in plane Couette flow. Phys. Fluids 25, 084103.Google Scholar
Eaves, T. S. & Caulfield, C. P. 2015 Disruption of SSP/VWI states by a stable stratification. J. Fluid Mech. 784, 548–564.CrossRefGoogle Scholar
Ellingsen, T. & Palm, E. 1975 Stability of linear flow. Phys. Fluids 18 (4), 487–488.Google Scholar
Farrell, B. F. & Ioannou, P. J. 1993a Optimal excitation of three-dimensional perturbations in viscous constant shear flow. Phys. Fluids A 5 (6), 1390–1400.CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 1993b Transient development of perturbations in stratified shear flow. J. Atmos. Sci. 50 (14), 2201–2214.2.0.CO;2>CrossRefGoogle Scholar
Garrett, C. 2003 Mixing with latitude. Nature 422, 477–478.CrossRefGoogle ScholarPubMed
Howard, L. N. 1961 Note on a paper of John W. Miles. J. Fluid Mech. 10, 509–512.Google Scholar
Ivey, G. N., Winters, K. B. & Koseff, J. R. 2008 Density stratification, turbulence, but how much mixing? Annu. Rev. Fluid Mech. 40, 169–184.Google Scholar
Kaminski, A. K., Caulfield, C. P. & Taylor, J. R. 2014 Transient growth in strongly stratified shear layers. J. Fluid Mech. 758, R4.Google Scholar
Kerswell, R. R., Pringle, C. C. T. & Willis, A. P. 2014 An optimization approach for analyzing nonlinear stability with transition to turbulence in fluids as an exemplar. Rep. Prog. Phys. 77, 085901.CrossRefGoogle Scholar
Klaassen, G. P. & Peltier, W. R. 1985 Evolution of finite amplitude Kelvin–Helmholtz billows in two spatial dimensions. J. Atmos. Sci. 42 (12), 1321–1339.2.0.CO;2>CrossRefGoogle Scholar
Kuhlbrodt, T., Griesel, A., Montoya, M., Levermann, A., Hofmann, M. & Rahmstorf, S. 2007 On the driving processes of the Atlantic meridional overturning circulation. Rev. Geophys. 45, RG2001.CrossRefGoogle Scholar
Kunze, E., Williams, A. J. III & Briscoe, M. G. 1990 Observations of shear and vertical stability from a neutrally buoyant float. J. Geophys. Res. 95, 18127–18142.Google Scholar
Luchini, P. 2000 Reynolds-number-independent instability of the boundary layer over a flat surface: optimal perturbations. J. Fluid Mech. 404, 289–309.CrossRefGoogle Scholar
Mack, S. A. & Schoeberlein, H. C. 2004 Richardson number and ocean mixing: towed chain observations. J. Phys. Oceanogr. 34, 736–754.Google Scholar
Maffioli, A., Brethouwer, G. & Lindborg, E. 2016 Mixing efficiency in stratified turbulence. J. Fluid Mech. 794, R3.Google Scholar
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, 570–593.Google Scholar
Mashayek, A. & Peltier, W. R. 2012 The ‘zoo’ of secondary instabilities precursory to stratified shear flow transition. Part 1. Shear aligned convection, pairing, and braid instabilities. J. Fluid Mech. 708, 5–44.CrossRefGoogle Scholar
Mellor, G. L. & Yamada, T. 1982 Development of a turbulence closure model for geophysical fluid problems. Rev. Geophys. Space Phys. 20 (4), 851–875.CrossRefGoogle Scholar
Miles, J. W. 1961 On the stability of heterogeneous shear flows. J. Fluid Mech. 496, 496–508.Google Scholar
Orr, W. M’F. 1907 The stability or instability of the steady motions of a perfect liquid and of a viscous liquid. Part I: a perfect liquid. Proc. R. Irish Acad. A 27, 9–68.Google Scholar
Peltier, W. R. & Caulfield, C. P. 2003 Mixing efficiency in stratified shear flows. Annu. Rev. Fluid Mech. 35, 135–167.Google Scholar
Polzin, K. 1996 Statistics of the Richardson number: mixing models and finestructure. J. Phys. Oceanogr. 26, 1409–1425.Google Scholar
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.Google Scholar
Pralits, J. O., Bottaro, A. & Cherubini, S. 2015 Weakly nonlinear optimal perturbations. J. Fluid Mech. 785, 135–151.CrossRefGoogle Scholar
Price, J. F., Weller, R. A. & Pinkel, R. 1986 Diurnal cycling: observations and models of the upper ocean response to diurnal heating, cooling, and wind mixing. J. Geophys. Res. 91 (C7), 8411–8427.Google Scholar
Pringle, C. C. T. & Kerswell, R. R. 2010 Using nonlinear transient growth to construct the minimal seed for shear flow turbulence. Phys. Rev. Lett. 105, 154502.CrossRefGoogle ScholarPubMed
Rabin, S. M. E., Caulfield, C. P. & Kerswell, R. R. 2012 Triggering turbulence efficiently in plane Couette flow. J. Fluid Mech. 712, 244–272.CrossRefGoogle Scholar
Reddy, S. C., Schmid, P. J., Baggett, J. S. & Henningson, D. S. 1998 On stability of streamwise streaks and transition thresholds in plane channel flows. J. Fluid Mech. 365, 269–303.CrossRefGoogle Scholar
Riley, J. J. & deBruynKops, S. M. 2003 Dynamics of turbulence strongly influenced by buoyancy. Phys. Fluids 15 (7), 2047–2059.Google Scholar
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, 591–621.CrossRefGoogle Scholar
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, 178–223.Google Scholar
Smyth, W. D. & Moum, J. N. 2012 Ocean mixing by Kelvin–Helmholtz instability. Oceanography 25 (2), 140–149.CrossRefGoogle Scholar
Smyth, W. D. & Moum, J. N. 2013 Marginal instability and deep cycle turbulence in the eastern equatorial Pacific ocean. Geophys. Res. Lett. 40, 6181–6185.CrossRefGoogle Scholar
Smyth, W. D., Moum, J. N. & Caldwell, D. R. 2001 The efficiency of mixing in turbulent patches: inferences from direct simulations and microstructure observations. J. Phys. Oceanogr. 31, 1969–1992.Google Scholar
Taylor, J. R.2008 Numerical simulations of the stratified oceanic bottom boundary layer. PhD thesis, University of California, San Diego.Google Scholar
Tearle, M. O.2004 Optimal perturbation analysis of stratified shear flow. PhD thesis, University of Colorado.Google Scholar
Thorpe, S. A. 1973 Experiments on instability and turbulence in a stratified shear flow. J. Fluid Mech. 61 (4), 731–751.Google Scholar
Trefethen, L. N., Trefethen, A. E., Reddy, S. C. & Driscoll, T. A. 1993 Hydrodynamic stability without eigenvalues. Science 261, 578–584.Google Scholar
Venayagamoorthy, S. K. & Koseff, J. R. 2016 On the flux Richardson number in stably stratified turbulence. J. Fluid Mech. 798, R1.Google Scholar
Wunsch, C. & Ferrari, R. 2004 Vertical mixing, energy, and the general circulation of the oceans. Annu. Rev. Fluid Mech. 36, 281–314.Google Scholar