Skip to main content

Transient growth in strongly stratified shear layers

  • A. K. Kaminski (a1), C. P. Caulfield (a1) (a2) and J. R. Taylor (a1)

We investigate numerically transient linear growth of three-dimensional perturbations in a stratified shear layer to determine which perturbations optimize the growth of the total kinetic and potential energy over a range of finite target time intervals. The stratified shear layer has an initial parallel hyperbolic tangent velocity distribution with Reynolds number $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathit{Re}=U_0 h/\nu =1000$ and Prandtl number $\nu /\kappa =1$ , where $\nu $ is the kinematic viscosity of the fluid and $\kappa $ is the diffusivity of the density. The initial stable buoyancy distribution has constant buoyancy frequency $N_0$ , and we consider a range of flows with different bulk Richardson number ${\mathit{Ri}}_b=N_0^2h^2/U_0^2$ , which also corresponds to the minimum gradient Richardson number ${\mathit{Ri}}_g(z)=N_0^2/(\mathrm{d}U/\mathrm{d} z)^2$ at the midpoint of the shear layer. For short target times, the optimal perturbations are inherently three-dimensional, while for sufficiently long target times and small ${\mathit{Ri}}_b$ the optimal perturbations are closely related to the normal-mode ‘Kelvin–Helmholtz’ (KH) instability, consistent with analogous calculations in an unstratified mixing layer recently reported by Arratia et al. (J. Fluid Mech., vol. 717, 2013, pp. 90–133). However, we demonstrate that non-trivial transient growth occurs even when the Richardson number is sufficiently high to stabilize all normal-mode instabilities, with the optimal perturbation exciting internal waves at some distance from the midpoint of the shear layer.

Corresponding author
Email address for correspondence:
Hide All
Arratia, C., Caulfield, C. P. & Chomaz, J.-M. 2013 Transient perturbation growth in time-dependent mixing layers. J. Fluid Mech. 717, 90133.
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.
Bretherton, F. P. & Garrett, C. J. R. 1969 Wavetrains in inhomogeneous moving media. Proc. R. Soc. Lond. A 302, 529554.
Constantinou, N. C. & Ioannou, P. J. 2011 Optimal excitation of two-dimensional Holmboe instabilities. Phys. Fluids 23, 074102.
Drazin, P. G. & Reid, W. H. 2004 Hydrodynamic Stability, 2nd edn Cambridge University Press.
Ellingsen, T. & Palm, E. 1975 Stability of linear flow. Phys. Fluids 18 (4), 487488.
Farrell, B. F. & Ioannou, P. J. 1993a Optimal excitation of three-dimensional perturbations in viscous constant shear flow. Phys. Fluids A 5 (6), 13901400.
Farrell, B. F. & Ioannou, P. J. 1993b Transient development of perturbations in stratified shear flow. J. Atmos. Sci. 50 (14), 22012214.
Foures, D. P. G., Caulfield, C. P. & Schmid, P. J. 2012 Variational framework for flow optimization using seminorm constraints. Phys. Rev. E 86, 026306.
Garrett, C. 2003 Mixing with latitude. Nature 422, 477478.
Holmboe, J. 1962 On the behaviour of symmetric waves in stratified shear layers. Geophys. Publ. 24, 67113.
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.
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.
Landahl, M. T. 1980 A note on an algebraic instability of inviscid parallel shear flows. J. Fluid Mech. 98 (2), 243251.
Lindborg, E. 2006 The energy cascade in a strongly stratified fluid. J. Fluid Mech. 550, 207242.
Luchini, P. & Bottaro, A. 2014 Adjoint equations in stability analysis. Annu. Rev. Fluid Mech. 46, 493517.
Mack, S. A. & Schoeberlein, H. C. 2004 Richardson number and ocean mixing: towed chain observations. J. Phys. Oceanogr. 34 (4), 736754.
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, 968.
Polzin, K. & Ferrari, R. 2004 Isopycnal dispersion in NATRE. J. Phys. Oceanogr. 34 (1), 247257.
Rabin, S. M. E., Caulfield, C. P. & Kerswell, R. R. 2012 Triggering turbulence efficiently in plane Couette flow. J. Fluid Mech. 712, 244272.
Riley, J. J. & de Bruyn Kops, S. M. 2003 Dynamics of turbulence strongly influenced by buoyancy. Phys. Fluids 15 (7), 20472059.
Schmid, P. J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129162.
Spedding, G. R., Browand, F. K. & Fincham, A. M. 1996 Turbulence, similarity scaling and vortex geometry in the wake of a towed sphere in a stably stratified fluid. J. Fluid Mech. 314, 53103.
Taylor, J. R.2008 Numerical simulations of the stratified oceanic bottom boundary layer. PhD thesis, University of California, San Diego.
Tearle, M. O.2004 Optimal perturbation analysis of stratified shear flow. PhD thesis, University of Colorado.
Wunsch, C. & Ferrari, R. 2004 Vertical mixing, energy, and the general circulation of the oceans. Annu. Rev. Fluid Mech. 36, 281314.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

JFM classification


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