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Irreversible mixing by unstable periodic orbits in buoyancy dominated stratified turbulence

Published online by Cambridge University Press:  26 October 2017

Dan Lucas Open the ORCID record for Dan Lucas [Opens in a new window]  and
C. P. Caulfield Open the ORCID record for C. P. Caulfield [Opens in a new window]
Dan Lucas*
Affiliation:
School of Computing and Mathematics, Keele University, Staffordshire, ST5 5BG
C. P. Caulfield
Affiliation:
BP Institute, University of Cambridge, Madingley Rise, Madingley Road, Cambridge CB3 0EZ, UK Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK
*
†Email address for correspondence: [email protected]
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Abstract

We consider turbulence driven by a large-scale horizontal shear in Kolmogorov flow (i.e. with sinusoidal body forcing) and a background linear stable stratification with buoyancy frequency $N_{B}^{2}$ imposed in the third, vertical direction in a fluid with kinematic viscosity $\unicode[STIX]{x1D708}$. This flow is known to be organised into layers by nonlinear unstable steady states, which incline the background shear in the vertical and can be demonstrated to be the finite-amplitude saturation of a sequence of instabilities, originally from the laminar state. Here, we investigate the next order of motions in this system, i.e. the time-dependent mechanisms by which the density field is irreversibly mixed. This investigation is achieved using ‘recurrent flow analysis’. We identify (unstable) periodic orbits, which are embedded in the turbulent attractor, and use these orbits as proxies for the chaotic flow. We find that the time average of an appropriate measure of the ‘mixing efficiency’ of the flow $\mathscr{E}=\unicode[STIX]{x1D712}/(\unicode[STIX]{x1D712}+{\mathcal{D}})$ (where ${\mathcal{D}}$ is the volume-averaged kinetic energy dissipation rate and $\unicode[STIX]{x1D712}$ is the volume-averaged density variance dissipation rate) varies non-monotonically with the time-averaged buoyancy Reynolds numbers $\overline{Re}_{B}=\overline{{\mathcal{D}}}/(\unicode[STIX]{x1D708}N_{B}^{2})$, and is bounded above by $1/6$, consistently with the classical model of Osborn (J. Phys. Oceanogr., vol. 10 (1), 1980, pp. 83–89). There are qualitatively different physical properties between the unstable orbits that have lower irreversible mixing efficiency at low $\overline{Re}_{B}\sim O(1)$ and those with nearly optimal $\mathscr{E}\lesssim 1/6$ at intermediate $\overline{Re}_{B}\sim 10$. The weaker orbits, inevitably embedded in more strongly stratified flow, are characterised by straining or ‘scouring’ motions, while the more efficient orbits have clear overturning dynamics in more weakly stratified, and apparently shear-unstable flow.

JFM classification

Nonlinear Dynamical Systems: Nonlinear Dynamical Systems Turbulent Flows: Stratified turbulence Mixing: Turbulent mixing
Type
Rapids
Information
Journal of Fluid Mechanics , Volume 832 , 10 December 2017 , R1
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Copyright
© 2017 Cambridge University Press 

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Lucas supplementary movie 1

Movie showing the streamwise velocity, u, for the unstable periodic orbit UPOo1. Time steps are 0.5 time units. Note animation is made in the frame travelling with the relative periodic orbit in order to visualise the periodicity in time.

Download Lucas supplementary movie 1(Video)
Video 3.2 MB

Lucas supplementary movie 2

Movie showing the total density, ρtot=ρ-z, for the unstable periodic orbit UPOo1. Time steps are 0.5 time units. Note animation is made in the frame travelling with the relative periodic orbit in order to visualise the periodicity in time.

Download Lucas supplementary movie 2(Video)
Video 2.7 MB

Lucas supplementary movie 3

Movie showing the streamwise velocity, u, for the unstable periodic orbit UPOl1. Time steps are 0.5 time units. Note animation is made in the frame travelling with the relative periodic orbit in order to visualise the periodicity in time.

Download Lucas supplementary movie 3(Video)
Video 5.4 MB

Lucas supplementary movie 4

Movie showing the total density, ρtot=ρ-z, for the unstable periodic orbit UPOl1. Time steps are 0.5 time units. Note animation is made in the frame travelling with the relative periodic orbit in order to visualise the periodicity in time.

Download Lucas supplementary movie 4(Video)
Video 4.7 MB

Lucas supplementary movie 5

Movie showing the perturbation density, ρ, for the unstable periodic orbit UPOl1. Time steps are 0.5 time units. Note animation is made in the frame travelling with the relative periodic orbit in order to visualise the periodicity in time.

Download Lucas supplementary movie 5(Video)
Video 6.7 MB