Stratified turbulence in the 21st century – new insights on an increasingly important problem

09:05-09:40 Stratified turbulent mixing in the ocean: patterns, processes, and parameterisation

Dr Jennifer MacKinnon, University of California San Diego, Scripps Institution of Oceanography, USA

Abstract

Though average observed diapycnal mixing rates in the ocean interior are consistent with values required by inverse models, recent focus has been on the dramatic spatial variability of mixing rates in both the upper and deep ocean, which spans several orders of magnitude. Global ocean models have been shown to be very sensitive not only to the overall level but to the detailed distribution of mixing. Some of these patterns are driven by the geography of generation, propagation and destruction of internal waves, which are thought to supply much of the power for turbulence in the ocean interior. I will briefly review some results from the last five years of a Climate Process Team tasked with improving representations of internal-wave driven mixing in the oceanic component of climate models. Another set of recent and ongoing work has turned to the poorly understood role of mesoscale and sub-mesoscale features in stratified oceanic turbulence. In some situations, the interplay between internal waves and mesoscale vorticity can noticeably enhance turbulent mixing rates. In other situations, sub-mesoscale instabilities act to re-stratify the ocean, a counter-balance of sorts to one-dimensional vertical mixing schemes. Recent observational examples of both situations will be presented, and discussed in the broader framework of global mixing rates.

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09:40-10:20 Mixing in density-stratified, free shear flows and the implications for mixing in the ocean

Professor Greg Ivey, University of Western Australia, Australia

Abstract

To first order, in the interior of the ocean the mean density and horizontal velocity fields can be considered a function of the vertical coordinate z. For this simple shearing flow, we introduce a new mixing length model to describe the mixing of density and momentum. We introduce a mixing length L_p controlling mixing in the density field and a mixing length L_m controlling mixing in the momentum field. There are no undetermined coefficients in the model, and no need to make any assumptions about the value of the flux Richardson number Ri_f. The model determines Ri_f and demonstrates Ri_f is dependent on the relative magnitudes of three length scales: L_p, L_o, and Ls , where L_o is the Ozmidov scale and Ls the Corrsin shear scale. The model predictions are in good agreement with published laboratory observations. We discuss the implications of the model for the interpretation of oceanic turbulent microstructure measurements and the description of mixing in numerical ocean models.

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10:50-11:30 Parameterising mixing in the stably stratified ocean interior

Dr Sonya Legg, Princeton University, USA

Abstract

Vertical mixing is suppressed in the stable density stratification of the ocean interior, yet the vertical turbulent diffusion of heat and salt still plays a significant role in the thermohaline circulation. Ocean climate models cannot explicitly resolve the mixing processes, so must employ parameterisations relating the mixing to resolved parameters. Such mixing requires a source of energy, supplied by sheared flow; when this shear is resolved, for example in large-scale ocean currents, the parameterised turbulent diffusion can be expressed in terms of the resolved flow and stratification. However, in much of the ocean, the shear responsible for mixing is due to internal waves, which are rarely simulated in climate models. These waves are generated by the tides and wind, propagate around the ocean, and eventually lead to mixing when they break. Parameterisation of the mixing due to breaking internal waves must account for the generation, propagation and dissipation of wave energy. High resolution simulations can be used to examine the mechanisms of wave breaking, extending understanding gained from observations. Here I will describe different internal wave breaking mechanisms, including nonlinear wave-wave interactions, wave reflection from sloping and shoaling topography, and transient hydraulic jumps, as well as recent efforts to combine this understanding into a global model of the tidally-driven internal wave energy budget leading to an energetically consistent parameterisation of mixing. The impact of different geographical distributions of wave-breaking on global ocean circulation will be demonstrated using coupled climate models.

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11:30-12:10 Mixing processes in the oil and gas industry

Dr Simon Bittleston, Schlumberger Gould Research, UK

Abstract

Oil wells are very long and skinny (with aspect ratios of the order 10^5) and in many operations a tube sits within the wellbore with fluids pumped down the centre of this pipe, and back up the annulus between the pipe and rock. Flow rates vary significantly leading to some operations being performed in laminar flows, whilst others are turbulent. The fluids can be Newtonian or non-Newtonian. When the tube is not concentric in the wellbore, it is possible that laminar, transitional and turbulent flow can coexist in the annular space at any particular depth. The simplest case of dispersion of a tracer in a single phase flow is already of interest as the time evolution of fluid properties of the outlet of a well can give an indication of earlier events near the bottom. Taylor dispersion calculations show how sensitive the outlet distribution can be to eccentricity of the inner tube, and how rotation of the inner tube can counteract this. In some cases it is also useful to understand how a tracer distribution approaches the Taylor limit. As many of the fluids used are non-Newtonian, even these relatively simple cases exhibit unusual behaviours. More complex cases involve pumping a sequence of fluids of varying densities and rheological properties down through the tube and up the annular space. In these cases mixing process are complex, with a variety of instabilities possible. An approximate model system can be derived for certain geometrical configurations leading to realistic prediction of mixing processes. Laminar flow problems are already challenging; adding the complexity of turbulent, or partially turbulent flows, leads to a range of problems which deserve greater study. This talk will lead from the single phase to the multi-fluid and from the laminar to the turbulent, to explain the broad range of rich mixing processes that can occur in an important practical application.

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13:10-13:50 Periodic orbits theory of turbulent flows

Professor Predrag Cvitanović, Georgia Institute of Technology, USA

Abstract

Partial differential equations are in principle infinite-dimensional dynamical systems. However, recent studies offer strong numerical evidence that the turbulent solutions of spatially extended dissipative systems evolve within a manifold spanned by a finite number of 'entangled' modes, dynamically isolated from the residual set of isolated, transient degrees of freedom. Initial studies, based on numerical simulations of long ergodic trajectories, yield no intuition about the geometry of such attractors. That is attained by studying the hierarchies of unstable periodic orbits, invariant solutions which, together with their Floquet vectors, provide an effective description of both the local hyperbolicity and the global geometry of an attractor embedded in a high-dimensional state space. Dynamical systems with translational or rotational symmetry arise frequently in studies of spatially extended physical systems, such as Navier-Stokes flows on periodic domains, with each fluid state having an infinite number of equivalent solutions obtained from it by a translation or a rotation. This multitude of equivalent solutions tends to obscure the dynamics of turbulence, and the crucial step in the analysis of such a system is symmetry reduction. We offer several implementations of 'method of slices' applicable to very high-dimensional problems and show that after application of the method, hitherto unseen global structures, for example to pipe flow, relative periodic orbits and their unstable manifolds are uncovered. Whether the periodic orbit theory of computing expectation values of measurable observables is applicable to such high-dimensional flows remains an open question.

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13:50-14:30 The zombie vortex instability – a new, fast, robust instability in rotating, horizontally-shearing, vertically-stratified flows

Professor Philip Marcus, University of California, Berkeley, USA

Abstract

Without instabilities, gas around a forming protostar remains in orbit, and the final star cannot form; dust grains cannot accumulate to form planets; and the compositions of meteorites cannot be explained. Unfortunately, the Keplerian motion within a disk is assumed by most astrophysicists to be stable by Rayleigh’s theorem because the angular momentum of the disk increases with increasing radius. We show that there is a new purely hydrodynamic instability that is violent and destabilises the protoplanetary disk, filling it with turbulence. The essential ingredients of the new instability are rotation, shear, and vertical density stratification, so the instability can occur in stratified Boussinesq (or fully compressible) Couette flows. Our new instability occurs at critical layers where neutrally-stable eigenmodes are singular in the inviscid limit (but finite, with a width that scales as the Reynolds number Re to the -1/3 power when viscosity is present) and requires an initial finite-amplitude perturbation. In a flow initialised with weak Kolmogorov noise with initial Mach number Ma, when Ma > Re^-1/2 (~10^-7 in a protoplanetary disk) the instability will be triggered and create turbulence and large-volume and large-amplitude vortices that fill the disk. When the initial perturbation is an isolated vortex, the vortex triggers a new generation of vortices at the nearby critical layers. After this second generation of vortices grows large, it triggers a third generation. The triggering of subsequent generations continues ad infinitum in a self-similar manner creating a 3D lattice of turbulent 3D vortices.

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15:00-15:40 The turbulent/non-turbulent interface in stably stratified fluids

Professor James Riley, University of Washington, USA

Abstract

The results of a study, employing direct numerical simulations, of the turbulent/nonturbulent interface of a wake in a stably-stratified fluid will be discussed. It is found that thresholds for both enstrophy and potential enstrophy are needed to identify the interface. Using conditional averaging relative to the location of the interface, various quantities of interest are examined. The thickness of the interface is found to scale with the Kolmogorov scale. From an examination of the Ozmidov and Kolmogorov length scales as well as the buoyancy Reynolds number, it is found that the buoyancy Reynolds number decreases and becomes of order 1 near the interface, indicating the suppression of the turbulence there by the stable stratification. Finally the overall rate of loss of energy due to internal wave radiation is found to be comparable to the overall rate of loss due to turbulent kinetic energy dissipation.

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15:40-16:20 Mixing and critical entrainment phenomena in stratified fluids

Professor Roberto Camassa, University of North Carolina, USA

Abstract

Density stratification can add a considerable layer of complexity to the dynamics of fluids: coherent structures, such as vortex rings, bodies or jets may move ambient fluids into regions where buoyancy forces can arise creating strong flows unless mitigated by mixing or viscosity. This can give rise to critical phenomena in which bodies and buoyant fluids may escape or become trapped as parameters (such as the propagation distance) are varied. This talk will present an overview of our theoretical, computational, and experimental studies on a class of these critical phenomena, focusing on the associated vertical transport and mixing.

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09:00-09:40 Experiments with mixing in stratified flow over a ridge

Professor Ross Griffiths, Australian National University, Australia

Abstract

Interaction of geostrophic balanced flow with bottom topography is thought to generate lee waves that can carry energy into the overlying ocean and lead to turbulent mixing. Topography will also cause local mixing, particularly instabilities in the wake. Ocean models require parameterisation of these effects and currently assume 30% of the energy removed from the mean flow is consumed in local mixing and turbulent dissipation, the remainder being radiated and causing turbulent mixing elsewhere. I will discuss laboratory experiments with a ridge towed through uniform density stratification, or a mixed layer under a uniform gradient, in a long channel, creating a mean flow over the ridge. The total mixing rate is measured across three parameter regimes including linear lee waves, nonlinear flow and an evanescent regime in which wave radiation is weak. Measurements provide the depth-dependence of turbulent mixing, allowing separation of the local and remote contributions to mixing. Remote mixing is dominant only for a narrow band of Froude numbers; under other conditions local mixing is dominant. The results suggest that mixing by local nonlinear mechanisms close to abyssal ocean topography may be much greater than remote mixing by lee waves.

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09:40-10:20 Rotating plane Couette flow – instabilities, structures and turbulence

Professor Henrik Alfredsson, KTH, Sweden

Abstract

Shear flows subjected to system rotation, where the rotation axis is parallel with the mean-flow vorticity, are influenced by a Coriolis force, which may have a strong effect on the flow field even at low rotation rates. For such flows, as pointed out by Bradshaw in 1969, there is an analogy with stratified flows. Here we will discuss plane Couette flow (PCF) under anticyclonic rotation both in the laminar and turbulent regimes. Without rotation, PCF is linearly stable for all Reynolds numbers, however, in experiments transition to turbulence is observed around Re=350. With anticyclonic rotation, the critical Reynolds number is as low as 20.6, at which point the flow bifurcates to a flow with streamwise-oriented roll cells. With increasing Reynolds number and/or rotation rate, the laminar roll cells develop various types of other complex instabilities and at higher Reynolds numbers the flow enters a turbulent regime, although it is still dominated by streamwise roll cells. 

We present experimental results where all three velocity components are measured with PIV, enabling determination of the mean flow and all four non-zero Reynolds stresses across the central parts of the channel. We discuss the resulting flow structures as well as an analysis of the Reynolds-stress equations and how they relate to the fact that the absolute vorticity, ie the sum of the averaged spanwise-flow vorticity and system rotation, tends to zero in the central region of the channel for high enough rotation rates.

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10:50-11:30 Transition in unstratified pipe flow

Professor Thomas Mullin, University of Manchester, UK

Abstract

Despite more than a century of research, the puzzle of why fluid motion along a pipe is observed to become turbulent as the flow rate is increased remains the outstanding challenge of hydrodynamic stability theory. The issue is both of deep scientific and engineering interest since most pipe flows are turbulent in practice, even at modest flow rates. All theoretical work indicates that the flow is linearly stable ie infinitesimal disturbances decay as they propagate along the pipe and the flow will remain laminar. In practice, finite amplitude perturbations are responsible for triggering turbulence and these become more important as the non-dimensional flow rate, the Reynolds number Re, increases. Transition is usually catastrophic and elucidating details of the processes involved is generally difficult. Here we show that the judicious choice of perturbation can be used to highlight important details and we also provide experimental evidence for long live edge states which are believed to exist on the boundary between laminar and turbulent flows. These new experimental results provide insights into the origins of the turbulent motion and suggest links can be made with recent theoretical work on the Navier Stokes equations.

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11:30-12:10 Stochastic parametrisation and inexact computing for weather and climate prediction

Professor Tim Palmer FRS, University of Oxford, UK

Abstract

There are good reasons to suppose the closure schemes for weather and climate models should be formulated stochastically, rather than deterministically as has been traditional. This puts into sharp focus a question of both theoretical and practical importance: in a multi-scale system such as weather and climate (with scale-dependent Lyapunov exponents), how many bits of real information are carried by the prognostic variables as a function of scale. There are good reasons to suppose that for many small-scale variables, information can be represented using significantly fewer bits than 64 (the standard default for scientific computing). Assessing this quantitatively may be a route to making much more efficient use of supercomputing resources and hence to increasing model resolution.

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13:10-13:50 From topographic internal gravity waves to instabilities and turbulence

Professor Sutanu Sarkar, University of California, San Diego, USA

Abstract

The substantial energy converted from the oscillating tide to internal waves at deep, rough topography is a key source of turbulence and mixing in the abyssal ocean. Some of this energy breaks down near the boundary to feed local turbulence and mixing while the remainder is radiated away to fuel remote turbulence. We have investigated the operative nonlinear processes through three-dimensional simulations that resolve the instabilities and ensuing turbulence. I will review local nonlinearities and their contribution to the baroclinic energy balance at a model triangular ridge and at a scaled-down model of realistic steep topography at Luzon Strait. The periodically stratified, oscillating boundary flow transitions to turbulence at sloping regions with near-critical angle where upslope bores form, at the upper portion of steep obstacles where downslope jets form and detach from the boundary, and through direct breaking of lee waves on or above the obstacle. Convective instability with high mixing efficiency occurs in the boundary layer as well as in breaking lee waves. High-mode internal wave beams launched from steep regions of the obstacle are turbulent in the near field. I will also touch upon the cascade of energy to waves with short vertical scales and eventually turbulence through parametric subharmonic instability (PSI) during refraction and after reflection.

13:50-14:30 Flavours of stratified turbulence

Professor W Richard Peltier, University of Toronto, Canada

Abstract

The role of stratified turbulence in effecting the vertical flux of mass required to enable the deep water that forms in the polar oceans to upwell to the surface of the southern ocean, and thereby lead to closure of the overturning circulation, is well known. What is as yet insufficiently well understood, however, is how this small scale process might best be represented in large scale models of the ocean general circulation. This is a complex issue because of the variety of mechanisms through which the small scale turbulence is produced. These mechanisms include a variety of breaking wave related processes in which the hydrodynamic waves of interest include those generated through initial shear instabilities of either Kelvin-Helmholtz or Holmboe type. Equally relevant breaking wave related mechanisms include those in which internal waves launched by stratified flow over bottom topography thereafter ‘break’ either near to or distant from their source of excitation. In the latter case the forcing could be related either to the barotropic tide or to the action of larger scale baroclinic eddies. An important issue, insofar as the parameterisation problem is concerned, is whether there might be generic properties shared by high Reynolds number stratified turbulence irrespective of the mechanism through which it is produced. In the case of shear generated breaking waves involving either of the above mechanisms it has proven possible to demonstrate that the post transition characteristics of the turbulence are rather generic though dependent upon knowledge of three distinct turbulence characteristics, respectively the buoyancy Reynolds number Re_b representative of turbulence intensity, a ‘bulk’ Richardson number Ri_b representative of the strength of the shear, and a turbulent Prandtl number Pr_t representing the ratio of turbulent  momentum diffusivity to that for density. A summary of recently proposed parameterisation schemes will be presented.

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15:00-15:40 Near-inertial energy propagation inside an anticyclone: a pathway to stratified turbulence

Dr Pascale Lelong, NorthWest Research Associates, USA

Abstract

Understanding the manner by which wind-generated, near-inertial energy leaves the oceanic mixed layer continues to be a topic of interest since it provides an important pathway from a surface energy source to eventual dissipation at depth. Whether most of the dissipation occurs at the base of the mixed layer or in the stratified interior, at critical layers or through shear instability remains an open question. The role of mesoscale eddies in trapping near-inertial energy is, by now, well established. Here, the focus is on the vertical propagation of near-inertial energy trapped in an isolated eddy with a detailed examination of the energy budget and scales of generated inertia-gravity waves as a function of eddy vorticity and spatial extent.

Waves with frequencies close to the inertial frequency break down primarily via shear instability, characterised by KH billows. Those with intermediate frequencies between inertial and buoyancy frequencies are subject to a hybrid shear/convective instability that gives rise to mushroom-like structures whereas high-frequency waves are prone to rapid convective instability. For near-inertial waves, the onset of instability occurs in the phase region of strongly stratified fluid in contrast to the scenario for convective instability where the density gradient is at its weakest. The energetics of breaking waves will be discussed as a function of their frequency and amplitude to establish whether the instability that leads to their breakdown is a factor in the resulting mixing efficiency.

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15:40-16:20 The turbulent transition of a supercritical downslope flow: sensitivity to downstream conditions

Professor Kraig Winters, University of California San Diego, Scripps Institution of Oceanography, USA

Abstract

Blocked, continuously stratified, crest controlled flows have hydraulically supercritical downslope flow in the lee of a ridge-like obstacle. The downslope flow separates from the obstacle and, depending on conditions further downstream, transitions to a subcritical state. A controlled, stratified overflow and its transition to a subcritical state are investigated here in a set of three-dimensional numerical experiments. The downslope flow is associated with an isopycnal and streamline bifurcation, which acts to form a nearly uniform density isolating layer and a sharp pycnocline that separates deeper blocked and stratified fluid between the ridges from the flow above. The height of the downstream obstacle is communicated upstream via gravity waves that propagate along the density interface and set the separation depth of the downslope flow. The penetration depth of the downslope flow, its susceptibility to shear instabilities, and the amount of energy dissipated in the turbulent outflow all increase as the height of a downstream ridge, which effectively sets the downstream boundary conditions, is reduced.

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