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

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.

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.

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.

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.

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.

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.

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.

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.