Session A23: Geophysical Fluid Dynamics: Stratified Turbulence I

Multiple instabilities in layered stratified plane Couette flow

We consider the linear stability and nonlinear evolution of a Boussinesq fluid consisting of three layers with density $\rho_a - \Delta \rho/2$, $\rho_a$ and $\rho_a + \Delta \rho/2$ of equal depth $d/3$ in a 2D channel where the horizontal boundaries are driven at a constant relative velocity $\Delta U$. Unlike unstratified flow, we demonstrate that for all $Ri_b=g \Delta \rho d/(\rho_a \Delta U)^2 > 0$, and for sufficiently large $Re=\Delta U d/(4 \nu)$, this flow is linearly unstable to normal mode disturbances of the form first considered by Taylor (1931). These instabilities, associated with a coupling between Doppler-shifted internal waves on the density interfaces, have a growth rate (maximised across wavenumber and $Ri_b$) which is a non-monotonic function of $Re$. Through 2D simulation, we explore the nonlinear evolution of these primary instabilities at various $Re$, demonstrating that the primary instabilities grow to finite amplitude as vortices in the intermediate fluid layer before rapidly breaking down, modifying the mean flow to become susceptible to strong and long-lived secondary instabilities of Holmboe (1962) type, associated with vortices now localised in the top and bottom layers.   

Experimental study of mixing mechanisms in stably stratified Taylor-Couette flow

We consider experimentally the mechanisms of mixing in stably stratified Taylor-Couette (TC) flow in a TC apparatus for which both cylinders can rotate independently. In the case for which only the inner cylinder rotates, centrifugal instability rapidly splits an initially linear density profile into an array of thin nearly homogeneous layers. Shadowgraph, PIV and density profiles measured by a moving conductivity probe allow us to characterise this process and the resulting flow. In particular, we observe turbulent intrusions of mixed fluid propagating relatively slowly around the tank at the interfaces between the layers, leading to a time-dependent variation in the sharpness and turbulent activity at these interfaces, whose period scales with (but is much larger than) the rotation period. Interestingly, the turbulent intrusions are anti-correlated between adjacent interfaces leading to snake-skin-like patterns in the spatio-temporal diagrams of the density profiles. We also explore how the presence of a density stratification modifies end effects at the top and bottom of the cylinders, in both the presence and absence of primary centrifugal instability.   

Turbulence and mixing from optimal perturbations to a stratified shear layer

The stability and mixing of stratified shear layers is a canonical problem in fluid dynamics with relevance to flows in the ocean and atmosphere. The Miles-Howard theorem states that a necessary condition for normal-mode instability in parallel, inviscid, steady stratified shear flows is that the gradient Richardson number, $Ri_g$ is less than 1/4 somewhere in the flow. However, substantial transient growth of non-normal modes may be possible at finite times even when $Ri_g>1/4$ everywhere in the flow. We have calculated the ``optimal perturbations'' associated with maximum perturbation energy gain for a stably-stratified shear layer. These optimal perturbations are then used to initialize direct numerical simulations. For small but finite perturbation amplitudes, the optimal perturbations grow at the predicted linear rate initially, but then experience sufficient transient growth to become nonlinear and susceptible to secondary instabilities, which then break down into turbulence. Remarkably, this occurs even in flows for which $Ri_g > 1/4$ everywhere. We will describe the nonlinear evolution of the optimal perturbations and characterize the resulting turbulence and mixing.   

Energy transfers, mixing efficiency and the internal structure of stratified Rayleigh-Taylor instability

Rayleigh--Taylor instability has been shown in experiments to have a high mixing efficiency ($\eta > 0.75$) when it occurs at an interface between two otherwise stably stratified layers. In this presentation, an implicit large eddy simulation (which uses numerical diffusion as a proxy for physical viscous diffusion) is used to model the instability and the resulting turbulent flow. The final state of simulations is shown to have an excellent match with experiments. The simulations allow the tracking of energy in the flow, revealing some interesting behavior with implications for the study of mixing in stratified flows more generally.   

Subharmonic instability during the off-critical reflection of an internal wave beam

Numerical simulations at laboratory scale are performed to study the reflection of an internal wave beam at a sloping bottom. When the incoming wave Froude number $Fr_i$ is small, the reflection process can be approximated by linear theory and almost all of the reflected energy is confined to the primary wave frequency. In cases where the incoming wave Froude number $Fr_i$ is sufficiently high ($\approx 0.07$ in the present study) and the internal wave angle is close to but greater than the slope angle, the reflected wave undergoes parametric subharmonic instability (PSI) resulting in the formation of two subharmonic waves with frequencies $0.33 \Omega$ and $0.67 \Omega $. The energy in the subharmonics is found to be of the same order as that in the primary reflected beam. PSI is not found during critical reflection ($\alpha=\beta$) at any incoming wave Froude number. Thus, reflection of internal waves at a near but off-critical slope provides a potential mechanism for mixing through the generation of subharmonic waves with smaller vertical scales that could break down into turbulence.   

Surface manifestation of internal waves emitted by an evolving stably stratified turbulent shear flow

Internal waves (IWs) from submerged turbulent sources may manifest themselves at the sea surface by generating coherent and persistent spatial features. Such IWs emitted by the turbulent wake of a towed sphere in a linearly stratified Boussinesq fluid are investigated numerically. The fully nonlinear three-dimensional simulations resolve both the wave-emitting turbulent wake at Reynolds number $Re\in[5\times10^{3},10^{5}]$ and Froude number $Fr\in[4,16,64]$, and the subsurface region where the IWs interact with the sea surface which is modeled by a free-slip rigid lid. As the wake evolves for up to 250 units of buoyancy timescales, IW characteristics such as wavelength and frequency are measured both near the source and at the surface for comparison; the statistics of magnitudes and orientations of IW-induced surface strains are reported. Various IW impacts at the surface, such as local enrichment of surfactant and dispersion of ocean surface tracers, are also discussed.   

Initially Isotropic Turbulence Subjected to Stabilizing Stratification

When turbulence in a stably stratified fluid decays, it often does so without a continuous source of energy. As a result, the turbulence time scale increases relative to the buoyancy time scale so that the Froude number $F$ deceases in time. In wakes, for instance, scaling arguments lead us to expect $F \sim O(1)$ one buoyancy period after the object has passed, and extensive studies have been carried out to understand how wakes evolve as the buoyancy force becomes increasingly important in time. Even in the unstratified case, though, a turbulent wake is a complicated flow to study. A much simpler configuration is isotropic homogeneous turbulence (IHT). For this study, simulated IHT that exhibits power-law decay is suddenly subjected to stabilizing stratification. The simulations use up to 8192x8192x4096 grid points to resolve the largest and smallest length scales of the flow over a span of at least 10 buoyancy periods. Two Reynolds numbers differing by an order of magnitude are considered, with the lower Reynolds number having a range of turbulence length scales comparable to that in laboratory experiments of stratified turbulent wakes. In this paper, the evolution of the flow as $F$ deceases with time is discussed, as is the effect of the initial Reynolds number.   

Lagrangian and Eulerian Acceleration Statistics in Turbulent Stratified Shear Flows

The Lagrangian and Eulerian acceleration statistics in homogeneous turbulence with shear and stratification are studied using direct numerical simulations. The Richardson number is varied from $Ri=0$, corresponding to unstratified shear flow, to $Ri=1$, corresponding to strongly stratified shear flow. In addition, the scale dependence of the acceleration statistics is studied using a wavelet-based approach. The probability density functions (pdfs) of both Lagrangian and Eulerian accelerations show a strong and similar influence on the Richardson number and extreme values for Eulerian acceleration are stronger than those observed for the Lagrangian acceleration. Similarly, the Eulerian time-rate of change of fluctuating density is observed to have larger extreme values than that of the Lagrangian time-rate of change. Hence, the time-rate of change of fluctuating density obtained at a fixed location by an Eulerian observer is mainly due to advection of fluctuating density through this location, while the time-rate of change of fluctuating density following a fluid particle is substantially smaller, and due to production and dissipation of fluctuating density.