Finite-amplitude manifestations of stratified shear flow instabilities and their spatio-temporal coherent structures are believed to play an important role in turbulent geophysical flows. Such shear flows commonly have layers separated by sharp density interfaces, and are therefore susceptible to the so-called Holmboe instability, and its finite-amplitude manifestation, the Holmboe wave. In this paper, we describe and elucidate the origin of an apparently previously unreported long-lived coherent structure in a sustained stratified shear flow generated in the laboratory by exchange flow through an inclined square duct connecting two reservoirs filled with fluids of different densities. Using a novel measurement technique allowing for time-resolved, near-instantaneous measurements of the three-component velocity and density fields simultaneously over a three-dimensional volume, we describe the three-dimensional geometry and spatio-temporal dynamics of this structure. We identify it as a finite-amplitude, nonlinear, asymmetric confined Holmboe wave (CHW), and highlight the importance of its spanwise (lateral) confinement by the duct boundaries. We pay particular attention to the spanwise vorticity, which exhibits a travelling, near-periodic structure of sheared, distorted, prolate spheroids with a wide ‘body’ and a narrower ‘head’. Using temporal linear stability analysis on the two-dimensional streamwise-averaged experimental flow, we solve for three-dimensional perturbations having two-dimensional, cross-sectionally confined eigenfunctions and a streamwise normal mode. We show that the dispersion relation and the three-dimensional spatial structure of the fastest-growing confined Holmboe instability are in good agreement with those of the observed confined Holmboe wave. We also compare those results with a classical linear analysis of two-dimensional perturbations (i.e. with no spanwise dependence) on a one-dimensional base flow. We conclude that the lateral confinement is an important ingredient of the confined Holmboe instability, which gives rise to the CHW, with implications for many inherently confined geophysical flows such as in valleys, estuaries, straits or deep ocean trenches. Our results suggest that the CHW is an example of an experimentally observed, inherently nonlinear, robust, long-lived coherent structure which has developed from a linear instability. We conjecture that the CHW is a promising candidate for a class of exact coherent states underpinning the dynamics of more disordered, yet continually forced stratified shear flows.

No abstract

We describe the long-term dynamics of sustained stratified shear flows in the laboratory. The Stratified Inclined Duct (SID) experiment sets up a two-layer exchange flow in an inclined duct connecting two reservoirs containing salt solutions of different densities. This flow is primarily characterised by two non-dimensional parameters: the tilt angle of the duct with respect to the horizontal, θ (a few degrees at most), and the Reynolds number Re, an input parameter based on the density difference driving the flow. The flow can be sustained with constant forcing over arbitrarily long times and exhibits a wealth of dynamical behaviours representative of geophysically-relevant sustained stratified shear flows. Varying θ and Re leads to four qualitatively different regimes: laminar flow; mostly laminar flow with finite-amplitude, travelling Holmboe waves; spatio-temporally intermittent turbulence with substantial interfacial mixing; and sustained, vigorous interfacial turbulence (Meyer & Linden, J. Fluid Mech., vol. 753, 2014, pp. 242-253). We seek to explain the scaling of the transitions between flow regimes in the two-dimensional plane of input parameters (θ, Re). We improve upon previous studies of this problem by providing a firm physical basis and non-dimensional scaling laws that are mutually consistent and in good agreement with the empirical transition curves we inferred from 360 experiments spanning θ ∈ [−1 • , 6 • ] and Re ∈ [300, 5000]. To do so, we employ state-of-the-art simultaneous volumetric measurements of the density field and the three-component velocity field, and analyse these experimental data using time-and volume-averaged potential and kinetic energy budgets. We show that regime transitions are caused by an increase in the non-dimensional time-and volume-averaged kinetic energy dissipation within the duct, which scales with θRe at high enough angles. As the power input scaling with θRe is increased above zero, the two-dimensional, parallelflow dissipation (power output) increases to close the budget through an increase in the magnitude of the exchange flow, incidentally triggering Holmboe waves above a certain threshold in interfacial shear. However, once the hydraulic limit of two-layer exchange flows is reached, two-dimensional dissipation plateaus and three-dimensional dissipation at small scales (turbulence) takes over, first intermittently, and then steadily, in order to close the budget and follow the θRe scaling. This general understanding of regime transitions and energetics in the SID experiment may serve as a basis for the study of more complex sustained stratified shear flows found in the natural environment.

An experimental study of the microstructural evolution of an immiscible basaltic emulsion shows that the Fe-rich liquid forms homogeneously nucleated droplets dispersed in a continuous Si-rich liquid, together with droplets heterogeneously nucleated on plagioclase, magnetite, and pyroxene. Heterogeneous nucleation is likely promoted by localised compositional heterogeneities around growing crystals. The wetting angle of Fe-rich droplets on both plagioclase and magnetite increases with decreasing temperature. Droplet coarsening occurs by a combination of diffusion-controlled growth and Ostwald ripening, with an insignificant contribution from coalescence. Characteristic microstructures resulting from the interaction of immiscible Fe-rich liquid with crystal grains during crystal growth can potentially be used as an indicator of liquid unmixing in fully crystallised natural samples. In magma bodies < ~ 10 m in size, gravitationally driven segregation of immiscible Fe-rich droplets is unlikely to be significant.

We present simultaneous two-dimensional velocity and scalar measurements on a central vertical plane in a pure turbulent plume. We use an edge-detection algorithm to determine the edge of the plume, and compare the data obtained in both a fixed Eulerian frame and a frame relative to local coordinates defined in terms of the instantaneous plume edge. In an Eulerian frame we observe that the time-averaged distributions of vertical and horizontal velocity are self-similar, the vertical velocity being well represented by a Gaussian distribution. We condition these measurements on whether fluid is inside or outside of the plume, and whether fluid inside is mixed plume fluid or engulfed ambient fluid. We find that, on average, 5% of the total vertical volume transport occurs outside the plume and this figure rises to nearly 14% at heights between largescale coherent structures. We show that the fluxes of engulfed fluid within the plume envelope are slightly larger than the vertical transport outside the plume -indicating that ambient fluid is engulfed into the plume envelope before being nibbled across the turbulent/non-turbulent interface (TNTI) and then ultimately irreversibly mixed. Our new measurements in the plume coordinate (following the meandering fluctuating plume)show the flow within the plume and in the nearby ambient fluid is strongly influenced by whether an eddy is present locally within the plume, or absent. When an eddy is present and the plume is wide, the vertical velocities near the plume edge are small and hence all vertical transport is inside the plume. In regions where the plume is narrow † Email address for correspondence: h.burridge@imperial.ac.uk 2 H. C. Burridge, D. A. Parker, E. S. Kruger, J. L. Partridge and P. F. Linden and there is no eddy, large vertical velocities and hence transport are observed outside the plume suggesting that pressure forces associated with the eddies accelerate ambient fluid which is then engulfed into the plume. Finally, we show that observing significant vertical velocities beyond the scalar edge of the plume does not suggest that the characteristic width of the velocity distribution is greater than that of the scalar field; on the contrary, we show our observations to be consistent with a buoyancy distribution that is up to 20% wider than that of the velocity. Measurements in the plume coordinates show that the mixing of momentum across the plume results in a distribution for which the differential entropy is close to maximal and the mixing of momentum is uninhibited (i.e. not bounded) by the TNTI of the plume. Furthermore, our measurements suggest that the scalar mixing across the plume may also result in a distribution for which the differential entropy is close to maximal but, in contrast to the momentum, the scalar mixing is strictly bounded by the plume edge.