The structure and origin of confined Holmboe waves

Abstract

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.

Layer formation and relaminarisation in plane Couette flow with spanwise stratification

“…The question then arises as to how these mechanisms compete, and also how turbulence is deformed by stratification in more generic situations. Just to mention one example, it has recently been observed that horizontal confinement in a duct, and hence the inevitable introduction of spanwise shear to the Holmboe instability of a principally vertically sheared (relatively 'sharp') density interface gives rise to a new robust long-lived nonlinear coherent structure in the form of a 'confined Holmboe wave' (Lefauve et al 2018). It remains to be seen how similar confinement might affect turbulent flows, even without the sharp density interface typically required for the initial occurence of Holmboe-type instabilities.…”

Regime transitions and energetics of sustained stratified shear flows

“…ML14 correctly recognised that the two-layer exchange flow was maximal because it is hydraulically controlled at both ends of the duct where it meets the reservoirs through a sharp change in geometry (an idea already present in Wilkinson (1986)). In other words, the flow is subcritical with respect to long interfacial waves inside the duct, and critical at either end, preventing the propagation of information (in particular of the exchange flow rate) from the exterior into the duct (see ML14, Lefauve et al 2018, § 3, and Lefauve 2018, § 1.3.2 for more details). The exchange flow is sustained in a quasi-steady state until the controls are 'flooded' by the accumulation of fluid of a different density coming from the other reservoir.…”

Analysis of one-dimensional models for exchange flows under strong stratification

“…Gravity acts in the negative z-direction. The setup, which is inspired by different experimental configurations (see, for example, Simpson and Linden 1989;Meyer and Linden 2014;Lefauve et al 2018), consists of two reservoirs connected by a channel of height h and length L, with L h (implicitly assuming that end effects at x = ±L/2 can be ignored). Each reservoir contains water with a different initial salt concentration, denoted by the salinities s 1 and s 2 in the left and right reservoirs, respectively.…”