An experimental investigation of the instability of a shear flow with multilayered density stratification

Abstract

The evolution of stratified shear flows with multilayer density distributions is discussed briefly from a theoretical perspective, generalizing the results of Caulfield [J. Fluid Mech. 258, 255 (1994)] to allow for asymmetry. Three distinct types of instability are predicted to occur according to linear theory. In the laboratory, we measure the density profile and the velocity profile continuously, and so are able to identify the flow characteristics that are applicable when each of the different instabilities grow. Knowledge of the bulk Richardson number is insufficient to predict the observed properties of the instabilities of the flow. The parameter that is most determinant of the selection of a particular type of instability is found to be the ratio R of the depth of the intermediate density layer to the depth over which the velocity varies, though any asymmetry in the flow (either in the velocity or density fields) also plays a role. If R is close to 1, and hence the layer of intermediate density occupies a significant portion of the shear layer, overturnings appear in the intermediate layer, which are long lived, and strongly two dimensional. These overturnings are the three layer stratified generalization of the Kelvin–Helmholtz instability first discussed by Taylor [Proc. R. Soc. London Ser. A 132, 499 (1931)]. Such modes inefficiently mix the background flow, and the major mixing mechanisms are found to consist of overturnings in the lower fluid layer (and, to a lesser extent, the upper layer). These overturnings are clearly manifestations of an (asymmetric) three layer generalization of the Holmboe [Geophys. Publ. 24, 67 (1962)] instability. In general, all three instabilities can be observed simultaneously at markedly different wavelengths and phase speeds for extended periods of time, even though linear theory may predict significantly different growth rates.

Marginally unstable Holmboe modes

“…Figure 1b shows the instability region for R = 3 and J 0 up to 80. We note that different unstable Holmboe modes have been found experimentally in [32] that were then interpreted in terms of the multi-layer model of [12]. The understanding however of the linear part of Holmboe instability for smooth shear and density profiles still remains conjectural and most of the results are based on numerical calculations and do not therefore constitute proofs.…”

“…Browand and Winant [25] first performed shear flow experiments in a stratified environment under conditions for which Holmboe's instabilities are present. Their investigation has been extended further by more recent experiments [11,26,27,28,29,30,31,32].…”

Optimal excitation of two dimensional Holmboe instabilities

“…T e −ikct , with −ikc an eigenvalue of the evolution operator in (8). The modal growth rate of the perturbation is kc i where c i = ℑ(c) and the flow is exponentially unstable to perturbations of wavenumber k if c i > 0.…”

“…Holmboe instabilities have been reproduced in laboratory experiments [5][6][7][8][9][10][11] and have been numerically simulated [12][13][14][15][16][17][18][19]. At finite amplitude these Holmboe modes * Electronic address: navidcon@phys.uoa.gr † Electronic address: pjioannou@phys.uoa.gr equilibrate into propagating waves and they can induce mixing in highly stratified environments [12,17,20].…”

A generalized action-angle representation of wave interaction in stratified shear flows

“…In fact, there are setups which are apparently counter-intuitive, e.g. Taylor-Caulfield instability (Taylor 1931;Caulfield et al 1995). In some cases we may use mathematical constrains providing necessary conditions for instability, like the ones of Rayleigh, Fjørtoft and Richardson (Drazin & Reid 2004).…”