Holmboe waves are long-lived traveling waves commonly found in environmental stratified shear flows in which a relatively sharp, stable density interface is embedded within a more diffuse shear layer. Although previous research has focused on their linear properties (the Holmboe instability), and on their turbulent properties (Holmboe wave turbulence), little is known about their finite-amplitude properties in the nonlinear but nonturbulent regime. In this paper we tackle this problem with a weakly nonlinear temporal stability analysis of Holmboe waves. We employ the rigorous and versatile amplitude expansion method recently proposed by Pham and Suslov [R. Soc. Open Sci. 5, 180746 (2018)], which remains well posed a finite distance away from the critical point of linear instability. Starting with the most amplified linear Fourier mode (order 1 in amplitude), we systematically derive the hierarchy of nonlinear modes (orders 2 and 3) required to obtain Landau coefficients that allow the instability to eventually saturate to a stable equilibrium amplitude. We introduce the algorithm step by step, first on a single-mode instability (suited to the weakly stratified Kelvin-Helmholtz instability), before extending it to the more subtle case of a double-mode instability (suited to the more strongly stratified Holmboe instability of interest). We present numerical solutions for the canonical stratified shear layer with hyperbolic-tangent symmetric profiles, shear-to-density thickness ratio $R=5$, and Prandtl (or Schmidt) number $\text{Pr}=700$. We select four locations on the linear stability boundary: three qualitatively distinct Holmboe cases (having widely different Reynolds numbers Re, Richardson numbers $J$, streamwise wave numbers $k$, and phase speeds), and one Kelvin-Helmholtz case to serve as a comparison. We produce supercritical bifurcation diagrams for each case, both in Re and $J$, and we find great differences in the scaling and magnitude of stable equilibrium branches, be it between Holmboe cases, between the Holmboe and Kelvin-Helmholtz cases, and between bifurcations in supercritical Re or $J$. We also study phase portraits to delve into the transient dynamics of the two counterpropagating Holmboe modes, and we find a special case in which specific initial conditions can lead to the nonmonotonic growth or decay of individual modes. Next, we deconstruct the perturbation expansions to investigate in detail the spatial structures of all the component modes (linear and nonlinear), and highlight the underlying saturation mechanisms. We again find differences between Holmboe cases, and Re or $J$ (such as the dominance of order-2 vs order-3 terms, and of various wave-number harmonics $0,k,2k,3k$). Finally, we discuss the potential relevance of our analysis to recent experimental measurements of supercritical Holmboe waves, and its possible extension to asymmetric Holmboe waves to tackle the question of mode selection. We believe these results provide a basis for a future fully nonlinear analysis of the Holmboe dynamical system.

• Received 2 November 2020
• Accepted 27 January 2021

DOI:https://doi.org/10.1103/PhysRevFluids.6.024803