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.

中文翻译：

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弱非线性Holmboe波

霍尔木波是在环境分层剪切流中常见的长寿命行波，其中相对尖锐，稳定的密度界面嵌入在更分散的剪切层中。尽管先前的研究集中在其线性特性（霍姆博不稳定）和湍流特性（霍姆波波动）上，但对于非线性但非湍流状态下的有限振幅特性知之甚少。在本文中，我们通过对Holmboe波进行弱非线性时间稳定性分析来解决此问题。我们采用了Pham和Suslov [ R. Soc。开放科学。 5，180746（2018）]，它与线性不稳定的临界点之间的距离有限。从放大最大的线性傅立叶模态（幅度为1阶）开始，我们系统地得出非线性模态的层次（阶数2和阶3），该阶跃是获得Landau系数所必需的，该系数允许不稳定性最终饱和到稳定的平衡幅值。我们首先针对单模不稳定性（适用于弱分层的Kelvin-Helmholtz不稳定性）逐步介绍算法，然后再将其扩展到更微妙的双​​模不稳定性（适用于更强的分层Holmboe不稳定性）出于兴趣）。我们提出了具有双曲正切对称轮廓的标准分层剪切层的数值解，剪切密度比$\mathrm{[R}=5$以及Prandtl（或Schmidt）编号 $\text{镨}=700$。我们在线性稳定性边界上选择四个位置：三个在质量上截然不同的Holmboe案例（雷诺数Re，理查森数有很大不同）$Ĵ$，流向波数 $ķ$和相速度），并用一个开尔文-海姆霍兹案例进行比较。我们分别针对Re和Re生成每种情况的超临界分叉图。$Ĵ$，并且我们发现稳定均衡分支的规模和大小存在很大差异，无论是在Holmboe案例之间，在Holmboe和Kelvin-Helmholtz案例之间，还是在超临界Re或分岔中 $Ĵ$。我们还研究了相像，以探究两个反向传播的Holmboe模式的瞬态动力学，并且我们发现了一种特殊情况，其中特定的初始条件可能导致各个模式的非单调增长或衰减。接下来，我们解构摄动展开，以详细研究所有分量模式（线性和非线性）的空间结构，并强调潜在的饱和机制。我们再次发现Holmboe案例与Re或$Ĵ$ （例如2阶和3阶项的优势以及各种波数谐波的优势 $0，ķ，2ķ，3ķ$）。最后，我们讨论了我们的分析与超临界Holmboe波的最新实验测量的潜在相关性，以及它可能扩展到非对称Holmboe波以解决模式选择的问题。我们相信这些结果为Holmboe动力学系统的未来完全非线性分析提供了基础。