Weakly nonlinear Holmboe waves

Abstract

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 & 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 (order 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 P r = 700. We select four locations on the linear stability boundary: three qualitatively distinct Holmboe cases (having widely different Reynolds numbers Re, Richardson numbers J, steamwise wavenumbers 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 counter-propagating Holmboe modes, and we find a special case in which specific initial conditions can lead to the non-monotonic 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 wavenumber 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.

Nonlinear spatiotemporal instabilities in two-dimensional electroconvective flows

“…As reported in Ref. [33], the linear operators here may be poorly conditioned near the critical point even if it is nonsingular but they found that it still can be inverted to give converged results. This is also the case in the present work.…”

“…where A is the complex amplitude of the disturbance and E = exp(iαx) (not to be confused with the symbol for the electric field E). Conventionally, in previous works [26,27,32,33], A is assumed to be only a function of time. As suggested by one of the reviewers, we also consider its spatial dependence for the analysis of spatiotemporal instability in the amplitude expansion scheme.…”

“…Among these six equations, L α,σ γ11 = 0 is exactly the linear eigenvalue problem corresponding to equation (17a). For T > T c near the linear critical conditions, the linear growth rate σ r > 0, and thus, the operators L 0,2σr and L 2α,2σ are non-singular (even when T → T c where σ r → 0), making the two corresponding equations readily solvable [32,33]. It should be noted that when T < T c (with σ r < 0), there will be resonances among decaying modes and special treatments are required [32,54].…”

“…It is noted that the amplitude expansion method has recently been improved by Suslov and his coworker [32] (see also Ref. [33]) to be more rigorous and versatile and this method will be utilised in the current work to probe the EHD flow when it is away from the linear criticality.…”

The effects of spanwise confinement on stratified shear instabilities

“…Therefore, here the phase velocity is −σ i (k)/k and the wave propagates in the the positive x-direction if it is positive. If σ r (k) > 0 the wave is unstable and grows as ∝ e σrt until nonlinearities come into play (see [8] for a treatment of these nonlinearities).…”