Non-normal weakly nonlinear analysis: asymptotic consistency and non-universality

Abstract

Non-normality can induce large transient growth in linearly stable systems. Determining whether this growth triggers a transition in the underlying nonlinear system, however, requires understanding the interaction between non-normality and nonlinearity. Here, we develop a weakly nonlinear theory for linearly-stable, non-normal systems subject to harmonic forcing, enabling a systematic analysis of this interaction. Following Ducimetière et al. (J. Fluid Mech., vol. 947, 2022, A43), we define a formal small parameter $\varepsilon$ as the reciprocal of the system's maximum linear amplification. However, we ensure asymptotic consistency by providing a framework that naturally adapts to the underlying structure of the system. The approach is applied to a harmonically forced channel flow and to a two-dimensional model mimicking the structure of the Orr-Sommerfeld-Squire equations. Unlike classical weakly nonlinear analysis near bifurcation points, the resulting amplitude equations are non-universal. In fact, a single linear mode amplified by the non-normality can nonlinearly excite a multi-modal and multi-frequency response at leading-order, which is system- or even regime-specific. Nevertheless, the method yields asymptotically consistent amplitude equations that capture this complexity provided a limit in which $\varepsilon\rightarrow0$ can be identified. As the forcing amplitude increases, the reduced equations capture stable nonlinear states emerging from the laminar flow, their subsequent bifurcations, and their eventual collision with the boundary of their basin of attraction. Thus, the amplitude equations can capture subcritical transitions driven by forcing and varied initial conditions and enable the identification of critical parameters beyond which no stable weakly nonlinear state exists.