Nonmodal amplitude equations

We consider fluid flows for which the linearized Navier-Stokes operator is strongly non-normal. The responses of such flows to external perturbations are spanned by a generically very large number of non-orthogonal eigenmodes. They are therefore qualified as ``nonmodal" responses, to insist on the inefficiency of the eigenbasis to describe them. In the aim of the article to reduce the system to a lower-dimensional one free of spatial degrees of freedom, (eigen)modal reduction techniques, such as the center manifold, are thus inappropriate precisely because the leading-order dynamics cannot be restricted to a low-dimensional eigensubspace. On the other hand, it is often true that only a small number (we assume only one) of singular modes is sufficient to reconstruct the nonmodal responses at the leading order. By adopting the latter paradigm, we propose a general method to analytically derive a weakly nonlinear amplitude equation for the nonmodal response of a fluid flow to a small harmonic forcing, stochastic forcing, and initial perturbation, respectively. In these last two problems, we assumed a parallel base flow with a spatially monochromatic external excitation. The present approach is simpler than the one we previously proposed and provides an explicit treatment of the sub-optimal responses. The derived amplitude equations are tested in three distinct flows. For sufficiently small excitation amplitudes, yet up to values large enough for the flow to depart from the linear regime, they can systematically predict the weakly nonlinear modification of the gains as the amplitude of the external excitation is increased. However, the proposed approach condemns the flow spatial structure to be mostly along the leading singular mode, and thus is often too simplistic to predict the occurrence of subcritical transitions as the excitation amplitude is increased to too large values.