This article illustrates the application of multiple scales analysis to two archetypal quasi-linear systems; i.e. to systems involving fast dynamical modes, called fluctuations, that are not directly influenced by fluctuation-fluctuation nonlinearities but nevertheless are strongly coupled to a slow variable whose evolution may be fully nonlinear. In the first case, fast waves drive a slow, spatially inhomogeneous evolution of their celerity field. Multiple scales analysis confirms that, although the energy E, the angular frequency ω and the modal structure of the waves evolve, the wave action E/ω is conserved in the absence of forcing and dissipation. In the second system, the fast modes undergo an instability that is saturated through a feedback on the slow variable. A new multi-scale analysis is developed to treat this case. The key technical point, confirmed by the analysis, is that the fluctuation energy and mode structure evolve slowly to ensure that the slow field remains in a state of near marginal stability. These two model systems appear to be generic, being representative of many if not all quasi-linear systems. In each case, numerical simulations of both the full and reduced dynamical systems are performed to highlight the accuracy and efficiency of the multiple scales approach. Python codes are provided as electronic supplementary material.
