  This article illustrates the application of multiple scales analysis to two
archetypal quasilinear 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 $\omega$, and the modal structure of the waves evolve, the
wave action $E/\omega$ 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 multiscale 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 quasilinear
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 supplementary material.

Fluid Dynamics