Modal reduction of a parametrically forced confined viscous flow

An exact reduction of parametrically and periodically forced linear dynamical systems to modal dynamics in low-dimensional subspaces is possible when the operators involved commute with one another. We describe how a modal analysis may still be possible in the noncommuting case. The approach is illustrated with the determination of neutral curves for a stably and linearly stratified fluid in a square cavity under harmonically modulated gravitational forcing. In this example, the noncommutation of operators in the resulting Mathieu system is a direct consequence of the combined action of diffusion and wall confinement effects. An ansatz for modal diffusion is proposed, which enables remarkably predictive estimates of neutral curves via superposition of modal responses.