Abstract: Smoke animations are hard to art-direct because simple changes in parameters such as simulation resolution often lead to unpredictable changes in the final result. Previous work has addressed this problem with a guiding approach which couples low-resolution simulations – that exhibit the desired flow and behaviour – to the final, high-resolution simulation. This is done in such a way that the desired low frequency features are to some extent preserved in the high-resolution simulation. However, the steady (i.e. constant) guiding used often leads to a lack of sufficiently high detail, and employing time-dependent guiding is expensive because the matrix of the resulting set of equations needs to be recomputed at every iteration. We propose an improved mathematical model for Eulerian-based simulations which is better suited for dynamic, time-dependent guiding of smoke animations through a novel variational coupling of the low- and high-resolution simulations. Our model results in a matrix that does not require re-computation when the guiding changes over time, and hence we can employ time-dependent guiding more efficiently both in terms of storage and computational requirements. We demonstrate that time-dependent guiding allows for more high frequency detail to develop without losing correspondence to the low resolution simulation. Furthermore, we explore various artistic effects made possible by time-dependent guiding.