  The modal decomposition techniques of proper orthogonal decomposition (POD)
and dynamic mode decomposition (DMD) have become a common method for analysing
the spatio-temporal coherence of dynamical systems. In particular, these
techniques are of interest for liquid injection systems due to the inherent
complexity of multiphase interactions and extracting the underlying flow
processes is desired. Although numerous works investigating flow processes have
implemented POD and DMD, the results are often highly interpretive with limited
link between the decomposition theory and the interpreted physical meaning of
the extracted modes. Here, we provide insight into the interpretation of POD
and DMD modes in a hierarchical structure. The interpretation of modes for
simple canonical systems is validated through knowledge of the underlying
processes which dominate the systems. We show that modes which capture true
underlying phenomena produce subsequent modes at higher harmonics, up until the
Nyquist limit, whose modal structure scales decrease proportionally with
increasing modal frequency. These higher harmonics primarily encode motion
information and may or may not capture additional structural information, which
is dependent on the system. We demonstrate these findings first on canonical
liquid injection systems to enhance the interpretation and understanding of
results extracted from practical jet in crossflow systems.
