  We consider the frequency domain form of proper orthogonal decomposition
(POD) called spectral proper orthogonal decomposition (SPOD). Spectral POD is
derived from a space-time POD problem for statistically stationary flows and
leads to modes that each oscillate at a single frequency. This form of POD goes
back to the original work of Lumley (Stochastic tools in turbulence, Academic
Press, 1970), but has been overshadowed by a space-only form of POD since the
1990s. We clarify the relationship between these two forms of POD and show that
SPOD modes represent structures that evolve coherently in space and time while
space-only POD modes in general do not. We also establish a relationship
between SPOD and dynamic mode decomposition (DMD); we show that SPOD modes are
in fact optimally averaged DMD modes obtained from an ensemble DMD problem for
stationary flows. Accordingly, SPOD modes represent structures that are dynamic
in the same sense as DMD modes but also optimally account for the statistical
variability of turbulent flows. Finally, we establish a connection between SPOD
and resolvent analysis. The key observation is that the resolvent-mode
expansion coefficients must be regarded as statistical quantities to ensure
convergent approximations of the flow statistics. When the expansion
coefficients are uncorrelated, we show that SPOD and resolvent modes are
identical. Our theoretical results and the overall utility of SPOD are
demonstrated using two example problems: the complex Ginzburg-Landau equation
and a turbulent jet.
