  We use spectral proper orthogonal decomposition (SPOD) to extract and analyze
coherent structures in the turbulent wake of a disk at Reynolds number $Re = 5
\times 10^{4}$ and Froude numbers $Fr$ = $2, 10$. We find that the SPOD
eigenspectra of both wakes exhibit a low-rank behavior and the relative
contribution of low-rank modes to total fluctuation energy increases with
$x/D$. The vortex shedding (VS) mechanism, which corresponds to $St \approx
0.11-0.13$ in both wakes, is active and dominant throughout the domain in both
wakes. The continual downstream decay of the SPOD eigenspectrum peak at the VS
mode, which is a prominent feature of the unstratified wake, is inhibited by
buoyancy, particularly for $Fr = 2$. The energy at and near the VS frequency is
found to appear in the outer region of the wake when the downstream distance
exceeds $Nt = Nx/U = 6 - 8$. Visualizations show that unsteady internal gravity
waves (IGWs) emerge at the same $Nt = 6 - 8$. A causal link between the VS
mechanism and the unsteady IGW generation is also established using the
SPOD-based reconstruction and analysis of the pressure-transport term. These
IGWs are also picked up in SPOD analysis as a structural change in the shape of
the leading SPOD eigenmode. The $Fr = 2$ wake shows layering in the wake core
at {$Nt > 15$} which is captured by the leading SPOD eigenmodes of the VS
frequency at downstream locations $x/D > 30$. The VS mode of the $Fr = 2$ wake
is streamwise-coherent, consisting of V-shaped structures at $x/D \gtrsim 30$.
Overall, we find that the coherence of wakes, initiated by the VS mode at the
body, is prolonged by buoyancy to far downstream. Also, this coherence is
spatially modified by buoyancy into horizontal layers and IGWs. Low-order
truncations of SPOD modes are shown to efficiently reconstruct important
second-order statistics.
