Coherence in turbulent stratified wakes through the lens of modal analysis*

Coherent structures are extracted in stratified disk wakes and analyzed using spectral proper orthogonal decomposition (SPOD) at moderately high $Re  = 5 \times 10^{4}$ and $Fr = 2, 10$. The obtained SPOD modes vary with modal index ($n$), frequency ($St$), and streamwise distance ($x/D$). Both wakes exhibit a low-rank behavior and the relative contribution of the low-rank modes to the turbulent kinetic energy increases with $x/D$. The vortex shedding (VS) mechanism, which corresponds to $St \approx 0.11-0.13$ in both wakes, is dominant throughout the domain, unlike their unstratified counterpart. The energy around the VS  frequency appears in the outer region of the wake in the form of internal gravity waves (IGWs) beyond $Nt = Nx/U =  6 - 8$. Overall, we find that the coherence of wake turbulence, initiated by the VS mode at the body, is prolonged to the far downstream owing to the buoyancy.  Also, this coherence is spatially modified by buoyancy into horizontal layers and IGWs. Low-order truncations of SPOD modes efficiently reconstruct important second-order statistics.