  The coherent structures in the turbulent wake of a disk at a moderately high
Reynolds number ($\Rey$) of $50,000$ are examined using spectral proper
orthogonal decomposition (SPOD) which considers all three velocity components
in a numerical database. The SPOD eigenvalues at a given streamwise ($x$)
location are functions of azimuthal wavenumber ($m$), frequency ($\Str$), and
SPOD index ($n$). By $x/D =10$, two specific modes dominate the fluctuation
energy: (i) the vortex shedding (VS) mode with $m=1, \Str =0.135, n=1$, and
(ii) the double helix (DH) mode with $m=2, \Str \rightarrow 0, n=1$. The VS
mode is more energetic than the DH mode in the near wake but, in the far wake,
it is the DH mode which is dominant. The DH mode, when scaled with local
turbulent velocity and length scales, shows self-similarity in eigenvalues and
eigenmodes while the VS mode, which is a global mode, does not exhibit strict
self-similarity. Modes $m = 0$, 3 and 4, although subdominant, also make a
significant net contribution to the fluctuation energy, and their eigenspectra
are evaluated. The reconstruction of TKE and Reynolds shear stress, $\langle
u'_{x} u'_{r} \rangle$, is evaluated by varying $(m,\Str,n)$ combinations.
Higher SPOD modes contribute significantly to the TKE, especially near the
centerline. In contrast, reconstruction of $\langle u'_{x}u'_{r}\rangle $
requires far fewer modes: $|m| \leq 4 $, $|\Str| \leq 1$ and $n \leq 3$. Among
azimuthal modes, $m=1$ and $2$ are the leading contributors to both TKE and
$\langle u'_{x}u'_{r} \rangle $. While $m=1$ captures the slope of the
shear-stress profile near the centerline, $m=2$ is important to capture
$\langle u'_{x}u'_{r} \rangle $ at and near its peak. SPOD is also performed in
the vicinity of the disk to describe the modal transition to the principal
contributors in the wake.
