  Linear stability analysis has proven to be a useful tool in the analysis of
dominant coherent structures, such as the von K\'{a}rm\'{a}n vortex street and
the global spiral mode associated with the vortex breakdown of swirling jets.
In recent years, linear stability analysis has been applied successfully to
turbulent time-mean flows, instead of laminar base-flows,
\textcolor{black}{which requires turbulent models that account for the
interaction of the turbulent field with the coherent structures. To retain the
stability equations of laminar flows, the Boussinesq approximation with a
spatially nonuniform but isotropic eddy viscosity is typically employed. In
this work we assess the applicability of this concept to turbulent strongly
swirling jets, a class of flows that is particularly unsuited for isotropic
eddy viscosity models. Indeed we find that unsteady RANS simulations only match
with experiments with a Reynolds stress model that accounts for an anisotropic
eddy viscosity. However, linear stability analysis of the mean flow is shown to
accurately predict the global mode growth rate and frequency if the employed
isotropic eddy viscosity represents a least-squares approximation of the
anisotropic eddy viscosity. Viscosities derived from the $k-\epsilon$ model did
not achieve a good prediction of the mean flow nor did they allow for accurate
stability calculations. We conclude from this study that linear stability
analysis can be accurate for flows with strongly anisotropic turbulent
viscosity and the capability of the Boussinesq approximation in terms of
URANS-based mean flow prediction is not a prerequisite.
