  Recent data-driven efforts have utilized spectral decomposition techniques to
uncover the geometric self-similarity of dominant motions in the logarithmic
layer, and thereby validate the attached eddy model. In this paper, we evaluate
the predictive capability of the stochastically forced linearized Navier-Stokes
equations in capturing such structural features in turbulent channel flow at
$Re_\tau=2003$. We use the linear coherence spectrum to quantify the
wall-normal coherence within the velocity field generated by the linearized
dynamics. In addition to the linearized Navier-Stokes equations around the
turbulent mean velocity profile, we consider an enhanced variant in which
molecular viscosity is augmented with turbulent eddy-viscosity. We use
judiciously shaped white- and colored-in-time stochastic forcing to generate a
statistical response with energetic attributes that are consistent with the
result of direct numerical simulation (DNS). Specifically, white-in-time
forcing is scaled to ensure that the two-dimensional energy spectrum is
reproduced and colored-in-time forcing is shaped to match normal and shear
stress profiles. We show that the addition of eddy-viscosity significantly
strengthens the self-similar attributes of the resulting stochastic velocity
field within the logarithmic layer and leads to an inner-scaled coherence
spectrum. We use this coherence spectrum to extract the energetic signature of
self-similar motions that actively contribute to momentum transfer and are
responsible for producing Reynolds shear stress. Our findings support the use
of colored-in-time forcing in conjunction with the dynamic damping afforded by
turbulent eddy-viscosity in improving predictions of the scaling trends
associated with such active motions in accordance with DNS-based spectral
decomposition.
