  In this paper, we address the problem of how to account for second-order
statistics of turbulent flows using low-complexity stochastic dynamical models
based on the linearized Navier-Stokes equations. The complexity is quantified
by the number of degrees of freedom in the linearized evolution model that are
directly influenced by stochastic excitation sources. For the case where only a
subset of velocity correlations are known, we develop a framework to complete
unavailable second-order statistics in a way that is consistent with
linearization around turbulent mean velocity. In general, white-in-time
stochastic forcing is not sufficient to explain turbulent flow statistics. We
develop models for colored-in-time forcing using a maximum entropy formulation
together with a regularization that serves as a proxy for rank minimization. We
show that colored-in-time excitation of the Navier-Stokes equations can also be
interpreted as a low-rank modification to the generator of the linearized
dynamics. Our method provides a data-driven refinement of models that originate
from first principles and captures complex dynamics of turbulent flows in a way
that is tractable for analysis, optimization, and control design.
