  A linear dynamical model for the development of the turbulent energy cascade
was introduced in Apolin\'ario \emph{et al} (J. Stat. Phys. \textbf{186}, 15
(2022)). This partial differential equation, randomly stirred by a forcing term
which is smooth in space and delta-correlated in time, was shown to converge at
infinite time towards a state of finite variance, without the aid of viscosity.
Furthermore, the spatial profile of its solution gets rough, with the same
regularity as a fractional Gaussian field. We here focus on the temporal
behavior and derive explicit asymptotic predictions for the correlation
function in time of this solution and observe that their regularity is not
influenced by the spatial regularity of the problem, only by the correlation in
time of the stirring contribution. We also show that the correlation in time of
the solution depends on the position, contrary to its correlation in space at
fixed times. We then investigate the influence of a forcing which is correlated
in time on the spatial and time statistics of this equation. In this situation,
while for small correlation times the homogeneous spatial statistics of the
white-in-time case are recovered, for large correlation times homogeneity is
broken, and a concentration around the origin of the system is observed in the
velocity profiles. In other words, this fractional velocity field is a
representation in one-dimension, through a linear dynamical model, of the
self-similar velocity fields proposed by Kolmogorov in 1941, but only at fixed
times, for a delta-correlated forcing, in which case the spatial statistics is
homogeneous and rough, as expected of a turbulent velocity field. The
regularity in time of turbulence, however, is not captured by this model.
