Space-time statistics of a linear dynamical energy cascade model

A linear dynamical model for the development of 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 space and delta-correlated time, shown to converge at infinite time towards state finite variance, without aid viscosity. Furthermore, spatial profile its solution gets rough, with same regularity as fractional Gaussian field. We here focus on temporal behavior derive explicit asymptotic predictions correlation function this observe that their not influenced problem, only stirring contribution. also show depends position, contrary fixed times. then investigate influence correlated statistics equation. In situation, while small times homogeneous white-in-time case are recovered, large homogeneity broken, concentration around origin system observed velocity profiles. other words, field representation one-dimension, through model, self-similar fields proposed Kolmogorov 1941, but times, forcing, expected The turbulence, however, captured model.