In this chapter, the origin of hydrodynamic turbulence in accretion disks is analyzed in the presence of temporally correlated noise to understand the origin of non-magnetic instability in Rayleigh-stable cold protoplanetary objects. In such flows, hydromagnetic/magnetohydrodynamic instability and the corresponding turbulent viscosity are expected to be sluggish or may even be ruled out. Previously, such flows were found to exhibit large transient hydrodynamic growth in two-dimension under linear perturbations, but insignificant growth in the presence of vertical columnar growth. To understand the origin of such instabilities that were not accounted for by conventional hydrodynamic (long-ranged) modes, in a series of articles between 2013 and 2014, we proposed thermal fluctuations as drivers of such instability and showed that an Orr-Sommerfeld-Squire model with spatially correlated noise can indeed explain this conundrum. In this chapter, we extend this analysis to temporally correlated noise, focusing on a regime where the Flory-type scaling in spatially correlated noise breaks down. The target is to understand how non-spatially correlated non-magnetic accretion may still amount to instability while being Rayleigh stable. A range of angular velocity profiles ( \(\Omega \) ) as functions of radial coordinate (r) have been explored: \(\Omega \propto r^{-q}\) ( \(1\leq q<2\) ). The model leads to a key understanding of non-magnetic accretion flows as a temporal noise-controlled instability. Additionally, we find remarkable differences between spatial and temporally correlated noise spectra, also with varying Reynolds numbers, that exhibit distinct universality classes.