Dynamical cascade models for Kolmogorov's inertial flow

Abstract

To resolve possible fluctuations about the mean motion of the Desnyansky-Novikov model for Kolmogorov's inertial flow, we have investigated two dynamical systems of the cascade process which are formally derivable from Burgers’ equation. The first cascade model produced no fluctuations, for its trajectory was identical with the Desnyansky–Novikov model's. Disappointingly, the second cascade system, which is similar to the Kerr–Siggia model, has also proved unable to engender fluctuations. This is because the second model when truncated consistently maps an arbitrary initial point into the attainable phase space of the first cascade model. However, when truncated inconsistently the trajectory of second model can exhibit a quite erratic and somewhat sporadic motion, thereby reflecting the apparently random motion of inviscid equilibrium solutions. Therefore, the observation of temporally intermittent fluctuations by a stationary Kerr–Siggia model is due to the inconsistent truncation produced by restricting energy dissipation for all but the upper truncation mode in their model.