Fluctuations around Bjorken Flow and the onset of turbulent phenomena

Abstract

We study how fluctuations in fluid dynamic fields can be dissipated or amplified within the characteristic spatio-temporal structure of a heavy ion collision. The initial conditions for a fluid dynamic evolution of heavy ion collisions may contain significant fluctuations in all fluid dynamical fields, including the velocity field and its vorticity components. We formulate and analyze the theory of local fluctuations around average fluid fields described by Bjorken's model. For conditions of laminar flow, when a linearized treatment of the dynamic evolution applies, we discuss explicitly how fluctuations of large wave number get dissipated while modes of sufficiently long wave-length pass almost unattenuated or can even be amplified. In the opposite case of large Reynold's numbers (which is inverse to viscosity), we establish that (after suitable coordinate transformations) the dynamics is governed by an evolution equation of non-relativistic Navier-Stokes type that becomes essentially two-dimensional at late times. One can then use the theory of Kolmogorov and Kraichnan for an explicit characterization of turbulent phenomena in terms of the wave-mode dependence of correlations of fluid dynamic fields. We note in particular that fluid dynamic correlations introduce characteristic power-law dependences in two-particle correlation functions.