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Bratanov, Vasil (2015): Nonuniversal features of turbulent systems. Dissertation, LMU München: Faculty of Physics



Turbulence is one of the most widespread phenomena in nature occurring in fluids and plasmas at all scales - from the blood flow in the human body, via the Earth's atmosphere to the remnants of supernovas at astrophysical scales. Despite its frequent occurrence, constructing a theory of turbulent motion, which provides reliable quantitative predictions, represents one of the unsolved problems of classical physics. Most of the research efforts in the past have been focused on studying the Navier-Stokes model of simple fluids and trying to understand the Fourier spectrum of velocity fluctuations. Due to this common restriction to the Navier-Stokes equations, turbulence is usually associated with power-law spectra of universal form which arise only at scales where both driving and dissipation mechanisms are inactive. However, recent studies reveal that many active systems which do not possess a true inertial range can, nevertheless, exhibit power-law spectra. Furthermore, those spectra are not of universal form which contradicts the classical theory of turbulence. One of the turbulent models we shall consider in this work derives from the Kuramoto-Sivashinsky equation. It describes a simple one-dimensional active system where energy is injected at large scales and dissipated at small scales. Based on observations from plasma physics we modify the linear part of the equation such that the large scales remain practically intact but the damping rate at high wave numbers approaches a constant. We construct a semi-analytical approximation for the modified equation which predicts a power-law form for the energy spectrum in the range where the ratio between the characteristic linear and nonlinear frequencies is scale-independent. Furthermore, we conclude that the steepness of this power law is not universal but depends on the frequency ratio. These results are confirmed by numerical simulations. Our analysis could also be relevant for kinetic Alfven-wave turbulence in the solar wind where similar conditions might occur. Further in this work we present the first systematic study of another active system which provides a continuum model aimed at the coarse-grained description of the dynamics observed in dense bacterial suspensions. The model extends the framework of the familiar Navier-Stokes equations by including additional linear and nonlinear terms in order to emulate energy injection and dissipation as well as the flocking tendency of bacteria. The resulting dynamics has been described as 'low-Reynolds-number turbulence' and the corresponding energy spectrum exhibits nonuniversal power laws at large scales. With the aid of extensive numerical simulations we study the scale-to-scale energy flow in spectral space. The physical insight gained this way helped us to develop an approximation for the spectral energy balance equation. Its solution provides an energy spectrum of a power-law form at small wave numbers. Furthermore, we derive a functional dependence of the steepness of this power law on the system parameters. A comparison with data from numerical simulations verifies our results.