Burgers turbulence and passive random advection.

摘要

The thesis is devoted to development of new methods in the theory of strong turbulence. These methods are illustrated with the so-called Burgers' model of turbulence, i.e., the Navier-Stokes equation without pressure, supplemented by a Gaussian, short-time correlated external force. The main goal of the theory is to describe the statistics of the velocity field. Since the Navier-Stokes equation is nonlinear, the problem is highly nontrivial; it is sometimes referred to as the “Ising model” of strong turbulence. The importance of the problem for plasma physics, astrophysics, physics of self-organized criticality, disordered systems, etc., is discussed in Chapter 1.; In this thesis a new self-consistent theoretical approach to the problem is developed. The problem is treated from the field-theoretical point of view, and, therefore, appropriate methods such as regularization, operator product expansion, and an assumption about scaling invariance are employed.; The scheme “from particular to general” is adopted. The main ideas of the approach are first developed in detail for the one-dimensional Burgers model in Chapter 2 and then generalized to the multidimensional case in Chapter 3. In all of the cases the velocity-difference and velocity-gradient probability density functions are obtained. Their derivation is based on the self-consistent conjecture about the operator product expansion for the dissipative term, introduced by Polyakov [1995]. Comparison of the obtained results with the available direct numerical simulations shows a very good agreement. The practically important longitudinal velocity-difference PDF and div v PDF in the multidimensional case are discussed within the approach.; In Chapter 4 the statistics of passive quantities (such as temperature, concentration, magnetic field) “frozen” into the turbulent fluid are obtained by using the methods developed in Chapters 2 and 3. The velocity field is assumed to be Gaussian, and short-time correlated, and the diffusivity is neglected. These considerations illustrate that even with simple statistics of the velocity field, the statistics of advected quantities are nontrivial due to nonlinear interactions of different spatial directions.; The last Chapter 5 summarizes the results and discusses future directions of research.