8 research outputs found
Two-Loop Calculation of the Anomalous Exponents in the Kazantsev--Kraichnan Model of Magnetic Hydrodynamics
The problem of anomalous scaling in magnetohydrodynamics turbulence is
considered within the framework of the kinematic approximation, in the presence
of a large-scale background magnetic field. Field theoretic renormalization
group methods are applied to the Kazantsev-Kraichnan model of a passive vector
advected by the Gaussian velocity field with zero mean and correlation function
. Inertial-range anomalous scaling for the
tensor pair correlators is established as a consequence of the existence in the
corresponding operator product expansions of certain "dangerous" composite
operators, whose negative critical dimensions determine the anomalous
exponents. The main technical result is the calculation of the anomalous
exponents in the order of the expansion (two-loop
approximation).Comment: Presented in the Conference "Mathematical Modeling and Computational
Physics" (Stara Lesna, Slovakia, July 2011
Anomalous scaling in two and three dimensions for a passive vector field advected by a turbulent flow
A model of the passive vector field advected by the uncorrelated in time
Gaussian velocity with power-like covariance is studied by means of the
renormalization group and the operator product expansion. The structure
functions of the admixture demonstrate essential power-like dependence on the
external scale in the inertial range (the case of an anomalous scaling). The
method of finding of independent tensor invariants in the cases of two and
three dimensions is proposed to eliminate linear dependencies between the
operators entering into the operator product expansions of the structure
functions. The constructed operator bases, which include the powers of the
dissipation operator and the enstrophy operator, provide the possibility to
calculate the exponents of the anomalous scaling.Comment: 9 pages, LaTeX2e(iopart.sty), submitted to J. Phys. A: Math. Ge
Symmetry Breaking in Stochastic Dynamics and Turbulence
Symmetries play paramount roles in dynamics of physical systems. All theories of quantum physics and microworld including the fundamental Standard Model are constructed on the basis of symmetry principles. In classical physics, the importance and weight of these principles are the same as in quantum physics: dynamics of complex nonlinear statistical systems is straightforwardly dictated by their symmetry or its breaking, as we demonstrate on the example of developed (magneto)hydrodynamic turbulence and the related theoretical models. To simplify the problem, unbounded models are commonly used. However, turbulence is a mesoscopic phenomenon and the size of the system must be taken into account. It turns out that influence of outer length of turbulence is significant and can lead to intermittency. More precisely, we analyze the connection of phenomena such as behavior of statistical correlations of observable quantities, anomalous scaling, and generation of magnetic field by hydrodynamic fluctuations with symmetries such as Galilean symmetry, isotropy, spatial parity and their violation and finite size of the system
Symmetry and Mesoscopic Physics
Symmetry is one of the most important notions in natural science; it lies at the heart of fundamental laws of nature and serves as an important tool for understanding the properties of complex systems, both classical and quantum. Another trend, which has in recent years undergone intensive development, is mesoscopic physics. This branch of physics also combines classical and quantum ideas and methods. Two main directions can be distinguished in mesoscopic physics. One is the study of finite quantum systems of mesoscopic sizes. Such systems, which are between the atomic and macroscopic scales, exhibit a variety of novel phenomena and find numerous applications in creating modern electronic and spintronic devices. At the same time, the behavior of large systems can be influenced by mesoscopic effects, which provides another direction within the framework of mesoscopic physics. The aim of the present book is to emphasize the phenomena that lie at the crossroads between the concept of symmetry and mesoscopic physics
Fast Magnetic Reconnection and Spontaneous Stochasticity
Magnetic field-lines in astrophysical plasmas are expected to be frozen-in at
scales larger than the ion gyroradius. The rapid reconnection of magnetic flux
structures with dimensions vastly larger than the gyroradius requires a
breakdown in the standard Alfv\'en flux-freezing law. We attribute this
breakdown to ubiquitous MHD plasma turbulence with power-law scaling ranges of
velocity and magnetic energy spectra. Lagrangian particle trajectories in such
environments become "spontaneously stochastic", so that infinitely-many
magnetic field-lines are advected to each point and must be averaged to obtain
the resultant magnetic field. The relative distance between initial magnetic
field lines which arrive to the same final point depends upon the properties of
two-particle turbulent dispersion. We develop predictions based on the
phenomenological Goldreich & Sridhar theory of strong MHD turbulence and on
weak MHD turbulence theory. We recover the predictions of the Lazarian &
Vishniac theory for the reconnection rate of large-scale magnetic structures.
Lazarian & Vishniac also invoked "spontaneous stochasticity", but of the
field-lines rather than of the Lagrangian trajectories. More recent theories of
fast magnetic reconnection appeal to microscopic plasma processes that lead to
additional terms in the generalized Ohm's law, such as the collisionless Hall
term. We estimate quantitatively the effect of such processes on the
inertial-range turbulence dynamics and find them to be negligible in most
astrophysical environments. For example, the predictions of the
Lazarian-Vishniac theory are unchanged in Hall MHD turbulence with an extended
inertial range, whenever the ion skin depth is much smaller than the
turbulent integral length or injection-scale Comment: 31 pages, 5 figure
Astrophysical magnetic fields and nonlinear dynamo theory
The current understanding of astrophysical magnetic fields is reviewed,
focusing on their generation and maintenance by turbulence. In the
astrophysical context this generation is usually explained by a self-excited
dynamo, which involves flows that can amplify a weak 'seed' magnetic field
exponentially fast. Particular emphasis is placed on the nonlinear saturation
of the dynamo. Analytic and numerical results are discussed both for small
scale dynamos, which are completely isotropic, and for large scale dynamos,
where some form of parity breaking is crucial. Central to the discussion of
large scale dynamos is the so-called alpha effect which explains the generation
of a mean field if the turbulence lacks mirror symmetry, i.e. if the flow has
kinetic helicity. Large scale dynamos produce small scale helical fields as a
waste product that quench the large scale dynamo and hence the alpha effect.
With this in mind, the microscopic theory of the alpha effect is revisited in
full detail and recent results for the loss of helical magnetic fields are
reviewed.Comment: 285 pages, 72 figures, accepted by Phys. Re