170 research outputs found
An improved \eps expansion for three-dimensional turbulence: two-loop renormalization near two dimensions
An improved \eps expansion in the -dimensional () stochastic
theory of turbulence is constructed at two-loop order which incorporates the
effect of pole singularities at in coefficients of the \eps
expansion of universal quantities. For a proper account of the effect of these
singularities two different approaches to the renormalization of the powerlike
correlation function of the random force are analyzed near two dimensions. By
direct calculation it is shown that the approach based on the mere
renormalization of the nonlocal correlation function leads to contradictions at
two-loop order. On the other hand, a two-loop calculation in the
renormalization scheme with the addition to the force correlation function of a
local term to be renormalized instead of the nonlocal one yields consistent
results in accordance with the UV renormalization theory. The latter
renormalization prescription is used for the two-loop renormalization-group
analysis amended with partial resummation of the pole singularities near two
dimensions leading to a significant improvement of the agreement with
experimental results for the Kolmogorov constant.Comment: 23 pages, 2 figure
Anomalous scaling of a passive scalar advected by the Navier--Stokes velocity field: Two-loop approximation
The field theoretic renormalization group and operator product expansion are
applied to the model of a passive scalar quantity advected by a non-Gaussian
velocity field with finite correlation time. The velocity is governed by the
Navier--Stokes equation, subject to an external random stirring force with the
correlation function . It is shown that
the scalar field is intermittent already for small , its structure
functions display anomalous scaling behavior, and the corresponding exponents
can be systematically calculated as series in . The practical
calculation is accomplished to order (two-loop approximation),
including anisotropic sectors. Like for the well-known Kraichnan's rapid-change
model, the anomalous scaling results from the existence in the model of
composite fields (operators) with negative scaling dimensions, identified with
the anomalous exponents. Thus the mechanism of the origin of anomalous scaling
appears similar for the Gaussian model with zero correlation time and
non-Gaussian model with finite correlation time. It should be emphasized that,
in contrast to Gaussian velocity ensembles with finite correlation time, the
model and the perturbation theory discussed here are manifestly Galilean
covariant. The relevance of these results for the real passive advection,
comparison with the Gaussian models and experiments are briefly discussed.Comment: 25 pages, 1 figur
Anomalous scaling, nonlocality and anisotropy in a model of the passively advected vector field
A model of the passive vector quantity advected by a Gaussian
time-decorrelated self-similar velocity field is studied; the effects of
pressure and large-scale anisotropy are discussed. The inertial-range behavior
of the pair correlation function is described by an infinite family of scaling
exponents, which satisfy exact transcendental equations derived explicitly in d
dimensions. The exponents are organized in a hierarchical order according to
their degree of anisotropy, with the spectrum unbounded from above and the
leading exponent coming from the isotropic sector. For the higher-order
structure functions, the anomalous scaling behavior is a consequence of the
existence in the corresponding operator product expansions of ``dangerous''
composite operators, whose negative critical dimensions determine the
exponents. A close formal resemblance of the model with the stirred NS equation
reveals itself in the mixing of operators. Using the RG, the anomalous
exponents are calculated in the one-loop approximation for the even structure
functions up to the twelfth order.Comment: 37 pages, 4 figures, REVTe
An improved \eps expansion for three-dimensional turbulence: summation of nearest dimensional singularities
An improved \eps expansion in the -dimensional () stochastic
theory of turbulence is constructed by taking into account pole singularities
at in coefficients of the \eps expansion of universal quantities.
Effectiveness of the method is illustrated by a two-loop calculation of the
Kolmogorov constant in three dimensions.Comment: 4 page
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
Pressure and intermittency in passive vector turbulence
We investigate the scaling properties a model of passive vector turbulence
with pressure and in the presence of a large-scale anisotropy. The leading
scaling exponents of the structure functions are proven to be anomalous. The
anisotropic exponents are organized in hierarchical families growing without
bound with the degree of anisotropy. Nonlocality produces poles in the
inertial-range dynamics corresponding to the dimensional scaling solution. The
increase with the P\'{e}clet number of hyperskewness and higher odd-dimensional
ratios signals the persistence of anisotropy effects also in the inertial
range.Comment: 4 pages, 1 figur
Calculation of the anomalous exponents in the rapid-change model of passive scalar advection to order
The field theoretic renormalization group and operator product expansion are
applied to the model of a passive scalar advected by the Gaussian velocity
field with zero mean and correlation function \propto\delta(t-t')/k^{d+\eps}.
Inertial-range anomalous exponents, identified with the critical dimensions of
various scalar and tensor composite operators constructed of the scalar
gradients, are calculated within the expansion to order
(three-loop approximation), including the exponents in
anisotropic sectors. The main goal of the paper is to give the complete
derivation of this third-order result, and to present and explain in detail the
corresponding calculational techniques. The character and convergence
properties of the expansion are discussed; the improved
``inverse'' expansion is proposed and the comparison with the
existing nonperturbative results is given.Comment: 34 pages, 5 figures, REVTe
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