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Title: Renormalization of tracer turbulence leading to fractional differential operators

Abstract

For many years quasilinear renormalization has been applied to numerous problems in turbulent transport. This scheme relies on the localization hypothesis to derive a linear transport equation from a simplified stochastic description of the underlying microscopic dynamics. However, use of the localization hypothesis narrows the range of transport behaviors that can be captured by the renormalized equations. In this paper, we construct a renormalization procedure that manages to avoid the localization hypothesis completely and produces renormalized transport equations, expressed in terms of fractional differential operators, that exhibit much more of the transport phenomenology observed in nature. This technique provides with a first step towards establishing a rigorous link between the microscopic physics of turbulence and the fractional transport models proposed phenomenologically for a wide variety of turbulent systems such as neutral fluids or plasmas.

Authors:
 [1];  [1];  [2];  [1];  [3]
  1. ORNL
  2. University of Alaska
  3. Asociacion EURATOM-CIEMAT
Publication Date:
Research Org.:
Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
Sponsoring Org.:
USDOE Laboratory Directed Research and Development (LDRD) Program
OSTI Identifier:
1003630
DOE Contract Number:
DE-AC05-00OR22725
Resource Type:
Journal Article
Resource Relation:
Journal Name: Physical Review E; Journal Volume: 74; Journal Issue: 1
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; HYPOTHESIS; PHYSICS; RENORMALIZATION; TRANSPORT; TURBULENCE; turbulence; fractional transport; quasilinear renormalization

Citation Formats

Sanchez, Raul, Carreras, Benjamin A, Newman, David E, Lynch, Vickie E, and van Milligen, B. Ph. Renormalization of tracer turbulence leading to fractional differential operators. United States: N. p., 2006. Web. doi:10.1103/PhysRevE.74.016305.
Sanchez, Raul, Carreras, Benjamin A, Newman, David E, Lynch, Vickie E, & van Milligen, B. Ph. Renormalization of tracer turbulence leading to fractional differential operators. United States. doi:10.1103/PhysRevE.74.016305.
Sanchez, Raul, Carreras, Benjamin A, Newman, David E, Lynch, Vickie E, and van Milligen, B. Ph. Sun . "Renormalization of tracer turbulence leading to fractional differential operators". United States. doi:10.1103/PhysRevE.74.016305.
@article{osti_1003630,
title = {Renormalization of tracer turbulence leading to fractional differential operators},
author = {Sanchez, Raul and Carreras, Benjamin A and Newman, David E and Lynch, Vickie E and van Milligen, B. Ph.},
abstractNote = {For many years quasilinear renormalization has been applied to numerous problems in turbulent transport. This scheme relies on the localization hypothesis to derive a linear transport equation from a simplified stochastic description of the underlying microscopic dynamics. However, use of the localization hypothesis narrows the range of transport behaviors that can be captured by the renormalized equations. In this paper, we construct a renormalization procedure that manages to avoid the localization hypothesis completely and produces renormalized transport equations, expressed in terms of fractional differential operators, that exhibit much more of the transport phenomenology observed in nature. This technique provides with a first step towards establishing a rigorous link between the microscopic physics of turbulence and the fractional transport models proposed phenomenologically for a wide variety of turbulent systems such as neutral fluids or plasmas.},
doi = {10.1103/PhysRevE.74.016305},
journal = {Physical Review E},
number = 1,
volume = 74,
place = {United States},
year = {Sun Jan 01 00:00:00 EST 2006},
month = {Sun Jan 01 00:00:00 EST 2006}
}
  • For many years quasilinear renormalization has been applied to numerous problems in turbulent transport. This scheme relies on the localization hypothesis to derive a linear transport equation from a simplified stochastic description of the underlying microscopic dynamics. However, use of the localization hypothesis narrows the range of transport behaviors that can be captured by the renormalized equations. In this paper, we construct a renormalization procedure that manages to avoid the localization hypothesis completely and produces renormalized transport equations, expressed in terms of fractional differential operators, that exhibit much more of the transport phenomenology observed in nature. This technique provides amore » first step toward establishing a rigorous link between the microscopic physics of turbulence and the fractional transport models proposed phenomenologically for a wide variety of turbulent systems such as neutral fluids or plasmas.« less
  • We study spectral asymptotics for a large class of differential operators on an open subset of R{sup d} with finite volume. This class includes the Dirichlet Laplacian, the fractional Laplacian, and also fractional differential operators with non-homogeneous symbols. Based on a sharp estimate for the sum of the eigenvalues we establish the first term of the semiclassical asymptotics. This generalizes Weyl's law for the Laplace operator.
  • We present, in the context of dimensional regularization, a presscription to renormalize Feynman diagrams with an arbitrary number of external fermions. This prescription, which is based on the original {close_quote}t Hooft{endash}Veltman proposal to keep external particles in four dimensions, is particularly useful to define the {open_quote}{open_quote}renormalization{close_quote}{close_quote} (in the context of effective Lagrangian) of physical four-quark operators without introducing any evanescent operator. The results obtained for {ital b}{implies}{ital s} processes agree with those from the so-called naive prescription, but disagree with the ones with the introduction of evanescent operators in a renormalization group analysis. We also present an explicit two loopmore » calculation of the mixing of the evanescent operators with the physical dimension-five operators for the same processes. Particular attention is paid to the unboundedness nature of such mixing and how a formal finite transformation is effected to decouple. The inevitable mass dependence of one of these schemes in the literature is pointed out as the cause for the difference mentioned. {copyright} {ital 1995 The American Physical Society.}« less