Structure and turbulence in the complex Ginzburg-Landau equation with a nonlinearity of arbitrary order.

Persistent Link:
http://hdl.handle.net/10150/187363
Title:
Structure and turbulence in the complex Ginzburg-Landau equation with a nonlinearity of arbitrary order.
Author:
Stark, Donald Richard.
Issue Date:
1995
Publisher:
The University of Arizona.
Rights:
Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
Abstract:
Numerical and analytical studies are undertaken for the "inviscid" limit of the complex Ginzburg-Landau (CGL) equation with the objective of studying the applicability of paradigms from finite dimensional dynamical systems and statistical mechanics to the case of an infinite dimensional dynamical system. The analytical results rely on exploiting the structure of this limit, which becomes the nonlinear Schrodinger (NLS) equation. In the NLS limit the CGL equation can exhibit strong spatio-temporal chaos. The initial growth of the bursts closely mimics the blowup solutions of the NLS equation. The study of this turbulent behavior focuses on the inertial range of the time-averaged wavenumber spectrum. Analytical estimates of the decay rate are constructed assuming both structure driven and homogeneous turbulence, and are compared with numerical simulations. The quintic case is observed to have a stronger decay rate than what is predicted by either theory. This reflects the dominance of dissipation in the dynamics. In the septic case, two distinct inertial ranges are observed. This combination suggests that the evolution of a single burst, on average, is predominantly due to the self-focusing mechanism of blowup NLS in the initial stage, and regularization effects of dissipation in the final stage. Because the initial stage is primarily influenced by the NLS structure, the rate of decay for this range is close to the decay predicted for the structure driven turbulence. In a numerical experiment it is observed that some NLS solutions survive the deformation due to a CGL perturbation. In some cases the question of persistence can be addressed analytically using an averaging technique similar to a Melnikov method for pde's.
Type:
text; Dissertation-Reproduction (electronic)
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Applied Mathematics; Graduate College
Degree Grantor:
University of Arizona
Committee Chair:
Levermore, C. David

Full metadata record

DC FieldValue Language
dc.language.isoenen_US
dc.titleStructure and turbulence in the complex Ginzburg-Landau equation with a nonlinearity of arbitrary order.en_US
dc.creatorStark, Donald Richard.en_US
dc.contributor.authorStark, Donald Richard.en_US
dc.date.issued1995en_US
dc.publisherThe University of Arizona.en_US
dc.rightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.en_US
dc.description.abstractNumerical and analytical studies are undertaken for the "inviscid" limit of the complex Ginzburg-Landau (CGL) equation with the objective of studying the applicability of paradigms from finite dimensional dynamical systems and statistical mechanics to the case of an infinite dimensional dynamical system. The analytical results rely on exploiting the structure of this limit, which becomes the nonlinear Schrodinger (NLS) equation. In the NLS limit the CGL equation can exhibit strong spatio-temporal chaos. The initial growth of the bursts closely mimics the blowup solutions of the NLS equation. The study of this turbulent behavior focuses on the inertial range of the time-averaged wavenumber spectrum. Analytical estimates of the decay rate are constructed assuming both structure driven and homogeneous turbulence, and are compared with numerical simulations. The quintic case is observed to have a stronger decay rate than what is predicted by either theory. This reflects the dominance of dissipation in the dynamics. In the septic case, two distinct inertial ranges are observed. This combination suggests that the evolution of a single burst, on average, is predominantly due to the self-focusing mechanism of blowup NLS in the initial stage, and regularization effects of dissipation in the final stage. Because the initial stage is primarily influenced by the NLS structure, the rate of decay for this range is close to the decay predicted for the structure driven turbulence. In a numerical experiment it is observed that some NLS solutions survive the deformation due to a CGL perturbation. In some cases the question of persistence can be addressed analytically using an averaging technique similar to a Melnikov method for pde's.en_US
dc.typetexten_US
dc.typeDissertation-Reproduction (electronic)en_US
thesis.degree.namePh.D.en_US
thesis.degree.leveldoctoralen_US
thesis.degree.disciplineApplied Mathematicsen_US
thesis.degree.disciplineGraduate Collegeen_US
thesis.degree.grantorUniversity of Arizonaen_US
dc.contributor.chairLevermore, C. Daviden_US
dc.contributor.committeememberFlaschka, Hermannen_US
dc.contributor.committeememberHyman, Jamesen_US
dc.identifier.proquest9620421en_US
All Items in UA Campus Repository are protected by copyright, with all rights reserved, unless otherwise indicated.