Green's function-stochastic methods framework for probing nonlinear evolution problems: Burger's equation, the nonlinear Schroedinger's equation, and hydrodynamic organization of near-molecular-scale vorticity

Journal Article · Fri Apr 15 00:00:00 EDT 2011 · Annals of Physics (New York)

DOI:https://doi.org/10.1016/j.aop.2010.12.004· OSTI ID:21579872

Keanini, R.G., E-mail: [e-mail] ^[1]

---

1. Department of Mechanical Engineering and Engineering Science, University of North Carolina at Charlotte, 9201 University City Blvd., Charlotte, NC 28223-0001 (United States)

Research Highlights: > Systematic approach for physically probing nonlinear and random evolution problems. > Evolution of vortex sheets corresponds to evolution of an Ornstein-Uhlenbeck process. > Organization of near-molecular scale vorticity mediated by hydrodynamic modes. > Framework allows calculation of vorticity evolution within random strain fields. - Abstract: A framework which combines Green's function (GF) methods and techniques from the theory of stochastic processes is proposed for tackling nonlinear evolution problems. The framework, established by a series of easy-to-derive equivalences between Green's function and stochastic representative solutions of linear drift-diffusion problems, provides a flexible structure within which nonlinear evolution problems can be analyzed and physically probed. As a preliminary test bed, two canonical, nonlinear evolution problems - Burgers' equation and the nonlinear Schroedinger's equation - are first treated. In the first case, the framework provides a rigorous, probabilistic derivation of the well known Cole-Hopf ansatz. Likewise, in the second, the machinery allows systematic recovery of a known soliton solution. The framework is then applied to a fairly extensive exploration of physical features underlying evolution of randomly stretched and advected Burger's vortex sheets. Here, the governing vorticity equation corresponds to the Fokker-Planck equation of an Ornstein-Uhlenbeck process, a correspondence that motivates an investigation of sub-sheet vorticity evolution and organization. Under the assumption that weak hydrodynamic fluctuations organize disordered, near-molecular-scale, sub-sheet vorticity, it is shown that these modes consist of two weakly damped counter-propagating cross-sheet acoustic modes, a diffusive cross-sheet shear mode, and a diffusive cross-sheet entropy mode. Once a consistent picture of in-sheet vorticity evolution is established, a number of analytical results, describing the motion and spread of single, multiple, and continuous sets of Burger's vortex sheets, evolving within deterministic and random strain rate fields, under both viscous and inviscid conditions, are obtained. In order to promote application to other nonlinear problems, a tutorial development of the framework is presented. Likewise, time-incremental solution approaches and construction of approximate, though otherwise difficult-to-obtain backward-time GF's (useful in solution of forward-time evolution problems) are discussed.

• OSTI ID: 21579872
• Journal Information: Annals of Physics (New York), Vol. 326, Issue 4; Other Information: DOI: 10.1016/j.aop.2010.12.004; PII: S0003-4916(10)00218-6; Copyright (c) 2010 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; ISSN 0003-4916
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ADVECTION
DIFFUSION
ENTROPY
FLUCTUATIONS
FOKKER-PLANCK EQUATION
GREEN FUNCTION
MATHEMATICAL SOLUTIONS
NONLINEAR PROBLEMS
PROBABILISTIC ESTIMATION
RANDOMNESS
SCHROEDINGER EQUATION
SHEAR
SOLITONS
STOCHASTIC PROCESSES
STRAIN RATE
STRAINS
VORTICES
CALCULATION METHODS
DIFFERENTIAL EQUATIONS
EQUATIONS
FUNCTIONS
MASS TRANSFER
PARTIAL DIFFERENTIAL EQUATIONS
PHYSICAL PROPERTIES
QUASI PARTICLES
THERMODYNAMIC PROPERTIES
VARIATIONS
WAVE EQUATIONS