1
|
Düring G, Krstulovic G. Exact result in strong wave turbulence of thin elastic plates. Phys Rev E 2018; 97:020201. [PMID: 29548166 DOI: 10.1103/physreve.97.020201] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/17/2017] [Indexed: 06/08/2023]
Abstract
An exact result concerning the energy transfers between nonlinear waves of a thin elastic plate is derived. Following Kolmogorov's original ideas in hydrodynamical turbulence, but applied to the Föppl-von Kármán equation for thin plates, the corresponding Kármán-Howarth-Monin relation and an equivalent of the 4/5-Kolmogorov's law is derived. A third-order structure function involving increments of the amplitude, velocity, and the Airy stress function of a plate, is proven to be equal to -ɛℓ, where ℓ is a length scale in the inertial range at which the increments are evaluated and ɛ the energy dissipation rate. Numerical data confirm this law. In addition, a useful definition of the energy fluxes in Fourier space is introduced and proven numerically to be flat in the inertial range. The exact results derived in this Rapid Communication are valid for both weak and strong wave turbulence. They could be used as a theoretical benchmark of new wave-turbulence theories and to develop further analogies with hydrodynamical turbulence.
Collapse
Affiliation(s)
- Gustavo Düring
- Facultad de Física, Pontificia Universidad Católica de Chile, Casilla 306, Santiago, Chile
| | - Giorgio Krstulovic
- Université de la Côte d'Azur, OCA, CNRS, Lagrange, Boîte Postale 4229, 06304 Nice Cedex 4, France
| |
Collapse
|
2
|
Kawagoe K, Huber G, Pradas M, Wilkinson M, Pumir A, Ben-Naim E. Aggregation-fragmentation-diffusion model for trail dynamics. Phys Rev E 2017; 96:012142. [PMID: 29347086 DOI: 10.1103/physreve.96.012142] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/15/2017] [Indexed: 06/07/2023]
Abstract
We investigate statistical properties of trails formed by a random process incorporating aggregation, fragmentation, and diffusion. In this stochastic process, which takes place in one spatial dimension, two neighboring trails may combine to form a larger one, and also one trail may split into two. In addition, trails move diffusively. The model is defined by two parameters which quantify the fragmentation rate and the fragment size. In the long-time limit, the system reaches a steady state, and our focus is the limiting distribution of trail weights. We find that the density of trail weight has power-law tail P(w)∼w^{-γ} for small weight w. We obtain the exponent γ analytically and find that it varies continuously with the two model parameters. The exponent γ can be positive or negative, so that in one range of parameters small-weight trails are abundant and in the complementary range they are rare.
Collapse
Affiliation(s)
- Kyle Kawagoe
- Kavli Institute for Theoretical Physics, University of California Santa Barbara, California 93106, USA
- Department of Physics, University of Chicago, Chicago, Illinois 60637, USA
| | - Greg Huber
- Kavli Institute for Theoretical Physics, University of California Santa Barbara, California 93106, USA
| | - Marc Pradas
- Kavli Institute for Theoretical Physics, University of California Santa Barbara, California 93106, USA
- School of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes MK7 6AA, England
| | - Michael Wilkinson
- Kavli Institute for Theoretical Physics, University of California Santa Barbara, California 93106, USA
- School of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes MK7 6AA, England
| | - Alain Pumir
- Kavli Institute for Theoretical Physics, University of California Santa Barbara, California 93106, USA
- Laboratoire de Physique, Ecole Normale Supérieure de Lyon, CNRS, Université de Lyon, F-69007 Lyon, France
| | - Eli Ben-Naim
- Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
| |
Collapse
|
3
|
Klimas AJ, Uritsky VM. Criticality and turbulence in a resistive magnetohydrodynamic current sheet. Phys Rev E 2017; 95:023209. [PMID: 28297949 DOI: 10.1103/physreve.95.023209] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/05/2016] [Indexed: 11/07/2022]
Abstract
Scaling properties of a two-dimensional (2d) plasma physical current-sheet simulation model involving a full set of magnetohydrodynamic (MHD) equations with current-dependent resistivity are investigated. The current sheet supports a spatial magnetic field reversal that is forced through loading of magnetic flux containing plasma at boundaries of the simulation domain. A balance is reached between loading and annihilation of the magnetic flux through reconnection at the current sheet; the transport of magnetic flux from boundaries to current sheet is realized in the form of spatiotemporal avalanches exhibiting power-law statistics of lifetimes and sizes. We identify this dynamics as self-organized criticality (SOC) by verifying an extended set of scaling laws related to both global and local properties of the current sheet (critical susceptibility, finite-size scaling of probability distributions, geometric exponents). The critical exponents obtained from this analysis suggest that the model operates in a slowly driven SOC state similar to the mean-field state of the directed stochastic sandpile model. We also investigate multiscale correlations in the velocity field and find them numerically indistinguishable from certain intermittent turbulence (IT) theories. The results provide clues on physical conditions for SOC behavior in a broad class of plasma systems with propagating instabilities, and suggest that SOC and IT may coexist in driven current sheets which occur ubiquitously in astrophysical and space plasmas.
Collapse
Affiliation(s)
- Alexander J Klimas
- University of Maryland, Baltimore County at NASA Goddard Space Flight Center, Greenbelt, Maryland 20771, USA
| | - Vadim M Uritsky
- Catholic University of America at NASA Goddard Space Flight Center, Greenbelt, Maryland 20771, USA
| |
Collapse
|
4
|
Connaughton C, Rajesh R, Zaboronski O. Constant flux relation for diffusion-limited cluster-cluster aggregation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:041403. [PMID: 18999423 DOI: 10.1103/physreve.78.041403] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/30/2008] [Indexed: 05/27/2023]
Abstract
In a nonequilibrium system, a constant flux relation (CFR) expresses the fact that a constant flux of a conserved quantity exactly determines the scaling of the particular correlation function linked to the flux of that conserved quantity. This is true regardless of whether mean-field theory is applicable or not. We focus on cluster-cluster aggregation and discuss the consequences of mass conservation for the steady state of aggregation models with a monomer source in the diffusion-limited regime. We derive the CFR for the flux-carrying correlation function for binary aggregation with a general scale-invariant kernel and show that this exponent is unique. It is independent of both the dimension and of the details of the spatial transport mechanism, a property which is very atypical in the diffusion-limited regime. We then discuss in detail the "locality criterion" which must be satisfied in order for the CFR scaling to be realizable. Locality may be checked explicitly for the mean-field Smoluchowski equation. We show that if it is satisfied at the mean-field level, it remains true over some finite range as one perturbatively decreases the dimension of the system below the critical dimension, d_{c}=2 , entering the fluctuation-dominated regime. We turn to numerical simulations to verify locality for a range of systems in one dimension which are, presumably, beyond the perturbative regime. Finally, we illustrate how the CFR scaling may break down as a result of a violation of locality or as a result of finite size effects and discuss the extent to which the results apply to higher order aggregation processes.
Collapse
Affiliation(s)
- Colm Connaughton
- Centre for Complexity Science, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, United Kingdom.
| | | | | |
Collapse
|