Time scale for energy equipartition in a two-dimensional FPU model

2005 ◽  
Vol 15 (1) ◽  
pp. 015108 ◽  
Author(s):  
Giancarlo Benettin
Keyword(s):  
Time Scale ◽  
Two Dimensional ◽  
Energy Equipartition

scholarly journals On energetics and inertial-range scaling laws of two-dimensional magnetohydrodynamic turbulence

2012 ◽  
Vol 703 ◽  
pp. 238-254 ◽  
Author(s):  
Luke A. K. Blackbourn ◽  
Chuong V. Tran
Keyword(s):  
Energy Transfer ◽  
Kinetic Energy ◽  
Energy Flux ◽  
Scaling Laws ◽  
Magnetic Energy ◽  
Inertial Range ◽  
Two Dimensional ◽  
Magnetohydrodynamic Turbulence ◽  
Direct Energy ◽  
Energy Equipartition

AbstractWe study two-dimensional magnetohydrodynamic turbulence, with an emphasis on its energetics and inertial-range scaling laws. A detailed spectral analysis shows that dynamo triads (those converting kinetic into magnetic energy) are associated with a direct magnetic energy flux while anti-dynamo triads (those converting magnetic into kinetic energy) are associated with an inverse magnetic energy flux. As both dynamo and anti-dynamo interacting triads are integral parts of the direct energy transfer, the anti-dynamo inverse flux partially neutralizes the dynamo direct flux, arguably resulting in relatively weak direct energy transfer and giving rise to dynamo saturation. This result is consistent with a qualitative prediction of energy transfer reduction due to Alfvén wave effects by the Iroshnikov–Kraichnan theory (which was originally formulated for magnetohydrodynamic turbulence in three dimensions). We numerically confirm the correlation between dynamo action and direct magnetic energy flux and investigate the applicability of quantitative aspects of the Iroshnikov–Kraichnan theory to the present case, particularly its predictions of energy equipartition and ${k}^{\ensuremath{-} 3/ 2} $ spectra in the energy inertial range. It is found that for turbulence satisfying the Kraichnan condition of magnetic energy at large scales exceeding total energy in the inertial range, the kinetic energy spectrum, which is significantly shallower than ${k}^{\ensuremath{-} 3/ 2} $, is shallower than its magnetic counterpart. This result suggests no energy equipartition. The total energy spectrum appears to depend on the energy composition of the turbulence but is clearly shallower than ${k}^{\ensuremath{-} 3/ 2} $ for $r\approx 2$, even at moderate resolutions. Here $r\approx 2$ is the magnetic-to-kinetic energy ratio during the stage when the turbulence can be considered fully developed. The implication of the present findings is discussed in conjunction with further numerical results on the dependence of the energy dissipation rate on resolution.

scholarly journals Oscillation for a Nonlinear Dynamic System with a Forced Term on Time Scales

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Xinli Zhang ◽  
Shanliang Zhu
Keyword(s):  
Dynamic System ◽  
Time Scale ◽  
Nonlinear Dynamic ◽  
Sufficient Conditions ◽  
Nonlinear Dynamic System ◽  
Two Dimensional ◽  
Differential Systems ◽  
Difference Systems ◽  
All Solutions ◽  
Forced Term

We consider a class of two-dimensional nonlinear dynamic system with a forced term on a time scale𝕋and obtain sufficient conditions for all solutions of the system to be oscillatory. Our results not only unify the oscillation of two-dimensional differential systems and difference systems but also improve the oscillation results that have been established by Saker, 2005, since our results are not restricted to the case whereb(t)≠0for allt∈𝕋andg(u)=u. Some examples are given to illustrate the results.

Elementary derivation of the kinetic equation for the two-dimensional guiding centre plasma

1978 ◽  
Vol 19 (1) ◽  
pp. 121-133 ◽  
Author(s):  
Michael Mond ◽  
Georg Knorr
Keyword(s):  
Kinetic Equation ◽  
Langevin Equation ◽  
Time Scale ◽  
Equations Of Motion ◽  
Planck Equation ◽  
Stochastic Equations ◽  
Iteration Scheme ◽  
Two Dimensional ◽  
Hopf Equation ◽  
Rigorous Solution

A kinetic equation for a two-dimensional inviscid hydrodynamic fluid is derived in two ways. First, the equations of motion for the modes of the fluid are interpreted as stochastic equations resembling the Langevin equation. To lowest order a Fokker–Planck equation can be derived which is the kinetic equation for one mode. Secondly, a suitable iteration scheme is applied to the Hopf equation which results in the same kinetic equation. A parameter describing the time scale is arbitrary and cannot be determined by the applied methods alone. It is shown that the kinetic equation satisfies the conservation requirements and relaxes to an equilibrium which is a rigorous solution of the Hopf equation.

scholarly journals Wavelet Transform in Vibration Analysis for Mechanical Fault Diagnosis

1996 ◽  
Vol 3 (1) ◽  
pp. 17-26 ◽  
Author(s):  
W.J. Wang
Keyword(s):  
Wavelet Transform ◽  
Time Scale ◽  
Vibration Analysis ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Vibration Signals ◽  
Time Frequency ◽  
Special Cases ◽  
Mechanical Faults ◽  
Short Time

The wavelet transform is introduced to indicate short-time fault effects in associated vibration signals. The time-frequency and time-scale representations are unified in a general form of a three-dimensional wavelet transform, from which two-dimensional transforms with different advantages are treated as special cases derived by fixing either the scale or frequency variable. The Gaussian enveloped oscillating wavelet is recommended to extract different sizes of features from the signal. It is shown that the time-frequency and time-scale distributions generated by the wavelet transform are effective in identifying mechanical faults.

scholarly journals Three-dimensional visualization of the interaction of a vortex ring with a stratified interface

2017 ◽  
Vol 820 ◽  
pp. 549-579 ◽  
Author(s):  
Jason Olsthoorn ◽  
Stuart B. Dalziel
Keyword(s):  
Vortex Ring ◽  
Time Scale ◽  
Richardson Number ◽  
Experimental Studies ◽  
Three Dimensional ◽  
Vortex Rings ◽  
Mixing Efficiency ◽  
Two Dimensional ◽  
Displacement Mode ◽  
Dimensional Instability

The study of vortex-ring-induced stratified mixing has long played a key role in understanding externally forced stratified turbulent mixing. While several studies have investigated the dynamical evolution of such a system, this study presents an experimental investigation of the mechanical evolution of these vortex rings, including the stratification-modified three-dimensional instability. The aim of this paper is to understand how vortex rings induce mixing of the density field. We begin with a discussion of the Reynolds and Richardson number dependence of the vortex-ring interaction using two-dimensional particle image velocimetry measurements. Then, through the use of modern imaging techniques, we reconstruct from an experiment the full three-dimensional time-resolved velocity field of a vortex ring interacting with a stratified interface. This work agrees with many of the previous two-dimensional experimental studies, while providing insight into the three-dimensional instabilities of the system. Observations indicate that the three-dimensional instability has a similar wavenumber to that found for the unstratified vortex-ring instability at later times. We determine that the time scale associated with this instability growth has an inverse Richardson number dependence. Thus, the time scale associated with the instability is different from the time scale of interface recovery, possibly explaining the significant drop in mixing efficiency at low Richardson numbers. The structure of the underlying instability is a simple displacement mode of the vorticity field.

Viscous selection of an elliptical dipole

2010 ◽  
Vol 658 ◽  
pp. 492-508 ◽  
Author(s):  
ZIV KIZNER ◽  
RUVIM KHVOLES ◽  
DAVID A. KESSLER
Keyword(s):  
Aspect Ratio ◽  
Asymptotic Analysis ◽  
Ideal Fluid ◽  
Time Scale ◽  
Initial Conditions ◽  
High Accuracy ◽  
Asymptotic State ◽  
Two Dimensional ◽  
Slow Time ◽  
Selection Of

A theory of viscous evolution and selection of symmetric two-dimensional dipoles is suggested, based on a combination of numerical simulations and an asymptotic analysis, where the slow time scale associated with the vorticity diffusion due to viscosity is incorporated. It is shown that viscosity first brings a dipole to an intermediate asymptotic state, which is independent of the initial conditions, and then slowly takes the dipole away from this state. We demonstrate that, among the variety of possible ideal-fluid dipole solutions, viscosity going to zero selects a unique solution, which is described to high accuracy by the elliptical dipole solution with a separatrix aspect ratio of 1.037.

scholarly journals Transport and instability in driven two-dimensional magnetohydrodynamic flows

2016 ◽  
Vol 799 ◽  
pp. 541-578 ◽  
Author(s):  
Sam Durston ◽  
Andrew D. Gilbert
Keyword(s):  
Magnetic Field ◽  
Time Scale ◽  
Large Scale ◽  
Body Force ◽  
Passive Scalar ◽  
Fast Time ◽  
Two Dimensional ◽  
Mean Flow ◽  
The Mean ◽  
Background Shear

This paper concerns the generation of large-scale flows in forced two-dimensional systems. A Kolmogorov flow with a sinusoidal profile in one direction (driven by a body force) is known to become unstable to a large-scale flow in the perpendicular direction at a critical Reynolds number. This can occur in the presence of a ${\it\beta}$-effect and has important implications for flows observed in geophysical and astrophysical systems. It has recently been termed ‘zonostrophic instability’ and studied in a variety of settings, both numerically and analytically. The goal of the present paper is to determine the effect of magnetic field on such instabilities using the quasi-linear approximation, in which the full fluid system is decoupled into a mean flow and waves of one scale. The waves are driven externally by a given random body force and move on a fast time scale, while their stress on the mean flow causes this to evolve on a slow time scale. Spatial scale separation between waves and mean flow is also assumed, to allow analytical progress. The paper first discusses purely hydrodynamic transport of vorticity including zonostrophic instability, the effect of uniform background shear and calculation of equilibrium profiles in which the effective viscosity varies spatially, through the mean flow. After brief consideration of passive scalar transport or equivalently kinematic magnetic field evolution, the paper then proceeds to study the full magnetohydrodynamic system and to determine effective diffusivities and other transport coefficients using a mixture of analytical and numerical methods. This leads to results on the effect of magnetic field, background shear and ${\it\beta}$-effect on zonostrophic instability and magnetically driven instabilities.

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