Universal Dependence of the Mean Square Displacement in Equilibrium Point Vortex Systems without Boundary Conditions

Yoshida, Takeshi

doi:10.1143/jpsj.78.024004

Cited by 5 publications

(7 citation statements)

References 27 publications

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“…This claim is corroborated by the fact that Kawahara & Nakanishi [96] observe anomalous diffusion which means that the evolution of the flow is more complex than it seems. These results have been confirmed recently by Yoshida [98] who showed that the mean square displacements exhibit a universal time dependence [r(t) − r(0)] 2 ∼ t µ with µ = 1.75 ± 0.1. These authors attribute anomalous diffusion to occasional long jumps of the particles convected by long living large vortices.…”

Section: Discussionsupporting

confidence: 74%

Kinetic equations for systems with long-range interactions: a unified description

Chavanis¹

2010

J. Stat. Mech.

179

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We complete the existing literature on the kinetic theory of systems with long-range interactions. Starting from the BBGKY hierarchy, or using projection operator technics or a quasilinear theory, a general kinetic equation can be derived when collective effects are neglected. This equation (which is not well-known) applies to possibly spatially inhomogeneous systems, which is specific to systems with long-range interactions. Interestingly, the structure of this kinetic equation bears a clear physical meaning in terms of generalized Kubo relations. Furthermore, this equation takes a very similar form for stellar systems and two-dimensional point vortices providing therefore a unified description of the kinetic theory of these systems. If we assume that the system is spatially homogeneous (or axisymmetric for point vortices), this equation can be simplified and reduces to the Landau equation (or its counterpart for point vortices). Our formalism thus offers a simple derivation of Landau-type equations. We also use this general formalism to derive a kinetic equation, written in angle-action variables, describing spatially inhomogeneous systems with long-range interactions. This new derivation solves the shortcomings of our previous derivation [P.H. Chavanis, Physica A 377, 469 (2007)]. Finally, we consider a test particle approach and derive general expressions for the diffusion and friction (or drift) coefficients of a test particle evolving in a bath of field particles. We make contact with the expressions previously obtained in the literature. As an application of the kinetic theory, we argue that the relaxation time is shorter for inhomogeneous (or high-dimensional) systems than for homogeneous (or low-dimensional) systems because there are potentially more resonances. We compare this prediction with existing numerical results for the HMF model and 2D point vortices. For the HMF model, we argue that the relaxation time scales as N for inhomogeneous distributions and as e N for permanently homogeneous distributions. Phase space structures can reduce the relaxation time by creating some inhomogeneities and resonances. Similar results are expected for 2D point vortices.

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Section: Discussionsupporting

confidence: 74%

Kinetic equations for systems with long-range interactions: a unified description

Chavanis¹

2010

J. Stat. Mech.

179

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“…This is based on recent observations suggesting that the deviations of the trajectory of generic vortices or TCs from a mean trajectory can be modeled with a universal law 9 for their so called mean square displacement. This law appears for the random movement of generic vortices in two dimensions as has been shown in experiments and numerical simulations 9 10 11 . In particular we suggest that track forecast cones available today can be linked directly to this measure of the trajectory deviations around a mean and that unless these deviations can be understood, and the factors giving rise to them are fully taken into account in models and simulations, reducing such uncertainty will be a difficult task.…”

supporting

confidence: 60%

“…It is this observation that has guided us to look for such a behavior in the movement of TCs, which are large scale single vortices. Indeed, for generic vortices in two dimensional turbulent flows 10 11 and for TCs 9 , the trajectory shows important deviations from a mean track. In the case of TCs, a displacement along a preferred direction with a non zero velocity is usually present.…”

Section: Resultsmentioning

confidence: 99%

Hurricane track forecast cones from fluctuations

Meuel

Prado

Seychelles

et al. 2012

Sci Rep

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Trajectories of tropical cyclones may show large deviations from predicted tracks leading to uncertainty as to their landfall location for example. Prediction schemes usually render this uncertainty by showing track forecast cones representing the most probable region for the location of a cyclone during a period of time. By using the statistical properties of these deviations, we propose a simple method to predict possible corridors for the future trajectory of a cyclone. Examples of this scheme are implemented for hurricane Ike and hurricane Jimena. The corridors include the future trajectory up to at least 50 h before landfall. The cones proposed here shed new light on known track forecast cones as they link them directly to the statistics of these deviations.

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“…Point vortices will feel correlations due to finite N effects and will (with the reserve that we have given in this paper) relax towards the Boltzmann distribution while a continuous vorticity field will feel inherent viscosity and will decay to zero. situation since it has been found numerically that point vortices can exhibit long jumps (Lévy flights) and strong correlations [82,83].…”

Section: Discussionmentioning

confidence: 99%

Kinetic theory of Onsager’s vortices in two-dimensional hydrodynamics

Chavanis

2012

Physica A: Statistical Mechanics and its Applications

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Starting from the Liouville equation, and using a BBGKY-like hierarchy, we derive a kinetic equation for the point vortex gas in two-dimensional (2D) hydrodynamics, taking two-body correlations and collective effects into account. This equation is valid at the order 1/N where N ≫ 1 is the number of point vortices in the system (we assume that their individual circulation scales like γ ∼ 1/N ). It gives the first correction, due to graininess and correlation effects, to the 2D Euler equation that is obtained for N → +∞. For axisymmetric distributions, this kinetic equation does not relax towards the Boltzmann distribution of statistical equilibrium. This implies either that (i) the "collisional" (correlational) relaxation time is larger than N tD, where tD is the dynamical time, so that three-body, four-body... correlations must be taken into account in the kinetic theory, or (ii) that the point vortex gas is non-ergodic (or does not mix well) and will never attain statistical equilibrium. Non-axisymmetric distributions may relax towards the Boltzmann distribution on a timescale of the order N tD due to the existence of additional resonances, but this is hard to prove from the kinetic theory. On the other hand, 2D Euler unstable vortex distributions can experience a process of "collisionless" (correlationless) violent relaxation towards a non-Boltzmannian quasistationary state (QSS) on a very short timescale of the order of a few dynamical times. This QSS is possibly described by the Miller-Robert-Sommeria (MRS) statistical theory which is the counterpart, in the context of two-dimensional hydrodynamics, of the Lynden-Bell statistical theory of violent relaxation in stellar dynamics.

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Universal Dependence of the Mean Square Displacement in Equilibrium Point Vortex Systems without Boundary Conditions

Cited by 5 publications

References 27 publications

Kinetic equations for systems with long-range interactions: a unified description

Kinetic equations for systems with long-range interactions: a unified description

Hurricane track forecast cones from fluctuations

Kinetic theory of Onsager’s vortices in two-dimensional hydrodynamics

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