Nonaxisymmetric thermal equilibria of a cylindrically bounded guiding-center plasma or discrete vortex system

Smith, R. C.; O’Neil, T. M.

doi:10.1063/1.859362

Cited by 125 publications

(125 citation statements)

References 31 publications

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“…At equilibrium, ensembles are equivalent for vortices in a disk (or other simple domains) when the Hamiltonian is postulated to be the only constraint [22,69]. However, they can be inequivalent in other kinds of domains [69] or also in a disk when the conservation of the angular momentum is taken into account [70,23]. Out-of-equilibrium, the kinetic equations describing Hamiltonian (microcanonical) and Brownian (canonical) point vortices are very different in any case.…”

Section: Resultsmentioning

confidence: 99%

Two-dimensional Brownian vortices

Chavanis

2008

Physica A: Statistical Mechanics and its Applications

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We introduce a stochastic model of two-dimensional Brownian vortices associated with the canonical ensemble. The point vortices evolve through their usual mutual advection but they experience in addition a random velocity and a systematic drift generated by the system as a whole. The statistical equilibrium state of this stochastic model is the Gibbs canonical distribution. We consider a single species system and a system made of two types of vortices with positive and negative circulations. At positive temperatures, like-sign vortices repel each other ("plasma" case) and at negative temperatures, like-sign vortices attract each other ("gravity" case). We derive the stochastic equation satisfied by the exact vorticity field and the Fokker-Planck equation satisfied by the N -body distribution function. We present the BBGKY-like hierarchy of equations satisfied by the reduced distribution functions and close the hierarchy by considering an expansion of the solutions in powers of 1/N , where N is the number of vortices, in a proper thermodynamic limit. For spatially inhomogeneous systems, we derive the kinetic equations satisfied by the smooth vorticity field in a mean field approximation valid for N → +∞. For spatially homogeneous systems, we study the two-body correlation function, in a Debye-Hückel approximation valid at the order O(1/N ). The results of this paper can also apply to other systems of random walkers with long-range interactions such as self-gravitating Brownian particles and bacterial populations experiencing chemotaxis. Furthermore, for positive temperatures, our study provides a kinetic derivation, from microscopic stochastic processes, of the Debye-Hückel model of electrolytes.

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Section: Resultsmentioning

confidence: 99%

Two-dimensional Brownian vortices

Chavanis

2008

Physica A: Statistical Mechanics and its Applications

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“…5 It was further developed by several groups. [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] In these studies, it is surprising to see that the ideal Euler mean-field predictions fit the Navier-Stokes (NS) results. The patch theory was put forward since the late 1980s.…”

Section: Introductionmentioning

confidence: 97%

On final states of two-dimensional decaying turbulence

Yin

2004

Physics of Fluids

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Numerical and analytical studies of final states of two-dimensional (2D) decaying turbulence are carried out. The first part of this work is trying to give a definition for final states of 2D decaying turbulence. The functional relation of -, which is frequently adopted as the characterization of those final states, is merely a sufficient but not necessary condition; moreover, it is not proper to use it as the definition. It is found that the method through the value of the effective area S covered by the scatter -plot, initially suggested by Read, Rhines, and White ["Geostrophic scatter diagrams and potential vorticity dynamics," J. Atmos. Sci. 43, 3226 (1986)] is more general and suitable for the definition. Based on this concept, a definition is presented, which covers all existing results in late states of decaying 2D flows (including some previous unexplainable weird double-valued -scatter plots). The remaining part of the paper is trying to further study 2D decaying turbulence with the assistance of this definition. Some numerical results, leading to "bar" final states and further verifying the predictive ability of statistical mechanics [Yin, Montgomery, and Clercx, "Alternative statistical-mechanical descriptions of decaying two-dimensional turbulence in terms of patches and points," Phys. Fluids 15, 1937 (2003)], are reported. It is realized that some simulations with narrow-band energy spectral initial conditions result in some final states that cannot be very well interpreted by the statistical theory (meanwhile, those final states are still in the scope of the definition).

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“…The prediction of such a dependence, in the context of a mean-field treatment of ideal line vortices ͑or guiding-center plasma rods͒, had been given 30 years ago 4,5 and has since been extended and refined in a series of investigations by several groups: [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] in every case referring to ideal, nonviscous systems. The system is Hamiltonian with a finite phase space, and it is natural to apply Boltzmann statistics to its dynamics, as originally suggested by Onsager 21 ͑see also Lin 22 ͒.…”

Section: Introductionmentioning

confidence: 99%

Alternative statistical-mechanical descriptions of decaying two-dimensional turbulence in terms of “patches” and “points”

2003

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Numerical and analytical studies of decaying, two-dimensional (2D) Navier-Stokes (NS) turbulence at high Reynolds numbers are reported. The effort is to determine computable distinctions between two different formulations of maximum entropy predictions for the decayed, late-time state. Though these predictions may be thought to apply only to the ideal Euler equations, there have been surprising and imperfectly-understood correspondences between the long-time computations of decaying states of NS flows and the results of the maximum-entropy analyses. Both formulations define an entropy through a somewhat ad hoc discretization of vorticity to the "particles" of which statistical mechanical methods are employed to define an entropy, before passing to a mean-field limit. In one case, the particles are delta-function parallel "line" vortices ("points" in two dimensions), and in the other, they are finite-area, mutually-exclusive convected "patches" of vorticity which in the limit of zero area become "points." The former are taken to obey Boltzmann statistics and the latter, Lynden-Bell statistics. Clearly, there is no unique way to reach a continuous differentiable vorticity distribution as a mean-field limit by either method. The simplest method of taking equal-strength points and equal-strength, equal-area patches is chosen here, no reason being apparent for attempting anything more complicated. In both cases, a non-linear partial differential equation results for the stream function of the "most probable" or maximum-entropy state, compatible with conserved total energy and positive and negative vorticity fluxes. These amount to generalizations of the "sinh-Poisson" equation which has become familiar from the "point" formulation. They have many solutions, which must be obtained numerically, and only one of which maximizes the entropy on the basis of which it was derived. These predictions can differ for the point and patch discretizations. The intent here is to use time-dependent, spectral-method direct numerical simulation of the Navier-Stokes equations to see if initial conditions which should relax to different late-time states under the two formulations actually do so.

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Nonaxisymmetric thermal equilibria of a cylindrically bounded guiding-center plasma or discrete vortex system

Cited by 125 publications

References 31 publications

Two-dimensional Brownian vortices

Two-dimensional Brownian vortices

On final states of two-dimensional decaying turbulence

Alternative statistical-mechanical descriptions of decaying two-dimensional turbulence in terms of “patches” and “points”

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