Steady-State Distributions of Interacting Discrete Vortices

Book, D. L.; Fisher, Shalom; McDonald, B. Edward

doi:10.1103/physrevlett.34.4

Cited by 51 publications

(22 citation statements)

References 11 publications

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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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Section: Introductionmentioning

confidence: 97%

On final states of two-dimensional decaying turbulence

Yin

2004

Physics of Fluids

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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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“…Equation (37) The sinh-Gordon equation arises also in the study of an infinite row of counter-rotating vortices in an inviscid and incompressible 2-dimensional fluid [22]. Finally, it appears in statistical mechanical models of two-dimensional vortices systems in a bounded magnetized plasma [23,24]. The essential point in the theory of the sinh-Gordon equation is its complete integrability [25,26], like in the case of the sine-Gordon equation [27] (which can be obtained from (37) Transform [26].…”

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confidence: 99%

Chern-Simons gauge field theory of two-dimensional ferromagnets

Martina

Pashaev

Soliani³

1993

Phys. Rev. B

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A Chern-Simons gauged Nonlinear Schrödinger Equation is derived from the continuous Heisenberg model in 2+1 dimensions. The corresponding planar magnets can be analyzed whithin the anyon theory. Thus, we show that static magnetic vortices correspond to the self-dual Chern -Simons solitons and are described by the Liouville equation. The related magnetic topological charge is associated with the electric charge of anyons. Furthermore, vortex -antivortex configurations are described by the sinh-Gordon equation and its conformally invariant extension. Physical consequences of these results are discussed.Lecce, June 1993 1

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Steady-State Distributions of Interacting Discrete Vortices

Cited by 51 publications

References 11 publications

On final states of two-dimensional decaying turbulence

On final states of two-dimensional decaying turbulence

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

Chern-Simons gauge field theory of two-dimensional ferromagnets

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