dorsal/arxiv
View SchemaIso-topological relaxation, coherent structures, and Gaussian turbulence in two dimensional magnetohydrodynamics
| Authors | M. B. Isichenko, A. V. Gruzinov |
|---|---|
| Categories | |
| ArXiv ID | plasm-ph/9609002 |
| URL | https://arxiv.org/abs/plasm-ph/9609002 |
| DOI | 10.1063/1.870634 |
| Journal | Phys. Plasmas 1(6), 1802--1816 (1994) |
Abstract
The long-time relaxation of ideal two dimensional magnetohydrodynamic turbulence subject to the conservation of two infinite families of constants of motion---the magnetic and the "cross" topology invariants--is examined. The analysis of the Gibbs ensemble, where all integrals of motion are respected, predicts the initial state to evolve into an equilibrium, stable coherent structure (the most probable state) and decaying Gaussian turbulence (fluctuations) with a vanishing, but always positive temperature. The non-dissipative turbulence decay is accompanied by decrease in both the amplitude and the length scale of the fluctuations, so that the fluctuation energy remains finite. The coherent structure represents a set of singular magnetic islands with plasma flow whose magnetic topology is identical to that of the initial state, while the energy and the cross topology invariants are shared between the coherent structure and the Gaussian turbulence. These conservation laws suggest the variational principle of iso-topological relaxation which allows us to predict the appearance of the final state from a given initial state. For a generic initial condition having X points in the magnetic field, the coherent structure has universal types of singularities: current sheets terminating at Y points. These structures, which are similar to those resulting from the 2D relaxation of magnetic field frozen into an ideally conducting viscous fluid, are observed in the numerical experiment of Biskamp and Welter. The Gibbs ensemble method developed in this work admits extension to other Hamiltonian systems with invariants not higher than quadratic in the highest-order-derivative variables. The turbulence in two dimensional Euler fluid is of a different nature: there the coherent structures are also formed, but the fluctuations about these structures are non-Gaussian.
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"abstract": "The long-time relaxation of ideal two dimensional magnetohydrodynamic\nturbulence subject to the conservation of two infinite families of constants of\nmotion---the magnetic and the \"cross\" topology invariants--is examined. The\nanalysis of the Gibbs ensemble, where all integrals of motion are respected,\npredicts the initial state to evolve into an equilibrium, stable coherent\nstructure (the most probable state) and decaying Gaussian turbulence\n(fluctuations) with a vanishing, but always positive temperature. The\nnon-dissipative turbulence decay is accompanied by decrease in both the\namplitude and the length scale of the fluctuations, so that the fluctuation\nenergy remains finite. The coherent structure represents a set of singular\nmagnetic islands with plasma flow whose magnetic topology is identical to that\nof the initial state, while the energy and the cross topology invariants are\nshared between the coherent structure and the Gaussian turbulence. These\nconservation laws suggest the variational principle of iso-topological\nrelaxation which allows us to predict the appearance of the final state from a\ngiven initial state. For a generic initial condition having X points in the\nmagnetic field, the coherent structure has universal types of singularities:\ncurrent sheets terminating at Y points. These structures, which are similar to\nthose resulting from the 2D relaxation of magnetic field frozen into an ideally\nconducting viscous fluid, are observed in the numerical experiment of Biskamp\nand Welter. The Gibbs ensemble method developed in this work admits extension\nto other Hamiltonian systems with invariants not higher than quadratic in the\nhighest-order-derivative variables. The turbulence in two dimensional Euler\nfluid is of a different nature: there the coherent structures are also formed,\nbut the fluctuations about these structures are non-Gaussian.",
"arxiv_id": "plasm-ph/9609002",
"authors": [
"M. B. Isichenko",
"A. V. Gruzinov"
],
"categories": [
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],
"doi": "10.1063/1.870634",
"journal_ref": "Phys. Plasmas 1(6), 1802--1816 (1994)",
"title": "Iso-topological relaxation, coherent structures, and Gaussian turbulence in two dimensional magnetohydrodynamics",
"url": "https://arxiv.org/abs/plasm-ph/9609002"
},
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