### Abstract

The long-time relaxation of ideal two-dimensional (2-D) magnetohydrodynamic (MHD) 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 nondissipative 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 isotopological relaxation that allows one 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 2-D relaxation of magnetic field frozen into an ideally conducting viscous fluid, are observed in the numerical experiment of D. Biskamp and H. Welter [Phys. Fluids B 1, 1964 (1989)] and are likely to form during the nonlinear stage of the kink tearing mode in tokamaks. 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 2-D Euler fluid is of a different nature: there the coherent structures are also formed, but the fluctuations about these structures are non-Gaussian.

Original language | English (US) |
---|---|

Pages (from-to) | 1802-1816 |

Number of pages | 15 |

Journal | Physics of Plasmas |

Volume | 1 |

Issue number | 6 |

State | Published - 1994 |

### Fingerprint

### ASJC Scopus subject areas

- Physics and Astronomy(all)
- Condensed Matter Physics

### Cite this

*Physics of Plasmas*,

*1*(6), 1802-1816.

**Isotopological relaxation, coherent structures, and Gaussian turbulence in two-dimensional (2-D) magnetohydrodynamics (MHD).** / Isichenko, M. B.; Gruzinov, A. V.

Research output: Contribution to journal › Article

*Physics of Plasmas*, vol. 1, no. 6, pp. 1802-1816.

}

TY - JOUR

T1 - Isotopological relaxation, coherent structures, and Gaussian turbulence in two-dimensional (2-D) magnetohydrodynamics (MHD)

AU - Isichenko, M. B.

AU - Gruzinov, A. V.

PY - 1994

Y1 - 1994

N2 - The long-time relaxation of ideal two-dimensional (2-D) magnetohydrodynamic (MHD) 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 nondissipative 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 isotopological relaxation that allows one 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 2-D relaxation of magnetic field frozen into an ideally conducting viscous fluid, are observed in the numerical experiment of D. Biskamp and H. Welter [Phys. Fluids B 1, 1964 (1989)] and are likely to form during the nonlinear stage of the kink tearing mode in tokamaks. 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 2-D Euler fluid is of a different nature: there the coherent structures are also formed, but the fluctuations about these structures are non-Gaussian.

AB - The long-time relaxation of ideal two-dimensional (2-D) magnetohydrodynamic (MHD) 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 nondissipative 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 isotopological relaxation that allows one 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 2-D relaxation of magnetic field frozen into an ideally conducting viscous fluid, are observed in the numerical experiment of D. Biskamp and H. Welter [Phys. Fluids B 1, 1964 (1989)] and are likely to form during the nonlinear stage of the kink tearing mode in tokamaks. 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 2-D Euler fluid is of a different nature: there the coherent structures are also formed, but the fluctuations about these structures are non-Gaussian.

UR - http://www.scopus.com/inward/record.url?scp=36449008527&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=36449008527&partnerID=8YFLogxK

M3 - Article

VL - 1

SP - 1802

EP - 1816

JO - Physics of Plasmas

JF - Physics of Plasmas

SN - 1070-664X

IS - 6

ER -