skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Dynamical behavior for the three-dimensional generalized Hasegawa-Mima equations

Abstract

The long time behavior of solution of the three-dimensional generalized Hasegawa-Mima [Phys. Fluids 21, 87 (1978)] equations with dissipation term is considered. The global attractor problem of the three-dimensional generalized Hasegawa-Mima equations with periodic boundary condition was studied. Applying the method of uniform a priori estimates, the existence of global attractor of this problem was proven, and also the dimensions of the global attractor are estimated.

Authors:
;  [1];  [2]
  1. College of Mathematics and Information Science, Henan University, Kaifeng 475001 (China)
  2. (China)
Publication Date:
OSTI Identifier:
20929616
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Mathematical Physics; Journal Volume: 48; Journal Issue: 1; Other Information: DOI: 10.1063/1.2424559; (c) 2007 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
70 PLASMA PHYSICS AND FUSION TECHNOLOGY; ATTRACTORS; BOUNDARY CONDITIONS; BOUNDARY LAYERS; EQUATIONS; ION ACOUSTIC WAVES; MATHEMATICAL SOLUTIONS; PERIODICITY; PLASMA; PLASMA DRIFT; PLASMA INSTABILITY; PLASMA SIMULATION; THREE-DIMENSIONAL CALCULATIONS

Citation Formats

Zhang Ruifeng, Guo Boling, and Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088. Dynamical behavior for the three-dimensional generalized Hasegawa-Mima equations. United States: N. p., 2007. Web. doi:10.1063/1.2424559.
Zhang Ruifeng, Guo Boling, & Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088. Dynamical behavior for the three-dimensional generalized Hasegawa-Mima equations. United States. doi:10.1063/1.2424559.
Zhang Ruifeng, Guo Boling, and Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088. Mon . "Dynamical behavior for the three-dimensional generalized Hasegawa-Mima equations". United States. doi:10.1063/1.2424559.
@article{osti_20929616,
title = {Dynamical behavior for the three-dimensional generalized Hasegawa-Mima equations},
author = {Zhang Ruifeng and Guo Boling and Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088},
abstractNote = {The long time behavior of solution of the three-dimensional generalized Hasegawa-Mima [Phys. Fluids 21, 87 (1978)] equations with dissipation term is considered. The global attractor problem of the three-dimensional generalized Hasegawa-Mima equations with periodic boundary condition was studied. Applying the method of uniform a priori estimates, the existence of global attractor of this problem was proven, and also the dimensions of the global attractor are estimated.},
doi = {10.1063/1.2424559},
journal = {Journal of Mathematical Physics},
number = 1,
volume = 48,
place = {United States},
year = {Mon Jan 15 00:00:00 EST 2007},
month = {Mon Jan 15 00:00:00 EST 2007}
}
  • The mechanism of four nonlinearly interacting drift or Rossby waves is used as the basic process underlying the turbulent evolution of both the Charney-Hasegawa-Mima-equation (CHME) and its generalized modification (GCHME). Hasegawa and Kodama's concept of equivalent action (or quanta) is applied to the four-wave system and shown to control the distribution of energy and enstrophy between the modes. A numerical study of the GCHME is described in which the initial state contains a single finite-amplitude drift wave (the pump wave), and all the modulationally unstable modes are present at the same low level (10{sup -6} times the pump amplitude). Themore » simulation shows that at first the fastest-growing modulationally unstable modes dominate but reveals that at a later time, before pump depletion occurs, long- and short-wavelength modes, driven by pairs of fast-growing modes, grow at 2{gamma}{sub max}. The numerical simulation illustrates the development of a spectrum of turbulent modes from a finite-amplitude pump wave.« less
  • Reduced magnetohydrodynamics consists of a set of simplified fluid equations which has become a principal tool in the interpretation of plasma fluid motions in tokamak experiments. The Hasegawa--Mima equation is applied to the study of electrostatic fluctuations in turbulent plasmas. The relations between these two nonlinear models is elucidated. It is shown that both models can be obtained from appropriate limits of a third, inclusive, nonlinear system. The inclusive system is remarkably simple.
  • The dual cascade is generally represented as a conservative cascade of enstrophy to short wavelengths through an enstrophy similarity range and an inverse cascade of energy to long wavelengths through an energy similarity range. This picture, based on a proof due to Kraichnan [Phys. Fluids [bold 10], 1417 (1967)], is found to be significantly modified for spectra of finite extent. Dimensional arguments and direct measurement of spectral flow in Hasegawa--Mima turbulence indicate that for both the energy and enstrophy cascades, transfer of the conserved quantity is accompanied by a nonconservative transfer of the other quantity. The decrease of a givenmore » invariant (energy or enstrophy) in the nonconservative transfer in one similarity range is balanced by the increase of that quantity in the other similarity range, thus maintaining net invariance. The increase or decrease of a given invariant quantity in one similarity range depends on the injection scale and is consistent with that quantity being carried in a self-similar transfer of the other invariant quantity. This leads, in an inertial range of finite size, to some energy being carried to small scales and some enstrophy being carried to large scales.« less
  • A Kolmogorov-type analysis of the energy- and enstrophy-cascading ranges of a forced Hasegawa-Mima equation allows one to derive a criterion for the threshold of the transition between the weak-turbulence and the strong-turbulence regimes. Contrary to general belief, it is found that due to the inverse energy cascade the large-scale portion of the inertial range is in the strong-turbulence regime in the limit of infinite Reynolds-like numbers for any finite amount of forcing.
  • The quasigeostrophic model is a simplified geophysical fluid model at asymptotically high rotation rate or at small Rossby number. We consider the quasigeostrophic equation with no dissipation term which was obtained as an asymptotic model from the Euler equations with free surface under a quasigeostrophic velocity field assumption. It is called the Hasegawa-Mima-Charney-Obukhov equation, which also arises from plasmas theory. We use a priori estimates to get the global existence of strong solutions for an Hasegawa-Mima-Charney-Obukhov equation.