Self-Organization of Plasmas with Flows ICNSP 2003/ 9/10 Graduate School of Frontier Sciences,, National Institute for Fusion Science R. NUMATA, Z. YOSHIDA, T. HAYASHI ICNSP 2003/ 9/10 p.1/14
Abstract We have developed a three dimensional nonlinear Hall-Magnetohydrodynamics (Hall-MHD) code, and obtained the double Beltrami Equilibrium, which is a theoretically predicted structure in the Hall-MHD plasma. ICNSP 2003/ 9/10 p.2/14
!! & % $ # # " ' 9 5 4 8. 5 6. 5 Double Beltrami Equilibrium Incompressible two-fluid MHD equations can be cast into coupled vortex equations (electron inertia, displacement current are neglected) (1) where are the generalized vorticities and are the corresponding flows, (2) ). Combining two Beltrami conditions yield ( : ion collisionless skin depth, : speed of light, : ion plasma frequency, : system size). A simplest equilibrium solution is given by the Beltrami conditions ( 2nd-order equation for and, " % ',/. + *) (,- + *) ( (3) where,10 2 3 4 7 4. ICNSP 2003/ 9/10 p.3/14
0 0 0,. 5 -. 5. ( Double Beltrami Equilibrium The general solution is described by a linear combination of eigenfunctions of the curl operator ( ) :, 0 - -. -,..(4) The Beltrami conditions give a special class of solution of the general steady state solution such that (5) where is a certain scalar field corresponding the energy density. The latter equality, called the generalized Bernoulli condition, demands the energy density is uniform in space. From the original electron and ion momentum equations, we find that the generalized Bernoulli condition is given by the relation, (6) ICNSP 2003/ 9/10 p.4/14
(7) (8) + ( (9) (10) (11) (12) ICNSP 2003/ 9/10 p.5/14 Relaxation Process Global Invariants * + ( * + ( Variational Principle Z.Yoshida and S.M.Mahajan, Phys. Rev. Lett., 88, 095001 (2002) are Lagrange multipliers.
3 8 Simulation Model We consider compressible Hall-MHD equations (13) (14) (15) 9 (16) (17) : where : density, : number density, : ion pressure, : Reynolds number, magnetic Reynolds number, : ratio of specific heat. We have assumed the quasi-neutrality, and. ICNSP 2003/ 9/10 p.6/14
Simulation Model Simulation Domain 3 dimensional rectangular domain Boundary Condition Periodic in direction Rigid perfect conducting wall in 5 5 direction 5 5 (18) (19) where is a unit normal vector, is a normal derivative, and and tangential component of the magnetic field, respectively. To assure tangential components of the electric field vanish, we set is the normal at the wall. Numerical Method 2nd order central difference in space, the Runge-Kutta-Gill method for time integration 2nd order smoothing to suppress short wave length mode add a diffusion in the equation of continuity to keep density positive ICNSP 2003/ 9/10 p.7/14
6 3 6 8 Dispersion Relation In order to check the validity of the code, we have solved the dispersion relation of purely transversal wave in a periodic domain. In an ideal limit, the dispersion relation for small perturbations around a uniform equilibrium, are constants, is given by 7 9 (20) which describes the Alfvén whistler wave. ω 18 16 14 12 10 8 6 4 2 0 numerical ε=0 ε=0.1 (+) ε=0.1 (-) ε=0.05 (+) ε=0.05 (-) analitical ε=0 ε=0.1 (+) ε=0.1 (-) ε=0.05 (+) ε=0.05 (-) 0 2 4 6 8 10 k ICNSP 2003/ 9/10 p.8/14
(21) (22) (23) ICNSP 2003/ 9/10 p.9/14 Initial Condition 2D force free equilibrium ( ) R.Horiuchi and T.Sato, Phys. Rev. Lett., 55, 211 (1985) ( ( ( ( ( ( ( ( t/ta= Flow (24) Density and Pressure are uniform Bz isosurface
ICNSP 2003/ 9/10 p.10/14 Equilibrium State 5 Aspect Ratio : Grid :,
5 4 Beltrami Conditions (25) (a) distribution of a (b) distribution of b (26) (27) Beltrami Parameters 0.1 0-0.1-0.2-0.3-0.4-0.5-0.6-0.7-0.8-0.9 a b σ 0 20 40 60 80 100 120 t/τa (c) distribution of σ (d) temporal behavior of a,b,σ ICNSP 2003/ 9/10 p.11/14
( Bernoulli Conditions (28) (a) V (b) p ICNSP 2003/ 9/10 p.12/14
Variational Principles 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 Energy Electron Helicity Ion Helicity Enstrophy 0 20 40 60 80 100 120 t/τa ICNSP 2003/ 9/10 p.13/14
Summary We have developed a nonlinear 3D Hall-MHD simulation code. Flow with perpendicular components remains in the two-fluid relaxed state, while it disappears in the single-fluid MHD model relaxed state. Beltrami & Bernoulli conditions are investigated for two-fluid relaxed state. High resolution simulation is necessary to see the scale separation. Further investigation of parameter dependence, compressibility effect, etc. ICNSP 2003/ 9/10 p.14/14