On the Turbulence Spectra of Electron Magnetohydrodynamics E. Westerhof, B.N. Kuvshinov, V.P. Lakhin...
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Transcript of On the Turbulence Spectra of Electron Magnetohydrodynamics E. Westerhof, B.N. Kuvshinov, V.P. Lakhin...
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FO M -Instituut vo o r PlasmafysicaA sso ciatio n EU RATO M -FO M
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On the Turbulence Spectra ofElectron Magnetohydrodynamics
E. Westerhof, B.N. Kuvshinov, V.P. Lakhin1, S.S. Moiseev*, T.J. Schep
FOM-Instituut voor Plasmafysica ‘Rijnhuizen’, Associatie Euratom-FOMTrilateral Euregio Cluster, Postbus 1207, 3430 BE Nieuwegein, The Netherlands
* Institute of Space Research of the Russian Academy of Sciences117810, Moscow, Russia
1 On leave from RRC Kurchatov Institute, Moscow, Russia
26th EPS Conference on Controlled Fusion and Plasma Physics, 14-18 June 1999, Maastricht, The Netherlands
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Overview
• 2D electron magnetohydrodynamics EMHD
• ideal statistical equilibrium spectra
• scaling symmetries and spectral laws of decaying turbulence
• finite density perturbations• invariants
• cascade directions• energy partitioning
• a temporal decay law
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• magnetic field representation: B = B0 ((1+b) ez + ez)
• generalized vorticity = b de2
2b + (1-neq(x)/n0)
generelized flux = de2
2
• evolution equations
2D EMHD
• with inertial skin depth de = c/pe
• with = 1 + (ce / pe)2
• [f,g] = ez • (f g)
],[],b[t
2
],b[t
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Finite Density Perturbations
• finite is the origin of the parameter = 1 + (ce / pe)2en~
• divergence of e momentum balance
• Poisson’s law . . . . . . . . . . . . . . . . . .
• and Ampere’s law . . . . . . . . . . . . . . BV
en4
c
0e
)( ec
1 BVE
en~4 e E
bdn
n~ 22e2
pe
2ce
0
e
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The Invariants
• Energy . . . . . . . . . .
• generalized Helicity
f arbitrary function of
• generalized Flux . . .
g arbitrary function of
222
e222
e22
2
1 )(dbdbxdE
Eb E
magnetic kinetic + internal
)f(xdH 2
)g(xdF 2
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• application of equilibrium statistical mechanics requires 1 finite dimensional system 2 Liouville theorem (conservation of phase space volume)
• achieved by truncated Fourier series representation of fields
‘detailed’ Liouville theorem for all kx ky
• invariants of the truncated system: only quadratic ones energy E; helicity H; mean square flux F
Ideal Equilibrium Spectra
y)kxi(kk
kk
k
kkkkkk
yxmax
minx
max
miny
yxyx)e~,b
~(b,
0b~b~
yx
yx
kk
kk
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TECIdeal Equilibrium Spectra
The Canonical Equilibrium Distribution
• Equilibrium probability density = (1/Z) exp( E H F )
Lagrange multipliers (‘inverse temperatures’)
fixed by Etot Htot Ftot and kmin kmax
• Equilibrium Spectra
E(kx,ky) = (4k2 + 2(1+de2k2)) / D
H(kx,ky) = 2 (1+de2k2) (1+de
2k2) / D
F(kx,ky) = 4(1+de2k2) / D
D = 4k2 +(1+de2k2)) 2(1+de
2k2) (1+de2k2)
convergence requires D > 0, and > 0
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de = 0.1 de = 0.01
squared flux cascade
Ideal Equilibrium Spectra
Examples of Equilibrium SpectraEnergy
Flux
Helicity
= = 10, = 1 = = 10, = 1000
energy cascade
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Energy Partitioning
• Ratio of energies Eb and E:
• small scales, kde >> 1, Eb / E = (1 + de2/)
• large scales, kde << 1, Eb / E = (1 + /k2)
• numerical calculations of decaying turbulence
Eb(kx,ky)
E(kx,ky)
(1+de2k2) k2 +(1+de
2k2))
k2 (1+de2k2)
=
• Eb / E << 1 initially: fast evolution to near equipartition
• Eb / E > 1 initially: ratio increases on dissipation time scale
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TECIdeal Equilibrium Spectra
Energy Partitioning
• spectra for Eb and E from simulations of decaying turbulence
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Scale Invariance and Spectra
• both kde << 1, kde >> 1: 2D EMHD invariant for transformations
r’ = r, t’ = 1 t, ’ = 1+ , ’ = 2+
• kde << 1: E b2 (magnetic) perturbations on scale r: b(r) = r 1+ F with F function of invariant(s)
• ’ = 3+1 = 1/3• thus: b(r) b(r) r4/3 and E(k) 2/3 k7/3
• kde >> 1: E v2 (kinetic) perturbations on scale r: v(r) = r F with F function of invariant(s)
• ’ = 31 = +1/3• thus: v(r) v(r) r2/3 and E(k) 2/3 k5/3
• a la Kolmogorov: only invariant is energy dissipation rate
• agrees with Biskamp et al. (1996) (1999)
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Energy Decay Law
• integrating over inertial range one obtains dE / dt = E3/2
• solution
• numerical results agree
data from case de = 0.3
20L2
t0
)( E1
E
E
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Summary and Conclusions
• applied equilibrium statistics to ideal 2D EMHD
• confirm normal energy cascade • confirm inverse mean square flux cascade, but kde < 1
• studied energy partitioning evolution to equipartition only for Eb < E initially
• derived spectral laws from scaling symmetries of 2D EMHD
• confirm Biskamp et al.: kde >> 1, E (k) k5/3
kde << 1, E (k) k7/3
• obtained temporal decay law, confirmed by simulations