Trilateral Euregio Cluster
TEC
On the Turbulence Spectra of
Electron Magnetohydrodynamics
E. Westerhof, B.N. Kuvshinov, V.P. Lakhin1, S.S. Moiseev*, T.J. Schep
FOM-Instituut voor Plasmafysica ‘Rijnhuizen’, Associatie Euratom-FOM
Trilateral Euregio Cluster, Postbus 1207, 3430 BE Nieuwegein, The Netherlands
* Institute of Space Research of the Russian Academy of Sciences
117810, Moscow, Russia
1 On leave from RRC Kurchatov Institute, Moscow, Russia
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FOM-Instituut voor Plasmafysica
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
• finite density perturbations
• invariants
• ideal statistical equilibrium spectra
• cascade directions
• energy partitioning
• scaling symmetries and spectral laws of decaying turbulence
• a temporal decay law
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Trilateral Euregio Cluster
2D EMHD
• magnetic field representation: B = B0 ((1+b) ez + y  ez)
• generalized vorticity
generelized flux
W = b - L de2 2b + (1-neq(x)/n0)
Y = y - de2 2y
• with inertial skin depth de = c/wpe
• with L = 1 + (wce / wpe)2
• evolution equations W  -[b, W] - [y,  2 y]
t
Y
 -[b, Y ]
t
• [f,g] = ez • (f  g)
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2D EMHD
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Finite Density Perturbations
• finite ~
n e is the origin of the parameter L = 1 + (wce / wpe)2
• divergence of e- momentum balance   E  - 1   (Ve  B)
c
• Poisson’s law . . . . . . . . . . . . . . . . . .   E  -4~
n ee
• and Ampere’s law . . . . . . . . . . . . . .
2
~
n e wce
 2 d e2 2 b
n 0 wpe
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Ve  -
c
B
4n 0e
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2D EMHD
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The Invariants

• Energy . . . . . . . . . . E  1  d 2 x b 2  Ld e2   b 2   y 2  d e2( 2y) 2
2
Eb
magnetic
• generalized Helicity H   d 2 x Wf(Y)
f arbitrary function of Y
• generalized Flux . . . F   d 2 x g(Y)
g arbitrary function of Y
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Ey
kinetic + internal

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Ideal Equilibrium Spectra
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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
k max
b, y   
k max
~
i(k x x  k y y)
~
(
b
,
y
)e

kxk y
kxk y
k x  k min k y  k min
~
bk x k y
 ‘detailed’ Liouville theorem ~
 0 for all kx ky
bk x k y
• invariants of the truncated system: only quadratic ones
energy E; helicity H; mean square flux F
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Ideal Equilibrium Spectra
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The Canonical Equilibrium Distribution
• Equilibrium probability density r = (1/Z) exp( - aE - bH - gF )
Lagrange multipliers a b g (‘inverse temperatures’)
a b g fixed by Etot Htot Ftot and kmin kmax
• Equilibrium Spectra
E(kx,ky) = (4ak2 + 2g (1+de2k2)) / D
H(kx,ky) = 2b (1+de2k2) (1+Lde2k2) / D
F(kx,ky) = 4a (1+de2k2) / D
D = 4aak2 + g (1+de2k2)) - b2(1+de2k2) (1+Lde2k2)
convergence requires D > 0, and a > 0
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Ideal Equilibrium Spectra
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Examples of Equilibrium Spectra
Energy
de = 0.1
de = 0.01
Flux
Helicity
energy cascade
squared flux cascade
a = g = 10, b = 1
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a = g = 10, b = 1000
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Ideal Equilibrium Spectra
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Energy Partitioning
• Ratio of energies Eb and Ey:
Eb(kx,ky)
Ey(kx,ky)
=
(1+Lde2k2) ak2 + g (1+de2k2))
ak2 (1+de2k2)
• small scales, kde >> 1, Eb / Ey = L (1 + g de2/a)
• large scales, kde << 1, Eb / Ey = (1 + g /ak2)
• numerical calculations of decaying turbulence
• Eb / Ey << 1 initially: fast evolution to near equipartition
• Eb / Ey > 1 initially: ratio increases on dissipation time scale
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Ideal Equilibrium Spectra
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Energy Partitioning
• spectra for Eb and Ey from simulations of decaying turbulence
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Scale Invariance and Spectra
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• both kde << 1, kde >> 1: 2D EMHD invariant for transformations
r’ = a r,
t’ = a1-b t, W’ = a1+b W,
• kde << 1: E  b2 (magnetic)
perturbations on scale r:
b(r) = r 1+b F
with F function of invariant(s)
Y’ = a2+b Y
• kde >> 1: E  v2 (kinetic)
perturbations on scale r:
v(r) = r b F
with F function of invariant(s)
• a la Kolmogorov: only invariant is energy dissipation rate e
• e’ = a3b+1 e  b = -1/3
• thus: b(r) b(r)  r4/3 and
E(k)  e2/3 k-7/3
• e’ = a3b-1 e  b = +1/3
• thus: v(r) v(r)  r2/3 and
E(k)  e2/3 k-5/3
• agrees with Biskamp et al. (1996) (1999)
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Scale Invariance and Spectra
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Energy Decay Law
• integrating over inertial range one obtains dE / dt = - e  -E3/2
• solution E 
E0
(1  2tL
E 0 )2
• numerical results agree
data from case de = 0.3
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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 < Ey initially
• derived spectral laws from scaling symmetries of 2D EMHD
• confirm Biskamp et al.:
kde >> 1, E (k)  k-5/3
kde << 1, E (k)  k-7/3
• obtained temporal decay law, confirmed by simulations
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