On the Turbulence Spectra of

1
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
26th EPS Conference on Controlled Fusion and
Plasma Physics, 14-18 June 1999, Maastricht, The
Netherlands
2

• 2D electron magnetohydrodynamics EMHD
• ideal statistical equilibrium spectra
• scaling symmetries and spectral laws of decaying
  turbulence

Overview

• finite density perturbations
• invariants
• cascade directions
• energy partitioning
• a temporal decay law

3
2D EMHD

• magnetic field representation B B0 ((1b) ez
  ??y ? ez)
• generalized vorticity W b - L de2 ??2b
  (1-neq(x)/n0)
• generelized flux Y y - de2 ??2y
• evolution equations

• with inertial skin depth de c/wpe
• with L 1 (wce / wpe)2
• f,g ez (?f ? ?g)

4
2D EMHD
Finite Density Perturbations

• finite is the origin of the parameter L 1
  (wce / wpe)2

• divergence of e- momentum balance
• Poissons law . . . . . . . . . . . . . . . . .
  .
• and Amperes law . . . . . . . . . . . . . .

5
2D EMHD
The Invariants

• Energy . . . . . . . . . .
• generalized Helicity
• f arbitrary function of Y
• generalized Flux . . .
• g arbitrary function of Y

Eb
Ey
magnetic kinetic internal
6
Ideal Equilibrium Spectra

• 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

7
Ideal Equilibrium Spectra
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 (1de2k2)) / D
• H(kx,ky) 2b (1de2k2) (1Lde2k2) / D
• F(kx,ky) 4a (1de2k2) / D
• D 4a(ak2 g (1de2k2)) - b2(1de2k2)
  (1Lde2k2)
• convergence requires D gt 0, and a gt 0

8
Ideal Equilibrium Spectra
Examples of Equilibrium Spectra
de 0.1
de 0.01
energy cascade
squared flux cascade
a g 10, b 1 a g
10, b ?1000
9
Ideal Equilibrium Spectra
Energy Partitioning

• Eb / Ey ltlt 1 initially fast evolution to near
  equipartition
• Eb / Ey gt 1 initially ratio increases on
  dissipation time scale

10
Ideal Equilibrium Spectra
Energy Partitioning

• spectra for Eb and Ey from simulations of
  decaying turbulence

11
Scale Invariance and Spectra

• both kde ltlt 1, kde gtgt 1 2D EMHD invariant for
  transformations

r a r, t a1-b t, W a1b W,
Y a2b Y

• kde ltlt 1 E ? b2 (magnetic)
• perturbations on scale r
• b(r) r 1b F
• with F function of invariant(s)
• e a3b1 e ? b -1/3
• thus ?b(r) b(r)? ? r4/3 and
• E(k) ? e2/3 k-7/3

• kde gtgt 1 E ? v2 (kinetic)
• perturbations on scale r
• v(r) r b F
• with F function of invariant(s)
• e a3b-1 e ? b 1/3
• thus ?v(r) v(r)? ? r2/3 and
• E(k) ? e2/3 k-5/3

• a la Kolmogorov only invariant is energy
  dissipation rate e

• agrees with Biskamp et al. (1996) (1999)

12
Scale Invariance and Spectra
Energy Decay Law

• integrating over inertial range one obtains dE /
  dt - e ? -E3/2
• solution
• numerical results agree
• data from case de 0.3

13
Summary and Conclusions

• applied equilibrium statistics to ideal 2D EMHD
• confirm normal energy cascade
• confirm inverse mean square flux
  cascade, but kde lt 1
• studied energy partitioning
• evolution to equipartition
  only for Eb lt Ey initially
• derived spectral laws from scaling symmetries of
  2D EMHD
• confirm Biskamp et al.
• kde gtgt 1, E (k) ?
  k-5/3
• kde ltlt 1, E (k) ?
  k-7/3
• obtained temporal decay law,
  confirmed by simulations