On the Turbulence Spectra of Electron Magnetohydrodynamics E. Westerhof, B.N. Kuvshinov, V.P. Lakhin...

Trilateral Eureg io Cluster

FO M -Instituut vo o r PlasmafysicaA sso ciatio n EU RATO M -FO M

TEC

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



TEC2D EMHD

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



TEC2D EMHD

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



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



TEC

de = 0.1 de = 0.01

squared flux cascade

Ideal Equilibrium Spectra

Examples of Equilibrium SpectraEnergy

Flux

Helicity

= = 10, = 1 = = 10, = 1000

energy cascade




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




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)



TECScale Invariance and Spectra

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

On the Turbulence Spectra of Electron Magnetohydrodynamics E. Westerhof, B.N. Kuvshinov, V.P. Lakhin...

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