On the Turbulence Spectra of Electron Magnetohydrodynamics E. Westerhof, B.N. Kuvshinov, V.P. Lakhin 1, 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 , Moscow, Russia 1 On leave from RRC Kurchatov Institute, Moscow, Russia 26 th EPS Conference on Controlled Fusion and Plasma Physics, June 1999, Maastricht, The Netherlands

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

magnetic field representation: B = B 0 ((1+b) e z +     e z ) generalized vorticity  = b   d e 2   2 b + (1-n eq (x)/n 0 ) generelized flux  =   d e 2   2  evolution equations 2D EMHD with inertial skin depth d e = c/  pe with  = 1 + (  ce /  pe ) 2 [f,g] = e z (  f   g)

2D EMHD Finite Density Perturbations finite is the origin of the parameter  = 1 + (  ce /  pe ) 2 divergence of e  momentum balance Poisson’s law and Ampere’s law

2D EMHD The Invariants Energy generalized Helicity f arbitrary function of  generalized Flux... g arbitrary function of  EbEb EE magnetic kinetic + internal

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 k x k y invariants of the truncated system: only quadratic ones energy E ; helicity H; mean square flux F Ideal Equilibrium Spectra

The Canonical Equilibrium Distribution Equilibrium probability density  = (1/Z) exp(   E   H   F ) Lagrange multipliers    (‘inverse temperatures’)    fixed by E tot H tot F tot and k min k max Equilibrium Spectra E(k x,k y ) = (4  k  (1+d e 2 k 2 )) / D H(k x,k y ) = 2  (1+d e 2 k 2 ) (1+  d e 2 k 2 ) / D F(k x,k y ) = 4  (1+d e 2 k 2 ) / D D = 4  k 2 +  (1+d e 2 k 2 ))   2 (1+d e 2 k 2 ) (1+  d e 2 k 2 ) convergence requires D > 0, and  > 0

d e = 0.1 d e = 0.01 squared flux cascade Ideal Equilibrium Spectra Examples of Equilibrium Spectra Energy Flux Helicity  =  = 10,  = 1  =  = 10,  =  1000 energy cascade

Ideal Equilibrium Spectra Energy Partitioning Ratio of energies E b and E  : small scales, kd e >> 1, E b / E  =  (1 +  d e 2 /  ) large scales, kd e << 1, E b / E  = (1 +  /  k 2 ) numerical calculations of decaying turbulence E b (k x,k y ) E  (k x,k y ) (1+  d e 2 k 2 )  k 2 +  (1+d e 2 k 2 ))  k 2 (1+d e 2 k 2 ) = E b / E  << 1 initially: fast evolution to near equipartition E b / E  > 1 initially: ratio increases on dissipation time scale

Ideal Equilibrium Spectra Energy Partitioning spectra for E b and E  from simulations of decaying turbulence

Scale Invariance and Spectra both kd e > 1: 2D EMHD invariant for transformations r’ =  r, t’ =  1  t,  ’ =  1+  ,  ’ =  2+   kd e << 1 : E  b 2 (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)   r 4/3 and E(k)   2/3 k  7/3 kd e >> 1 : E  v 2 (kinetic) perturbations on scale r : v(r) = r  F with F function of invariant(s)  ’ =  3  1    = +1/3 thus:  v(r) v(r)   r 2/3 and E(k)   2/3 k  5/3 a la Kolmogorov: only invariant is energy dissipation rate  agrees with Biskamp et al. (1996) (1999)

Scale Invariance and Spectra Energy Decay Law integrating over inertial range one obtains dE / dt =  E 3/2 solution numerical results agree data from case d e = 0.3

Summary and Conclusions applied equilibrium statistics to ideal 2D EMHD confirm normal energy cascade confirm inverse mean square flux cascade, but kd e < 1 studied energy partitioning evolution to equipartition only for E b < E  initially derived spectral laws from scaling symmetries of 2D EMHD confirm Biskamp et al.: kd e >> 1, E (k)  k  5/3 kd e << 1, E (k)  k  7/3 obtained temporal decay law, confirmed by simulations