Dynamics of Electron Vortices E. Westerhof FOM-Instituut voor Plasmafysica “Rijnhuizen” Postbus 1207, 3430 BE Nieuwegein, Netherlands In collaboration with: B.N. Kuvshinov, J. Rem, and T.J. Schep Workshop on Nonlinear Structures in Magnetized Plasmas, Tarusa, Kaluga Region, Russia, 6-9 September 2000 E. Westerhof

Topics 2D dipole vortex solutions in EMHD perpendicular propagation oblique propagation numerical techniques stability properties dipole interactions conclusions Workshop on Nonlinear Structures in Magnetized Plasmas, Tarusa, Kaluga Region, Russia, 6-9 September 2000 E. Westerhof

2D Electron Magnetohydrodynamics magnetic field representation: B = B0 ((1+b) ez + y  ez) generalised vorticity W = b - L de2 2b + kn x generalised 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) Workshop on Nonlinear Structures in Magnetized Plasmas, Tarusa, Kaluga Region, Russia, 6-9 September 2000 E. Westerhof

A numerical code for 2D EMHD equilibrium flux: yeq = - by x double periodic boundary conditions standard pseudo-spectral for non-linear terms typically, 512  512 Fourier modes de-aliassed using 2/3 rule hyper-viscosity with n=3: hn(-2)n b,y Workshop on Nonlinear Structures in Magnetized Plasmas, Tarusa, Kaluga Region, Russia, 6-9 September 2000 E. Westerhof

Stationary propagating dipoles Solutions of F = FF(b - uy x) W = FW(b - uy x) - F’F(b - uy x) using method described by Kuvshinov: f(x,y) = f(r) cos  internal to separatrix, r < rs: B1 = J1 (k2<0), I1(k2>0), external to separatrix, r > rs: B1 = K1(k2>0) Workshop on Nonlinear Structures in Magnetized Plasmas, Tarusa, Kaluga Region, Russia, 6-9 September 2000 E. Westerhof

Perpendicular propagation by = 0 external scale k12 = ke2 = (1+kn/uy)/L  uy<-kn; uy>0  = 0 in internal region identical to Larichev-Reznik modon of Charney-Hasegawa-Mima equation   0 in internal region superposition of two Larichev-Reznik modes  additional separatrix in ‘stream function’ b - uy x Workshop on Nonlinear Structures in Magnetized Plasmas, Tarusa, Kaluga Region, Russia, 6-9 September 2000 E. Westerhof

  0, dipole with 2nd separatrix Parameters: L = 1.8, kn = .002, uy = -.02, ke = 0.5 Workshop on Nonlinear Structures in Magnetized Plasmas, Tarusa, Kaluga Region, Russia, 6-9 September 2000 E. Westerhof

Evolution of dipole with 2nd separatrix: instability t = 100 t = 200 contours of b - L2b, steps: .20 Workshop on Nonlinear Structures in Magnetized Plasmas, Tarusa, Kaluga Region, Russia, 6-9 September 2000 E. Westerhof

Oblique propagation by  0 External scales: conditions: ke2 > 0, and (1- ke)2 > by2  by2 /Luy2 no real k(e) solutions in incompressible, homo-geneous case (i.e. case of pure whistler modes) extension of Larichev-Reznik modons to by  0 consequently, dynamics is very similar to these Workshop on Nonlinear Structures in Magnetized Plasmas, Tarusa, Kaluga Region, Russia, 6-9 September 2000 E. Westerhof

uy > 0, tilt-stable dipole P.M. Workshop on Nonlinear Structures in Magnetized Plasmas, Tarusa, Kaluga Region, Russia, 6-9 September 2000 E. Westerhof

uy < 0, tilt-unstable dipole Parameters: L = 1.8; kn = .01 by = .0022; uy = -.02 orbit:diamonds show Dt =100 shown are contours of b - L2b steps: .05 Workshop on Nonlinear Structures in Magnetized Plasmas, Tarusa, Kaluga Region, Russia, 6-9 September 2000 E. Westerhof

Higher order dipoles with 2nd sep... t = 0 L = 1.8; kn = .01; by = .0022; uy = .02 t = 80 contours of b - L2b, steps: .10 Workshop on Nonlinear Structures in Magnetized Plasmas, Tarusa, Kaluga Region, Russia, 6-9 September 2000 E. Westerhof

… with 2nd separatrix are unstable t = 120 t = 200 Workshop on Nonlinear Structures in Magnetized Plasmas, Tarusa, Kaluga Region, Russia, 6-9 September 2000 E. Westerhof

Head-on dipole collisions t = 0 L = 1.8; kn = .01; by = .0022; u1 = -.015; u2 = .02 t = 100 contours of b - L2b, steps: .05 Workshop on Nonlinear Structures in Magnetized Plasmas, Tarusa, Kaluga Region, Russia, 6-9 September 2000 E. Westerhof

… collisions appear ‘soliton-like’, ... t = 400 t = 600 Workshop on Nonlinear Structures in Magnetized Plasmas, Tarusa, Kaluga Region, Russia, 6-9 September 2000 E. Westerhof

…‘soliton like’, but for [y,2y] t = 700 t = 700 y - 2y, steps: .001 b - L2b, steps: .10 Workshop on Nonlinear Structures in Magnetized Plasmas, Tarusa, Kaluga Region, Russia, 6-9 September 2000 E. Westerhof

Summary and Conclusion 2D EMHD stationary propagating dipoles by = 0: solutions with finite y perturbation have additional separatrix and are unstable by  0: extensions of Larichev-Reznik modon with behaviour that is very similar to the latter dominant effect in dynamics is ‘vorticity’ transport, with weak effect of [y,2y] term (except 0 ‹ uy « kn) double separatrices (‘vorticity shields’) unstable in all cases Workshop on Nonlinear Structures in Magnetized Plasmas, Tarusa, Kaluga Region, Russia, 6-9 September 2000 E. Westerhof