1. 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 E. Westerhof

2. 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 E. Westerhof

3. 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 E. Westerhof

4. 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 E. Westerhof

5. 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 E. Westerhof

6. 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 E. Westerhof

7.   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 E. Westerhof

8. 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 E. Westerhof

9. 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 E. Westerhof

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

11. 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 E. Westerhof

12. 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 E. Westerhof

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

14. 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 E. Westerhof

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

16. …‘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 E. Westerhof

17. 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 E. Westerhof