Transcript No Slide Title

Trilateral Euregio Cluster
TEC
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
Association EURATOM-FOM
FOM-Instituut voor Plasmafysica
Workshop on Nonlinear Structures in Magnetized Plasmas, Tarusa, Kaluga Region, Russia, 6-9 September 2000
E. Westerhof
Trilateral Euregio Cluster
TEC
Topics
•
•
•
•
•
•
•
2D dipole vortex solutions in EMHD
perpendicular propagation
oblique propagation
numerical techniques
stability properties
dipole interactions
conclusions
Association EURATOM-FOM
FOM-Instituut voor Plasmafysica
Workshop on Nonlinear Structures in Magnetized Plasmas, Tarusa, Kaluga Region, Russia, 6-9 September 2000
E. Westerhof
Trilateral Euregio Cluster
TEC
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
• with inertial skin depth de = c/wpe
• with L = 1 + (wce / wpe)2
• evolution equations
W
 -[b, W] - [y, 2y]
t
Y
 -[b, Y]
t
• [f,g] = ez • (f  g)
Association EURATOM-FOM
FOM-Instituut voor Plasmafysica
Workshop on Nonlinear Structures in Magnetized Plasmas, Tarusa, Kaluga Region, Russia, 6-9 September 2000
E. Westerhof
Trilateral Euregio Cluster
TEC
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
Association EURATOM-FOM
FOM-Instituut voor Plasmafysica
Workshop on Nonlinear Structures in Magnetized Plasmas, Tarusa, Kaluga Region, Russia, 6-9 September 2000
E. Westerhof
Trilateral Euregio Cluster
TEC
Stationary propagating dipoles
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 
• Solutions of
(b - u y x )(r )  c1B1(k1r )  c2B1(k 2r )  c3r
y(r )  c4B1(k1r )  c5B1(k 2r )  c6r
• internal to separatrix, r < rs: B1 = J1 (k2<0), I1(k2>0),
external to separatrix, r > rs: B1 = K1(k2>0)
Association EURATOM-FOM
FOM-Instituut voor Plasmafysica
Workshop on Nonlinear Structures in Magnetized Plasmas, Tarusa, Kaluga Region, Russia, 6-9 September 2000
E. Westerhof
Trilateral Euregio Cluster
TEC
Perpendicular propagation by = 0
• external scale k12 = ke2 = (1+kn/uy)/L  uy<-kn; uy>0
• y = 0 in internal region
• identical to Larichev-Reznik modon of CharneyHasegawa-Mima equation
• y  0 in internal region
• superposition of two Larichev-Reznik modes
 additional separatrix in ‘stream function’ b - uy x
Association EURATOM-FOM
FOM-Instituut voor Plasmafysica
Workshop on Nonlinear Structures in Magnetized Plasmas, Tarusa, Kaluga Region, Russia, 6-9 September 2000
E. Westerhof
Trilateral Euregio Cluster
y  0, dipole with 2nd separatrix
TEC
Parameters: L = 1.8, kn = .002, uy = -.02, ke = 0.5
Association EURATOM-FOM
FOM-Instituut voor Plasmafysica
Workshop on Nonlinear Structures in Magnetized Plasmas, Tarusa, Kaluga Region, Russia, 6-9 September 2000
E. Westerhof
Trilateral Euregio Cluster
t = 100
Evolution of dipole with
2nd separatrix: instability
TEC
t = 200
contours of b - L2b, steps: .20
Association EURATOM-FOM
FOM-Instituut voor Plasmafysica
Workshop on Nonlinear Structures in Magnetized Plasmas, Tarusa, Kaluga Region, Russia, 6-9 September 2000
E. Westerhof
Trilateral Euregio Cluster
Oblique propagation by  0
TEC
• External scales:
( e) 2 1 
2
2
2
2 2
2
k1,2  1  ke - b y  1  ke - b y - 4ke 
2



• conditions: ke2 > 0, and (1- ke)2 > by2  by2 /Luy2
• no real k(e) solutions in incompressible, homogeneous case (i.e. case of pure whistler modes)
• extension of Larichev-Reznik modons to by  0
• consequently, dynamics is very similar to these
Association EURATOM-FOM
FOM-Instituut voor Plasmafysica
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
Trilateral Euregio Cluster
uy < 0, tilt-unstable dipole
TEC
Parameters:
L = 1.8;
kn = .01
by = .0022; uy = -.02
orbit:diamonds show
Dt =100
shown are contours of
b - L2b
steps: .05
Association EURATOM-FOM
FOM-Instituut voor Plasmafysica
Workshop on Nonlinear Structures in Magnetized Plasmas, Tarusa, Kaluga Region, Russia, 6-9 September 2000
E. Westerhof
Trilateral Euregio Cluster
Higher order dipoles with 2nd sep...
t=0
L = 1.8; kn = .01; by = .0022; uy = .02
TEC
t = 80
contours of b - L2b, steps: .10
Association EURATOM-FOM
FOM-Instituut voor Plasmafysica
Workshop on Nonlinear Structures in Magnetized Plasmas, Tarusa, Kaluga Region, Russia, 6-9 September 2000
E. Westerhof
Trilateral Euregio Cluster
… with 2nd separatrix are unstable
t = 120
TEC
t = 200
Association EURATOM-FOM
FOM-Instituut voor Plasmafysica
Workshop on Nonlinear Structures in Magnetized Plasmas, Tarusa, Kaluga Region, Russia, 6-9 September 2000
E. Westerhof
Trilateral Euregio Cluster
TEC
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
Association EURATOM-FOM
FOM-Instituut voor Plasmafysica
Workshop on Nonlinear Structures in Magnetized Plasmas, Tarusa, Kaluga Region, Russia, 6-9 September 2000
E. Westerhof
Trilateral Euregio Cluster
… collisions appear ‘soliton-like’, ...
t = 400
TEC
t = 600
Association EURATOM-FOM
FOM-Instituut voor Plasmafysica
Workshop on Nonlinear Structures in Magnetized Plasmas, Tarusa, Kaluga Region, Russia, 6-9 September 2000
E. Westerhof
Trilateral Euregio Cluster
…‘soliton like’, but for [y,2y]
t = 700
b - L2b, steps: .10
TEC
t = 700
Association EURATOM-FOM
FOM-Instituut voor Plasmafysica
y - 2y, steps: .001
Workshop on Nonlinear Structures in Magnetized Plasmas, Tarusa, Kaluga Region, Russia, 6-9 September 2000
E. Westerhof
Trilateral Euregio Cluster
TEC
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
Association EURATOM-FOM
FOM-Instituut voor Plasmafysica
Workshop on Nonlinear Structures in Magnetized Plasmas, Tarusa, Kaluga Region, Russia, 6-9 September 2000
E. Westerhof