Hall-MHD simulations of counter-helicity spheromak merging
by E. Belova
PPPLOctober 6, 2005
CMSO General Meeting
Model and Parameters
• Simulations are done using the HYM code.• Very simple physical model: resistive MHD equations plus the Hall
term in the Ohm’s law (the only 2-fluid effect which is included).
• Zero electron inertia is assumed.• Hyperresistivity is used to stabilize Hall effects on small scales.• HYM code uses (Z x R x φ)= 385 x 127 x 16 grid.
Length is normalized to ion skin depth: di=1, ∆Z=∆R=0.2.• Perfectly conducting boundary conditions.• Numerical scheme: 4th-order finite difference, explicit.
• Dimensionless parameters: η=0.001, µ=0.002, di/R=0.03.
JB/JBvE ηn
Hall-MHD simulations with different Bφ polarity
Normal Bφ Reversed Bφ
-10 -5 0 5 10
-10 -5 0 5 10
ψ ψR
Z
R
Z
In HMHD simulations, the X-point position shifts outward by about 2-3di when direction of toroidal field is reversed (t= 5 tA).
Hall-MHD simulations with different Bφ polarity.
JR (normal Bφ) JR (reversed Bφ)
-20 -10 -5 0 5 10 20
R R
0
28
0
28
-20 -10 -5 0 5 10 20
V -shape current contours /\ -shape current contours
Z Z
t= 5 tA
Difference in radial current contours.
Hall-MHD simulations with different Bφ polarity.
Jφ (normal Bφ) Jφ (reversed Bφ)
R
t= 5 tA
Toroidal current contours
Hall-MHD simulations with different Bφ polarity.Velocity profiles
t= 10 tA
Normal Bφ
VR(R)
Vφ(R)
R
0.15
0.0
0.3
0.0
-0.1
Reversed Bφ
VR(R)
Vφ(R)
R
0.0
-0.15
0.2
0.0
-0.2
MHD Results (no Hall Effect)
-10 -5 0 5 10
Ψ (reversed Bφ)
Z
JR (reversed Bφ)
R
VR(R)
(reversed Bφ)
R
Ψ (normal Bφ) JR (normal Bφ) (normal Bφ)
VR(R)
-10 -5 0 5 10Z
0.06
0.0
-0.04
0.06
0.0
-0.04
3D plots of magnetic field lines (HMHD)
R
Zφ
Normal Bφ direction Reversed Bφ direction
Field lines near x-point are bent by the electron flows. The local field line structure explains the observed shift in x-point position, and the ion flow profiles.
t= 10 tA
3D plots of magnetic field lines (MHD)
For comparison, field line bending is not seen in the MHD simulations – reconnection occurs in a plane => current and flow profiles are approximately symmetric (up-down), and there is no radial shift in X-point position.
t= 10 tA
Normal Bφ direction Reversed Bφ direction
Same effect in 2D HMHD reconnection results in “quadrupole” field
Ve
• In 2D reconnection everything remains symmetric (no guide field).
• In counter-helicity reconnection, X-point shifts radialy because the reconnection ‘plane’ is tilted relative to R-Z plane. It shifts inward or outward depending on the sign of radial component of Ve. The X-point shift should also depend on Bφ/Bpol ratio.
Time evolution/reconnection rates in HMHD and MHD simulations (S=1000, di/R=0.03)
MHD
HMHD
HMHD(R)
Time evolution of toroidal field energy (and reconnection rates) are very similarin MHD and Hall-MHD simulations and for different initial field polarity –> it is determined by global (ion) dynamics, and does not depend on the local fieldstructure near the X-point.
t/tA t/tA
Time evolution/reconnection rates in MHD simulations (S=1000-20,000)
• Driven reconnection with η-independent peak reconnection rate for range of η>2·10-4
• Reconnection slows down for smaller η due to magnetic field pressure build up and “sloshing”, similar to island coalescence problem [Biskamp’80 and others].
t / tA t / tA
η=10-3
η=2·10-4
η=5·10-4
η=5·10-5
η=10-4
η=10-3
η=2·10-4
η=5·10-4
η=5·10-5
η=10-4
Ex
Reconnection rate
Summary
• In the counter-helicity spheromak merging new signatures of Hall reconnection have been identified:
- inward/outward radial shift of the x-point
- nearly unidirectional radial ion flow (positive/negative)
- ‘V’ or ‘/\’ -shaped radial current contours
- dependence of the above signatures on the Bφ polarity
- dependence on Bφ/Bpol ratio (not studied yet)• These effects are related to generation of a quadrupole field in Hall-
MHD.
• For the same set of parameters (S=1000, di/R=0.03), the global dynamics/reconnection rates are not modified significantly by the Hall effects and/or by Bφ polarity.
Hall-MHD simulations of counter-helicity spheromak merging E. Belova, PPPL
•In counter-helicity spheromak merging simulations new signatures of Hall reconnection have been identified:
- inward/outward radial shift of the x-point
- nearly unidirectional radial ion flow (positive/negative)
- ‘V’ or ‘/\’ -shaped radial current contours
- dependence of the above signatures on the Bφ polarity•These effects are related to generation of a quadrupole field in Hall-MHD.•Similar effects are observed in MRX.
Normal Bφ
Reversed Bφ
ψ
ψ
+
+
-
-