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Hall-MHD simulations of counter-helicity spheromak merging

CMSO General Meeting. Hall-MHD simulations of counter-helicity spheromak merging. by E. Belova PPPL October 6, 2005. Model and Parameters. Simulations are done using the HYM code.

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Hall-MHD simulations of counter-helicity spheromak merging

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  1. CMSO General Meeting Hall-MHD simulations of counter-helicity spheromak merging by E. Belova PPPL October 6, 2005

  2. 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.

  3. Hall-MHD simulations with different Bφpolarity Normal Bφ Reversed Bφ ψ ψ R R -10 -5 0 5 10 Z -10 -5 0 5 10 Z In HMHD simulations, the X-point position shifts outward by about 2-3di when direction of toroidal field is reversed (t= 5 tA).

  4. Hall-MHD simulations with different Bφpolarity. Difference in radial current contours. JR (normal Bφ) JR (reversed Bφ) 28 28 R R 0 0 -20 -10 -5 0 5 10 20 -20 -10 -5 0 5 10 20 Z Z V -shape current contours /\ -shape current contours t= 5 tA

  5. Hall-MHD simulations with different Bφ polarity. Toroidal current contours Jφ(normal Bφ) Jφ (reversed Bφ) R t= 5 tA

  6. Hall-MHD simulations with different Bφpolarity.Velocity profiles Normal Bφ Reversed Bφ 0.15 VR(R) VR(R) 0.0 0.0 -0.15 0.3 0.2 Vφ(R) Vφ(R) 0.0 0.0 -0.2 -0.1 R R t= 10 tA

  7. MHD Results (no Hall Effect) Ψ(normal Bφ) JR(normal Bφ) (normal Bφ) 0.06 VR(R) R 0.0 -0.04 -10 -5 0 5 10 Z Ψ(reversed Bφ) JR(reversed Bφ) (reversed Bφ) 0.06 VR(R) R 0.0 -0.04 -10 -5 0 5 10 Z

  8. 3D plots of magnetic field lines (HMHD) Normal Bφdirection Reversed Bφdirection t= 10 tA R φ Z 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.

  9. 3D plots of magnetic field lines (MHD) Normal Bφdirection Reversed Bφdirection t= 10 tA 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.

  10. 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.

  11. Time evolution/reconnection rates in HMHD and MHD simulations (S=1000, di/R=0.03) MHD HMHD HMHD(R) t/tA t/tA Time evolution of toroidal field energy (and reconnection rates) are very similar in 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 field structure near the X-point.

  12. 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]. Reconnection rate η=5·10-4 η=2·10-4 η=10-3 η=5·10-4 η=10-4 Ex η=5·10-5 η=2·10-4 η=5·10-5 η=10-3 η=10-4 t / tA t / tA

  13. 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 Bφpolarity - 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 Bφpolarity.

  14. Hall-MHD simulations of counter-helicity spheromak merging E. Belova, PPPL Normal Bφ • 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 Bφpolarity • These effects are related to generation of a quadrupole field in Hall-MHD. • Similar effects are observed in MRX. - + ψ - + ψ Reversed Bφ

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