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

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Model and Parameters Simulations are done using the HYM code. Very simple physical model: resistive MHD equations plus the Hall term in the Ohms 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: d i =1, Z=R=0.2. Perfectly conducting boundary conditions. Numerical scheme: 4 th -order finite difference, explicit. Dimensionless parameters: η=0.001, µ=0.002, d i /R=0.03.

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Hall-MHD simulations with different B φ polarity Normal B φ Reversed B φ ψ ψ R Z R Z In HMHD simulations, the X-point position shifts outward by about 2-3d i when direction of toroidal field is reversed (t= 5 t A ).

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Hall-MHD simulations with different B φ polarity. J R (normal B φ )J R (reversed B φ ) RR V -shape current contours/\ -shape current contours ZZ t= 5 t A Difference in radial current contours.

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Hall-MHD simulations with different Bφ polarity. J φ (normal B φ )J φ (reversed B φ ) R t= 5 t A Toroidal current contours

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Hall-MHD simulations with different B φ polarity. Velocity profiles t= 10 t A Normal B φ V R (R) V φ (R) R Reversed B φ V R (R) V φ (R) R

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MHD Results (no Hall Effect) Ψ (reversed B φ ) Z J R (reversed B φ ) R V R (R) (reversed B φ ) R Ψ (normal B φ ) J R (normal B φ ) (normal B φ ) V R (R) Z

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3D plots of magnetic field lines (HMHD) R Z φ Normal B φ directionReversed B φ 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 t A

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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 t A Normal B φ directionReversed B φ direction

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Same effect in 2D HMHD reconnection results in quadrupole field VeVe 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 V e. The X-point shift should also depend on B φ /B pol ratio.

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Time evolution/reconnection rates in HMHD and MHD simulations (S=1000, d i /R=0.03) MHD HMHD HMHD(R) 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. t/t A

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Time evolution/reconnection rates in MHD simulations (S= ,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 [Biskamp80 and others]. t / t A 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 ExEx Reconnection rate

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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 B φ polarity - dependence on B φ /B pol 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, d i /R=0.03), the global dynamics/reconnection rates are not modified significantly by the Hall effects and/or by B φ polarity.

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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 B φ polarity These effects are related to generation of a quadrupole field in Hall-MHD. Similar effects are observed in MRX. Normal B φ Reversed B φ ψ ψ

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