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

+

+

-

-