Analytical and numerical models of field line behaviour in 3D reconnection David Pontin Solar MHD Theory Group, University of St. Andrews Collaborators: Klaus Galsgaard (Copenhagen) Gunnar Hornig (St Andrews) Eric Priest (St Andrews) Magnetic Reconnection Theory, Isaac Newton Institute, 17 th August 2004

Overview Review: properties of 3D kinematic rec. See: Review: properties of 3D kinematic rec. See:  Priest, E.R., G. Hornig and D.I. Pontin, On the nature of three- dimensional magnetic reconnection, J. Geophys. Res., 108, A7, SSH6 1-8,  G. Hornig and E.R. Priest, Evolution of magnetic flux in an isolated reconnection process, Physics of Plasmas 10(7), (2003)  Pontin, D. I., G. Hornig and E. R. Priest, Kinematic reconnection at a magnetic null point: Spine-aligned current, Geophys. Astrophys. Fluid Dynamics, in press (2004a)  Pontin, D. I., G. Hornig and E. R. Priest, Kinematic reconnection at a magnetic null point: Fan-aligned current, Geophys. Astrophys. Fluid Dynamics, submitted (2004b) motivating…… motivating…… Numerical simulation on 3D reconnection in the absence of a magnetic null (work in progress!!) Numerical simulation on 3D reconnection in the absence of a magnetic null (work in progress!!)

2D Reconnection: basic properties Bline velocity w, s.t. Bline velocity w, s.t. Bline mapping not continuous: break in diffusion region at X-point only. Bline mapping not continuous: break in diffusion region at X-point only. 1-1 correspondence of reconnecting Blines and flux tubes. 1-1 correspondence of reconnecting Blines and flux tubes.

3D Rec.: No w for isolated non-ideal region (D) in general no w s.t. in general no w s.t. Blines followed through D do not move at v outside D. Blines followed through D do not move at v outside D. Blines continually change their connections in D. Blines continually change their connections in D.

Analytical examples Solve kinematic steady resistive MHD eq.s: Solve kinematic steady resistive MHD eq.s: Resistive Ohm’s law Resistive Ohm’s law Maxwell’s eq.s; Maxwell’s eq.s; t-independent t-independent Impose B, then deduce E, v. Impose B, then deduce E, v. Assume localised Assume localised

rec Counter-rotating flows Diffusion region Impose Impose Source of rotation: consider pot. drop round loop Source of rotation: consider pot. drop round loop

Flux tube rec: Splitting and flipping, Splitting and flipping, no rejoining of flux tubes no rejoining of flux tubes

2D vs 3D Occurs: Occurs: at nulls at nulls or not Bline Bline mapping mappingdiscontinuous continuous (except at separatrices) Unique Bline Unique Bline velocity velocity exists (singular at X-point) does not exist Hence: Change of Change of connections connections Blines break at one point (X) continual & continuous through D Beyond D: Beyond D: Blines move at v move at 1-1 Bline rec. 1-1 Bline rec.yes in general, no

New properties to look for in dynamical numerical expt: Blines split immediately on entering non-ideal region (D). Blines split immediately on entering non-ideal region (D). Blines continually & continuously change connections in D. Blines continually & continuously change connections in D. : mismatching : mismatching Counter-rotating flows above and below D. Counter-rotating flows above and below D. Non-existence of perfectly-reconnecting Blines. Non-existence of perfectly-reconnecting Blines.

Numerical Experiment Code: HPF Code: HPF Eqs. Eqs. Staggered grids: b.c., f.c., e.c. Staggered grids: b.c., f.c., e.c. 6 th order derivative algorithm (+ 5 th order interp.) 6 th order derivative algorithm (+ 5 th order interp.) 3 rd order predictor-corrector in time 3 rd order predictor-corrector in time BCs dealt with using ghost zones, periodic b.c.`s in 2 dir.s & driven in other BCs dealt with using ghost zones, periodic b.c.`s in 2 dir.s & driven in other

Initial setup : two flux patches on top and bottom, rotated w.r.t. each other + background : two flux patches on top and bottom, rotated w.r.t. each other + background Calculate potential field in domain Calculate potential field in domain Driving on boundaries moves patches to joining lines; sinusoidal profile Driving on boundaries moves patches to joining lines; sinusoidal profile

B in domain in volume ‘hyperbolic flux tube’ in volume ‘hyperbolic flux tube’ 2D X-pt, uni-dir. comp. 2D X-pt, uni-dir. comp. Generalisation of separator intersection of 2 QSLs Generalisation of separator intersection of 2 QSLs Topologically simple Topologically simple Geometrically complex Geometrically complex Twist induces strong Twist induces strong V.S. Titov, K. Galsgaard and T. Neukirch, Astrophys. J. 582, , V.S. Titov, K. Galsgaard and T. Neukirch, Astrophys. J. 582, , K. Galsgaard, V.S. Titov and T. Neukirch, Astrophys. J. 595, , K. Galsgaard, V.S. Titov and T. Neukirch, Astrophys. J. 595, , 2003.

Different expts. Cold plasma / full MHD Cold plasma / full MHD 1. Fixed localised resistivity: 1. Fixed localised resistivity: 2. `Anomalous resistivity’, dep on J 2. `Anomalous resistivity’, dep on J

Induced plasma velocity Stagnation pt. v in central plane: Stagnation pt. v in central plane: `Pinching’ `Pinching’ Also have up/down flow ~1/3 of strength Also have up/down flow ~1/3 of strength Strong outflows suggest rec. Strong outflows suggest rec.

Induced Current (central plane) concentration, centred on axis, grows steadily. concentration, centred on axis, grows steadily. `Wings’ mark outflow jets `Wings’ mark outflow jets remains well resolved

Behaviour of lines Following circular X-sections of Blines in inflow: Following circular X-sections of Blines in inflow:

-flowlines Choose Blines initially joined & follow from both ends. Paths of intersections with central plane. cf. Choose Blines initially joined & follow from both ends. Paths of intersections with central plane. cf. Blines split on entering D. Blines split on entering D. Flow lines coloured to show speed. Flow lines coloured to show speed.

Rot flows w maps similar to kin. solns: background rot? w maps similar to kin. solns: background rot? Calc above/below D Calc above/below D sign agrees with kinematic model for J dir

Importance of Parallel Electric Field v. important for Bline rec v. important for Bline rec movie moviemovie Flow lines coloured with

Importance of Parallel Electric Field II Surface of in central plane. Surface of in central plane. Profile of along selection of Blines Profile of along selection of Blines Localisation in plane elongated along conc. Localisation in plane elongated along conc. Struc simple- monotonic decay away from O Struc simple- monotonic decay away from O

Expt 2 Initial/boundary cond.s same Initial/boundary cond.s same = 1.5 / 2.5 / 3.5 = 1.5 / 2.5 / 3.5

J As before, but wings develop only when As before, but wings develop only when rec. delayed until system sufficiently stressed. rec. delayed until system sufficiently stressed. Width of J conc same as before Width of J conc same as before

Non-ideal regions Isosurf.s of at 25% of max: Fixed Switch-on

v Stag flow, but jets only develop later Stag flow, but jets only develop later up/down flow marks J wings up/down flow marks J wings large extent in plane large extent in plane rot flows still present- also larger extent rot flows still present- also larger extent

Bline rec pattern of rec similar pattern of rec similar not as well localised along B not as well localised along B

w-flows region of w-splitting & squashed & stretched region of w-splitting & squashed & stretched Nature of mis- matching same Nature of mis- matching same

Conclusions Shows rec in HFT Shows rec in HFT Full MHD simulation- qualitative agreement with kinematic model for rec. Full MHD simulation- qualitative agreement with kinematic model for rec.  rotational flows  nature of w mis-matching Qualitatively similar for fixed/`anomalous’ Qualitatively similar for fixed/`anomalous’ Bline rec continual & continuous in non-id region Bline rec continual & continuous in non-id region Flux evolution requires TWO Bline velocities Flux evolution requires TWO Bline velocities

Null rec: spine Induced plasma flow rotational: Induced plasma flow rotational: Blines rec. in `shells’ Blines rec. in `shells’ Source of rot. flows Source of rot. flows

Flux tube rec.: J parallel to spine Add ideal stag. flow to see global effect Add ideal stag. flow to see global effect Tubes split entering D, Tubes split entering D, flip, but do not rejoin. flip, but do not rejoin. No v across spine/fan No v across spine/fan

Null rec: fan has different signs for x +ve / -ve. So unidirectional across fan. has different signs for x +ve / -ve. So unidirectional across fan. across fan, opposite sign for y +ve / -ve across fan, opposite sign for y +ve / -ve

Flux tube rec.: J parallel to fan Plasma crosses spine and fan Plasma crosses spine and fan Split, flip, no rejoin. Split, flip, no rejoin. Flux transported across fan at finite rate. Flux transported across fan at finite rate.

Results so far Confirmed: Confirmed:  splitting of lines on entering non-ideal region  continual reconnection in non-ideal region: non-existence of unique Bline velocity  existence of rot flows Importance of parallel electric field for process, simple structure of profile. Importance of parallel electric field for process, simple structure of profile.