Analytical and numerical models of field line behaviour in 3D reconnection

Analytical and numerical models of field line behaviour in 3D reconnection

David PontinDavid PontinSolar MHD Theory Group, University of St. AndrewsSolar MHD Theory Group, University of St. Andrews

Collaborators: Klaus Galsgaard (Copenhagen)Collaborators: Klaus Galsgaard (Copenhagen)Gunnar Hornig (St Andrews)Gunnar Hornig (St Andrews)

Eric Priest (St Andrews) Eric Priest (St Andrews)

Magnetic Reconnection Theory,Magnetic Reconnection Theory,

Isaac Newton Institute, 17Isaac Newton Institute, 17thth August 2004 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, Priest, E.R., G. Hornig and D.I. Pontin, On the nature of three-On the nature of three-dimensional magnetic reconnection, dimensional magnetic reconnection, J. Geophys. Res.J. Geophys. Res., , 108108, A7, SSH6 1-8, , A7, SSH6 1-8, 2003.2003.

G. Hornig and E.R. Priest, Evolution of magnetic flux in an isolated G. Hornig and E.R. Priest, Evolution of magnetic flux in an isolated reconnection process, reconnection process, Physics of PlasmasPhysics of Plasmas 1010(7), 2712-2721 (2003) (7), 2712-2721 (2003)

Pontin, D. I., G. Hornig and E. R. Priest, Kinematic reconnection at a Pontin, D. I., G. Hornig and E. R. Priest, Kinematic reconnection at a magnetic null point: Spine-aligned current, magnetic null point: Spine-aligned current, Geophys. Astrophys. Fluid Geophys. Astrophys. Fluid Dynamics,Dynamics, in press (2004a) in press (2004a)

Pontin, D. I., G. Hornig and E. R. Priest, Kinematic reconnection at a Pontin, D. I., G. Hornig and E. R. Priest, Kinematic reconnection at a magnetic null point: Fan-aligned current, magnetic null point: Fan-aligned current, Geophys. Astrophys. Fluid Geophys. Astrophys. Fluid Dynamics,Dynamics, submitted (2004b) submitted (2004b)

motivating……motivating…… Numerical simulation on 3D reconnection in the Numerical simulation on 3D reconnection in the

absence of a magnetic null absence of a magnetic null (work in progress!!)(work in progress!!)

2D Reconnection: basic properties BBline velocity line velocity ww, ,

s.t.s.t.

BBline mapping line mapping notnot

continuous: break continuous: break in diffusion region in diffusion region at X-point at X-point onlyonly..

1-1 correspondence 1-1 correspondence of reconnecting of reconnecting BBlines and flux lines and flux tubes.tubes.

$

0E w B+ ´ =

3D Rec.: No w for isolated non-ideal region (D) in general in general nono ww s.t. s.t.

BBlines followed lines followed through through DD do do notnot move at move at vv outside outside DD..

BBlines lines continuallycontinually change their change their connections in connections in DD..

$0E w B+ ´ =

Þ

Analytical examples Solve kinematic steady resistive MHD eq.s:Solve kinematic steady resistive MHD eq.s: Resistive Ohm’s lawResistive Ohm’s law

Maxwell’s eq.s;Maxwell’s eq.s; t-independentt-independent

Impose Impose BB, then deduce , then deduce EE,, vv. .

Assume localisedAssume localised

0

0

0

E v B J

E

B

B J

h

m

+ ´ =

Ñ´ =

Ñ× =

Ñ´ =

h

h

0 0( , , ),B B y kx b=

recrec

Counter-rotating flowsCounter-rotating flows

2 2 20( )

0A x y B seh h - + +=

Diffusion regionDiffusion region

ImposeImpose

0B ¹

Source of rotation: consider Source of rotation: consider pot. drop round looppot. drop round loop

Flux tube rec:

Splitting and Splitting and flipping,flipping,

no rejoining of no rejoining of flux tubesflux tubes

0B ¹

2D vs 3D Occurs:Occurs: at nullsat nulls at nulls or notat nulls or not

BBlineline

mappingmapping

discontinuousdiscontinuous continuous (except at continuous (except at separatrices)separatrices)

Unique Unique BBlineline

velocityvelocity

exists (singular at exists (singular at X-point)X-point)

does not existdoes not exist

Hence:Hence:

Change of Change of

connectionsconnections

BBlines break at one lines break at one point (X)point (X)

continual & continual & continuous through Dcontinuous through D

Beyond D:Beyond D: BBlines move at lines move at vv move at move at

1-1 1-1 BBline rec.line rec. yesyes in general, noin general, no

iw v¹

New properties to look for in dynamical numerical expt:

BBlines split immediately on entering non-ideal lines split immediately on entering non-ideal region (region (DD).).

BBlines continually & continuously change lines continually & continuously change connections in connections in DD..

: mismatching: mismatching Counter-rotating flows above and below Counter-rotating flows above and below DD.. Non-existence of perfectly-reconnecting Non-existence of perfectly-reconnecting BBlines.lines.

,in outw w

Numerical Experiment Code: HPFCode: HPF Eqs.Eqs.

Staggered grids: b.c., f.c., e.c.Staggered grids: b.c., f.c., e.c. 66thth order derivative algorithm (+ 5 order derivative algorithm (+ 5 thth order interp.) order interp.) 33rdrd order predictor-corrector in time order predictor-corrector in time BCs dealt with using ghost zones, periodic b.c.`s BCs dealt with using ghost zones, periodic b.c.`s

in 2 dir.s & driven in otherin 2 dir.s & driven in other

,er ,B vr ,E J

.( )vt

rr

¶=- Ñ

¶B

Et

¶=- Ñ´

¶J B=Ñ´

E v B Jh+ ´ = ( ).v

vv P J Bt

rr t

¶=- Ñ + - Ñ + ´

¶.( ) . visc joule

eev P v Q Q

t

¶=Ñ - Ñ + +

¶

Initial setup : two flux : two flux

patches on top and patches on top and bottom, rotated bottom, rotated w.r.t. each other + w.r.t. each other + backgroundbackground

B

90

vr^

3 3 3(64 /128 / 256 )

Calculate potential field in Calculate potential field in domaindomain

Driving on boundaries Driving on boundaries moves patches to moves patches to joining lines; sinusoidal joining lines; sinusoidal profile profile

B in domain in volume ‘hyperbolic in volume ‘hyperbolic

flux tube’flux tube’ 2D X-pt, uni-dir. comp.2D X-pt, uni-dir. comp. Generalisation of separator Generalisation of separator

intersection of 2 QSLsintersection of 2 QSLs Topologically simpleTopologically simple Geometrically complexGeometrically complex Twist induces strongTwist induces strong

B

J

V.S. Titov, K. Galsgaard and T. Neukirch, V.S. Titov, K. Galsgaard and T. Neukirch, Astrophys. J.Astrophys. J. 582582, 1172-1189, 2003., 1172-1189, 2003.

K. Galsgaard, V.S. TitovK. Galsgaard, V.S. Titov and T. Neukirch,and T. Neukirch, Astrophys. J.Astrophys. J. 595595, 506-516, 2003. , 506-516, 2003.

^

Different expts.

2

0Areh h -=

Cold plasma / full MHDCold plasma / full MHD 1. Fixed localised resistivity:1. Fixed localised resistivity:

2. `Anomalous resistivity’, dep on 2. `Anomalous resistivity’, dep on JJ2 2

00 02

,

0,

J JJ J

Jotherwise

hh

ìï -ïï >ï=íïïïïî

0e

t t

ræ ö¶ ¶ ÷ç = = ÷ç ÷çè ø¶ ¶

Induced plasma velocity

Stagnation pt. Stagnation pt. vv in central in central plane:plane:

`Pinching’`Pinching’ Also have Also have

up/down flow up/down flow ~1/3 of ~1/3 of strengthstrength

Strong Strong outflows outflows suggest rec.suggest rec.

Induced Current (central plane)

concentration, centred on concentration, centred on axis, grows steadily.axis, grows steadily.

`Wings’ mark outflow jets`Wings’ mark outflow jets

J

remains remains well well

resolvedresolved

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

B

-flowlines Choose Choose BBlines initially joined & follow from both lines initially joined & follow from both

ends. Paths of intersections with central plane. cf.ends. Paths of intersections with central plane. cf.

w

BBlines split on lines split on entering entering DD..

Flow lines Flow lines coloured to coloured to show speed.show speed.

Rot flows ww maps similar to kin. solns: background rot? maps similar to kin. solns: background rot? Calc above/below Calc above/below DDv dl×òÑ

0.01r =

0.1r =

0.2r =

sign agrees with kinematicsign agrees with kinematic

model for model for JJ dir dir

Importance of Parallel Electric Field v. important for v. important for BBline recline recE

movie

Eò

Flow lines Flow lines

coloured withcoloured with

Importance of Parallel Electric Field II Surface of in Surface of in

central plane.central plane. Profile of along Profile of along

selection of selection of BBlineslines

Eò

E

Localisation in plane Localisation in plane elongated along conc.elongated along conc.

Struc simple- monotonic Struc simple- monotonic decay away from Odecay away from O

J

Expt 2 Initial/boundary cond.s sameInitial/boundary cond.s same

= 1.5 / 2.5 / 3.5= 1.5 / 2.5 / 3.5

2 20

0 02,

0,

J JJ J

Jotherwise

hh

ìï -ïï >ï=íïïïïî

0J

J As before, but As before, but

wings develop only wings develop only whenwhen

rec. delayed until rec. delayed until system sufficiently system sufficiently stressed.stressed.

Width of Width of JJ conc conc same as before same as before

0J J?

®

® ®

0 2.5J =

0 1.5J =

0 3.5J =

fixed h-

Non-ideal regionsIsosurf.s of at 25% of max:Isosurf.s of at 25% of max:

Fixed Fixed Switch-onSwitch-onh hJh

v Stag flow, but Stag flow, but

jets only jets only develop laterdevelop later

up/down flow up/down flow marks marks JJ wings wings

large extent in large extent in planeplane

rot flows still rot flows still present- also present- also larger extentlarger extent

Bline rec

pattern of pattern of rec similarrec similar

not as not as well well localised localised along along BB

E

w-flows

region of region of ww-splitting -splitting & & squashed squashed & & stretchedstretched

Nature of Nature of mis-mis-matching matching samesame

E

Conclusions Shows rec in HFTShows rec in HFT Full MHD simulation- qualitative agreement with Full MHD simulation- qualitative agreement with

kinematic model for rec.kinematic model for rec. rotational flowsrotational flows nature of nature of ww mis-matching mis-matching

Qualitatively similar for fixed/`anomalous’Qualitatively similar for fixed/`anomalous’ BBline rec continual & continuous in non-id regionline rec continual & continuous in non-id region Flux evolution requires TWO Flux evolution requires TWO BBline velocitiesline velocities

0B ¹

h

Null rec: spine Induced plasma flow Induced plasma flow

rotational:rotational:

Blines rec. in `shells’Blines rec. in `shells’ Source of rot. flowsSource of rot. flows

J P

Flux tube rec.: J parallel to spine

Add ideal stag. Add ideal stag. flow to see flow to see global effectglobal effect

Tubes split Tubes split entering D, entering D,

flip, but do flip, but do notnot rejoin.rejoin.

No No vv across across spine/fanspine/fan

Null rec: fan

,J BdshF = ×ò J J x= $ Þ F has different signs for has different signs for xx +ve / -ve. So +ve / -ve. So unidirectional across fan.unidirectional across fan.

0zvÞ ¹

xE

J P

across fan, across fan, opposite sign opposite sign for for yy +ve / -ve +ve / -ve

Flux tube rec.: J parallel to fan

Plasma Plasma crosses spine crosses spine and fanand fan

Split, flip, no Split, flip, no rejoin.rejoin.

Flux Flux transported transported across fan at across fan at finite rate.finite rate.

Results so far Confirmed:Confirmed:

splitting of lines on entering non-ideal regionsplitting of lines on entering non-ideal region continual reconnection in non-ideal region: continual reconnection in non-ideal region:

non-existence of unique non-existence of unique BBline velocityline velocity existence of rot flowsexistence of rot flows

Importance of parallel electric field for process, Importance of parallel electric field for process, simple structure of profile.simple structure of profile.

B