Analytical and numerical models of field line behaviour in 3D reconnection David Pontin David Pontin Solar MHD Theory Group, University of St. Andrews Solar MHD Theory Group, University of St. Andrews Collaborators: Klaus Galsgaard Collaborators: Klaus Galsgaard (Copenhagen) (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, 17 Isaac Newton Institute, 17 th th August 2004 August 2004
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Analytical and numerical models of field line behaviour in 3D reconnection
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, - PowerPoint PPT Presentation
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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..
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
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
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.