Current sheet formation and magnetic reconnection at 3D null points

David PontinDavid Pontin

18th October 200618th October 2006

“ “There is a theory which states that if anyone There is a theory which states that if anyone ever discovers exactly what the Universe is ever discovers exactly what the Universe is for and why it is here, it will instantly for and why it is here, it will instantly disappear and be replaced by something even disappear and be replaced by something even more bizarre and inexplicable.more bizarre and inexplicable.

There is another theory which states that this There is another theory which states that this has already happened”has already happened”

Douglas Adams, The Restaurant at the Douglas Adams, The Restaurant at the End End of the of the UniverseUniverse

Collaborators: A. Bhattacharjee, K. Galsgaard (U. of Copenhagen)Collaborators: A. Bhattacharjee, K. Galsgaard (U. of Copenhagen)

Complex 3D magnetic fields

Complex, everchanging Complex, everchanging BB Very low dissipationVery low dissipation Where do Where do JJ sheets form? sheets form? locations of heating, energy locations of heating, energy

release, dynamic phenomenarelease, dynamic phenomena®

e.g. e.g. the the SunSun

Common approximation: 2D JJ sheets form at X-type nulls of sheets form at X-type nulls of BB in 2D: in 2D:

But nature is 3D!But nature is 3D!

Magnetic reconnection:Magnetic reconnection:

Sites of J sheet formation in 3D

(1) In absence of (1) In absence of nullsnulls

(2) 3D nulls – (2) 3D nulls – BB=0=0

(3) Separators(3) Separators

joining 2 such nulls joining 2 such nulls – often cited– often cited

Longcope & Cowley, 1996Longcope & Cowley, 1996

`Parker Mechanism’`Parker Mechanism’

3D null structure

Determine local structure of null by examining JacobianDetermine local structure of null by examining Jacobian Eigenvalues/eigenvectors determine spine/fan orientation Eigenvalues/eigenvectors determine spine/fan orientation

(Fukao (Fukao et al.et al. 1975; Parnell 1975; Parnell et al.et al. 1996) 1996)

3D nulls in nature Solar corona: 7-15 coronal nulls Solar corona: 7-15 coronal nulls

expected for every 100 photospheric expected for every 100 photospheric flux conc.s flux conc.s (Schrijver & Title 2002; Longcope (Schrijver & Title 2002; Longcope et al.et al. 2003; Close 2003; Close et et al.al. 2004) 2004)

Earth’s magnetosphere: standard Earth’s magnetosphere: standard model contains 2 nulls (dayside rec at model contains 2 nulls (dayside rec at separator??)separator??)

Also recent observations of nulls in Also recent observations of nulls in tail tail JJ sheet sheet (Xiao (Xiao et al.et al. 2006) 2006)

The laboratory The laboratory (Bogdanov (Bogdanov et al.et al. 1994) 1994)

Dome Dome topology:topology:

Ideal vs. non-ideal evolution

Curl Ohm’s law:Curl Ohm’s law:

Pure advection of Pure advection of BB: field: field

““frozen-in” to plasmafrozen-in” to plasma““Non-ideal” term: v. small Non-ideal” term: v. small

in almost all of Universein almost all of Universe

Plasma trapped Plasma trapped like beads on a wirelike beads on a wire

field lines cannot break or pass through since field lines cannot break or pass through since vv regular regular Energy stored in B by twisting/braiding of field lines, & Energy stored in B by twisting/braiding of field lines, &

also by stretching at hyperbolic field structuresalso by stretching at hyperbolic field structures Only when extremely strong currents build up can field Only when extremely strong currents build up can field

lines `slip’ through plasma & so break and `reconnect’lines `slip’ through plasma & so break and `reconnect’

Kinematics – non-ideal evolution

Evolution can be viewed as Evolution can be viewed as ideal if ideal if anyany ww exists satisfying exists satisfying

(Despite recent claims (Despite recent claims otherwise) can show that otherwise) can show that certain evolutions are certain evolutions are prohibited, e.g. prohibited, e.g.

For one special choice of BC’s, For one special choice of BC’s, ww only non-smooth only non-smooth

For all other BC’s, For all other BC’s, ww singular at spine or fan singular at spine or fan

®®

Study kinematic limit. Dynamics not included, but Study kinematic limit. Dynamics not included, but singularities point to singularities point to JJ sheets in dynamic regime. sheets in dynamic regime.

Kinematic solns. II -

(Pontin (Pontin et al.et al. 2005) 2005)

Field lines reconnect Field lines reconnect round spine / across fanround spine / across fan

Rec rate:Rec rate:

steady-state,steady-state,

localisedlocalised h

3D resistive MHD simulations Code developed by Nordlund, Galsgaard and co at Univ. Code developed by Nordlund, Galsgaard and co at Univ.

of Copenhagenof Copenhagen (Nordlund & Galsgaard 1997; Pontin & Galsgaard (Nordlund & Galsgaard 1997; Pontin & Galsgaard 2006)2006)

Initial equilibriumInitial equilibrium Boundaries line-tied, y Boundaries line-tied, y

& z ‘far’& z ‘far’

Current evolution

JJ associated with disturbance focuses at null associated with disturbance focuses at null

z=0 planez=0 plane

J contd.

JJxx solid line solid line

JJyy dotted dotted

JJzz dashed dashed

JJ component which grows component which grows is Jis Jzz - to fan, to shear - to fan, to shear

planeplane

JJ profile dependent on profile dependent on driving strengthdriving strength

^

vv00=0.001=0.001 v v00=0.01=0.01 vv00=0.04=0.04

Magnetic struc of J sheet Retain single nullRetain single null 3D sheet: 3D sheet: BBlines lines

exactly anti- along exactly anti- along z=0z=0

For For z>0z>0, , BByy

`discontinuous’; `discontinuous’; BBzz

smoothsmooth

Stagnation pt Stagnation pt flowflow

2D-like2D-like

Lorentz force Lorentz force accelerates flow; accelerates flow; pressure force pressure force opposes collapseopposes collapse

Flow - collapse

t=1.6t=1.6 t=2.4 t=2.4

t=3.0t=3.0 t=5.0 t=5.0

and reconnection Localised concentration Localised concentration

develops, centred on z-axisdevelops, centred on z-axis Peaks close to Peaks close to JJ peak peak J sheet & reconnection?J sheet & reconnection? `spine rec’ &`spine rec’ &

`fan rec’`fan rec’

EE

E ®

J sheet properties Peak current and Peak current and

rec rate scale rec rate scale linearly with linearly with driving vel (vdriving vel (v00))

Sheet dimensions Sheet dimensions also scale linearly also scale linearly with vwith v00..

Sheet properties II

Scaling of Scaling of JJ v.important v.important Does sheet continually grow Does sheet continually grow

when continually driven, when continually driven, reaching system size, or self reaching system size, or self regulated somehow?regulated somehow?

Seems to continually growSeems to continually grow

Sweet-Parker-likeSweet-Parker-like®

Effect of compressiblity – analytical solns Analytical incompressible Analytical incompressible

solns. – fan and spinesolns. – fan and spine

(Craig et al 1995; Craig & Fabling (Craig et al 1995; Craig & Fabling 1996, 1998)1996, 1998)

Assuming Assuming simplifies Eq.s.simplifies Eq.s.

Further simplification used:Further simplification used: fan casefan case

Background field 3D, disturbance fields of low Background field 3D, disturbance fields of low dimensionalitydimensionality

get J sheets of infinite extent – straight tubes along spine get J sheets of infinite extent – straight tubes along spine or infinite planes coincident with fanor infinite planes coincident with fan

Þ

Effect of compressibility - simulations Incompressible limit is (ideal gas )Incompressible limit is (ideal gas ) Even for , geometry of flow and Even for , geometry of flow and JJ sheet v.different: sheet v.different:

g®¥ 5 / 3g=10g=

Dynamic accessibility of incompressible solutions

Results for spine driving imply that incompressible `fan Results for spine driving imply that incompressible `fan current’ solns current’ solns areare dynamically accessible dynamically accessible

Driving the fan:Driving the fan: Very similar current concentration Very similar current concentration Increasing has similar effect to Increasing has similar effect to before; spine & fan do not collapse before; spine & fan do not collapse to same extentto same extent

BUTBUT

current spreads more along current spreads more along fanfan

g

Implies `spine current’ solns Implies `spine current’ solns are notare not dynamically dynamically accessible accessible (see also (see also Titov et al. 2005Titov et al. 2005))

Expect spine currents to result from rotations, spine Expect spine currents to result from rotations, spine J (Pontin & Galsgaard, 2006)(Pontin & Galsgaard, 2006)

Why are current sheets not linear & infinite extent for compressible case?

Analytical solns use 1D disturbance Analytical solns use 1D disturbance fieldsfields

J B´ IntenseIntense JJ in sheet generates massive in sheet generates massive

force perp. to sheet force perp. to sheet Must be balanced by Must be balanced by In steady state in sheetIn steady state in sheet t-dep solution: in sheet t-dep solution: in sheet

This pressure force implausibly This pressure force implausibly large, cannot be maintained in large, cannot be maintained in plasmas with realisticplasmas with realistic

PÑ

1/P h®Ñ

tP eaÑ

h

Summary

3D nulls may be important sites of rec and energy release 3D nulls may be important sites of rec and energy release in complex 3D magnetic fieldsin complex 3D magnetic fields

Certain evolutions of Certain evolutions of BB prohibited under ideal MHD prohibited under ideal MHD Under (shear) boundary driving, Under (shear) boundary driving, JJ sheet forms at null sheet forms at null Null spine and fan close up – 3D sheet forms at null with B Null spine and fan close up – 3D sheet forms at null with B

lines exactly anti-parallel at nulllines exactly anti-parallel at null Development of - reconnectionDevelopment of - reconnection Qualitative nature of sheet is Sweet-Parker-likeQualitative nature of sheet is Sweet-Parker-like In incompressible limit, morphology of sheet changesIn incompressible limit, morphology of sheet changes Analytical fan sheet solns realised, but not spine sheetAnalytical fan sheet solns realised, but not spine sheet

E

Thanks for listening!Thanks for listening!

“ “On the planet Earth, man had always assumed that he was more On the planet Earth, man had always assumed that he was more intelligent than dolphins because he had achieved so much – the wheel, intelligent than dolphins because he had achieved so much – the wheel, New York, wars and so on – whilst all the dolphins had ever done was New York, wars and so on – whilst all the dolphins had ever done was muck about in the water having a good time. But conversely, the muck about in the water having a good time. But conversely, the dolphins has always believed that they were more intelligent than man – dolphins has always believed that they were more intelligent than man – for precisely the same reasons.”for precisely the same reasons.”

Douglas Adams, The Hitchhiker’s Guide to the GalaxyDouglas Adams, The Hitchhiker’s Guide to the Galaxy

Kinematic solutions. I -

(Pontin (Pontin et al.et al. 2004) 2004)

Rotational flows and recRotational flows and rec

required by structure of required by structure of BB & & JJ¬