1. Current sheet formation and magnetic reconnection at 3D null points David Pontin 18th October 2006 “There is a theory which states that if anyone ever discovers exactly what the Universe is for and why it is here, it will instantly disappear and be replaced by something even more bizarre and inexplicable. “There is a theory which states that if anyone ever discovers exactly what the Universe is for and why it is here, it will instantly disappear and be replaced by something even more bizarre and inexplicable. There is another theory which states that this has already happened” There is another theory which states that this has already happened” Douglas Adams, The Restaurant at the End of the Universe Collaborators: A. Bhattacharjee, K. Galsgaard (U. of Copenhagen)

2. Complex 3D magnetic fields Complex, everchanging B Complex, everchanging B Very low dissipation Very low dissipation Where do J sheets form? Where do J sheets form? locations of heating, energy release, dynamic phenomena locations of heating, energy release, dynamic phenomena e.g. the Sun

3. Common approximation: 2D J sheets form at X-type nulls of B in 2D: J sheets form at X-type nulls of B in 2D: But nature is 3D! But nature is 3D! Magnetic reconnection: Magnetic reconnection:

4. Sites of J sheet formation in 3D (1) In absence of nulls (2) 3D nulls – B=0 (3) Separators joining 2 such nulls – often cited Longcope & Cowley, 1996 `Parker Mechanism’

5. 3D null structure Determine local structure of null by examining Jacobian Determine local structure of null by examining Jacobian Eigenvalues/eigenvectors determine spine/fan orientation Eigenvalues/eigenvectors determine spine/fan orientation (Fukao et al. 1975; Parnell et al. 1996)

6. 3D nulls in nature Solar corona: 7-15 coronal nulls expected for every 100 photospheric flux conc.s Solar corona: 7-15 coronal nulls expected for every 100 photospheric flux conc.s (Schrijver & Title 2002; Longcope et al. 2003; Close et al. 2004) Earth’s magnetosphere: standard model contains 2 nulls (dayside rec at separator??) Earth’s magnetosphere: standard model contains 2 nulls (dayside rec at separator??) Also recent observations of nulls in tail J sheet (Xiao et al. 2006) The laboratory (Bogdanov et al. 1994) The laboratory (Bogdanov et al. 1994) Dome topology:

7. Ideal vs. non-ideal evolution Curl Ohm’s law: Curl Ohm’s law: Pure advection of B: field “frozen-in” to plasma “Non-ideal” term: v. small in almost all of Universe Plasma trapped like beads on a wire Plasma trapped like beads on a wire field lines cannot break or pass through since v regular Energy stored in B by twisting/braiding of field lines, & also by stretching at hyperbolic field structures Energy stored in B by twisting/braiding of field lines, & also by stretching at hyperbolic field structures Only when extremely strong currents build up can field lines `slip’ through plasma & so break and `reconnect’ Only when extremely strong currents build up can field lines `slip’ through plasma & so break and `reconnect’

8. Kinematics – non-ideal evolution Evolution can be viewed as ideal if any w exists satisfying Evolution can be viewed as ideal if any w exists satisfying (Despite recent claims otherwise) can show that certain evolutions are prohibited, e.g. (Despite recent claims otherwise) can show that certain evolutions are prohibited, e.g. For one special choice of BC’s, w only non-smooth For all other BC’s, w singular at spine or fan Study kinematic limit. Dynamics not included, but singularities point to J sheets in dynamic regime. Study kinematic limit. Dynamics not included, but singularities point to J sheets in dynamic regime.

9. Kinematic solns. II - (Pontin et al. 2005) Field lines reconnect round spine / across fan Field lines reconnect round spine / across fan Rec rate: Rec rate: steady-state,localised

10. 3D resistive MHD simulations Code developed by Nordlund, Galsgaard and co at Univ. of Copenhagen (Nordlund & Galsgaard 1997; Pontin & Galsgaard 2006) Code developed by Nordlund, Galsgaard and co at Univ. of Copenhagen (Nordlund & Galsgaard 1997; Pontin & Galsgaard 2006) Initial equilibrium Initial equilibrium Boundaries line-tied, y & z ‘far’ Boundaries line-tied, y & z ‘far’

11. Current evolution J associated with disturbance focuses at null J associated with disturbance focuses at null z=0 plane

12. J contd. J x solid line J y dotted J z dashed J component which grows is J z - to fan, to shear plane J component which grows is J z - to fan, to shear plane J profile dependent on driving strength J profile dependent on driving strength v 0 =0.001 v 0 =0.01v 0 =0.04

13. Magnetic struc of J sheet Retain single null Retain single null 3D sheet: Blines exactly anti- along z=0 3D sheet: Blines exactly anti- along z=0 For z>0, B y `discontinuous’; B z smooth For z>0, B y `discontinuous’; B z smooth

14. Stagnation pt flow Stagnation pt flow 2D-like 2D-like Lorentz force accelerates flow; pressure force opposes collapse Lorentz force accelerates flow; pressure force opposes collapse Flow - collapse t=1.6 t=2.4 t=3.0 t=5.0

15. and reconnection Localised concentration develops, centred on z-axis Localised concentration develops, centred on z-axis Peaks close to J peak Peaks close to J peak J sheet & reconnection? J sheet & reconnection? `spine rec’ & `spine rec’ & `fan rec’

16. J sheet properties Peak current and rec rate scale linearly with driving vel (v 0 ) Peak current and rec rate scale linearly with driving vel (v 0 ) Sheet dimensions also scale linearly with v 0. Sheet dimensions also scale linearly with v 0.

17. Sheet properties II Scaling of J v.important Scaling of J v.important Does sheet continually grow when continually driven, reaching system size, or self regulated somehow? Does sheet continually grow when continually driven, reaching system size, or self regulated somehow? Seems to continually grow Seems to continually growSweet-Parker-like

18. Effect of compressiblity – analytical solns Analytical incompressible solns. – fan and spine Analytical incompressible solns. – fan and spine (Craig et al 1995; Craig & Fabling 1996, 1998) Assuming simplifies Eq.s. Assuming simplifies Eq.s. Further simplification used: Further simplification used: fan case Background field 3D, disturbance fields of low dimensionality Background field 3D, disturbance fields of low dimensionality get J sheets of infinite extent – straight tubes along spine or infinite planes coincident with fan

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

20. Dynamic accessibility of incompressible solutions Results for spine driving imply that incompressible `fan current’ solns are dynamically accessible Results for spine driving imply that incompressible `fan current’ solns are dynamically accessible Driving the fan: Driving the fan: Very similar current concentration Very similar current concentration Increasing has similar effect to before; spine & fan do not collapse to same extent Increasing has similar effect to before; spine & fan do not collapse to same extentBUT current spreads more along fan Implies `spine current’ solns are not dynamically accessible (see also Titov et al. 2005) Implies `spine current’ solns are not dynamically accessible (see also Titov et al. 2005) Expect spine currents to result from rotations, spine Expect spine currents to result from rotations, spine (Pontin & Galsgaard, 2006)

21. Why are current sheets not linear & infinite extent for compressible case? Analytical solns use 1D disturbance fields Analytical solns use 1D disturbance fields Intense J in sheet generates massive force perp. to sheet Intense J in sheet generates massive force perp. to sheet Must be balanced by Must be balanced by In steady state in sheet In steady state in sheet t-dep solution: in sheet t-dep solution: in sheet This pressure force implausibly large, cannot be maintained in plasmas with realistic This pressure force implausibly large, cannot be maintained in plasmas with realistic

22. Summary 3D nulls may be important sites of rec and energy release in complex 3D magnetic fields 3D nulls may be important sites of rec and energy release in complex 3D magnetic fields Certain evolutions of B prohibited under ideal MHD Certain evolutions of B prohibited under ideal MHD Under (shear) boundary driving, J sheet forms at null Under (shear) boundary driving, J sheet forms at null Null spine and fan close up – 3D sheet forms at null with B lines exactly anti-parallel at null Null spine and fan close up – 3D sheet forms at null with B lines exactly anti-parallel at null Development of - reconnection Development of - reconnection Qualitative nature of sheet is Sweet-Parker-like Qualitative nature of sheet is Sweet-Parker-like In incompressible limit, morphology of sheet changes In incompressible limit, morphology of sheet changes Analytical fan sheet solns realised, but not spine sheet Analytical fan sheet solns realised, but not spine sheet

23. Thanks for listening! “On the planet Earth, man had always assumed that he was more 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 muck about in the water having a good time. But conversely, the dolphins has always believed that they were more intelligent than man – for precisely the same reasons.” “On the planet Earth, man had always assumed that he was more 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 muck about in the water having a good time. But conversely, the dolphins has always believed that they were more intelligent than man – for precisely the same reasons.” Douglas Adams, The Hitchhiker’s Guide to the Galaxy

25. Kinematic solutions. I - (Pontin et al. 2004) Rotational flows and rec Rotational flows and rec required by structure of B & J