Current sheet formation and magnetic reconnection at 3D null points

1. Collaborators: A. Bhattacharjee, K. Galsgaard (U. of Copenhagen) Current sheet formation and magnetic reconnection at 3D null points “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” Douglas Adams, The Restaurant at the End of the Universe David Pontin 18th October 2006

2. Complex 3D magnetic fields • Complex, everchanging B • Very low dissipation • Where do J sheets form? 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: Magnetic reconnection: But nature is 3D!

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

5. 3D null structure • Determine local structure of null by examining Jacobian • 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 (Schrijver & Title 2002; Longcope et al. 2003; Close et al. 2004) Dome topology: 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)

7. Ideal vs. non-ideal evolution • 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 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 Only when extremely strong currents build up can field lines `slip’ through plasma & so break and `reconnect’

8. Kinematics – non-ideal evolution Study kinematic limit. Dynamics not included, but singularities point to J sheets in dynamic regime. • Evolution can be viewed as ideal if anyw exists satisfying • (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

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

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

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

12. J contd. J component which grows is Jz - to fan, to shear plane J profile dependent on driving strength Jx solid line Jy dotted Jz dashed v0=0.001 v0=0.01 v0=0.04

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

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

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

16. J sheet properties • Peak current and rec rate scale linearly with driving vel (v0) • Sheet dimensions also scale linearly with v0.

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

18. Effect of compressiblity – analytical solns • Analytical incompressible solns. – fan and spine (Craig et al 1995; Craig & Fabling 1996, 1998) • Assuming simplifies Eq.s. • Further simplification used: fan case 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 ) • 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 • Driving the fan: • Very similar current concentration • Increasing has similar effect to before; spine & fan do not collapse to same extent BUT current spreads more along fan Implies `spine current’ solns are not dynamically accessible (see also Titov et al. 2005) Expect spine currents to result from rotations, spine (Pontin & Galsgaard, 2006)

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

22. Summary • 3D nulls may be important sites of rec and energy release in complex 3D magnetic fields • Certain evolutions of B prohibited under ideal MHD • 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 • Development of - reconnection • Qualitative nature of sheet is Sweet-Parker-like • In incompressible limit, morphology of sheet changes • 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.” Douglas Adams, The Hitchhiker’s Guide to the Galaxy

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