Scaling of Magnetic Reconnection in Collisional and Kinetic Regimes

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1 Scaling of Magnetic Reconnection in Collisional and Kinetic Regimes William Daughton Los Alamos National Laboratory Collaborators: Vadim Roytershteyn, Brian Albright H. Karimabadi, Lin Yin & Kevin Bowers LA-UR

2 Scaling Approach in Physics Physicists like to identify relevant dimensionless parameters and use these to develop scalings Very powerful - can collapse an infinite number of physical problems into a single dimensionless problem However, there are potential pitfalls: 1. Failure to identify all dimensionless parameters 2. Dangerous to extrapolate too far -- new physics?

3 Example - Flow drag on a sphere Stokes Flow - Re < 0.5 Laminar Wake Turbulent boundary layer Re > 10 5 Experimental Data log C D The Drag Force on a Sphere (UMAP Journal, Volume 12, no. 1, Spring 1991, pp ), log Re

4 Why is this relevant for reconnection? Scaling even a single parameter can be complicated as new dynamical regimes are encountered New regimes may profoundly alter scalings Reconnection has many dimensionless parameters Equations may actually change - fluid vs kinetic Scaling studies are quite limited and only 2D We are close for some applications; while others are quite hopeless for any foreseeable computer

5 Two Key Parameters in Scaling Studies Earth s magnetosphere Solar Corona & Stellar Flares accretion disks, magnetars Lundquist - S L/d i Computer Simulations Laboratory Experiments < 10 5 < 400 < Which parameter is most relevant? It depends on the regime - collisional vs kinetic

6 Length of diffusion region is central issue We don t have a rigorous theory of how this is set in either collisional or kinetic regimes Various Ideas: 1. Two-fluid physics (Hall, electron pressure, dispersive waves) 2. Secondary magnetic islands (plasmoids) 3. Form of non-ideal terms in Ohm s law P e 4. Kinetic instabilities in layer or outflow jet 5. MHD turbulence 6. Intrinsically 3D effects 7. Boundary conditions - macroscopic driver Unlikely there is only one right answer - perhaps all have some merit in certain regimes?

7 Is Nature Subtle or Malicious? Mass Ratio Plasma Regime Scale Separation Dispersive Waves Non-Ideal Term Rate m i m e B g B o P Yes Whistler ~0.1 m i m e B g B o Yes Kinetic Alfven P ~0.1 m i = m e T i = T e B g B o P No No ~0.1 m i = m e T i = T e B g B o P No No ~0.1 m i = m e T i T e P Yes No ~0.1 Is a Unified Theory of reconnection possible?

8 Focus of this Talk: What is scaling within resistive MHD? When does MHD model break down? New approach to bridge between regimes What role do plasmoids play? Emerging parameter space map of regimes Real world is 3D - many new uncertainties

9 Scaling of Resistive MHD

10 Resistive MHD is the usual starting point Sweet-Parker Model L sp 2δ sp U in V A = δ sp L sp = 1 S 1/2 S 4πV AL sp ηc 2 = τ R τ A Slow scaling has motivated research for 50+ years - Petschek, anomalous resistivity, turbulence, two-fluid & kinetic effects, 3D,...

11 Petschek Reconnection ~ 1964 U in U in V A δ V A U in B D B Slow-Shocks V A 1 log(s) Rates consistent with observations Widely accepted for 20+ years Not correct for uniform or Spitzer resistivity Requires localized anomalous dissipation Ugai and Tsuda, 1977 Biskamp, 1986 Yan, Lee, and Priest,1992 Uzdensky and Kulsrud, 2000

12 Sweet-Parker is Not Correct for S > 10 4 MHD Simulations - Mattheaus 1985, Biskamp 86, Yan 92, Malara, 92 Linear theory - Loureiro 07, Bhattacharjee 09, Samtaney 09 N p S 3/8 Increasingly violent for large S τ A γ S 1/4 Two important consequences: 1. Faster reconnection - Lapenta 08, Bhattacharjee 09, Cassak Push towards kinetic scales - Shibata 01, Daughton 09, Cassak 09

13 When does MHD break down? i.e. for basic physics studies

14 Two ways in which resistive MHD may break 1. Resistive layers approach ion kinetic scale 2. Electric fields approach Dreicer runaway limit E cr (m e T e ) 1/2 ν ee /e Using collisional SP scaling E y E cr ( ρi δ sp )( me m i ) 1/2 1 β 1/2 Runaway requires: 1. Electron scale layers or 2. Extremely large guide field β (m e /m i )

15 Clear evidence that MHD fails for ion scale layers Sweet-Parker thickness 2L sp 2δ sp δ sp L sp S 1/2 (L sp η) 1/2 δ sp > d i,ρ s δ sp d i,ρ s Collisional Regime Kinetic Regime Theory & Simulation Aydemir, 1992 Ma & Bhattacharjee, GRL, 1996 Cassak et al, PRL, 2005 Uzdensky, PRL, 2007 Simakov & Chacon, PRL, 2008 Malyshkin, PRL, 2008 Lab. Experiments Ren et al, PRL, 2005 Egedal et al, PRL, 2007 Critical Resistivity ˆη d i L sp 4d i L x ˆη c Cassak et al, 05, 06 Uzdensky, 07 Length & role of electron layers?

16 Transition well Verified in Two-Fluid Simulations Cassak et al, 2006 Sweet-Parker Kinetic

17 Can we study transition with kinetic approach? f s t + v f s x + q s m s ( E + v B ) c f s v = s C ss + Maxwell s Equations Fokker-Planck Collision Operator Advantages of this approach: 1. Rigorous treatment of transition & super Dreicer fields 2. Full treatment of ion and electron kinetic effects 3. Better understand laboratory experiments 4. Influence of plasmoids on transition to kinetic regimes 1. Phys. Plasmas 16, (2009) 2. Phys Rev. Lett. 103, (2009) 3. Daughton et al, 2010

18 Transition is Clearly Observed in Collisional PIC z d i L x = 100 ˆη o =0.1 tω ci =50 Sweet-Parker Regime J y tω ci = 120 x/d i d i J y Kinetic Regime x/d i

19 Strong confirmation that Hall physics is playing a central role in this transition However, there are still some outstanding discrepancies between two-fluid & kinetic

20 Electron layers are much shorter in two-fluid z d i m i /m e = 300 Rogers et al, 2001 L 0.3d i x/d i m i /m e =25 Cassak et al, 2001 L 0.2d i Sullivan al, 2009 m i /m e =25 L 0.8d i

21 Collisional PIC simulations have much longer layers Unstable to plasmoids z d i z d i tω io = 140 tω io =160 Dramatically different than two-fluid simulations z d i tω io =180.. but very similar to collisionless PIC results z d i tω io =190 Daughton et al, 2006 Karimabadi et al, 2007 Klimas et al, 2008 z d i tω io =200 x/d i

22 Cluster observations support notion of elongated electron layers 1. Li-Jen Chen et al, Secondary islands within reconnecting electron layer L 5d i 2. Phan et al, Highly extended electron jets > 60d i

23 Runaway Fields Require Collisionless Mechanisms Can balance non-ideal field with either 1. P e 2. Plasma instabilities 1 See Roytershteyn et al, PoP Rei/FNI ν E y /E crit

24 Runaway Fields Require Collisionless Mechanisms Can balance non-ideal field with either 1. P e 2. Plasma instabilities 1 See Roytershteyn et al, PoP Rei/FNI e NI ν E y /E crit

25 Influence of Plasmoids on the Transition to Kinetic Regimes n-1 n-1 Really insightful paper by Shibata & Tanuma, 2001 n n+1 n n+1

26 10 tω ci = 200 J y z Time Evolution for our largest Case d i tω ci io = = J y z L x = 800d i d i tω ci =425 J y z Daughton et al, Phys Rev. Lett., 2009 d i 10 New plasmoids 0 x/d i 800

27 Transitions much sooner than expected from simple estimate that neglects plasmoids ER L x = 200d i L x = 400d i L x = 800d i Repeated formation of new plasmoids ˆη ˆη c Transition Resistivity 4x larger than simple estimate δ sp d i tω io

28 Can we construct new transition estimates? Assume: 1. Plasmoid scaling N p ~ ( S /S crit ) 2. New layers obey SP scaling Cassak,09 Bhattacharjee,10 Rate in collisional regime is much faster Transition to kinetic regime occurs much sooner E R S (1)/ 2 ˆ c ei 1/(1+ ) ce /(1+ ) S crit d i L x 1 +1 Various simulations show 0.6 1

29 The limit ~1 is particularly interesting Transition resistivity for i ˆ c ei ce 1/2 1/2 S crit independent of system size Plasmoids will push evolution to kinetic scales if S > S crit ~10 4 ei ce < S crit 1/2 ~ 1/2 100

30 Emerging Map of Reconnection Regimes log(s) Collisionless c~ 1/L (stable SP SP Laye Layer) c 1/ Kinetic i c~ 1/Sc 1/2 -P Plasmoid induced c Runaway a fields R~0.1 Collisional MHD with Plasmoids e~e Collisional - Sweet-Parker Regime Assume Np~S log(l/d i )

31 Future Uncertainties: Scaling studies have all been limited to 2D!

32 Island formation is more complicated in 3D Drift Tearing - Coppi et al, 1979, Catto et al, 1974, Gladd, 1990, Daughton et al, 2005 Percolation - Galeev, Kuznetsova, Zeleny, 1986 Volume filling islands - Drake et al, Nature, 2006 Galeev et al, 1986 Drake et al, 2006

33 2D vs 3D Dynamics is Quite Different J m i = m e 2D 10 6 cells J 3D cut z 10 9 cells x 240d i 1024 cells

34 Primary & secondary islands form a spectrum interacting oblique flux ropes See movies m i m e =1

35 Electron layers that form along separatrices are also unstable to secondary islands 2D Simulation z d i m i m e = 100 in-plane current z d i x/d i

36 Under certain conditions, theory & simulations suggest a spectrum of oblique flux ropes m i /m e Flux ropes may interact differently than islands in 2D models