Quasi-separatrix layers and 3D reconnection diagnostics for linetied tearing modes John M. Finn, LANL A. Steve Richardson, NRL CMSO, Oct 2011 Operated by Los Alamos National Security, LLC for the U.S. Department of Energy s NNSA U N C L A S S I F I E D
Outline Review of Magnetohydrodynamics (MHD) and Reconnection Reconnection Diagnostics Squashing degree Q and quasi-separatrix layer (QSL) Ideal MHD electrostatic potential (Δϕi) and resistive MHD potential (Δϕr) Examples of the Diagnostics Q: simple doublet field Δϕ's & Q: tearing mode in an infinite cylinder Line Tied Modes Two mode approximation for line tied modes Compare Q and Δϕ's for a line tied mode Short case (resistive diffusion) and long case (tearing mode) Summary and Conclusions U N C L A S S I F I E D Slide 2 /25 Operated by Los Alamos National Security, LLC for the U.S. Department of Energy s NNSA
Magnetohydrodynamics (MHD) Visco-resistive MHD Force free (zero β) plasma Mass Conservation: Momentum and Ohm's law: Maxwell: U N C L A S S I F I E D Slide 3 /25 Operated by Los Alamos National Security, LLC for the U.S. Department of Energy s NNSA
Ideal MHD If plasma resistivity is zero, (ideal Ohm s law) This approximation is good for hot (low resistivity), strongly magnetized plasmas Also good for long length scale, fast plasma dynamics E.g., outer layer for tearing modes; reconnection only occurs in the tearing layer. Outer layer: Ideal MHD is a good From Ideal Ohm s law, the perpendicular fluid velocity is The EXB drift of a particle. U N C L A S S I F I E D Slide 4 /25 Operated by Los Alamos National Security, LLC for the U.S. Department of Energy s NNSA
Frozen-in-flux Consider the flux through an arbitrary surface ΔS: U N C L A S S I F I E D Slide 5 /25 Operated by Los Alamos National Security, LLC for the U.S. Department of Energy s NNSA
Reconnection If the plasma has finite resistivity (or other non-ideal effects), then frozen flux no longer holds -- the plasma and field lines can slip past one another For large η, this happens globally -- resistive diffusion For very small η, this can only occur locally, associated with changing the magnetic topology -- reconnection Squashing of flux tubes At the origin U N C L A S S I F I E D Slide 6 /25 Operated by Los Alamos National Security, LLC for the U.S. Department of Energy s NNSA
Outline Review of Magnetohydrodynamics (MHD) and Reconnection Reconnection Diagnostics Squashing degree Q and quasi-separatrix layer (QSL) Ideal MHD electrostatic potential (Δϕi) and resistive MHD potential (Δϕr) Examples of the Diagnostics Q: simple doublet field Δϕ's & Q: tearing mode in an infinite cylinder Line Tied Modes Two mode approximation for line tied modes Compare Q and Δϕ's for a line tied mode Short case (resistive diffusion) and long case (tearing mode) Summary and Conclusions U N C L A S S I F I E D Slide 7 /25 Operated by Los Alamos National Security, LLC for the U.S. Department of Energy s NNSA
Reconnection Diagnostics: the squashing degree Geometrically, reconnection is associated with the stretching and squashing of flux tubes Squashing degree is defined using a field line mapping Field lines define mapping: Jacobian J maps tangent vectors: Max and min eigenvalues of minor axis of ellipse from z=-l to z=+l (Singular values of J) 2 give lengths of major and Since we are interested in squashing and not flux tube expansion, take ratio of singular values to find aspect ratio of ellipse Aspect ratio is a function of squashing degree Q [Priest Démoulin 95, Titov 07] Large Q -- Quasi-separatrix layers (QSLs) U N C L A S S I F I E D Slide 8 /25 Operated by Los Alamos National Security, LLC for the U.S. Department of Energy s NNSA
Reconnection Diagnostics: potentials Δϕi and Δϕr Reconnection is due to a localized breakdown of the ideal Ohm s law (consider resistive MHD as a special case) Given magnetic fields B and vector potential A(x,t), we can compute two potentials: These equations can be integrated along field lines from one end of the system to the other to obtain Δϕi and Δϕr The ideal potential Δϕi was considered in Lau and Finn, ApJ, 350:672 691, Feb. 1990. The difference between these is the integrated parallel current, or quasi potential : Hesse, Forbes, and Birn, ApJ, 631(2):1227, 2005. Regions where these two quantities differ significantly are reconnection sites U N C L A S S I F I E D Slide 9 /25 Operated by Los Alamos National Security, LLC for the U.S. Department of Energy s NNSA
Outline Review of Magnetohydrodynamics (MHD) and Reconnection Reconnection Diagnostics Squashing degree Q and quasi-separatrix layer (QSL) Ideal MHD electrostatic potential (Δϕi) and resistive MHD potential (Δϕr) Examples of the Diagnostics Q: simple doublet field Δϕ's & Q: tearing mode in an infinite cylinder Line Tied Modes Two mode approximation for line tied modes Compare Q and Δϕ's for a line tied mode Short case (resistive diffusion) and long case (tearing mode) Summary and Conclusions U N C L A S S I F I E D Slide 10/25 Operated by Los Alamos National Security, LLC for the U.S. Department of Energy s NNSA
Examples of Diagnostics: Q for Doublet Field Doublet-like field contains X-point, flux tube stretching, and real separatrix when L goes to infinity Magnetic field: x2 1 0.5 0 Integrating field lines gives mapping M 0.5 1 1 0 1 Regions of large Q define the quasiseparatrix layer (QSL) on the stable manifold (a) 1 (b) 1 x 1 10 3 0.5 40 0.5 L=1 L=2 x2 0 x2 20 1 0.5 0.5 0 2 1 1 0 1 1 1 0 1 0 x 1 U N C L A S S I F I E D x 1 Slide 11/25 Operated by Los Alamos National Security, LLC for the U.S. Department of Energy s NNSA
Examples of Diagnostics: Q for Doublet Field Q is a function of field line, so can be plotted as a function of initial or final point (integrating backwards -- unstable manifold) (a) 1 10 4 2 (b) 1 10 4 2 x2 0.5 0 1.5 1 X2 0.5 0 1.5 1 L=2.5 0.5 0.5 0.5 0.5 1 1 0 1 x...or as an isosurface 1 in 3D space In 3D, the QSL looks like a hyperbolic flux tube, a structure studied in solar physics in models of reconnection in solar flares 0 x2 1 0.5 0 1 0 1 X 1 0 0.5 U N C L A S S I F I E D 1 x 1 0 1 2 0 z 2 Slide 12/25 Operated by Los Alamos National Security, LLC for the U.S. Department of Energy s NNSA
Second Example: Bifurcation of an arcade QSLs could also indicate the onset of reconnection, just before a bifurcation causes the creation of closed flux surfaces Example: Magnetic Arcade to plasmoid bifurcation L/L 0 20 15 10 5 L/L 0 6 4 2 0.35 0.36 x ini 1 0.8 0.6 1 0.8 0.6 0 0 0.1 0.2 0.3 0.4 0.5 1 0.8 0.6 x ini y y y 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0.5 1 x 0 0 0.5 1 x 0 0 0.5 1 x U N C L A S S I F I E D Slide 13/25 Operated by Los Alamos National Security, LLC for the U.S. Department of Energy s NNSA
Examples of Diagnostics: m=1 Tearing Mode in a Periodic Cylinder Solve linearized force-free MHD equations -- tokamak-like, cylindrical Zero β Tearing mode is localized near the mode rational surface, where and goes like, growth rate γ 1 0.5 B r B q B z 0.1 0.05 v r v q v z 0 B 0 ṽ -0.05 0.5-0.1 1 0 0.5 1 1.5 2 U N C L A S S I F I E D Operated by Los Alamos National Security, LLC for the U.S. Department of Energy s NNSA -0.15 0 0.5 1 1.5 2 Slide 14/25
Examples of Diagnostics: Periodic Tearing Mode For infinitesimal tearing mode, can estimate Δϕi Integrating z=-l to z=l gives The argument of the sinc function is approximately linear in r, centered at the mode rational surface The width of the sinc function is then approximately U N C L A S S I F I E D Slide 15/25 Operated by Los Alamos National Security, LLC for the U.S. Department of Energy s NNSA
Examples of Diagnostics: Periodic Tearing Mode Mode amplitude = 1x10-3 (a) r Numerical results are in good agreement with the sinc function calculation Δϕideal Δϕresistive Difference 10 4 10 4 10 4 1.4 (b) 1.4 (c) 1.4 5 4 2 1.3 1.3 1.3 2 0 0 0 1.2 1.2 1.2 2 2 5 4 1.1 1.1 1.1 0 p/2 /p/ 3p/2/2p/ 0 p/2 /p/ 3p/2/2p/ 0 p/2 /p/ 3p/2/2p/ r r q q U N C L A S S I F I E D Operated by Los Alamos National Security, LLC for the U.S. Department of Energy s NNSA Mode amplitude = 1x10-2 (b) 1.4 Δϕideal 10 3 5 (c) 1.3 r 0 1.2 1.1 5 0 p/2 /p/ 3p/2/2p/ q r q Difference 1.4 1.3 1.2 1.1 0 p/2 /p/ 3p/2/2p/ q 10 3 4 2 0 2 4 Slide 16/25
Examples of Diagnostics: Periodic Tearing Mode Calculation of Q shows peak near stable manifold, which indicates presence of a QSL (mode amplitude = 1x10-3 ) log(q) θ=π/2 As amplitude increases, Q becomes more peaked at the stable manifold. U N C L A S S I F I E D Slide 17/25 Operated by Los Alamos National Security, LLC for the U.S. Department of Energy s NNSA
Outline Review of Magnetohydrodynamics (MHD) and Reconnection Reconnection Diagnostics Squashing degree Q and quasi-separatrix layer (QSL) Ideal MHD electrostatic potential (Δϕi) and resistive MHD potential (Δϕr) Examples of the Diagnostics Q: simple doublet field Δϕ's & Q: tearing mode in an infinite cylinder Line Tied Modes Two mode approximation for line tied modes Compare Q and Δϕ's for a line tied mode Short case (resistive diffusion) and long case (tearing mode) Summary and Conclusions U N C L A S S I F I E D Slide 18/25 Operated by Los Alamos National Security, LLC for the U.S. Department of Energy s NNSA
Line Tied Modes: The Two-Mode Approximation Consider the same cylindrical system, but with line-tied boundary conditions Satisfy BCs by expanding in radial eigenfunctions of the infinite cylinder [Evstatiev et al. 2006] Expansion coefficients εk chosen so that the BCs are satisfied For large L, two modes with very similar k will have nearly the same radial form BCs then imply a matrix equation for εk, k1, k2, and L, which gives U N C L A S S I F I E D Slide 19/25 Operated by Los Alamos National Security, LLC for the U.S. Department of Energy s NNSA
Line Tied Modes: Two-Mode Approximation Compute growth rate for a range of k s, and pick k1 and k2 with the same γ Modes have similar r dependance Distance between mode rational surfaces gives the geometric width wg of the LT mode: Delzanno & Finn, 2008 Huang & Zweibel, 2009 U N C L A S S I F I E D Slide 20/25 Operated by Los Alamos National Security, LLC for the U.S. Department of Energy s NNSA
Line Tied Modes: Two Cases Examine two cases Case I: wt>wg (Large L), this mode should behave like a tearing mode (with reconnection). Case II: wt<wg, (Small L) this mode should just have global resistive diffusion (not reconnection) Apply reconnection diagnostics to these, to see if the two cases can be distinguished Previous work showed that Case I has reconnection, but Case II has only resistive diffusion... Delzanno & Finn, 2008; Huang & Zweibel, 2009 U N C L A S S I F I E D Slide 21/25 Operated by Los Alamos National Security, LLC for the U.S. Department of Energy s NNSA
Case I (wt>wg, large L): Diagnostics for amplitude = 1x10-3 (a) 1.35 10 4 5 (c) (b) 1.35 10 4 3.5 r 1.3 0 Δϕideal r 1.3 3 Q 1.25 1.2 0 p/2 /p/ 3p/2/2p/ 5 1.25 1.2 0 p/2 /p/ 3p/2/2p/ 2.5 q q Dots: surface of section U N C L A S S I F I E D Slide 22/25 Operated by Los Alamos National Security, LLC for the U.S. Department of Energy s NNSA
Larger amplitude -- nonlinear... r r (a) (b) 1.35 1.3 1.25 1.2 1.35 1.3 1.25 0 p/2 /p/ 3p/2/2p/ q 10 3 2 0 Δϕideal 2 10 4 5 0 Δϕresistive r (c) 1.35 1.3 1.25 1.2 0 p/2 /p/ 3p/2/2p/ q - Surface of section shows chaotic region 10 4 6 4 2 Q 1.2 0 p/2 /p/ 3p/2/2p/ q U N C L A S S I F I E D Slide 23/25 Operated by Los Alamos National Security, LLC for the U.S. Department of Energy s NNSA
Case II (wt<wg, small L): Two-mode results Amplitude=1x10-3 for calculation of Δϕ's Range of amplitudes used for Q calculations Neither Q nor alone can distinguish between Case I and Case II U N C L A S S I F I E D Slide 24/25 Operated by Los Alamos National Security, LLC for the U.S. Department of Energy s NNSA
Summary and Conclusions Reviewed two reconnection diagnostics Electrostatic potentials Δϕi (~ideal MHD) and Δϕr (resistive MHD). Better than just the difference -- Squashing degree Q Applied these to two simple examples Doublet-like field: Q becomes localized along the separatrix for long systems Q measures geometry and topology of field lines Single tearing mode: Δϕ becomes localized at the mode rational surface Δϕ measures mode dynamics through A/ t, as well as field line geometry Δϕ good even for infinitesimal modes Q measures equilibrium field rather than tearing mode for small amplitude modes Applied to two cases (L large; L small) for 'line tied' modes Q and Δϕ s both detect tearing mode By comparing (Δϕi - Δϕr)= to Δϕi the cases can be distinguished U N C L A S S I F I E D Slide 25/25 Operated by Los Alamos National Security, LLC for the U.S. Department of Energy s NNSA
References Lau Finn, 1990 Priest Démoulin, 1995 Titov, 2007 Delzanno Finn, 2008 Huang Zweibel, 2009 U N C L A S S I F I E D Slide 26/25 Operated by Los Alamos National Security, LLC for the U.S. Department of Energy s NNSA