Gyrokinetic Simulations of Tearing Instability
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1 Gyrokinetic Simulations of Tearing Instability July 6, 2009 R. NUMATA A,, W. Dorland A, N. F. Loureiro B, B. N. Rogers C, A. A. Schekochihin D, T. Tatsuno A rnumata@umd.edu A) Center for Multiscale Plasma Dynamics, Univ. Maryland B) EURATOM/UKAEA Fusion Associateion, Culham Science Centre C) Dept. Phys. & Astron., Dartmouth College D) Rudolf Peierls Centre for Theor. Phys., Univ. Oxford 2009 SIAM Annual Meeting, Denver, Colorado, July 6-10 (2009) p.1/18
2 Outline Magnetic Reconnection and Tearing Instability Basic Theory for Tearing Instability Gyrokinetics Collisions and Resistivity AstroGK gyrokinetics code Numerical Simulations Electron Physics Ion Physics Kinetic effect (ion temperature) MHD limit Summary 2009 SIAM Annual Meeting, Denver, Colorado, July 6-10 (2009) p.2/18
3 Magnetic Reconnection Magnetic reconnection is ubiquitous in plasmas Plasma Flows 7 Global Scale (~ 10 m) Diffusion Region Field Lines Reconnect! Microscale physics (<1m) Magnetic reconnection converts magnetic field energy into high speed plasma flows, heating of plasmas, and energetic particles. Sawtooth oscillations, island growth due to tearing mode, disruptions in fusion experimental devices (Degradation of confinement). Magnetospheric substorms, solar flares in astrophysical situation SIAM Annual Meeting, Denver, Colorado, July 6-10 (2009) p.3/18
4 Magnetic Reconnection Magnetic reconnection is a classic example where multiple physics (scales) are involved Physics to break flux-freezing is necessary for field lines to change its topology: primarily by collisions (resistivity). In collisionless environments, time scale of reconnection based on resistivity is far too slow to explain realistic explosive phenomena. Resistive spatial scale falls below kinetic scales (MHD theory is not valid). Global structure drastically changes depending on microscopic processes. Diffusion Region Electron Pressure comes in Questions What determines time scale of reconnection? What provides mechanism for field lines to reconnect? Electron Inertia Ion Sound Ion Inertia (Hall effect) ρe de ρs di 2009 SIAM Annual Meeting, Denver, Colorado, July 6-10 (2009) p.4/18
5 Tearing Instability Tearing instability is a resistive instability of current sheet configuration Spontaneous onset of reconnection process Linear stage Standard boundary layer or singular perturbation problem Background current profile By0(x)=ψ0 scale ~ a Without resistivity and inertia, solution is singular d 2 «ψ dx 2 ky 2 + ψ 0 ψ 0 ψ = 0 (1) Inner Region x Outer Region (Ideal) Flux Flow 2009 SIAM Annual Meeting, Denver, Colorado, July 6-10 (2009) p.5/18
6 Tearing Instability Tearing instability is a resistive instability of current sheet configuration Spontaneous onset of reconnection process Linear stage Standard boundary layer or singular perturbation problem Background current profile By0(x)=ψ0 scale ~ a Without resistivity and inertia, solution is singular d 2 «ψ dx 2 ky 2 + ψ 0 ψ 0 ψ = 0 (1) Inner Region x Outer Region (Ideal) Flux Flow 2009 SIAM Annual Meeting, Denver, Colorado, July 6-10 (2009) p.5/18
7 Tearing Instab. Theory (resistive MHD) a Time Scales (normalized by τ A = a/v A ) Alfvén time (parallel to the reconnecting field): τ H /τ A 1/ka Resistive time scale: τ R /τ A µ 0 av A /η S: Lundquist number Assumptions Time scale separations 1/τ R γ 1/τ H (2) Scale separations l a (3) where l is the resistive current layer width Dispersion relation (matching two solutions) a = π 8 γ5/4 τ 1/2 Γ((λ 3/2 1)/4) H τ3/4 R Γ((λ 3/2 + 5)/4) (4) λ = γτ 2/3 H τ1/3 R, is a jump of ψ providing instability criterion ( > 0). a Furth, Killeen, and Rosenbluth, Phys. Fluids 6, 459 (1963) 2009 SIAM Annual Meeting, Denver, Colorado, July 6-10 (2009) p.6/18
8 Tearing Instab. Theory (More Physics) Hall effect (Ion inertia) a d i Alfvén wave dispersion (Whistler) Pressure effect b (Ion Sound) ρ s Alfvén wave couples to sound wave Finite Larmor Radius (FLR) c ρ i(e) Alfvén wave dispersion (KAW) Tensorial pressure ρ e(i) break flux-freezing Electron inertia d d e break flux-freezing Some of the effects can exist in fluid models, while some are essentially kinetic. Gyrokinetics No ad-hoc inclusion of detailed physics validation of various fluid models Collision physics Do not assume resistivity Disadvantages: heavy, a e.g. Terasawa, GRL (1983), Fitzpatrick & Porcelli, PoP (2004) b Coppi et al., Nuclear Fusion (1966). c Porcelli, PRL (1991) d Schep et al. (1994) 2009 SIAM Annual Meeting, Denver, Colorado, July 6-10 (2009) p.7/18
9 Gyrokinetics Reduced kinetic model 5 dimensional phase space Strong guide field (B 0 ) allows to separate out fast cyclotron motion Already multiscale Note: two-dimensional magnetic reconnection or tearing instability dynamics primarily independent of the guide field 2009 SIAM Annual Meeting, Denver, Colorado, July 6-10 (2009) p.8/18
10 Gyrokinetics: Basic equations The distribution function of particles is given by f = 1 qφ T 0 f 0 + h, where f 0 = n 0 /( πv th ) 3 exp( v 2 /vth 2 ) is the Maxwellian, and the thermal velocity is given by v th = p 2T 0 /m. The equations to solve are the gyrokinetic equation for h = h(r, V, V ), h t + V h Z + 1 { χ R, h} C(h) R = q f 0 χ R, (5) B 0 T 0 t χ = φ v A and the field equations for φ(r), A (r), and δb (r), n i =n e X s» q2 sn 0s φ T 0s + q s ( B) =µ 0 j 2 A X Z = µ 0 q s P +I B 0δB µ 0 «=0 B 0 δb = µ 0 X Z s s Z h s r dv = 0, h s r v dv mv v h s r dv. (6) (7) (8) 2009 SIAM Annual Meeting, Denver, Colorado, July 6-10 (2009) p.9/18
11 Collision Operator Recently, linearized collision operators for gyrokinetic simulations, which satisfies physical requirements are established and implemented in AstroGK a. The operators are the pitch-angle scattering (Lorentz), the energy diffusion, and moments conserving corrections to those operators for like-particle collisions. Electron-ion collisions consists of pitch angle scattering by background ions and ion drag are also included. We, here, mainly discuss the electron-ion collisions since it contributes to resistivity. The operator is given by (in Fourier space) C ei (h e,k ) = ν ei vth,e V ξ (1 ξ2 ) h e,k ξ 1 4 (1 + ξ2 ) V 2 v 2 th,e + 2V J 0 (α e )u,i,k v 2 th,e k 2 ρ2 e h e,k f 0e! (9) We examine how this collision operator relates with resistivity which decays the current. a Abel et al, Phys. Plasmas 15, (2008), arxiv: ; Barnes et al, accepted Phys. Plasmas (2009), arxiv: v SIAM Annual Meeting, Denver, Colorado, July 6-10 (2009) p.10/18
12 Spitzer Resistivity From the fluid picture current decays due to collisional resistivity as j t = η µ 0 k 2 j, (10) and the decay rate is τ 1 decay = (η/µ 0)k 2. Using the Spitzer resistivity given by η = m e /(1.98τ e n e e 2 ) where τ e = 3 π/(4ν ei ), the decay rate is casted into the following form, τ 1 decay = Cν ei(d e k) 2 (11) where the constant C = 4/( π) Figures show dependence of decay rate on ν and d e k. Numerical estimated proportionality constant agrees with Spitzer s value within 5% error. Decay Rate ( Resistivity) Decay Rate ( Resistivity) AstroGK Fit with C ν (d e k) 2 C= d e k= Collision Frequency ν AstroGK Fit with C ν (d e k) 2 C= ν= d e k 2009 SIAM Annual Meeting, Denver, Colorado, July 6-10 (2009) p.11/18
13 AstroGK Publicly available at ghowes/astrogk/ (paper in preparation) Eulerian (not particle) Fourier spectral in coordinate space (x,y) Periodic boundary in x and y Gaussian quadrature for velocity space integral Time integral: Implicit Euler for collisions Adams-Bashforth (3rd) for nonlinear terms Parallelized using MPI library NERSC (Franklin), NCSA (Jaguar), TACC (Ranger), etc secs/step Franklin Ranger linear processors SIAM Annual Meeting, Denver, Colorado, July 6-10 (2009) p.12/18
14 Simulation Setting Parameters r λ i /a, σ m e /m i, (12) τ T 0i /T 0e, β e n 0 T 0e /(B 2 g /2µ 0) (13) λ i is typical ion scale, B g is the guide magnetic field «ρ i /λ i =τ 1/2, d i /λ i =βe 1/2 Γ 1/2, ρ s /λ i = (1 + τ), (14) 2 ρ e /λ i =σ 1/2, d e /λ i =β 1/2 e σ 1/2, (15) l(ν) Change collisionality ν to vary current layer width l Electron Phys.Ion Phys. de λi a MHD 2009 SIAM Annual Meeting, Denver, Colorado, July 6-10 (2009) p.13/18
15 Electron Physics r =0.2, σ =0.01, τ =1, β e =0.3 ρe de ρiρsdi a 5 Growth Rate: γτ A τ H /τ A =0.486, a= AstroGK 1F MHD 2F MHD Current Sheet Width: l/a τ H /τ A =0.486, a= d e /a Lundquist Number: τ R /τ A Lundquist Number: τ R /τ A Transition to collisionless reconnection independent of collisions Electron inertia mediated d e sets lower bound of scale Difference between GK & 2F MHD may be ascribed to the treatment of pressure (Γ 5/3) 2009 SIAM Annual Meeting, Denver, Colorado, July 6-10 (2009) p.14/18
16 Ion Physics r =0.02, σ =0.01, τ =1, β e =0.3 ρiρsdi β a 50 Growth Rate: γτ A τ H /τ A =0.486, a= β e =0.3 β e =0.075 β e = F MHD 2F MHD Current Sheet Width: l/a τ H /τ A =0.486, a= Lundquist Number: τ R /τ A Lundquist Number: τ R /τ A Not MHD even if ion scale a β e slows down tearing growth Approaches to MHD limit as β decreases 2009 SIAM Annual Meeting, Denver, Colorado, July 6-10 (2009) p.15/18
17 Ion temperature Cold ion (τ = 10 4 ) for different r (r = 0.2: electron phys., r = 0.02: ion phys.) ρ i becomes irrelevant (< d e ) Growth Rate: γτ A τ H /τ A =0.486, a=23.2 r=0.2, τ=1 r=0.2, τ=10-4 r=0.02, τ=1 r=0.02, τ=10-4 1F MHD Lundquist Number: τ R /τ A Ion temperature has no effect (ρ i ρ s ) 2009 SIAM Annual Meeting, Denver, Colorado, July 6-10 (2009) p.16/18
18 MHD limit Low beta β e = 10 3, Cold ion τ = 10 4 case τ H /τ A =0.486, a=23.2 τ H /τ A =0.486, a=23.2 Growth Rate: γτ A AstroGK 1F MHD Current Sheet Width: l/a Lundquist Number: τ R /τ A Lundquist Number: τ R /τ A 2009 SIAM Annual Meeting, Denver, Colorado, July 6-10 (2009) p.17/18
19 Summary Magnetic reconnection and tearing instability are very good example of multiscale physics. We have performed collisionless and collisional tearing instability simulations, and have scanned for ν ei. We have observed transition from collisional regime (collision dependent growth rate) to collisionless regime (collision independent growth rate). Equation of state need to be considered carefully not simply p ρ Γ Understanding ion physics is rather difficult pressure effect and Hall effect are tangled Ion temperature seems not playing significant role in the regimes considered GK simulation can also capture correct resistive MHD limit Plasma Beta β Regimes of Tearing Instability S increase MHD EMHD Hall Effect α Strong Hall Low β Collisionless Electron Phys. Ion Phys. MHD limit Regimes of two fluid tearing instability obtained by Ahedo and Ramos [PPCF (2009)] 2009 SIAM Annual Meeting, Denver, Colorado, July 6-10 (2009) p.18/18
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