Momentum transport from magnetic reconnection in laboratory an. plasmas. Fatima Ebrahimi

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1 Momentum transport from magnetic reconnection in laboratory and astrophysical plasmas Space Science Center - University of New Hampshire collaborators : V. Mirnov, S. Prager, D. Schnack, C. Sovinec Center for Magnetic Self Organization in Laboratory and Astrophysical Plasmas (CMSO) US-Japan Reconnection Workshop Oct. 07, 2009

2 Plasma rotation flattens during a reconnection event (MST experiment) 30 Parallel velocity (km/s) Parallel Velocity (km/s) Core (toroidal) MST 0 Edge (poloidal) Time (ms)

3 Momentum transport in accretion disk Angular momentum is radially transported (L r 1/2 ) Can not be explained by classical process.

4 Question: What is the role of magnetic reconnection in momentum transport in astrophysical and laboratory plasmas?

5 Helical magnetic field in tori ϕ B R θ B = B θ ˆθ + Bφ ˆφ Consider perturbations: Ṽ, B e ik r e i(mθ nφ/r) periodic in φ and θ Resonant surfaces where k B = mb θ /r nb φ /R = 0 q = rb φ /RB θ = m/n

6 Reconnection occurs at many radii in the Reversed Field Pinch (RFP) q = rb T RB p = m n m=1 n=6 n=7 n=8 m=0 m=1 core tearing modes are linearly unstable. m=0 tearing modes are nonlinearly driven. k 1 ± k 2 = k 3 k=(m,n) (1,6) (1,7) (0,1)

7 m=0 bursts are obtained at high S MHD computations. sawtooth crashes = sudden reconnection events due to resistive MHD activities m=1, n= 6 m=1, n= 7 m=0, n=1 m=0 modes are driven nonlinearly during large reconnection events - Choi et al. PRL 2006 Computations at low S don t show pronounced sawteeth and m=0 bursts.

8 Outline Momentum transport in RFP transport from a single tearing mode transport from multiple tearing modes - nonlinear coupling momentum generation - nonlinear single helicity NIMROD simulations Momentum transport in a Keplerian disk transport from a single tearing mode transport from multiple tearing modes - nonlinear coupling

9 Quasilinear calculations Find the structure of stresses using quasilinear theory, constructed from linearized eigenfunctions Calculate stresses in the ideal (outer) MHD region in the presence of mean flow < J B > z γ[kv z (r s ) kv z (r)] Solve resistive MHD equations near reconnection layer with small shear flow Use CGJ tearing ordering (γ η 3/5 ) and consider small flow of the order V 0 η 1/5 In the absence of mean flow < r B r Bz > = 0 and < rṽr Ṽz > = 0 In the presence of mean flow In the inner layer, stresses are nonzero.

10 Quasilinear inner layer calculations show that the MHD stresses from tearing modes are nonzero. The inner layer equations with flow shear: η (6) B r γ ργ 2 (1 + ig x )ξ(2) r = γ where g = (k V 0 ) (r s) = (1 + ig x γ [ (B )γ η (4) ) B r (B )ξ (2) r (1 + ig x γ ) + ik (B )2 γ ξ (2) r + 2B2 θ k 2 η r B(4) b s In the absence of mean flow < r B φ Br > = 0 and < rṽr Ṽφ > = 0. B φ and B r are out of phase. (J B) ] B B (4) r In the presence of mean flow, B φ and B r are in phase, and the stresses are nonzero in the inner region. F. Ebrahimi et al. PRL 2007, F. Ebrahimi et al. POP 2008

11 nonlinear single mode computation (with mode numbers m and k) mode coupling between the (m,k) mode and mean (0,0)

12 Single tearing mode does transport momentum during nonlinear evolution flow profile flattening Single helicity < J B > from computation agrees with the quasilinear analytical calculations. Flow profile is flattened around the resonant surface. The flow flattens very rapidly, in about one thousandth of a viscous diffusion time ( or 0.05 resistive diffusion times).

13 Multi-mode computations Many tearing modes are resonant at different locations Tearing modes are nonlinearly coupled

14 Flow profile is affected by the tearing fluctuations during sawtooth oscillations sawtooth crash Total fluctuation-induced Lorentz force is bi-directional Cause momentum transport as in experiment Many tearing modes are resonant at different locations Nonlinear coupling shifts the phase between J k1 and B k1

15 In the absence of m=0 modes (nonlinear mode coupling), fluctuation-induced forces are reduced. without m=0 modes without fluctuations The flattening of the flow profile does not occur. Momentum transport is greatly reduced, also shown experimentally.

16 Two-fluid relaxation with finite pressure Momentum generation Extended-MHD NIMROD code is employed. Full density and temperature equation with isotropic heat conduction are evolved. Pressure gradient is included in the equilibrium. m=0 is linearly unstable. MST parameters are used.

17 2-fluid nonlinear evolution of pressure driven mode 2fl MHD 2fl n=1 n=0 MHD E+ < Ṽ B >= ηj+ < J B > /en 2-fluid effect (Hall term) suppresses the MHD growth rate of m=1 core mode, however the growth rate of m=0 mode increases in Hall model (S= ). Deeper field reversal is obtained for higher S (S=10 6 ) computation.

18 Self-consistent mean flows are generated due to the two-fluid physics. x VP Re Vphi vs. i x VZ Re Vz vs. i ix/mx ix/mx The magnitude of these flows are about 1% of the Alfven velocity. Flows are mainly concentrated in the outer half of the plasma volume due to the m=0 mode perturbation.

19 Nonlinear evolution with Hall term has signature of diamagnetic rotation Z Re VR and VZ Re VR and VZ R Z Z Re VR and VZ R R

20 Momentum transport in disks

21 Motivations In astrophysical disks stresses from flow-driven MRI instabilities can contribute to momentum transport (a well-developed model). Does current-driven reconnection transport momentum in disks? Strong magnetic field results from the process of star formation. Magnetized disks around young stars can carry significant current. Current-driven instabilities may become important and contribute to momentum transport.

22 Motivations B Star accretion disk a disk threaded by magnetic field flux accumulated by the process of star formation Large scale magnetic field in disks 1-Mean magnetic field from interstellar medium can be advected by the infalling material leading to a flux concentration in the inner disk region 2- Local dynamo action in the disk F. Shu et al. ApJ 2007

23 We perform linear and nonlinear computations in a thick disk configuration.

24 Current-driven reconnection in a disk Flow-driven modes are stable (strong field). Force balance eqs: JxB = 0 and P = ρv 2 φ /r Both axial and azimuthal magnetic fields are imposed Ω2 B ϕ Bz Ω1 B z B ϕ

25 Resistive tearing modes become unstable. == J. B/B 2 λmax Consider perturbations: Ṽ, B e ik r e i(mφ kzz) periodic in φ and z directions { } m=0 -- 4, n=1,-1 =2π r Bz/ L Bφ Reconnection occurs at resonant surfaces where K B = mb φ /r k z B z = 0 Field line winding number q q = 2πrB z /LB φ = m/n k z = 2πn/L, m and n are azimuthal and axial mode numbers. modes with 5 q 5 linearly unstable

26 Tearing growth rates are increased with current gradient (λ max ) for a given Keplerian flow. λ max =17 n=1 n=-1 λ max =10 Tearing modes with m=0 3 (with n=1, n=-1) are linearly unstable with Keplerian flow profile V φ r 1/2. The growth rates are increased with λ max and the modes become double tearing modes.

27 The mode structure becomes double tearing with increasing current gradient Eigenmode structure for tearing modes single tearing (λ max =10) double tearing (λ max =17) = (B r + rs B r rs)/b r rs is the jump in the logarithmic derivative of B r across the resistive layer. For tearing mode to be unstable, > 0

28 MHD stresses from tearing modes are localized around the reconnection layer (nonlinear single mode computations). single tearing (λ max =10) double tearing (λ max =17)

29 Tearing modes are unstable in a disk configuration (multi-helicity computation). Modal magnetic energy (λ max = 14) m=0, 1 and 2 modes are linearly unstable. High m modes, m > 2, are nonlinearly driven. Total magnetic energy increases with λ max.

30 Maxwell stresses from multiple tearing modes transport momentum outward. Total stresses at time t2 during nonlinear phase Evolution of toroidal rotation with multiple tearing modes.

31 Summary Current-driven reconnection transports momentum in lab geometry (established through quasilinear theory and nonlinear computation). momentum transport much more rapid than explained by classical viscous forces nonlinear mode coupling enhances transport Self-consistent 2-fluid computations are performed to understand the origin of mean flows. Current-driven reconnection transports momentum in thick disk geometry. MHD stresses from tearing modes are localized around the reconnection layer.