Electron inertial effects & particle acceleration at magnetic X-points

1. Electron inertial effects & particle acceleration at magnetic X-points • Presented by K G McClements1 • Other contributors: A Thyagaraja1, B Hamilton2, L Fletcher2 • 1 EURATOM/UKAEA Fusion Association, • Culham Science Centre • 2 University of Glasgow • Work funded jointly by United Kingdom Engineering & Physical Sciences Research Council & by EURATOM • 8th IAEA Technical Meeting on Energetic Particles in Magnetic Confinement Systems, San Diego, October 6 2003

2. Introduction (1) • Magnetic X-points frequently occur in both fusion & astrophysical plasmas: • in tokamak divertor operation at plasma boundary; in tokamaks generally, due to classical & neo-classical tearing modes • energy release in solar flares1 • X-points have weakly-damped eigenmode spectrum, with ~Alfvén range1,2 - channel for dissipation of free energy; could affect evolution of X-point configuration, redistribute/ accelerate energetic particles &/or affect turbulent transport • 1 Craig & McClymont Astrophys. J. 371, L41 (1991) • 2 Bulanov & Syrovatskii Sov. J. Plasma Phys. 6, 661 (1981)

3. Introduction (2) • Craig & McClymont studied small amplitude oscillations of current-free 2D X-point in limit of incompressible resistive MHD: equilibrium B-field • B0 - field at boundary • R=(x2+y2)1/2 =R0 • Linearised MHD equations  discrete spectrum of damped modes in Alfvén range • y • x

4. Eigenvalue problem with electron inertia (1) • For reconnection events in tokamaks it is often appropriate to include e- inertia in Ohm’s law: • Writing where Bzis constant, putting • & linearising induction/momentum equations 

5. Eigenvalue problem with electron inertia (2) • Put r = R/R0, normalise time to R0/cA0where cA0 = B0/(0)1/2 • (in case of magnetic islands R0 should be « island width) • Introduce Lundquist number S = 0R0 cA0/ & dimensionless e- skin depth e =c/(peR0)  • seek azimuthally symmetric solutions  • Boundary conditions at r = 0 & r =1 • Solutions obtained numerically using shooting method & analytically in terms of hypergeometric functions

6. Discrete & continuum eigenmodes (1) • r • S=103, e=0.01 • Upper plots: discrete mode • Lower plots: continuum mode • Discrete spectrum: frequency 0& damping  increase with number of radial nodes • Continuum modes singular but field energy is finite • 0 = 0.8 •  = 0.2 • 0 = 0.8 •  = 0.2 • r • 0 = 5.0 •  = 5.0 • 0 = 5.0 •  = 5.0 • r • r

7. Discrete & continuum eigenmodes (2) • No finite 0 continuum exists in MHD model, except in ideal limit - shear Alfvén continuum • If e  0 finite 0 continuum exists for finiteS • 2 characteristic dimensionless length scales: • inertial length 0 e • resistive length (0 /S)1/2 • For 0 e < (0 /S)1/2 non-singular eigenfunctions exist; eigenfunctions become singular & spectrum continuous when inertial length ~ resistive length

8. S = 103 • 104Im(2) • e Discrete & continuum eigenmodes (3) • Consider • Problem becomes singular if Im(2) = 0 • 0 &  computed in limit e= 0for lowest frequency discrete mode; this mode is tracked as eincreases • Im(2) approaches 0, then remains there • discrete mode merges with continuum: but continuum exists for e below that at which curve crosses eaxis

9. Discrete & continuum eigenmodes (4) • Im(2) vanishes if • - contrasts with much weaker (logarithmic) scaling with S found by Craig & McClymont in resistive MHD case: • reconnection is Petschek-like (“fast”) • Continuum: 0  1/e [in physical units 0  min(pecA0/c,i)] • At sufficiently high 0 discrete spectrum does not exist; field energy must be dissipated at rate   1/S • reminiscent of Sweet-Parker (“slow”) reconnection: but absolute reconnection rate is extremely fast • Initial value problem of reconnection at X-points, taking into account e- inertial effects, addressed by Ramos et al.3 • 3 Ramos et al. Phys. Rev. Lett. 89, 055002 (2002)

10. Energetic particle production • Hamilton et al.4 - eigenmode analysis unaffected by presence of longitudinal (toroidal) field •  accelerating Ezfield; ion trajectories computed for solar flare parameters using full orbit CUEBIT code5 • Perturbed field computed using MHD eigenfunction • Acceleration found to be extremely efficient when (as in tokamak case) strong toroidal field is present  due to high E & suppression of drifts • 4 Hamilton et al. Solar Phys. 214, 339 (2003) • 5 Wilson et al. IAEA Fusion Energy FT/1-5 (2002)

11. Discussion • Existence of continuous spectrum for finite S &earises from interior singularity of eigenmode equation &is thus independent of boundary conditions • Intrinsic damping  = 1/(2Se2) of continuum modes distinct from continuum damping • Other physical effects (e.g. equilibrium currents, pressure gradients, flows) could drive instability & introduce gaps & gap modes in continuum (cf. TAEs) • Further details: see McClements & Thyagaraja UKAEA FUS Report 496 (2003), available on http://www.fusion.org.uk/

12. Conclusions • Spectrum of current-free magnetic X-point determined, taking into account resistivity & electron inertia • For finite collisionless skin depth, spectrum has discrete & continuous components; continuum modes arise from interior singularities that are not resolved by resistivity & have intrinsic damping • Eigenmodes have frequencies typically in Alfvén range - could redistribute or accelerate energetic particles & affect turbulent transport processes • Test particle simulations with fields corresponding to discrete resistive MHD X-point mode  efficient production of energetic particles if longitudinal B field is present