1. Electron inertial effects & particle acceleration at magnetic X-points Presented by K G McClements 1 Other contributors: A Thyagaraja 1, B Hamilton 2, L Fletcher 2 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 8 th 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 flares 1 X-points have weakly-damped eigenmode spectrum, with  ~Alfvén range 1,2 - 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 B 0 - field at boundary R=(x 2 +y 2 ) 1/2 =R 0 Linearised MHD equations  discrete spectrum of damped modes in Alfvén range x y

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 B z is constant, putting & linearising induction/momentum equations 

5. Eigenvalue problem with electron inertia (2) Put r = R/R 0, normalise time to R 0 /c A0 where c A0 = B 0 /(  0  ) 1/2 (in case of magnetic islands R 0 should be « island width) Introduce Lundquist number S =  0 R 0 c A0 /  & dimensionless e - skin depth  e =c/(  pe R 0 )  –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  0 = 0.8  = 0.2  0 = 0.8  = 0.2 r  0 = 5.0  = 5.0 r  0 = 5.0  = 5.0 r S=10 3,  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

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 finite S 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. Discrete & continuum eigenmodes (3) Consider Problem becomes singular if Im(  2 ) = 0  0 &  computed in limit  e = 0 for lowest frequency discrete mode; this mode is tracked as  e increases Im(  2 ) approaches 0, then remains there 10 4 Im(  2 ) ee S = 10 3  discrete mode merges with continuum: but continuum exists for  e below that at which curve crosses  e axis

9. 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(  pe c A0 /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 Discrete & continuum eigenmodes (4) 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 E z field; ion trajectories computed for solar flare parameters using full orbit CUEBIT code 5 4 Hamilton et al. Solar Phys. 214, 339 (2003) 5 Wilson et al. IAEA Fusion Energy FT/1-5 (2002) 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

11. Discussion Existence of continuous spectrum for finite S &  e arises from interior singularity of eigenmode equation & is thus independent of boundary conditions Intrinsic damping  = 1/(2S  e 2 ) 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