Electron inertial effects

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(x2y2)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 Ohms law
• Writing where Bz is constant,
  putting
• linearising induction/momentum equations ?

5
Eigenvalue problem with electron inertia (2)

• Put r R/R0, normalise time to R0/cA0 where 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)

• S103, ?e0.01
• 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

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 finite S
• 2 characteristic dimensionless length scales
• inertial length ?0 ?e
• resistive length (?0 /S)1/2
• For ?0 ?e lt (?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

• discrete mode merges with continuum but
  continuum exists for ?e below that at which curve
  crosses ?e axis

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 Ez field 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
  ?e arises 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