1. Karl Schindler, Bochum, Germany
1. MHD-Stability of magnetotail equilibria
2. Remarks
on perturbed Harris sheet (resonance)
on Bn-stabilization of collisionless tearing
Cooperation: Joachim Birn, Michael Hesse

2. Motivation and Background
Quasistatic
evolution
TCS
reconnection
Topology conserving
Dynamics
(instability)
Quasistatic
evolution
TCS
reconnection
Here:
Ideal MHD instabilities in magnetotail configurations
Restricted set of modes
Complex equilibrium
All modes
Restricted set of equilibria
Here:

3. Magnetohydrostatic
Equilibria
Constant background pressure included
Aspect ratio:
Does the strong curvature at the vertex cause instability?
Does the background pressure destabilize?

4. The MHD variational principle (Bernstein et al. 1958) reduces to
Boundary condition: vanishing displacement vector, implying A1=0
w(A1) positive for all perturbations A1 and all field lines A
Is necessary and sufficient for stability w.r.t. arbitrary ideal-MHD perturabations
(Schindler, Birn, Janicke 1983, de Bruyne and Hood 1989, Lee and Wolf 1992)

5. p0 constant background pressure pm maximum pressure
Numerical minimizations
Model 1 (Voigt 1986)
symmetric modes
(antisymmeric modes stable)
v1 vertex position
unstable
stable
p0 constant background pressure
pm maximum pressure
full line: marginal Interchange criterion: ,

6. ( ) Model 2 (Liouville 1853) 2 For small aspect ratio
pressure on x-axis:
( ) 2
Stable in all cases studied, consistent with entropy criterion

7. Model 3
pressure on x-axis:
Choice:
Symmetric modes: stable for n<10
Stabilization by background pressure for n=14

8. Conclusions from numerical examples
Symmetric modes on closed field lines:
Stability consistent with entopy criterion: dS/dp<0 nec. and suff. for stability.
Unstable parameter regions are stabilized by a small background pressure.
Instabilities arise from rather rapid pressure decay with x. Realistic configurations
are found stable.
The antisymmetric modes were found stable, except for model 3 when n > 6.
Again, realistic cases ( ) are stable.
Open field lines, which are present in models 1 and 2, were found to be
stable in all cases.

9. Analytical approach
Euler-Lagrange
Reinterpreted as
(Hurricane 1997)

10. The function
Singularities at
The function for model 3 with n = 2; x10 = 1; v1 = 2 and = 0:3 (curve a) and = 0:1 (curve b)
All three models give
, General property?

11. 5. Discussion
Why does the strong curvature at the vertex not cause instability in typical cases?
: leading terms cancel each other
with
Why does the background pressure stabilize, although increasing pressure often destabilizes?
The pressure gradient destabilizes (through )
while p0 stabilizes (through the compressibility term)

12. Present results (2D equilibria) support
Quasistatic
evolution
TCS
reconnection
rather than
Quasistatic
evolution
Ideal MHD
instability
TCS
reconnection
3D equilibria? Kinetic effects?

13. Quasi-static growth phase: conservation of mass, magnetic flux, entropy, topology
Thin current sheet, loss of equilibrium

14. References
Schindler, K. and J. Birn, MHD-stability of magnetotail eqilibria including a background pressure,
J. Geophys. Res. In press, 2004

15. Remarks on perturbed Harris sheet
Model:
Quasistatic deformation with p(A) kept constant
p(A) A
Conservation of topology or continuity (Hahm&Kulsrud)

16. Linear perturbation of Harris sheet continuous, topology changed
P(A) fixed
Field lines

17. Linear perturbation of Harris sheet not continuous, topology conserved
P(A) fixed
Field lines
Surface current density

18. Perturbation of Harris sheet not continuous, topology conserved
P(A) fixed
Linear approximation
Tail-approximation
Field lines
Surface current density

24. Kinetc variational principle for 2D Vlasov stability,
ergodic particles
H,Py 1
<0

25. (Te finite)
Finite electron mass required? (Hesse&Schindler 2001)

26. K normalized compressibility term
z normalized electron gyroradius in Bn
Hesse&Schindler 2001

27. W i
t = 80
t = 100
t = 116
Hesse&Schindler 2001