1. MHD-Stability of magnetotail equilibria 2. Remarks

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 andBackground Quasistatic evolution reconnection TCS Topology conserving Dynamics (instability) Quasistatic evolution reconnection TCS 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. A 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. Numerical minimizations Model 1 (Voigt 1986) symmetric modes (antisymmeric modes stable) v1 vertex position p0 constant background pressure pmmaximum pressure unstable stable full line: marginal Interchange criterion: ,

6. () 2 Model 2 (Liouville 1853) For small aspect ratio pressure on x-axis: 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 for model 3 with n = 2; x10 = 1; v1 = 2 and = 0:3 (curve a) and = 0:1 (curve b) The function Singularities at All three models give , General property?

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

12. Present results (2D equilibria) support Quasistatic evolution TCS reconnection rather than Ideal MHD instability Quasistatic evolution 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. 1 <0 Kinetc variational principle for 2D Vlasov stability, ergodic particles H,Py

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