1. MHD-Stability of magnetotail equilibria 2. Remarks
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1. MHD-Stability of magnetotail equilibria
2. Remarks on perturbed Harris sheet (resonance)
on Bn-stabilization of collisionless tearing
Cooperation: Joachim Birn, Michael Hesse
Karl Schindler, Bochum, Germany
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Motivation and Background
Quasistatic evolution TCS
reconnectionQuasistatic evolution
Topology conservingDynamics(instability)
TCS
reconnection
Ideal MHD instabilities in magnetotail configurations
Restricted set of modesComplex equilibrium
All modesRestricted set of equilibriaHere:
Here:
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Magnetohydrostatic Equilibria
Aspect ratio:
Constant background pressure included
Does the strong curvature at the vertex cause instability?
Does the background pressure destabilize?
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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)
Boundary condition: vanishing displacement vector, implying A1=0
The MHD variational principle (Bernstein et al. 1958) reduces to
A
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Model 1 (Voigt 1986)
full line: marginal Interchange criterion:
v1 vertex position
p0 constant background pressure
pm maximum pressure
,
symmetric modes(antisymmeric modes stable)
unstable stable
Numerical minimizations
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Model 2
Stable in all cases studied, consistent with entropy criterion
For small aspect ratio pressure on x-axis:
(Liouville 1853)
( )2
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Model 3
pressure on x-axis:
Choice:
Symmetric modes: stable for n<10
Stabilization by background pressure for n=14
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Conclusions from numerical examples
1. 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 configurationsare found stable.
2. The antisymmetric modes were found stable, except for model 3 when n > 6.Again, realistic cases ( ) are stable.
3. Open field lines, which are present in models 1 and 2, were found to be stable in all cases.
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Analytical approach
Euler-Lagrange
Reinterpreted as
(Hurricane 1997)
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The function
Singularitiesat
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?
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: leading terms cancel each otherwith
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?
The pressure gradient destabilizes (through )
while p0 stabilizes (through the compressibility term)
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Quasistatic evolution TCS
reconnectionQuasistatic evolution
Ideal MHDinstability
TCS
reconnection
Present results (2D equilibria) support
rather than
3D equilibria? Kinetic effects?
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Jmax
1
10
100
1000
104
0.01 0.02 0.03 0.04 0.05 0.06
p=0.1 p=0.2
perturbation amplitude |a 1|
3
2
1
0
2
0
-2
Z
0 -10 X
critical state
-20
-15
-10
-5
0
5
10
15
20
Z
-20
-15
-10
-5
0
5
10
15
20
-60-50-40-30-20-100
critical state
Z
-20
Jy
initial state
Quasi-static growth phase:conservation of mass, magnetic flux, entropy, topology
0.0
0.2
0.4
0.6
0.8
1.0
1.2
-1 -0.99 -0.98 -0.97 -0.96 -0.95 -0.94 -0.93 -0.92
p
A
Thin current sheet, loss of equilibrium
S ln( pV ) V ds /B
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References
Schindler, K. and J. Birn, MHD-stability of magnetotail eqilibria including a background pressure,J. Geophys. Res. In press, 2004
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Remarks on perturbed Harris sheet
Model:
Quasistatic deformation with p(A) kept constant
Conservation of topology or continuity (Hahm&Kulsrud)
A
p(A)
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Linear perturbation of Harris sheetcontinuous, topology changed
P(A) fixed
Field lines
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Linear perturbation of Harris sheetnot continuous, topology conserved
P(A) fixed
Surface current density
Field lines
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Perturbation of Harris sheetnot continuous, topology conserved
P(A) fixed
Surface current density
Linear approximationTail-approximation
Field lines
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<0
Kinetc variational principle for 2D Vlasov stability,ergodic particles
1
H,Py
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(Te finite)
Finite electron mass required? (Hesse&Schindler 2001)
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K normalized compressibility termnormalized electron gyroradius in Bn
Hesse&Schindler 2001
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it = 80it = 80
it = 100it = 100
it = 116it = 116
Hesse&Schindler 2001
1.000
10.000
100.000
0 10 20 30 40 50
electron Larmor radius in B z , x-axis(t=88)
x