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

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:

Magnetohydrostatic Equilibria

Aspect ratio:

Constant background pressure included

Does the strong curvature at the vertex cause instability?

Does the background pressure destabilize?

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

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

Model 2

Stable in all cases studied, consistent with entropy criterion

For small aspect ratio pressure on x-axis:

(Liouville 1853)

( )2

Model 3

pressure on x-axis:

Choice:

Symmetric modes: stable for n<10

Stabilization by background pressure for n=14

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.

Analytical approach

Euler-Lagrange

Reinterpreted as

(Hurricane 1997)

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?

: 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)

Quasistatic evolution TCS

reconnectionQuasistatic evolution

Ideal MHDinstability

TCS

reconnection

Present results (2D equilibria) support

rather than

3D equilibria? Kinetic effects?

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

References

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

Remarks on perturbed Harris sheet

Model:

Quasistatic deformation with p(A) kept constant

Conservation of topology or continuity (Hahm&Kulsrud)

A

p(A)

Linear perturbation of Harris sheetcontinuous, topology changed

P(A) fixed

Field lines

Linear perturbation of Harris sheetnot continuous, topology conserved

P(A) fixed

Surface current density

Field lines

Perturbation of Harris sheetnot continuous, topology conserved

P(A) fixed

Surface current density

Linear approximationTail-approximation

Field lines

<0

Kinetc variational principle for 2D Vlasov stability,ergodic particles

1

H,Py

(Te finite)

Finite electron mass required? (Hesse&Schindler 2001)

K normalized compressibility termnormalized electron gyroradius in Bn

Hesse&Schindler 2001

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