PROBLEMS OF GEOCOSMOS

Proceedings of the 7th International Conference "Problems of Geocosmos" (St. Petersburg, Russia, 26-30 May 2008)
Tearing instability. Tearing instability is a
symmetrical mode of plasma perturbation (perturbed
component of vector potential A1 ( z) = A1 ( − z)
) which
is periodical along magnetic field direction
~ exp( ikx x − iωt ) . Growing of tearing perturbation is
supported by resonance interaction of unmagnetized
particles with plasma waves in the central region of CS
( B x ~ 0 ). For the first time Harris CS model (Harris
1962) was used for instability investigations where
electrons were considered as resonance particles
(Coppi et al. 1966). But the nonzero normal
component of magnetic field Bz ≠ 0 always is present
in Earth’s magnetosphere. Electrons become
Figure 1. Electron and ion current
magnetised by B z magnetic component and resonance
density and plasma density
interaction is dominated by ions (Schindler 1974,
Galeev and Zelenyi 1976). The stabilization effect of magnetized electrons (Lembége and Pellat 1982)
makes Harris CS stable to tearing perturbation (Pellat et al. 1991).
There exist several models of CS equilibrium which are alternative to Harris CS with Bz ≠ 0 (Sitnov et al.
2006, Birn et al. 2004, Zelenyi et al. 2004). Also one can investigate tearing-like perturbation along
directions which are not coinciding with magnetic field lines and propagate under the angle
θ = arctan ( k y kx
) to it. In our work we study tearing perturbation in model of TCS (Zelenyi et al. 2004) for
which lager store of free energy was found (Zelenyi et al. 2008).
Kink and sausage instability. Symmetric wave perturbation exp{ i − iωt} two values of arctan ( k y kx
)
kr has two well known modes for
θ = . If θ = 0 (direction along magnetic field) perturbation is named tearing and
if θ = π 2 (direction along the current j y ) perturbation is named sausage. Sausage mode was investigated
for Harris CS (Lapenta and Brackbill 1997, Daughton 1999). But several features of structure can be a reason
of the difference in behaviour of this mode in TCS .
Asymmetric mode ( A1 y ( − z) = − A1 y ( z)
) with θ = π 2 is named kink perturbation. This mode is
investigated for Harris CS (Kuznetsova and Zelenyi 1985, Daughton 1999). It was shown that for TCS this
perturbation has larger growth rate than for Harris one and larger period of oscillation (Artemyev et al.
2008a). Because of substantial storage of free energy in TCS model with Bz ≠ 0 the modes with angle not
only θ = π 2 can exist in the neutral plane of this CS. Therefore, it this work we take into account various
modes of instability with different values of angel θ .
Energy principle. Necessary conditions of CS instability. In this section we consider energy function of
( 2)
the second order of perturbation W for TCS and for Harris CS with Bz ≠ 0 . We start from standard
( 2)
equation for W (Schindler 2006):
2
2
( 2) B1 1 f̃ 1 j
W = ∫ dr − ∑ 8π 2 ∫ j ∂f0 j ∂H 0 j
1 ∂j
q
0 2
j res
drdp −∑ 1d d −∑
1 f1 j d d
j 2c
∫ A r p
∂ 0
j c ∫ vA r p
A
(3)
= z i − iωt f̃ = f − ∂f ∂A A
res
− f . The resonance part of perturbed
Here we use ( ) { }
A1 A1 exp kr and 1 j 1 j ( 1 j 0 ) 1 1 j
res
distribution function f 1 j can be obtained for each models of CS independently. In the presence of the
normal component of magnetic field B z electrons are magnetized near the neutral plane and perturbation of
their density corresponds with magnetic field perturbation
Schwartz inequality to rewrite equation (4):
n1 = n0 ( B1z Bz
) . In this case one can use
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