PROBLEMS OF GEOCOSMOS

Proceedings of the 7th International Conference "Problems of Geocosmos" (St. Petersburg, Russia, 26-30 May 2008)
Sufficient condition of CS instability. To obtain sufficient condition of instability one should find
res res 3
nontrivial solution of equation (5) with resonance term j = e∫ v fi d v . Resonance part of perturbed
res
distribution function f i can be fund by standard way from linearized Vlasov equation (Lapenta et al. 1997,
Daughton 1999, Silin et al. 2002). Also we applies Coulomb calibration for the perturbed vector potential,
div A 1 = 0 . One could consider two different polarizations of perturbations of vector potential A 1 . First
polarization is presented in the form A1 = A1 xe x + A1
ye y (Galeev and Zelenyi 1976, Silin et al. 2002).
Coulomb calibration then imposes the following condition of components of the perturbed vector potential:
A cosθ + A sinθ = 0 . Perturbations with such polarization are suppressed when θ → π 2 . Perturbation
1x y
with another polarization A1 = A1 ye y + A1
ze z (Lapenta and Brackbill 1997) can be developed also for
θ = π 2 . In this paper we will consider below both kinds of polarization.
For the polarization of vector potential 1 A in a form A1 = A1 xe x + A1
ye y , it is more convenient to consider
single equation for its magnitude
A = A + A instead of the system of two equations for each of these
2 2
1 1x 1y
components. It is straight forward, because 1x A and 1y A are linearly coupled (i.e. 1 1 tan
x y
Coulomb calibration:
− −
res
{ ( 1 4π z cos θ ) 0.5 ( y ) cos θ} ( , θ,
, )
2 2 2 2 4 1 2
1 0 0 1 1
A = − A θ ) by
d A dz − k + p B − c ∂j ∂ A A = − j z A t (6)
For another type of polarization A1 = A1 ye y + A1
ze z one could solve the single equation for A 1y like it was
done by Lapenta and Brackbill (1997) for the sausage mode ( θ = π 2 ). Contrary to previous works we
extended our results for perturbations at arbitrary angles (not only θ π 2
= )neutral plane ( , )
− −
res
{ ( 1 0.5 cos θ ) 0.5 ( ) } ( , θ,
, )
2 2 2 2 2 1
1y 0 z y 0 1y 1y
x y propagating:
d A dz − k + p B − c ∂j ∂ A A = − j z A t (7)
Because the resonant current densities,
t
∫
0
( − ) ( )
res
j ~ K t t′ A t′ dt′
1
, depends on
time, equations (6) and (78) could be
considered as evolutionary equations and
could be solved by method of finite
elements (Lapenta and Brackbill 1997,
Daughton 1999). Then one could obtain
corresponding eigenfrequencies and
growth rates as a function of TCS
parameters, wavenumbers k and
propagation angles θ .
The resulting growth rates as a function of
propagation angles θ for both
polarizations are shown in figure 3. As
one can see, both symmetric polarization
modes have equal positive values of Figure 3. Growth rate as a function of angle θ for two
growth rate in the case of tearing polarizations. Parameters have following values:
instability ( θ = 0 ), but asymmetric modes τ = 3 , L = 0.8ρi , kL = 0.3,
b n = 0.1 .
are suppressed when θ → 0 . While the
perturbation angle θ increases perturbation with the polarization A1 = A1 xe x + A1
ye y become suppressed.
But perturbation having another polarization ( A1 = A1 ye y + A1
ze z ) become correspondingly sausage or kink
instabilities ( θ = π 2 ). In the range θ ~ π 2 the perturbation growth rate become larger than the one of
tearing mode ( θ ~ 0 ). Thus the vector potential perturbations in propagating along the direction of the
currents are more probable than perturbation of the tearing mode type (i.e. waves moving along x direction).
Also the asymmetric perturbations at θ = π 2 have larger growth rate than the symmetric one like it was
obtained in Harris CS (Daughton 1999).
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