Chapter 4 NEUTRAL POINT INSTABILITY AND PLASMA PARAMETER 4.1 Introduction 4.2 Coalesence (neutral point) instability and tearing mode instability 4.3 Results and discussion a Growth rate of coalesence and tearing mode instability b Particle behaviour around an X type neutral point c Plasmoid particle energy d Plasma f3 parameter dependence on neutral point instability 4.4 Conclusion Cd
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Chapter 4
NEUTRAL POINT INSTABILITY AND PLASMA ~ PARAMETER
4.1 Introduction
4.2 Coalesence (neutral point) instability and tearing mode instability
4.3 Results and discussion
a Growth rate of coalesence and tearing mode instability
b Particle behaviour around an X type neutral point
c Plasmoid particle energy
d Plasma f3 parameter dependence on neutral point instability
4.4 Conclusion
Cd
4.1 Introduction
This chapter deals with the linear stability of a periodic structure, that
occurs in the tail of the magnetosphere as well as the plasma Bparameter
dependence on the growth rate of coalescence instability. The tearing mode
instability which leads to the formation of a plasmoid is also studied.
A tearing mode instability (TMI) can dissipate energy stored in the current
systems and magnetic field of the earth's magnetotail. The occurrence of the
collisionless tearing mode instability provides the first stage in the development of
a large scale magnetic reconnection in col1isionless magnetospheric plasmas. Coppi
et al.(1%6) and Schindler (1974) studied the collisionless tearing instability, in
which the dissipation that permits reconnection is the electron and / or ion Landau
damping resulting from finite particle inertia. They have offered an appealing
mechanism to explain the onset of magnetospheric substorrns in the near-earth PS.
Later collisionless tearing instahility was studied by many authors (Galeev et a\.,
1986, Wang et aL 1990, Pritchett et al.. 1991, Ding et al ",1992). Baumjohann
and Haerendel (1987) suggested that the most likely instability which leads to the
onset of tail reconnection, formation of near-earth X line and plasmoid ejection
was the ion tearing mode instability. The process of magnetic island coalescence
were studied by many authors (Biskamp and Schindler. 1971. Pritchett and Wu,
1979, Biskamp and Welter, 1980, Richardson et al ., 1989)..
Our study reveals that plasmoid is not a stable structure and propagates in
a field of changing magnetic field. It is found that particles should spend a fairly
long time near an X type neutral point and higher energy particles are not largely
confined in a tailward moving PMD. Coalescence of islands are favoured by
electrons whi Ie the onset of reconnection is favoured by ions (H+). Growth rate of
coalescence instability for low and high ~c showed that growth rate is high for
islands having smaller length as well as high ~ parameter which leads the islands
to coalesce (Renuka et aI., 1992).
4.2 Coalescence (neutral point) and tearing mode instability
TMI leads to the formation of a magnetic neutral Itne in the near-earth
region, whose earthward side belongs to an X type neutral line while tailward side
represents an 0 type neutral line. The 0 type neutral line is surrounded by closed
magnetic loops which contribute a magnetic island or a PMD.
We have studied the neutral point instability (NPI) of a periodic structure
(plasmoid) based on the theory of Biskamp and Schindler (1971). They discussed
the stability of two dimensional plasmas with neutral points in the frame work of
Vlasov theory. The stability criteria for the periodic structure (Fig 4.1) is derived in
which a typical electron larmor radius (away from the neutral points) is much
smaller than the characteristic length of the magnetic field.
They lIsed the ergodic behaviour of the system and also the condition
that particles can oscillate between two neighbouring X points and can reach an
adjacent X point. Biskamp and Schindler (\971) evaluated the gro~th rate of
NPI as
'Ie = k V h de (\00 ~tl/2e11_ t-d
~s e4.1
where kc = 2Jt fA, J~ being the wavelength of the periodic structure, Vlh is the
thermal velocity of particles in the PS and de is the average excursion of particles
around the neutral point given by ,
de = (Lps P)1/2 4.2
where Lps is the PS extent and p (d'~~c, v.L is thevQloc~':'J) is the gyroradius of
the particles.
To study the particle behaviour around an X type neutral point, the
fraction of the time the particle spends within the diffusion region and the total time
'to taken by the particles to traverse between two consecutive X type neutral points
are considered. The expression for 11 and 'to are obtained by considering an X
type neutral point where field lines are represented by a set of parabolas given by
I11 =
(100 ~)e de
and 'toI
=(kcYth11)
4.3
4.4
The growth rate can be used to study the effect of NPI on a single particle.
The particle energy in terms of its thermal energy is given by
W....::.L -W1hj -
4.5
where j =i,e .... and r = I if the instability changes the topology of the magnetic
field.
The growth rate of tearing mode (Schindler 1974) is given by
Yjtv . 3/2 L-5/2
thJ Pj z 4.6
where Lz is the PS half thickness.
4.3 Results and discussion
a Growth rate of coalescence and tearing mode instability
Using equation 4.1 we have studied the variation of growth rate of (Yc )
NPI of PMOs in the tail region with different PS extent Lps• gyroradius p and
longitudinal extent of the PMO 'A. Equation 4.6 is used to study the growth rate
ofTMI. The numerical values of the parameters used are
Lps =20,25,30.35,40 and 45 RE
T e = 3 X lOt> K, Ti = 107 K
Figure 4.2 shows the variation of the growth rate of electron NPI Yo.:
(solid line) and ion NPI Yic (dashed line) with different PS extent and plasmoid
extent I... It is evident from Fig 4.2 that the growth rate of electron NPI is greater
than that of ion NPI. Both Yo..: and YIC decreases with the increase of Lps and II. .
For example when I.. and Lps increases from 56-95 RE and 20-45 RE respectively
Yo.: and YIC decreases as indicated in Fig 4.2. For a constant II. (eg, II. = 56 RE) Yo.:
and YIC decreases when Lps increases from 20-45 RE. This confirms the result of
Nishida et al.( 1986) that the growth rate uf TM is enhanced in the region earthward
Figure 4.1. A sheet pinchwith a periodic structure
2L = lR EPS
0.921- --....---__-
4
3 /
....... 2""'iIII 2 IQlV}
~
Ia""'i"-
0.1o . 61<':-_~----,--_....___ __"T"-
0.06 0.08
! p (R E)
Figure 4.3(a). Variation of growth rate of tearing modeins tab i 1 i t Y (it) V5 pro ton gy r 0 r ad ius (~).
4. 4 ~ ----j
... -
.......... -- - ............. ---... ....
... ....... ............... ... y3R
E.............
... ........ ...
77R; '.. .............
.......... _9 5... RE
...
MIUOJtil
cpoM
'-,.IJ 2. 2 ' ......
-- ....
45403525 30L(RE)
o•61--_---. ---.- ---.-__--'
20
Figure 4.2. Variation of growth rate of neutralpoint instability ( --Ie ) with the plasma sheetextent (L) for different plasmoid length( A )
3
L pS = lRE
........ 0.8..;I
LlQj
l/)..;tI0..;"-~.a
0.3
6
2R E........
..;I
LlQj 2l/)
1.0I0..;'-'
"l>-"
0.6
2
0·5
0.0008 0.001
Ie (R E)
0.003
Figure 4.3(b). Variation of growth rate of tearing modeins tab iIi t Y (-{ ) V5 e 1 e c t ron gy r 0 r ad ius ('e ).
f;
of the original neutral1ine and reconnection starts in the middle tail.
Figure 4.3 (a)-(b) shows the electron and ion tearing mode growth rate vs
particle gyroradi us with different PS thickness. The growth rate of ions is found to
be greater than that of electrons and as the PS becomes thinner and thinner growth
rate increases. This result shows that the thinning of the PS favours the growth of
TM and hence the onset of reconnection and PMD formation. In the case of
coalescence instability, the growth rate of electrons are fo~nd to be greater than
ions. Thus it can be concluded that if ion tearing mode instability favours the onset
of reconnection, coalescence of islands are favoured by electrons.
b Particle behaviour around the X type neutral points
The stability of the island structure using coalescence growth rate
is discussed in this section. It is obvious from our results that plasmoid is not a
stable structure and the growth rate of smaller wavelength island structure have
larger growth rates that subsequently tend to coalesce.
In the small - B region of an X point, particles are not strictly trapped, but
have a certain probability of escape. moving along the separatrix. Using equations
4.2,4.3 and 4.4 the average excursion of electrons in the region around the neutral
point dc, the fraction of the time the particle spends in the small magnetic field
region or diffusion region 11 and the total time taken by a particle to traverse the
distance between two consecutive X type neutral points 'to are evaluated for chosen
Lps and l'v val ues.
For an electron gyroradius of 0.02 RE, as the PS extent becomes larger
68
and larger, the average excursion of electrons in the region around the neutral point
de increases for a fixed distance (I-.) between two X points as depicted in Table
4.1. For eg, when Lps increases from 20-45 RE, de varies from 0.63245 RE to
0.9487 RE for I-. = 30 RE.
Table 4.2 shows the variation of 11 with Lps =20-45 RE. P =0.02 RE
and with different p (= 0.018 - 0.028 RE) and constant Lps (= 45 RE). The time
an electron spends in the diffusion region 11 decreases with the increase of Lps and
hence de. As an example, when P = 0.02 RE and for a 10 RE change in Lps, 11
changes from 0.6296 -0.6059 sec. At Lps= 45RE, when p varies form 0.02
0.026 RE, 1\ changes from 0.596-0.6176 sec respectively. Hence.ll increases with
the increase of gyroradi us at a constant Lps val ue.
It is obvious that near an X type neutral line, particles should spend a
fairly long time. When the magnetic field remains the same and if we increase the
PS thickness TI is observed to decrease. This also shows the tendency of particles
to spend more time near the X type neutral points.
The total time L() taken by the electrons to traverse the distance between the
neighbouring X point is found to increase with the increase of I. and Lps. Table
4.3 shows this variation for a constant Pc ( =0.02 RE) and Lps =20-45 RE. For
example. when Lps =20 RE and in, changes from 30-105 RE. "tl) is 4.786 - J6.75
sec and if Lps changes from 20-45 REo and I. =30 RE, "to varies between 4.786
5.3479 sec respectively as depicted in Table 4.3. The increase in p or decrease in
magnetic field lead to a reduction of "to. These results show that the plasmoids
propagates in a field of changing magnetic field (different p) which agrees with the
result of Ugai ( 1985).
Table 4.1 - Variation of de with Lps at A. = 30 RE and Pe=0.02RE
Lps de
(RE) (RE)
20 0.6324
25 0.707]
30 0.7745
35 0.8366
40 0.8944
45 0.9487
Table 4.2 - Variation of 11 with Lp s at Pe =0.02 RE and
with Pe at Lps = 45 RE
Pc =0.02 RE Lps = 45 RE
Lps Yl Pc II(RE) (sec) (RE) (sec)
20 0.6660 0.018 0.5885
25 0.6458 0.020 0.5966
30 0.62% 0.022 0.6041
35 0.6167 0.024 0.6111
40 0.6059 0.026 0.6176
45 0.5966 0.028 0.6238
Table 4.3 - Variation of 'to with Lps and,,- at Pe =0.02 RE