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

Lps 'to (sec)

(RE) A=30 53 60 63 73 83 93 105

(RE)

20 4.7860 8.4554 9.5719 10.0509 11.6400 13.2410 14.8360 16.7507

25 4.9406 8.7286 9.8811 10.3755 12.0220 13.6687 15.3160 17.2918

30 5.0671 8.9520 10.1340 10.6411 12.3298 14.0186 15.7081 17.7345

35 5. 1737 9.1404 10.3472 10.8651 12.5893 14.3]36 16.0386 18.1077

40 5.2663 9.3039 10.5323 I 1.0594 12.8145 14.5696 16.3255 18.4315

45 5.3479 9.4486 10.6955 11.2308 13.0131 ]4.7954 16.5785 18.7172

The increase in II as well as the decrease in 'to, which is caused by the

reduction in the magnetic field (or increase in p), contributes to the enhanced

growth rate of NPI for smaller wavelength structure. If the magnetic field is

constant, 'to increases with the increase in PS extent. This confirms that the neutral

points are unstable or in other words the PMD is not a stable structure.

The most fundamental possibility about the stability of the island

configuration, is either the break up of islands into smaller ones or the coalescence

of two islands into a larger one depending on the wavelength of the original

islands. The latter process is the more frequent one, since shorter wavelength

tearing modes usually have larger growth rates. These results are in agreement

with those of Biskamp and Welter ( 1980).

c Plasmoid particle energy

The effect of NPI on particles in the event A IJan 28 (-0825 - 0847

UT) I and G IMarch 22 (-03 l4 - 03 16 UT) I in CDAW 8 plasmoids (Richardson et

al (l 989) are studied in this section. Our study shows that higher energy particles

are not largely confined inside a tailward moving PMD.

Eqn 4.6 help us to study the effect of NPI on a single particle. Table

4.4(a)-(b) indicate Ye..:, Yic and the corresponding particle energies respectively for

the A and G plasmoids. For the event A plasmoid, the electron energies We are in

the range 1.14 - 1.83 keY at Lps =20 RE and 0.97 - 1.03 keY at Lps =45 RE

respectively. While for the event G plasmoid, We is 2.44 - 2.69 keY at Lps =20

RE and 2.08 - 2.26 keV at Lps =45 RE. The decrease in growth rate as well as the

increase in PMD extent reduce the electron and ion energies. As an example,

70

Table 4.4 (a ) - Particle energy for the event A plasmoid

0 .. = 95 RE)

Particle Lps=20RE Lps=45RE

P de y W Y w(RE) (RE) lO-4sec-1 (keY) 10-4 sec- I (keY)

.0087 .1988 6.9550 1.1389 2.8507 0.9710

.0093 .2050 7.1948 1.1504 2.9466 0.9765

.0010 .2191 7.7464 1.1664 3.1698 0.9887

Electron .0011 .2274 8.0718 1.1758 3.3009 0.9955

.0012 .'2367 8.4394 1.1865 3.4488 1.0032

.0013 .2473 8.8535 1.1984 3.6173 1.0116

.0016 .2733 9.9068 1.8332 4.0381 1.0315

.0685 1.7562 3.6033 7.2116 1.:Y869 504121

.0728 1.8103 3.7363 7.3081 1.4363 5.4637

.0832 1.9353 4.0524 7.5157 1.5517 5.5787

Proton .0896 2.0083 4.2391 7.6368 1.6198 5.6438

.0971 2.0903 4.4510 7.7737 1.6966 5.7192

.1059 2.1832 4.6944 7.9243 1.7851 5.8009

.1295 2.4137 5.3117 8.3013 2.0071 6.0003

Table 4.4 (b) - Particle energy for the event G plasmoid

(A = 65RE)

Particle Lps=20RE Lps=45RE

p de. Y W Y W(R

E) (RE) lO-4sec-l (keY) 10-4 -1 (keY)sec

.0008 .2001 10.2324 2.4421 4.1945 2.0775

.0009 .2089 10.7379 2.4653 4.3987 2.0943

.0011 .2191 11.3194 2.4914 4.6331 2.1131

Electron .0012 .2310 12.0043 2.5209 4.8506 2.1250

.0013 .2450 12.8174 2.5546 5.2361 2.1583

.0018 .2829 15.0481 2.6113 6.1313 2.1946

.0020 .3025 16.2147 2.6829 6.5982 2.2491

.0021 .3099 16.6637 2.6986 6.7777 2.2602

.0693 1.7666 5.3040 15.4518 2.0409 11.5815

.0756 1.8452 5.5904 15.7340 2.1461 11.7393

.0833 1.9359 5.9253 16.0573 2.2688 11.9184

Proton .0925 2.0399 6.3147 16.4254 2.4108 12.1125

.1040 2.1636 6.7859 16.8605 2.5817 12.3552

.1387 2.4984 8.1021 18.0266 3.0544 12.9701

.1585 2.6709 8.8040 18.6243 3.3037 13.2766

.1664 2.7368 9.0765 18.8526 3.3999 13.3919

for A. =65 RE (G p1asmoid), if Ycr: increases from 1.023 x 10-3 sec-1 to 1.666 x

10-3 sec- 1 We changes from 2.44 - 2.69 keV respectively. With the increase of PS

extent as well as PMD extent the particle energy decreases. This shows that higher

energy particles are not confined to a tail ward moving PMD. This result is

in close analogy with Scholer et al. (1984) that after substorm onset higher energy

particles are not accelerated in the PMDs. However, Scholer and Jamitzky (1987)

reported that higher energy particles are largely confined to an emerging PMD.

Energy of ions is found to be greater than that of electrons. The energy

obtained by an individual particle due to reconnection depends on the time it

spends in the region of rapid magnetic field variation as well as the distance

traveled in the direction of the electric force. The present study also shows that the

ions have spend a relatively longer time near the diffusion region and hence the

higher energy. Lui et al.( 1984) considered the acceleration region a's an X type

magnetic neutral line where magnetic field merging occurs, accelerating particles

to energies in the keV to MeV range in agreement with our results. Because of

reconnection in the PS, plasma is accelerated away from the X line on both sides.

This acceleration is directly proportional to the external driving force (Usadi et aI.,

1993).

d Plasma f3 parameter dependence on NPI

We studied the electron plasma ~ parameter dependence of the growth rate

of NPI for the two PMD regions considered above.

71

Ding et a1. (1992) studied the ~ dependence ofthe collisionless tearillg

instability at the dayside magnetopause and showed that it is controlled by the

magnetic field profile across the magnetopause current sheet and larger plasma ~

may not always lead to a smaller tearing growth rate.

The growth rate of electrons using EqnA.I for the NPI of events A and G

plasmoids have calculated to study the plasma ~ parameter dependence on the

growth rate and hence the island coalescence. Figure 4A(a)-(b) shows the

variation of growth rate of NPI with ~e at constant number density (nd and

temperature (Tc) of events A and G respectively at Lps = 20 RE and 45 RE

respectively. The growth rate shows an enhancement with an increase of Be at

constant nc and T e.

If the magnetic pressure .is greater than the kinetic pressure (~« I), the

plasma is compressed to a higher density and temperature. In low Bcases, Scholer

and David (1987) found an immediate coalescence of two PMDs due to the higher

compressibility of the plasma.

The calculated values of growth rate for low and high ~e show that Yc is

high for islands having smaller length as well as high ~ parameter which leads the

islands to coalesce. This conflrms the result of Biskamp and Welter (1980) and

Richardson et a1. (1989) that small islands have larger growth rates which

subsequently tend to coalesce.

_.,I':"

(a)

0.57

(b)

0.320.16

(je

6T =3.4x10 K

e

0.04

9

7..- -

(a)-

r-iI

UQ)

Ul 4(V)

Ior-i

0.27

~e

5T =6. 4x10 K

e

0.11

.-II

U(1) 7(/) (b)

("')

Io.-I

~u4

10

.-IIU(1)(/)6

<:I'Io.-I

.....u3

0.07

-...,.P1

0.064 0.12 0.015 0.035 0.036

f3e e

10- r-i

n ==. 5 -3 I

1.5xlO m Ue Q)

Ul2xlO 5m- 3

"'" 7 n =I e0r-i

~ "'?-o

3

0.06 0.08 0.05 0.11 0.27

f3e ~e

o0.04

1l..-::::======::=====:J0.025

.-I 9I

U(1)(/)

<:I'Io24

-..,.U

Figure 4.4. variation of growth rate of neutral pointinstabi1i ty ( {, ) vs plasma ~~ parameter

(a) at constant temperature (T )(b) at constant number densitye(n )e

4.4 Conclusion

The present study shows that plasmoid is not a stable structure and

propagates in a field of changing magnetic field. 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 while the onset of reconnection are favoured by ions (H+). Growth rate

of coalescence instability for low and high ~c show that growth rate is high for

islands having smaller length as well as high ~ parameter which leads the islands

to coalesce. Particle energy is found to decrease with an increase in PMD extent.

~ .-)

/ ...,