theunited nations

educational, scientificand culturalorganization

abdus salaminternational centre for theoretical physics

international atomicenergy agency

SMR 1331/17

AUTUMN COLLEGE ON PLASMA PHYSICS

8 October - 2 November 2001

DYNAMIC CURRENT SHEETS - II

Lev Zelenyi

Russian Acedemy of Sciences, Space Research InstituteMoscow, Russia

These are preliminary lecture notes, intended only for distribution to participants.

strada costiera, I I - 34OI4trieste italy - tel. +39 04022401 I I fax +39 040224I63 - scijnfo@ictp.trieste.it - www.ictp.trieste.it

I N T E R N A . T I O N A I A T O M I C E N E R G Y A G E N C YUNITED NATIONS EDUCATIONAL, SctKiwnciiyf^CULTiJRAL ORGANIZATIONINTERNATIONAL CENTRI: FOR IMORETICAL PHYSICSI.C.T.P., P.O. BOX 586, 34100 TRIESTE, ITALY, CABLE: CENTRATOM TRIESTE

Lecture 1.

CURRENT SHEETS IN SPACE PLASMA

Non adiabatic particle dynamics andnumerical simulations

Lecture 2.

EQUILIBRIUM CONFIGURATIONS OFFORCED SHEETS

Role of trapped and transient populations.

Lecture 3.

FRACTAL STRUCTURES IN CURRENT SHEETS

Multiscale turbulence and current branching

Acknowledgements:

UCLA MPAE, LindauV.Peroomian J.BUchnerM.Ashour-Abdalla

Maryland University Space Research InstituteD.Schriver A.GaleevM.Sitnov A.MilovanovS.Sharma H.Malova

E.GrigorenkoCalabria UniversityG.Zimbardo Georgian ObservatoryP.Veltri A.TaktakishviliA.Greko

I N T E R N A T I O N A ! A T O M I C E N E R G Y A G E N C YUNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION

INTERNATIONAL CENTRE Fot H»EORETICAL PHYSICSI.C.T.P., P.O. BOX 586, 34100 TRIESTE, ITALY, CABLE: CENTRATOM TRIESTE

Lecture 2.

EQUILIBRIUM CONFIGURATION OF FORCED

(ANISOTROPIC) CURRENT SHEETS

1. Introduction.1. Isotropic and Anisotropic CS.2. Experimental observations of thin CS in magnetosphere.

2. Kinetic properties of CS1. Structure of the phase space2. Nonadiabatic trajectories and structure of currents.3. DIA - and PARAMAGNETIC currents in plasma.4. Phase mixing and trapped population.

3. Equilibrium solutions in quasiadiabatic approximation1. Limits of quasiadiabaticity2. Plasma sources (mantle)3. Basic equation and Liouville theorem4. Grad-Shafranov equations for plasma equilibrium.5. Limits of weak and strong anisotropies.6. Nonadiabatic effects for superstrong anisotropy.7. Scaling of CS thickness.

4. Influence of electric fields and Speiser acceleration.1. Nonadiabatic acceleration.2. Thinning of CS

5. Trapped population and windows for the existence ofequilibrium solutions

6. Conclusions: Possible scenarios of Disruption

Formation of current shett structure due tointeraction of the impinging plasma beams

Models of sheet

* Eastwood, [1972,1974]First studies of structure;

A lexeev and Kropotkin, [1971]Step-like field model;

* Francfort and Pellat, [1976]Analytically: z, = p(vT/vD)4/3;

* Pritchett and Coroniti, [1992]Numerically: L = p(vT/vD)4/\L*p(BjB0f

3 (6<bn).

* Burkhartetal, [1992]Strong ani^otropy i~(vT/vD)4/3

catastrophy at K—>1;

* Holland and Chen, [1993],Harold, Chen, [1996]Numerically: weak/strong

anisotropy, role of M9 JD;

Ashour-Abdalla et ah, [1994]Estimate L~p for weakanisotropy;

Kropotkin et al, [1997],Analitically: L~

Related worksSpeiser, [1965]Nonadiabatic orbits;

Sonnerup, [1971]Calculation of Iz;

Rich etal, [1972]Stress balance;

Cowley, [1978]Effect of pressure anisotropyon the equilibrium structure;

Lyons and Speiser, [1985]Conductivity;

Chen and Palmadesso, [1986]

Buchner and Zelenyi, [1986];Role of chaos;

Zelenyi and Savenkov, [1993]Estimate of AIZ for J Z « K ;

Kaufmann etal, [1993-1998]Taking into account trappedparticles;

Delcourtetal, [1993-199*]Centrifugally-driven effects;

Hesse, Kusnezova et al., [1997-1999]The role of electron effects.

EXPERIMENT:Existence of thin current sheets have been manifested in numerous experimentaldata:1. The thin current sheet 0.2 RE thick is embedded inside a plasma sheet with athickness about several RE:

• Sergeev, V.A., V. Angelopoulos, C. Carlson, and P. Sutcliffe,Current sheet measurements within a flapping plasma sheet, J. Geophys.Res., 103,9177, 1998.

• Sanny, J., R.L. McPherron, C.T. Russell, D.N. Baker, T.I.Pulkkinen, and A. Nishida, Growth phase thinning of the near-Earthcurrent sheet during the CDAW-6 substorm, J. Geophys. Res., 99, 5805,1994.

2. Current sheets with a thickness about ion gyroradius (-400 km) are observedin the near Earth tail during the growth and expansion substorm phases.

• Mitchell, D. et al. Geophys. Res. Lett., 17, 583, 1990. ~"~ ' ""*• Sergeev, V.A., D.G. Mitchell, C.T. Russell, and D.J.

Williams, Structure of the tail plasma/current sheet at 11 Re and itschanges in the course of a substorm, J. Geophys. Res., 98, 17345, 1993

• Pulkkinen, T. I., et al., In: Substorms 1, ESA-SP 335, Noordwijk, TheNetherlands, p. 131, 1992.

3. Thin current sheets are also observed in the distant tail (X>200 Re).• Pulkkinenn, T.I., et al. Geophys. Res.Lett., 20, 2427,1993. "• Gosling, J.Geophys.Res., 90, 6354, 1985.

4. It wa.s found that the thin current sheet is carried by cross tail quasi- adiabaticions [Pulkkinen et al.,1992; Lui A., JGR,98, 423, 1993].5. In several experiments TCS are found to display the overshoot structure

overshoot

Essential experimentalproperties of CS:small thickness,anisotropy, embeddingstructure,ions as carrier, presumablyovershoot structure.

\ ISEE-3 MAGNETOMETER 3 SEC AVEX . 2 1 JAhl 1983 83021 GSM COORDINATES

07:43:00•203.91

07 46:00•203.91

•9.1?•3.92

07A7.O0-203.92

•9.17•3 91

07:48:00

• 3 9 !

07:49:00•203.92

-9 17•3 91

BAKER AND PUUCXINEN 157

KK m '

300

200

100

n

-

. .Total

•

.Tail

a)

-

-

10 20 30

z[REl

10 15

--5

20

K [ mA n

~STJ

300

200

100

uc

rs. Total

/ VR i nsi- / / \

/ • • < \

/..-• VAdded

) 10

Radial distance

b)

sheet

20 30

[RE]

5

0

-5

(

1 ' ! 1

d)

^ K < 3

-Lon = 146" Modified field

) 5 10

Radial distance

15

[RE]

- 5Z[RE]

0

-5

20

Fig. 9. Left: Current intensities integrated over the current sheet thickness (in mA/m) in the ISEF. 1 meridian a) inthe basic T89 model, and b) in the modified model of the Pulkkinen et al. (1991). Total current (solid line), tailcurrent (dotted line), and ring current (dashed line) are shown for both cases. The additional current sheet is shownby dash-dotted line. Right: Field line projections to the ISEE 1 meridian computed using c) basic T89 model and d)the modified model. The region of chaotic electron motion for 1 keV electrons (see text) is shown hatched.

magnetic field by gradually increasing the cross-tail current,thinning the current sheet, and adding another localized tailcurrent system with increasing intensity in the basicTsyganenko model. This gives a global growth phase fieldmodel which compares favorably with measurements fromseveral spacecraft in the magnetotail. The modified, time-dependent Tsyganenko model allows a systematic mapping ofauroral features recorded by the auroral imagers into the near-tail region. These results, and a mapping study utilizingauroral images from Viking and the Tsyganenko 1987 model(Elphinstone et al., 1990), suggest that the auroralbrightenings correspond to regions of thinned and intensifiedcurrent sheet remarkably close to the Earth.

Specifically, Pulkkinen et al. (1991) include growth phaseeffects for the event- under study by varying the Tsyganenkomodel parameters and by modification of the functional form ofthe field components. The current sheet is thinned locally bymodification of the X and Y dependent function determining the

current sheet thickness in the model (see Pulkkinen 1991) withthe minimum thickness, location of the minimum thickness,and size of the thinned region as free parameters. The cross-tailcurrent is intensified by enhancement of the model tail currentby a constant factor. However, because the tail current peaksbeyond 10 Rg, its enhancement cannot represent stretching ofthe field in the near-Earth region between 6 and 10 RE. In orderto further enhance stretching close to the Earth, a new thincurrent sheet is added to the model. The form of the currentdistribution is similar to the model ring current term, but thepeak intensity, location of current maximum, and the currentsheet thickness are set according to the observed degree of fieldstretching.

A time-dependent model for the growth phase was constructedby linearly varying the parameters describing the enhancementof the currents during growth phase until substorm onset. Thefield at the beginning of the growth phase is represented by theunmodified Tsyganenko model using the highest level of

Parameter of Adiabaticity

(Buchner, Zelenyi: 1986,89)

Ouasiadiabatic approximation

(Ti « 5keV, L « 3000 km, Bn = lnT, Bo - 20nT,

- parameter of smallness

(opposite limit to the Guiding Center Theory:

Motion could be roughly separated on slow (x) and fast (z) parts

Action for fast variableT «i. £ J

- natural invariant of motion (Speiser 1965, Sonnerup 1970, BZ 1989)

Iz, - is conserved only approximately

QUASIADIBATIC APPROXIMATION :

J

• • • • ' - * 1 •'"*-*• ;

\rmts :<

i cucumber orbits

Speiser's vrMtsI ' I r I ^ I

0.0 0.5 1.0 S.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

1^=371/80

The principal types of"son trajectories in TCS

(X-Z plane)and corresponding

current densities |V(Z)

0.01.2

1.0

0 , 8 -

1.0 2.1) 3.0 4.0

ring orbits

5.01.2

y- 0J8

a . 4 -

fli

0.4

2 .00 - .

0 . 0 0 -

-2.00 ^ 1 ' I ' I r T2.00 3.00 4.00 5.00 6.00 7.00

- - X1 . 0 0 - ,

0.00

-1.00

-2.00 -1.00 0.00z

1.00 2.00

! i ! ' I ' l '

10.0 - ,

5.0 -15 IJ 1,5 2.0 2.5 3.0 3.5 4J 15 if)

a=sqrt(L|))

-2010

120

140

X

160

180

1100

2.00

1.00-

0.00 -jc*

-8.00 -4.00 0.00z

4.00 8.00

THE PARTIAL CUHRENTS ANDMAGNETIC FIELDS OF THE TRAPPED

POPULATION

Partial current densities of transient and trapped populationsS=10, K=l

2.00 —i

16,00

-2.00

0.35 —

0.30 —

0.25 —

0.20 —

0.15 —

0.10 —9 0.05 —

0.00

-0.05 -^

-0.10 —

-0.15 —

-0.20 —

-0.25 —

-0.30 —

\/ XTRAPPED PARTICl£f

i n^n-crossingregion

TCSmeanderingregion

10

PLASMA

IS DIAMA6A/FTIC

11

z

M

'a 6'• 9- K p a c u oft it a D a Ma r i i tmr bi f| TO K H n I I yiipy-

ocrpbiMyrvitOM.

R = H

12

BASICS

Forced current sheet is formed by interactionof impinging ion beams. Principal difference fromHARRIS equilibrium. - ^

Assumptions:

1DA

0v

d _ ddx dy j

Ion dominated CSBn=const^4) (quasineutral sheet)

Quasiadiabaticity Iz = — jzdz = const2TT

Contibutions from the transient and trappedorbitsPhase mixing of trapped population

GRAD-SHAFRANOV description for self-consistent equilibria

meanderingtrajectoryrotatingin Bn field

13

IEOR ¥ OF THIN CURRENTSHEETS (TCSs)

injected ions

tC/)<DO

IID

B={Bx(z),0,Bn]

IJ y ( /_ / " IJX

^ i j - / \ D is the

measure of ionstreaming;

v / is the thermal

velocity^

y;} is the flow driftvelocity

Equations of particle motion,

mr - [v x Bj

meBn

me(vzB(z)-vxBn)

v = vv5(z)me y

(2)

14

EXACT INTEGRALS OF MOTION:

1. Total energy-

2. Generalized momentum

mv<\ 2 2 2 2 / o \

mVv =mv — A (x,z\cy nx (4)

o

=> (^z>/(^x)»^

Motions could be decoupled

Therefore action integral of fast motion Iz isconserved:

1. = — q mv7dz « const (5)

Substituting (3) and (4) in (5) we receive

r ~j _ m r

- m ci~2TT^

' 2 :0 x

2 :

I ^me y )2dz'-c,z)j

2

Jz'

(6)

15

CALCULATION of lz

Due to homogeneity of TCS along X direction wehave from (4)

Ay(x,z) - Ay(xtz') =

Eq.(7) does not depend explicitly on X.Finally, expression for adiabatic invariant Iz is:

(8))

Limits of integration

^. = 2 JV^; limits of integration in (8) z}(z,vy,vjZ2

and z2(z, yy,v^ are determined accordingly to Eq.(8)v =o (9)

2 2y

v mc z'JB(z")dz"

J

(10)

and z2 are the solutions of the equation

\B{z")dz"=-vy±Jv2y+v

16

DISTRIBUTION FUNCTION

Distribution function of the injected ions (source) isa shifted Maxwellian

(12)

and vj_ are parallel and perpendicular components ofalong and across B:

(13)2 2 2V / / + V l = v 0

Two examples of the source distributionfunctions

1. 8i=VTi/vni<l (strong anisotropy),2. 82=V-I:/VD2>1 (weak anisotropy)

17

EXPRESSION FOR f(Iz ,v0)

In the region of strong r I the Iz can be easily calculated:

m

In

mIn

jvzdz

2K

V_L J rd(j) - m v

o In coo

mv

'o

Therefore,v, «

m(14)

/(v))

exp v mV

\D

Jmv

(15)

For \z\~L , where B is large, the motion is magnetized and

the adiabatic invariant Iz is equivalent to the magnetic2mvimoment ju = ——. Therefore

IB2mc

Iz=—ju (16)

However, during crossings of the neutral sheet (z=0) Iz isapproximately conserved, otherwise, // experiences strongjumps.

18

LIOUVILLE'S THEOREM

Distribution function in phase space

Pi

dF dfJ=ltS V dq, J dp.

= 0

f(r, v, 0

at ot m I c

19

APPLICA TION OF THE LIOUVILLETHEOREM TO ID TCS MODEL

Vlasov-Maxwell system for one-dimensional TCSdB Am r 3

]f(z9v)vyd v (19)dz vWe substitute in (19) the distribution function (16)taking into account two independent integrals ofmotion {v0, Iz}:

/ (z , v) = /(v0 ,12 (z, v)) = const (20)

Introducing the new variables

= z'0

.4/3, b = BIBo, w- v

1

vD 'n v(21)

D

we obtain from (19):

db r \2

.3/2

VD

\VAJ\wy exp<w

'I \

2/3 j yd w

(22)where j is the dimensionless adiabatic invariant Iz

wy-e2l3lb(C")dC"

V J (23)

20

GRA D-SHA FRA NO V EQ UA TION FORCOLLISIONLESS PLASMA EQUILIBRIA

Using in (22) the new variable, i.e. vector potential

we obtain Grad-Shafranov

equation for plasma equilibrium"2

1/3 \| 2b (TJ) =

(24)

whereM l

7] 00 00

= ±\di\dwx \wydwy J0 0 0 H \

Boundary condition:

D{'[boundary/ (25)

21

Undetermined parameter VA/VD could be found fromboundary condition (25):

! v 2V2 1/6

IVD3

n

1}

/ 4

it]

h erf (i

f ) ( 7 »

?"1 )

W x ) + F ( " ) ( 7 7 >"*))

(26)

Simplified form of pressure balance along X axiswas found by Burkhart et al. (1992):

A

VD

1 +s exp

1/2!

V

(27)

This is equivalent to marginal firehose stabilitycondition by Rich et ah (1972)

^ 1 = (28)

and/?^ are the parallel and perpendicularcomponents of the plasma pressure outside thecurrent sheet.

22

10

Equation for the pressure balance

Flux of momentum px out of the sheet is balanced withthe field line tension force Fx = (Bz I4TT)CIBX Idz

This balance after integrating over the height of TCSBXBZ = lvxvjind*v+ ]vxvzfoutd

3vVJJ <0 Vn >0

where A signifies the normalized variables (see BurkhartetaL, 1992) v = v/vT, 6 = B/

In the absence of scatteringfi«=fo», ~ exp{-v\ - (29)

Converting to pitch angle, gyrophase coordinates andintegrating over phase on can obtain:

B2 = jv1dv±dvII 12)fin , vnv;/<0

12)fooul , vu) = pnvn>0

Taking into account the incoming and outcomingdistribution functions (29) we find that

B2=v2D

VD

V n 1/2

23

11

VA _

VD

1 +-2 \ l / 2

V erf(e-1

COMPARISON OF THE PRESSURE BALANCECONDITIONS (analytical Eq. (26) and numerical

Eq. (27)).

2.50-1

0.00

• numerical solutions

0.0 2.0 40 6.0 8.0 10.0

24

Limit of strong anisotropys=vT/vD«l

3/2

7t

s)

\cT-d0_2V2

cos da,„

s PL

Scaling of TCS

L/pL~s4/3

(Francfort and Pellat, 1976)

Limit of weak anisotropy£=VT/VD»I

a+2y

- J

a 0

= max{o,a'-y- + z1 j

Scaling

L/pL~ 1(Ashour-Abdalla,Zelenyi et al.,1994)

1.0-1

0 8 -

0 6 -

0 4 -

0 . 2 -

00

_™//«4/3^=z/(s-"p0)

i I i I i I i I r - | i |

0.0 1.0 2.0 3.0, 4.0 5.0 6.0

I.O-1

0 . 8 -

0 6 -

OO1 ' I • r

3.0 4.0 5.0 6.0

25

s1

8

i

« / :

> / :

10.0 —

.-r -

20.0 —

0.0

L

L

%\\v\\\

Scaling of TCS

*pDs4/i' = pTe'/J

(Francfort and Pellat, 1976)

&PT (Ashour-Abdalla, Zelenyi et al., 1994)

8

0.1

\ — 0.25 8.(1 —

\ \ — • •—— 0.5

\ \ \\ \ ^ \

i

2.0 —

1.5 —

-

1 (1 —

0 S

—

0.3 —

-

0.2 —

-

Ill —

-

11.10 .

() MX

—

-

1 ' 1 ^1.0 2 . 0 , . 3.0 4.0 5.0

26

THE SELF-CONSISTENT TCS MAGNETIC FIELDSAND CURRENTS AT DIFFERENT VALUES OF

PARAMETER s=VT/VD

l.O—i

0 . 8 -

0 . 6 -

0.0

60.0-]

40.0-

r_>-

2 0 . 0 -

\

\

\

\

i

0.0

\

\— — - — .

1 ""1

1.0

— —

2.0 3.0

8

0.1

0.25

0.5

10

I4.0

T "I5.0

27

PLASMA DENSITY PROFILES

0.0 1.0

EMBEDDING of TCS into PLASMA SHEET0.35

0.30

0.25

8=2

0.05 H

0.00paramagnetic current prevails

1.20

1.15

1.10

1.05

1.00

0.95

0.0 1.0 2.0 3.0 4.0 5.0

28

THE UNIFIED PICTURE OF THIN ION CURRENT SHEET SCALING AT

DIFFERENT 8.

Non-jyliabati Quasi-adiabatic CS

A l z « IZ

L/R weakanisotropy

stronganisotropy

Bn/BO =Vt/Vd

Two kinds of thin ion sheets1. Quasi-adiabatic CS (s>Bn/B0)2. Non-adiabatic CS (s<Bn/B0).The analytical model works good in the quasi-adiabatic approximationand doesn't work in non-adiabatic case.

Lst

The thickness of PS and CSThe scaling of PS and CS is presented in this picture.Weak anisotropy (s>l). The role of diamagnetic currents M I great.The embedded structure is characteristic for the profiles of PS and CS.The thickness of PS is about RL [Ashour Abdalla, Zelenyi et al., 1994].Thethickness of CS is about 0.5RL due to opposite currents competition.

29

Uni-CUd scheme oi. t L *•/£* •+ +j uy une selfconsistent Current

thickness L ( £>n 54'*m^^& J V i s

L/Ro SURFACE

YW&

• x

30

Current Sheet Acceleration(Shabansky, 1971; Lyons and Speiser, 1982)

Ey > con\action —>Bn(x) field

Nonadiabatic(current sheet)acceleration

B \'o

7Qnasi-adiabaticacceleration

2pn

Vo

cEy

deHoffmann- Teller frameout-Win=2EAy

Ay = 2 pn

v'=vo+vc

_ VC> VC =

/77 0 C

AtV =Nonadiabatic energy gain

Problem: Wdirected

^.

Stationary frameAv = 2v vc vc

^ ( 2 v c ) 2 = 2 m / v c2

(Reflection from the Mirmi

moving with Vc(x))

= 2micE;

Role of scattering

31

Ey=O.U

Ey=0.25 Ey=0.5

Ey=LO

32

Influence of TRAPPED population

No transfer ]

transient population

source: Plasma mantle

trapppilk=l

lb Imaxsource: LLBL

Initial distribution of the transient population:

F ~ exp \-(VII-VDY + v± tvT

s=vt/vD-measure of source anisotropyv, is the thermal velocity,V£> is the drift flow velocity

The initial distribution function of the trapped

particles F~kexp(-vl/V>)

Transient (Speiser) orbits Trapped orbits

Iz « const

• Create cross-tail currentby competition of

dia- and para-magnetic effects

Iz = const

Redistribute cross-tailcurrent

Jv = 0y

33

INFLUENCE OF THE QUASI-TRAPPED IONPOPULA TfON ON THE TCS EQUILIBRIUM

Trapped population changes:•the amplitude of the self-consistent B(z)• spatial current structure

°decrease of Jdia at z » L°smearing of Jpara at z » 0

k=5

whole population

transient one

trapped one

8.0

i10.O

-0.20

Influence of trapped population (k=0.5; 1; 1.3; 1.4)

0.25-,

0.20

1.6 - ,

1.2 -

£> 0.8 -

0.4 -

0.5

1.3

1.4

0.0-f 1 1 r

16.0 0.0 4.0 8.0 12.0 16.(1

-0.05 -J

34

CO I

-1010

5

0

-5

-1010.0

7.5

5.0

2.5

0.0UTGSMXGSMYGSMZ

overshooti -+*

07:43:00-203.91

-9.16-3.94

07:44:00-203.91

-9-16-3-93

07:45:00-203.91

-9.17-3 93

07:46:00-203.91

-9.17-3.92

07:47:00-203.92

-9.17-3.91

D 7:48:00-203.92

-9.17-3.91

07:49:00-203.92

-9.17-3.91

35

Flickering ofCS structure

A =B

V/7

Vp±

t flickering

10 7* 40 6.0 8.0 10.0 12.0

"WINDOW" of SOLUTIONS

is the measure of ion streaming;

K is the measure of trapped population

K

20.0 —.

18.0 -

16.0 —

14.0 -

12.0 —

10.0 -

8.0 —

A - Equilibrium solutionsB - Quasi-equilibrium (flickering) solutionsC — No solutions

37

THE STRUCTURE OF BASIC EQUATIONS (l)-(4)

F(±) functions are the contributions of partial currents along the positive andnegative Y-axis.

1) The negative current ("diamagnetic"), is due to magnetization currentand ion drifts (gradient and centrifugal) in the non-uniform magnetic field, it iscreated by non-crossing parts of particle orbits and directed along the negativedirection (Figure 4).

2)The positive current ("paramagnetic") is carried by meandering part ofion orbits crossing the plane of magnetic field reversal.In figure below the main constituents of the cross-tail current are illustrated:A. Non-crossing region, where guiding center approximation could be applied.Schematically are shown: Larmor rotation of particles and their gradient,curvature and magnetization drifts. Net current in this region is negative whichillustrates the natural diamagnetism of magnetized plasma.B. Meandering region of non-guiding center motion. Positive paramagnetic(edge) current is carried by meandering segments of the transit (Speiser) orbits.Particles trapped within CS on specific meandering orbits carry the negativecurrent near the plane $z=0$ and positive current at the edges of their trappingregion. At the edges of CS partial compensation of paramagnetic current bydiamagnetic drift currents could occur

Anon-crossingregion

Bmeanderingregion

The segment of trapped orbit The segment of meandering (transient orin the negative Y direction trapped) orbit in the positive Y direction

38

Harris equilibrium.

ch2(z/L)exp

-F 0) 2 + (yy -

V27}

Te T, 'pi

Bulk velocity (VQ) along B does not influence current sheet structure.

Vollit(inh(z'• i',}

n V

uj - diamagnetic drift velocity

PROFILES of current and density COINCIDE

\z

-Bo

Bx = Bo tanh :0 U

20

n(z) = nc

£>

0

- duunamietic current

39

DIAMAGNETIC versus PARAMAGNETICcurrent sheets

/

= B/B

Harris sheet:

DIAMAGNET1C • Guiding center particles: ANISOTROPICCURRENT CURRENT

Non-adiabatic particles: PARAMAGNETICCURRENT

r 2!zi

made from Diamagnetic currents (supported by LOCAL pressuregradients)

Filamentation Kinking

JD J

FORCED CURRENT SHEETSsupported by paramagnetic currents

(due to pressure anisotropy)could be more STABLE vs. TEARING/KINK MODES

Harris CS -» Anisotropic CS

• Even infinitely small value of B,, drastically changes the structureof equilibrium.

• Meandering region becomes dynamically accessible from the lobes.

• Current sheet structure becomes dependent on vy <-> Vj_ anisotropy.

• Existence of Bn-»0 transforms the properties of equilibrium solution(even if Bn does not explicitly enter the final equations)

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14

CONCLUSIONSl.We consider the class of self-consistent equilibriums

of thin current sheets (L<pOj) with magnetic tensionbalanced by centrifugal force(i?V)Z?.

2. CS structure is determined by the competition ofIMA- and FARAmagnetic effects

=> The overshoot structure of the magnetic fieldprofile is characteristic for weak anisotropic TCS.

3. Flux of TRAPPED particles is a free parameterwhich essentially control the structure of CS,although <jy>TR=0

4. TRAPPED population suppresses* Paramagnetic current within CS* Diamagnetic current of transient particles at the edgesofCS

4a. As a result steady solutions exist in a window ofparameters (s,£)

c - measure of anisotropy, k - measure of the trappedpopulation

5. Trapped population occupies the well-defineddomains in the phase space which could beidentified by the in situ measurements of iondistribution function fj(v) in the vicinity of the fieldreversal.

(>. There are many indications on the existence of the(tanamics (flickering) quasi-equilibriums.

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REFERENCES for LECTURE 2

A. Results of Lecture 2 are published in following papers:

1. Zelenyi L.M., M.I.Sitnov, H.V.Malova, S.Sharma. Thin and superthin ion currentsheets. Quasi-adiabatic and nonadiabatic models. J. Nonlinear Processes in Geophysics, 7,3/4, 127-139, 2000

2. Sitnov M.I., L.M.Zelenyi, H.V.Malova, S.Sharma. Thin current sheet embedded withina thicker plasma sheet: Self-consistent kinetic theory. J.Geophys.Res., 105, A-6, 13.029-13.044, 2000

3. Malova H.V., M.I.Sitnov, L.M.Zelenyi, S.Sharma. Self-consistent model ID currentsheet: the role of drift, magnetization and diamagnetic currents. AGU, Geophys.Monograph 118, Magnetospheric Current Systems, 313-322, 2000.

4. Zelenyi, L.M., A.Galeev, and C.F. Kennel, Ion Precipitation From the Inner PlasmaSheet Due to Stochastic Diffusion, J. Geophys. Res., 95, 3871, 1990.

5. Ashour-Abdalla, M., L. M. Zelenyi, V. Peroomian, and R. L. Richard, On the structureof the magnetotail current sheet, Geophys. Res. Lett., 20, 2019, 1993.

6. Zelenyi, L. M., B. V. Savenkov. Violating of Quasiadiabaticity During the ParticleMotion in Magnetic Field Configurations with Strongly Curved Field Lines. Plasma.Phys. Rep. (Transl.from Russian), 19, 712, 1993.

7. Savenkov, B.V., L.M. Zelenyi. Application of Separatrix Crossing Theory toNondiffusion Model of Current Sheet Resonance. Geophys. Res. Lett., 23, 3255 , 1996.

8. Vainshtein D.L., L.M. Zelenyi, A.I. Neishtadt, B. V. Savenkov. Jumps in an adiabaticinvariant with small initial values. Plasma Phys. Rep., 25, 4, 290-303, 1999, Transl. fromFizika Plazmy, 25, 4, 333-337, 1999.

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B. Additional important publications:

1. Harris, E. G., On a Plasma Sheath Separating Regions of Oppositely Directed Magneticfields, Nuovo Chimento, 23, 115, 1962.

2. Eastwood, J. W. Consistency of fields and particle motion in the 'Speiser' model of thecurrent sheet. Planet. Space Sci., 20, 1555-1568, 1972.

3. Speiser T. W. Conductivity without collisions or noise. Planet. Space Sci., 18, 613,1970.

4. Sonnerup B.U.O. Adiabatic particle orbits in a magnetic null sheet. J. Geophys. Res.,76,8211,1971.

5. Syrovatsky, S.I. On the Appearance of Current Sheet in Plasma with Frozen-in StrongMagnetic Field. Sov. Phys. JETP, 60, 1727, 1971.

6. Burkhart G.R., Drake J. F., Dusenbery P. B., Speiser T. W. A particle model formagnetotail neutral sheet equilibria. J. Geophys. Res., 97, 13799-13815, 1992.

7. Rich F.J., Vasyliunas V.M., Wolf R. A. On the balance of stresses in the plasma sheet.J. Geophys. Res., 77, 4670-4676, 1972.

8. Pritchett P. L., Coroniti F. V. Formation and stability of the self-consistent one-dimensional tail current sheet. J. Geophys. Res., 97, 16773-16787, 1992.

9. Lyons, L. R., T. W. Speiser. Evidence for current sheet acceleration in the geomagnetictail. J. Geophys. Res., 87, 2276, 1982.

10. Frankfort P, R.Pellat. Merging in collisionless plasma. Geophys. Res. Lett. 3, 433,1976.

11. Pulkkinen T. I., Baker D. N., Mitchell D. G., McPherron R. L., Huang C. Y., Frank L.A., Thin current sheets in the magnetotail during substorms: CD AW 6 revisited, J.Geophys. Res., 99, 5793, 1994.

12. Holland, D.L., J. Chen. Self-consistent current sheet structures in the quiet-timemagnetotail. Geophys. Res. Lett., 20, 1775-1778, 1993.

13. Kaufmann, Richard L., Ball, Bryan M., Paterson, W. R., Frank, L. A. Plasma sheetthickness and electric currents. J. Geophys. Res., 106, A4, 6179-6194, 2001

14. Shabansky V.P. Space Sci. Res., 3,12, 1971.

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C. TEXTBOOKS

1. Bateman G., MHD instability, The MIT Press Cambridge, Massachusetts, and London,England, 1979.

2. Dendy R. Plasma Physics: an Introductory Course, Published by Press Syndicate of theUniv. Cambridge, Cambridge University Press , N.-Y., USA, 1993.

3. Krall N.A., A.W. Trivelpiece. Principles of plasma physics, McGraw-Hill BookCompany, N.-Y., 1973

4. Nicholson, D.R., Introduction to plasma theory, Ed. By J.Wiley&Sons, Inc., N.-Y.,USA, 1983

5. Sturrock P. A., Plasma Physics, Published by the Press Syndicate of the University ofCambridge, Cambridge University Press , N.-Y., USA, 1994.

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