A model for thinning of the plasma sheet

Planet Space Sci., Vol. 25, pp. 703 lo 710. Perpmon

A MODEL

Press, 1977. Printed in Northern Ireland

FOR THINNING

OF THE PLASMA

SHEET

J. K.cm0 Cooperative Institute for Research in Environmental Sciences, University of Colorado, Boulder, CO 80302, U.S.A. and J. IL KAN. A. T. Y. LUI aud S.-I. AKASOFIJ Geophysical Institute, University of Alaska, Fairbanks, AK 99701, U.S.A.

(Received 9 December 1976) Abstrac&A one-dimensional model for thinning of the plasma sheet is developed on the basis of launching a fast mode MHD rarefaction wave propagating in the tailward direction along the plasma sheet. Behind the rarefaction wave the pressure is reduced, leading to thinning of the plasma sheet and also to an Earthward plasma flow with a speed on the order of the sound speed ac. The plasma sheet thickness is reduced by a factor of 2 if an Earthward plasma flow speed of 0.8~ is induced. The predictions of the model are in reasonable agreement with observations.

1. INTRODUCTION

One of the most dramatic

changes in the plasma sheet during substorms is plasma sheet thinning. This feature has been studied most extensively by Hones and his colleagues (Hones et al., 1967; Hones ef al., 1971a,b; Akasofu et al., 1971; Meng ef al., 1971; Lui et al., 1975a,b) and by several others (Lazarus ef al., 1968; Fairfield and Ness 1970; Aubry and McPherron, 1971; Nishida and Lyon, 1972). In understanding the cause of plasma sheet thinning, it is important to know how the plasma sheet becomes deformed during thinning. It has been suggested by Hones (1973) that plasma sheet thinning observed at the Vela satellite distance (- 18R,) can be explained qualitatively in terms of the formation of a magnetic neutral line in the near-Earth plasma sheet, (namely within the Vela satellite distance) and the subsequent loss of plasma tailward of the neutral line. However, an extensive search for the proposed neutral line by Lui et al. (1976, 1977a,b) indicates that there is no clear indication of the predicted large-scale changeinthemagneticfieldtopologywith. in the lunar distance. Further, Lui et al. (1977c) found that a Sunward plasma flow is most frequently detected during plasma sheet thinning in the midnight sector within X- -3OR,, namely at distances well beyond the predicted location of the neutral line. Therefore, Lui et al. (1977a) concluded that thinning of the plasma sheet can best be described in terms of dejktion. Figure 1 shows schematically 703

how the thinning process develops in the plasma sheet. Plasma sheet thinning is initiated in the nearEarth magnetosphere and proceeds rapidly tailward at a speed of a few hundred km/set. At the onset of thinning (Stage l), the inclination of the magnetic field at the inner portion of the plasma horn (location A) decreases while the field dips slightly southward at the outer portion of the plasma horn (location C). The field near the equatorial plane becomes dipole-like (location B), iu association with the Earthward injection of plasma sheet particles (Vasyliunas, 1968; DeForest and McIlwain, 1971). As the plasma sheet thinning proceeds tailward (Stage 2), the field at C starts to rotate slightly northward while northward rotations of the field at A and B continue. Southward dipping of the field occurs at location D and E at this time; the largest dipping is experienced near the time of the crossing of the boundary of the plasma sheet. The third stage depicts the plasma sheet thinning reaching X= -3OR,. Positive B, values are observed at the midplane (location F), in spite of a large southward dipping occurring further down the tail (location G). Further, thinning begins at about the onset time of substorms (Hones et al., 1971a; Hones et al., 1976) and continues until about the maximum epoch; it has been reported by a number of workers that the plasma sheet recovers suddenly at about the maximum epoch or during the recovery phase. Thus, thinning of the plasma sheet is the most important feature in the plasma sheet during

3. K. CHAO,3. R. KAN, A. T. Y. LUI and S.-I. AKASORI

704

the idea of launching a rarefaction wave into the tail. The latter suggested that a slow mode rarefaction wave “propagates into the tail accelerating plasma and flux toward the nightside, relaxing the tail magnetic con~guration, and producing an expansion of the plasma sheet.” Apparently they were trying to explain expansion of the plasma sheet in terms of a slow mode rarefaction wave. 2. A PL.ASMA SHEET MODEL

_. PtASMA

A

SWEET

THtNNiNG : STAGE 2

c

B

D

F

E

dF=_._z

;=

;\=I<

i-p

FIG.

1.

RESSMZ DURING

A

SCHEMATIC

CHANGE PLASMA

IN

DIAGRAM

TO IL.LUSTRA~

THF3 PLASMA

SHEET

THINNING

SHEET AT

G

THE PROG-

CONFIGURATION THE

SUBSTORM

EXPANSION

Points A-G are used to show the magnetic fieid changes at various locations of the magnetosphere. The thin lines mark the plasma sheet boundary in the previous stage. The dotted region is the plasma sheet and its horn. The variations of the magnetic field direction at each of the four stages are further illustrated at the bottom of the figure (Taken from Lui et al., 1977a). the expansive phase when the aurorai buige extends rapidly poleward and an asymmetrical ring current is formed by an ‘injection’ process of plasma from the plasma sheet into the Van Allen belt, namely the inflation of the inner magnetosphere, which accounts for a sign&ant portion of the substorm energy (Akasofu, 1968). These findings suggest that thinning of the plasma sheet results from Earthward displacement of plasma from the plasma sheet to the inner magnetosphere, resulting in the inflation of the inner magnetosphere and aurora1 activity. In this paper we examine thinning of the plasma sheet in terms of deflation of the plasma sheet due to the propagation of a fast mode rarefaction wave in the anti-solar direction. It is shown that characteristics of plasma sheet during thinning are consistent with such an idea. Kropotkin (1972) and Coroniti and Kennel (1973) are the first to consider

We assume a one-dimensional model of the plasma sheet which probably is valid in the distant tail of the magnetosphere. Further discussion of this assumption will be given in the last section. The boundary of the plasma sheet is a tangential discontinuity. The reasons for this are: (i) The magnetic field across the boundary changes magnitude; (ii) no plasma flow of speed greater than 100 kmlsec across the boundary has been observed; (iii) the plasma density increases discontinuously by an order of magnitude across the boundary, from the tail lobe to the plasma sheet. It follows from (iii) that the plasma sheet boundary cannot be a shock wave, because the density jump for an MHD shock wave with r=$ cannot be larger than 4 (Landau and Lifshitz, 1959). Nor can it be an Alfven discontinuity which requires that the density remain unchanged. This leaves only one possibility, namely that the boundary is a tangential discontinuity. For a tangential discontinuity, total pressure has to be balanced across the boundary. In the lobe, the plasma pressure is small compared with its magnetic pressure. Therefore, the lobe pressure PL can be approx~ated by the magnetic pressure BT2/Sw,where BT denotes the magnetic field intensity in the lobe. On the other hand, inside the plasma sheet, the parameter /3 = Po/(Bo2/8?r) is at least one order of magnitude larger, where PO and B,, denote the plasma pressure and magnetic field in the plasma sheet, respectively. In Fig. 2, the pressure balance in the cross-section of the magnetotail is sketched. Figure 3 illustrates the one-dimensional plasma sheet configuration. The magnetic field is parallel to the neutral sheet along.the X-axis. The Earthward end of the plasma sheet is assumed to be bounded by a “piston” while the other end is bounded by the X-type reconnection line. The purpose of the hypotheticat piston is to launch a low’ pressure region in the near-Earth plasma sheet. Further discussion on the “piston” will be given in the last Section. Before the onset of a substorm, the plasma sheet

A model for thinning of the plasma sheet solar

wind

p ssure

PS kconstant1

705

3. GENERATION

OF A FAST MHD RAREFACTION WAVE IN THE PLASMA SJXEET

At time Earthward

piston

t = 0, the with

a

speed

U

is assumed (in

the

to move

negative

T?

direction)

3 ‘f I

FIG.

2. A

CROSS-SECTION OF THE DISTANT PHERICTAIL.

MAGNETOS-

plasma sheet, magnetospheric lobe and solar wind are in static pressure balance across their boundaries.

The

is in static equilibrium, P,O = PO + g

i.e.

(1)

= PL = constant,

where the plasma pressure is assumed isotropic. The subscript ‘i’ denotes the total pressure and the subscript ‘0’ indicates the initial state before any disturbance is generated in the plasma sheet. The pressure PL responds only to changes of the lateral pressure of solar wind P, which is assumed to be constant. Thus, PL is also constant. T / -- ---------

A

-iii

% _-___________ FE

pio =PL la)

-2

as shown in Fig. 3ib). As a result; a rarefaction wave is generated and propagates tailward along the plasma sheet. The front F1 of this rarefaction wave propagates at a characteristic speed a0 in the T? direction. Since /3 (= 8?rP,,/BoZ) is greater than unity in the plasma sheet, the rarefaction wave along B must be the fast mode of MHD waves (Kantrowitz and Petschek, 1966). The slow and AlfvCn modes in this case are transverse waves. The solution of the MHD fast rarefaction wave propagating along the magnetic field in a 6 > 1 plasma is identical to the rarefaction wave of ordinary gas dynamics (Landau and Lifshitz, 1959). Therefore, the characteristic wave speed for the fast mode must be the sound speed, Q~, i.e. a,= m where y is the ratio of specific heats, P,, and p0 are the pressure and density, respectively, in the unperturbed plasma sheet. Behind the rarefaction wave, a plasma flow in the negative X direction is induced. The pressure PI and the density p1 behind the rarefaction wave are lower than the ambient pressure and density. Here, the subscript “0” and “1” indicate the stata: before and after the rarefaction wave, respectivtl;jr! If the piston moves at a speed U where UC 2a,/y - 1, then the solution for the plasma flow speed behind the rarefaction wave is shown in Fig. 4(a). If the piston moves faster than 2a,/y - 1, then the solution for the speed of the plasma flow is shown in Fig. 4(b) where the region between the piston and the trailing edge of the wave is a vacuum. The other parameters behind the wave are given by (Landau and Lifshitz, 1959) p1= p,[l-;(YPI = P,[l-

1) (u(/QJ’(Y-l)

(2)

;(r - 1) Jul/aJY’(+),

(3)

where y = 2 for the rarefaction wave in an isotropic plasma. Although the plasma in the unperturbed plasma sheet is collisionless, the plasma pressure has been tb) assumed isotropic in our model. This assumption is FIG. 3. (a) THE PLASMA SHEET MODELED AS A ONEjustifiable on the grounds that instabilities tend to DIMENSIONAL SLAB. reduce the degree of anisotropy. Behind the The magnetic field vanishes in the center of the slab. The rarefaction wave, the pressure along the Xpressure is in static balance where PL is the lobe magnetic direction will be reduced more than that in the pressure. (b) A RAREFACTIONWAVE GENERATED INTHEPLASMA SHEET perpendicular direction according to the CGL douDUE TO EARTHWARD MOTION OF THE INNER EDGE EDGE AT ble adiabatic variation (Chew et al., 1956). The A SPEED u. resulting pressure anisotropy is limited to a low F~ is the front of the rarefaction wave propagating along level in a high /3 plasma by the mirror instability the tailward direction.

J. K. CHAo, J. R. KAN, A. T. Y. Lur and S.-I. AKAsoru

706

From the frozen field condition, pendicular to B leads to

compression

The pressure balance condition

per-

can be written as

(P,Jl = p,,+g=

p,,.

From (6), (7) and (8), one can solve for Bz, Pz and pz. The results are given by

Bz* Pm -=8~ O+BJ P,O p”=(l+&) I%. 4. (a) Tuls RAREFACTION

VBL.OClTY

WAVE

FOR

OF THB A

SPEED

PLASMA

BEHIND

OF

INNBR

THB

U12a,/y-1. The flow is in the Earthward direction. (b) Trm vnLoclTy OF THE PLASMA IS THE SAME AS FOR Ur2a,/y-1.

FIG.

EDGE

3(a)

(Spitxer, 1965). Therefore, we assume that the plasma is isotropic behind the rarefaction wave. OF PLASMA SHEET DUE PERPENDKULAR COMPRESSION

4. THINNING

TO

Since the pressure behind the rarefaction wave becomes lower than that across the plasma sheet boundary, the boundary is forced to move toward the neutral sheet. Thus, the plasma sheet behind the rarefaction wave will be thinned by the perpendicular compression. The front of the rarefaction wave continues to propagate tailward while the plasma sheet is thinned behind the wave as shown in Fig. 1. During the perpendicular compression of the plasma sheet, the plasma is assumed to follow the CGL double adiabatic variation. The total pressure PC2after compression becomes anisotropic, i.e.

(Pt2)= PLZ+g

O1

THE

,

where the subscript ‘2’ denotes the state after thinning and Bz is the magnetic field in the thinned plasma sheet. Since the compression is perpendicular to the B direction, the first adiabatic invariant (the magnetic moment) is conserved, i.e.

(6)

01) where

The equation of state for P2 during a perpendicular compression is given by (Thompson, 1962, page 164) PL2242 -_Pz5

_

PI3 PI5

(12) .

Hence, using (lo), (11) and (12) we find 42

=

(13)

PI

The pressure anisotropy in the thinned plasma sheet can be obtained from (10) and (13) as given by z=

(_%)vz(P!?)-l”.

(14)

Using the results obtained above, the properties of the thinned plasma sheet will be studied and compared with observations. In particular, we examine two cases corresponding to U = 0.2a, and U= 0.8a,. Figure 5(a) shows plots of the ratios of B2*/BoZ, P121Po, PII~IP~.pzlpo and PdPlz

as func$ons

of 13

for lJ= 0.2a,. Figure 5(b) is identical to Fig. 5(a) except U= 0.8a,. Figure 5(c) plots the same ratios as in Fig. 5(a), but as functions of V/a, for /3 = 1. It can be seen from these figures that the rarefaction and compression change the unperturbed state by

707

A model for thinning of the plasma sheet

0:/B’,

92’Po

%2%

. p2 ‘4

%2’P2

(a)

FIG. 5. (a) (=

8d’JBo2)

PLOT.3 OF THE

(b)

RATIO OF

B22/BoZ, ~H/~IJ, ~~~z/po,

V=

0.2ao.

(b) F’LQTS

lo-20% for U= 0.2~~~. However, for U= 0.8ao, the changes from the unperturbed state are 50% or more. The magnetic field in the plasma sheet can be described by B. = B, tanh (43) where z is the distance from the neutral sheet in units of Earth radius RE, the half width of the plasma sheet is 3R, and B, is determined by the value of B,, at the plasma sheet boundary. Figure 6 shows the profiles of the magnetic field before and after thinning. The solid curve is for the unperturbed magnetic field profile, while the dashed curve is the magnetic field profile in the thinned plasma sheet for U= 0.8~. The discontinuity in each profile represents the plasma sheet boundary which has been assumed a tangential discontinuity. The half thickness of the thinned plasma sheet is reduced from 3 to 1.6R,. In obtaining the thickness of the thinned plasma sheet, we have assumed that the change in thickness is proportional to the change in the magnetic flux (i.e. Sz - SB) under the assumption of the conservation only

of magnetic

P21Po

AND

42/p12

AS

OF THE SAME RATIOS AS IN FIG. 5(a) FOR U = OFTHESA~%ERATIOSASIN~G.~(~)WTASFUN(TIIONSOFU/~,FOR~=~.

FOR

flux in a one-dimensional

plasma

sheet

FUNCTIONS

OF fi

0.8ao. (c) Prors

Table 1 shows the profiles of various quantities in the thinned plasma sheet as a function of z2. The coordinate z in the first column gives the corresponding positions of z2 in the unperturbed plasma sheet. model.

FIG. 6. Tnn MAGNETIC FIELD PROFILES BEFORE THINNING (SOLID CURVES) AND THE SAME PROFILE AFlER THINNING (DASHED CURVE).

Note

the half thickness reduces from 3 to 1.6& case of U = 0.8~.

in this

708

J. K. CHAO,J. R. KAN, A. T. Y. Lur TABLE

0.6 1.2 1.8 2.4 3.0

0.28 0.58 0.92 1.28 1.67

4.4 3.7 3.1 2.6 2.3

1

0.93 0.79 0.65 0.55 0.48

0.44 0.41 0.37 0.34 0.32

0.83 0.76 0.69 0.63 0.59

0.48 0.52 0.57 0.62 0.66

DISCUSSIONS

We have presented a model for thinning of the plasma sheet on the basis of launching a fast mode MHD rarefaction wave in the tailward direction along the plasma sheet. For simplicity, the rarefaction wave has been assumed to be a plane wave in the above analysis. This assumption can be seen to hold approximately in the distant plasma sheet from the following considerations. A non-planar fast MI-ID rarefaction wave is assumed being launched in the Earthward end of the plasma sheet and is propagating in a model field of the plasma sheet (Kan, 1973) as shown in Fig. 7. The wave fronts are constructed by using Huygen’s principle. It is seen that the wave front becomes increasingly more planar and more perpendicular to the magnetic field as the wave propagates tailward. The piston speed U appropriate for plasma sheet thinning can be estimated from previous observa-

and

S.-I. AICASOFU

tions. Taking the proton temperature of the plasma sheet during thinning to be 1 keV (Hones er al., 1971b) which corresponds to a sound speed a,= 400 km/set, and noting the observed plasma flow speed V during thinning to be 100-500 km/set (Lui ef al., 1976a), we find that the flow speed V is less than 2a,/(y- l), and thus must be equal to the piston speed U. The observed flow speed suggests that the ratio V/a, ranges from 0.25 to 1.25. Therefore, the assumption U= 0.8a. as the piston speed in the previous ‘calculation is justified. At the piston speed of 0.8ao, the plasma sheet thickness is reduced by a factor of 2. From Vela observations, Hones et al. (1971b) have deduced that the plasma sheet in the midnight sector is reduced from a thickness of 5 or 6R, to 1 or 2R, during thinning. This result has been confirmed by Lui et al. (1975b) who have further noted that a similar but less drastic reduction in the plasma sheet thickness occurs in the evening and morning sectors during plasma sheet thinning. Thus, the predicted reduction in the plasma sheet thickness is adequate for the evening and morning sectors of the plasma sheet but is somewhat less than the observed values in the midnight sector. However, this discrepancy may be attributed to the assumption of our model in which the magnetic flux is conserved during thinning. This assumption leads to a rather large increase in the magnetic field

1

0

-1

FIG.~.THEFRONTS OFTHEFAST

MHD RAREFAC~ONWAVE(THINCIJRVES)PROPAGATINGINAMODEL TAILMAGNETICFmXD(HEAb'YCURVR).

z*

709

A model for thinning of the plasma sheet

intensity, as shown in Table 1. Since observations indicate no large increase in the magnetic field intensity during thinning, an appreciable amount of magnetic flux must have been removed from the plasma sheet during thinning. Therefore, the amount of thinning predicted by the model is underestimated. To estimate an upper limit on the reduction of the plasma sheet thickness by taking into account the magnetic flux leakage from the plasma sheet, we assume (i) that @= 1 at the boundary of the plasma sheet before thinning (ii) that the magnetic flux is completely removed from the plasma sheet after thinning. The resulting reduction of the plasma sheet thickness is given in Table 2. At V/u0 = 0.8, the plasma sheet thickness is reduced by about a factor of 4. This reduction can be considered as an upper limit of thinning and can indeed account for the observed reduction in the midnight sector. The findings of Lui et al. (1976d), namely, that plasma flow during thinning is directed mostly Sunward, and the success of the present model suggest strongly that thinning of the plasma sheet results from loss of plasma from the plasma sheet towards the Earth. Further, since the inflation of the inner magnetosphere accounts for a significant portion of substorm energy (Akasofu, 1968), the expansive phase of the magnetospheric substorm may be described in terms of a deflation of the plasma sheet, namely, in terms of Earthward displacement of plasma, from the plasma sheet in the magnetotail to the inner magnetosphere, resulting in the inflation of the inner magnetosphere and aurora1 activity. In this connection, it may be noted that Kan and Akasofu (1976) suggested that the source of auroral energy is the kinetic energy of the Earthward flow of plasma from the magnetotail. We shall mention only briefly a possible mechanism of generating the rarefaction wave. The launching of a rarefaction wave can be considered as a consequence of a reduction of the cross-tail current which is a part of the two solenoidal circuits generating the magnetic fields in the magnetotail. A reduction of the current in such inductive circuits will induce a large voltage which can in turn cause a high speed (Ex B) drift motion of plasma. This TABLE 2 LJ/ao 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

L&o 0.60 0.54 0.48 0.43 0.38 0.34 0.30 0.26 0.23 0.20 Lo = thickness of the plasma sheet before thinning. L2 = thickness of the plasma sheet after thinning.

means that the magnetic energy in the inductive circuit is converted into the kinetic energy of the plasma. A detailed discussion of the generation mechanism for the rarefaction wave in the plasma sheet will be presented elsewhere. Acknowledgements-This work was supported in part bv the United- States Energy Research &id Development Administration Contract number E (4.5-l) 2229 and bv the National Science Foundation, Atmospheric Sciences Section Grant ATM 74-23832 and Grant ATM 75031614AOl. REFERENCES Akasofu,

S.-I. (1968). Polar and Magnetospheric

Sub-

storms. Reidel, Dordrecht.

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710

J. K. CHAo, J. R. KAN, A. Tt. --r. -LUI ana* -Y.-I. - AKASON A

Landau, L. D. and Lifshitx, E. M. (1959). Fluid Mechanics, Chapter X, Addison-Wesley, New York. Lazarus, A. J., Siscoe, G. L. and Ness, N. F. (1968). Plasma and magnetic field observations during the magnetosphere passage of Pioneer 7, J. geophys. Res. 73, 2399. Lui, A. T. Y., Hones, E. W., Jr., Venkatesan, D., Akasofu, S.-I. and Bame, S. J. (1975a). Response of the plasma sheet at -18R, to sudden southward turnings of the interplanetary magnetic field, J. geophys. Res. 80,929. Lui, A. T. Y., Hones, E. W., Jr., Venkatesan, D., Akasofu, S.-I. and Bame, S. J. (1975b). Complete plasma dropouts at Vela satellites during thinning of the plasma sheet, J. geophys. Res. 80, 4649. Lui, A. T. Y., Meng, C.-I. and Akasofu, S.-I. (1976). Search for the magnetic neutral line in the near-earth plasma sheet: 1. Critical re-examination of earlier studies on magnetic field observations, J. geophys. Res. 81, 5934. Lui, A. T. Y., Meng, C.-I. and Akasofu, S.-I. (1977a). Search for the magnetic neutral line in the near-earth plasma sheet: 2. Systematic study of IMP-6 magnetic field observations. J. geophs. Res. In press.

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