Inference of the Angular Velocity of Plasma in the Jovian

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 98, NO. A1, PAGES 67-79, JANUARY 1, 1993

Inference of the Angular Velocity of Plasma in the Jovian Magnetosphere From the Sweepback of Magnetic Field

KRISH• K. KHtmANA AND MARGARET G. KIVELSON •

Institute of Geophysics and Planetary Physics, University of California at Los Angeles

Particle observations from the dayside magnetosphere of Jupiter have shown that the plasma subcoro- rates beyond a radial distance of •20 Rj. However, no information is available on the azimuthal velocity of plasma on the nightside of Jupiter owing to the unfavorable viewing georr•tries of the Voyager spacecraft. In this paper, we follow the approach of Vasyliunas (1983) to calculate the torque applied by the magnetic field on the plasma and calculate the resulting plasma angular velocity for various assumed outflow rates. Thus for a given value of the outflow rate we can calculate the sector-averaged angular velocity of plasma from the observed values of the azimuthal component of the magnetic field (B•) in the lobes and the normal component (B,) in the plasma sheet. We develop new techniques to determine the normal and azimuthal components of the magnetic field in a plasma sheet coordinate systerm From the Voyager 2 data, it a•s that the corotarion in the postmidnight quadrant of the magnetosphere can be mainmlned up to a radial distance of •50 Rj if the outflow ram in that quadrant does not exceed 2.5 x 10 •9 amu/s. For outflow rates exceeding 2.5 x l0 •9 amu/s in that quadrant, partial corotation would result. Our conclusions differ from those of Vasyliunas •cause we have reanalyzed the Voyager data and have improved the estimates of B• and Bz. A comparison of results from Voyager 1 and 2 is presented.

•NTRODUCTION

It is well known that the rapid rotation of Jupiter confines the magnetospheric plasma to a radially extended plasma sheet and stretches the magnetic field lines through the action of centrifugal force on the magnetospheric particles. However, it may not be fully appreciated that the rapid rotation of Jupiter may also be responsible for a moderately large sweepback of its magnetic field lines. Viewed from above the north pole of Jupiter, the magnetic field lines would be seen to spiral out of their meridians making an angle of ~40 ø within a radial distance of 100 R•. The sweepback of the magnetic field lines is thought to result mainly from the radial flow of plasma moving outward toward lower centrifugal potential. Conservation of angular momentum requires that outward moving plasma reduce its angular velocity. Because the plasma is "frozen" to the magnetic field, it drags the field lines with it, bending them in the process. The field in turn applies stress on the plasma to make it corotate with Jupiter. This interaction results in a net outward transfer of angular momentum in the Jovian magnetosphere, the angular momentum being extracted from the Jovian rotation [Hill, 1979; Vasyliunas, 1983; Huang and Hill, 1989].

An alternative but equivalent picture of the outward transfer of the angular momentum can be presented in terms of electric fields and currents imposed on the outflowing magnetospheric plasma. If the ion-neutral collisional frequency is sufficienfiy large in the ionosphere, the plasma in the ionosphere corotares with the neutral atmosphere. If the Pedersen conductivity is also large in the ionosphere, E?, the ionospheric electric field,

•Also at Department of Earth and Space Sciences, University of California, Los Angeles.

Copyright 1993 by the American Geophysical Union.

Paper number 9ZIA01890. 0148-0227/93/9ZIA-01890505.00

will be varfishingly small. Here the star denotes a quantity measured in the corotating frame. Magnetospheric plasma displaced outward across field lines initially lags behind the ionospheric plasma at the foot of its flux tube. A v xB electromotive force develops in the magnetospheric plasma which produces outward radial currents earfled by the relative radial displacement of the positive and negative charges. The associated J xB forces accelerate the plasma in the sense of eorotafion. The currents must close through the ionosphere, and the effectiveness of the acceleration is determined in part by how much current can flow through the ionospheric part of the circuit. If the ionospheric conductivity is high, the currents flow freely and corotarion is easily maintained. If, on the other hand, the ionospheric conductivity is negligibly small, there may be no path available to the closure currents. As a result, a polarization electric field builds up in the ionosphere and maps to the magnetosphere, building up in magnitude until it exacfiy cancels the effect of the v*x B electromotive force on the particles. In such a case the plasma is free to move at any velocity consistent with the laws of conservation of linear and angular momentum. Even in this case, currents do flow initially to establish the surface charges in the ionosphere nec- essary to develop the polarization electric field, but once the surface charge concentrations have been set up in the ionosphere, the ionosphere ceases to control the motion of plasma in the equatorial plane. Our interest is not in either of the limiting cases described above but in cases with interrn•ate values of ionospheric conductivity for which the ionosphere imposes partial ocroration on the equatorial plasma.

Particle observations from the Voyager spacecraft [Bridge et al., 1979; McNutt et al., 1981; Belcher, 1983; Krimigis et al., 1981; Sands and McNutt, 1988] have shown that corotation is not strictly enforced in the magnetospheric plasma on the dayside beyond a radial distance of 20 R•. Kaiser and Desch [ 1980] pointed out the presence of a Jovian radio source near Io's toms which rotates at a velocity 3-5% slower than system IH. Kaiser and Desch and Pontius and Hill [ 1982] have related this apparent slowing down of the plasma to the local mass

67

68 KHURANA AND KIVELSON: JOVIAN PLASMA ANGULAR VELOCITY

loading in the torus. Pontius and Hill specifically ruled out outward plasma transport as the cause of the slowing down of plasma in this localized region. However, no direct infor- marion on the azimuthal velocity of plasma on the nightside of Jupiter is available because of the unfavorable viewing ori- entations of the Voyager plasma detector (PLS) cups [McNutt et el., 1981]. Two theoretical treatments show that indirect information can be obtained from the magnetic data because the angular velocity of plasma must be consistent with the stress balance. Hill [1979] solved the problem for a dipole magnetic field model assuming that the angular velocity of the plasma is constant everywhere on an L shell. The model takes into account self-consistently the ionospheric conductivity, whose chief role is to regulate the current flowing through the ionosphere-magnetosphere system. Although Hill used a dipole magnetic field in order to obtain analytical results, it is known that beyond 10 Rj, the magnetic field of Jupiter differs significantly from that of a dipole. It is therefore important to assess the way in which quantitative results are changed when a realistic field configuration is analyzed. Vasyliunas [ 1983] approached the problem somewhat differently. He used the

measured azimuthal component of the magnetic field (Bqs) in system HI coordinates to calculate the radial currents flowing through the plasma sheet, thus avoiding the need to assume a value of the ionospheric conductivity. To calculate the torque applied by the magnetic field on the plasma, the component normal to the plasma sheet (B•) is also required. As By is difficult to obtain from system III measurements, Vasyliunas used the By from an empirical field model derived by Goertz et al. [ 1976] from the Pioneer measurements. He then used the concept of stress balance in the azimuthal direction to calculate the radial mass flux-weighted average of the angular velocity of the plasma in the presence of magnetic torques for various assumed outflow rates. Although the ionospheric conductivity does not enter the equations directly, it is taken to be high enough to carry the required currents in the ionosphere. (Vasyliunas [ 1983] has calculated the requirements this assumption imposes on the ionospheric conductivity.)

In this work we follow the theoretical framework of Vasyli- unas [1983] but we introduce an approach to data analysis which changes the quantitative conclusions. As the data analysis requires the knowledge of various assumptions made in deriving the final equations, we have recapitulated the deriva- tion of Vasyliunas [1983] in the Appendix and made various assumptions explicit. In particular, we point out the value of using a special plasma sheet coordinate system to examine the stress on the plasma. The final equation that relates the angular velocity of plasma to the magnetic data is equation (A10):

This equation relates {fl,,•), the radial mass flux weighted average of the angular velocity of the magnetospheric plasma in the segment A•5 (of an angular width of •r/2) located at a distance p from the rotation axis to S, the mass outflow rate B•t, the azimuthal component of magnetic field in the lobes B,•, the vertical component in the plasma sheet and Q,•, the angular velocity of plasma at the inward boundary p•.

In the following section we present new techniques to obtain the estimates of the azimuthal and vertical components

of the magnetic field in the special plasma sheet coordinate system. Unlike Vasyliunas, we conclude that for reasonable outflow rates (•2.5 x 1029 amu/s spread over the postmidnight quadrant) corotation could have been maintained up to a radial distance of 50 Rj during the Voyager 2 flyby but not during the Voyager 1 flyby. A comparison of results from Voyager 1 and 2 is presented.

DATA

The currents responsible for the outward transfer of angular momentum are the radial currents flowing through the plasma sheet [Vasyliunas, 1983]. Before we present the observations, let us first consider how the data should reflect the effects of the radial currents (see Figure 1 for a schematic of the expected current path geometry). Figure 2a shows a cartoon of the field lines (solid lines) of the Jovian dipole field and current sheet as they would appear in the absence of any radial outflow of plasma when viewed from above the north pole of Jupiter. As Jupiter rotates, a spacecraft located at the position Rx on the Jovigraphic equator would move up and down through the dipole equator because of the 9.6 ø tilt between the Jupiter's rotational and dipole axes. Near the equator, the spacecraft would see positive Br (system m) above the plasma sheet and negative Br below it. When a strong outflow of plasma and the associated radial currents are present, the field lines would bend backward (represented by shaded lines in the figure). The skewed field applies stress on the plasma to make it corotate. The B• resulting from the sweepback (called B qs(swp) hereafter) is positive (westward) in system III (S 1]]) coordinates above the plasma sheet and negative below it, thus in phase with the distance from the equator as shown in Figure 2b. (Although the equations in the Appendix are derived in a right-handed coordinate system, data are usually discussed in the left-handed "system III" coordinate system. Our data- related remarks refer to this left-handed system in which B• is positive westward, and B•(swp) at the spacecraft is positive above the equator.) Notice that at large distances from the

current sheet, B qs(swp) reaches a saturation level because no significant radial currents are present in the low-plasma density region that we refer to as the lobes.

retum Birkeland currents

(via magnetopause)

...... Field lines

Meridional View

Fig. 1. The field line and current path geometry for the assurr• model. The currents axe indicated by open arrows. Sizes are not to scale, and the field lines are merely sketched. The size of Jupiter has been greatly exaggerated. Figure adapted from Vasyliunas [1983].

KI-KIRANA AND KIVEI.30N: JOVIAN PLASMA ANG• VELOCITY 69

Field line for an axial dipole No outflow

Field line with o•tflo•• View From Above

(a)

S/C trajectory

B • at S/C from Plasma Outflow

• S/C trajectory

(c)

B q• at S/C from Dipole Tilt

Fig. 2a Field lines of Jupiter in the absence of a plasma outflow (solid lines) and when a substantial plasma outflow is present (shaded lines). The view is from above the ecliptic plane. (b) The azimuthal component of the magnetic field generated at the $/C location by the radial currents induced by the outflowing plasma and the distance of an inertial $/C from the plasma sheet treasured nonml to it. (c) The azimuthal magnetic field resulting from the representation of the internal field of Jupiter and the field of its plasma sheet in the system Ill coordinates. Also plotted is the trajectory of the spacecraft with restx•t to th• plasma sheet.

There are several additional sources of B•6 in the system !II coordinate data. One source is any nonazimuthal symmetry of the internal planetary field arising from higher-order multipoles. Additional B• in system HI results when inter- hal field components are azimuthally symmetric about a tilted axis. Perturbations related to an azimuthal current sheet symmetric about a tilted axis also contribute to B6 in S HI. To illustrate this point, let us return to the dipole geometry with no radial outflow of plasma (solid lines in Figure 2a). Let us assume that at a time t, the Jovian dipole is pointing toward the spacecraft which is at the point R• on the Jovigraphic equator. The spacecraft would be at its farthest distance from the magnetic equator and the plasma sheet at this time. The field lines through the spacecraft would lie in a Jovigraphic meridian plane, and the space, aft would not see any B4 (S HI). A quarter of a Jovian rotation later, the dipole would point toward R2, and the spacecraft, still at point R• would be located on the magnetic equator. However, the field lines through the spacecraft would tilt out of the local Jovigraphic meridian, and

a firrite B• would be present in system Ill coordinates. No- tice that unlike B•(swp), this additional contribution to B•

is in quadrature with the distance of the spacecraft from the magnetic equator (see Figure 2e). The B• contribution arises purely from the representation of the internal field of Jupiter and the azimuthal current sheet in Jovigraphie coordinates and has nothing to do with the sweepback of the field lines. These contributions to B•5 therefore must be removed before the data are used to analyze the consequences of plasma outflow. We use the abbreviation B qs(extran) to designate the sam of the two extraneous contributions discussed above.

Now, we mm to actual data from the nightside magnetosphere of Jupiter. Figures 3a and 3b show sunrotary plots of Br, B 0, and B• in system HI coordinates along the Voyager 1 and 2 trajectories, respectively. These plots were obtained by smoothing the 48-s resolution data of the Voyager flux gate magnetometers (principal investigator N. F. Ness) obtained from the National Space Science Data Center (NSSDC) by calculating second-order polynomial fits to running data windows of 60-rain duration. After each fit to the data, the window was advanced by 48 s, and the process was repeated. It can be seen that beyond r = 40 R j, Br and B• vary in phase as the spacecraft moves in and out of the plasma sheet. Thus the field lines do appear to be swept back in the outer magnetosphere in accordance with a plasma outflow model.

A closer inspection of the observed B• in system HI for r < 35 Rj (see the solid curves of Figures 4a and 4b drawn at

48-s resolution without smoothing) reveals that B06 and z (the distance of the spacecraft from the center of the plasma sheet) are not in phase but are nearly in quadrature. This suggests

that the observed B05 has a large contribution from B•(extran) introduced above. To obtain B• (swp), either the magnetic data shotfid be analyzed in a coordinate system in which B•(extran) is absent or B qs(extran) should be subtracted from the system IT[ data. If the magnetic field were entirely due to a centered dipole, then rotation into the dipole coordinate system would be a logical choice for this purpose. However, inside of 20 R•, higher-order multipole terms distort the magnetic equator, and a simple rotation to a dipole magnetic equator-centered plasma sheet would not remove all of the "extraneous" B qs. In addition, close to Jupiter B•6 (swp) is very small (our estimates indicate that it is smaller than 8 nT inside of 30 R•) compared to the dominant component B0(S ILl') (which may be as large as 500 nT at r = 10 R•) and small errors in the calculation of the nomxal to the local magnetic equator can produce uncer- tainties in the azimuthal component comparable in magnitude to the expected values of B{(swp). The local normal to the actual magnetic equator (and therefore the current sheet) often deviates from a model-derived normal by several degrees because of plasma sheet flapping and errors in fitting a global model of the plasma sheet structure. For all these reasons, B•_ (swp) cannot be found reliably by rotating the data. A rotation scheme that calculates components of B directly in the local plasma sheet coordinate' . system by 'means of a copla- narity technique was used by Hairston and Hill [1985, 1986] to obtain B• (swp). However, as stressed by Hairston and Hill [1985], their technique finds the sheet normal only in places where the field gradients axe large, that is, close to a current sheet crossing point and not in the lobes where the field gradients are very small. The technique also fails when there is a moderate amount of noise in the magnetic data. Here we introduce a new approach to the evaluation of B (swp) which, though still not without problems, should give better estimates than those previously obtained.

70 Kzr•• • •o•: $ov•,• P•s• A•• V•err•

5o

3o

10

-10

-3o

-50 10 20 30 40 50 60 70 80 90 100

Radial Dis(anee (R•)

5O

3O

-10

-30

-50 10 20 30 40 50 60 70 80 90 100

Radial Distance (R•) Fig. 3. Summary plots of magnetic data in system IH coordinates for (a) Voyager 1 and (b) Voyager 2 as a function of (spherical) radial distance measured in Rz. Shown in this spherical coordinate system are the radial component B• (solid curve), the colatitudinal component B 8 (dotted curve) and the azimuthal component B• (dashed curve). The data were smoothed by a technique described in the text.

To obtain B½(swp) inside of r = 35 R.t, we first calculate the azimuthal contribution to the system 111 observations of the in- temal field plus pcrrarbafions produced by an external current system that flows azimuthally in the dipole equator [Connerhey et al., 1981 ]. This azimuthal contribution (B• (extran)) calculated from the Goddard O4 + current sheet model is shown by open circles on Figures 4a and 4b and dominates the observed Be for r _< 15 Rj for Voyager 1 and r _ 25 Rj for Voyager 2. The O4 model includes nonaxisymmetric inter- hal moments, and therefore their effect is properly removed

from the data. Notice that in general, B½5(extran ) is larger for Voyager 2 compared to its values during the Voyager 1 flyby. This is understandable because the magnetic field was more dipole like (and therefore larger in the plasma sheet) during the Voyager 2 flyby. A bigger component normal to the plasma sheet gives a larger projection in the azimuthal direction in

system 11I coordinates. We subtract this contribution from the data and the remaining azimuthal component of the magnetic field is what we identify as B½(swp). Because. we measure this quantity in the ½ direction of the system llI coordinates (rather than in the • direction of the plasma sheet coordinate system), we underestimate it by as much as 1 - cos(9.6 ø) • 1.4%. We will ignore this small difference. Figures 5a and 5b show B• (swp) along with the spacecraft distance from the dipole magnetic equator. The azimuthal component is much reduced in its magnitude, and beyond 25 Rj it is almost in phase with z as would be anticipated for an azimuthal field induced by radial currents. Inside of 25 R j, the phase relations between z and B½(swp) are not systematic for Voyager 1 and still in quadrature for Voyager 2. This suggests that fartha corrections are required especially for the Voyager 2 estimates which we will discuss below. The figures also show several

• • •ON: Jov•tN PL, SMX A•• V•xn• 71

50 ' ' ..... • ' " , " • ' 10

,; : Vo rager 1 ,---. , ,, ] -I " 5

I' ,.._ / ', ,I , : I / , ',, ; !_ ,

/' / ,,' ' / o I

l/ ,,, OXo/- ø ',,

' -30 -5

-50 -10 10 20 3O

Distance

5O

Voyager 2 3O

I 1, / ',.,d, • 'd : ..',J ./..

-- /• ',. q / _• '.l"'k ,,/.oOø ,'• •_,,ooS,•

P ,, •"•o : .... •:• -10 ' ' ' c• " \/' ' ' I / -3o o Z '" ',-"-

-50 ' ' , I .... 10 3O

lO

5

-5

,,, i ......... • , -lO 20

Radial distance (Rj)

Fig. 4. The ob•-•cd B•(S lid (soUd curve) and the conuibution to it from the 04 + plasma she, ct model of Cofiaeraey el al. [1981] (open circles) as functions of radial distance for (a) Voyager 1 and (b) Voyager 2. The dashed curves show the distance of the spacecraft from the dipole magnetic equator.

large localized enhancements in B• which occur at rimes when the spacecraft are immersed in the plasma sheet and B•b(swp) is expected to be zero. We have nor identified.the source(s) of these localized enhancements, but these data points from inside the plasma sheei can be excluded from an analysis that see• to estimate B• within the lobes. Also, the approach to remove B• (extran) should be modified outside of p = 35 R•, because the Conhomey et al. model is not valid beyond • distance. As B•(extran) is essentially the projection Of B, (the magnetic componem normal to the magnetic equator) onto the plane of the rotational equator, it 'dunigishes with increasing distance from Jupiter ami is insignificant outside of p = 35 R•. Thus we estimate B f0(swp) using only lobe values of B• in system HI with the Conncmcy el al. model values subtracted inside of 35 R,• but nor outside.

The values (only within the lobes) found from the above m•lysis for the Voyager 1 and Voyager 12 outbound passes are

shown by solid line segments in Figures 6a and 6b, respectively. The location of the spacecraft with respect to the cmrcnt sheet was found from the Khurana [1992] model. Outside of p = 30 R•r, only those values of B•(swp) (i.e., the B•(swp) values in the lobes) are shown for'which the spacecrift was at least 4.5 R• above or below the center of the current sheex. Inside of 30 R•, the spacecraft did not make large excursiom with respect to the current sheet (see Figures 5a and 5b* for the relevant information about the spacecraft trajectories); therefore the condition of lobe entry was relaxed, and the spacecraft were assumed •o be located in the lobes when they were at least 2.:5 R• away from the center of the current sheet Voyager 1 did not satisfy even this relaxed criterion inside of 20'R•; therefore we will nor use data from inside of this distance for Voyager 1. We would like to mention here that the magnetic field modeling work of Coanerney et aL [1981] hu shown • the plasma sheer half thickness in the region $ R• < p < 50 R• is

72 I•A AND KIVET•ON: JOVIAN PLASMA ANG• VELOC•

50

.--. 30

lO

-10

-30

-50 10

Vo,,y..%ger 1

I , I [

20 30

Distance (R•) 50

•, 3o

' I ' I

Voyager 2 .

, ,,

"'t 10 ;

i i

o

-lO

lO

-50 ,' i ,, , , ,, I -!0 lO 20 30

Radial distance (R•)

Fig. 5. The residual B•(sweepback) as a function of radial distance over a portion of the outbound (a) Voyager 1 and (b) Voyager 2 trajectoncs. The dashed curves show the distance of the spa•t from the dipole magnetic equator. Note that many of the largest Sl:•k• in the data conespond to minimum values of SIC distance from the magnetic equator and not to probable lobe encounters.

•2.5 Rj. Therefore the above relaxed criterion should still be

adequate in obtaining B• observations in the lobes. As mentioned above, inside of 25 R;, the B•z.(extran) contri-

bution is still present in some of the data. However, we can use our knowledge of the phase differences between B•(extran) and B (swp) to fttnaher improve our estimates of B (swp) . ' Figures 23 and 2c show that m the lobes B 4(swp) attains a plateau value, whereas B (extran) remains positive for half of the time spent in the lobes and negative for the other half. The average B•(extran) from the dominant dipole field and current sheet field is zero when averaged over all of the data obtained in a lob• traversal. Therefore we can further improve our esti-

mates of B•L(swp) by averaging the B• values obtained in the lobes. These averages are shown by Solid circles in Figures

In order to carry out the integration in equation (A 1 O) one needs to know B•.(swp) for all distances between pz and

100 R•. We have interpolated B4L(swp) by fitting the data to sixth amt fourth-order linear polynomials for Voyager 1 and Voyager 2, respectively. The interpolated values (assumed to be antisymmetric in the two hemispheres) are shown by the dashed envelopes plotted in Figures 6a and 6b. As system is a left-handed coordinate system, and the final calculations are performed in a right-handed coordinate system, the signs of B q•.(swp) shown in Figures 6a and 6b should be changed before using it in the final calculations.

In Figure 7 we compare our estimates of B•t.(swp) with the previously published results. The B4L(swp' ) data have been traditionally presented in the literature as Bqb/pB p ratios. Therefore to facilitate the comparison, we have converted our data also into these ratios. It can be seen from the figure, that the Behannon et al. [!981] estimate is too .high inside of ,40 Ra compared to all Of the other estimates. Our estimate of B•z(swp) for Voyager 1 is similar to that of Goertz et

IZOtLrRANA AND KIVEI•ON: JOVIAN PI•SMA ANG• VELOC1TY 73

I5

I0

-10

_i5 0 '

(a)

15

•o (b)

0

m-5

-10

-15 ' 0

Voyager 1

......... 8'0 40 60


100

Voyager 2

•o 6o •o 1 oo


Fig. 6. B• values observed in the lobes (thick solid lines) as a function of radial distance to 100 R/. The averages of B• values seen during each of the lobe traversal are shown by solid circles. The be. st fit curves to these averaged f •œ(swp) and their negative images a• plotted by broken lines. (a) voyager 1 and (b) Voyager 2.

at. [1976] and Vasyliunas [1983]. However, our estimates of B•t;(swp) between 10 R/< p < 30 R/are much smaller for Voyager 2 than any of the previously published values. This result affects the final calculations sig•ficantly.

To calmlate the radial mass flux-weighted average of the angular velocity (fl.,•) from equation (A10), we also require the component of B normal to the plasma sheet which we call B,. The z direction differs from the 19 direction by at most 10 ø. B0(S lid is always positive (southward) along the spacecraft trajectory and in the lobes is the smallest of the three components beyond r m 30 Rj. Therefore B: is also small beyond 30 R•. Because other components are'much larger in the lobes, even small errors in the estimates of'spacecraft position or of the orientation of the plasma sheet can produce large fractional errors in the calculated B,. The errors in the calculation of the spacecraft position relative to the current sheet may be large both because the location of the spacecraft is based on imperfect global models of the plasma sheet structure (see, for example, Behannon et •il. [198!], Khurana and Kivelson [1989], or Khurana, [1992]) and because the plasma sheet flaps on time scales much shorter than the Jovian day. (Note that the plasma sheet also flaps with a period of a

Jovian day in inertial coordinates because of its tilt with respect to the levigraphic equator. This flapping can be calculated and is of no concern here.) Because direct measurement of B, in the plasma sheet is so difficult, we adopted an indirect but relatively robust approach to estimate B, in the plasma sheet.

By definition, at the center of the plasma sheet, only the locally normal component of the field is finite. Observations show (and the sweepback of the field lines requires) that both B p and B• are zero at the center of the plasma sheet at large radial distances, say beyond 25 R/. B, itself varies slowly through the plasma sheet. At the center of the plasma sheet, the field strength lB[ is minimum and is equal to IB,!. Values of lB[ are independent of the coordinate system, which makes this identity very useful. However, because of the presence of large-amplitude waves and the flapping of the plasma sheet, good estimates of average lB[ near the center of the plasma sheet cannot be obtained from the raw data. To obtain an

average IBI at the center of the plasma sheet, we first smoothed all three components of the system 111 magnetic data along the spacecraft trajectory by using second-order polynomial fits over running windows of 1-hour duration (see Figure 8). We then calculated the smoothed [BI from these smoothed components. The minimum value of smoothed I BI seen in each lobe to lobe traversal of the spacecraft provides a good estimate of IBz I at the center of the plasma sheet for that pass. We fitted a power law

IB•[.:o(nT ) - a [p(Rj)] -• (1)

to these values of IB.I (se Figures 9a and 9b for Voyager 1 and 2, respectively). The best fit curves yield a = 5.4x 10 • and b = 2.71 for Voyager 1 and a = 4.3 x I0 • and b = 2.56 for Voyager 2. Shown also on the figures are the Jovian dipole values of B,. Inside of 100 R/, the observed B, is generally smaller than that of the Jovian dipole. Nishida [1983] reached the same conclusion, though he did not calculate the best fit curves.

RESULTS

The Voyager 1 and Voyager 2 outbound data cover a portion of the postmidnight quadrant (0000-0600 LT) of the magne-

•e•a. øoonet aL 1(]2 Vasyliunas

.Bq• .... Goert•' ' - - - - -a•.- - - - --• P Bp / Voyager 1

10

' 2'0 ' 10 Radial distance (Rj)

F•. 7. A •m•mn of • q•fi•, B •/pBp, o•• f• Vo• 1 •d Voyag• 2 •m •o p•sent work • p•vio•y

74 KHURANA AND •N: JOVIAN P!.ASMA ANGUIAR ¾ELDCI•

Br,Br $III

Bo,B o SIII

SIII

IBI, IBI 8HI

(nT)

16

8

0

-8

8

0

-8

8

0

-8

2000

July 12, 1979

220O 0000 i .... , t , , ,

0200 0400 0600

Universal Time

F•E. S. Illusion of the method used for deterrrfinau'on of B, in tlz plasma sheet coordinates. Shown •xe the system HI magnetic d_at_a at 48-s resolution (thin curves) and running second-order averages of the components (solid thick curves) for the Voyager 2 pass. The smoothed [•[ (solid thick curve in the last panel) was obtained from t!• relation

-- + + each plasma sheet crossing, B. is taken as the minimum value of the smoothed •o.•.

tosphere. We take each to be representative of Lhat quadrant during the epoch of their pass. We thus evaluate the radial mass flux weighted average of the angular velocity ((P-m)) over a quadrant of r/2 in •, and in Lifts we again follow Va- syliunas [1983]. We use equation (A10) adopting the above es6mates of B, and B with the boundary condition fl,(20 R•) = fl•. The results • shown by solid lines in Figures 10a and 10b for Voyager 1 and 2, respectively, for 'a number of assumed outflow ra•r• As discussed in the Appendix and in more derail by Vasyliunas [1983], if the plasma motion is in the form of a predominam outflow, (f•,,,) can be interpreted as the average angular velocity of plasma in the magne[osphcre. However, if circulating flows dominaœe over the radial outflow,

then this quantity is a radial mass flux-weighted average of the angular velocity and can exceed the corotational velocity.

In Figure 10c we have used the magnetic field paramem3 used by Vasyliufias [1983] (for C = 10) and reproduced his results for comparison with our work. Our Voyager ! angular velocity profiles (Figure 10a) differ by a factor of •.,2 from Vasyliunas [1983]. Our results for Voyager 2 (Figure 10b) differ considerably from the solutions obtained by Vasyliunas [1983] which show either supercororation or subcorormion bur no broad region of near cororation seen in our solution (for $ = 2.5 x 102o arnu/s) in the quadrant inside of ,o = 50 R•. For this outflow rate, the breakdown of corotar. ion even ouraide of r • 50-60 R• is nor very severe; the partial comraQon velocity is

Voyager 1 Voyager

10 ø

10 -1

Bz = 5.4X104,D -2'7!

Dipole

(a)

Observed " .... •.

10 3

10 2

10 ø

Bz = 4.3X104p -2'56

Dipole

Observed ß • &'-e... _

10-! , , , , , oo 1 o 1 oo

Fig. 9. Plot of Bdr-'-O) (soUd circles) obts/• from the Or••Om of Figure 3 versus cylindrical radial distance in the plasma sheet coordinate system and rite best fit to it (solid line). For comparison, B, of the Jovian dipole at the equator is aim shown. (a) Voyager 1 ar•l (b) Voyager 2.

AND •ON: JOVIAN PI•SMA ANGULAR VELOCrYY 75

1

o lO

<Ilrn>

-1

-2 I0

o

(a) S amu/s

Voyager 1

20 40 60 80


1 oo

ld

o lO

<rim > nj

-1

-2 !o

o

, ,,

(b)

: lc? S amu/s

Voyager 2 i i i • , ,i , ,, I .....

20 40 60 80 100


<a__m_> •J -1 -1 lO s = •'-•.........• lO

4[ , Vasyliunas [1983] -2 10 ß ..... 10 0 20 4'0 '" {•0 ' •0 ' 100


<i'l m >

S arnu/s

• - arnt•ls Voyager, 2 , S:

20 40 60 80


Fig. 10. The calculated nonnali• angular velocity of magnetospheric plasma (solid lines) as functions of radial distance for four different outflow rates (taken over ,..X• = •r/2) when the average B•t ' values in the lobes in the postmidnight quadrant were used. (a) Curves based on analysis of Voyager 1 data. (b) Curves based on analysis of Voyager 2 magnetic field data. The baseline at 1 represents corotarion. (c) Reproduction of the results of Vasyliunas [1983]. (d) The nonmlized angular velocity during the Voyager 2 flyby when the integrations are started from p

100

not much below half of the full corotarion velocity near r = 100 Rj. In addition, Vasyliunas found that to avoid supercorotation solutions, plasma outflow rates would have to exceed 2.5 x l(P ø amu/s. Vasyliunas reached his conclusions using p-dependent forms of B, and B• that differ from the values used here. In particular, he used the empirical field model of Goertz et al. [1976] to represent B, in the plasma sheet. Even though, the Goertz et al. model provides an excellent fit to the Pioneer 10 data, we have found that this empirical model overestimates the values of B, obtained from direct analysis of Voyager data by approximately a factor 6f 2 (1.5) near p = 20 Rj for Voyager 1 (Voyager 2) and underestimates B, by a factor of 1.5 (3) for Voyager 1 (Voyaggr 2) near p = 100 Rj. In addition, Vasyliunas used B•(S ILl) without modification in the calculation of torque•. For both B• and B,, our modified estimates are smaller than those used by Va, sylitmas inside of 30 Rj. Consequently, although Vasyliunas found that for S = 2.5x 102• amu/s (9.,,.,} > O,j in the inner and the middle Jovian magnetosphere, the smaller torques that we infer for Voyager 2 do not produce supercorotation for such source rates. Figure

10d shows the angular velocity profile for Voyager 2 when the integration is started from p = 10 Rj. Once again, it is seen that for $ = 2.5 x 1099 amu/s near-corotation solutions can be obtained between p = 10 and 50 Rj. No reliable data for

B q6•;(swp) are available for Voyager 1 inside of 20 One can invert the question of whether the plasma corotarion

can be maintained in the presence of a substantial outflow and ask instead: How much torque from the magnetic lines of force would keep the plasma corotating? And how much torque is actually being applied by the magnetic field? The answers to these questions can be obtained by rewriting the two sides of equation (A9) for Ad = •r/2 as

- asahi? - pt] (2)

o)

76 •• •'• •ot•: $OVLa• Pt•SM• ANa• VF. LrX:ITY

where T• is the integrated torque required to keep plasma in corotarion everywhere inside of the radial distance p and T•,• is the integrated torque applied by the magnetic field. Figures 11a and 1 !b show T,,• (thin solid lines) as functions of radial distance for Voyager I and Voyager 2 for various assumed outflow rates. The observed T•,• calculated from equation (3) by using fits to the magnetic data is shown by the thick lines on these figures. Once again, notice that during Voyager 2 flyby within ,-,50 R., the observed magnetic torque may be sufficient to maintain coreration in an outflowing plasma with an outflow rate of 2.5 x 1029 amuds. It must, however, be stressed that just because there are solutions consistent with coreration does not mean that the plasma actually corotates in the inner to middle magnetosphere. The solutions are critically dependent on the value of the plasma outflow rate. We believe that $ = 2.5x 1029 amu/s per quadrant is a reasonable figure. On the other hand, if future observations show that the plasma outflow rates are much higher than 2.5x 1029 amu/s, then the plasma must be subcorotating in the postmidnight quadrant beyond 20 R•.

The radial gradients of T,,• and Tm• provide further information on the additional torques required to maintain corotafion

1019 (a)

1 01 61 Voyager 1 1015 , 0 20 40 10 •0 100 Radial distance (Rj)

ld 9

18

•,lo

1016 Voyager 2

1015 t , I • ! .... i,, o :o 4'0 ' ;o'" Radial distance (Rj)

Fig. 11. The torque (T•) requiavxi to keep plasma in corotarion (solid thin lines) for three different outflow rates (taken over A& = •r]2) and the obsea-ved magnetic torque (Tma•) (solid thick line) as a function of radial distance. Data from (a) Voyager 1 and (b) %yaget 2 are used to determine

and the amount actually provided by the magnetic field, respectively. The gradients are shown by solid thin lines (To,) and solid thick lines (T=,•) in Figures 12a and 12b, respectively, and were obtained by evaluating the two sides of equation (A8). The figures show that the torque applied by the magnetic field 'increases with the radial distance (•Tm•g/i9p > O) even though it cannot keep up with the torque demanded by corotation. The fact that the mag. netic field is able to exert ever increasing (with radial distance) torque on the plasma suggests that there are no abrupt latitudinal changes in the conductivity of the ionosphere on the nightside of the Planet. However, at higher latitudes, the currents flowing through the ionosphere- magnetosphere circuit are not sufficient to enforce corotarion.

DiscussIoN

In this section we will focus on some of the implicit assumptions used to obtain equation (A10). We will alsb discuss the re!evance and implications of some of our results.

In the discussion of equation (AS), we remarked that the assumptions of azimuthal symmetry and no field-aligned currents in the lobe require that B qb be proportional to lip in the lobes. This prediction can now be tested. Figure 7 which reveals confined fluctuations in the outer magnetosphere appears roughly consistent with a p4 falloff beyond a radial distance of 40 R.t. This suggests that beyond this critical distance the lobes are essentially free of field-aligned current; that is, no ap- preciable field-aligned currents feed the plasma sheet at large distances. Between 10 and 40 Rj, Bq•r. actually increases with radial distance for Voyager 2. An argument advanced by Con- nerney [1981] provides an interpretation. Connerney showed that in the presence of field-aligned currents, B•t ' can have any value between 0 and #ol•f2•rp, where I•a is the axial current (rerum Birkeland curren0 enclosed between 0 and p.

The evidence that B• (lobe) for Voyager 2 increases with radial distance inside of 4ORj, would be consistent with • increasing with radial distance. This suggests that the field-aligned currents being fed to the magnetosphere increase in magnitude with p inside of -,,40 R•. This inference is consistent with the torque requirements of an outward flowing corotating plasma. The torque requirements of a corerating plasma increase as p2 (see left-hand side of equation (A9)). If the plasma corotares, B•L must increase with radial distance sufficiently fast that the magnetic torque applied on the plasma (the right-hand side of equation (A9)) will be proportional to p2.

Another point that must be mentioned here is that we have not allowed for particle losses in our calculations. To first order, the particle losses can be accommodated in our scheme by postulating that the primary effect of losses into the ionosphere is to reduce the number density and therefore the outflow rate of the plasma. One could then adopt an ouffiow rate profile that is a function of the radial distance p and carry out the integration over p in equation (A9) with a varying $. For ex- amp!e, to get fl,, = fl• everywhere in the magnetosphere for Voyager 2 (that is full coreration), more than 50% of the particles would have to be lost to the ionosphere between p = 20 Rj and p = 100 Rj. Such large losses seem unlikely from the observations of the Jovian aurorae [Broadfoot et al., 1981] and therefore it is likely that plasma subcorotated beyond p = 50 R• during the Voyager flyby, if the source rate was at least 2.5x !029 amuds in the pertinent quadrant.

Finally we shall consider the significanqe of the differences seen in the angular velocity profiles of Vo•,ager I and 2. For

AND KIVELSON: JOVIAN PLASMA ANG• VEIX•ITY 77

I0

10 9

(a)

Voyager 1

Observed


100

10 10

,-, 10 9

z

(b)

Voyager 2

Observed

20 46 ' 6• ' 8• 100 Radial distance (Rj)

Fig. 12. The radial gradients of Tmag (solid thick line) and Te•r (solid thin lines) versus radial distance. Results for (a) Voyager 1 and (b) •yager 2. The assumed outflow rates (over •r/2) are labeled.

Voyager 1, our results though different in detail are similar to the angular velocity profiles obtained by Vasyliunas [1983]. Solutions for Voyager 1 and those of Vasyliunas [1983] show either supercorotafion or subcorotafion of the radial mass flux- weighted average of the angular velocity depending on the assumed outflow rate. However, Voyager 2 data admit near- eorotation solution inside of p = 50 R• for S = 2.5x 1029 amu/s. If it is assumed that the mass transport is in the form of a dominant radial outflow, a simple interpretation for these differences can be advanced in terms of the time variability of the plasma source. It is well known that Io's volcanism, which is the principal source of heavy ions for the Jovian magnetosphere, is a sporadic phenomenon with time scales of hours to weeks. During the Voyager 1 encounter, Pele, one of the principal volcanoes on !o, was active but was found to be inactive during the Voyager 2 flyby. Therefore it. is likely that the mass outflow rate during Voyager 1 encounter was very high and resulted in the breakdown of corotafion inside of 20 R•. We suggest that during the Voyager 2 flyby the outflow rate was lower, that is just enough to keep the plasma in near corotation out to 50 R•.

SUMMARY AND CONCLUSIONS

We have evaluated the averaged angular velocity of plasma from magnetic observations using plasma outflow rate as a parmeter. New techniques have been developed to calculate the normal and azimuthal components of' the magnetic field in and near to the plasma sheet in a plasma sheet coordinate system. The revised field components differ substantially (especially during the flyby of Voyager 2) from the quantities used in previous analyses. With the revised field values it appears that during the Voyager 2 flyby for an outflow rate of 2.5x 1029 ainu/s, the observed magnetic torque may be sufficient to keep the plasma in corotafion to radial distances of 50 R• in the postrnidnight quadrant. This result differs from previous results which showed that there were no broad regions of space in the middle magnetosphere where solutions consistent with corotafion were possible. The results from Voyager 1 and Voyager 2 data are substantially different from each other and we suggest that temporal variations occurred in the outflow rate of plasma between the two flybys.

APPENnIX: CALCULnaXON OV •O• V••

•OM • •G•C • DATA

•e follow•g •fion re•ac• •e •reficfl •ework of V•liu• [1983]. We have •ed to m•e explicit • of •e steps •d •smpfio• •at •e n•• to d•ve •e •uafio• •at we u• • o• •lysis. •e Jovi• pl• sh•t is filt• wi• re•t to •e Jovi•ap•c •uator by up' to •10 • •d is w•• beyond a fatal •st•ce of m 40R•. [Be•n et al., 1981; Khurana • Kive•on, 1989]. •erefore •e malysis of torques is faciHta• by ••uc•g a c•r•te system a•• l•ally wi• •s s•a•. We • a H•t-•• cyl•ca! c•r•ate system •, •, •), w•ch we ref• to • •e pl•ma sheet c•r•te system, wi• ) a•gn• pmallel to •e pl• sh•t •d po•g ra•ally outw•d, • pe•en•culm to ß e pl•ma •eet •d • comple•g •e

•e momen• •uafion for a pl•ma c• be •tten • • •• ff•e as

du

p• • - -Vp + a x B (A1) where • • •o pl•ma demity, u is •e pl•ma v•tor re- llim, p is •e pla• pr•s•e •d J x B is •e b•y for• card by c•enm J flow•g •ough •e plm• •e •- •u•al component of •e memomum equation • cyl•c• c•rd•at• b•omes

37 + (u. v) - (^2) where the pressure gradient term vanishes if the system is axially symmetric. As the interest is in a steady state stress balance situation, all quantities involving time derivatives are dropped. Expanding the inertial term on the left-hand side, one gets

,o Or) : O z .[ (A3) The second term on the left-hand side vanishes again on the grounds of axial symmetry. As the azimuthal velocity of the plasma is not believed to be a strong function of z, the third

78 Kmm,•^ XND KntV.•ON: JOV•XN P•SMn AsG• VELocrrY

term on the left-hand side can also be neglected in comparison to the first term. Thus using Ampere's law, we rewrite the azimuthal component of the momentum equation as

/9 B•, -IV x = Iv x

[10 10Bo]Bo [ 1• 3B, OB•.]B, (A4) = p In the model being considered (see Figure 1), the only currents present in the plasma sheet and its boundary layer are the radial currents which are responsible for the magnetic torque exerted on the plasma, and the field-aligned currents which are force- free. In the lobes (by which we mean the portions of the flux tubes with very low plasma density) only field-aligned currents are allowed. If one were to assume that even field-

aligned currents are absent in the lobes, then B qb would be cr 1/p. (This statement follows from Ampere's law with J = 0 and the assumption of azimuthal symm6try.) Therefore the functional form of B•(p) in the lobes gives information about the current paths in the magnetosphere. This issue is further addressed in the body of the paper in the discussion section.

Ignoring the terms involving derivatives with respect to • and noting that B , B , and their radial gradients are small inside of the plasma sheet (thin sheet approximation, see Vasyliunas [1983] for more details), one can simplify the above equation in the plasma sheet to

(A5)

where •")•m is the angular velocity of the plasma and uqb = pft.. To calculate the net outward transport of the angular mo-

mentum, one needs to integrate this equation over the volume occupied by the outflowing plasma. Equation (A5) was writ- ten for a local plasma sheet coordinate system which changes with distance from the planet. It is importam to evaluate the local stresses in such a coordinate frame. However, for the calculation of the angular momentum transport, the errors introduced by integrating the equations, as if the plasma sheet were a plane surface, are small.

To calculate the stress imparted by the magnetic field to the plasma over the whole thickness of the plasma sheet, integrate the above equation over z (-h to h where h is the half width of the plasma sheet) at fixed p.

o - ............. (^6) Po

where a = 2p.h is the plasma column density. B•L = B• (north lobe) = -B•(south lobe) and B, is approximated as constant over the wxdth of the plasma sheet (by virtue of thin sheet approximation). The above equation expresses the requirement that the rate of change of angular momentum of the plasma is equal to the torque exerted on it by the magnetic field.

To calculate the magnetic torque on a shell of plasma of azimuthal extent A• located at a radial distance of p, integrate over •5 to obtain

AS •p((n.)p 2) = 2peAq•B•B•' (A7) #o

where AS = p f dq3aup is the integrateA mass flux across the segment Aq3, and (fl..•) = p f aupfl.•dck/AS is the dial mass flux-weighted average of the angular velocity of the magnetospheric plasma in the segment A•5 at p. Vasyliunas [1983] has shown that (fl.,,,) can exceed f>.• if a large fraction of the magnetospheric plasma is recirculating in the magnetosphere. However, Hill et al. [ 1982] have shown that a simple large-scale two-cell convection pattern that. transports plasma radially is not in accord with the Pioneer and Voyager observations. If the outflow is everywhere positive, the calculations can be further simplified by assuming that the radial flux aUp is azimuthally symmetric. In this case (•m) becomes the same as a simple average over of Q,,, over

Let AS=apupAck denote the total outflow rate of plasma (often expressed in amu/s) in the postmidnight sector (A• = •r/2). Then,

zas W) - &' s. (^8) Po

Finally, integrate over p to calculate the (integrated) torque T, applied to a shell of plasma as it moves (and expands in volume) from p• to p.

p

Po P•

(A9)

which yields p

f (½)' ;r p• B,•IBz dp + = Sp • #o

(AlO)

where fl.,,,a is the angular velocity of plasma at p•. This equation identical to equation (11.138) of Vasyliunas [1983] (with A•b replaced by •r/2) enables us to calculate the sector- averaged angular velocity of plasma as a function of p from the

observed values of B•5 in the lobes and B• in the postmidrfight quadrant of the plasma sheet. (The calculations can be done separately for other quadrants if future measurements show pronounced asymmetries.) In principle, the distances p and p• should now be measured along the hinged and warped plasma sheet surface. However, in practice for measurement of distances one can ignore the distortion from a planar surface because the corrections are small.

Acknowledgments. We gratefully acknowledge the generous help of Vytenis Vasyliunas during the revision of this paper. We would also like to thank R. J. Walker for many helpful discussions. The Voyager MAG data were obtained from the NSSDC. This work was supported in pan by JPL under contract 955232, by the National Aeronautics and Space Administration Division of Space Physics under grant NAGW- 2047 and by the Atmospheric Science Division of the National Science Foundation under grant ATM89-13342. UCLA IGPP publication 3312.

The Editor thanks the two referees and G. Zimbardo for their assistance in evaluating this paper.

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(Rt. ceived March 2, 1992; •evised June 4, 1992;

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Inference of the Angular Velocity of Plasma in the Jovian