Field-aligned currents at the dayside magnetopause, reconnection and magnetic helicity

Adv. Space Res. Vol. 13, No.4, pp. (4)45—(4)50, 1993 Printed in GreatBritain. All rights reserved.

0273—1177193 $24.00 Copyright © 1993 COSPAR

FIELD-ALIGNED CURRENTS AT THE DAYSIDE MAGNETOPAUSE, RECONNECTION AND MAGNETIC HELICITY J. M. Finch and M. J. Rycroft College ofAeronautics, CranfleldInstitute of Technology, Cra4field, Bedfordshire, MK43 OAL~U.K

ABSTRACF The magnetosphere is a complex spatially and temporally dependent plasma physical system which links, in both directions, the solar wind and the ionosphere. A crucial linking mechanism is due to geomagnetic fieldaligned currents, driven by electric fields; changes in these two quantities determine the energy flow. Dawn, noon and dusk Region I currents link the northern and southern auroral ovals just inside the magnetopause, and Region H currents close just beyond the plasmapause. Attention is focused on mechanisms for the generation of field-aligned currents under non steady-state conditions, in particular magnetic field twisting during the process of reconnection. INTRODUCTION Dungey’s reconnection model Ill is an elegantly simple explanation of macroscopic magnetospheric observations. The model did not, indeed never intended to, supply an explanation for either the nature or means of the reconnection process; it merely proposed a method that could be tested against, and has agreed with, many observed phenomena. To its opponents’ chagrin, it is still used when discussing many of the larger scale magnetospheric phenomena. No other model, to date, can match reconnection’s success rate at explaining different observations. Reconnection has been taken as the method by which the magnetic fields convey mass, momentum and energy of plasma across the magnetopause. This paper is concerned with the geometry of merging magnetic fields and with their helicity before and after reconnection. Helicity is a topological quantity of a vector field and its associated vector potential. In a magnetic field, having a vector potential, defined by B=curlA, the magnetic helicity, K ,is defined by equation (1); ~,

~,

K=J(~BJdV

(1)

where the total helicity for volume dV is a gauge invariant only if ~ where ~ is the outwardly directed normal to the magnetic surface, see Song and Lysak 121. If ~.ij is not zero, as is often the case, then the boundary motions of volume dV redistribute helicity between itself and the volume outside 131. Quoting Berger and Field 141, “helicity has been used successfully in such diverse fields as the study of DNA structure, the description of three-dimensional excitation of chemical and biological media and the study of polymer chains”. Applied to solar physics, helicity has begun to help in understanding the contorted nature of sunspot magnetic fields 151. It has only been in the last eight years that helicity has been applied to the reconnection process atthe dayside magnetopause, notably to understand better flux transfer events, FFEs 121, and, more recently, to consider a current dynamo effect during three-dimensional time dependent reconnection 161. It is by using helicity concepts that some conclusions may be reached concerning not only the generation of fieldaligned currents in the newly reconnected flux tubes but also the propagation of Alfven waves along the same flux tubes. THEORY Geometryof Reconnection Figure 1 shows a two-dnnensional geometrical projection of a magnetopause section viewed from the Sun, across which an interplanetary magnetic field, ~ (with southward and dawn to dusk components), lies over a tenestrial magnetic field (TMF), Taking spatially uniform fields away from the reconnection region, such ~.

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1. M. Finch and M. J. Rya~oft

= 2), we allow reconnection to take place within definite bounds that I~El=32nTand lB. I=8nT (or r~/r~ noted by the four field lines in Figurel

-

de-

2r~,

~

/~\

Fig. 1. Two-dimensional projection of dayside reconnection between two flux tubes viewed from the Sun. In some ways Figure 1 is an ad hoc approach: it includes no recognised merging theory such as component or tangent merging, 171 and 181 respectively. Yet, Figure 1 is inherently logical and has atleast some physical foundation, such as conservation of magnetic flux. Figure 1 allows reconnection, irrespective of the plasma kinetic processes at woit, to occur through a randomly chosen section of the magnetopause. Because of the boundary conditions, reconnected flux only appears in the parallelogram with the merging line being a diagonal of the parallelogram, of length, L. Working through the geometry of Figure 1, equation (2) can be derived. (Lsina)2

(!!!!~)2= 1

=

+ (~)

+

2cosaiJ~

(2)

Figure 2 shows a plot of merging line clock angle, 0, versus IMF clock angle, a, for various realistic ratios of magnetic flux density; it gives some insight into the relationship between the parameters of equation (2). 90~ 80~

__________ mçnsOc

a

70•

..00

—10 2.0

~60 .

~4O 30-

—

20-

—

10-

—

—.

00

20

40

I

I

I

I

I

I

I

60

80

100

120

140

160

180

IMPclod angle/de9reea

Fig. 2 The merging line clock angle, 0, as a function of the IMF clock angle, a, for several magnetic ratios, BE/Bi.

Reconnection and Magnetic Heicity

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It is worth noting that equation (2) is equivalent to equation (4) of Crooker et al. 181, only when the ratio of magnetic flux densities, I~l/l~l = L/P = 1, where L is the amount of magnetospheric flux lost to the magnetosheath and P is the amount of magnetosheath flux penetrating into the magnetosphere. However, since the two viewpoints have very different assumptions, their disagreement at other ratio values is only to be expected. To produce a more realistic merging geometry, Figure 1 is extended to a three-dimensional representation of the magnetopause, incorporating a dipole-like ~E with north and south cusps, Figure 3a-b. t

3

t2 ti t2 t3

a. b. Figure 3. Extension ofFigure 1 into 3-D with elongated cusps. Figure 3a) is a picture of the dayside magnetopause as seen from the Sun. It attempts to show the 3-D magnetopause in 2-D, with curved terrestrial magnetic field lines leaving (entering) the southern (northern) elongated cusp. Because of the projection, field lines towards dawn and dusk are distorted more than shown here. The IMF is shown impinging onto the magnetopause in the Y-Z plane (in Geocentric Solar Magnetospheric, GSM, coordinates) and reconnecting at the points of first contact. This is a tenet used by Crooker et al. 181 in their tangent merging line model. Figure 3a does not show conditions for an instant in time, but rather shows field lines connecting at three differing times, t1
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J. M. Finch and M. J. Ry~oft

tween the two original flux tubes, C1 and C2, and the two newly created flux tubes D1 and D2, see Figure 4. This removes the need to understand the complex processes of reconnection itself. Extending Figure 1 slightly but still keeping the 2-D geometry simple, let C1 and C2 both be straight flux tubes with axial magnetic fields, see Figure 4a. When the tubes meet at the magnetopause, there is reconnection on one field line per tube. As reconnection proceeds, the field lines on either side of this initially reconnected field line will reconnect between tubes C1 and C2. Ultimately, the half of tube C1 entering the reconnection region will merge into the half of C2 leaving the reconnection region to produce, say, D1 whilst the two remaining halves of C1 and C2 merge to produce tube D2, see Figure 4b.

Fig. 4. Line diagram to visualise reconnection between two flux tubes. What now is the topology of tubes C1, C2, D1 and D2? To answer this question we need to introduce several topological definitions: MUTUAL HELICITY : here there is a topological interaction between two flux tubes, as in the case of C1 and C2 which cross each other. SELF HELICITY : there are two phenomena associated with an individual flux tube; a flux tube can have, simultaneously: a. Knot Helicity where the axis crosses itself, e.g. a figure-8. It says nothing about the nature of the field lines, see Figure 5a. b. Twisted Helicity where field lines have an azimuthal component so that they twist around the axis. It says nothing about the nature of the tube axis, see Figure 5b. -

-

Fig. 5. A knotted and a twisted flux tube, respectively. Before the reconnection site, ideal magnetohydrodynamics (MIlD) applies to the magnetosheath plasma flow and therefore helicity is conserved 141. During reconnection, the ideal case breaks down allowing relative “motion” of the magnetic field with respect to the plasma and hence the magnetic topology changes. Beyond the reconnection site, in the magnetosphere, ideal MHD again applies: helicity, after reconnection, essentially equals the helicity before. There is an intrinsic half twist in the magnetic field lines (ie 180 degree twist) as a result of the reconnection process, not intuitively obvious to the observer. Tubes D1 and D2 do not appear to be twisted in Figure 4b. However, if we could physically hold both ends of tube D1, say, and straighten the tube horizontally in the plane of the page, we would see that the field lines have a one-half twist.

Reconnection and Magnetic Heicity

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The half twist of a particular field line is ~QLa direct result of its two parent field lines reconnecting but is, rather, a necessary condition to enable further field lines to reconnect; subsequently the reconnected field lines are moved out of the way by the remaining part of the flux tube. Once a new field line has been formed from two parent field lines, it is twisted around the flux tube; the twist propagates away from the reconnection site along both anns of the flux tubes D1 and D2. This pulse in the magnetic field is characterised as a torsional Alfven wave 1101. Aconsequence of the magnetic pulse is that a field-aligned current, J11, is created in agreement with Ampere’s law. This, in turn, suffers a resistive loss along the field line which dissipates the twist helicity. Three points that need to be considered further concerning the effectiveness of helicity are as follows: 1. The pulse contained by the flux tube may reflect off the auroral ionosphere, with the possibility of setting up standing waves within the magnetospheric cavity. 2. Tubes D1 and D2 will be dragged tailwards by the solar wind streaming past the magnetosphere due to their interplanetary anns being frozen-in to the solar wind flow. This will vary the radius of curvature of the flux tube and have an effect on both the propagation of the pulse and the forces acting upon the flux tube. 3. The tube diameter will decrease from the magnetopause ~ 25nT) to the ionosphere (~ 5000nT), effectively altering the pitch of the pulse. Two relevant time constants, which can be used to set the limits on the helicity analysis, are: i) the time for reconnection, Tr~2 minutes, and b) the time for the flux tube to be dragged across the polar regions, ~ 2 hours. —

Taylor 1111 stated that the net dissipation of helicity will be small on reconnection timescales in high magnetic Reynolds number plasmas. Wright and Berger 1121 produced an upper bound on the helicity dissipation for reconnection at the dayside magnetopause of AK/K < 0.09; a more typical value might be —0.01. In other words, helicity is expected to be conserved within a few percent within the reconnection time of —2 minutes. This enables pulses to propagate along both arms of each reconnected flux tube for a considerable distance before they 4), this corresponds to a travel appreciably decay.2RE. Taking the pulse to the travel atthe Alfven speedas(vA=l2Okms distance of about Needless to say, pulse relaxes further it travels toward the auroral ionosphere. DISCUSSION Saunders 1131 discusses recent developments in the study of quasi-steady reconnection (QSR) sheets of accelerated plasma flow when the reconnection rate is constant for atleast 1 minute and poses important remaining questions. The aim of much work in the field is to link the magnetopause reconnection processes, through Region I field-aligned currents (FACs), to the ionospheric plasma flow patterns surrounding the dayside cusps. The occurrence of FTEs at the dayside magnetopause injects helicity into the flux tubes, thus generating the required FACs that enter the ionosphere around the cusps, and quite possibly produce the cusp aurorae. -

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These coupling processes are mentioned in passing only, as this paper is more concerned with understanding the relationship between why and where reconnection occurs. The where, we have assumed, is dependent upon conditions mentioned in the tangent merging model 181. The why concerns the growth rate of instabilities along/across the magnetopause boundary. The crux of our hypothesis is that the IMP, changing direction almost constantly, will cause the point of first contact between the IMP and the TMF, and hence the direction of the merging line, always to be in a state of change. However, if the IMP direction remains roughly constant for a certain time interval (1 minute), there will be an increasing central region of near “steady-state” reconnection during this interval whilst, further out, older merging line geometries continue their life cycle. An abrupt change in the IMF direction means that older open structures, continuing their life cycle, wax and wane until absorbed into the open flux producedby the new IMP orientation. Depending upon the variability of the IMF direction, we perceive a transition between classic steady-state and transient, FFE, reconnection, with quasi-steady reconnection perhaps being an intermediate state. The variability in the IMF direction (and magnitude), controlled, first, by the level of solar magnetic activity (dependent upon the solar cycle) and secondly, by the variability of the solar wind particle pressure (due to the IMF being frozen into the solar wind) may well be a convenient tool for differentiating between the conflicting views of the magnetic reconnection and pressure pulse generation mechanisms of FTE signatures.

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J. M. Finch and M~I. Rya~oft

REFERENCES 1. J.W. Dungey, Interplanetary Magnetic Field and the Auroral Zones, Phys. Rev. Len., 6,47, (1961). 2. Y. Song and R.L. Lysak, Evaluation of Twist Helicity of Flux Transfer Event Flux Tubes, J. Geophys. Res., 94, 5273, (1989). 3. M.A. Berger and A.N. Wright, The Generation of Twisted Flux Ropes During Magnetic Reconnection, in: Physics ofMagnetic Flux Ropes, eds. C.T. Russell, E.R. Priest and L.C. Lee, Geophys. Monogr. 58, Mn. Geophys. Union, 1990, p. 521. 4. M.A. Berger and G.B. Field, The Topological Properties of Magnetic Helicity, J. Fluid Mech., 147, 133, (1984). 5. E.R. Priest, The Equilibrium of Magnetic Flux Ropes, in: Physics ofMagnetic Flux Ropes, eds. C.T. Russell, ER. Priest and L.C. Lee, Geophys. Monogr. 58, Mn. Geophys. Union, 1990, p. 1.

6. Y. Song and R.L. Lysak, The Current Dynamo and its Statistical Description during 3-D Time-dependent Reconnection, in: Physics of Magnetic Flux Ropes, eds. C.T. Russell, E.R. Priest and L.C. Lee, Geophys. Monogr. 58, Am. Geophys. Union, 1990, p. 533. 7. B.U.O. Sonnerup, Magnetopause Reconnection Rate, J. Geophys. Res., 79, 1546, (1974). 8. N.U. Crooker, G.L. Siscoe and FR. Toffoletto, A Tangent Subsolar Merging Line, J. Geophys. Res., 95, 3787, (1990). 9. H.E. Petschek, in: The Physics of Solar Flares, ed. W.N. Hess, NASA SP-50, US Gov’t Printing Office, Washington DC, 1964, pp. 425-439. 10. A.N. Wright, The Evolution of an Isolated Reconnected Flux Tube, Planet. Space Sd., 35, 813, (1987). 11. J.B. Taylor, Relaxation of Toroidal Plasma and Generation of Reverse Magnetic Fields, Phys. Rev. Leu., 33, 1139, (1974). 12. A. N. Wright and M.A. Berger, The Effect of Reconnection upon the Linkage and Interior Structure of Magnetic Flux Tubes, J. Geophys. Res., 94, 1295, (1989). 13. M. Saunders, Quasi-steady Dayside Reconnection, J. Geomag. Geoelectr., 43, Suppl., 141, (1991).