JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 92, NO. A1, PAGES 109-116, JANUARY 1, 1987
MagnetosphericInterchange Instability
DAVID J. SOUTHWOOD1 AND MARGARET G. KIVELSON2
Instituteof Geophysicsand Planetary Physics, University of California,Los An•teles
The interchangeinstability is reviewedwith particularemphasis on its relevanceto planetarymag- netosphericprocesses. Results of earlierwork are reviewedand drawn togetherby the derivationof a generalstability condition, based on a smallperturbation approach, for a plasmawith isotropicpressure containedin a curvedmagnetic field in the presenceof gravity. The resultsare not restrictedto low pressure(low •) or specialfield geometry,and the straightfield resultis recoveredin the appropriate limit. We use a small perturbationapproach for the instabilityanalysis. Conditions under which an effective gravity may be introduced to simulate rotational effects are discussed.Recent criticism of the Jovianring currentimpoundment hypothesis which was based on the straightfield line resultsis shown to be ill founded.
INTRODUCTION by Vasyliunas [1972], Jaggi and Wo/f [1973], and Southwood Since Gold's [1959] original paper, interchangemotions [1977] has very similar physics to the instability. Detailed have been regardedas being of great importancein terrestrial studies of magnetosphericinterchange instability were done magnetospheric physics. The term interchange describes by Sonnerupand Laird [1963]. Since the recognition that the motion in which flux tubesmove without affectingthe curva- processesthat set up the ring current are likely to be inter- ture at any fixed point. Few would quarrel with a first-order change stable (first tested by Nakada et al. [1965]), there has description of the terrestrial magnetosphericcirculation been particular interest in the instability's role in the forma- system as a driven interchange motion. It has to be driven for tion and stability of the plasmapause [Richmond, 1973; Le- two reasons. In the inner magnetosphere, where the magnetic maire, 1974]. field is dipolar, the net motion of the ring current particles is The plasma circulation processes of the Jovian mag- thought to be from flux tubes at high latitudes to low. netosphere are less well defined. Centrifugal forces are more Charged particles gain energy in the process,e.g., by conserv- important, sourcesof plasma are more complex, and far less ing the first two adiabatic invariants, and that energy must information has been recorded by spacecraft. As there is a ultimately be provided by the driving source of the circulation substantial interior equatorial source of plasma (namely, the system.In addition, the ionosphere forms a collisional bound- satellite, Io), a net outward transport of plasma is expected. ary layer at the feet of the magnetospheric flux tubes, and Melrose [1967] first examined interchange instability for the energy must also be expended in moving the flux tube feet Jovian system. Interchange instability, perhaps rather like through the ionosphere. Through the bulk of the earth's mag- Gold originally envisaged,could provide an important means netospherethe motion is magnetohydrodynamic and proceeds of redistributing plasma radially. Arguments that we shall without large magnetic field distortion. For the most part the review below show that in a dipolelike field configuration it magnetic forces dominate the gas pressure,the dynamic pres- can be energetically favorable for plasma to move outward sure of the flow, and any other forces(e.g., gravity) that act. under interchange. However, after the Voyager Jovian en- Gold's [1959] original hypothesiswas that interchange mo- counters, Siscoe et al. [1981] pointed out that outward diffu- tions would take place spontaneously; inward pressure gradi- sion of Iogenic (cold) plasma through interchange would be ents in the plasma would provide a source of energy for con- stopped by the steep outward gradient of the hotter (ring vective overturning motion in the earth's outer ionized envi- current) particles found at the outer edge of the Io torus. They ronment. It is now clear that the solar wind controls the circu- called this effectring current impoundment of the Io plasma. lation and that much of the charged particle population orig- Recently, Chen•l [1985] has called the theory of Io torus- inates from the distant parts of the system rather than the ring current impoundment into doubt and furthermore has inner radiation zones. Thus Gold's hypothesis is no longer cast doubt on the operation of the interchangeinstability as tenable. understood by many previous authors from Chandrasekhar Gold's original idea may not be applicable to the high- [1958] onward. The purpose of this paper is not so much to pressure ring current region, but the interchange instability refute Cheng's result (in fact, we show that his instability con- may still occur as a transient process when the system is se- dition is correct in appropriate circumstances)but rather to verely disturbed. The ring current shielding processdiscussed rehabilitate the previous work. The condition derived by Cheng was derived earlier by Tser- kovnikov [1960] and also by Newcomb [1961]. Newcomb showedthat the conditionfound by •Cheng was of limiteduse •Also at Department of Physics,Imperial College of Scienceand in that there existsa classof "quasi-interchange"motions that Technology, London. 2Also at Department of Earth and SpaceSciences, University of very slightly bend the field for which the stability condition is California, Los Angeles. independentof the field (i.e.,whose stability is governedby the familiar Schwarzschildcondition for an unmagnetizedstrati- Copyright 1987 by the American Geophysical Union. fied atmosphere [Schwarzschild,1906]). The implication of Paper number 6A8530. Newcomb's result is that the straight field interchangemotion 0148-0227/87/006A- 8530505.00 is singular and, in general, the quasi-interchangecondition,
109 1 l0 SOUTHWOOD AND KIVELSON: MAGNETOSPHERIC INTERCHANGE INSTABILITY the Schwarzschild condition, is the more relevant. When the after a little manipulation that the field pressure change is undisturbed equilibrium has field lines that are curved, the given by singularity is removed. In particular, when gravity is negligi- aBB(•)/•o = --(V ßu)B2/kto -- u ßVB2/2kto -- u ßcB2/kto (5) ble, the stability condition proposed by Gold [1959] holds. When the field is curved, we shall see that interchange mo- It is also appropriate at this point to use equation (4) to derive tions are possible that conservethe field energy, and in these constraintson the allowed variations of u along and acrossthe circumstances the interchange stability condition does not field that are inherent in the assumptionof interchange depend on the field strength. motion. We introduce temporarily a local coordinate system The standard treatment of the interchange instability (see, based on the undisturbedfield. We choosea coordinate, xx, for example, Newcomb [1961] and Cheng [1985]) is based on along the ambient field, such that an energy principle [Bernstein et al., 1958]. The energy has to dx• - ds/B (6) be calculated to second order, and there has been some con- fusion over the precise form of the flux tube energy [Cheng, where ds is an elemental length along the field. 1985]. Our approach is exactly equivalent to the many earlier Now we choose the second coordinate such that it is con- works, but we adopt a small amplitude perturbation approach stant along field lines. At each point along the field, x 2 points which makes clear what dynamical constraints are satisfied. in the direction of a potential interchange motion. After an interchangedisplacement the locus of all particles that started INTERCHANGE MOTIONS AND EQUILIBRIUM on a given field line correspondsto a field line of the original We shall consider small amplitude departures from an equi- field. A third orthogonal coordinate direction, x3, is then librium in which a plasma embedded in a magnetic field is uniquely specifiedat each point of the tube. We choose the supported by field and gas pressuregradients against a gravi- coordinatesx2, x 3 suchthat they are flux conserving,i.e., tational field. In the initial equilibrium one has B dS = dx2 dx 3 (7) V p-j x B=nmg where dS is the elemental area perpendicular to the field at or alternatively, each point. V(p + B2/2/•o)-- b ßVbB2/•0 = nmg Let us suppose that the scale factors for the three coordi- natesare hx, h2, h3, and unit vectorsare 6x, 62, 63, respectively. or Evidently, VPr -- B2/!•oc -- nmg= 0 (1) h• = B h2h3 - B- • (8) where j is the electrical current density, b is the unit vector It also follows that parallel to the field B, c is the field curvature,b ß¾b, and Pr is the total (plasma plus magnetic)pressure. Vx• = 6x/hx (9) We now consider displacing the plasma from equilibrium. and there are similar relationships for the other coordinates. In an interchange motion, flux tubes retain their shape; the The introduction of such a coordinate system allows the field strength may change, but the field is not bent. Accord- vector interchange motion to be representedon the local flux ingly, we consider displacementsthat are perpendicular to the tube by unperturbed field. Furthermore, we shall have to place con- straints on the manner in which the displacement varies both U: U262 (10) along and acrossthe background magnetic field direction. Two important properties of interchange motions can be We shall proceed by considering a perturbation velocity derived using the local coordinate system,(xx, x 2, x3). By field, u, and a correspondingplasma displacement,u/a (i.e., we vector manipulation of the scalar product of ¾x2 with equa- shall take the displacementto vary with time as exp (at)). It is tion (4) one may show that easier to discussstability rather than instability directly. To demonstrate stability, we shall have to show that there are no B. V(u2/h2) = 0 (11) displacementsfor whicha 2 is real and positive.(In dissipation- freeMHD thereis no possibilityof cr2 beingimaginary.) Thus u2/h2 is independentof position along a given tube. The requirement expressesthe fact that the displacement is pro- GOVERNING EQUATIONS portional at each point along the field to the spacing between The first-orderchanges in density,n {•), and pressure, the field lines in the equilibrium. A subtler relation holds con- are given by cerning the divergenceof the flow field and the displacement: an(J)= -n(V ßu) -- u ßVn (2) V ßU = J-1 •(Ju2/h2)/•x2 (12) ap{•)= --7p(V' u)- u. Vp (3) where 0r is the Jacobian of the coordinate transformation, i.e., h•h2h3,from the Cartesianto the field-alignedsystem. How- where 7 is the ratio of specific heats. Equation (3) is derived ever, by construction,0r is unity, and as u2/h2 is independent from the adiabatic change of pressure in the plasma rest of position along the field, it follows also that div u is also frame. independent of position along B. The field change is computed from the frozen field equation The dynamical requirement on the displacementis that the first-order momentum equation be satisfied.In an interchange aB(•)= V x (u x B) (4) motion the field is not bent; thus to first order one has Taking the scalar product of B/• 0 with equation(4), one finds nmau= --V(p(•) + BB(•)/•o)+ 2cBB(•)/•o + n(•)mgñ (13) SOUTHWOODAND KIVELSON: MAGNETOSPHERICINTERCHANGE INSTABILITY where the subscript _t_signifies the component perpendicular to the ambient magnetic field. f nmo'2udV=- ; {PyE(V ßu) +u ß(VP r+ cB2/lto)/Py] 2 QUADRATICFORM -- u ß(Vp -- Vpcr)E2B2/!•o(Ußc) + nmgß u]/P• We next develop an expressionfor the total energy associ- + mg. u(u. ¾n- u. Vnc0} dV (21) ated with the perturbation. The initial step is to take the scalar product of equation (13) with u. One finds where the "critical" presure and density gradients,Pc•, nc•, are defined such that nmau2 = -- u ßV(p {x) + BB(x)/!•o) Vpc• = yp[(VB)/B + c] (22) + 2u ßcBB(x)//•o + n(x)mgß u (14) and Equation (14) expressesenergy conservationpoint by point. The right-hand side is the rate of work done (power) by the Vnc•= n[(VB)/B + c] (23) body force within the plasma; the left-hand side is the rate of Now considerthe expression(21). For stability, there must be changeof kinetic energy. no perturbation possiblefor which the right-hand side of (21) Next we integrate equation (14) throughout the volume in is positive.The first term is a squareand thus is alwaysstabil- which the plasma is displaced.One has izing; the worst caseis a perturbationfor which it is zero, i.e., for which ;nmau2dV=--;{u.V[p(X)+BB(X)/!•o] (V. u) + u. (VPT + eB2/i•o)/Py= 0 (24) -- 2u. cBB{X)/!.•o -- n{X)rngßu} dV (15) Inspection of equations(3) and (5) showsthat condition (24) where the integralsare understoodto be taken over the entire correspondsto the requirement that the total perturbation volume in which the perturbationsare present. pressure(plasma plus magnetic)is balanced.It has a ready Now we make use of the vector identities interpretationin the dynamicalpicture of instability.Consider equation (15). The curvature and gravity forcesare the sole V. [up(x) ] = u. Vp(x) + p(X)V. u (16) forcesacting when the total pressurevanishes. In a field where V ß [uBB{x)] = u. V[BB(x)] + BB{X)Vß u (17) the external (gravity or other) force, the curvature,and the pressuregradients are coplanar,i.e., act in meridiansin a plan- and Gauss'stheorem to give etary context, the total perturbation pressurebalance will en- couragethe fastestgrowth of motionsthat are restrictedto the meridian. Driving flow out of the meridian consumesenergy j"nmau2dV=-yu(p(X'+BB(X'/llo).dS without releasing any and is thus energeticallyless efficient. The remaining terms in (21) are proportional to the square +f [(Vßu)(p (x)+BB(X)/•o) of the amplitude of the displacement.Consider the pressure gradient terms.The worst casefrom the point of view of sta- + 2u. eBB(X)/•o+ n{X)rngßu] dV (18) bility is when the gradient in pressurepoints in the same Substitutingfrom equations(2), (3), and (5) and the equilibri- direction as the curvature vector, e, and exceedsthe critical um condition, (1), gives pressuregradient given by (22) over somepart of the flux tube. The final terms involve gravity. Evidently, a gradient in den- sity in the oppositedirection to the gravity field reducesthe stability of the system. ;nma2u2dV=--•{(V'u)2py In the context of planetary magnetospheresit is reasonable + 2(V ßu)u ß [VP r + cB2//1o] to assumethat the gravity field pointsin the samesense as the + 2u ßc[u. VB2/2/to+ u ßcB2//to] field curvature vector (toward the planet). It follows that nor- mally the gravitationalforce will stabilizemotions where the + (mg. u)(u. Vn)} dV (19) field curvature is destabilizing. wherePy = (7P+ B2//•o)and we haveremoved the contri- ROTATING SYSTEMS bution from the surfaceintegral by taking the boundariesto be outsidethe region in which the plasma displacementstake In a rotating system the centrifugal force enters the basic place. equations as an effectivegravitational force that points away For stability,a 2 must be negativefor all possibledisplace- from the planetary rotation axis. In addition, in a rotating ments.Let us rearrange(19) suchthat the crossterms in V ß u system,Coriolis force will enter the dynamics.However, there and u on the right appear in a perfectsquare. We obtain are some further subtleties. If the system is rotating, there must be some force acting to maintain the rotation. In the caseof a magnetospherein steady state the field stressesact to • nma2u2dV=-- ; {P•[(V .u) +u . (VPT +CB2/•o)/P•]2 enforcecorotation with the planetary ionosphere.There may thus be "hidden" forces to be accounted for when one consid- + 2u ße[u ßVB2/2/1o + u. eB2//1o] ers interchangemotions in a rotating system. -- [u. (VPT + ee2/l•o)]2/Py+ (rag. u)(u. Vn)} dV (20) We shall assumethat rigid corotation is maintained on the time scale of the interchange displacementswe consider. After further algebra and substitutionof the equilibrium con- (Other possible assumptionsare examined in the discussion dition, (1), we can recastthe equation in the following form: sectionlater in the paper.) As a result,angular momentum will 112 SOUTHWOOD AND KIVELSON: MAGNETOSPHERICINTERCHANGE INSTABILITY not be conserved when plasma moves at right angles to the rotation axis. The forces in the system that counter the Co- riolis force in the azimuthal senseare transmitted through the • nmrr2u2dx• =-- f dx• uß(Vp - VPc•) Vßu (31) plasma by the field. The energy associatedwith spinning flux Now recall our earlier demonstration that V. u is indepen- tubes up or down is provided by the ionosphere of the planet. dent of position along the flux tube. It may thus be moved With the assumption of rigid corotation there is no azi- outside the integral. Similarly, u-(¾p) can be removed from muthal perturbation velocity. In the meridian there is no Co- the integral becausep is constant on each flux tube, and using riolis force component, and the perpendicular equation of the local coordinates introduced earlier, motion takes the form u-(Vp) = (u2/h2)•p/•x2 (32) nmau= -- V(p(•) + BB(•)/po) + 2eBB(•)/po Let us choosethe coordinate x 2 to increasein the direction of + n(•)mg•_-- n(•)m[fl x (fl x r)]•_ (25) curvature. The condition for stability then becomes where fl is the angular velocity of the plasma. A modified energy equation which replaces (13) is obtained c3p/c3x2<[f d,x, (h2)-•(&pc•/&x2)]/(f dx•)(33) by dotting (25) with u. It is evident that the set of equations Equation (33) can be further simplified once it is noted that governing the instability remain unchanged if g is replaced by Vpc• is related to the local gradient in flux tube volume. Con- an effective acceleration: sider an elemental length, ds, along the field. The local gradi- gE =g- f! x (fl x r) (26) ent of the volume,ds/B (or dxa), in the direction of •2 is given by In an idealized rotating systemwith straight field lines [Cheng, 1985], one or the other of the terms in (26) is usually dolni- f•2' V(ds/B)= ds•2 ø[--(VB)/B 2 --c/B] (34) nant. In magnetosphericapplications, although the centrifugal and thus (33) can be rewritten as acceleration may dominate near the equatorial portion of a flux tube, it may be necessaryto retain both terms of gE because of the importance of the gravitational contribution •p/c•x2+[ypr3(;dXl)/r3x2]/(fdxl)
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