Abstract Many widely used methods for describing and understanding the magnetosphereare based on balance conditions for quasi-static equilibrium (this is particularly true of theclassical theory of magnetosphere/ionosphere coupling, which in addition presupposes theequilibrium to be stable); they may therefore be of limited applicability for dealing withtime-variable phenomena as well as for determining cause-effect relations. The large-scalevariability of the magnetosphere can be produced both by changing external (solar-wind)conditions and by non-equilibrium internal dynamics. Its developments are governed bythe basic equations of physics, especially Maxwell’s equations combined with the uniqueconstraints of large-scale plasma; the requirement of charge quasi-neutrality constrains theelectric field to be determined by plasma dynamics (generalized Ohm’s law) and the elec-tric current to match the existing curl of the magnetic field. The structure and dynamics ofthe ionosphere/magnetosphere/solar-wind system can then be described in terms of three in-terrelated processes: (1) stress equilibrium and disequilibrium, (2) magnetic flux transport,(3) energy conversion and dissipation. This provides a framework for a unified formulationof settled as well as of controversial issues concerning, e.g., magnetospheric substorms andmagnetic storms.
The magnetosphere is observed to be continually varying on many different time scales;some of the characteristic variations in the form of particular types of events (e.g., magneticstorms, magnetospheric substorms) are among the most challenging problems to explain,as well as often of great practical significance (e.g., for space weather). The large-scalevariability of the magnetosphere can be the result both of changing external (solar-wind)
V. M. Vasyliunas (�)Max-Planck-Institut für Sonnensystemforschung, 37191 Katlenburg-Lindau, Germanye-mail: [email protected]
conditions and of non-equilibrium internal dynamics; separating the contributions of the twois not always simple, since the solar wind is likewise continually varying on many differenttime scales.
This review discusses the variability of the magnetosphere primarily from the point ofview of physical understanding, of trying to see how the complex variable phenomena ofthe solar wind/magnetosphere/ionosphere/atmosphere system follow from the basic laws ofphysics. Essential observational results as currently understood are taken into account, butthis is not intended as a review of observations as such. The aim is rather a systematic phys-ical description/formulation, in which observations may suggest and guide explanations butshould not appear as explicit premises, and in which absolute primacy is given to the basicequations (including conservation laws) in their complete form, with any approximationsexplicitly introduced and justified.
Section 2 summarizes the most important conventional methods, which mostly assume(explicitly or implicitly) an equilibrium situation and hence may need to be re-examinedwhen applied to time-varying cases. Section 3 reviews some important differences betweenelectrodynamics in space and in the ordinary laboratory and discusses the modificationsthey impose on methods mentioned in Sect. 2. Section 4 presents definitions and physi-cal descriptions of the two major types of events: magnetospheric substorms and magneticstorms. (Length limitations preclude a discussion of sawtooth and steady-magnetospheric-convection events, which might be viewed as variants of the substorm.) Finally, Sect. 5attempts to interpret the principal types of magnetospheric-variability events in terms of acoherent set of fundamental physical processes or, where that is not yet possible, to formu-late a set of essential physical questions.
The emphasis is on understanding the physical processes that underlie distinct individualevents, of large spatial scale, on various time scales: substorm onset (minutes or less), sub-storm phases and similar events (tens of minutes to an hour or so), recurrence tendencies ofsame (typically hours), magnetic storms (hours to days).
2 Conventional (Quasi-Equilibrium) Methods and Their Limitations
2.1 Magnetic Field Configuration
The configuration of the magnetic field is what in essence defines the magnetosphere. Ourknowledge of it in the Earth’s magnetosphere is very extensive but is derived almost entirelyfrom observations, often represented by empirical models (e.g. Tsyganenko 2001, and ref-erences therein) which can be quite sophisticated; physical understanding, however, in thesense of seeing how the models follow from the basic equations, is somewhat limited. Nu-merous theoretical models that describe specific regions of the configuration and its changesdo exist, but they represent rather a patchwork, each model applied to a different aspect andoften derived in a different way. In recent years global numerical simulations have comeinto widespread use, to calculate magnetic fields and other properties of a model magne-tosphere; they do constitute a unified treatment but can be as incomprehensible as the realmagnetosphere.
A common custom in magnetospheric and ionospheric physics, following elementaryE&M textbook usage, is to preferentially describe the magnetic field configuration, whetherdeduced from observations or from theory, by specifying the electric currents that wouldproduce the configuration via Ampère’s law
4πJ = c∇ × B (1)
Physics of Magnetospheric Variability
(Gaussian units are used throughout this paper); changes of the configuration are likewisedealt with as corresponding changes of the current system. It may be noted, however, thatessentially our entire empirical knowledge about the magnetic configuration of the mag-netosphere and its changes has been derived from observations of the magnetic field, anystatements about currents being inferred therefrom by invoking (1). Direct determinationsof J from charged-particle observations have been attempted in the magnetosphere, but,leaving aside questions of how reliable they are in view of spacecraft charging constraints,their relative contribution to our knowledge of current systems in the magnetosphere hasbeen negligible. In the ionosphere, no direct measurements of J have ever been reported,to my knowledge, nor do I know of any practical method for making them. The relationbetween magnetic field and electric current, under conditions appropriate to space plasmas,is further examined in Sect. 3.2.
2.1.1 Dayside Magnetosphere
The one case of a magnetic field configuration that is almost completely understood in phys-ical terms is the dayside magnetosphere, under conditions where penetration of magneticfield and of plasma across the magnetopause can be neglected (idealized as a magneticallyclosed magnetosphere). Unable to penetrate to any significant extent into the geomagneticfield, the solar-wind plasma is initially slowed down and compressed until the pressure hasincreased sufficiently so that its lateral gradient deflects the flow around the magnetosphere;the exterior pressure also deforms and compresses the magnetic field inside the magne-tosphere. In equilibrium, both the exterior plasma flow and the interior magnetic field aretangent to the boundary surface, the magnetopause; furthermore, the total pressure (plasmaplus magnetic) is the same on both sides at any point of the boundary. With the idealiza-tions of negligible magnetic pressure outside and negligible plasma pressure inside, thisbecomes what is generally known as the Chapman-Ferraro model, which has a well-definedmathematical formulation extensively investigated in the 1960’s (see, e.g., Siscoe 1988, fordetailed review and references).
Calculations based on the Chapman-Ferraro model predict the location and shape of themagnetospheric boundary, the magnetic field line pattern within the magnetosphere (visu-alizable as a compressed dipole field), and the resulting geomagnetic disturbances. For themost part, with the exception primarily of intense magnetic storm periods, these predictionsare in reasonable agreement with what is observed on the day side of the magnetosphere, inparticular the decreasing distance of the subsolar magnetopause with increasing solar winddynamic pressure ρswVsw
2 and the northward jumps of the low-latitude geomagnetic field(sudden commencements and sudden impulses) when ρswVsw
2 suddenly increases.The magnetic field and its changes in the Chapman-Ferraro model are usually described
in terms of a current system, the Chapman-Ferraro current. Fundamentally, however, theinterior magnetic field is calculated from the condition that it be tangent to the magnetopausesurface, the location of which is adjusted to satisfy the condition of equal total pressurebalance; the current is then obtained from the magnetic field change via (1). (The conditionthat the exterior flow be tangent to the magnetopause surface is satisfied by adjusting thelocation of the bow shock.)
When the Chapman-Ferraro model was first proposed in the 1930’s, it was more or lesstaken for granted that there was no interplanetary magnetic field. The Chapman-Ferraro cur-rent, which served solely to contain the interior magnetic field within the volume of themagnetosphere, was confined to the boundary surface (magnetopause) since any sources ofthe interior field except for the Earth’s dipole were assumed negligible. This was a magnet-ically closed magnetosphere (to my knowledge, the first clear concept of a magnetosphere
V.M. Vasyliunas
with a definite magnetic topology). By the time of the 1960’s work reviewed by Siscoe(1988), the existence of the interplanetary magnetic field had been established, but a closedmagnetosphere was still widely assumed, at least as a first approximation.
As long as the total exterior pressure is not greatly modified, the properties of the cal-culated Chapman-Ferraro model are not significantly affected by the presence of the inter-planetary magnetic field. The Chapman-Ferraro current, however, is now required not onlyto confine the interior field within the magnetosphere but also to exclude the exterior fieldfrom the magnetosphere; the magnetic field is assumed tangent to the boundary surface onboth sides. The exterior field, carried by the plasma flow in the magnetosheath, becomesdraped around the magnetopause, as suggested by Piddington (1964) and modeled quan-titatively by Spreiter et al. (1966) and Alksne (1967). The draping configuration impliesa current system that is not confined to the magnetopause but extends from it through themagnetosheath to the bow shock Although clearly implied by (1), this current system wasnot explicitly mentioned by Spreiter and coworkers (a rare instance of describing a magneticfield without referring to the associated current). It is, however, a necessary feature of theChapman-Ferraro closed magnetosphere in the presence of a non-zero interplanetary mag-netic field and may thus be called the “exterior Chapman-Ferraro current”; the direction ofthe current depends of course on the orientation of the interplanetary magnetic field. Whatappears to be a (misnamed) example of it is the so-called “reconnection current” deduced bySiebert and Siscoe (2002) from an MHD simulation of the magnetosphere interacting witha southward interplanetary magnetic field: it has the geometry expected from the draping ofthe field (which may be reduced but in general not eliminated by reconnection).
2.1.2 Nightside Magnetosphere and Magnetotail
Unlike those for the day side, the predictions of the Chapman-Ferraro model for the nightside disagree completely with even the most basic aspects of the observed magnetosphere,as sketched in Fig. 1: on the day side, the magnetic field lines are compressed, while onthe night side they are observed to be stretched out to form an extended magnetotail. TheChapman-Ferraro model, on the contrary, although its magnetopause shape superficially re-sembles the observed tail boundary, has field lines that are compressed at all local times;they are just compressed much more on the day side than on the night side. This is seenparticularly clearly in the model calculation of the distorted field by Mead (1964): the field
Fig. 1 Schematic view of a(magnetically closed)magnetosphere, cut in thenoon-midnight meridian plane.Open arrows: solar wind bulkflow. Solid lines withinmagnetosphere: magnetic fieldlines
Physics of Magnetospheric Variability
strength at the equator is everywhere larger than the dipole value (his Fig. 1), and the equa-torial crossing distance of every field line is smaller than the nominal dipole L (his Figs. 4and 5).
Contrary to a widespread impression that the magnetotail was first found in the observa-tions (Ness 1965) and was a surprise to theorists, the existence of the magnetotail was in factpreviously suggested on theoretical grounds (Parker 1958; Piddington 1960; Dessler 1964;Axford et al. 1965); the magnetotail field lines sketched in Fig. 3 of Piddington (1960) orFig. 1 of Dessler (1964) are virtually identical to those sketched with the label “experi-mental” in Fig. 14 of Ness (1965). The physical reason for the magnetotail is the assumedexistence of some process by which a force exerted on plasma within the nightside mag-netosphere pulls it (and with it the magnetic field lines) away from the Earth in the anti-sunward direction, to form the configuration of quasi-uniform antisunward (northern lobe)or sunward (southern lobe) magnetic fields, extending to large distances (� distance todayside magnetopause). Proposed as the primary tailward force have been (a) tangential(viscous-like) drag across the magnetopause by the external flow (Axford and Hines 1961),(b) magnetic tension of field lines connected across the magnetopause (Dungey 1961), partof the open-magnetosphere concept discussed in Sect. 2.1.4 and now widely considered thedominant effect, (c) internal pressure of plasma or of hydromagnetic waves (Dessler 1964),now generally considered unimportant at Earth (but possibly significant at the giant planetsJupiter and Saturn). The global role of the tailward force has been emphasized by Siscoe(1966) (see also Vasyliunas 2009).
The reversal of the magnetic field between the northern and southern lobes implies theexistence of a current sheet, with the associated plasma sheet, in the equatorial region of themagnetotail. What is called the magnetotail current system consists of the equatorial currentsheet plus its closing currents on the tail magnetopause north and south (forming in cross-section the familiar theta pattern) and possibly also via Birkeland (magnetic-field-aligned)currents to the ionosphere. Temporal variations of the magnetotail field configuration or cur-rent system are among the primary aspects of substorms and related events (Sect. 4.1). Boththe configuration and its changes have been studied very extensively, but primarily from anobservational/empirical point of view. Numerous theoretical models have been developedfor special aspects (see, e.g., Schindler 2007, and references therein), but there is little fun-damental global understanding at a level comparable to that of the Chapman-Ferrro model.
2.1.3 Inner Magnetosphere
In the inner region of the magnetosphere, at distances well below those of the magnetopauseand the inner edge of the magnetotail current sheet, the magnetic field is dominated by theinternal field of the Earth, adequately represented as a dipole for most purposes of magne-tospheric physics. In addition, the effects of plasma within the magnetosphere are of interest,primarily in relation to disturbances of the geomagnetic field. This is one of the few instancesin magnetospheric physics where the electric current seemingly can be determined (at leastapproximately) independently of its relation (1) to the magnetic field: charged-particle driftsin the inhomogeneous magnetic field depend on the sign of the charge and thus give rise toa current. According to a fundamental result derived by Parker (1957), the current density Jgiven by the sum of all the single-particle drifts satisfies the plasma momentum equation
∂
∂t(ρV) + ∇ · κ − ρg = 1
cJ × B (2)
where κ = ρVV + P is the kinetic tensor. Gradient, curvature, and magnetization drifts addup to the pressure tensor term; the time-derivative and inertial terms come from the so-called
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polarization drifts (see, e.g. Northrop 1963); the gravitational term, neglected as unimportantin most plasma contexts (with the possible exception of vertical structure of the ionosphere),comes from the g × B drift. (E × B drifts carry no net current.)
Equation (2) plays an essential role in magnetosphere/ionosphere coupling, further dis-cussed in Sects. 2.2 and 3.5. The current system in the inner magnetosphere, governed by (2),is generally referred to as the ring current; its variations resulting from enhancements of thelocal plasma pressure constitute the primary aspect of magnetic storms (Sect. 4.2).
2.1.4 The Open Magnetosphere
After the Chapman-Ferraro model had explicitly (even if perhaps by default, through ne-glect of the interplanetary magnetic field) introduced the concept of the magnetically closedmagnetosphere, the essential next step was the explicit proposal by Dungey (1961) that themagnetosphere can be magnetically open, with magnetic field lines from the Earth’s dipoleconnecting across the magnetopause with those of the interplanetary magnetic field whenthe relative orientation of the two fields is favorable for reconnection. This concept and thegeneral concept of tangential drag across the magnetopause, introduced at almost the sametime by Axford and Hines (1961), constitute the two basic ideas from which most of thephysics of magnetospheric variability derives, directly or indirectly.
Figure 2 illustrates schematically the main features of the simplest magnetically openmagnetosphere, for the case when the interplanetary magnetic field is southward, i.e., an-tiparallel to the dipole magnetic field in the equatorial plane. The topology of the magneticfield lines and the plasma bulk flow is shown in three projections: noon-midnight meridianplane, equatorial plane, and projected along field lines to the ionosphere. The primary newfeature is the existence of the separatrix surface with two branches: one (topologically atorus, or a doughnut with the Earth as the hole) separates open and closed field lines, the
Fig. 2 Schematic topological view of a magnetically open magnetosphere. (a) Upper left: noon-midnightmeridian plane (solid lines: magnetic field lines, open arrows: plasma bulk flow directions). (b) Lower left:equatorial plane (lines: plasma flow streamlines, line of x’s: magnetic X-line = closed/interplanetary fieldline boundary). (c) Right: projection on ionosphere (lines: plasma flow streamlines, line of x’s: open/closedfield line boundary = projection of magnetic X-line = polar cap boundary). The sunward direction is alwaysto the left
Physics of Magnetospheric Variability
other (topologically a cylinder surrounding and touching the torus) separates open and in-terplanetary field lines (thick lines in Fig. 2a represent cuts through both branches of theseparatrix). The intersection of the two branches is the magnetic X-line, also called the sep-arator line; note that in general it is not a magnetic neutral line—at the X-line, all magneticfield components perpendicular to the line vanish by definition, but the component along theline is nonzero except at isolated points (and in some idealized geometries).
As illustrated in Fig. 2b, the X-line forms a closed ring around the Earth’s dipole axis;this is a topological necessity (the ring, though, need not lie in the equatorial plane exceptfor cases of special symmetry). Magnetic field reconnection, however, occurs on only thosesegments of the X-line where plasma flows across the separatrix (both branches). As canbe seen from the figure, there are two distinct such segments, generally referred to as thedayside and the nightside reconnection region, respectively (for a more general discussionof possible geometries, see, e.g., Vasyliunas 1984); the terms dayside X-line and nightsideX-line are also widely used but tend to obscure the fact that the X-line of the open mag-netosphere must be a complete ring (in addition, there is another type of nightside X-line,discussed in Sect. 2.1.5).
Quantitative properties of an open magnetosphere include the total amount of open mag-netic flux ΦM , leaving one hemisphere to connect to the interplanetary magnetic field (equalto the dipole flux through the polar cap, defined as the region of open field lines at the Earth’ssurface and illustrated in Fig. 2c). The time rate of change of the open flux is equal to therate of flux addition by dayside reconnection minus the rate of flux removal by nightsidereconnection:
d
dtΦM = cEd − cEn (3)
where Ed and En are the line integrals of the electric field along the dayside and the nightsidereconnection segments of the X-line, respectively. Ed can be related to solar wind parametersby
Ed � 1
cLXVswBs (4)
where LX is the length of the dayside reconnection segment projected along plasma flowstreamlines back into the undisturbed solar wind (as illustrated in Fig. 2b—not the length atthe X-line itself), and Bs is the reconnecting component of the interplanetary magnetic field,frequently approximated as the southward component. In a quasi-equilibrium state, Ed isequal to maximum line integral of the electric field across the polar cap, usually called thepolar cap potential (or transpolar potential) ΦPC (the use of Φ both for magnetic flux andfor potential, although perhaps confusing, is traditional). The open flux can be expressed as
ΦM � LX LMT Bs (5)
where LMT (also illustrated in Fig. 2b) is the distance between the dayside and the night-side locations of the outer (interplanetary/open) branch of the separatrix, extended into theundisturbed solar wind; less precisely but more understandably, LMT may be viewed as theeffective length of the magnetotail.
All three parameters of the open magnetosphere can be inferred from observations. Thearea of the polar cap can be estimated by a variety of auroral and precipitating particle ob-servations, and from it the open flux ΦM . The polar cap potential can be estimated fromionospheric plasma flow measurements, either by particle or by radar techniques, and com-parison with solar-wind flow and magnetic field measurements then yields an estimate of
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LX . In quasi-equilibrium, the solar wind flow time across the distance LMT should equal theionospheric plasma flow time across the polar cap, from which, as pointed out by Dungey(1965), LMT can be estimated. The empirical description of what the basic properties ofthe open magnetosphere are, how they relate to solar wind parameters, and how they varyin connection with various types of magnetospheric events has become quite extensive, butphysical understanding remains rather limited. The following are some of the fundamentalquestions for which there are as yet no generally accepted answers:
1. Can the quantitative properties of the open magnetosphere, in particular the values of ΦM ,ΦPC , LX , and LMT , be derived in a physically understandable way from first principles?
2. In case the solar wind parameters remain steady for a sufficiently long period (manyhours at least, or perhaps days), does the magnetosphere settle into an essentially steadystate, with constant values for the properties enumerated above?
3. If the answer to question 2 is yes, are these constant values unique functions of the solar-wind parameters? i.e., do they depend on the parameter values at the present time only (asdistinct from depending on their past history, or else in some non-deterministic fashion)?
4. What is the physical definition of the magnetopause in an open magnetosphere?
Numerical simulations of the magnetosphere on a global scale are increasingly used asa tool to answer questions such as the above. In addition to the well-known general limita-tions and caveats (e.g., Post and Votta 2005; Greenwald 2010), there is the problem that theoutput of an individual simulation often may be as complex and as difficult to interpret asthe observational data from the actual magnetosphere, with no obvious way of extracting thephysical insight which is what question 1 really is about. The remark “the theorist knowslittle, understands much; the experimenter knows much, understands little; the computersimulator knows everything, understands nothing” (Vasyliunas 2008) undoubtedly is exag-gerated, but sometimes perhaps not all that much. To my knowledge, no definite answers toquestions 2 and 3 have been forthcoming from numerical simulations to date. Question 2is particularly difficult to deal with, both from observations (the solar wind is rarely, if atall, steady on the time scales in question) and from simulations (the running time is oftenlimited by numerical problems).
Question 4 has received little attention, perhaps because in practice there is not muchdifficulty, either in observational data or in simulations, about identifying what may reason-ably be considered a magnetopause. There is, however, a conceptual issue: a closed magne-tosphere has a boundary surface not crossed by any field lines, which uniquely defines themagnetopause, whereas in an open magnetosphere there is no such unique surface but therecan be several distinct waves and discontinuities at the interface to the solar wind.
2.1.5 Topological Changes in the Magnetotail
Besides the overall topology of the open magnetosphere described in Sect. 2.1.4, changesof the magnetic field configuration suggestive of additional topological structures occur inthe magnetotail, primarily in association with magnetospheric substorms. Observations ofmagnetic fields and plasma flows (e.g., Nishida and Nagayama 1973; Hones et al. 1973;Russell and McPherron 1973) indicated a reconnection magnetic X-line located at dis-tances of some 15–30RE , much closer than the typical distance (generally estimated as100–200RE or more) of the nightside reconnection X-line of the open magnetosphere. Sis-coe and Cummings (1969) were the first (as far as I know) to propose explicitly that a newmagnetic X-line, distinct from the distant nightside X-line, forms near the Earth; they basedtheir argument on breakdown of stress balance that maintains the magnetotail and visual-ized the X-line as forming at or near the interface between the dipolar inner magnetosphere
Physics of Magnetospheric Variability
Fig. 3 Possible changes of themagnetic field topology in themagnetotail of an openmagnetosphere (Vasyliunas1976)
and the magnetotail, ∼ 6–8RE . Subsequent work (e.g., Schindler 1974) interpreted X-lineformation as the result of tearing instability of the magnetotail current sheet and placed itfarther out, well within the magnetotail stretched-out field line region.
The temporal development of a possible three-dimensional magnetic topology associatedwith formation of a new X-line sketched by Vasyliunas (1976) is shown here in Fig. 3. Pan-els 1 through 5 show schematically the topology of magnetic field lines at five sequentialinstants of time. Shown in each panel are the noon-midnight meridian plane, the equato-rial plane, and the projection on the ionosphere, in the same format as in Fig. 2; a newfeature is the magnetic O-line (marked by a line of o’s in the equatorial plane). The pre-event state (panel 1) is topologically identical with Fig. 2. Within the region of closed fieldlines, the onset of magnetic reconnection creates an initially localized “plasmoid” (panel 2),containing field lines confined within it and not connected either to the Earth or to the in-terplanetary magnetic field; topologically, the X-line is a short segment joined to an O-linesegment to form a complete ring. The plasmoid grows (panel 3), expanding both in themeridian plane (as the result of continuing reconnection) and in X-line extent (for no clearlyidentified reason), until it reaches the closed/open field line separatrix (panel 4, onset oflobe reconnection). After this, the plasmoid is contained within the region of interplanetaryfield lines (panel 5). The net result of the entire sequence is to “break off” a segment of theopen-magnetosphere X-line ring and effectively shorten the magnetotail.
The highly idealized and simplified plasmoid model of the original Fig. 3, intended to il-lustrate just the topology, was redrawn (meridian plane projection only) with the addition ofmany “realistic” details of size and shape by Hones (1976, 1977), and in this version became
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the iconic image of what is now called the near-Earth X-line (NEXL) model of the substorm(or near-Earth neutral line, NENL; the caveat in Sect. 2.1.4 concerning the neutral line ofthe open magnetosphere applies here, too). The highly controversial question of how thismodel is related to the actual substorm is discussed in Sect. 4.1; here I am concerned onlywith what can be said about the magnetic configuration by itself. At present, understandingthe topological changes in the magnetotail is in pretty much the same state as understandingthe open magnetosphere (Sect. 2.1.4) or the magnetotail itself (Sect. 2.1.2): there is veryextensive empirical knowledge, there is a patchwork of models for many individual aspects,and more recently a patchwork of individual numerical simulations, but the physical under-standing of the process as a whole remains limited. Among the unanswered fundamentalquestions are the following:
1. Why and how does the onset of reconnection ocurr?2. What is the real topology of the structure commonly referred to as the plasmoid?3. What determines the location of the newly formed magnetic X-line?
The study of magnetic field line reconnection (see, e.g., Schindler 2007; Longcope 2009;Forbes 2009, and references therein) is a major branch of space and astrophysical plasmaphysics in its own right. Question 1 addresses a very specific issue of how reconnectioncomes to be where it wasn’t before (thus, studies of the properties and evolution of alreadyoccurring reconnection, however informative otherwise, do not provide an answer). The nu-merous studies of reconnection brought about by tearing-mode instability of the magnetotail(Schindler 2007, and references therein) do address question 1 directly to some extent; thereremains, however, the problem of how to set up the initial configuration, assumed to be inequilibrium but unstable, as well as how to differentiate between that and straightforwardevolution to non-equilibrium, e.g., as the result of changing boundary conditions.
Question 2, concerned with the three-dimensional structure of the plasmoid, has not beenhelped by the widespread tendency to consider only the meridian plane view of the topo-logical sequence in Fig. 3 and hence to think of the plasmoid magnetic field as basicallytwo-dimensional, to be modified (or even destroyed) by the addition of an out-of-plane com-ponent. The plasmoid in Fig. 3 as originally conceived is an intrinsically three-dimensionalobject with a well-defined magnetic topology: a closed volume, the field lines within whichdo not connect to anything outside the volume. What has not been established, to my knowl-edge, is whether such a closed volume does exist, or whether it even can exist topologically;what is lacking so far is a conceptual demonstration of the possibility of the closed sin-gular ring that is part X-line, part O-line (for the open magnetosphere, the closed X-linering mentioned in Sect. 2.1.4 can be demonstrated with the simple example of uniform fieldsuperposed on a tilted dipole).
An alternative interpretation of the observed plasma and magnetic field structures, par-ticularly in case of an appreciable dawn-dusk component of B, is in terms of a magneticflux rope (see, e.g., Sibeck 1990, and references therein). Although flux ropes have beendiscussed in a variety of contexts for decades, a precise definition of what a flux rope is has,to my knowledge, been first given only recently by Moldwin et al. (2009): a flux rope is atwisted flux tube, a flux tube being defined in turn as “the volume enclosed by a set of fieldlines that intersect a simple closed curve.” The implied topology of the magnetic field is notclear.
Question 3, a quantitative sub-aspect of question 1, plays a significant role in understand-ing substorm physics and is further discussed in Sect. 4.1.
Historical Note A curious fact about the origin of the topological sketch of Fig. 3 is thatthe initial motivation for it had nothing to do with substorms. I was trying to understand
Physics of Magnetospheric Variability
what would happen to the open magnetosphere if the nightside reconnection X-line wereto be continually carried antisunward by the plasma mantle (or magnetosheath) flow: wasthere anything to prevent the magnetotail length LMT from increasing indefinitely whilethe open flux ΦM remained constant? Only after finding the topological sequence did I seethe (in retrospect obvious) relation to substorm phenomenology, and published the result inthat context. (Historians of science have repeatedly remarked upon the difference betweenresearch as formally published and research as actually done.)
2.2 Magnetosphere/Ionosphere Coupling Theory
The description of plasma dynamics on the basis of self-consistent coupling between magne-tosphere and ionosphere is arguably one of the most successful theories in magnetosphericphysics. It accounts for the pattern of magnetospheric convection at auroral and low lati-tudes, the distribution of Birkeland currents, and the variations of all these in response tochanging orientation of the interplanetary magnetic field. The basic scheme of calculationis illustrated in Fig. 4 in the form in which it was first proposed by Vasyliunas (1970) andWolf (1970), generalizing earlier work by Fejer (1964); since then it has been elaboratedconsiderably in detail, but neither the logic nor the equations have changed appreciably (fora review see, e.g., Wolf 1983, and references therein). If the electric field in the magne-tosphere is known, particle motions can be calculated from E × B drifts (plus other drifts ifconsidered significant), and the continuity/transport equations (together with boundary con-ditions on particle concentrations) then determine the plasma pressure in the magnetosphere.
From plasma pressure gradients, J⊥ can be calculated with the use of (2) (provided thetime-derivative and inertial terms can be neglected). Current continuity ∇ · J = 0 then deter-mines J‖, the Birkeland currents flowing between the magnetosphere and the ionosphere;from the requirement that these currents close through the ionosphere, the ionosphericOhm’s law (together with appropriate boundary conditions) serves to determine the elec-tric field in the ionosphere. Mapping this electric field to the magnetosphere along magneticfield lines (under the MHD approximation E · B � 0 or some more accurate version of thegeneralized Ohm’s law if deemed necessary) imposes the self-consistency requirement ofagreement with the initally assumed magnetospheric electric field and thus closes the sys-tem of equations.
The simple scheme of Fig. 4 can be implemented at differing levels of sophistication thatcan vary over an enormous range, from straightforward analytical models (e.g., Fejer 1964;Iwasaki and Nishida 1967; Vasyliunas 1970) to complex advanced versions of the Rice Con-vection Model (e.g., Toffoletto et al. 2003), for which nevertheless the computing effort ismuch smaller that required for a MHD simulation even when the physics in the latter isconsiderably less detailed. Subsidiary physical effects such as pressure anisotropies, inho-mogeneous and variable ionospheric conductances, influences of precipitating particles andmodels for parallel electric fields can be incorporated, with corresponding gains in ability toaccount for complicated aspects of observed phenomena. There is, however, a fundamentallimitation when the theory is applied to describe magnetospheric variability. In the diagramof Fig. 4, all the links involve equations in which the time derivatives have been neglected,with the sole exception of the transport equation that relates the pressure distribution inthe magnetosphere to the electric field; in particular, all the processes of mapping betweenthe ionosphere and the magnetosphere are treated explicitly as instantaneous, neglectingany propagation effects. The only time variability contained in the theory is the evolutionof pressure as the plasma is advected by the flow, plus any time variations in the imposedboundary conditions. As discussed in Sect. 3.5, this limitation is more restrictive than mightappear at first.
3 Electrodynamics of Large-Scale Plasmas
From the beginning, the magnetosphere/ionosphere coupling theory was formulated ina form chosen more for mathematical convenience than for physical understanding. Va-syliunas (1970) introduced the scheme of Fig. 4 with the words “. . . it proves convenientto formulate the convection problem not in terms of the dynamical concepts of flow andstress but in terms of electric field and current” and then added as a parenthetical remark“fundamentally, of course, these two modes of treating the problem are equivalent, sincein a plasma there is a close connection between the flow and the electric field and be-tween the stress and the electric current.” How close is the connection and what does itreally mean has more recently become the subject of what is sometimes called the B-V vs.J-E controversy (e.g., Parker 1996, 1997, 2000, 2007, Heikkila 1997; Lui 2000). Decid-ing whether B and V or, instead, J and E are to be treated as the primary variables hasan obvious bearing on the physics of magnetospheric variability: as long as we are deal-ing with a steady system, we may treat both sets of variables as equivalent, but as soon aswe consider time variations, the question which one is physically producing the change ofthe other becomes unavoidable. In this section I review some calculations (Buneman 1992;Vasyliunas 2001, 2005a, 2005b) that, in my view, unambiguously resolve the controversy.
Physics of Magnetospheric Variability
3.1 The Generalized Ohm’s Law
From the conventional E&M perspective of looking for currents that make magnetic fields,the first question about variability might be: what determines the time evolution of the cur-rent? With the current density J defined as
J =∑
a
qa
∫d3vvfa(v) (6)
where fa(v) is the velocity distribution function of charged particles of species a, the equa-tion for the time evolution of J, determined by summing the motions of all the chargedparticles, can be calculated from the appropriate sum of velocity-moment equations (see,e.g. Rossi and Olbert 1970; Greene 1973):
∂J∂t
=∑
a
{q2
ana
ma
(E + 1
cVa × B
)− qa
ma
∇ · κa + qanag}
+(
δJδt
)
coll
(7)
where qa , ma , na , Va , and κa are the charge, mass, concentration, bulk velocity, and ki-netic tensor, respectively, of species a, and (δJ/δt)coll represents the sum of all collisioneffects. (The gravitational acceleration g, included for completeness, mostly is unimportantin practice.)
Equation (7) is the complete and (except for neglect of relativistic effects) exact form ofwhat is commonly called the generalized Ohm’s law. For a quasi-neutral plasma of electronsand one species of singly charged ions with |ni − ne| � n and mi � me , (7) becomes
∂J∂t
= ne2
me
(E + 1
cV × B − J × B
nec
)
+ e
(∇ · κe
me
− ∇ · κ i
mi
)+ e (ni − ne)g +
(δJδt
)
coll
(8)
where V is the bulk flow velocity of the plasma. The collision term in the ionosphere(electron-ion, electron-neutral, and ion-neutral collisions) is
(δJδt
)
coll
= −(
νei + νen + me
mi
νin
)J + (νen − νin) ne (V − Vn) (9)
(see, e.g. Song et al. 2001, 2005; Vasyliunas and Song 2005), where Vn is the bulk flowvelocity of the neutral atmosphere and the ν’s are the collision frequencies. In the mag-netosphere, interparticle collisions are generally negligible, but effects of fluctuations andparticle-wave interactions may be represented by a term of the form
(δJδt
)
coll
=∑
a
{q2
a
ma
(〈δnaδE〉 + 1
c〈δ (naVa) × δB〉
)}(10)
where 〈 〉 = average over fluctuations. To simplify notation, introduce the symbol E∗ torepresent the sum of all the terms other than E on the right-hand side of (7) or (8); then (7)or (8) may be written as
∂J∂t
= ω2p
4π
(E − E∗) (11)
V.M. Vasyliunas
where
ω2p = 4π
∑
a
q2ana
ma
≈ 4πne2
me
is the (electron) plasma frequency. From (8), E∗ is given by
E∗ ≡ −1
cV × B + J × B
nec− ∇ · κe
ne+ me
mi
∇ · κ i
ne− me
ne2
(δJδt
)
coll
(12)
(the gravitational term, utterly negligible, has been left out). Equation (11) states that when-ever E �= E∗, the current density changes with time. The rate of change may be indepen-dently related to E by eliminating B from Maxwell’s equations to obtain
∂J∂t
= − 1
4π
{c2∇ × (∇ × E) + ∂2E
∂t2
}(13)
Combining (13) with (11) gives
E − E∗ = −λ2e∇ × (∇ × E) − 1
ω2p
∂2E∂t2
(14)
where λe ≡ c/ωp is the electron inertial length (also known as the collisionless skin depth).According to (14), E � E∗ unless E varies on spatial scales of order λe or smaller and/or
on time scales of order ωp−1 or shorter. On spatial and temporal scales L and τ defined by
L � λe and τ � 1/ωp (15)
what one may call in this context the large-scale limit, (14) implies that the ∂J/∂t term in(7), (8), or (11) can be neglected (a more detailed argument is given by Vasyliunas 2005a),and the generalized Ohm’s law reduces to E − E∗ � 0, or
E + 1
cV × B − J × B
nec+ ∇ · κe
ne− me
mi
∇ · κ i
ne+ me
ne2
(δJδt
)
coll
� 0 (16)
The scales λe and ωp−1 are those of electron plasma oscillations and associated electromag-
netic waves, in which charge-separation effects are important. The large-scale limit is thusthe regime in which the behavior of the plasma is strongly constrained by quasineutrality.The essential physics expressed by the generalized Ohm’s law (16) is that E assumes thevalue it must have in order to prevent a differential acceleration of ions and electrons thatwould separate charges too much.
3.2 Relation Between B and J
Ampère’s law (1) is not exact but is only an approximation to one of Maxwell’s equations
∂E∂t
= −4πJ + c∇ × B (17)
In a time-dependent situation, even if 4πJ and c∇ ×B are equal at one particular time, thereis no a priori reason why they should remain strictly equal at any other time: the evolu-tionary equations for charged particle motion and for the magnetic field, although coupled,
Physics of Magnetospheric Variability
are distinct. Any departure from equality, however, produces E evolving according to (17),which does two things: E acts to change B through Faraday’s law
∂B∂t
= −c∇ × E (18)
on a time scale τ1 ∼ L/c (light travel time across a typical spatial scale); simultaneously,E acts also to change J through the (exact) generalized Ohm’s law (11), on a time scale τ2 ∼ωp
−1 (inverse plasma frequency) (see Vasyliunas 2005b, for a more extended discussionwith quantitative examples). Equality of 4πJ and c∇ × B is reached on whichever timescale is the shorter of the two.
As long as τ1 and τ2 are both very short in comparison to time scales for phenomena ofinterest, Ampère’s law (1) remains an excellent approximation; its physical meaning, how-ever, depends critically on the ratio τ1/τ2 = Lωp/c = L/λe. In the ordinary E&M laboratoryenvironment, L/c � ωp
−1, hence B changes to make c∇ × B match the existing 4πJ. In thelarge-scale-limit (15) appropriate to much of the magnetosphere and ionosphere, on the otherhand, ωp
−1 � L/c and it is 4πJ that changes in order to match the existing c∇ × B. State-ments, found in many papers on magnetospheric variability, concerning change (disruption,diversion, wedge formation, etc.) of the electric current are thus merely descriptions of whatthe corresponding change in the magnetic field is, not explanations of how it comes about.
That J is determined by ∇ × B has long been a familiar concept within magnetohy-drodynamics (Cowling 1957; Dungey 1958; Parker 1996, 2000, 2007); what the work ofVasyliunas (2005a, 2005b) has shown is that this concept is valid well beyond MHD, downto the scales of electron plasma oscillations, where it is limited by the breakdown of chargequasineutrality.
3.3 What Determines the Magnetic Field
An exact equation, not limited to the approximate (1), relating B and J can be derived byeliminating E from Maxwell’s equations:
1
c2
∂2B∂t2
− ∇2B = 4π
c∇ × J (19)
which allows B at any point to be determined from J (more precisely, from ∇ × J) if the lat-ter is known over the complete backward light cone of the point. The discussion in Sect. 3.2shows that in a plasma, because J is subject to the additional constraint of satisfying thegeneralized Ohm’s law (11), it cannot be assumed to be known a priori, independently of B;in the large-scale limit (15), in fact, J is itself determined from B (more precisely, from∇ × B), raising the obvious question: if not from J, how then is the magnetic field to bedetermined?
Fundamentally, the evolution of B is determined by E through (18). In the large-scalelimit (15), E is determined in turn by (16) and thus by the bulk flow and other mechanicalproperties (kinetic tensors of the various species) of the plasma. The evolution of these isto be calculated from the plasma equations: the momentum equation (2) to determine Vsuffices in the simplest (MHD) approximation, but more precise approximations may beused, all the way to determining the bulk flow by numerical integration of individual iontrajectories. In (16) and (2), J is replaced by (c/4π)∇ × B. The above is a complete andself-consistent scheme for determining the evolution of B in the limit (15), correspondingmore or less to what in numerical simulations is called a hybrid code.
V.M. Vasyliunas
Fig. 5 Validity regions of various approximations. Tick marks are at factors of 10
Regions of validity of various approximations, as functions of spatial scale L and tem-poral scale τ , are illustrated in Fig. 5. The figure is drawn for VA = 800 km s−1; character-istic time scales ωp
−1, Ωi−1 (inverse of proton gyrofrequency), and Alfvén travel time τA
are marked, together with the corresponding characteristic lengths λe = c/ωp , λi = VA/Ωi ,and LFL = VAτA, the effective length of a field line (or any other macroscopic length scaleof interest). By the relativistic causality condition that no physical effect can propagatefaster than the speed of light, everything is limited to lie below the line L = cτ . The “hy-brid” approximation described above is valid for L � λe; note that, given the relativisticcausality condition, a separate limit on τ is redundant. The MHD approximation is validfor L � λi and τ � Ωi
−1. The Rice Convection Model (RCM), representative of mag-netosphere/ionosphere coupling theory calculations (Sect. 2.2), is limited in time scale toτ � τA but in spatial scale can be extended to L ∼ λi and even shorter if drift effects areincluded in sufficient detail. The “fully kinetic” treatment is in principle valid on all scaleswithout restriction, but in practice is difficult to apply except for highly idealized or simpli-fied systems. The “EM lab” refers to the laboratory of ordinary E&M devices, e.g., currentcoils, capacitors, resistors, etc., described for the most part by circuit theory; it is valid inthe time-scale range L/c � τ � ωp
−1, in Fig. 5 at the opposite extreme to MHD and RCM,suggesting that circuit analogies to the magnetosphere are unlikely to provide much physicalinsight.
In general, the equations of classical (non-quantum) physics determine only the timeevolution of B (or any other quantity) from initial conditions which can be specified arbi-trarily (subject only to the constraint ∇ · B = 0); there is no requirement or guarantee thatthere should ever be a steady state. If, however, a steady or near-steady state does exist, themomentum equation (2) can be written as
∇ · κ − ρg = 1
cJ × B = 1
4π(∇ × B) × B (20)
Physics of Magnetospheric Variability
and can be used to determine B from the requirement that the Lorentz force of the currentgiven by ∇ × B must balance the mechanical stresses; if B is predominantly the dipole field(or, more generally, a known field of external origin), (20) can be used to directly calculateJ itself, or at least its components J⊥ perpendicular to B. This stress balance calculation isthe basic method used in magnetosphere/ionosphere coupling theory to determine currentsin the magnetosphere, more recently applied also to determine B itself when the dipoleapproximation is no longer adequate, by numerical solution of (20) (e.g., Lemon et al. 2003,2004).
Not generally appreciated is the fact that currents in the ionosphere also are derived fromstress balance. They are not ohmic currents driven by an electric field in the conventionalsense: if we insert the collision term (9) into (16), we do not obtain what is usually calledthe ionospheric Ohm’s law
J = σP E∗⊥ + σH B × E∗ + σ‖E‖, E∗ ≡ E + 1
cVn × B (21)
(σP ,σH ,σ‖ are Pedersen, Hall, and parallel conductivities). Instead, we need to invoke themomentum equation (2) applied to the ionosphere, with additional terms for plasma-neutralcollisions (e.g., Song et al. 2001) on the right-hand side, and with the left-hand side usuallyneglected:
0 � 1
cJ × B − ne (miνin + meνen) (V − Vn) − me (νin − νen)
Je
(22)
using (22) and (16) (with kinetic-tensor terms neglected) to eliminate V then yields (21).This means that the ionospheric current density is determined primarily by the requirementthat its Lorentz force must balance the collisional drag force from the relative bulk mo-tion between plasma and neutrals (see, e.g., Vasyliunas and Song 2005, for a more detaileddiscussion).
Further physical insight is provided by considering how stress balance can be maintainedin a time-dependent situation. Analogously to the discussion of ∇ × B and J (Sect. 3.2),even if the Lorentz force and the mechanical stresses are equal at one particular time, thereis no general reason why they should remain strictly equal at other times; any departure fromequality, however, implies by (2) a changing bulk flow. If at some time t = 0, J × B/c is outof balance with the plasma stresses by δF , V begins to change roughly as
δV ∼ (δF/ρ) t (23)
By (16), change of V implies change of E, in general with δ(∇ × E) �= 0 and thus by (18)implying change of B:
δB ∼ (BδV/L) t ∼ (δF/ρ) t2 (B/L) (24)
where L is the spatial scale of ∇ × E. The resulting (∇ × B) × B/4π can come into stressbalance with δF after a time t defined in order of magnitude by
δF ∼ BδB/4π L ∼ (δF/ρ) t2(B2/4π L2
)(25)
or t2 ∼ (4πρ/B2)L2 = L2/VA2 ≡ τA
2. The physical process that establishes stress balanceand determines the corresponding current is thus one of force imbalance producing plasma
V.M. Vasyliunas
flow which deforms the magnetic field until the magnetic and mechanical stresses balance,on a time scale of order Alfvén wave travel time.
An essential pressuposition of the above argument is that the flow produced by a lackof equilibrium acts to bring the system toward equilibrium; this means, by definition, thatthe system is assumed to be stable. Inferring currents or magnetic field configurations fromstress balance is thus possible only if there are no large-scale instabilities playing a signifi-cant role.
3.4 What Determines the Plasma Flow
Fundamentally, the time evolution of the plasma bulk flow V is governed by the momen-tum equation (2); the actual value of V should be determined by the time integral of (2).As discussed in Sect. 3.1, in the large-scale limit (15) the electric field E is governed bythe generalized Ohm’s law in the form (16), which equates the actual value (not the timederivative) of E to a series of plasma quantities, of which an important (and in the MHDapproximation, the dominant) term is −V × B/c. From this formulation it is clear that theplasma flow is determined by the dynamics via (2), and the electric field is then determinedby the flow (plus other effects if significant) via (16).
There has been nevertheless (particularly in the ionospheric community) a persistent andwidespread view that the electric field physically produces the flow (or at least contributesto it) by creating the drift motion
V⊥ = cE × B
B2(26)
of any charged particle placed in crossed E and B fields. The simple textbook derivationsof (26), however, treat E and B as given and ignore the fields of the drifting particles them-selves. A self-consistent plasma calculation of the initial value problem by Buneman (1992)(on laboratory plasma pushed into a magnetic field, a paper that remained totally unknown inthe magnetospheric/ionospheric community) and by Vasyliunas (2001) shows, on the con-trary, that if one starts with plasma flow and no electric field, the flow continues and theelectric field appears (with the mean value corresponding to the MHD approximation) on atime scale defined essentially by the plasma frequency, but if one starts with an electric fieldand no plasma flow, the electric field simply dissolves into plasma waves (with nearly zeromean) and no appreciable flow appears; this result holds as long as VA
2 � c2, i.e., the inertiaof the plasma is dominated by the rest mass of the plasma particles and not by the relativisticenergy-equivalent mass of the magnetic field. The precise result for the final steady meanvalues Em, Vm, given the initial values E0, V0, is
Em = VA2
c2 + VA2 E0 − c2
c2 + VA2
1
cV0 × B (27)
Vm = c2
c2 + VA2 V0 + VA
2
c2 + VA2 c
E0 × BB2
(28)
Note that the condition VA2 � c2 is distinct from the large-scale limit (15), although both
have the property of holding for sufficiently high density if everything else is held fixed. Thetextbook result that E creates the drift (26) follows from (28) in the opposite limit VA
2 � c2,i.e., negligible plasma density, as expected.
There are several intuitive arguments to understand why flows produce electric fields butelectric fields do not produce flows.
Physics of Magnetospheric Variability
1. Most fundamental: Bulk flow carries linear momentum and thus can be produced onlyby adding linear momentum to the plasma, but the linear momentum density of the elec-tromagnetic field is smaller than that of the plasma by a factor VA
2/c2 (conserving thetotal linear momentum, under the assumption that the mean values satisfy the MHD ap-proximation, yields (27) and (28) directly, Vasyliunas 2001).
2. Simplest: as discussed in Sect. 3.1, on spatial and temporal scales large compared to thoseof electron plasma oscillations, the electric field is determined by the requirement that thedifferential acceleration of ions and electrons must not separate charges too much. Thebulk flow of the plasma is essentially that of the ions; since these are much heavier, theelectric field primarily changes the flow of the electrons to match that of the ions.
3. Drift motion (26) results from cycloidal trajectories of individual particles, of oppositesense for positive and negative particles and therefore tending to separate charges; theelectric field associated with this charge separation acts to reduce the initially imposed E.Conversely, in the absence of an imposed E, positive and negative particles initially mov-ing together will gyrate in opposite directions, again tending to separate charges, but nowwith the associated electric field acting to produce a drift (26) in the direction of the ini-tial motion. In both cases, the effects are dominant (imposed E reduced to nearly zero,or E nearly equal to −V × B/c created) if the concentration (number density) of plasmaparticles is sufficiently high; the quantitative criterion is easily shown to be equivalent toVA
2 � c2.
3.5 New Understanding of M-I Coupling Theory
The electrodynamics of space plasmas as summarized in Sects. 3.1, 3.2, 3.3, and 3.4 allowsa reformulation of the magnetosphere/ionosphere coupling theory with emphasis on physi-cal understanding instead of mathematical convenience. The new view of the calculationalscheme is illustrated in Fig. 6, to be compared and contrasted with the classical Fig. 4. Thefollowing are the principal differences.
1. Left-right link, top: In both magnetosphere and ionosphere, the plasma bulk flow appearsas the primary physical quantity, for which the electric field is merely a convenient proxy.The mutual adjustment of the flow in the two regions is established by Alfvén wavespropagating back and forth between the two; the traditional electric potential mappingalong field lines represents not the physical process but only its final result when and if asteady state is reached. It is now immediately obvious that the theory can be applied onlyon time scales long compared to wave propagation times.
2. Top-bottom link, left: No major differences.3. Top-bottom link, right: Here is perhaps the largest conceptual change. The coupling of
perpendicular and parallel currents by ∇ · J = 0 is merely the mathematical formulationof a physical process, the essence of which is that when the magnetic field is deformedby mechanical stresses in one location, the deformation extends necessarily to other lo-cations, by the structure of the Maxwell stress tensor (a point emphasized particularly byParker 2007). As shown in Sect. 3.2, current continuity is established on time scales of∼ ωp
−1, hence on longer scales it is satisfied automatically and need not be imposed as aseparate requirement (what is often discussed as “current closure” is really the couplingof the Maxwell stresses along different portions of a field line). The resulting magneticstress in the ionosphere accelerates the plasma until the collisional drag force from thedifferential flow between plasma and neutrals becomes equal to the J × B/c force; in theprocess, the entire magnetic stress exerted from the magnetosphere on the ionosphere is
V.M. Vasyliunas
Fig. 6 Revised schematic diagram of magnetosphere/ionosphere coupling calculations
transferred to, and must be balanced by, the neutral atmosphere (a point emphasized byVasyliunas and Song 2005).
4. Left-right link, bottom: Stress balance between the plasma and the Lorentz force remainsunderstood as the basic physical process here (with ∇ × B rather than J now viewedas the more appropriate description in terms of causes). The important change is that afundamental limitation is now explicitly recognized: the assumption of stress balance isvalid only if the system is stable and remains in slowly evolving quasi-equilibrium. Themagnetosphere/ionosphere coupling theory is thus intrinsically incapable of describingexplosive non-equilibrium developments (which, e.g., a substorm onset is generally as-sumed to be), and there is considerable uncertainty whether or how to apply it to themagnetospheres of Jupiter and Saturn, where interchange instabilities may play a domi-nant role in plasma transport.
3.6 Energy Conversion, Storage, and Dissipation
Magnetospheric variability, in events of all types, manifests itself through various phenom-ena of energy change and dissipation. The primary initial source of energy is the solar wind.The questions by what process and in what form does the energy enter the magnetosphere,what are its flow paths and conversions within the magnetosphere, what are its ultimatesinks, and what determines the time history of these developments constitute a major topicof magnetospheric physics in its own right, recently reviewed by Vasyliunas (2010). Here Ipresent a summary of the magnetospheric global energy budget in a quasi-steady or time-averaged context, as background for the discussion (Sects. 4 and 5) of time-varying energy
Physics of Magnetospheric Variability
Fig. 7 Energy flow chart for Earth’s magnetosphere. Rectangular boxes: energy reservoirs. Rounded boxes:energy sinks. Lines: energy flow/conversion processes (dotted line: process of less importance); numberskeyed to description in text (question mark: process uncertain). Only the energy-flow paths are shown, notthe mass-flow paths
conversion processes, many of which can be viewed as consequences of time offsets ordelays in particular branches of the average energy budget diagram.
A sketch of the principal energy reservoirs, conversion processes and dissipation/lossprocesses is shown in Fig. 7, adapted from the energy budget diagram for a general magne-tosphere in Vasyliunas (2010), expanded and specialized to the solar-wind-dominated mag-netosphere of Earth. The primary source of energy for the magnetosphere, shown by thedouble-lined box, is the kinetic energy of solar wind plasma bulk flow. The thermal and mag-netic energies of the solar wind can be neglected (Vasyliunas 2010); although the interplan-etary magnetic field does exert a dominant influence on energy conversion processes in theEarth’s magnetosphere, it does so primarily by control of magnetic reconnection processesand open field lines, not by supply of magnetic energy. Since any flow carries not only ki-netic energy but also linear momentum, extracting kinetic energy from the solar wind flownecessarily means also extracting linear momentum, which requires that a force be appliedto permanently slow down the flow. The mere deflection of solar wind flow around the mag-netospheric obstacle (not taking into account tangential forces at the magnetopause) extractsnet energy from the flow only because of entropy increase at the bow shock, without whichthe flow would return ultimately to its initial value; this, the irreversible heating of solarwind plasma, is labeled Process (0) in Fig. 7 and has no direct effect on the magnetosphere(see Vasyliunas 2010, for a more detailed discussion).
Process (1) is the principal process by which solar wind energy enters the magnetosphere.As suggested by Siscoe (1966) and Siscoe and Cummings (1969), the tangential force thatforms the magnetotail (see Sect. 2.1.2) acts against the magnetosheath flow, thereby ex-tracting kinetic energy and converting it to magnetic energy, stored predominantly in the
V.M. Vasyliunas
magnetic field of the magnetotail. Other formulations (e.g., MHD dynamo of solar windflow, solar wind potential acting against the nightside magnetopause current) describe whatis essentially the same process.
Process (2) are conversions of magnetic energy into mechanical form (kinetic energy ofeither bulk flow or thermal motions): in the plasma sheet (2a) by magnetic reconnection andadiabatic compression), in the ring current region (2b) by adiabatic compression), and inassociation with Birkeland currents (2c) by auroral acceleration and ionospheric ion-neutralcollisional dissipation (often called “Joule heating,” but see Vasyliunas and Song 2005).
Process (3) represent loss of energy (mechanical and magnetic) by outflow down thedistant magnetotail, e.g., by plasmoid formation and escape (Sect. 2.1.5). Process (4) areloss of mechanical energy from the plasma sheet and ring current, by precipitation intothe atmosphere (4a, 4b) and by charge exchange (4c) producing fast neutrals that escapefrom the system. Process (5) are loss of mechanical energy from the ionosphere and theaurora, into heating of the atmosphere by auroral precipitation and ion-neutral collisions(5a), and into electromagnetic emissions (5b) (auroral light, plasma waves—interesting astracers or indicators of processes, but generally representing amounts of energy insignificantin relation to the magnetospheric energy budget).
Process (6) represent the energy input into deformation of magnetic field by plasma pres-sure in the ring current and in the near-Earth plasma sheet (the distinction from magnetotailfield is not always clear-cut). Finally, process (7) allows for the possibility that the deformedmagnetic field in the outer magnetosphere may produce its own auroral phenomena, analo-gously to process 2(a) in the magnetotail.
Processes of minor importance on the scale of the entire magnetosphere have been leftout of Fig. 7: in particular, direct particle precipitation into the atmosphere from the mag-netosheath or the solar wind (precipitation from the polar cusp, solar particle entry intothe polar cap, “polar rain”). Electromagnetic radiation from the aurora, likewise of minorimportance, has been included solely because light emission is part of the concept “aurora.”
Of interest are the average energy flow rates (average power) in the various branchesof the energy budget diagram. The total power Ptotal from the solar-wind energy sourcecan be viewed as a known quantity, fixed by the solar wind parameters and the size of themagnetosphere; in order of magnitude it is equal to the flux of solar wind kinetic energythrough an area equal to the cross-section of the magnetotail, 1
2 ρsw(Vsw)3AT . The power inpaths (0) and (1) is fixed by force considerations: P(0) is ∼ Vsw × FMP , the solar wind forceon the magnetopause (Chapman-Ferraro force), and P(1) is ∼ Vsw × FMT , the magnetotailforce. Since FMP exceeds FMT typically by a factor ∼ 10, irreversible heating at the bowshock constitutes, as far as the amount of energy is concerned, the largest energy dissipationprocess in the solar wind interaction with the Earth system; the entire energy budget of themagnetosphere proper amounts to about 10% of that.
For the power P(2a) + P(2b), effectively the total power into the ring current, inner magne-tosphere, and ionosphere/atmosphere, there have been numerous empirical estimates, alongwith a search for a dependence on solar wind parameters, leading to empirical formulas ofwhich those of Burton et al. (1975) and Perrault and Akasofu (1978) (the “ε parameter”) arethe best known (review, Gonzalez 1990). The resulting P(2a) + P(2b) is in general nearly anorder of magnitude smaller than P(1) estimated from the magnetotail force for comparablesolar wind conditions; this implies that, at least on the average, a large part of the power P(1)
supplied to the magnetosphere escapes down the magnetotail, via paths (3a) and (3b), andonly a fraction enters near-Earth space. (Seemingly, what produces all the space weathereffects is something like a few percent of the total power in the solar wind interaction withthe Earth system.)
Physics of Magnetospheric Variability
4 Two Prototypical Examples of Magnetospheric Variability
4.1 Magnetospheric Substorm
Magnetospheric substorms are spectacular events of dynamic change and energy conver-sion/dissipation in the magnetosphere, long viewed as comparable in impressiveness (andpossibly analogous in physical processes as well) to solar flares. The observed phenomenaare wide-ranging and complex, and there is not much unanimity on what defines a magne-tospheric substorm (Rostoker et al. 1980, 1987). Probably the most spectacular phenomenonand the one most widely used as a unifying concept is the auroral substorm, summarized inthe well-known classic figure of Akasofu (1964). It is accompanied by intense geomagneticdisturbances at latitudes of the aurora and elsewhere. Within the magnetosphere, observedsubstorm phenomena include (1) greatly enhanced intensities and energies of charged parti-cles, (2) changes of the magnetic field in the nightside magnetosphere and magnetotail, thefield becoming initially more stretched (tail-like) and subsequently more dipolar (dipolar-ization), and (3) appearance of fast (∼ VA) bulk flows of plasma in the magnetotail. Attemptsat theoretical interpretation are also wide-ranging and complex, and some aspects (notablythe sudden onset of the substorm expansion phase) have proved to be singularly intractable,with no consensus after decades of research.
From the detailed phenomenology of the magnetospheric substorm, extensively de-scribed in the literature (e.g., Akasofu 1977; Kennel 1995; Syrjäsuo and Donovan 2007, andmany others), one may extract two primary aspects: (1) enhanced energy input and dissipa-tion, and (2) change of magnetic field configuration, in two distinct phases. An additionalaspect is that the primary controlling factor for the occurrence of substorms is the interplan-etary magnetic field, in particular its southward component—opposite to the Earth’s dipolefield in the equatorial plane. More specifically, the physical description can be summarizedas a two-phase process. Growth phase: as a consequence of a southward interplanetary mag-netic field, the configuration of the magnetosphere changes, its magnetic field becominghighly stretched (increased magnetic flux in the magnetotail, reduced flux in the nightsideequatorial region). Expansion phase, initiated by the onset: the magnetic field changes tomore nearly dipolar (increased flux on the nightside), and there is enhanced energy inputand dissipation to the inner magnetosphere and the ionosphere/atmosphere; the process oc-curs on dynamical time scales (comparable to or shorter than wave travel times) and isaccompanied (most probably) by changes of magnetic topology.
With reference to the energy flow paths of Fig. 7: during the growth phase, P(1) is en-hanced and is appreciably larger than the sum P(2a) + P(2b) + P(2c) + P(3a). During the ex-pansion phase, P(2a) and especially P(2c) are enhanced; P(3a) and P(3b) presumably are en-hanced in connection with plasmoid formation and the topological changes exemplified byFig. 3.
The increasing magnetic flux in the magnetotail during the growth phase implies that inthe magnetosphere the magnetic flux transport rate toward the dayside reconnection regionhas increased (to match the increased magnetic flux transport in the solar wind), but the fluxreturn rate from the nightside reconnection region has not increased correspondingly; why?is the essential physical question about the growth phase. Possible reasons include:
1. The nightside reconnection rate increases after a delay, e.g., by solar-wind flow time toreach the distant X-line (but communication from the dayside to the magnetotail can gomuch faster by wave propagation within the magnetosphere than by advection in the solarwind).
V.M. Vasyliunas
2. The return flow within the plasma sheet is opposed by adverse tailward stress (e.g., pres-sure gradient).
3. The amount of open flux depends directly on the interplanetary magnetic field (thispresumes a unique quasi-equilibrium configuration of the open magnetosphere, i.e., un-proved “yes” answers to questions 2 and 3 of Sect. 2.1.4.)
The essential physical questions about the substorm onset and expansion phase fall intotwo groups. First: the configuration of the magnetosphere at the end of the growth phase ischaracterized by
1. increased flux in the magnetotail2. increased magnetic tension force on the Earth3. thin current sheet4. deep penetration of current sheet (reaching some particular inner structure?)5. enhanced circulation of magnetospheric convection flow in the ionosphere
This configuration has become unsustainable; why?—has some threshold been exceeded? (ifso, which item in the above list is the critical one? several at once?), have external conditionschanged? Almost every proposed model can be matched to an item in the list: 1 or 2 isthe critical item for NEXL models, 3 for “current disruption” models, 4 for ballooning orinterchange instability models, 5 for MI coupling models, change of external conditions for“northward Bz turning” models.
Second: highly variable strong plasma flows are an ubiquitous observed feature. Includedare “bursty bulk flows” (relation to substorm not entirely clear), tailward and earthwardflows associated with near-Earth X-line, earthward flows implied by dipolarization. Essen-tial questions for understanding the temporal development of these flows:
1. What are the stresses that impart the linear momentum to create these flows?2. What creates these stresses (or brings them out of balance)?
4.2 Magnetic Storm
The concept of the geomagnetic storm is older and in many ways simpler than that of themagnetospheric substorm. In contrast to the substorm, there is a clear and widely accepteddefinition of what constitutes a magnetic storm, in terms of the time variation of the geo-magnetic Dst index: a prolonged (hours to days) interval of negative Dst values (Gonzalezet al. 1994). (The storm sudden commencement and the initial phase of positive Dst, whichaccompany many but not all storms, are no longer considered an integral part of the stormconcept.)
The physical understanding of what a magnetic storm is relies heavily on a remarkabletheoretical result, the Dessler-Parker-Sckopke theorem, which relates the external magneticfield at the location of a dipole to properties of the plasma trapped in the field of the dipole.First derived by Dessler and Parker (1959) for special pitch-angle distributions and extendedto any distribution by Sckopke (1966), the theorem states that b(0), the magnetic disturbancefield of external origin at the location of a dipole of moment μ, satisfies
μ · b(0) = 2UK (29)
where UK is simply the total kinetic energy content of plasma in the magnetosphere (in-dependent of the spatial distribution, the partition between bulk-flow and thermal energy,or any properties of the energy spectrum). Subsequently (Carovillano and Siscoe 1973;
Physics of Magnetospheric Variability
Vasyliunas 2006a, 2006b, and references therein) the theorem was derived from a virial-theorem argument (in place of the original Biot-Savart integration usable only for axiallysymmetric configurations) and thereby considerably generalized, with the addition of non-linear effects plus magnetopause and magnetotail terms. The primary use of the Dessler-Parker-Sckopke theorem remains, however, inferring the plasma energy content (the energywithin the box “ring current plasma” in Fig. 7) and its time history, from which two essentialaspects of magnetic storm physics have been established: the essential phenomenon of themagnetic storm is the addition of a large amount of plasma energy to the dipolar field regionof the magnetosphere, which results from a particular condition in the solar wind, namely“a sufficiently intense and long-lasting interplanetary convection electric field” (Gonzalezet al. 1994), meaning −V × B/c with the interplanetary magnetic field having a southwardcomponent.
Since geomagnetic storms, particularly the intense ones, are characterized by unusuallylarge amounts of energy stored as mechanical energy of plasma in the ring current region (incomparison to other storage regions), the implication for Fig. 7 is that during the develop-ment of an intense storm, P(2b) is unusually large, on the average, and exceeds the loss rateP(4a) + P(4c). Whether this enhanced conversion rate from magnetic energy into mechanicalenergy of ring current plasma results from some different interaction process or simply froma different time history of solar wind parameters is an unresolved question.
4.3 Is There an Essential Difference Between Storms and Substorms?
The presence of a southward component of the interplanetary magnetic field (opposite tothe dipole field in the Earth’s equatorial plane) is well established as a condition for theoccurrence both of storms and of substorms. This may suggest the question (put to me bya solar physicist): do magnetic storms and magnetospheric substorms really constitute twophysically distinct phenomena, or are they merely different-time-scale manifestations of asingle phenomenon? Leaving aside matters of time scale and sequence, the discussion inSects. 4.1 and 4.2 of the energy budget Fig. 7 points out one essential conceptual difference:the defining signature of a magnetic storm represents an enhanced storage of plasma energy,while that of a magnetospheric substorm represents in essence (independent of argumentsabout what it is in detail) an enhanced dissipation of energy.
5 Fundamental Processes of Magnetospheric Variability: Conclusions and Questions
1. Shaping of the magnetic field through stresses by and on the plasma determines the con-figuration of the solar-wind/magnetosphere/ionosphere system.
2. Stresses are changed primarily by plasma flow, through the associated transport of mag-netic flux and evolution of plasma pressure.
3. Variable advection of magnetic flux in the solar wind seems to be the primary factor forproducing large-scale variability of the Earth’s magnetosphere.
4. Magnetospheric substorm is a process of enhanced energy dissipation, connected withtransitions from equilibrium to non-equilibrium, in two steps (both in general with mag-netic topological changes):(a) Step 1: mechanical stresses deform the magnetic field into a configuration of in-
creased energy (plasma flow transports magnetic flux and, with field lines attachedto the massive Earth, increases the magnetic energy).
V.M. Vasyliunas
(b) Step 2: the magnetic configuration becomes unsustainable and changes quickly, re-leasing the energy (why the configuration becomes unsustainable and what causesthe quick change remain highly disputed questions).
5. Magnetic storm is a process of establishing and maintaining an enhanced quasi-equilibrium storage of mechanical energy. The relation, if any, to the substorm is notclear.
6. For given solar wind parameters, what determines the amount of open magnetic flux (or,equivalently, the effective length of the magnetotail)?
7. When and why is open magnetic flux not returned smoothly? (May be an essential factordifferentiating substorms from other types of events.)
8. What unbalanced stresses accelerate the fast plasma flows (including those associatedwith the formation of the near-Earth X-line)?
Acknowledgements I am grateful to Paul Song and George Siscoe for illuminating discussions.
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