Physics of Magnetospheric Variability

Space Sci Rev DOI 10.1007/s11214-010-9696-1

Vytenis M. Vasyliunas¯

Received: 23 July 2010 / Accepted: 3 September 2010 © Springer Science+Business Media B.V. 2010

Abstract Many widely used methods for describing and understanding the magnetosphere are based on balance conditions for quasi-static equilibrium (this is particularly true of the classical theory of magnetosphere/ionosphere coupling, which in addition presupposes the equilibrium to be stable); they may therefore be of limited applicability for dealing with time-variable phenomena as well as for determining cause-effect relations. The large-scale variability of the magnetosphere can be produced both by changing external (solar-wind) conditions and by non-equilibrium internal dynamics. Its developments are governed by the basic equations of physics, especially Maxwell’s equations combined with the unique constraints of large-scale plasma; the requirement of charge quasi-neutrality constrains the electric field to be determined by plasma dynamics (generalized Ohm’s law) and the electric current to match the existing curl of the magnetic field. The structure and dynamics of the 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 formulation of settled as well as of controversial issues concerning, e.g., magnetospheric substorms and magnetic storms.

Keywords Magnetic storms · Magnetospheric substorms · Solar-wind/magnetosphere interaction · Magnetosphere/ionosphere/thermosphere interaction

1 Introduction

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., magnetic storms, 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-scale variability of the magnetosphere can be the result both of changing external (solar-wind)

V. M. Vasyliunas¯ () Max-Planck-Institut für Sonnensystemforschung, 37191 Katlenburg-Lindau, Germany e-mail: [email protected] V.M. Vasyliunas¯ conditions and of non-equilibrium internal dynamics; separating the contributions of the two is not always simple, since the solar wind is likewise continually varying on many different time scales. This review discusses the variability of the magnetosphere primarily from the point of view of physical understanding, of trying to see how the complex variable phenomena of the solar wind/magnetosphere/ionosphere/atmosphere system follow from the basic laws of physics. Essential observational results as currently understood are taken into account, but this is not intended as a review of observations as such. The aim is rather a systematic physical description/formulation, in which observations may suggest and guide explanations but should not appear as explicit premises, and in which absolute primacy is given to the basic equations (including conservation laws) in their complete form, with any approximations explicitly 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-examined when applied to time-varying cases. Section 3 reviews some important differences between electrodynamics in space and in the ordinary laboratory and discusses the modifications they impose on methods mentioned in Sect. 2. Section 4 presents definitions and physical descriptions of the two major types of events: magnetospheric substorms and magnetic storms. (Length limitations preclude a discussion of sawtooth and steady-magnetospheric- convection events, which might be viewed as variants of the substorm.) Finally, Sect. 5 attempts to interpret the principal types of magnetospheric-variability events in terms of a coherent set of fundamental physical processes or, where that is not yet possible, to formulate a set of essential physical questions. The emphasis is on understanding the physical processes that underlie distinct individual events, of large spatial scale, on various time scales: substorm onset (minutes or less), substorm phases and similar events (tens of minutes to an hour or so), recurrence tendencies of same (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. Our knowledge of it in the Earth’s magnetosphere is very extensive but is derived almost entirely from observations, often represented by empirical models (e.g. Tsyganenko 2001, and references therein) which can be quite sophisticated; physical understanding, however, in the sense 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 changes do exist, but they represent rather a patchwork, each model applied to a different aspect and often derived in a different way. In recent years global numerical simulations have come into widespread use, to calculate magnetic fields and other properties of a model magnetosphere; they do constitute a unified treatment but can be as incomprehensible as the real magnetosphere. A common custom in magnetospheric and ionospheric physics, following elementary E&M textbook usage, is to preferentially describe the magnetic field configuration, whether deduced from observations or from theory, by specifying the electric currents that would produce 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 likewise dealt with as corresponding changes of the current system. It may be noted, however, that essentially our entire empirical knowledge about the magnetic configuration of the magnetosphere and its changes has been derived from observations of the magnetic field, any statements about currents being inferred therefrom by invoking (1). Direct determinations of 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 has been 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 relation between 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 physical terms is the dayside magnetosphere, under conditions where penetration of magnetic field and of plasma across the magnetopause can be neglected (idealized as a magnetically closed magnetosphere). Unable to penetrate to any significant extent into the geomagnetic field, the solar-wind plasma is initially slowed down and compressed until the pressure has increased sufficiently so that its lateral gradient deflects the flow around the magnetosphere; the exterior pressure also deforms and compresses the magnetic field inside the magnetosphere. In equilibrium, both the exterior plasma flow and the interior magnetic field are tangent to the boundary surface, the magnetopause; furthermore, the total pressure (plasma plus 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, this becomes what is generally known as the Chapman-Ferraro model, which has a well-defined mathematical formulation extensively investigated in the 1960’s (see, e.g., Siscoe 1988,for detailed review and references). Calculations based on the Chapman-Ferraro model predict the location and shape of the magnetospheric boundary, the magnetic field line pattern within the magnetosphere (visu- alizable as a compressed dipole field), and the resulting geomagnetic disturbances. For the most part, with the exception primarily of intense magnetic storm periods, these predictions are in reasonable agreement with what is observed on the day side of the magnetosphere, in particular the decreasing distance of the subsolar magnetopause with increasing solar wind 2 dynamic pressure ρswVsw and the northward jumps of the low-latitude geomagnetic field 2 (sudden commencements and sudden impulses) when ρswVsw 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, the interior magnetic field is calculated from the condition that it be tangent to the magnetopause surface, the location of which is adjusted to satisfy the condition of equal total pressure balance; the current is then obtained from the magnetic field change via (1). (The condition that the exterior flow be tangent to the magnetopause surface is satisfied by adjusting the location of the bow shock.) When the Chapman-Ferraro model was first proposed in the 1930’s, it was more or less taken for granted that there was no interplanetary magnetic field. The Chapman-Ferraro current, which served solely to contain the interior magnetic field within the volume of the magnetosphere, was confined to the boundary surface (magnetopause) since any sources of the interior field except for the Earth’s dipole were assumed negligible. This was a magnetically 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 closed magnetosphere 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 calculated Chapman-Ferraro model are not significantly affected by the presence of the interplanetary magnetic field. The Chapman-Ferraro current, however, is now required not only to confine the interior field within the magnetosphere but also to exclude the exterior field from the magnetosphere; the magnetic field is assumed tangent to the boundary surface on both sides. The exterior field, carried by the plasma flow in the magnetosheath, becomes draped around the magnetopause, as suggested by Piddington (1964) and modeled quan- titatively by Spreiter et al. (1966) and Alksne (1967). The draping configuration implies a current system that is not confined to the magnetopause but extends from it through the magnetosheath to the bow shock Although clearly implied by (1), this current system was not explicitly mentioned by Spreiter and coworkers (a rare instance of describing a magnetic field without referring to the associated current). It is, however, a necessary feature of the Chapman-Ferraro closed magnetosphere in the presence of a non-zero interplanetary magnetic field and may thus be called the “exterior Chapman-Ferraro current”; the direction of the current depends of course on the orientation of the interplanetary magnetic field. What appears to be a (misnamed) example of it is the so-called “reconnection current” deduced by Siebert and Siscoe (2002) from an MHD simulation of the magnetosphere interacting with a southward interplanetary magnetic field: it has the geometry expected from the draping of the 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 night side 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 on the night side they are observed to be stretched out to form an extended magnetotail. The Chapman-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 seen particularly 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 the noon-midnight meridian plane. Open arrows: solar wind bulk flow. Solid lines within magnetosphere: magnetic field lines Physics of Magnetospheric Variability strength at the equator is everywhere larger than the dipole value (his Fig. 1), and the equatorial crossing distance of every field line is smaller than the nominal dipole L (his Figs. 4 and 5). Contrary to a widespread impression that the magnetotail was first found in the observations (Ness 1965) and was a surprise to theorists, the existence of the magnetotail was in fact previously 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)or Fig. 1 of Dessler (1964) are virtually identical to those sketched with the label “experi- mental”inFig.14ofNess(1965). The physical reason for the magnetotail is the assumed existence of some process by which a force exerted on plasma within the nightside magnetosphere pulls it (and with it the magnetic field lines) away from the Earth in the antisunward direction, to form the configuration of quasi-uniform antisunward (northern lobe) or sunward (southern lobe) magnetic fields, extending to large distances ( distance to dayside 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), part of the open-magnetosphere concept discussed in Sect. 2.1.4 and now widely considered the dominant effect, (c) internal pressure of plasma or of hydromagnetic waves (Dessler 1964), now generally considered unimportant at Earth (but possibly significant at the giant planets Jupiter and Saturn). The global role of the tailward force has been emphasized by Siscoe (1966)(seealsoVasyliunas¯ 2009). The reversal of the magnetic field between the northern and southern lobes implies the existence of a current sheet, with the associated plasma sheet, in the equatorial region of the magnetotail. What is called the magnetotail current system consists of the equatorial current sheet 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 current system are among the primary aspects of substorms and related events (Sect. 4.1). Both the configuration and its changes have been studied very extensively, but primarily from an observational/empirical point of view. Numerous theoretical models have been developed for special aspects (see, e.g., Schindler 2007, and references therein), but there is little fundamental 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 magnetopause and the inner edge of the magnetotail current sheet, the magnetic field is dominated by the internal field of the Earth, adequately represented as a dipole for most purposes of magnetospheric 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 instances in magnetospheric physics where the electric current seemingly can be determined (at least approximately) independently of its relation (1) to the magnetic field: charged-particle drifts in the inhomogeneous magnetic field depend on the sign of the charge and thus give rise to a current. According to a fundamental result derived by Parker (1957), the current density J given by the sum of all the single-particle drifts satisfies the plasma momentum equation ∂ 1 (ρV) +∇·κ − ρg = J × B (2) ∂t c where κ = ρVV + P is the kinetic tensor. Gradient, curvature, and magnetization drifts add up to the pressure tensor term; the time-derivative and inertial terms come from the so-called V.M. Vasyliunas¯ polarization drifts (see, e.g. Northrop 1963); the gravitational term, neglected as unimportant in 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 discussed 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 the local 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 neglect of the interplanetary magnetic field) introduced the concept of the magnetically closed magnetosphere, the essential next step was the explicit proposal by Dungey (1961) that the magnetosphere can be magnetically open, with magnetic field lines from the Earth’s dipole connecting across the magnetopause with those of the interplanetary magnetic field when the relative orientation of the two fields is favorable for reconnection. This concept and the general concept of tangential drag across the magnetopause, introduced at almost the same time by Axford and Hines (1961), constitute the two basic ideas from which most of the physics of magnetospheric variability derives, directly or indirectly. Figure 2 illustrates schematically the main features of the simplest magnetically open magnetosphere, 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 magnetic field lines and the plasma bulk flow is shown in three projections: noon-midnight meridian plane, equatorial plane, and projected along field lines to the ionosphere. The primary new feature is the existence of the separatrix surface with two branches: one (topologically a torus, 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-midnight meridian 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 field line boundary). (c) Right: projection on ionosphere (lines: plasma flow streamlines, line of x’s: open/closed field line boundary = projection of magnetic X-line = polar cap boundary). The sunward direction is always to the left Physics of Magnetospheric Variability other (topologically a cylinder surrounding and touching the torus) separates open and interplanetary field lines (thick lines in Fig. 2a represent cuts through both branches of the separatrix). 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 magnetic field components perpendicular to the line vanish by definition, but the component along the line 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 except for cases of special symmetry). Magnetic field reconnection, however, occurs on only those segments of the X-line where plasma flows across the separatrix (both branches). As can be seen from the figure, there are two distinct such segments, generally referred to as the dayside and the nightside reconnection region, respectively (for a more general discussion of possible geometries, see, e.g., Vasyliunas¯ 1984); the terms dayside X-line and nightside X-line are also widely used but tend to obscure the fact that the X-line of the open magnetosphere 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 magnetic flux ΦM , leaving one hemisphere to connect to the interplanetary magnetic field (equal to the dipole flux through the polar cap, defined as the region of open field lines at the Earth’s surface and illustrated in Fig. 2c). The time rate of change of the open flux is equal to the rate of flux addition by dayside reconnection minus the rate of flux removal by nightside reconnection: d Φ = cE − cE (3) dt M d n where Ed and En are the line integrals of the electric field along the dayside and the nightside reconnection segments of the X-line, respectively. Ed can be related to solar wind parameters by 1 E L V B (4) d c X sw s where LX is the length of the dayside reconnection segment projected along plasma flow streamlines back into the undisturbed solar wind (as illustrated in Fig. 2b—not the length at the 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 is equal to maximum line integral of the electric field across the polar cap, usually called the polar cap potential (or transpolar potential) ΦPC (the use of Φ both for magnetic flux and for potential, although perhaps confusing, is traditional). The open flux can be expressed as

ΦM LXLMT Bs (5) where LMT (also illustrated in Fig. 2b) is the distance between the dayside and the nightside locations of the outer (interplanetary/open) branch of the separatrix, extended into the undisturbed solar wind; less precisely but more understandably, LMT maybeviewedasthe effective length of the magnetotail. All three parameters of the open magnetosphere can be inferred from observations. The area of the polar cap can be estimated by a variety of auroral and precipitating particle observations, and from it the open flux ΦM . The polar cap potential can be estimated from ionospheric plasma flow measurements, either by particle or by radar techniques, and comparison with solar-wind flow and magnetic field measurements then yields an estimate of V.M. Vasyliunas¯

LX. In quasi-equilibrium, the solar wind ﬂow time across the distance LMT should equal the ionospheric 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 of the open magnetosphere are, how they relate to solar wind parameters, and how they vary in connection with various types of magnetospheric events has become quite extensive, but physical understanding remains rather limited. The following are some of the fundamental questions 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,andLMT , be derived in a physically understandable way from first principles? 2. In case the solar wind parameters remain steady for a sufficiently long period (many hours at least, or perhaps days), does the magnetosphere settle into an essentially steady state, 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 (as distinct 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 as a tool to answer questions such as the above. In addition to the well-known general limitations and caveats (e.g., Post and Votta 2005; Greenwald 2010), there is the problem that the output of an individual simulation often may be as complex and as difficult to interpret as the observational data from the actual magnetosphere, with no obvious way of extracting the physical insight which is what question 1 really is about. The remark “the theorist knows little, understands much; the experimenter knows much, understands little; the computer simulator knows everything, understands nothing” (Vasyliunas¯ 2008) undoubtedly is exag- gerated, but sometimes perhaps not all that much. To my knowledge, no definite answers to questions 2 and 3 have been forthcoming from numerical simulations to date. Question 2 is particularly difficult to deal with, both from observations (the solar wind is rarely, if at all, steady on the time scales in question) and from simulations (the running time is often limited by numerical problems). Question 4 has received little attention, perhaps because in practice there is not much difficulty, 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 magnetosphere has a boundary surface not crossed by any field lines, which uniquely defines the magnetopause, whereas in an open magnetosphere there is no such unique surface but there can 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, changes of the magnetic field configuration suggestive of additional topological structures occur in the magnetotail, primarily in association with magnetospheric substorms. Observations of magnetic 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 distances of some 15–30 RE , much closer than the typical distance (generally estimated as 100–200 RE 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 new magnetic X-line, distinct from the distant nightside X-line, forms near the Earth; they based their 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 the magnetic ﬁeld topology in the magnetotail of an open magnetosphere (Vasyliunas¯ 1976)

and the magnetotail, ∼ 6–8RE . Subsequent work (e.g., Schindler 1974) interpreted X-line formation as the result of tearing instability of the magnetotail current sheet and placed it farther out, well within the magnetotail stretched-out field line region. The temporal development of a possible three-dimensional magnetic topology associated with 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 sequential instants of time. Shown in each panel are the noon-midnight meridian plane, the equatorial plane, and the projection on the ionosphere, in the same format as in Fig. 2;anew feature 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 field lines, 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 interplanetary magnetic field; topologically, the X-line is a short segment joined to an O-line segment to form a complete ring. The plasmoid grows (panel 3), expanding both in the meridian plane (as the result of continuing reconnection) and in X-line extent (for no clearly identified reason), until it reaches the closed/open field line separatrix (panel 4, onset of lobe reconnection). After this, the plasmoid is contained within the region of interplanetary field lines (panel 5). The net result of the entire sequence is to “break off” a segment of the open-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 of many “realistic” details of size and shape by Hones (1976, 1977), and in this version became V.M. Vasyliunas¯ 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 of the open magnetosphere applies here, too). The highly controversial question of how this model is related to the actual substorm is discussed in Sect. 4.1; here I am concerned only with what can be said about the magnetic configuration by itself. At present, understanding the topological changes in the magnetotail is in pretty much the same state as understanding the open magnetosphere (Sect. 2.1.4) or the magnetotail itself (Sect. 2.1.2): there is very extensive empirical knowledge, there is a patchwork of models for many individual aspects, and more recently a patchwork of individual numerical simulations, but the physical understanding of the process as a whole remains limited. Among the unanswered fundamental questions 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 plasma physics in its own right. Question 1 addresses a very specific issue of how reconnection comes to be where it wasn’t before (thus, studies of the properties and evolution of already occurring reconnection, however informative otherwise, do not provide an answer). The numerous 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; there remains, however, the problem of how to set up the initial configuration, assumed to be in equilibrium but unstable, as well as how to differentiate between that and straightforward evolution 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 been helped by the widespread tendency to consider only the meridian plane view of the topological sequence in Fig. 3 and hence to think of the plasmoid magnetic field as basically two-dimensional, to be modified (or even destroyed) by the addition of an out-of-plane component. The plasmoid in Fig. 3 as originally conceived is an intrinsically three-dimensional object with a well-defined magnetic topology: a closed volume, the field lines within which do not connect to anything outside the volume. What has not been established, to my knowledge, 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-line ring mentioned in Sect. 2.1.4 can be demonstrated with the simple example of uniform field superposed on a tilted dipole). An alternative interpretation of the observed plasma and magnetic field structures, particularly in case of an appreciable dawn-dusk component of B, is in terms of a magnetic flux rope (see, e.g., Sibeck 1990, and references therein). Although flux ropes have been discussed 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 a twisted flux tube, a flux tube being defined in turn as “the volume enclosed by a set of field lines that intersect a simple closed curve.” The implied topology of the magnetic field is not clear. Question 3, a quantitative sub-aspect of question 1, plays a significant role in understanding 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 that the 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 were to be continually carried antisunward by the plasma mantle (or magnetosheath) flow: was there anything to prevent the magnetotail length LMT from increasing indefinitely while the open flux ΦM remained constant? Only after finding the topological sequence did I see the (in retrospect obvious) relation to substorm phenomenology, and published the result in that context. (Historians of science have repeatedly remarked upon the difference between research 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 magnetosphere and ionosphere is arguably one of the most successful theories in magnetospheric physics. It accounts for the pattern of magnetospheric convection at auroral and low latitudes, the distribution of Birkeland currents, and the variations of all these in response to changing orientation of the interplanetary magnetic field. The basic scheme of calculation is illustrated in Fig. 4 in the form in which it was first proposed by Vasyliunas¯ (1970)and Wolf (1970), generalizing earlier work by Fejer (1964); since then it has been elaborated considerably in detail, but neither the logic nor the equations have changed appreciably (for a review see, e.g., Wolf 1983, and references therein). If the electric field in the magnetosphere is known, particle motions can be calculated from E × B drifts (plus other drifts if considered significant), and the continuity/transport equations (together with boundary conditions on particle concentrations) then determine the plasma pressure in the magnetosphere.

Fig. 4 Schematic diagram of self-consistent magnetosphere/ionosphere coupling calculations (after Va- syliunas¯ 1970) V.M. Vasyliunas¯

From plasma pressure gradients, J⊥ can be calculated with the use of (2) (provided the time-derivative and inertial terms can be neglected). Current continuity ∇·J = 0 then determines J, the Birkeland currents flowing between the magnetosphere and the ionosphere; from the requirement that these currents close through the ionosphere, the ionospheric Ohm’s law (together with appropriate boundary conditions) serves to determine the electric field in the ionosphere. Mapping this electric field to the magnetosphere along magnetic field lines (under the MHD approximation E · B 0 or some more accurate version of the generalized Ohm’s law if deemed necessary) imposes the self-consistency requirement of agreement with the initally assumed magnetospheric electric field and thus closes the system of equations. ThesimpleschemeofFig.4 can be implemented at differing levels of sophistication that can 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 is much smaller that required for a MHD simulation even when the physics in the latter is considerably less detailed. Subsidiary physical effects such as pressure anisotropies, inhomogeneous and variable ionospheric conductances, influences of precipitating particles and models for parallel electric fields can be incorporated, with corresponding gains in ability to account for complicated aspects of observed phenomena. There is, however, a fundamental limitation when the theory is applied to describe magnetospheric variability. In the diagram of 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 in the magnetosphere to the electric field; in particular, all the processes of mapping between the ionosphere and the magnetosphere are treated explicitly as instantaneous, neglecting any propagation effects. The only time variability contained in the theory is the evolution of pressure as the plasma is advected by the flow, plus any time variations in the imposed boundary conditions. As discussed in Sect. 3.5, this limitation is more restrictive than might appear at first.

3 Electrodynamics of Large-Scale Plasmas

From the beginning, the magnetosphere/ionosphere coupling theory was formulated in a form chosen more for mathematical convenience than for physical understanding. Va- syliunas¯ (1970) introduced the scheme of Fig. 4 withthewords“...itprovesconvenient to formulate the convection problem not in terms of the dynamical concepts of flow and stress 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, since in a plasma there is a close connection between the flow and the electric field and between the stress and the electric current.” How close is the connection and what does it really 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;Lui2000). Decid- ing whether B and V or, instead, J and E are to be treated as the primary variables has an obvious bearing on the physics of magnetospheric variability: as long as we are dealing with a steady system, we may treat both sets of variables as equivalent, but as soon as we consider time variations, the question which one is physically producing the change of the 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 current? With the current density J defined as 3 J = qa d vvfa(v) (6) a where fa(v) is the velocity distribution function of charged particles of species a, the equation for the time evolution of J, determined by summing the motions of all the charged particles, can be calculated from the appropriate sum of velocity-moment equations (see, e.g. Rossi and Olbert 1970; Greene 1973): ∂J q2n 1 q δJ = a a E + V × B − a ∇·κ + q n g + (7) ∂t m c a m a a a δt a a a coll where qa , ma , na , Va ,andκa are the charge, mass, concentration, bulk velocity, and kinetic tensor, respectively, of species a,and(δJ/δt)coll represents the sum of all collision effects. (The gravitational acceleration g, included for completeness, mostly is unimportant in practice.) Equation (7) is the complete and (except for neglect of relativistic effects) exact form of what is commonly called the generalized Ohm’s law. For a quasi-neutral plasma of electrons and one species of singly charged ions with |ni − ne|n and mi me,(7) becomes ∂J ne2 1 J × B = E + V × B − ∂t me c nec ∇·κe ∇·κi δJ + e − + e (ni − ne) g + (8) me mi δt coll 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 δJ me =− νei + νen + νin J + (νen − νin) ne (V − Vn) (9) δt coll mi

(see, e.g. Song et al. 2001, 2005;Vasyliunas¯ and Song 2005), where Vn is the bulk flow velocity of the neutral atmosphere and the ν’s are the collision frequencies. In the magnetosphere, interparticle collisions are generally negligible, but effects of fluctuations and particle-wave interactions may be represented by a term of the form δJ q2 1 = a δn δE + δ (n V ) × δB (10) δt m a c a a coll a a where =average over fluctuations. To simplify notation, introduce the symbol E∗ to represent 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 ω2 = p E − E∗ (11) ∂t 4π V.M. Vasyliunas¯ where q2n 4πne2 ω2 = 4π a a ≈ p m m a a e is the (electron) plasma frequency. From (8), E∗ is given by 1 J × B ∇·κ m ∇·κ m δJ E∗ ≡− V × B + − e + e i − e (12) 2 c nec ne mi ne ne δt coll (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 independently related to E by eliminating B from Maxwell’s equations to obtain ∂J 1 ∂2E =− c2∇×(∇×E) + (13) ∂t 4π ∂t2

Combining (13) with (11)gives

1 ∂2E E − E∗ =−λ2∇×(∇×E) − (14) e 2 2 ωp ∂t where λe ≡ c/ωp is the electron inertial length (also known as the collisionless skin depth). ∗ Accordingto(14), E E unless E varies on spatial scales of order λe or smaller and/or −1 on time scales of order ωp 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 1 J × B ∇·κ m ∇·κ m δJ E + V × B − + e − e i + e 0 (16) 2 c nec ne mi ne ne δt coll

−1 The scales λe and ωp are those of electron plasma oscillations and associated electromagnetic waves, in which charge-separation effects are important. The large-scale limit is thus the regime in which the behavior of the plasma is strongly constrained by quasineutrality. The essential physics expressed by the generalized Ohm’s law (16)isthatE assumes the value it must have in order to prevent a differential acceleration of ions and electrons that would 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 =−4πJ + c∇×B (17) ∂t In a time-dependent situation, even if 4πJ and c∇×B are equal at one particular time, there is no apriorireason 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 =−c∇×E (18) ∂t 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 ∼ −1 ωp (inverse plasma frequency) (see Vasyliunas¯ 2005b, for a more extended discussion with quantitative examples). Equality of 4πJ and c∇×B is reached on whichever time scale is the shorter of the two. As long as τ1 and τ2 are both very short in comparison to time scales for phenomena of interest, Ampère’s law (1) remains an excellent approximation; its physical meaning, however, depends critically on the ratio τ1/τ2 = Lωp/c = L/λe. In the ordinary E&M laboratory −1 environment, L/c ωp , hence B changes to make c∇×B match the existing 4πJ.Inthe large-scale-limit (15) appropriate to much of the magnetosphere and ionosphere, on the other −1 hand, ωp 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 what the 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 magnetohydrodynamics (Cowling 1957; Dungey 1958;Parker1996, 2000, 2007); what the work of Vasyliunas¯ (2005a, 2005b) has shown is that this concept is valid well beyond MHD, down to the scales of electron plasma oscillations, where it is limited by the breakdown of charge quasineutrality.

3.3 What Determines the Magnetic Field

An exact equation, not limited to the approximate (1), relating B and J can be derived by eliminating E from Maxwell’s equations:

1 ∂2B 4π −∇2B = ∇×J (19) c2 ∂t2 c which allows B at any point to be determined from J (more precisely, from ∇×J) if the latter is known over the complete backward light cone of the point. The discussion in Sect. 3.2 shows that in a plasma, because J is subject to the additional constraint of satisfying the generalized 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 be determined? Fundamentally, the evolution of B is determined by E through (18). In the large-scale limit (15), E is determined in turn by (16) and thus by the bulk flow and other mechanical properties (kinetic tensors of the various species) of the plasma. The evolution of these is to be calculated from the plasma equations: the momentum equation (2) to determine V suffices in the simplest (MHD) approximation, but more precise approximations may be used, all the way to determining the bulk flow by numerical integration of individual ion trajectories. In (16)and(2), J is replaced by (c/4π)∇×B. The above is a complete and self-consistent scheme for determining the evolution of B in the limit (15), corresponding more 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- −1 poral scale τ , are illustrated in Fig. 5. The figure is drawn for VA = 800 km s ; character- −1 −1 istic time scales ωp , Ωi (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 scale of interest). By the relativistic causality condition that no physical effect can propagate faster than the speed of light, everything is limited to lie below the line L = cτ. The “hybrid” approximation described above is valid for L λe; note that, given the relativistic causality condition, a separate limit on τ is redundant. The MHD approximation is valid −1 for L λi and τ Ωi . The Rice Convection Model (RCM), representative of magnetosphere/ionosphere coupling theory calculations (Sect. 2.2), is limited in time scale to τ τA but in spatial scale can be extended to L ∼ λi and even shorter if drift effects are included in sufficient detail. The “fully kinetic” treatment is in principle valid on all scales without restriction, but in practice is difficult to apply except for highly idealized or simplified systems. The “EM lab” refers to the laboratory of ordinary E&M devices, e.g., current coils, capacitors, resistors, etc., described for the most part by circuit theory; it is valid in −1 the time-scale range L/c τ ωp ,inFig.5 at the opposite extreme to MHD and RCM, suggesting that circuit analogies to the magnetosphere are unlikely to provide much physical insight. In general, the equations of classical (non-quantum) physics determine only the time evolution 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 that there should ever be a steady state. If, however, a steady or near-steady state does exist, the momentum equation (2) can be written as

1 1 ∇·κ − ρg = J × B = (∇×B) × B (20) c 4π Physics of Magnetospheric Variability and can be used to determine B from the requirement that the Lorentz force of the current given 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 calculate J itself, or at least its components J⊥ perpendicular to B. This stress balance calculation is the basic method used in magnetosphere/ionosphere coupling theory to determine currents in the magnetosphere, more recently applied also to determine B itself when the dipole approximation 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 from stress balance. They are not ohmic currents driven by an electric field in the conventional sense: if we insert the collision term (9)into(16), we do not obtain what is usually called the ionospheric Ohm’s law

∗ ˆ ∗ ∗ 1 J = σ E + σ B × E + σE, E ≡ E + V × B (21) P ⊥ H c n

(σP ,σH ,σ are Pedersen, Hall, and parallel conductivities). Instead, we need to invoke the momentum equation (2) applied to the ionosphere, with additional terms for plasma-neutral collisions (e.g., Song et al. 2001) on the right-hand side, and with the left-hand side usually neglected:

1 J 0 J × B − n (m ν + m ν )(V − V ) − m (ν − ν ) (22) c e i in e en n e in en e 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 requirement that its Lorentz force must balance the collisional drag force from the relative bulk motion between plasma and neutrals (see, e.g., Vasyliunas¯ and Song 2005, for a more detailed discussion). Further physical insight is provided by considering how stress balance can be maintained in 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, there is no general reason why they should remain strictly equal at other times; any departure from equality, however, implies by (2) a changing bulk ﬂow. If at some time t = 0, J × B/c is out of 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) t 2 (B/L) (24) where L is the spatial scale of ∇×E. The resulting (∇×B) × B/4π can come into stress balance with δF after a time t deﬁned in order of magnitude by δF ∼ BδB/4πL ∼ (δF/ρ) t 2 B2/4πL2 (25)

2 2 2 2 2 2 or t ∼ (4πρ/B )L = L /VA ≡ τA . The physical process that establishes stress balance and 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 lack of equilibrium acts to bring the system toward equilibrium; this means, by definition, that the system is assumed to be stable. Inferring currents or magnetic field configurations from stress balance is thus possible only if there are no large-scale instabilities playing a significant role.

3.4 What Determines the Plasma Flow

Fundamentally, the time evolution of the plasma bulk flow V is governed by the momentum 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 by the generalized Ohm’s law in the form (16), which equates the actual value (not the time derivative) of E to a series of plasma quantities, of which an important (and in the MHD approximation, the dominant) term is −V × B/c. From this formulation it is clear that the plasma flow is determined by the dynamics via (2), and the electric field is then determined by the flow (plus other effects if significant) via (16). There has been nevertheless (particularly in the ionospheric community) a persistent and widespread view that the electric field physically produces the flow (or at least contributes to it) by creating the drift motion E × B V⊥ = c (26) B2 of any charged particle placed in crossed E and B fields. The simple textbook derivations of (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 in the magnetospheric/ionospheric community) and by Vasyliunas¯ (2001) shows, on the contrary, that if one starts with plasma flow and no electric field, the flow continues and the electric field appears (with the mean value corresponding to the MHD approximation) on a time scale defined essentially by the plasma frequency, but if one starts with an electric field and no plasma flow, the electric field simply dissolves into plasma waves (with nearly zero 2 2 mean) and no appreciable flow appears; this result holds as long as VA c , i.e., the inertia of the plasma is dominated by the rest mass of the plasma particles and not by the relativistic energy-equivalent mass of the magnetic field. The precise result for the final steady mean values Em, Vm, given the initial values E0, V0,is

2 2 VA c 1 Em = E − V × B (27) 2 2 0 2 2 0 c + VA c + VA c 2 2 c VA E0 × B Vm = V0 + c (28) 2 2 2 2 2 c + VA c + VA B

2 2 Note that the condition VA c 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. The 2 2 textbook result that E creates the drift (26) follows from (28) in the opposite limit VA c , i.e., negligible plasma density, as expected. There are several intuitive arguments to understand why flows produce electric fields but electric fields do not produce flows. Physics of Magnetospheric Variability

1. Most fundamental: Bulk flow carries linear momentum and thus can be produced only by adding linear momentum to the plasma, but the linear momentum density of the elec- 2 2 tromagnetic field is smaller than that of the plasma by a factor VA /c (conserving the total linear momentum, under the assumption that the mean values satisfy the MHD approximation, yields (27)and(28) directly, Vasyliunas¯ 2001). 2. Simplest: as discussed in Sect. 3.1, on spatial and temporal scales large compared to those of electron plasma oscillations, the electric field is determined by the requirement that the differential acceleration of ions and electrons must not separate charges too much. The bulk flow of the plasma is essentially that of the ions; since these are much heavier, the electric 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 opposite sense for positive and negative particles and therefore tending to separate charges; the electric 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 now with the associated electric field acting to produce a drift (26) in the direction of the initial 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 plasma particles is sufficiently high; the quantitative criterion is easily shown to be equivalent to 2 2 VA c .

3.5 New Understanding of M-I Coupling Theory

The electrodynamics of space plasmas as summarized in Sects. 3.1, 3.2, 3.3,and3.4 allows a reformulation of the magnetosphere/ionosphere coupling theory with emphasis on physical understanding instead of mathematical convenience. The new view of the calculational scheme is illustrated in Fig. 6, to be compared and contrasted with the classical Fig. 4.The following are the principal differences. 1. Left-right link, top: In both magnetosphere and ionosphere, the plasma bulk flow appears as 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 waves propagating back and forth between the two; the traditional electric potential mapping along field lines represents not the physical process but only its final result when and if a steady state is reached. It is now immediately obvious that the theory can be applied only on 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 formulation of a physical process, the essence of which is that when the magnetic field is deformed by mechanical stresses in one location, the deformation extends necessarily to other locations, by the structure of the Maxwell stress tensor (a point emphasized particularly by Parker 2007). As shown in Sect. 3.2, current continuity is established on time scales of −1 ∼ ωp , hence on longer scales it is satisfied automatically and need not be imposed as a separate requirement (what is often discussed as “current closure” is really the coupling of the Maxwell stresses along different portions of a field line). The resulting magnetic stress in the ionosphere accelerates the plasma until the collisional drag force from the differential flow between plasma and neutrals becomes equal to the J × B/c force; in the process, 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 by Vasyliunas¯ and Song 2005). 4. Left-right link, bottom: Stress balance between the plasma and the Lorentz force remains understood as the basic physical process here (with ∇×B rather than J now viewed as the more appropriate description in terms of causes). The important change is that a fundamental limitation is now explicitly recognized: the assumption of stress balance is valid only if the system is stable and remains in slowly evolving quasi-equilibrium. The magnetosphere/ionosphere coupling theory is thus intrinsically incapable of describing explosive non-equilibrium developments (which, e.g., a substorm onset is generally assumed to be), and there is considerable uncertainty whether or how to apply it to the magnetospheres of Jupiter and Saturn, where interchange instabilities may play a dominant role in plasma transport.

3.6 Energy Conversion, Storage, and Dissipation

Magnetospheric variability, in events of all types, manifests itself through various phenomena 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 ultimate sinks, and what determines the time history of these developments constitute a major topic of magnetospheric physics in its own right, recently reviewed by Vasyliunas¯ (2010). Here I present 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); numbers keyedtodescriptionintext(question mark: process uncertain). Only the energy-flow paths are shown, not the mass-flow paths conversion processes, many of which can be viewed as consequences of time offsets or delays in particular branches of the average energy budget diagram. A sketch of the principal energy reservoirs, conversion processes and dissipation/loss processes is shown in Fig. 7, adapted from the energy budget diagram for a general magnetosphere in Vasyliunas¯ (2010), expanded and specialized to the solar-wind-dominated magnetosphere of Earth. The primary source of energy for the magnetosphere, shown by the double-lined box, is the kinetic energy of solar wind plasma bulk flow. The thermal and magnetic energies of the solar wind can be neglected (Vasyliunas¯ 2010); although the interplanetary magnetic field does exert a dominant influence on energy conversion processes in the Earth’s magnetosphere, it does so primarily by control of magnetic reconnection processes and open field lines, not by supply of magnetic energy. Since any flow carries not only kinetic energy but also linear momentum, extracting kinetic energy from the solar wind flow necessarily means also extracting linear momentum, which requires that a force be applied to permanently slow down the flow. The mere deflection of solar wind flow around the magnetospheric obstacle (not taking into account tangential forces at the magnetopause) extracts net energy from the flow only because of entropy increase at the bow shock, without which the flow would return ultimately to its initial value; this, the irreversible heating of solar wind 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 that forms the magnetotail (see Sect. 2.1.2) acts against the magnetosheath flow, thereby extracting 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 wind flow, solar wind potential acting against the nightside magnetopause current) describe what is essentially the same process. Process (2) are conversions of magnetic energy into mechanical form (kinetic energy of either bulk flow or thermal motions): in the plasma sheet (2a) by magnetic reconnection and adiabatic compression), in the ring current region (2b) by adiabatic compression), and in association with Birkeland currents (2c) by auroral acceleration and ionospheric ion-neutral collisional dissipation (often called “Joule heating,” but see Vasyliunas¯ and Song 2005). Process (3) represent loss of energy (mechanical and magnetic) by outflow down the distant magnetotail, e.g., by plasmoid formation and escape (Sect. 2.1.5). Process (4) are loss of mechanical energy from the plasma sheet and ring current, by precipitation into the atmosphere (4a, 4b) and by charge exchange (4c) producing fast neutrals that escape from the system. Process (5) are loss of mechanical energy from the ionosphere and the aurora, into heating of the atmosphere by auroral precipitation and ion-neutral collisions (5a), and into electromagnetic emissions (5b) (auroral light, plasma waves—interesting as tracers or indicators of processes, but generally representing amounts of energy insignificant in relation to the magnetospheric energy budget). Process (6) represent the energy input into deformation of magnetic field by plasma pressure in the ring current and in the near-Earth plasma sheet (the distinction from magnetotail field is not always clear-cut). Finally, process (7) allows for the possibility that the deformed magnetic field in the outer magnetosphere may produce its own auroral phenomena, analogously to process 2(a) in the magnetotail. Processes of minor importance on the scale of the entire magnetosphere have been left out of Fig. 7: in particular, direct particle precipitation into the atmosphere from the magnetosheath or the solar wind (precipitation from the polar cusp, solar particle entry into the polar cap, “polar rain”). Electromagnetic radiation from the aurora, likewise of minor importance, 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 branches of the energy budget diagram. The total power Ptotal from the solar-wind energy source can be viewed as a known quantity, fixed by the solar wind parameters and the size of the magnetosphere; in order of magnitude it is equal to the flux of solar wind kinetic energy 1 3 through an area equal to the cross-section of the magnetotail, 2 ρsw(Vsw) AT . The power in paths (0) and (1) is fixed by force considerations: P(0) is ∼ Vsw × FMP , the solar wind force on the magnetopause (Chapman-Ferraro force), and P(1) is ∼ Vsw × FMT , the magnetotail force. Since FMP exceeds FMT typically by a factor ∼ 10, irreversible heating at the bow shock constitutes, as far as the amount of energy is concerned, the largest energy dissipation process in the solar wind interaction with the Earth system; the entire energy budget of the magnetosphere proper amounts to about 10% of that. For the power P(2a) + P(2b), effectively the total power into the ring current, inner magnetosphere, and ionosphere/atmosphere, there have been numerous empirical estimates, along with a search for a dependence on solar wind parameters, leading to empirical formulas of which those of Burton et al. (1975) and Perrault and Akasofu (1978)(the“ parameter”) are the best known (review, Gonzalez 1990). The resulting P(2a) + P(2b) is in general nearly an order of magnitude smaller than P(1) estimated from the magnetotail force for comparable solar 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), and only a fraction enters near-Earth space. (Seemingly, what produces all the space weather effects is something like a few percent of the total power in the solar wind interaction with the 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 conversion/dissipation in the magnetosphere, long viewed as comparable in impressiveness (and possibly analogous in physical processes as well) to solar flares. The observed phenomena are wide-ranging and complex, and there is not much unanimity on what defines a magnetospheric substorm (Rostoker et al. 1980, 1987). Probably the most spectacular phenomenon and the one most widely used as a unifying concept is the auroral substorm, summarized in the well-known classic figure of Akasofu (1964). It is accompanied by intense geomagnetic disturbances at latitudes of the aurora and elsewhere. Within the magnetosphere, observed substorm phenomena include (1) greatly enhanced intensities and energies of charged particles, (2) changes of the magnetic field in the nightside magnetosphere and magnetotail, the field becoming initially more stretched (tail-like) and subsequently more dipolar (dipolarization), and (3) appearance of fast (∼ VA) bulk flows of plasma in the magnetotail. Attempts at theoretical interpretation are also wide-ranging and complex, and some aspects (notably the 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 described in the literature (e.g., Akasofu 1977; Kennel 1995; Syrjäsuo and Donovan 2007,and many others), one may extract two primary aspects: (1) enhanced energy input and dissipation, and (2) change of magnetic field configuration, in two distinct phases. An additional aspect is that the primary controlling factor for the occurrence of substorms is the interplanetary magnetic field, in particular its southward component—opposite to the Earth’s dipole field in the equatorial plane. More specifically, the physical description can be summarized as a two-phase process. Growth phase: as a consequence of a southward interplanetary magnetic field, the configuration of the magnetosphere changes, its magnetic field becoming highly stretched (increased magnetic flux in the magnetotail, reduced flux in the nightside equatorial region). Expansion phase, initiated by the onset: the magnetic field changes to more nearly dipolar (increased flux on the nightside), and there is enhanced energy input and dissipation to the inner magnetosphere and the ionosphere/atmosphere; the process occurs on dynamical time scales (comparable to or shorter than wave travel times) and is accompanied (most probably) by changes of magnetic topology. With reference to the energy flow paths of Fig. 7: during the growth phase, P(1) is enhanced and is appreciably larger than the sum P(2a) + P(2b) + P(2c) + P(3a). During the expansion phase, P(2a) and especially P(2c) are enhanced; P(3a) and P(3b) presumably are enhanced in connection with plasmoid formation and the topological changes exemplified by Fig. 3. The increasing magnetic flux in the magnetotail during the growth phase implies that in the magnetosphere the magnetic flux transport rate toward the dayside reconnection region has increased (to match the increased magnetic flux transport in the solar wind), but the flux return 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 to reach the distant X-line (but communication from the dayside to the magnetotail can go much faster by wave propagation within the magnetosphere than by advection in the solar wind). V.M. Vasyliunas¯

2. The return flow within the plasma sheet is opposed by adverse tailward stress (e.g., pressure gradient). 3. The amount of open flux depends directly on the interplanetary magnetic field (this presumes 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 into two groups. First: the configuration of the magnetosphere at the end of the growth phase is characterized by 1. increased flux in the magnetotail 2. increased magnetic tension force on the Earth 3. thin current sheet 4. 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? (if so, which item in the above list is the critical one? several at once?), have external conditions changed? Almost every proposed model can be matched to an item in the list: 1 or 2 is the critical item for NEXL models, 3 for “current disruption” models, 4 for ballooning or interchange 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. Included are “bursty bulk flows” (relation to substorm not entirely clear), tailward and earthward flows 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 the magnetospheric substorm. In contrast to the substorm, there is a clear and widely accepted definition of what constitutes a magnetic storm, in terms of the time variation of the geomagnetic Dst index: a prolonged (hours to days) interval of negative Dst values (Gonzalez et al. 1994). (The storm sudden commencement and the initial phase of positive Dst, which accompany many but not all storms, are no longer considered an integral part of the storm concept.) The physical understanding of what a magnetic storm is relies heavily on a remarkable theoretical result, the Dessler-Parker-Sckopke theorem, which relates the external magnetic field 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 extended to any distribution by Sckopke (1966), the theorem states that b(0), the magnetic disturbance field 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 (independent 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 axially symmetric 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 energy within the box “ring current plasma” in Fig. 7) and its time history, from which two essential aspects of magnetic storm physics have been established: the essential phenomenon of the magnetic storm is the addition of a large amount of plasma energy to the dipolar field region of the magnetosphere, which results from a particular condition in the solar wind, namely “a sufficiently intense and long-lasting interplanetary convection electric field” (Gonzalez et al. 1994), meaning −V × B/c with the interplanetary magnetic field having a southward component. Since geomagnetic storms, particularly the intense ones, are characterized by unusually large amounts of energy stored as mechanical energy of plasma in the ring current region (in comparison to other storage regions), the implication for Fig. 7 is that during the development of an intense storm, P(2b) is unusually large, on the average, and exceeds the loss rate P(4a) + P(4c). Whether this enhanced conversion rate from magnetic energy into mechanical energy of ring current plasma results from some different interaction process or simply from a 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 to the dipole field in the Earth’s equatorial plane) is well established as a condition for the occurrence both of storms and of substorms. This may suggest the question (put to me by a solar physicist): do magnetic storms and magnetospheric substorms really constitute two physically distinct phenomena, or are they merely different-time-scale manifestations of a single phenomenon? Leaving aside matters of time scale and sequence, the discussion in Sects. 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 arguments about 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 configuration of the solar-wind/magnetosphere/ionosphere system. 2. Stresses are changed primarily by plasma flow, through the associated transport of magnetic flux and evolution of plasma pressure. 3. Variable advection of magnetic flux in the solar wind seems to be the primary factor for producing large-scale variability of the Earth’s magnetosphere. 4. Magnetospheric substorm is a process of enhanced energy dissipation, connected with transitions from equilibrium to non-equilibrium, in two steps (both in general with magnetic topological changes): (a) Step 1: mechanical stresses deform the magnetic field into a configuration of increased energy (plasma flow transports magnetic flux and, with field lines attached to 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 causes the 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 not clear. 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 factor differentiating substorms from other types of events.) 8. What unbalanced stresses accelerate the fast plasma flows (including those associated with the formation of the near-Earth X-line)?

Acknowledgements I am grateful to Paul Song and George Siscoe for illuminating discussions.

References

S.I. Akasofu, The development of the auroral substorm. Planet. Space Sci. 12, 273–282 (1964) S.I. Akasofu, Physics of Magnetospheric Substorms (Dordrecht, Reidel, 1977) A.Y. Alksne, The steady-state magnetic field in the transition region between the magnetosphere and the bow shock. Planet. Space Sci. 15, 239–245 (1967) W.I. Axford, C.O. Hines, A unifying theory of high-latitude geophysical phenomena and geomagnetic storms. Can. J. Phys. 39, 1433–1464 (1961) W.I. Axford, H.E. Petschek, G.L. Siscoe, Tail of the magnetosphere. J. Geophys. Res. 70, 1231–1236 (1965) O. Buneman, Internal dynamics of a plasma propelled across a magnetic field. IEEE Trans. Plasma Sci. 20, 672–677 (1992) R.K. Burton, R.L. McPherron, C.T. Russell, An empirical relationship between interplanetary conditions and Dst.J.Geophys.Res.80, 4204–4214 (1975) R.L. Carovillano, G.L. Siscoe, Energy and momentum theorems in magnetospheric dynamics. Rev. Geophys. Space Phys. 11, 289–353 (1973) T.G. Cowling, Magnetohydrodynamics (Interscience, New York, 1957), Sect. 1.3 A.J. Dessler, Length of magnetospheric tail. J. Geophys. Res. 69, 3913–3918 (1964) A.J. Dessler, E.N. Parker, Hydromagnetic theory of geomagnetic storms. J. Geophys. Res. 24, 2239–2252 (1959) J.W. Dungey, Cosmic Electrodynamics (Cambridge University Press, London, 1958), p. 10 J.W. Dungey, Interplanetary magnetic field and the auroral zones. Phys. Rev. Lett. 6, 47–48 (1961) J.W. Dungey, The length of the magnetospheric tail. J. Geophys. Res. 70, 1753 (1965) J.A. Fejer, Theory of the geomagnetic daily disturbance variations. J. Geophys. Res. 69, 123–137 (1964) T.G. Forbes, Magnetic reconnection, in Heliophysics: Plasma Physics of the Local Cosmos, ed. by C.J. Schriver, G.L. Siscoe (Cambridge University Press, New York, 2009), pp. 113–138 W.D. Gonzalez, A unified view of solar wind-magnetosphere coupling functions. Planet. Space Sci. 38, 627– 632 (1990) W.D. Gonzalez, J.A. Joselyn, Y. Kamide, H.W. Kroehl, G. Rostoker, B.T. Tsurutani, V.M. Vasyliunas,¯ What is a geomagnetic storm? J. Geophys. Res. 99, 5771–5792 (1994) J.M. Greene, Moment equations and Ohm’s law. Plasma Phys. 15, 29–36 (1973) M. Greenwald, Verification and validation for magnetic fusion. Phys. Plasmas 17, 058101 (2010). doi:10.1063/1.3298884 W.J. Heikkila, Comment on “The alternative paradigm for magnetospheric physics” by E.N. Parker. J. Geo- phys. Res. 102, 9651–9656 (1997) E.W. Hones Jr., The magnetotail: its generation and dissipation, in Physics of Solar Planetary Environments, ed. by D.J. Williams (AGU, Washington, 1976), pp. 558–571 E.W. Hones Jr., Substorm processes in the magnetotail: comments on ‘On hot tenuous plasmas, fireballs, and boundary layers in the Earth’s magnetotail’ by L.A. Frank, K.L. Ackerson, and R.P. Lepping. J. Geo- phys. Res. 82, 5633–5640 (1977) E.W. Hones, J.R. Asbridge, S.J. Bame, S. Singer, Substorm variations of the magnetotail plasma sheet from XSM ≈−6RE to XSM ≈−60RE . J. Geophys. Res. 78, 109–132 (1973) Physics of Magnetospheric Variability

N. Iwasaki, A. Nishida, Ionospheric current system produced by an external electric field in the polar cap. Rept. Ionos. Space Res. Jpn. 21, 17 (1967) C.F. Kennel, Convection and Substorms (Oxford University Press, New York, 1995) C. Lemon, F. Toffoletto, M. Hesse, J. Birn, Computing magnetospheric force equilibria. J. Geophys. Res. 108(A6), 1237 (2003). doi:10.1029/2002JA009702 C. Lemon, R.A. Wolf, T.W. Hill, S. Sazykin, R.W. Spiro, F.R. Toffoletto, J. Birn, M. Hesse, Magnetic storm ring current injection modeled with the Rice Convection Model and a self-consistent magnetic field. Geophys. Res. Lett. 31, L21801 (2004). doi:10.1029/2004GL020914 D.W. Longcope, Magnetic field topology, in Heliophysics: Plasma Physics of the Local Cosmos, ed. by C.J. Schriver, G.L. Siscoe (Cambridge University Press, New York, 2009), pp. 77–112 A.T.Y. Lui, Electric current approach to magnetospheric physics and the distinction between current disruption and magnetic reconnection, in Magnetospheric Current Systems, ed. by S.-I. Ohtani, R. Fujii, M. Hesse, R.L. Lysak. AGU Geophysical Monograph, vol. 118 (AGU, Washington, 2000), pp. 31–40 G.D. Mead, Deformation of the geomagnetic field by the solar wind. J. Geophys. Res. 69, 1181–1195 (1964) M.B. Moldwin, G.L. Siscoe, C.J. Schriver, Structures of the magnetic field, in Heliophysics: Plasma Physics of the Local Cosmos, ed. by C.J. Schriver, G.L. Siscoe (Cambridge University Press, New York, 2009), pp. 139–162 N.F. Ness, The Earth’s magnetic tail. J. Geophys. Res. 70, 2989–3005 (1965) A. Nishida, N. Nagayama, Synoptic survey for the neutral line in the magnetotail during the substorm expansion process. J. Geophys. Res. 78, 3782–3798 (1973) T.G. Northrop, The Adiabatic Motion of Charged Particles (Interscience, New York, 1963) E.N. Parker, Newtonian development of the dynamic properties of ionized gases of low density. Phys. Rev. 107, 924–933 (1957) E.N. Parker, Interaction of the solar wind with the geomagnetic field. Phys. Fluids 1, 171–187 (1958) E.N. Parker, The alternative paradigm for magnetospheric physics. J. Geophys. Res. 101, 10587–10625 (1996) E.N.Parker,Reply.J.Geophys.Res.102, 9657–9658 (1997) E.N. Parker, Newton, Maxwell, and magnetospheric physics, in Magnetospheric Current Systems, ed. by S.-I. Ohtani, R. Fujii, M. Hesse, R.L. Lysak. AGU Geophysical Monograph, vol. 118 (AGU, Washing- ton, 2000), pp. 1–10 E.N. Parker, Conversations on Electric and Magnetic Fields in the Cosmos (Princeton University Press, Princeton, 2007) P. Perrault, S.-I. Akasofu, A study of geomagnetic storms. Geophys. J. R. Astron. Soc. 54, 547–573 (1978) J.H. Piddington, Geomagnetic storm theory. J. Geophys. Res. 65, 93–106 (1960) J.H. Piddington, Recurrent geomagnetic storms, solar M-regions and the solar wind. Planet. Space Sci. 12, 113–118 (1964) D.E. Post, L.G. Votta, Computational science demands a new paradigm. Phys. Today 58(1), 35–41 (2005) B. Rossi, S. Olbert, Introduction to the Physics of Space (McGraw–Hill, New York, 1970), Chaps. 10 and 12 G. Rostoker, S.I. Akasofu, J. Foster, R.A. Greenwald, Y. Kamide, K. Kawasaki, A.T.Y. Lui, R.L. McPherron, C.T. Russell, Magnetospheric substorms—definition and signatures. J. Geophys. Res. 85, 1663–1668 (1980) G. Rostoker, S.I. Akasofu, W. Baumjohann, Y. Kamide, R.L. McPherron, The roles of direct input of energy from the solar wind and unloading of stored magnetotail energy in driving magnetospheric substorms. Space Sci. Rev. 46, 93–111 (1987) C.T. Russell, R.L. McPherron, The magnetotail and substorms. Space Sci. Rev. 15, 205–266 (1973) N. Sckopke, A general relation between the energy of trapped particles and the disturbance field near the Earth. J. Geophys. Res. 71, 3125–3130 (1966) K. Schindler, A theory of the substorm mechanism. J. Geophys. Res. 79, 2803–2810 (1974) K. Schindler, Physics of Space Plasma Activity (Cambridge University Press, New York, 2007) D.G. Sibeck, Evidence for flux ropes in the Earth’s magnetotail, in Physics of Magnetic Flux Ropes, ed. by C.T. Russell, E.R. Priest, L.C. Lee. AGU Geophysical Monograph, vol. 58 (AGU, Washington, 1990), pp. 637–646 K.D. Siebert, G.L. Siscoe, Dynamo circuits for magnetopause reconnection. J. Geophys. Res. 107, A7 (2002). doi:10.1029/2001JA000237 G.L. Siscoe, A unified treatment of magnetospheric dynamics with applications to magnetic storms. Planet. Space Sci. 14, 947–967 (1966) G.L. Siscoe, The magnetospheric boundary, in Physics of Space Plasmas (1987), ed. by T. Chang, G.B. Crew, J.R. Jasperse (Scientific, Cambridge, 1988), pp. 3–78 G.L. Siscoe, W.D. Cummings, On the cause of geomagnetic bays. Planet. Space Sci. 17, 1795–1802 (1969) P. Song, T.I. Gombosi, A.J. Ridley, Three-fluid Ohm’s law. J. Geophys. Res. 106, 8149–8156 (2001) V.M. Vasyliunas¯

P. Song, V.M. Vasyliunas,¯ L. Ma, A three-fluid model of solar wind-magnetosphere-ionosphere-thermosphere coupling, in Multiscale Coupling of Sun-Earth Processes, ed. by A.T.Y. Lui, Y. Kamide, G. Consolini (Elsevier, Amsterdam, 2005), pp. 447–456 J.R. Spreiter, A.L. Summers, A.Y. Alksne, Hydromagnetic flow around the magnetosphere. Planet. Space Sci. 14, 223–250 (1966) F.R. Toffoletto, S. Sazykin, R.W. Spiro, R.A. Wolf, Modeling the inner magnetosphere using the Rice Con- vection Model. Space Sci. Rev. 108, 175–196 (2003) N.A. Tsyganenko, Empirical magnetic field models for the space weather program, in Space Weather, ed. by P. Song, H.J. Singer, G.L. Siscoe. AGU Geophysical Monograph, vol. 125 (AGU, Washington, 2001), pp. 339–345 M. Syrjäsuo, E. Donovan, Proceedings of the Eighth International Conference on Substorms, ICS-8,Univer- sity of Calgary, Alberta, Canada, 2007 V.M. Vasyliunas,¯ Mathematical models of magnetospheric convection and its coupling to the ionosphere, in Particles and Fields in the Magnetosphere, ed. by B.M. McCormack (Reidel, Dordrecht, 1970), pp. 60– 71 V.M. Vasyliunas,¯ An overview of magnetospheric dynamics, in Magnetospheric Particles and Fields, ed. by B.M. McCormac (Reidel, Dordrecht, 1976), pp. 99–110 V.M. Vasyliunas,¯ Steady state aspects of magnetic field line merging, in Magnetic Reconnection in Space and Laboratory Plasmas, ed. by E.W. Hones Jr. AGU Geophysical Monograph, vol. 30 (AGU, Washington, 1984), pp. 25–31 V.M. Vasyliunas,¯ Electric field and plasma flow: what drives what? Geophys. Res. Lett. 28, 2177–2180 (2001) V.M. Vasyliunas,¯ Time evolution of electric fields and currents and the generalized Ohm’s law. Ann. Geophys. 23, 1347–1354 (2005a) V.M. Vasyliunas,¯ Relation between magnetic fields and electric currents in plasmas. Ann. Geophys. 23, 2589– 2597 (2005b) V.M. Vasyliunas,¯ Ionospheric and boundary contributions to the Dessler-Parker-Sckopke formula for Dst. Ann. Geophys. 24, 1085–1097 (2006a) V.M. Vasyliunas,¯ Reinterpreting the Burton-McPherron-Russell equation for predicting Dst. J. Geophys. Res. 111, A07S04 (2006b). doi:10.1029/2005JA011440 V.M. Vasyliunas,¯ Understanding the magnetosphere: the counter-intuitive simplicity of cosmic electrodynamics. Van Allen lecture, presented at AGU Fall Meeting (2008) V.M. Vasyliunas,¯ Fundamentals of planetary magnetospheres, in Heliophysics: Plasma Physics of the Local Cosmos, ed. by C.J. Schriver, G.L. Siscoe (Cambridge University Press, New York, 2009), pp. 256–294 V.M. Vasyliunas,¯ Energy conversion in planetary magnetospheres, in Heliophysics: Space Storms and Radi- ation: Causes and Effects, ed. by C.J. Schriver, G.L. Siscoe (Cambridge University Press, New York, 2010), pp. 263–291 V.M. Vasyliunas,¯ P. Song, Meaning of ionospheric Joule heating. J. Geophys. Res. 110, A02301 (2005). doi:10.1029/2004JA010615 R.A. Wolf, Effects of ionospheric conductivity on convective flow of plasma in the magnetosphere. J. Geo- phys. Res. 75, 4677–4698 (1970) R.A. Wolf, The quasi-static (slow-flow) region of the magnetosphere, in Solar-Terrestrial Physics, ed. by R.L. Carovillano, J.M. Forbes (Reidel, Dordrecht, 1983), pp. 303–368