Auroral arcs as sites of magnetic stress release

Authors


Abstract

[1] An analytical model is presented for auroral arcs as the result of a fast release of magnetic shear stresses. The shear stresses are set up by a longitudinal convection that is driven by pressure forces in the outer magnetosphere against the frictional forces exerted in the lower ionosphere. A distorted-dipole geometry is employed allowing for high plasma beta near the equator. Steep ledges in the radial pressure distribution, extending along the direction of convection, are invoked as the sources of the auroral current sheets. The differential magnetic energy content of these narrow current sheets is released within a few Alfvén transit times by the decoupling of the magnetospheric plasma and field from the ionosphere, owing to the existence of field-aligned potential drops in the auroral acceleration region, and converted into kinetic energy of the primary auroral particles. A well-known current-voltage relation is employed for the formulation of the energy conversion process. This scenario has two important consequences. (1) The loss of magnetic energy creates a concomitant decrease of internal energy of the generator plasma and results in a progression of pressure ledge and auroral current sheet into the more highly stressed magnetic field region. This is the reason for the observed proper motion of auroral arcs with respect to the plasma frame. (2) Plasma and field undergo a rapid stress relief motion along the arc with large but mostly reversible displacements. The net displacement, equivalent to a small S-shaped contribution to the essentially U-shaped potential distribution above the auroral arc, is consistent with the transit of the field lines through the progressing current sheet. This scenario is cast into a set of simple relations expressing the key parameters of auroral arcs, such as width, energy flux, potential drop, and proper motion. The main ingredient herein is an auxiliary magnetic perturbation field into which the main properties of the large-scale current system are condensed. It corresponds to about twice the transverse magnetic perturbation field near the arc and thus to the total shear stresses. Two free parameters are the relative magnitude of the pressure jump at the ledge in the source plasma and the plasma beta. Matching the quantitative results of the relations for the arc properties with observed values suggests pressure jumps of order 10% and beta values between 1 and 5.

1. Introduction

[2] “Bright auroral displays during dark polar nights belong certainly to the most memorable natural phenomena a person can experience”. This statement, the first sentence in “Auroral Plasma Physics” edited by Paschmann et al. [2002], can serve to explain the great and continuing effort invested in the investigation of the origin of auroral arcs, both observationally and theoretically. A huge amount of data has been accumulated on the morphological structure, the dynamics, the relations to the magnetosphere, and the host of microphysical processes. Equivalent amounts of theoretical work have been devoted to their understanding. The volume cited above is an excellent summary of the state of the art.

[3] This paper deals solely with the macrophysical aspects, the origin and flow of energy and the ensuing auroral characteristics, whereas the microphysical processes are condensed into a well-known current-voltage relation [Knight, 1973]. The paper stands in the tradition of attempts, starting perhaps with Atkinson [1970], to identify sources of the field-aligned currents and derive various consequences from the interplay of ionospheric and magnetospheric plasmas. In the paper by Atkinson, as well as in the theories of Sato and Holzer [1973], Sato [1978] and Miura and Sato [1980], the field-aligned currents are generated by the diversion of part of the Pedersen current at ionospheric conductivity inhomogeneities. Thereby energy is being tapped from the storage in the large-scale magnetospheric-ionospheric current system and a feedback on the ionosphere exists through the acceleration of electrons by electric fields sustained by these currents.

[4] Another line of approach, and the one we shall follow in this paper, is to locate the source of field-aligned currents in the high magnetosphere, the near-Earth tail or the boundary layers. Hasegawa and Sato [1979] presented a general theory of the source strength distinguishing three sources, vorticity, cross-field currents in the presence of magnetic field gradients, and inertial currents with density gradients. An example of the first class is the theory of magnetospheric boundary layers by Sonnerup [1980], followed up by Lotko et al. [1987]. Vasyliunas [1972] formulated a general framework for the second class, which explained the Region 2 currents and became the foundation of the numerical models of the Rice University group on magnetospheric convection [Wolf, 1974; Harel et al., 1981]. Also to this class belong pressure-driven convection models [Haerendel, 1990]. The third class comprises, explicit- or implicitly, theories involving the launching of Alfvén waves by the magnetospheric plasma dynamics. Haerendel [1992] related the origin of the so-called substorm current wedge to the inertial currents generated by the braking of the reconnection flows in the near-Earth tail. Numerical modeling by Birn et al. [1999] clarified the relative importance of inertial versus pressure forces. Surprisingly, pressure forces have rarely been explicitly invoked as drivers of field-aligned currents.

[5] A necessary ingredient in most of auroral theories is the propagation of energy and momentum between ionosphere and the plasma of the outer magnetosphere or tail, the ultimate energy source of the aurora. Hence Alfvén waves have received much attention, not only as energy carriers but also as the accelerating agent of the primary auroral electrons [e.g., Goertz and Boswell, 1979; Lysak and Dum, 1983; Haerendel, 1983; Hui and Seyler, 1992; Thompson and Lysak, 1996; Vogt and Haerendel, 1998; Rankin et al., 1999; Chaston et al., 2002], whereby the origin of these waves was mostly left outside the considerations.

[6] The other important ingredient of any successful theory of auroral arcs is the physics of the conversion of electromagnetic into kinetic energy, somewhere above the ionosphere and besides the direct acceleration by resonant interactions of propagating Alfvén waves. One of the most successful tools became the kinetic theory of a current-voltage relation by Knight [1973], Fridman and Lemaire [1980], and Lyons [1980], which is based on the adiabatic motion of electrons in a converging magnetic field. However, a large variety of other processes has been investigated and applied ranging from anomalous resistivity [Kindel and Kennel, 1971; Papadopoulos, 1977], often employed in numerical simulations, to electrostatic double layers [Block, 1972; Swift, 1975; Sato and Okuda, 1981; Ergun et al., 2002], and various mechanisms for electron and ion heating by high-frequency or solitary waves and phase-space holes. Such processes, as important as they are for the full understanding of auroral physics, are outside the scope of this paper.

[7] The author of the present paper has a long tradition of invoking Alfvén waves as feeders of the auroral acceleration region [Haerendel, 1980, 1983, 1988, 1994, 2001]. His particular contribution was to identify the release of magnetic shear stresses as the immediate source of auroral energy, whereas the final source was to be sought in the forces (here pressure force) driving the magnetospheric convection and thereby creating these stresses. Their release is enabled by the “cutting” of field lines or, in more physical terms, the decoupling of magnetosphere from ionosphere by the parallel potential drops set up in sheets of intense field-aligned currents attending oblique standing or slowly propagating Alfvén waves. This paper tries to put previous outlines of this scenario into a coherent analytical model and derive quantitative conclusions. The perhaps most intriguing aspect of auroral physics, how the liberated magnetic energy is converted into the kinetic energy of the particles, is not explicitly addressed, but represented by the famous current-voltage relation. The emphasis is solely on the macroscopic energy balance and on the plasma dynamics.

2. Basic Concepts of the Model

[8] The plasma convection in and above the auroral oval, in particular during substorms, is generally directed away from midnight. The underlying reason is the existence of a maximum of the total pressure in the midnight sector of the near-Earth plasma sheet. This maximum is built up during the expansion phase of a substorm [e.g., Shiokawa et al., 1997] and is likely to persist through the recovery phase. The pressure forces exert shear stresses on the magnetic field by which momentum is transferred to the ionospheric plasma and finally, by ion-neutral friction, to the upper atmosphere. This is the origin of the large-scale Region 1/Region 2 field aligned current system [Iijima and Potemra, 1976; Haerendel, 1990]. Within this convection system, auroral arcs can appear. The author [Haerendel, 1980, 1988, 1994] has interpreted the arcs as narrow sheets of fast stress relief, faster than by the ambient convective motion and dissipation. The reason for the accelerated stress relief lies in the appearance of U-shaped electric potential distributions above the auroral arcs, which decouple the magnetospheric plasma motions from the ionosphere and, at the same time, generate the auroral particle beams by field-aligned acceleration. While the attention of auroral researchers has been mainly focused on the second aspect, the first one is of equal importance. It allows fast stress relief motions along narrow lanes above the arcs, unimpeded by ionospheric friction.

[9] The stress relief motions propagate from the point of origin of the decoupling, the auroral acceleration region, upward along the field lines and affect the pressure balance in the source region and thereby the field aligned currents. The result must be a reduction of the free energy content of the involved flux tubes, since the auroral process consumes previously stored energy. For this reason, the author has introduced the notion of “magnetic fractures” and has named the auroral acceleration region the “fracture zone” and the arcs “auroral pressure valves” [Haerendel, 1980, 1992]. However, no self-consistent model yet exists relating the stress release directly to the auroral energy conversion process. This is the goal of this paper.

[10] Irrespective of the frequently striking changes of auroral arcs in shape and brightness, we will adopt a quasi-stationary model. In order to cope with the loss of energy stored above the arc, we postulate that the auroral arc is the trace of a slowly propagating front of stress relief, greatly elongated in the direction of ambient convection and narrow in the perpendicular direction. The fact that auroral arcs are not “frozen” in the plasma, but possess a proper motion with respect to the plasma frame, has been verified by combined optical and radar measurements [Haerendel et al., 1993; Frey et al., 1996]. If liberation and conversion of magnetic shear stresses is the origin of auroral arcs, one should think that arcs are invariably moving into regions of higher magnetic shear. However, the cited observations have shown that the opposite can also be true. In other words, auroral arcs may not only feed on the excess energy of more strongly sheared fields, but can also be traces of a process increasing the shear. In this paper we will pursue only the first situation, while the second one will be treated in a follow-on paper.

[11] As the auroral arc propagates and releases magnetic shear stresses, it also reduces the pressure in the source region, i.e., it releases internal energy. Both, magnetic and internal, energy are feeding the auroral acceleration process. But why is all that happening? Like in a fracture of a solid body, where molecular bonds have to be broken, the “magnetic fractures” involve microphysical processes. The accelerating parallel electric fields owe their generation to the dynamics of the current carrying particles. It requires high current density so that either an anomalous resistivity is generated by plasma turbulence, or the current carrying electrons have to fight the mirror force of the converging magnetic field in order to maintain the current [Kindel and Kennel, 1971; Knight, 1973; Vedin and Rönnmark, 2006]. This implies that the field aligned current sheet separating regions of higher and lower shear stresses must be sufficiently narrow to allow the onset of such processes. This is the very reason for the narrowness of the auroral arc.

[12] The approach taken in this paper is to describe energy storage and release in a simple dipole geometry accounting only for shear stresses in the longitudinal direction. This means that upward and downward field-aligned currents are strictly balanced across the convection zone (auroral oval and plasma sheet), as in Boström's [1964] current sheet of Type 2 (Figure 1), and that divergences of longitudinal currents in magnetosphere and ionosphere are being ignored. Hence this model is not easily applicable to the complex morphology of arcs near midnight or inside the westward traveling surge, although the underlying physics may be similar. But the advantage of the model is that it generates a set of simple physical relations allowing the derivation of the basic auroral quantities, like energy flux, current strength, potential drop, arc width and propagation speed, and that it sheds new light on their interdependences as well as their dependence on the magnetospheric plasma parameters and the strength of the global current system. Naturally, the dipole approximation of the magnetic field begins to fail, when shear stresses exist strong enough to generate auroral arcs. A plasma beta exceeding unity is required at high altitudes. For this reason we will introduce a mathematically convenient, but not fully consistent modified dipole approximation.

Figure 1.

The Type II current system after [Boström, 1964] showing a pressure maximum in the midnight sector, the dynamo regions, the resulting longitudinal convection and frictional control of the ionosphere.

[13] Of course, other types of auroras exist. Not all are pressure gradient driven. An overview can be found in [Haerendel, 2001; Paschmann et al., 2002].

[14] The following four subsections outline the basic ingredients of the model.

2.1. Dipole Geometry

[15] The stress relief fronts or auroral current sheets are considered as extending along constant L-shells. This is consistent with the dominant orientation of auroral arcs, at least on the evening side. The coordinates are: y along the current sheet, i.e., in ϕ−direction; ℓ along a field line, counting downward from the equator; and q transverse to B and to the current sheet pointing outward. ζ = sinλ, where λ is the dipole latitude. We will use the following relations in the subsequently derived integral quantities and restrict ourselves to pure dipole geometry for the first three sections of the paper:

equation image
equation image
equation image
equation image
equation image

ϕ is the equatorial longitude of the undisturbed field line, and B0(ζ) designates the magnetic field strength unaffected by internal currents. All integrations along the field lines will use the unperturbed geometry for the integration path.

[16] As pointed out above, we are employing the dipole geometry because of its simplicity (as compared, for instance, with the Tsyganenko [2002] model). It will allow us to cast flux tube integrated quantities into simple algebraic relations with numerical factors which are expected not to alter significantly in realistic geometries. However, a pure dipole field will not be an appropriate tool for this model, since we will find that the equatorial plasma beta (βeq), which will appear in practically all key relations, must be greater than unity in order to generate quantities typical for auroral arcs. For greater transparency however, we will derive these relations first in the strict dipole geometry and apply the high beta corrections at a later stage (section 5).

2.2. Magnetospheric Current Sources

[17] The basic assumption underlying this treatment of auroral arcs is that they are means of releasing magnetic stresses set up by a high plasma pressure in the midnight sector of the near-Earth plasma sheet (or tail), as a consequence of hot plasma injections, for instance during substorms. In the context of the latter, auroral arcs play the role of localized “pressure valves” and may have an impact on the substorm dynamics [Haerendel, 1992]. In the present concept of auroral physics we will neglect the impulsive phase of the substorm, i.e., the near-Earth plasma injection and field dipolarization, but rather consider the subsequent phase of high pressure (high beta) in the midnight sector and large-scale pressure forces pointing away from midnight and driving plasma convection against the frictional force of the ionosphere. This is depicted in Figure 1 in which also are shown the resultant large-scale current sheets following the original cartoon of Boström [1964]. Auroral arcs are mainly located inside the regions of upward current, i.e., on the poleward side of the evening auroral oval or plasma sheet and on the low-latitude side of the morning oval. The regions of downward current, equally fascinating [Carlson et al., 1998], are not being treated in this paper in spite of many similarities with regard to their magnetic stress relief function.

[18] The view of Figure 1 is a very simplified one, as it neglects the longitudinal currents and their divergences. This is corrected in Figure 2, which shows the magnetic gradient and curvature currents in a low-latitude cut through the plasma sheet. In contrast to the magnetization or pressure-driven currents, these components can have a finite divergence and thus be sources of field-aligned currents, j. Static equilibrium can be expressed by the balance of three basic forces, the magnetic pressure force, M, the magnetic tension, T, and the pressure force, P:

equation image

with:

equation image
Figure 2.

Driving forces (thin arrows), magnetic gradient and curvature currents (double arrows) and their divergences (⊗ and Θ) for selected locations in a plane of low northern latitude assuming a pressure maximum in the midnight sector (solid lines are pressure contours).

[19] The sources of j are given by:

equation image

eB = B/B. The plasma pressure is assumed to be isotropic [cf. Haerendel, 1990].

[20] The above relations also apply under slow convection, as long as the inertial forces are negligible. This assumption underlies the model pursued in this paper.

[21] As is obvious from equation (7), there is no source or sink of j, if M, T, and P are collinear. Where this is not the case, foremost near the boundaries of the plasma sheet, are the locations of the field aligned current sources. One can express the magnetic gradient and curvature currents by the respective forces:

equation image

[22] Henceforward we will restrict ourselves to isotropic pressure. In using equation (6) we can write:

equation image

[23] This formulation reveals the importance of high beta for any field-aligned current source and in particular, as we will see, for auroral arcs to form.

[24] Figure 2 shows, in the equatorial plane, the respective driving forces for some locations of high current divergence, under the assumption of an appreciable pressure maximum in the midnight sector. B0 is directed out of the plane, the solid lines are pressure contours. The dominant magnetic gradient and curvature current, jB, (double arrows) is directed westward. However, there are substantial radial components due to the shear stresses opposing the longitudinal pressure forces. With the aid of the physical relations (7) and (9) one can easily verify positive and negative divergences of j and the directions of jB. (We note here that in the dynamic phase of a substorm an eastward current may be superimposed in the near-Earth tail whose attending field aligned currents close via a westward ionospheric current [Haerendel, 1992; Shiokawa et al., 1997; Birn and Hesse, 2005]. In this model we will not be concerned with the complex substorm dynamics and with the divergence of the longitudinal currents, but concentrate on the outflow regions of the near-Earth plasma sheet with westward convection in the evening sector or the eastward convection in the morning sector.)

[25] The main purpose of showing Figure 2 is to remind the reader that there are two nearly equal contributors to the field aligned currents in the off-midnight sectors, the gradient and curvature currents across and along the plasma sheet. The first class of transverse currents is widely distributed in local time, but has short radial divergence scales. These currents constitute the Lorentz force balancing the longitudinal pressure force. The second class of transverse currents is driven by the radial forces and is more concentrated, but has longer divergence scales (the longitudinal pressure gradients). This appears to create equal contributions to the sources of j. We will, however, neglect the latter, called Type I currents by Boström [1964], because they would introduce the longitudinal dimension explicitly into our model. It is for the sake of simplicity that we will restrict ourselves entirely to the Type 2 or meridional currents, although this will lead to reduced magnetic shear stresses. This is in keeping with the basic idealization of our model, which allows only for a strict east-west orientation of the pressure-driven convection, in which auroral arcs are imbedded. The role of radial forces and associated longitudinal currents will be the subject of a subsequent paper.

2.3. A Propagating Stress Relief Front

[26] We consider a simple pressure profile across the plasma sheet, p0(L), (Figure 3). The pressure is isotropic and falls off with a uniform scale length, ℓP, in longitude, whereby ℓP is measured at the equator. Since p0 is constant along B, ℓP must vary along the field lines due to their convergence. Whereas the longitudinal pressure gradient, −equation image, and the balancing magnetic shear stresses are the origin of the transverse current, j, the function of the radial pressure variation in this model is reduced to providing the divergence of j, and thus sources and sinks of the field-aligned currents. Actually, it is the product of plasma pressure and flux tube volume, V, that determines the strength of the field-aligned current [Vasyliunas, 1970; Heinemann and Pontius, 1990].

Figure 3.

Meridional cut through the Type II current system in the evening sector and profile of pressure times flux tube volume (pV) in the generator region, but displaced toward lower altitudes for better visibility. The ledge in the pressure profile is a source of concentrated field-aligned current through an auroral arc.

[27] In the evening sector, j is flowing equatorward, so that j × B counteracts the longitudinal pressure force. Thus j is directed upward in regions where p0V decreases with increasing L. Auroral arcs, being quite narrow in comparison with the width of the total upward current region, are attributed to steep local gradients in p0(L). A sketch is shown in Figure 3. At the pressure ledge there is a sink of upward j, feeding the enhanced j in the adjacent region of higher shear stresses. Although the total upward and downward field-aligned currents must be balanced in our model, there is no need for a local balance of the upward directed arc current by a nearby downward current.

[28] Within such a sheet of intense upward j, field-aligned potential drops can be generated in the low-altitude magnetosphere and thus allow the field lines to decouple from the ionosphere. While the “feet” of the field lines above the ionosphere are being displaced in the direction of decreasing pressure, consistent with a reduction of the shear stresses, the thermal energy content, i.e., the pressure, is also being reduced (j · E < 0). As a consequence, the ledge in the pressure profile is displaced toward the higher stressed field region without the plasma moving in that direction (Figure 4). By “eating” itself into the magnetic energy reservoir, the sheet of concentrated j can persist for times much longer than the energy release time of the current sheet, which is a few times the Alfvén transit time, τA, from equator to ionosphere. In this sense, we consider the auroral arc as a stress release front propagating slowly with respect to the ambient plasma frame.

Figure 4.

Cartoon of a propagating pressure ledge/auroral current sheet due to the release of internal energy of the generator plasma in response to magnetic stress reductions.

[29] We have separated the build-up of the pressure profile, the natural result of a substorm, from the energy release through auroral arcs. In reality, the two processes overlap in time. For instance, while arcs are reducing the internal energy content, new plasma is being supplied from the tail. Furthermore, the arc is not the only energy release mechanism, even more energy is normally consumed by the convection against the frictional ionosphere [Lu et al., 1995]. This background convection is ignored in this context as being slow in comparison with the stress relief motions.

2.4. Energy Flow

[30] When high j sets up a parallel potential drop and the magnetospheric section of a flux tube becomes decoupled from the ionosphere, the magnetic tensions become progressively reduced and the quasi-static friction-controlled equilibrium of forces is perturbed. The perturbation proceeds from the auroral acceleration region upward toward the generator region of the current. The source plasma reacts by starting to move fast in the direction of the driving pressure force. In our model, this motion, like the ambient convection, is directed strictly in longitude. This is quite consistent with the dominant orientation of isolated arcs in the evening sector and the ray motions therein. Associated with this stress-relief motion is a loss of internal energy, due to E · j < 0. This energy is carried away along the field lines as Poynting flux additionally feeding the particle acceleration in the fracture zone.

[31] The whole process of energy flow above auroral arcs is very complex. It will be greatly simplified in the here presented model by treating current sheets and flux tubes of finite width rather than resolving the current and energy flow inside. However, for certain aspects it will be helpful to throw a glance on the principal structure of the auroral current sheet.

[32] Figure 5 contains an illustration taken from an earlier paper [Haerendel, 1994, Figure 4]. It shows, in a straight-field-line geometry, oblique standing Alfvén waves attached to the progressing acceleration region (fracture zone), which is also being evacuated. The dotted equipotential lines are not continued upward, because the description as electrostatic fields is only approximate and limited to the strong magnetic field and low-density region within about one Earth radius above the ionosphere, where the Alfvén velocity is extremely high. Field lines are not equipotential lines above an auroral arc. In the leading oblique wave the higher-altitude plasma receives the information of the “fracture” below and responds by starting to move in the stress relief direction. (We will address this motion more quantitatively in section 7.1). Eventually, this motion will die out while the corresponding energy flows as Poynting flux, SP, into the acceleration region, indicated by the second oblique wave. It takes one τA before the signal from the onset of decoupling (fracture) has reached the generator plasma, another τA for the full signal to arrive there, and a third τA for the Poynting flux, initiated by the release of internal energy, to be absorbed in the fracture zone. The high incompressibility of the low-altitude magnetic field enforces a return flow or opposing electric field, as indicated by the U-shaped contribution to the potential. The plasma above will be dragged backwards, to a large extent reversing the stress relief. This will take at least one more τA bringing the total passage time of the Alfvénic front up to about 4 τA. The return flow needs energy. So, not all the energy flowing into the acceleration region is being converted there into kinetic energy of particles. According to [Haerendel, 1994] a fraction (of the order of one third) is being channeled upward again. This will become clearer by the analysis given in section 7.2, while the so-called interference zone (below the acceleration region in Figure 5) will be addressed in section 7.3.

Figure 5.

A model of the electromagnetic structure of the auroral current sheet in a straight-field-line geometry (Figure 4 of [Haerendel, 1994]). Slightly oblique Alfvén waves attached to a slowly propagating “fracture zone” (speed vn) communicate the decoupling from the ionosphere by field-aligned potential drops to the generator plasma. On the leading edge (right) magnetic shear stresses are relieved, but largely restored by the return flow on the trailing edge. Dotted lines are electric equipotential contours exhibiting an asymmetry between leading and trailing edge with partial continuation into the ionosphere in accordance with a net stress release. The “fracture zone” interacts with the ionosphere through small-scale damped Alfvén waves, the “interference region”.

[33] The above argumentation means nothing else than perfect matching between the Alfvénic wave impedance and the quasi-Ohmic resistance of the energy conversion region, or absence of partial reflection. While the existence of the latter is well-known and extensively treated in the literature on propagating Alfvén waves (see references in section 1), it must be absent in slowly propagating quasi-stationary oblique Alfvén waves, adopted for the present approach. Otherwise, quasi-stationarity would be violated and oscillatory solutions would have to be introduced. Those who have watched structured auroral arcs may favor the latter, but for a simple theoretical model, meant to reproduce some essential features of the arcs, the quasi-stationary approach offers strong advantages. Beyond that, perfect matching maximizes dissipation and therefore appears to be a state toward which cosmic plasmas like to evolve.

[34] What this discussion should show, besides the complexity of the stress relief process, are two items. The whole stress liberation process takes about four Alfvén transit times, and much of the temporarily liberated energy is not consumed in the auroral acceleration process but re-invested in the field (section 7.2). Subsequently, we will therefore adopt 4τA as the intrinsic time-scale of an auroral arc and, for simplicity reasons, set the liberated energy at twice the stored magnetic energy, thereby accounting for the contribution from the internal energy of the source plasma. An approximate analysis of the reaction of the source plasma to the fracture process, presented in section 7.1, supports this approach.

3. Field Aligned Current

[35] The basic concept of our auroral arc model is that the energy consumed in creating the primary particle beams is taken out of the magnetic energy stored in shear stresses set up by magnetospheric pressure forces working against the ionospheric friction. This means that there is a large-scale current system with the high-altitude plasma as generator, connecting to the ionosphere by field-aligned currents, and closing there by transverse Pedersen currents. This has been sketched in Figure 3, including a jump or ledge in the pressure profile providing an enhanced divergence of the transverse current and thus feeding the auroral current sheet. In this section we will derive a relation between pressure gradient and field-aligned current.

[36] First, we return to the gradient and curvature current (equation (9)), the part of the total transverse current with a non-vanishing divergence. Conforming to our model we are only interested in the radial component of this current

equation image

with β = 2μ0p/B2. In order to relate the current solely to the driving force, i.e., the pressure gradient, we have to make an assumption on the relative orientation and magnitude of the magnetic shear and normal stresses, T and M. There are two possibilities. The longitudinal components of M and T can either be co-aligned or oppose each other. In the first case, we can conclude that the magnetic normal stresses must be smaller than the shear stresses, because otherwise jB,rad and the resulting field aligned currents would be reversed and the convection would be directed toward instead of away from midnight. In the second case, T has to balance more than the plasma pressure force. Unfortunately, there is no simple way to assess which of the two cases is holding in the off-midnight plasma sheet. One may, however, argue that the forces compressing the near-Earth plasma sheet during substorms are also compressing the magnetic field. Hence B may be higher near midnight than at later and earlier local times at the same L-shell and oppose T. Not having a general clue or numerical model at hand clarifying the situation, we will assume that ∣M∣ ≪ ∣T∣. This is most likely a conservative approach underestimating the generator power. However, since there will be a range of beta values to choose from when finally evaluating the auroral parameters, such uncertainty appears to be acceptable. It has no effect on the principles of this model, but has the distinct advantage of providing a simple relation between pressure gradient and source current.

[37] With the above assumption, P ≈ -T, and equation (10) can be written:

equation image

or:

equation image

[38] The next step is to relate the pressure gradient to j. For this we need the 2D divergence of the currents in a meridional plane (from now on j = j⊥,rad):

equation image

[39] From this follows the integral form

equation image

[40] In evaluating equation image we replace ∇p by p0(L)/ℓP(ℓ). As discussed above, the pressure is considered as being isotropic and constant along B0, only dependent on L, while the variation of the transverse scale along B0 is explicitly taken into account:

equation image

P is the characteristic pressure scale at the equator in y-direction. The last term on the right-hand side projects this scale along the field line. In separating the dependences on L and ζ = sinλ, we write with help of equations (1) and (4):

equation image

[41] The divergence in the radial direction is:

equation image

[42] From equation (3) we thus obtain:

equation image

[43] Introducing this relation into equation (13) yields

equation image

[44] Abbreviating:

equation image

writing:

equation image

assuming that ℓP scales as L and Beq as L−3, and with

equation image

we obtain:

equation image

[45] The function F1 depends strongly on the field geometry. Since it will be modified in section 5, no numerical values will be shown here. A transparent writing for j in the ionosphere is:

equation image

[46] The term FL(L) is quite interesting. It represents the divergence of the transverse current. If one ignores the beta-dependence of the currents, one obtains the value 1.5 instead of 4 for the second term on the r.h.s. of equation (18), because the flux tube volume is proportional to L3 in a strict dipole geometry. However, the geometry of the near-Earth plasma sheet, especially at its outer boundary, is much too distorted to place much emphasis on the exact power of L. The signs in equation (20) have been chosen in a way that j is directed along Bo if FL(L) > 0. This applies at the equatorial side of the evening auroral oval. On the polar side, j must be opposed to the direction of B0 and

equation image

[47] A rather steep fall-off of the pressure with L is needed for a negative sign. Within the underlying dipole geometry, we can call this inequality the auroral condition on the evening side. The signs are reversed on the morning side, because the pressure gradient points in the negative ϕ-direction.

[48] Equation (19a) provides the basis for the subsequent calculation of the magnetic energy content of the global current circuit as well as that of the auroral current sheet. j is solely determined by the longitudinal (ℓP) and radial (FL) pressure profiles. There is no feedback from the ionosphere. In a more realistic magnetic field geometry the functions F1(ζ) and FL(L) would come out differently. But whatever the variation of p02/Beq3p will be, -FL of order unity will be needed for a reasonably concentrated upward current (see discussion in section 4).

4. Magnetic Energy Content

[49] The free magnetic energy per unit cross-section of a flux tube bent in the longitudinal direction is:

equation image

[50] The jump of By through a current sheet of width, w(ℓ′), is:

equation image

[51] We introduce wup, the ionospheric width of the upward directed current sheet, and obtain w(ℓ′) as function of altitude:

equation image

[52] Inserting j from equation (19a) as characteristic for the interior oval, not far from the to be considered arc, By becomes:

equation image

[53] With this definition, Wm in equation (21) is the energy content of a sheared flux tube adjacent to the current sheet:

equation image

with

equation image

[54] By integrating down to the ionosphere using equation image ≅ 2, we find

equation image

[55] It is interesting to realize why βeq enters Wm with the fourth power. One half of the exponent comes from the pressure force balancing the shear stresses, the other half results from the fraction of the transverse current that has a finite divergence (equations (11a) and (11b)). All of the above applies strictly only within the framework of the dipole approximation. In the real world, cross-section and effective length of the flux tube, over which the energy density is being integrated (equation (21)), change. This is probably the most critical deviation of the model from reality and has to be corrected for, as will be done in the following section.

5. High-Beta Corrections

[56] The appearance of bright and narrow auroral arcs is related to plasma injections in the magnetosphere. From what is known about the near-Earth plasma sheet during and following a substorm, the plasma beta assumes high values [e.g., Kistler et al., 1992]. Typical values seem to range near 3, but occasionally much higher values have been reported [Lui et al., 1992]. Kistler et al. compared the pressure distribution within the highly stretched and low β field during the pre-breakup phase of a substorm with that in the high-beta, post-breakup phase and found that the total pressure was nearly unchanged. If Bo denotes the pre-breakup field strength and B the field after plasma injection and depolarization, one can cast this finding into:

equation image

[57] Incorporation of high-beta effects into our formalism is essential because of the sensitive dependence of the key properties on βeq. This has two consequences. According to equations (11a), (11b), and (19a) the transverse and field aligned current densities increase with increasing beta, and the flux tube volume, which determines energy content and release, will also be significantly enhanced. In order to simulate these two effects, we will adopt the above relation for the description of the modified field, B, but interpret Bo as being the ideal dipole field and not the stretched and compressed field before plasma injection. The reason is just mathematical convenience, since there is no simple analytic model available for the pre-breakup field configuration in the near-Earth tail. The main effect of this shortcoming is that for the same beta the flux tube volume is somewhat exaggerated and the field strength underestimated.

[58] Equation (28) can be rephrased as

equation image

is a function of L and of the position along the field lines. Whereas βeq(L) will not be further specified here, we describe the variation of B and β along a field line in the dipole approximation exploiting our assumption of isotropic pressure. With Bs being the surface field at the equator (Bs = 0.3 G) we get:

equation image

with

equation image

[59] These relations describe a field that approaches a dipole field near Earth and becomes progressively weaker with increasing radial distance. In addition, we need consistent expressions for the spatial scales. The distorted field line is described by:

equation image

REL is the radial distance at the crest of the distorted field line, and B(L, ζ) can be expressed the same way as in the case of an ideal dipole field:

equation image

[60] For the derivation of the transverse scales, dy and dq, we ignore any curvature of the meridional planes due to a concentration of high-beta plasma in the midnight sector and obtain:

equation image
equation image

[61] Here we made use of the flux conservation: B(L, ζ)dydq = const. Equations (32) through (35) provide a full description of the modified dipole field, on which we will base our high beta corrections. Figure 6 shows the distortion of the field lines as a function of βeq for constant ionospheric foot points. In comparison with more realistic near-Earth tail models [Tsyganenko, 2002], the compression from the side of the tail lobes is missing and the flux tube volume becomes somewhat too large.

Figure 6.

Examples of field lines in the distorted dipole geometry adopted for the model.

[62] We can now repeat the previous derivations of equation image, By, and Wm with the modified dipole geometry. Since the modifying factor, (1 + βeq(L)/b2(ζ))n, also contains an L-dependence, the differentiation, equation image, would introduce somewhat unhandy expressions. However, we will make nowhere use of the L-dependence other than through the term FL, as defined by equation (18), whose sign determines whether j is up or down. The main impact of the distorted dipole geometry is on energy content and Alfvén transit time. Therefore, we neglect such geometrical contributions and maintain only the dominant ones arising from p0(L) and the term BsL−3. The thus simplified expression for equation image becomes:

equation image

[63] With this relation one derives readily the modified expressions for j, By, and Wm and for the auxiliary functions.

equation image

F1(βeq, ζ) is the now modified auxiliary function of equation (17):

equation image

[64] The further derivations can be found in Annex 1. For subsequent use, we will restrict ourselves to the expressions of the key quantities above the ionosphere:

equation image
equation image
equation image

with:

equation image

[65] In contrast to the corresponding expressions in sections 3 and 4, the distorted dipole field leads to reasonable beta-dependences of the key physical quantities. The ζ- and βeq-dependences of F1(βeq, ζ) and F2(βeq, L, ζ) can be found in Annex 1. Figure 7 shows the beta-dependent expressions, βeq2 · equation image · F1(βeq, ζion) and βeq4 · equation image · F2(βeq, ζion), which determine the magnitudes of j, By, and Wm just above the ionosphere. They differ from the preceding equations by a factor (1 + βeq)−1/3 which arises from FL(L), as demonstrated below (equations (51) and (71)). Even though our distorted dipole field does not depict too well the field derived from databased numerical models [e.g., Tsyganenko, 2002], Figure 7 shows clearly that a beta of the order of unity and slightly above, is required for generating realistic quantities. The relations (39) to (41) constitute the basis of our auroral arc model.

Figure 7.

The βeq-dependent factors in the expressions for By,arc and equation imagearc according to equations (52) and (73).

6. Auroral Arcs

[66] With the preparations of the preceding sections we can now address the central subject of this paper, the auroral arc. A key ingredient is the introduction of jumps or ledges in the radial pressure profile, discussed in section 2.3 (Figure 4), as sources of the concentrated field aligned currents above an arc. The origin of such ledges is not subject of this paper, but one can easily imagine that they are a consequence of intermittent or pulsed plasma injections into the midnight sector of the outer magnetosphere and that the resulting inhomogeneties are carried with and stretched by the convective flow. It is in this sense that one may call auroral arcs of the here-discussed type a “by-product of magnetospheric convection”.

[67] Given a narrow current sheet, the arc model rests on three postulates, which have been discussed in detail in sections 2.3 and 2.4:

[68] 1. The differential magnetic energy content of the auroral current sheet and an equal amount supplied by the source plasma are released within four Alfvén transit times, τA.

[69] 2. The corresponding energy (Poynting) flux into the auroral acceleration region is largely converted into kinetic energy of the primary auroral particles.

[70] 3. The high-pressure side of the ledge is being eroded, the ledge moves into the region of higher shear stresses.

[71] In converting these postulates into concrete equations we will proceed in five steps. First, we address the auroral current sheet, then the energy conversion and its consequences, and thirdly condense the results into a set of simple and transparent physical relations. They are subsequently evaluated for a selected L-value and a consistent set of ambient parameters. Finally, we address the convection, in which the auroral arcs are imbedded.

6.1. Auroral Current Sheet

[72] The key aspect of this theory of pressure-driven auroral arcs is that gradients or ledges in the radial pressure profile constitute the sources of intensified field-aligned currents, strong enough to initiate the auroral energy conversion process in the low-altitude magnetosphere. The free energy resides in the excess of stored magnetic energy on the high-pressure side of the ledge, which is to be emptied in the stress release and energy conversion process. Therefore, we must introduce the radial pressure gradient at the ledge into the above derived expressions for j, By and Wm. Fortunately, by assuming pressure isotropy, we could essentially separate the L- from the ζ-dependence by introducing the function FL(L). Strictly speaking, the function F1(βeq, ζ) of equation (38) depends on L as well. But it contains the large-scale field geometry, on which the small pressure jump has little effect.

[73] One condition on the size of the pressure jump across the ledge must be that j clearly exceeds the ambient current density. This means that the pressure gradient contained in the function FL(L) must have a local spike. Without specifying the contour of this radial ledge in the magnetospheric pressure distribution (see Figure 4), we only introduce the relative pressure jump, δ, and write:

equation image

with:

equation image

[74] We saw before that the square bracket must be of order unity. Furthermore, with an arc width of order 10 km, corresponding to ΔLarc ≈ 0.12 for L = 10, LLarc can be of the order of 102. So we require only δ ≥ 0.01. With equation (39) we can then express the upward field aligned current above an auroral arc approximately as:

equation image

[75] The factor in brackets would appear the same way in the equation for the full ζ-dependence of j (equation (37)).

6.2. Auroral Energy Conversion

[76] The energy flux into the auroral acceleration region stems from the difference in magnetic energy content upstream and downstream of the moving current sheet. According to equation (21) this difference is

equation image
equation image

where equation imageequation image〉 is the average of this ratio over the flux tube integral of the magnetic energy density. The separability of the L-dependence allows to carry the ledge factor in equation (44) into the expression for By. Combining equations (40) and (44) we obtain:

equation image

[77] With equation image−1equation image 1, one finds 〈equation image〉 ≈ δ and

equation image

[78] It has to be noted that the factor of 2 stems from the assumption of a small pressure jump (δ ≪ 1).

[79] As discussed in section 2.4, the energy liberation process takes about 4τA, hence:

equation image

[80] This relation, which is the foundation of our auroral arc model, takes into account that not only magnetic energy is being released, but also internal energy of the source plasma (E · j < 0) as a consequence of the stress relief process. Thus the total energy available should be about twice Wm(ζion). On the other hand, about one third of that is reinvested in the return flow on the rear side of the current sheet.

[81] Before we address the quantitative implications of equation (48), we go a little further in specifying the parameter wup, which enters the definition of the magnetic perturbation field, By. In equation (22) we defined By by multiplying an intermediate value of the upward field-aligned current with the equivalent width of the total upward current sheet. We can proceed another way and integrate the maximum transverse current, j, over the length of a field line:

equation image

[82] This integrated transverse current must eventually enter the ionosphere and must be identical, by magnitude, with the height-integrated ionospheric closure current:

equation image

[83] Using equations (14) and (24) with the appropriate high-beta modifications, one obtains:

equation image

[84] With L = 8, corresponding to ζion = 0.9343, we find: wup · ∣FL(L)∣ ≈ 607 km · equation image, a not unreasonable number. The greatest advantage of the preceding manipulation is however that we get rid of an explicit evaluation of FL(L) and replace it by a geometric quantity.

[85] A first quantitative evaluation of equation (48) may be in place. Replacing in equation (41) the product wup · FL(L) by the above, choosing τA = 32 sec (L = 8) and setting ℓp = 5RE, we find:

equation image

[86] If we choose δ = 0.1, i.e., a pressure jump of 10% above the arc, we receive for βeq = 1: equation imagem = 2.6 mW/m2, for βeq = 2: equation imagem = 9.6 mW/m2, and for βeq = 3: equation imagem = 42 mW/m2. Such energy fluxes are not uncommon in active structured auroral arcs. The high sensitivity to the equatorial beta value shown in Figure 7 is clearly seen.

[87] Equation (48) expresses the energy stored in a flux tube of unit cross-section in the ionosphere plus internal energy from the source plasma to be released within 4τA. Now we apply postulate 2 and assume that the whole liberated energy is converted into kinetic energy of auroral electrons and ions. This is an overestimate, since even with perfect matching between the incoming Poynting flux and the conversion process some fraction (estimated to be of order one third) of the available energy is to be re-invested into the magnetic field. However, in view of the uncertainty with regard to the equatorial beta values, we ignore this minor correction.

[88] The energy conversion is adopted from the well-known and often successfully applied theory of Knight [1973], Fridman and Lemaire [1980], and Lyons [1980] in its simplest version, by invoking the existence of a quasi-Ohm's law:

equation image

[89] A parallel potential drop, Φ, arises, if the current density, due to the magnetic field convergence, becomes so high that acceleration of the current carrying particles is needed in order to maintain the continuity of the current imposed from above by the plasma dynamics (here, the pressure force). We will have to check later whether the field-aligned current density implied by postulate 2 actually falls into the range of magnitudes, where the simple quasi-Ohm's law for large mirror ratios between the source plasma and the energy conversion region applies [Lyons, 1980].

[90] However, the field-line convergence is not the only cause for the parallel potential drop. It appears to consist of two parts, a distributed one at medium altitudes, where the mirror effect dominates, and a rather localized one at the interface between the topside ionosphere. This transpires from the observations of high E by Hull et al. [2003] and of oblique double layers by Ergun et al. [2004]. The 1-D simulations of Ergun et al. [2002] also exhibit the existence of two different types of parallel potential drops. Since there is no simple way to separate between the two contributions and since observations exhibited general agreement between experimental and theoretical values of K, we will in the following work with the theoretical definition. Recent particle-fluid simulations by Vedin and Rönnmark [2006] have shown that the Knight's current-voltage relation is approached within 200 s, during which the magnetospheric electrons are being strongly heated.

[91] The conductance, K, after the above references is defined as:

equation image

nH is the number density of the hot electrons in the source region and ceH their thermal velocity. The other quantities have the conventional meaning. If the current is prescribed, as is the case in our model, the inverse of K becomes the crucial quantity. It can be seen as the impedance exerted on the current flow due to the mirror force. Typical values of K fall into the range from 10−10 S/m2 to 4 · 10−9 S/m2 [Olsson et al., 1998; Haerendel, 1999].

[92] As discussed in detail in section 2.4, we will pursue the ideal case that the downward Poynting flux flows with Alfvén speed into the energy conversion region or fracture zone and does not suffer from substantial reflections. This means that wave impedance and quasi-Ohmic resistance inside the potential drop match [Lysak, 1981; Vogt and Haerendel, 1998]. The resulting relation:

equation image

provides an expression for the width, warc, of an auroral arc or, better, of the auroral current sheet or an ‘inverted-V event’ in the original definition of Frank and Ackerson [1971]. Optical structures can be substantially thinner [Haerendel, 1999]. We insert equation imagearc from equations (48) and (41) and By(ζion) from equation (40) and obtain:

equation image

with:

equation image

for the whole range of beta values covered in Figure 7. As first found by Lysak [1981], (μ0VAK)−1/2 is the characteristic scale for matching the wave impedance, μ0VA, of the incoming Poynting flux with the end resistance, K−1, of the energy conversion region. As we are dealing here with flux tube integrated quantities, VA is quite appropriately replaced by REL/τA [cf. Haerendel, 1999]. δ appears in this context as a relative measure of the incoming Poynting flux. For a first orientation what kind of quantities are implied by our model, we evaluate warc by choosing K = 0.6 (μA · m−2)/kV, τA = 32 sec and L = 8. With δ = 0.1 this gives an arc width of 10 km, a very realistic value.

[93] Our next step is to relate equation image to the energy inflow equation imagearc via the quasi-Ohm's law (equation (53)):

equation image

[94] Using warc from equation (56) we get:

equation image

[95] With the previously chosen values for the external parameters and equation (51), equation image = 1.25 kV for βeq = 1 and Φ = 2.9 kV for βeq = 2. Also these values fall into the observed range. We notice again the very sensitive dependence of Φ as well as of equation imagearc on βeq. Although our model for the high-beta field distortions exaggerates the flux tube volume and thereby the stored energy, this appears to be a significant finding which will be discussed below.

[96] Hidden in the writing of ϕ is the somewhat surprising result that ϕ depends on K−1/2 (through warc and equation (56)), rather than on K−1, as suggested by the quasi-Ohm's law (equation (55)). The reason is that j is not an independent quantity, but depends already on K1/2 by the matching condition of Poynting flux and energy conversion rate.

[97] Finally, it follows from the preceding treatment, in particular from postulate 3, that the stress relief front propagates relative to the plasma frame with the speed:

equation image

[98] This manifests itself as a proper motion of auroral arcs in the plasma frame, which has been found to be of the order of 100 m/s [Haerendel et al., 1993; Frey et al., 1996]. The magnitudes implied by equation (60) are quite consistent with these observations.

6.3. Summary of Crucial Parameters

[99] In summing up the above-derived relations for the basic auroral parameters, we introduce the auxiliary magnetic perturbation field strength,

equation image

[100] It is about twice the maximum horizontal perturbation due to the large-scale current system in the ionosphere. For βeq = 2.0, equation image = 1.27 · 103 nT. With the help of equation image one can express the other quantities as follows:

equation image
equation image
equation image
equation image

with Γ in equation (57).

[101] The simplicity of the above relations is striking. Once equation image has been calculated, the other quantities follow from transparent combinations of the integral Alfvén speed, REL/4τA, with equation image, warc, and the smallness parameter, δ. While equation image contains the information of the global magnetospheric-ionospheric current circuit, warc contains essentially the ratio of mirror impedance, K−1, and wave impedance, μ0REL/4τA, in its integral form. The main accomplishment of this model is the identification of the energy reservoir that is tapped by the auroral process, namely the differential magnetic energy between the sheared fields on either side of the auroral current sheet. By introducing postulate 1, as expressed in equation (48), we not only couple the state of the magnetosphere with the energy conversion rate above the auroral arc, but also circumvent the tricky subject of the interaction of an Alfvén wave with the ionosphere. Perfect matching is the postulate underlying the quasi-stationarity of the model. Instead of the local Alfvén speed the integral speed, REL/2τA, appears.

[102] There is a further advantage. Instead of calculating the auxiliary magnetic perturbation field strength, equation image, from equation (61), with its underlying mathematical simplifications, one can choose a value for the local ionospheric perturbation field, By, and determine equation image from:

equation image

[103] So, one can treat the characteristics of an arc independently of concrete knowledge of the large-scale structure of the magnetosphere, by choosing By and searching for the value of δ giving the best fit to the observed properties. Or one determines δBy,arc, equation image, and equation imagearc directly from satellite data and checks the consistency of

equation image

with the observations. However, here one should remember that our model deals with single straight arcs imbedded in a longitudinal convection system.

6.4. Numerical Values

[104] In order to convey an impression of the capability of the above set of physical relations to generate realistic values for the characteristics of an auroral arc, we determine first a set of ambient parameters for the definition of conductance, K, and Alfvénic transit time, τA, consistent with the chosen values of βeq. From equation (30) we derive:

equation image

[105] Since L measures the actual equatorial distance of a distorted field line, we keep the ionospheric latitude fixed (sin λ = ζion) and derive L from:

equation image

[106] For the calculation of K, which we take literally as in equation (54), we need electron density and temperature in the source region. To this end we make use of the observation [Christon et al., 1991; Baumjohann, 1993] that the ratio of ion and electron temperatures in the near-Earth plasma sheet is typically 7, i.e.,

equation image

[107] Having chosen n and Te for a given value of βeq, we also calculate τA from:

equation image

[108] This relation contains the assumption of constant density along a field line above the ionosphere. A convenient expression for the auxiliary magnetic perturbation, equation image, can be derived from equations (51) and (61):

equation image

and with equation (57):

equation image

[109] Table 1 contains a set of environmental parameters for a range of beta values and a constant magnetospheric plasma density, ne = 0.5 cm−3, calculated for ζion = 0.9343, i.e., λion = 69.1°, and lp = 5 RE.

Table 1. Ambient Parameters
βeqLequation image, nTTe, keVτA, sK, μA · m−2/kV
0.38.2217.470.4217.30.714
1.08.83166.90.5923.80.602
2.09.45597.10.5235.90.639
3.09.9112570.4447.10.695
5.010.6132190.3376.70.809
10.011.73116300.20146.31.047

[110] With these parameters and δ = 0.1, one derives from the relations (56) and (60)()()()()(65) the auroral characteristics contained in Table 2.

Table 2. Auroral Characteristics for δ = 0.1
βeqwarc, kmvn, m/sΦ, kVequation imagearc, mW/m2j, μA/m2
0.36.7961.840.020.0
1.07.9861.471.120.89
2.09.5664.3111.92.76
3.010.2547.7942.15.4
5.011.63815.0181.612.1
10.013.42336.3137537.9

[111] The prime conclusion to draw from Table 2 is that for beta between 1 and 5 and pressure jumps of order 10% in the source plasma, one obtains quite realistic values for the arc characteristics. This means that already 10% of the overall energy content of a flux tube is sufficient to supply the energy dumped in an auroral arc during four Alfvénic transit times. Ten percent may even be a conservative assumption. There is, of course, a wide range of choices for the ambient parameters, such as plasma density and pressure gradient. One could also wonder whether the theoretical expression for the conductance, K, in equation (54) should be taken literally.

[112] Our example indicates that below about βeq = 1, the field-aligned currents are too small to cause aurora, and that above βeq = 5 the underlying concept of smooth plasma convection may not be fulfilled any more. The source plasma may be more dynamic and inertial forces may come into play. It is, however, remarkable that the magnitudes of j fall at all into the observed range of a few to several μA/m2 and thus fulfill the conditions for the validity of the Knight-relation in the approximation of Lyons [1980]. The way j is calculated, this is not a trivial result, but is another hint to the validity of the concept that what feeds the aurora it is stored magnetic energy.

[113] In discussing the above quantities one should also remember that the transverse current, jB,rad, might actually be higher by up to a factor of 3 than what was assumed in the derivation of equations (11a) and (11b). A stronger jB,rad would increase the magnitudes of equation image, By, and Φ by the same factor and equation imagearc by the square and thus shift the range of realistic beta values downward. Reducing this uncertainty would need numerical modeling of the magnetospheric plasma convection.

6.5. Comparison With Joule Dissipation

[114] It has been pointed out in the beginning of this paper that we are not postulating that the here presented theory applies to all structured auroral arcs. In the framework of this model, the auroral arc appears as a by-product of magnetospheric convection. The work done by the magnetospheric plasma pressure against the friction of the ionosphere/upper atmosphere is transferred downward via magnetic shear stresses. Thereby energy is temporarily stored in the stressed field and can be partially tapped by the auroral process, owing to the fact that in high-density current sheets field aligned potential drops can locally decouple the magnetic field from the ionosphere. For this reason it makes sense to compare the arc-related energy consumption with the Ohmic energy dissipation, equation imageion. Within the framework of our model, the ionospheric closure current is given by J = μ0−1 · By(ζion) and the convection electric field by E = ΣP−1 · J, with the height integrated Pedersen conductivity, ΣP. This implies:

equation image

[115] Comparison with equation imagearc yields:

equation image

[116] The second term on the r.h.s., which is the product of the integral Alfvén wave impedance and the height-integrated ionospheric conductivity, is a number of the order of 10. It shows how much more efficient the auroral energy conversion is in comparison ohmic dissipation. However, since the auroral arcs typically tap only a small fraction of the stored energy (δ ≪ 1), the auroral particle precipitation does not add much to the energy flow into the ionosphere, in spite of its spectacular optical consequences. Inside the arc, however, Σp is enhanced and the ohmic dissipation reduced. But the sum of auroral energy flux and Joule heating may just be the same as outside the arc. This theoretical result is quite consistent with comparisons of the measured energy depositions by Lu et al. [1995] and Baker et al. [1997].

[117] The near-equality of the two dissipation rates means that, unless δ ≫ 0.1, the time-scale of the ohmic dissipation is similar to that of the auroral energy conversion. Since this time-scale, of order (5–10)τA, is relatively short, there is a conflict with the concept implicitly underlying our model that a stress relief front is propagating through a quasi-static ionosphere. This conflict can be solved by assuming that a continuous, not necessarily constant energy supply exists along the auroral oval, maintaining the convective flow steady. In reality it is not even necessary that the arcs propagate into the region of higher shear stresses. Frey et al. [1996] also observed arcs with reverse proper motions, so as to enlarge the magnetic energy reservoir besides feeding the auroral particle flux. That requires an additional supply of energy. For reasons of simplicity, we will not try to incorporate such effects in this paper. It will be the topic of a follow-on paper.

7. Stress Relief Motion

[118] The key concept underlying the here presented approach to the macrophysics of auroral arcs is the decoupling of magnetic flux tubes from the ionosphere due to localized field aligned potential drops. This leads to a release of magnetic shear stresses and a conversion of the free magnetic energy into kinetic and thermal energy of the primary auroral particles. While the first is enabled by the transverse electric field components, the parallel components create the second. The typical U- or S-shaped potential distributions ideally characterizing the auroral acceleration region are consistent with this concept.

[119] Our treatment ignores what happens inside the current sheet by simply integrating over its width. Section 2.4 offers at least a qualitative discussion of its internal structure in terms of standing oblique Alfvén waves and of the upward progression of the stress relief initiated at low altitudes. In response to that, the internal energy of the source plasma decreases at the pressure ledge. In total this is a complex process whose proper treatment is outside the scope of this paper. However, we will discuss two of these aspects, the local response of the source plasma and the decoupling motions at the “feet” of the auroral field lines.

7.1. High Altitude Plasma Response

[120] As the stress relief front propagates into the region of higher shear stresses and progressively decouples the magnetospheric field from the ionosphere, the previously established balance of forces is being disturbed. What happens is that the Lorentz force, j × B, is being reduced because the inertial current of the front is oppositely directed to the transverse current above the front (cf. Figure 5). As a consequence, the plasma is being accelerated in the −∇p0 direction as sketched in Figure 8. Following the derivations of By(24), j(19a), and j ((14), (11b)), it can be seen that the reduction of the net transverse current in the source region is given by

equation image
Figure 8.

Principles of the stress release motions: At the leading edge of the current sheet, plasma and field overshoot in the “fracture zone”, but much of the previous magnetic tension becomes restored through the return flow. The generator plasma expands and transfers internal energy to Poynting flux adding to the magnetic stress release.

[121] The resulting acceleration of the plasma is approximately:

equation image

[122] We are deliberately unspecific about the altitude where this equation applies. It refers to the main source region in the neighborhood of the equator, i.e., β ≈ 1. More critical is that so far we did not consider the internal structure of the stress relief front and are thus unable to specify the temporal dependence of the reduction of the Lorentz force. What is written on the right-hand side of equation (77) is a net force after full exposure of the plasma to the stress relief current sheet. This takes an Alfvénic transit time, τA, to arrive. For this reason we apply a simple linear ansatz for the build-up of the accelerating force:

equation image

[123] At t = τA the total displacement, Δy, experienced by the plasma is:

equation image

where we have introduced the acoustic time-scale, τp = ℓp/cs, and the adiabatic constant, γ. For β > 1, vA < cs. This applies locally. But also for the integral quantity, τA, which is mainly determined by the low Alfvén speed at high altitudes, one can expect: τA > τp. Consistent with βeq = 2, we set τA/τp = equation image and find with γ = 5/3:

equation image

[124] At t > τA, the transverse current reduction begins to fade, the flow velocity slows down, and energy is carried downward as Poynting flux. The maximum displacement reached by the source plasma should then be about twice the value obtained from equation (80). Maintaining the previously chosen values, δ = 0.1, βeq = 2 and ℓP = 5RE, this amounts to about 0.4 RE. Considering the roughness of our estimates one can conclude that the displacement is of the order of δ times the pressure gradient scale. This result lends support to our estimate that the energy derived from the internal energy of the source plasma approximately equals the liberated magnetic energy.

[125] At this point, the return flow at low altitudes begins to be felt by the source plasma (cf. section 2.4). The electric field in the trailing part of the auroral acceleration region or fracture zone, which opposes the field direction in the leading edge (cf. Figure 5), is propagating upward and enforces a return into the direction of the undisturbed pressure gradient, ∇p0. However, since most of the energy supplied by the stress release is consumed in the auroral acceleration process, the upgoing Alfvén wave, which carries the electric field, is strongly damped. One should therefore expect the return flow to die out in the source plasma.

[126] The net displacement of the source plasma in the direction of diminishing pressure causes a net loss of internal energy, as (total) transverse current and electric field are opposed to each other, and the current carrying particles, dominantly ions, lose energy. This constitutes the postulated erosion process.

7.2. Relaxation of the Shear Stresses

[127] It is easy to see how the relaxation of shear stresses proceeds in the case of pressure driven and friction controlled quiet time convection. As the internal energy of the source plasma becomes exhausted, the field-aligned currents fade, the field lines straighten, the energy flow into the ionosphere dies out, and the convective motion (including that of the source plasma) comes to a halt. Much more difficult to visualize is the sequence of events in the case of the auroral energy conversion. It is accompanied by high transverse electric fields, creating fast counter-flows, while in the source region the stress release causes a net displacement in the direction of the pressure force. How can these two seemingly incongruent motions be made consistent?

[128] There is a simple conclusion that can be safely drawn. After passage of the auroral current sheet, the field-lines must be less strongly stressed than before, since a current sheet has passed through them. Using equations (33), (34), and (A1) we calculate the net ionospheric displacement of the same field lines prior to and after the stress release, by integrating along a field line the differential displacements projected into the ionosphere:

equation image

with:

equation image

[129] The value of the last integral is 0.856. With δ = 0.1, βeq = 2, ζion = 0.9343, L = 9.45, and ℓp = 5RE, as chosen above, we get a displacement near the ionosphere, δyion ≅ 62 km. How is that achieved?

[130] We must compare with the longitudinal excursions a field-line undergoes due to the decoupling from the ionosphere. The average electric field just above the fracture zone (projected into the ionosphere) is:

equation image

[131] It is very interesting that the electric field does not depend on the smallness parameter, δ. The reason is that the related plasma motions respond to the relief of the full, not the differential shear stresses. The magnitudes of the electric fields can be as high as 1 V/m, quite in agreement with satellite measurements above auroral arcs [e.g., Mozer et al., 1977]. Their directions are pointing toward the center of the current sheet (convergent fields). At the leading side of the moving arc the corresponding motions are in the direction of stress release and reversed on the trailing side. These motions manifest themselves visibly as motions of rays, folds or curls in active arcs [Davis, 1978]. Their magnitudes range between a few and about 25 km/s [Davis and Hicks, 1964; Haerendel et al., 1996]. Our above example (for βeq = 2 and L = 9.45) yields equation image = 910 mV/m and for the average flow velocity 16.8 km/s, both quantities projected into the ionosphere.

[132] Accordingly, the displacement during one half of the passage time of the arc, 2τA, is

equation image

[133] With 1204 km it is substantially farther than the net field line displacement δyion. Quite alike to the breaking of a stick of wood, which will overshoot before assuming the relaxed position, the decoupled field lines overshoot as well.

[134] But there is a decisive asymmetry in the plasma motions as well as in the brightness distributions. The potential is not exactly U-shaped. The leading side contains more pronounced and brighter structures (folds or curls) and is wider in latitudinal extent than the trailing side [Haerendel et al., 1996]. This has been worked into the cartoon of Figure 5 and is hidden in equation (83). It becomes apparent when we replace equation image in equations (82) and (83) by the transverse field component By(ζion), following equation (66):

equation image

[135] The shear stresses, and thus By, are stronger by the fraction δ in front of the moving arc than behind. This implies that the return flow causes a displacement smaller by

equation image

[136] Rewriting equation (81), so that By appears on the r.h.s., and evaluating the βeq-dependent terms, one finds a factor of 1.6 instead of 3.3 in equation (86). It means that the field lines, decoupled from the ionosphere, do not return to the point of origin, but displaced downstream twice as much as expected from the mere relaxation of the magnetic field. There is a simple reason for that. In the source plasma, the field lines are also being displaced by a distance equivalent to the low-altitude field relaxation. This must be added to the total displacement. Herewith we demonstrate, at least within the framework of our model, that the plasma motions corresponding to the converging electric fields in and above the acceleration region are indeed stress relief motions. Of course, this model has been built on this concept. But the fact that it also produces quantitatively the right auroral properties is a strong argument for auroral arcs being traces of fast magnetic stress release. It validates also the analogy with a fracture process, first proposed by Haerendel [1980].

[137] We can draw another conclusion. The stress relief process proceeds in the way that initially much more stored energy is being released than finally converted into energy of the auroral particles. Most of the energy is immediately reinvested into the system. With our choice of δ of order 0.1, the converted energy is only a small fraction of the latter. This insight is in contrast to the earlier conclusion of the author [Haerendel, 1994] that one third of the incoming Poynting flux is being reinvested.

[138] Figure 8 is a cartoon depicting the whole process, magnetic stress release and the additional input into the Poynting flux from the generator plasma, and Figure 9 [Haerendel, 1994] shows the ground tracks of the stress relief motions in the “fracture zone” viewed from the reference frame of the moving arc. Figure 10 summarizes the key quantities involved in the stress relief, the magnetic jump through the current sheet, δBy, the net displacement, dy, and the mean speed of auroral rays, vray, as a function of the plasma beta for the case contained in Tables 1 and 2.

Figure 9.

Plasma flow lines just above the fracture zone viewed along the magnetic field direction in the reference frame of the arc propagating with velocity, vn, (Figure 3 of [Haerendel, 1994]). The incomplete return flow on the trailing side is visible.

Figure 10.

Stress relief motions, vray, manifested by fast moving structures like auroral rays or folds, the resulting net displacement, δy, and the jump, ΔBy,arc, of the transverse magnetic field through the current sheet for various values of βeq.

[139] In the process of stress release, the plasma in or just above the acceleration region is moving along the arc direction one order of magnitude farther than the final displacement given in equation (86). This implies that our model arc must at least have the corresponding length of several 100 km to a few 1000 km, for the range of beta between 1 and 3. In other words, the fast stress relief motions are the reason for the wide longitudinal extent of auroral arcs. On the other hand, we get another argument against beta values well above 3. They would imply arc lengths that are hard to accommodate in the convection system.

[140] The above derived relations between shear stresses and stress relief motions contain implicitly the need for the existence of parallel electric fields with the property that Bxcurl{B(B · E)/B2} ≠ 0. It means that a section of a field-line, small or large, will not transform into a section of a neighboring field-line under the E × B drift, and E cannot be canceled by a space charge [Longmire, 1963]. Being set up by a quasi-stationary hydromagnetic wave and not being really electrostatic, the electric field in the fracture zone fulfills this condition. The most important issue is, however, that the current-voltage relation is such that only highly concentrated currents lead to such parallel fields on narrow transverse scales. The same sheet current with much larger transverse scale would not create decoupling, and thus would not extract energy from the stressed field other than by ohmic dissipation in the ionosphere.

7.3. Slippage

[141] The asymmetry of the transverse voltages on the leading and trailing sides of the arc implies that there must be an S-shaped contribution to the U-shaped potential. This is shown in Figure 5. Such S-shaped potentials have been frequently postulated in the literature [e.g., Chiu et al., 1981; Marklund, 1984; Marklund et al., 2007] without recognizing their true function. The reason for that is that in most previous studies of auroral electric fields the arc was (mostly implicitly) considered as not moving, i.e., “frozen” in the plasma. But when its proper motion is taken into account, there is automatically a front and a rear side, a current sheet is being traversed, and the “foot points” of the field lines must become displaced.

[142] However, there is a problem. The S-shaped contribution to the potential is distributed over the whole arc width underneath the auroral cavity, and the electric field is (cf equation (83)):

equation image

[143] The problem arises from the fact that it does not necessarily match with the ionospheric field, E⊥,ion. The ratio of the two fields is precisely the one given in equation (75) for the ratio of the particle energy flux inside an arc to the Joule dissipation in the conducting ionosphere, with one proviso. The height-integrated Pedersen conductivity must have the value applicable to the interior of the arc:

equation image

[144] Any arc of appreciable intensity enhances ΣP,arc by collisional ionization, and the ratio may well exceed unity, in particular for δ ≫ 0.1. In order to assess this enhancement, we use the relations derived by Robinson et al. [1987]:

equation image

[145] Wel is the mean energy of the auroral electrons in keV, and JE the auroral energy flux measured in mW/m2. We identify the first with equation image and the latter with equation imagearc. With the numbers from Tables 1 and 2, the ratio in equation (88) turns out to be of order 3 to 5 for δ = 0.1 and βeq = 2-3. The two fields do not match. If it were applied to the ionosphere, the high-altitude field would be short-circuited by the high ionospheric conductivity. The “foot point” motion above would be slowed down and the displacement reduced. The slippage of the magnetospheric field relative to the ionosphere would not be consistent with the requirements resulting from the transit through the auroral current sheet. It would draw into question the basic concept of our auroral arc model.

[146] To solve this problem we suggest that the S-shaped potential is not fully applied to the ionosphere, for the following reason. The interaction between the acceleration region, or auroral cavity, with the lower ionosphere must be highly structured, as noticed by the author in 1983 and elaborated later-on [Haerendel, 1983, 1988, 1994]. The cause lies in the combination of the arc's proper motion and the short traveltime of an Alfvén wave between the bottom of the auroral cavity and the region of current closure (E- and lower F regions), which is of the order of 1 sec. As the auroral current sheet propagates into the more highly stressed magnetic field, a strong transverse electric field will be applied propagating not only upward toward the source region, as discussed above, but also downward (cf. Figure 5). Here the signal will be reflected within such a short time that the front has progressed by only about 0.1 km or less. This will continue throughout the passage time of the arc. The exact consequences cannot be elaborated in the framework of this paper. Haerendel [1994] suggested the set-up of an electric interference pattern, largely damping the large-scale electric field applied from above. Dubinin et al. [1988] proposed the generation of strong Alfvénic turbulence by nonlinear wave-wave interactions. However, Alfvén waves with transverse scales of the order of 100 m must be highly damped owing to the finite parallel conductivity in the lower ionosphere [Lessard and Knudsen, 2001].

[147] Without trying to dig deeper into this complex subject, we conclude that the ionosphere above an auroral arc and in the lower acceleration region are structured by Alfvénic turbulence with scales down to 100 m and below. This turbulence is thought to provide an effective shielding of the S-shaped potential contribution from the highly conducting lower ionosphere. At this point it is just a conjecture that requires theoretical work and analysis of relevant electric field data.

[148] In summary, our model implies the existence of two slippage zones. One is the auroral cavity or “fracture zone”, where the liberated energy is temporarily converted into the kinetic energy of the fast motions along the arc and finally consumed by the particle acceleration process or re-invested in the return flow. The other lies below the fracture zone in the upper ionosphere and enables the net slippage of the magnetic field as required by the reduction of the shear stresses.

8. Summary

[149] The here presented analytical model of auroral arcs is based on the concept that the energy converted in the auroral acceleration process is derived from the energy stored in magnetic shear stresses and internal energy of the generator plasma in the high-latitude magnetospheric convection system. Longitudinal pressure forces act as drivers of the convection. Based on an isotropic pressure model and a simplified expression for the current divergence, a large-scale magnetospheric-ionospheric current system is derived, for which the magnetic energy content of the convecting flux tubes can be easily calculated. A fundamental feature of the model is the existence of ledges of the radial pressure distribution, elongated in longitude by the convection. They constitute locations of enhanced divergences of the transverse currents and are thus sources of the auroral current sheets. The auroral process in this model is the release of excess magnetic energy stored in these current sheets and of internal energy of the source plasma owing to the decoupling from the ionosphere by parallel potential drops and their conversion into kinetic energy of the primary auroral particles. It is postulated that the magnetic energy release is completed within a few Alfvén transit times. The conversion process itself is simply described by a current-voltage relation. As a consequence of the energy release at the pressure ledge in the source plasma, ledge and auroral current sheet propagate into the more strongly sheared field region. This imparts a proper motion to the auroral arc.

[150] On the basis of these ingredients, a simple set of physical relations is derived for sheet current (magnetic perturbation field), energy flux carried by the auroral electrons, field-aligned potential drop, width of current sheet or arc, and proper motion (equations (56), (60)(65)). There are three basic system parameters. Most important is an auxiliary magnetic perturbation field, equation image, into which the pressure model, the distorted dipole geometry, and the properties of the large-scale current system are condensed. The second is an integral Alfvén speed, REL/2τA. The third is the conductance parameter, K, in the quasi-Ohm's law, which appears here as the inverse, or mirror resistance. These three system parameters are calculated in accordance with common knowledge of the magnetosphere. There are only two free parameters, the relative magnitude, δ, of the pressure jump at the ledge and the plasma beta of the source region. They appear in various combinations in a set of simple and transparent physical relations. With δ of order 10−1 and βeq between 1 and 5 they produce realistic quantities for all characteristic parameters of isolated auroral arcs.

[151] This leads to some obvious conclusions. (1) The auroral arcs of this model are by-products of magnetospheric convection and consume only a small fraction of the energy stored in the convective system. (2) Typical auroral energy fluxes require a plasma beta above unity for an effective generator of auroral currents. (3) The narrow width of arcs implies a matching of the Alfvén wave impedance and the effective resistance experienced by the field-aligned current in the auroral acceleration region. (4) Proper motions of the arcs with respect to the ambient plasma are a consequence of the magnetic energy release and therefore a fundamental property. (5) The fast plasma motions, enabled by the decoupling from the ionosphere, are indeed stress relief motions. The model further provides, for the first time, an interpretation of the significance of the S-shaped contribution to the total electric potential in the acceleration region. It corresponds to the net displacements of the flux tubes that are consistent with the stress reduction during the transit of current sheet and arc. The mismatch of this potential with the one determined by ionospheric current and (enhanced) conductivity is thought to be overcome by damping in the “interference region” below the auroral acceleration region.

[152] The simplicity of such a model for a complex plasma phenomenon is achieved by certain sacrifices of generality and mathematical rigor. The most important ones are: (1) the lack of a self-consistent formulation of the undisturbed equilibrium of plasma and magnetic pressures in the presence of convection; (2) the implicit assumption of stationarity of the convection system; (3) the restriction to only longitudinal pressure forces and meridional feeder currents; and (4) some mathematical simplifications in deriving the divergence of the transverse current. All of these deficiencies have been made visible in the text and motivated. None of them appears to be of the kind as to invalidate the conclusions of the model. For instance, a different choice of the force balance in the generator region would only change some numerical factors that could be absorbed by slightly different values of βeq.

[153] The here presented model deals with isolated arcs imbedded in the magnetospheric convection system. But even though it may also be applicable to more complex situations, a host of unexplained phenomena remains, like the origin of multiple arcs, of arc splitting, of spirals, torches, folds, curls and the like. The contribution of this paper, irrespective of the idealizations applied, may be to spread the recognition that magnetic stress release is a key process in creating the beauty and the sometimes breath-taking dynamics of auroral displays.

Appendix A:: Latitude Dependence of Current and Energy

[154] The expressions for By(L, ζ) and Wm(L, ζ) in the distorted dipole field follow from the definitions in equations (21) and (22) and equations (30) to (36):

equation image
equation image

[155] The two auxiliary functions, F1(βeq, ζ) and F2(βeq, ζ), defined in section 5, are displayed in Figures A1 and A2. They depend strongly on beta. Referring to equations (32) and (34), one sees that the function F1(βeq, ζ) describes the growth of By(L, ζ) and J(L, ζ) along a field line, projected to the ionosphere. The lower sections of the converging flux tube (λ > 170) only concentrate the current, but add little because of the rapid decrease of the gradient and curvature currents with decreasing beta. Except for a constant factor, F2(βeq, ζ) is the energy stored in a flux tube above the position r(L, ζ). Comparison of the functions F1 and F2 for the same βeq shows that the stored magnetic energy still grows toward lower altitudes, although little new field-aligned current is added for ζ > 0.3.

Figure A1.

Auxiliary function F1(βeq, ζion) from equation (38) as function of βeq.

Figure A2.

Auxiliary function F2(βeq, ζion) from equation (42) as function of βeq.

Acknowledgments

[156] This work was in part performed while visiting the Space Sciences Laboratory of the University of California, Berkeley. I would like to express my thanks to Profs. Robert P. Lin and Forrest S. Mozer for their outstanding hospitality and support of my research and to all my colleagues at SSL for many stimulating discussions.

[157] Wolfgang Baumjohann thanks Christopher C. Chaston and another reviewer for their assistance in evaluating this paper.

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