Engineering topics

Farrugia, C. J.

Publications and source records attributed to Farrugia, C. J..

At least 19 records

NASA NTRS2016

Magnetospheric Multiscale Mission Observations and Non-Force Free Modeling of a Flux Transfer Event Immersed in a Super-Alfvenic Flow

We analyze plasma, magnetic field, and electric field data for a flux transfer event (FTE) to highlight improvements in our understanding of these transient reconnection signatures resulting from high-resolution data. The approximate 20 s long, reverse FTE, which occurred south of the geomagnetic equator near dusk, was immersed in super-Alfvnic flow. The field line twist is illustrated by the behavior of flows parallel perpendicular to the magnetic field. Four-spacecraft timing and energetic particle pitch angle anisotropies indicate a flux rope (FR) connected to the Northern Hemisphere and moving southeast. The flow forces evidently overcame the magnetic tension. The high-speed flows inside the FR were different from those outside. The external flows were perpendicular to the field as expected for draping of the external field around the FR. Modeling the FR analytically, we adopt a non-force free approach since the current perpendicular to the field is nonzero. It reproduces many features of the observations.

Farrugia, C. J.↗

NASA NTRS2014

A Vortical Dawn Flank Boundary Layer for Near-Radial IMF: Wind Observations on 24 October 2001

We present an example of a boundary layer tailward of the dawn terminator which is entirely populated by rolled-up flow vortices. Observations were made by Wind on 24 October 2001 as the spacecraft moved across the region at the X plane approximately equal to −13 Earth radii. Interplanetary conditions were steady with a near-radial interplanetary magnetic field (IMF). Approximately 15 vortices were observed over the 1.5 hours duration of Wind's crossing, each lasting approximately 5 min. The rolling up is inferred from the presence of a hot tenuous plasma being accelerated to speeds higher than in the adjoining magnetosheath, a circumstance which has been shown to be a reliable signature of this in single-spacecraft observations. A blob of cold dense plasma was entrained in each vortex, at whose leading edge abrupt polarity changes of field and velocity components at current sheets were regularly observed. In the frame of the average boundary layer velocity, the dense blobs were moving predominantly sunward and their scale size along the X plane was approximately 7.4 Earth radii. Inquiring into the generation mechanism of the vortices, we analyze the stability of the boundary layer to sheared flows using compressible magnetohydrodynamic Kelvin-Helmholtz theory with continuous profiles for the physical quantities. We input parameters from (i) the exact theory of magnetosheath flow under aligned solar wind field and flow vectors near the terminator and (ii) the Wind data. It is shown that the configuration is indeed Kelvin-Helmholtz (KH) unstable. This is the first reported example of KH-unstable waves at the magnetopause under a radial IMF.

Magnetosheath↗

NASA NTRS2012

Deep Solar Activity Minimum 2007-2009: Solar Wind Properties and Major Effects on the Terrestrial Magnetosphere

We discuss the temporal variations and frequency distributions of solar wind and interplanetary magnetic field parameters during the solar minimum of 2007 - 2009 from measurements returned by the IMPACT and PLASTIC instruments on STEREO-A.We find that the density and total field strength were significantly weaker than in the previous minimum. The Alfven Mach number was higher than typical. This reflects the weakness of magnetohydrodynamic (MHD) forces, and has a direct effect on the solar wind-magnetosphere interactions.We then discuss two major aspects that this weak solar activity had on the magnetosphere, using data from Wind and ground-based observations: i) the dayside contribution to the cross-polar cap potential (CPCP), and ii) the shapes of the magnetopause and bow shock. For i) we find a low interplanetary electric field of 1.3+/-0.9 mV/m and a CPCP of 37.3+/-20.2 kV. The auroral activity is closely correlated to the prevalent stream-stream interactions. We suggest that the Alfven wave trains in the fast streams and Kelvin-Helmholtz instability were the predominant agents mediating the transfer of solar wind momentum and energy to the magnetosphere during this three-year period. For ii) we determine 328 magnetopause and 271 bow shock crossings made by Geotail, Cluster 1, and the THEMIS B and C spacecraft during a three-month interval when the daily averages of the magnetic and kinetic energy densities attained their lowest value during the three years under survey.We use the same numerical approach as in Fairfield's empirical model and compare our findings with three magnetopause models. The stand-off distance of the subsolar magnetopause and bow shock were 11.8 R(sub E) and 14.35 R(sub E), respectively. When comparing with Fairfield's classic result, we find that the subsolar magnetosheath is thinner by approx. 1 R(sub E). This is mainly due to the low dynamic pressure which results in a sunward shift of the magnetopause. The magnetopause is more flared than in Fairfield's model. By contrast the bow shock is less flared, and the latter is the result of weaker MHD forces.

solar minimum↗

NASA NTRS2012

An Analysis of the Origin and Propagation of the Multiple Coronal Mass Ejections of 2010 August 1

On 2010 August 1, the northern solar hemisphere underwent significant activity that involved a complex set of active regions near central meridian with, nearby, two large prominences and other more distant active regions. This activity culminated in the eruption of four major coronal mass ejections (CMEs), effects of which were detected at Earth and other solar system bodies. Recognizing the unprecedented wealth of data from the wide range of spacecraft that were available-providing the potential for us to explore methods for CME identification and tracking, and to assess issues regarding onset and planetary impact-we present a comprehensive analysis of this sequence of CMEs.We show that, for three of the four major CMEs, onset is associated with prominence eruption, while the remaining CME appears to be closely associated with a flare. Using instrumentation on board the Solar Terrestrial Relations Observatory spacecraft, three of the CMEs could be tracked out to elongations beyond 50◦; their directions and speeds have been determined by various methods, not least to assess their potential for Earth impact. The analysis techniques that can be applied to the other CME, the first to erupt, are more limited since that CME was obscured by the subsequent, much faster event before it had propagated far from the Sun; we discuss the speculation that these two CMEs interact. The consistency of the results, derived from the wide variety of methods applied to such an extraordinarily complete data set, has allowed us to converge on robust interpretations of the CME onsets and their arrivals at 1 AU.

Harrison, R. A.↗

NASA NTRS2011

Crater Flux Transfer Events: Highroad to the X Line?

We examine Cluster observations of a so-called magnetosphere crater FTE, employing data from five instruments (FGM, CIS, EDI, EFW, and WHISPER), some at the highest resolution. The aim of doing this is to deepen our understanding of the reconnection nature of these events by applying recent advances in the theory of collisionless reconnection and in detailed observational work. Our data support the hypothesis of a stratified structure with regions which we show to be spatial structures. We support the bulge-like topology of the core region (R3) made up of plasma jetting transverse to reconnected field lines. We document encounters with a magnetic separatrix as a thin layer embedded in the region (R2) just outside the bulge, where the speed of the protons flowing approximately parallel to the field maximizes: (1) short (fraction of a sec) bursts of enhanced electric field strengths (up to approximately 30 mV/m) and (2) electrons flowing against the field toward the X line at approximately the same time as the bursts of intense electric fields. R2 also contains a density decrease concomitant with an enhanced magnetic field strength. At its interface with the core region, R3, electric field activity ceases abruptly. The accelerated plasma flow profile has a catenary shape consisting of beams parallel to the field in R2 close to the R2/R3 boundary and slower jets moving across the magnetic field within the bulge region. We detail commonalities our observations of crater FTEs have with reconnection structures in other scenarios. We suggest that in view of these properties and their frequency of occurrence, crater FTEs are ideal places to study processes at the separatrices, key regions in magnetic reconnection. This is a good preparation for the MMS mission.

Farrugia, C. J.↗

NASA NTRS2005

Geotail Observations of the Spatial Dependence of Kelvin-Helmholtz Waves on an Inbound Passage through the Dusk Flank Boundary Layer

On August 1, 1998, the Geotail spacecraft made an inbound passage perpendicular to the dusk magnetopause at the dusk terminator when the interplanetary magnetic field had been very northward for more than 10 hours. Typical 3-minute-period Kelvin-Helmholtz waves were observed and the density in the boundary layer and magnetopause was observed to have an unusually high value near 5 /cc. Compressible MHD calculations using the measured values at Geotail yield substantial growth rates that support the idea that the magnetopause was Kelvin-Helmholtz unstable. In contrast to many previous events where a spacecraft remained in the boundary layer, this passage allowed study of how the waves varied with distance inward from the magnetopause. In a layer adjacent to the magnetosheath, rapid magnetic field fluctuations were seen with variations of at least 50 nT/s. Initially the boundary waves led to transitions between the magnetosheath and the fluctuating region with magnetosheath-like densities and tailward velocities, but as the spacecraft moved inward, the transitions were more likely to be between the fluctuating region and a hotter region with magnetosphere-like densities of 5kc. Gradually the velocity perturbations began to exhibit 360 degree rotations. Such rotations are similar to the vortices seen earlier by the ISEE spacecraft throughout the magnetotail which were suspected of being caused by Kelvin-Helmholtz instability of the boundary.

Fairfield, D. H.↗

NASA NTRS2003

Polar, Cluster and SuperDARN Evidence for High-Latitude Merging during Southward IMF: Temporal/Spatial Evolution

Magnetic merging on the dayside magnetopause often occurs at high latitudes. Polar measured fluxes of accelerated ions and wave Poynting vectors while skimming the subsolar magnetopause. The measurements indicate that their source was located to the north of the spacecraft, well removed from expected component merging sites. This represents the first use of wave Poynting flux as a merging discriminator at the magnetopause. We argue that wave Poynting vectors, like accelerated particle fluxes and the Walen tests, are necessary, but not sufficient, conditions, for identifying merging events. The Polar data are complemented with nearly simultaneous measurements from Cluster in the northern cusp, with correlated observations from the SuperDARN radar, to show that the locations and rates of merging vary. Magnetohydrodynamic (MHD) simulations are used to place the measurements into a global context. The MHD simulations confirm the existence of a high-latitude merging site and suggest that Polar and SuperDARN observed effects are attributable to both exhaust regions of a temporally varying X-line. A survey of 13 merging events places the location at high latitudes whenever the interplanetary magnetic field (IMF) clock angle is less than approximately 150 degrees. While inferred high-latitude merging sites favor the antiparallel merging hypothesis, our data alone cannot exclude the possible existence of a guide field. Merging can even move away from equatorial latitudes when the IMF has a strong southward component. MHD simulations suggest that this happens when the dipole tilt angle increases or when IMF B(sub X) increases the effective dipole tilt.

Maynard, N. C.↗

NASA NTRS1995

Magnetic flux rope versus the spheromak as models for interplanetary magnetic clouds

Magnetic clouds form a subset of interplanetary ejecta with well-defined magnetic and thermodynamic properties. Observationally, it is well established that magnetic clouds expand as they propagate antisunward. The aim of this paper is to compare and contrast two models which have been proposed for the global magnetic field line topology of magnetic clouds: a magnetic flux tube geometry, on the one hand, and a spheromak geometry (including possible higher multiples), on the other. Traditionally, the magnetic structure of magnetic clouds has been modeled by force-free configurations. In a first step, we therefore analyze the ability of static force-free models to account for the asymmetries observed in the magnetic field profiles of magnetic clouds. For a cylindrical flux tube the magnetic field remains symmetric about closest approach to the magnetic axis on all spacecraft orbits intersecting it, whereas in a spheromak geometry one can have asymmetries in the magnetic field signatures along some spacecraft trajectories. The duration of typical magnetic cloud encounters at 1 AU (1 to 2 days) is comparable to their travel time from the Sun to 1 AU and thus magnetic clouds should be treated as strongly nonstationary objects. In a second step, therefore, we abandon the static approach and model magnetic clouds as self-similarly evolving MHD configurations. In our theory, the interaction of the expanding magnetic cloud with the ambient plasma is taken into account by a drag force proportional to the density and the velocity of expansion. Solving rigorously the full set of MHD equations, we demonstrate that the asymmetry in the magnetic signature may arise solely as a result of expansion. Using asymptotic solutions of the MHD equations, we least squares fit both theoretical models to interplanetary data. We find that while the central part of the magnetic cloud is adequately described by both models, the 'edges' of the cloud data are modeled better by the magnetic flux tube. Further comparisons of the two models necessarily involve thermodynamic properties, since real magnetic configurations are never exactly force-free and gas pressure plays an essential role. We consider a polytropic gas. Our theoretical analysis shows that the self-similar expansion of a magnetic flux tube requires the polytropic index gamma to be less than unity. For the spheromak, however, self-similar, radially expanding solutions are known only for gamma equal to 4/3. This difference, therefore, yields a good way of distinguishing between the two geometries. It has been shown recently that the polytropic relationship is applicable to magnetic clouds and that the corresponding polytropic index is approximately 0.5. This observational result is consistent with the self-similar model of the magnetic flux rope but is in conflict with the self-similar spheromak model.

Farrugia, C. J.↗

NASA NTRS1995

Nonlinear evolution of magnetic flux ropes. 2: Finite beta plasma

In this second paper on the evolution of magnetic flux ropes we study the effects of gas pressure. We assume that the energy transport is described by a polytropic relationship and reduce the set of ideal MHD equations to a single, second-order, nonlinear, ordinary differential equation for the evolution function. For this conservative system we obtain a first integral of motion. To analyze the possible motions, we use a mechanical analogue -- a one-dimensional, nonlinear oscillator. We find that the effective potential for such an oscillator depends on two parameters: the polytropic index gamma and a dimensionless quantity kappa the latter being a function of the plasma beta, the strength of the azimuthal magnetic field relative to the axial field of the flux rope, and gamma. Through a study of this effective potential we classify all possible modes of evolution of the system. In the main body of the paper, we focus on magnetic flux ropes whose field and gas pressure increase steadily towards the symmetry axis. In this case, for gamma greater than 1 and all values of kappa, only oscillations are possible. For gamma less than 1, however, both oscillations and expansion are allowed. For gamma less than 1 and kappa below a critical value, the energy of the nonlinear oscillator determines whether the flux rope will oscillate or expand to infinity. For gamma less than 1 and kappa above critical, however, only expansion occurs. Thus by increasing kappa while keeping gamma fixed (less than 1), a phase transition occurs at kappa = kappa(sub critical) and the oscillatory mode disappears. We illustrate the above theoretical considerations by the example of a flux rope of constant field line twist evolving self-similarly. For this example, we present the full numerical MHD solution. In an appendix to the paper we catalogue all possible evolutions when (1) either the magnetic field or (2) the gas pressure decreases monotonically toward the axis. We find that in these cases critical conditions can occur for gamma greater than 1. While in most cases the flux rope collapses, there are notable exceptions when, for certain ranges of kappa and gamma, collapse may be averted.

Osherovich, V. A.↗

NASA NTRS1995

The response of ionospheric convection in the polar cap to substorm activity

We report multi-instrument observations during an isolated substorm on 17 October 1989. The European Incoherent Scatter (EISCAT) radar operated in the SP-UK-POLI mode measuring ionospheric convection at latitudes 71 deg Lambda - 78 deg Lambda. Sub-Auroral Magnetometer Network (SAMNET) and the EISCAT Magnetometer Cross provide information on the timing of substorm expansion phase onset and subsequent intensifications, as well as the location of the field aligned and ionospheric currents associated with the substorm current wedge. Interplanetary Monitoring Platform-8 (IMP-8) magnetic field data are also included. Evidence of a substorm growth phase is provided by the equatorward motion of a flow reversal boundary across the EISCAT radar field of view at 2130 MLT, following a southward turning of the interplanetary magnetic field (IMF). We infer that the polar cap expanded as a result of the addition of open magnetic flux in the tail lobes during this interval. The flow reversal boundary, which is a lower limit to the polar cap boundary, reached an invariant latitude equatorward of 71 deg Lambda by the time of the expansion phase onset. We conclude that the substorm onset region in the ionosphere, defined by the westward electrojet, mapped to a part of the tail radially earthward of the boundary between open and closed magnetic flux, the distant neutral line. Thus the substorm was not initiated at the distant neutral line, although there is evidence that it remained active during the expansion phase.

Lester, M.↗

NASA NTRS1994

Auroral activity associated with Kelvin-Helmholtz instability at the inner edge of the low-latitude boundary layer

Auroral activity occurred in the late afternoon sector (approx. 16 MLT) in the northern hemisphere during the passage at Earth of an interplanetary magnetic cloud on January 14, 1988. The auroral activity consisted of a very dynamic display which was preceded and followed by quiet auroral displays. During the quiet displays, discrete rayed arcs aligned along the geomagnetic L shells were observed. In the active stage, rapidly evolving spiral forms centered on magnetic zenith were evident. The activity persisted for many minutes and was characterized by the absence of directed motion. They were strongly suggestive of intense filaments of upward field-aligned currents embedded in the large-scale region 1 current system. Distortions of the flux ropes as they connect from the equatorial magnetosphere to the ionosphere were witnessed. We assess as possible generating mechanisms three nonlocal sources known to be associated with field-aligned currents. Of these, partial compressions of the magnetosphere due to variations of solar wind dynamic pressure seem an unlikely source. The possibility that the auroral forms are due to reconnection is investigated but is excluded because the active aurora were observed on the closed field line region just equatorward of the convection reversal boundary. To support this conclusion further, we apply recent results on the mapping of ionospheric regions to the equatorial plane based on the Tsyganenko 1989 model (Kaufmann et al., 1993). We find that for comparable magnetic activity the aurora map to the equatorial plane at X(sub GSM) = approx. 3 R(sub E) and approx. 2 R(sub E) inward of the magnetopause, that is, the inner edge of the boundary layer close to dusk. Since the auroral forms are manifestly associated with magnetic field shear, a vortical motion at the equatorial end of the flux rope is indicated, making the Kelvin-Helmholtz instability acting at the inner edge of the low-latitude boundary layer the most probable generating source.

Farrugia, C. J.↗

NASA NTRS1994

Cusp/cleft auroral activity in relation to solar wind dynamic pressure, interplanetary magnetic field B(sub z) and B(sub y)

Continuous optical observations of cusp/cleft auroral activities within approximately equal to 09-15 MLT and 70-76 deg magnetic latitude are studied in relation to changes in solar wind dynamic pressure and interplanetary magnetic field (IMF) variability. The observed latitudinal movements of the cusp/cleft aurora in response to IMF B(sub z) changes may be explained as an effect of a variable magnetic field intensity in the outer dayside magnetosphere associated with the changing intensity of region 1 field-aligned currents and associated closure currents. Ground magnetic signatures related to such currents were observed in the present case (January 10, 1993). Strong, isolated enhancements in solar wind dynamic pressure (Delta p/p is greater than or equal to 0.5) gave rise to equatorward shifts of the cusp/cleft aurora, characteristic auroral transients, and distinct ground magnetic signatures of enhanced convection at cleft latitudes. A sequence of auroral events of approximately equal to 5-10 min recurrence time, moving eastward along the poleward boundary of the persistent cusp/cleft aurora in the approximately equal to 10-14 MLT sector, during negative IMF B(sub z) and B(sub y) conditions, were found to be correlated with brief pulses in solar wind dynamic pressure (0.1 is less than Delta p/p is less than 0.5). Simultaneous photometer observations from Ny Alesund, Svalbard, and Danmarkshavn, Greenland, show that the events often appeared on the prenoon side (approximately equal to 10-12 MLT), before moving into the postnoon sector in the case we study here, when IMF B(sub y) is less than 0. In other cases, similar auroral event sequences have been observed to move westward in the prenoon sector, during intervals of positive B(sub y). Thus a strong prenoon/postnoon asymmetry of event occurence and motion pattern related to the IMF B(sub y) polarity is observed. We find that this category of auroral event sequence is stimulated bursts of electron precipitation that originate from magnetosheath plasma that has accessed that dayside magnetosphere in the noon or near-noon sector, possibly at high latitudes, partly governed by the IMF orientation as well as by solar wind dynamic pressure pulses.

Sandholt, P. E.↗

NASA NTRS1994

The solar wind flow along the subsolar line in the magnetic barrier and reconnection at the magnetopause

The magnetic barrier is a significant fraction of the magnetosheath region where magnetic forces react back on the plasma flow. We use a magnetohydrodynamic approach to calculate the solar wind flow around the magnetosphere. The use of so-called frozen-in coordinates, where the flow lines and the magnetic field are coordinate axes, is found to be most appropriate. Numerical solutions are presented for the problem of the flow around a blunt body. The results near the magnetopause are used as input parameters for a model of the reconnection of magnetic field lines based on shock-type structures.

Erkaev, N. V.↗

NASA NTRS1994

Observations in the sheath region ahead of a magnetic cloud and in the dayside magnetosheath during magnetic cloud passage

We present magnetic field and particle (protons and electrons) observations in the sheath region behind an interplanetary shock driven by a magnetic cloud, and in the magnetic cloud itself. We also discuss observations in the dayside terrestrial magnetosheath during cloud passage. We find that the region ahead of the cloud is in pressure balance. Further, throughout its extent (greater than 0.06 AU), the magnetic field strength is anticorrelated with the plasma density, with the latter decreasing steadily as the cloud is approached. This behavior is indicative of magnetic forces influencing the flow topology and highlights a large-scale breakdown of predictions based solely on gas dynamical considerations. We also study density structures inside the cloud which result in an undulating dynamic pressure being applied to the magnetopause causing it to oscillate with amplitudes of approximately 1-3 Re and period approximately 2h.

Farrugia, C. J.↗

NASA NTRS1994

Solar activity and coronal mass ejections on the western hemisphere of the Sun in mid-August 1989: Association with interplanetary observations at the ICE and IMP 8 spacecraft

During the 10-day period from August 12 to 21, 1989, a sequence of coronal mass ejections (CMEs) was observed above the west limb of the Sun by the Solar Maximum Mission (SMM) coronagraph. Most of these CMEs apparently originated in the vicinity of one particularly active region during its passage from near central meridian to behind the west limb of the Sun. We present observations made at 1 AU during this period by the International Cometary Explorer (ICE) (formerly International Sun Earth Explorer-3 (ISEE 3)) and Interplanetary Monitoring Platform (IMP 8) spacecraft which were separated by approximately 75 deg in heliolongitude. Following CMEs on August 12 associated with solar events at approximately W40 deg, IMP 8 (in Earth orbit) detected a strong shock followed by signatures in magnetic field, solar wind plasma, and energetic ion data which suggest that CME-related material ('ejecta') forming the shock driver engulfed the spacecraft. This spacecraft only observed weak shocks, and no ejecta, from later CMEs originating further west of the spacecraft. In contrast ICE, off the west limb at approximately W75 deg, observed the shock from the W40 deg event but failed to encounter the shock driver, whereas clear ejecta signatures were observed following events further west, closer to the spacecraft heliolongitude. The disappearance of these signatures (which include bidirectional energetic ion flows, bidirectional solar wind heat fluxes, quiet, enhanced magnetic fields and anomalously cool plasma) at IMP 8 and their emergence at ICE as the solar source region moved westward supports the association of such signatures with ejecta related to CMEs. The dual-spacecraft observations are also consistent with the conclusion of Richardson and Cane (1993) that ejecta at 1 AU typically extend approximately 50 degs in longitude from the solar source. Some plausible associations between particular intervals of ejecta signatures at ICE and individual CMEs are made. However, these associations are complicated by the large number of CMEs present, by intermittent ICE data coverage, and by uncertainties in the ejecta propagation speeds to the spacecraft.

Richardson, I. G.↗

NASA NTRS1993

Polytropic relationship in interplanetary magnetic clouds

High time-resolution data from the ISEE 3 and IMP 8 spacecraft are presented for the magnetic field and the proton and electron populations of a number of magnetic clouds, in order to investigate such clouds' thermodynamics and the relationship between their magnetic and thermodynamic structures. It is judged on the basis of these data that while the magnetic flield of the cloud expands, the ions are cooled. Hot electrons are trapped by the magnetic field in the magnetic cloud's core. These conditions are favorable for the generation of ion-acoustic waves.

Osherovich, V. A.↗

NASA NTRS1993

Simultaneous observations of solar MeV particles in a magnetic cloud and in the earth's northern tail lobe - Implications for the global field line topology of magnetic clouds and for the entry of solar particles into the magnetosphere during cloud passage

Simultaneous ISEE 3 and IMP 8 spacecraft observations of magnetic fields and flow anisotropies of solar energetic protons and electrons during the passage of an interplanetary magnetic cloud show various particle signature differences at the two spacecraft. These differences are interpretable in terms of the magnetic line topology of the cloud, the connectivity of the cloud field lines to the solar surface, and the interconnection between the magnetic fields of the magnetic clouds and of the earth. These observations are consistent with a magnetic cloud model in which these mesoscale configurations are curved magnetic flux ropes attached at both ends to the sun's surface, extending out to 1 AU.

Farrugia, C. J.↗

NASA NTRS1993

Nonlinear evolution of magnetic flux ropes. I - Low-beta limit

We study the nonlinear self-similar evolution of a cylindrical magnetic flux tube with two components of the magnetic field, axial and azimuthal. We restrict ourselves to the case of a plasma of low beta. Introducing a special class of configurations we call 'separable fields', we reduce the problem to an ordinary differential equation. Two cases are to be distinguished: (1) when the total field minimizes on the symmetry axis, the magnetic configuration inexorably collapses, and (2) when, on the other hand, the total field maximizes on the symmetry axis, the magnetic configuration behaves analogously to a nonlinear oscillator. Here we focus on the latter case. The effective potential of the motion contains two terms: a strong repulsive term and a weak restoring term associated with the pinch. We solve the nonlinear differential equation of motion numerically and find that the period of oscillations grows exponentially with the energy of the oscillator. Our treatment emphasizes the role of the force-free configuration as the lowest potential energy state about which the system oscillates.

Osherovich, V. A.↗