Mirror Force Research Articles

Contributions to the magnetospheric parallel electric field

[1] Upward field-aligned currents and their associated parallel electric fields couple the ionosphere to the magnetosphere. It is desirable to know how such a potential drop is distributed along the flux tube, what controls its variation, and how it is balanced by the plasma. By considering the motion of the ionospheric and magnetospheric electrons and ions, under the influence of electrostatic and magnetic mirror forces, a quasi steady state, quasi-neutral electric field distribution along the magnetic flux tube can be obtained. A feature of the potential profiles is the occurrence of a potential jump that splits the profile into three distinct regions: below the jump, within the jump, and above the jump. Within a kinetic framework, we analyze how the plasma velocity distributions evolve along the flux tube, taking into account ionospheric, magnetospheric, mirroring, and precipitating electron populations. By calculating the moments of the governing Vlasov equation, we ascertain what balances the parallel electric field (E∥) and how it is maintained, establishing a dynamical equilibrium. Our calculations show that (1) earthward of the jump E∥ ≈ −(p⊥/enB)∇∥B associated with the ionospheric electrons, except for at the base of the F region where p∥ contributions become more significant; (2) within the jump magnetosphere electrons dominate and E∥ ≈ −(1/en)∇∥p∥; and (3) above the jump mirroring magnetospheric electrons make a principal contribution of E∥ ≈ −(1/en)∇∥p∥, with a secondary contribution of −(p⊥ − p∥)∇∥B/(ne) becoming comparable beyond ≈3 RE. Additionally, we found that although the precipitating electrons carry the field-aligned current, it is the mirroring population that determines where E∥ is concentrated and hence where precipitating electrons are accelerated.

SUPPRESSION OF REFLECTED ELECTRONS BY KINETIC ALFVÉN TURBULENCE IN A QUASI-PERPENDICULAR SHOCK: PARTICLE-IN-CELL SIMULATIONS

Shock drift acceleration is one of the important mechanisms for electron acceleration associated with magnetic mirror reflection along the magnetic field in a quasi-perpendicular collisionless shock. We study the influence of a rippled shock surface in the in-plane magnetic field on the magnetic mirror reflection of electrons. Simulation results show that the number of reflected electrons reduces after generation of the rippled shock surface. Electric and magnetic wavenumber spectra of the generated fluctuations in the shock transition region indicate the existence of kinetic Alfven turbulence. The kinetic Alfven turbulence decreases the electron pitch angle by parallel scattering, which reduces the magnetic mirror force acting on the electrons. These results suggest that the shock-generated kinetic Alfven turbulence suppresses the magnetic mirror reflection of electrons during the shock drift acceleration.

A KINETIC MODEL OF SOLAR WIND GENERATION BY OBLIQUE ION-CYCLOTRON WAVES

The fast solar wind is generated by extended perpendicular ion heating in coronal holes, but the kinetic mechanism responsible for this heating has not been determined. One long-standing possibility is the resonant-cyclotron dissipation of ion-cyclotron waves, replenished from a turbulent cascade of interacting counter-propagating Alfven waves. We present results of a kinetic model for proton heating by the quasilinear resonant-cyclotron wave-particle interaction in a coronal hole. The resonant wave spectrum is taken as a power law in wavenumber, uniformly distributed in propagation direction between 0° and 60° with respect to the large-scale radial magnetic field. We obtain the steady-state solution of the kinetic guiding-center equation for the proton distribution in an expanding coronal hole, including the effects of large-scale forces of gravity, charge-separation electric field, Alfven wave ponderomotive force, and mirror force, along with the small-scale scattering from the wave dissipation. We find that plausible wave intensities can yield reasonable flow speeds and temperatures in the heliocentric radial range between 2 and 6 solar radii. We address the claim in earlier work that dissipation of parallel-propagating ion-cyclotron waves cannot provide enough acceleration and show that claim to be incorrect. We find that the combined action of the large-scale forces and the resonant-cyclotron scattering produces proton distribution functions with a characteristic structure: compressed in the sunward half of velocity space with a high-density shell separate from the origin, and relatively expanded in the anti-sunward half of velocity space. We suggest that qualitatively similar proton distributions would result from the kinetic evolution of any sufficiently effective perpendicular heating mechanism operating in an expanding coronal hole.

Magnetic mirror effects on a collisionless plasma in a convergent geometry

Several plasma thruster concepts, as well as ion engine chambers, use magnetic cusps to protect walls and to throttle electron flow to anodes. We present a kinetic model of the plasma in the vicinity of one cusp. Electrons, strongly confined by the electrostatic presheath and sheath, are assumed isotropic. Collisionless ions are either magnetically guided or completely nonmagnetized, thus bracketing conditions of interest. For magnetized ions, electrostatic and magnetic mirror forces compete, and the resulting self-consistent potential is found by imposing quasineutrality. A similar competition occurs for nonmagnetized ions, this time as a result of the convergence of equipotential lines. Analytical solutions are found for monoenergetic ions, and these are generalized to the case of an initially Maxwellian population, for which some numerical iteration is required. The presheath potential drop is in all cases of the order of 0.6–0.75 times the electron temperature, and ions enter the sheath at a sonic velocity, according to Bohm’s criterion. Contrary to intuition, the cusp does not reduce the ion flux (per unit area) to the wall, only the size of the wall area section that carries this flux by virtue of its connection to the distant plasma. These kinetic results are verified by checking the conservation of relevant moments of the ion distribution, including two new quantities that generalize the average magnetic moment and the total ion enthalpy by accounting for the nonzero ion heat fluxes.

Anomalous transport effects on the parallelEfield in downward auroral current regions of the Earth's magnetosphere

[1] The physical processes that determine the fluid quantities and the self-consistent electric field (E∥) parallel to the magnetic field have been an unresolved problem in magnetospheric physics for over 40 years. We review a recently developed kinetic and multimoment fluid theory for inhomogeneous, nonuniformly magnetized plasma with temperature anisotropy in the guiding-center and gyrotropic approximation and apply the theory to solve for the quasi steady state in the long-range potential region of a downward Birkeland current sheet when electrostatic ion cyclotron turbulence is dominant. We find that an electron, bump-on-tail-driven ion cyclotron instability produces the turbulence and that a large enhancement in ∣E∥∣ by nearly a factor of 40 occurs when the turbulence is present compared to the case when it is absent. Anomalous momentum transfer (anomalous resistivity) by itself has a very small effect on E∥; however, the presence of the turbulence and the anomalous energy transfers (anomalous heating and cooling) that result have a very large effect on the entire solution. In the electron and ion momentum balance equations for E∥, the turbulence enhances the magnitude of E∥ by reducing the effect of the generalized parallel pressure gradients and thereby enhancing the effect of the mirror forces. A new, nonlinear formula for the current-voltage relationship in downward Birkeland current regions is also given.

The effect of spectral anisotropy of fast magnetosonic turbulence on the plasma heating at the proton kinetic scales

The energy transport associated with the fast magnetosonic turbulence in a low-beta plasma is one of the possible driving forces of the solar wind. Two different kinds of the turbulence spectra are discussed in this paper: a unidirectional spectrum of the fluctuations propagating away from the sun and a bidirectional spectrum composed of the sunward and antisunward modes. The evolution of the system is analyzed with the help of hybrid numerical simulations at the proton kinetic scales. It is shown that in the unidirectional case, the heating is always preferentially across the background magnetic field. It is this property that allows the solar wind plasma to be propelled away from the sun by the mirror force of the inhomogeneous magnetic field. Adding the counterpropagating modes reverses the sense of the energy deposition with the parallel heating now dominating the process. However, the parallel heating is produced mostly by fast modes propagating along the magnetic field. If these modes are eliminated, the heating becomes perpendicular again.

Gyrokinetic simulations including the centrifugal force in a rotating tokamak plasma

Tokamak experiments operate with a rotating plasma, with toroidal velocity which can be driven externally but can also arise spontaneously. In the frame that corotates with the plasma, the effects of the centrifugal force are felt through a centrifugal drift and an enhanced mirror force [Peeters et al., Phys. Plasmas 16, 042310 (2009)]. These inertial terms become important in the case of strong rotation, as is common in spherical devices, and are also important for heavy impurity ions even at small toroidal velocities. In this work, the first gyrokinetic simulations including the centrifugal force in a strongly rotating plasma are presented. The enhanced mirror force redistributes density over a flux surface and modifies the trapping condition, destabilizing trapped electron modes. At intermediate scales this can result in promotion of the trapped electron mode over the ion temperature gradient (ITG) mode as the dominant instability, which under marginal conditions could result in an enhanced electron heat flux. The centrifugal drift acts to damp the residual zonal flow of the geoacoustic mode, while its frequency is increased. For nonlinear ITG dominated turbulence, increased trapped electron drive and reduced zonal flow lead to an increase in ion heat diffusivity if the increased rotation is not accompanied by rotational shear stabilization. An increased fraction of slow trapped electrons enhances the convective particle pinch, leading to an increase in the steady state density gradient with strong rotation. Linear ITG mode results show an increased pinch of heavy trace impurities due to their strong centrifugal trapping.

The self-consistent parallel electric field due to electrostatic ion-cyclotron turbulence in downward auroral-current regions of the Earth’s magnetosphere. IV

The physical processes that determine the self-consistent electric field (E∥) parallel to the magnetic field have been an unresolved problem in magnetospheric physics for over 40 years. Recently, a new multimoment fluid theory was developed for inhomogeneous, nonuniformly magnetized plasma in the guiding-center and gyrotropic approximation that includes the effect of electrostatic, turbulent, wave-particle interactions (see Jasperse et al. [Phys. Plasmas 13, 072903 (2006); Jasperse et al., Phys. Plasmas13, 112902 (2006)]). In the present paper and its companion paper [Jasperse et al., Phys. Plasmas 17, 062903 (2010)], which are intended as sequels to the earlier work, a fundamental model for downward, magnetic field-aligned (Birkeland) currents for quasisteady conditions is presented. The model includes the production of electrostatic ion-cyclotron turbulence in the long-range potential region by an electron, bump-on-tail-driven ion-cyclotron instability. Anomalous momentum transfer (anomalous resistivity) by itself is found to produce a very small contribution to E∥; however, the presence of electrostatic, ion-cyclotron turbulence has a very large effect on the altitude dependence of the entire quasisteady solution. Anomalous energy transfer (anomalous heating and cooling) modifies the density, drift, and temperature altitude profiles and hence the generalized parallel-pressure gradients and mirror forces in the electron and ion momentum-balance equations. As a result, |E∥| is enhanced by nearly a factor of 40 compared to its value when turbulence is absent. The space-averaged potential increase associated with the strong double layer at the bottom of the downward-current sheet is estimated using the FAST satellite data and the multimoment fluid theory.

Cosmic ray transport in non-uniform magnetic fields: consequences of gradient and curvature drifts

AbstractLarge-scale spatial variations of the guide magnetic field of interplanetary and interstellar plasmas give rise to the mirror force −(p⊥2/2mγB)∇B). The parallel component of this mirror force causes adiabatic focusing of the cosmic ray guiding center whereas the perpendicular component of the mirror force gives rise to the gradient and curvature drifts of the cosmic ray guiding center. Adiabatic focusing and the gradient and curvature drift terms additionally enter the Fokker–Planck transport equation for the gyrotropic cosmic ray particle phase space density in partially turbulent non-uniform magnetic fields. For magnetohydrodynamic turbulence with dominating magnetic fluctuations, the diffusion approximation is justified, which results in a modification of the diffusion–convection transport equation for the isotropic part of the gyrotropic phase space density from the additional focusing and drift terms. For axisymmetric undamped slab Alfvenic turbulence we show that all perpendicular spatial diffusion coefficients are caused by the non-vanishing gradient and curvature drift terms. For a specific (symmetric in μ) choice of the pitch-angle Fokker–Planck coefficients we show that the ratio of the perpendicular to parallel spatial diffusion coefficients apart from a constant is determined by the spatial first derivatives of the non-constant cosmic ray Larmor radius in the non-uniform magnetic field.

Interpretive modelling of scrape-off plasmas on the MAST tokamak

Electrical currents in the scrape-off layer (SOL) of MAST are modelled using an interpretive Onion-Skin Model (OSM) constrained with experimental data from MAST diagnostics. The model was extended to include the effects of the magnetic mirror force, which has a strong influence on the particle and momentum balance in spherical tokamaks, such as MAST [1]. These modifications serve to more accurately model the parallel electric fields present in the MAST SOL, which can alter plasma dynamics via the E × B drift. Simulations show that the electrical current at the divertor targets is predominantly thermoelectric, whereas Pfirsch–Schlüter currents have a greater contribution to the total current in the bulk of the SOL plasma.

Frequency stabilization and output power undulations of diode lasers with feedback by volume holographic gratings

The mode-hop behavior and the power characteristics of a laser diode with wavelength-selective optical feedback are experimentally investigated. The feedback is provided by external volume holographic gratings, also called ‘Bragg mirrors’, at normal incidence. We demonstrate that a Bragg mirror forces a laser diode to operate only within a narrow wavelength range, and that the emission wavelength of the laser diode is stabilized against variations of the injection current. Moreover, we present periodic undulations of the power characteristics of the laser, depending on the driving current. They can be qualitatively explained with a simple model which takes into account that the threshold gain in the laser system strongly depends on the wavelength.

Electron thermal effects in standing shear Alfvén waves

Recent hybrid MHD‐kinetic electron simulations of Field Line Resonances have illustrated that the acceleration of electrons to carry the field aligned current can dissipate a significant amount of wave energy over only half an Alfvén cycle. This was done in the limit of colder electron temperatures, and in the present study we extend this to consider temperatures of up to several hundred eV. It is found that mirror force effects enhance both the parallel electric field needed to support the given current as well as the dissipation associated with the acceleration of the electrons to carry it. The current‐voltage (C‐V) relation appears consistent with the Knight relation for a portion of the evolution but then saturates with the decline of electrons within the loss cone.

Nonlinear saturation of mirror instability

Mechanism of the nonlinear saturation of compressible electromagnetic mirror instability is studied using the gyrokinetic particle simulation. Phase‐space particle trapping due to the mirror force is found to be the dominant saturation mechanism in the simulation of a single mirror mode with relatively weak drive. At the nonlinear saturation, the phase‐space island of the distribution function is formed. The oscillation frequency of the saturated perturbation amplitude is close to the bounce frequency of the trapped particles, which is comparable to the linear growth rate of the mirror mode. Scaling of the saturation amplitude is consistent with the onset of the particle trapping. With strong instability drive, relaxation toward marginal stability dominates the nonlinear saturation of the mirror instability. Phase‐space trapping, however, persists after the saturation and continues to regulate the nonlinear evolution of the mirror mode.

New description of Io's cold plasma torus

Despite more than 25 years of study of the Io plasma torus, its generation, dynamics, and even its spatial structure are still poorly understood, especially in the case of the inner, cold region of the torus. To remedy this lack, we analyzed ground‐based coronagraphic images of the torus in S+ 6371 Å emission. We derived cold torus properties by modeling and removing these images' inherent line‐of‐sight integration and atmospheric blurring, using new deconvolution techniques, obtaining high‐spatial‐resolution estimates of the three‐dimensional (3‐D) S+ distributions. From these 3‐D distributions, we discovered that the cold torus is washer‐shaped, with a roughly constant vertical thickness ≤0.25 Jovian radius (RJ), and a radial width that varies from 0.6 to 0.9 RJ. The cold torus is separated by a 0.1–0.2 RJ‐wide low‐density region, or “gap,” from the “ribbon” region which lies just outside it. The small, approximately constant washer height implies an ion parallel temperature (T∥) of ∼3 eV, compared with a ribbon T∥ that varies from about 20 to 50 eV as a function of Jovian magnetic longitude (λIII). The washer has a distinct inner edge, not seen before, whose jovicentric distance varies with λIII so as to create the variable cold torus width. Thus this inner edge is concentric with neither Jupiter nor the rest of the torus. We also confirm the existence of a tilt between the midplanes of the ribbon and cold torus, with an orientation that cannot be produced by the magnetic mirror force acting on ion temperature anisotropy. The structure and composition of the gap and cold torus are best explained by a model in which a small amount of warm S+ plasma diffuses inwards while radiatively cooling. While still warm, its distribution over a large scale height keeps its density small, forming the gap. After sufficient cooling, it collapses to the centrifugal equator, where its higher density and continued inward diffusion make it more visible as the cold torus washer. However, its low electron temperature (probably ≤ T∥) must be kept from further decline by a hitherto‐unsuspected energy source that powers the observed visible wavelength radiation from the cold torus and fluctuates on timescales less than the plasma diffusion time. The formation of the abrupt cold torus inner edge might indicate the loss there of either this energy source or the plasma itself.

Electron reflectometry in the martian atmosphere

The technique of electron reflectometry, a method for remote estimation of planetary magnetic fields, is expanded from its original use of mapping crustal magnetic fields at the Moon to achieving the same purpose at Mars, where the presence of a substantial atmosphere complicates matters considerably. The motion of solar wind electrons, incident on the martian atmosphere, is considered in detail, taking account of the following effects: the electrons' helical paths around the magnetic field lines to which they are bound, the magnetic mirror force they experience due to converging field lines in the vicinity of crustal magnetic anomalies, their acceleration/deceleration by electrostatic potentials, their interactions with thermal plasma, their drifts due to magnetic field line curvature and perpendicular electric fields and their scattering off, and loss of energy through a number of different processes to, atmospheric neutrals. A theoretical framework is thus developed for modeling electron pitch angle distributions expected when a spacecraft is on a magnetic field line which is connected to both the martian crust and the interplanetary magnetic field. This framework, along with measured pitch angle distributions from the Mars Global Surveyor (MGS) Magnetometer/Electron Reflectometer (MAG/ER) experiment, can be used to remotely measure crustal magnetic field magnitudes and atmospheric neutral densities at ∼180 km above the martian datum, as well as estimate average parallel electric fields between 200 and 400 km altitude. Detailed analysis and full results, concerning the crustal magnetic field and upper thermospheric density of Mars, are left to two companion papers.

Simulation of auroral electron acceleration by inertial Alfvén waves

A two‐dimensional particle code, similar to that used in the work of Swift (2007), is used to model the acceleration of auroral electrons by inertial Alfvén waves. The simulation domain is 20,000 km parallel to the magnetic field and some tens of kilometers across. Effects of a variable magnetic field and magnetic mirror force are included. The Alfvén wave is launched by a moving bipolar potential waveform with peak amplitudes of ±200 V imposed at the top of the simulation domain. Electrons are accelerated to 3 keV energy. Two types of acceleration processes are seen: one is impulsive acceleration from shock structures, and the other is from weaker parallel electric fields extending over several thousand kilometers. The shock structures are very small, and electrons accelerated by these structures can produce auroral forms with thicknesses considerably smaller than an electron inertial length. Electrons also tend to undergo multiple acceleration events. Electrons are more strongly accelerated upward. Half of these are destined to precipitate in the opposite hemisphere, and half are destined to become trapped. This implies that much of the aurora seen in the Northern Hemisphere is due to electrons accelerated in the Southern Hemisphere. The simulations also suggest that auroral acceleration processes provide a likely source for radiation belt particles.

Generation of short‐burst radiation through Alfvénic acceleration of auroral electrons

A comparison of modeled simulation results with in situ observations is presented in an effort to better understand the generation mechanism of short‐burst radiation (AKR) observed in the Alfvénic acceleration region. The simulation is performed using a linear gyrofluid Alfvén model to generate propagating inertial Alfvén waves with a test particle scheme to study the behavior of electrons under the influence of waves. The observations are taken from the wave and plasma instruments on board the Fast Auroral Snapshot (FAST) satellite. We find that single‐ and multiple‐electron shell distributions are produced as a result of parallel electric fields generated from inertial Alfvén waves. Electrons outside the loss cone are reflected back up the flux tube because of the mirror force forming electron conics. Distributions obtained from this simulation resemble the observed electron distributions in the Alfvénic auroral acceleration region where short‐burst radiations were present. Multiple‐shell distributions as well as electron conics are unstable, making them candidates for triggering the electron‐cyclotron maser instability. Hence we suggest that these unstable distributions, resulting from the effects of inertial Alfvén wave, could be the source of the observed short‐burst radiation in the terrestrial auroral zone magnetosphere‐ionosphere coupling region. We also briefly discuss the application of this general mechanism to explain the Io‐associated S‐bursts observed emanating from the Jovian magnetosphere.

A two‐dimensional particle code simulation of inertial Alfvén waves and auroral electron acceleration

A two‐dimensional particle code used to simulate the propagation of inertial Alfvén waves down auroral field lines. The simulation domain is 20,000 km parallel to the magnetic field and some tens of kilometers perpendicular to the field. Electrons are tied to the magnetic field. Effects of a variable magnetic field and magnetic mirror force are included. A spatially narrow Alfvén pulse is launched by application of a variable potential with peak amplitude of 100 V at the top of the simulation domain. If the thickness of the pulse is less than the electron inertial length, the pulse tends to broaden as it propagates down the field line. The result is a parallel electric field considerably smaller than might be expected, assuming a wave whose perpendicular wavelength maps along magnetic field lines. In this model the strongest interaction between the electrons and the Alfvén wave occurs in the 3000 to 7000 km altitude range with an energy gain of the order 100 eV. The downward acceleration was strongest when the applied excitation consisted of a moving positive pulse followed by a negative pulse. Strongest upward acceleration was seen when the polarity was reversed. When the magnitude of the driving potential was doubled, there appeared to be some acceleration in the 9000 to 18,000 km altitude range and acceleration of some particles to 1.5 keV.

On the behavior of O+5 in coronal holes: Importance of sunward propagating waves

The high thermal anisotropy of O+5 in coronal holes, as observed by the Ultraviolet Coronagraph Spectrometer (UVCS) on the Solar and Heliospheric Observatory (SOHO), suggests that these ions are being heated by the cyclotron resonance. The observations indicate that the O+5 temperature steadily increases from r ≈ 1.5 rS (where the ions become almost collisionless) out to r ≈ 3.5 rS (the outer limit of the observations). Previous models have not been able to reproduce even the qualitative result of a steady temperature rise. We suggest that the problem has been that previous models have considered O+5 resonating only with outward propagating waves. Once the ions are heated perpendicularly to the background magnetic field, they are accelerated to high outward flow speeds by the mirror force. As a result, they resonate with outgoing waves having higher (normalized) wave numbers, where there is presumably less power; they may even drop out of resonance altogether. We suggest here that resonances with inward (i.e., sunward) propagating waves may be the key to explaining the observed O+5 temperature rise. In that case, as the ions are accelerated by the mirror force, they never drop out of resonance, and they resonate with ingoing waves having lower (normalized) wave numbers where there is presumably more power. We offer a very simple strawman model to illustrate the differences between oxygen resonances with ingoing and outgoing waves and to show that the UVCS/SOHO results can be approximately reproduced if the ingoing wave power spectrum in the resonant range varies as k−γ, with γ ≈ 5/3. We point out that it is really necessary to take into account the fact that O+5 (and other heavy ions) can resonate with ingoing and outgoing waves simultaneously, which can only be studied via full kinetic solutions for the ion distribution functions; however, it is possible that once the ions are accelerated by the mirror force, the resonances with sunward propagating waves will be dominant.

Monte Carlo modeling of toroidal ion distributions and ion temperatures at high altitudes equatorward of the cusp: Effect of finite gyroradius

We report that the effect of finite gyroradius is responsible for production of the H+ and O+ toroids at high altitudes equatorward of the cusp that are observed by TIDE and TIMAS ion instruments on board the polar spacecraft. The energization of charged particles, owing to interaction with electromagnetic turbulence, has an important influence on the plasma outflow in space. The effect of wave‐particle interactions (WPI) on H+ and O+ outflow at high altitudes equatorward of the cusp was investigated by using Monte Carlo method. The Monte Carlo model includes the effect of WPI, gravity, polarization electrostatic field, and the divergence of the geomagnetic field within the simulation tube (1.2–10 Earth radii, RE). As the ions drift upward along the geomagnetic field lines, they interact with the electromagnetic turbulence and consequently get heated in the direction perpendicular to the geomagnetic field. The mirror force converts some of the gained ion energy in the perpendicular direction into parallel kinetic energy. These effects combine to form an ion‐conic velocity distribution. However, as the ions are heated and move to higher altitudes, the ion gyroradius ρi may become comparable to the perpendicular wavelength of the electromagnetic turbulence λ⊥. As the ratio ρi/λ⊥ becomes >1, then the heating rate turns to be self‐limited and the ion velocity distribution displays toroidal features. A comparison has been made between the Monte Carlo calculations obtained in this study and observations of H+ and O+ ion velocity distributions and temperatures. The comparison showed a remarkably close agreement in the corresponding results for the ion velocity distribution and its temperature. As a result of the comparison, we were able to predict the characteristic value of the perpendicular wavelength of the electromagnetic turbulence λ⊥ at high altitudes equatorward of the cusp. To our knowledge, this represents the first successful comparison of observed toroids with a theoretical model.