Results for “FORCE-FREE MAGNETIC FIELD” — Page 9

NASA NTRS1992

A comparative study of dynamically expanding force-free, constant-alpha magnetic configurations with applications to magnetic clouds

We contrast two different solutions of the constant alpha, force-free MHD equation, both of which have been suggested as models for magnetic clouds: a solution in cylindrical coordinates and one in spherical coordinates. In line with the observation that magnetic clouds expand, we generalize these static models and construct their expanding counterparts. We find that expansion introduces in both cases a large asymmetry in the field strength signature which is in the same sense as that seen the the data, i.e. towards the leading edge of the cloud. We then do a least squares fit of the respective models to one-spacecraft data on a magnetic cloud. We find that the fitting routine converges in both cases. However, while purely formally we cannot distinguish between the two models using data from one spacecraft, the field components in the 'spherical' model have features not compatible with data on magnetic clouds.

Farrugia, C. J.↗

NASA NTRS2012

Towards a Realistic Pulsar Magnetosphere

We present the magnetic and electric field structures as well as the currents ami charge densities of pulsar magnetospberes which do not obey the ideal condition, E(raised dot) B = O. Since the acceleration of particles and the production of radiation requires the presence of an electric field component parallel to the magnetic field, E(sub ll) the structure of non-Ideal pulsar magnetospheres is intimately related to the production of pulsar radiation. Therefore, knowledge of the structure of non-Ideal pulsar maglletospheres is important because their comparison (including models for t he production of radiation) with observations will delineate the physics and the parameters underlying the pulsar radiation problem. We implement a variety of prescriptions that support nonzero values for E(sub ll) and explore their effects on the structure of the resulting magnetospheres. We produce families of solutions that span the entire range between the vacuum and the (ideal) Force-Free Electrodynamic solutions. We also compute the amount of dissipation as a fraction of the Poynting flux for pulsars of different angles between the rotation and magnetic axes and conclude that tltis is at most 20-40% (depending on t he non-ideal prescription) in the aligned rotator and 10% in the perpendicular one. We present also the limiting solutions with the property J = pc and discuss their possible implicatioll on the determination of the "on/ off" states of the intermittent pulsars. Finally, we find that solutions with values of J greater than those needed to null E(sub ll) locally produce oscillations, potentially observable in the data.

Kalapotharakos, Constantinos↗

NASA NTRS1975

The calculation of force-free fields from discrete flux distributions

This paper presents particularly simple mathematical formulas for the calculation of force-free fields of constant alpha from the distribution of discrete sources on a flat surface. The advantage of these formulas lies in their physical simplicity and the fact that they can be easily used in practice to calculate the fields. The disadvantage is that they are limited to fields of 'sufficiently small alpha'. These formulas may be useful in the study of chromospheric magnetic fields by the comparison of high-resolution H-alpha photographs and photospheric magnetograms.

Sheeley, N. R., Jr.↗

NASA NTRS2011

Magnetic Clouds at/near the 2007 - 2009 Solar Minimum: Frequency of Occurrence and Some Unusual Properties

Magnetic clouds (MCs) have been identified for the period 2007 2009 (at/near the recent solar minimum) from Wind data, then confirmed through MC parameter fitting using a force-free model. A dramatic increase in the frequency of occurrence of these events took place from the two early years of 2007 (with five MCs) and 2008 (one MC) compared to 2009 (12 MCs). This pattern approximately mirrors the occurrence-frequency profile that was observed over a three-year interval 12 years earlier, with eight events in 1995, four in 1996, and 17 in 1997, but decreased overall by a factor of 0.62 in number. However, the average estimated axial field strength taken over all of the 18 events of 2007 - 2009 (called the "recent period" here) was only 11.0 nT, whereas |BO| for the 29 events of 1995 - 1997 (called the "earlier period" ) was 16.5 nT. This 33% average drop in |BO| is more or less consistent with the decreased three-year average interplanetary magnetic field intensity between these two periods, which shows a 23% drop. In the earlier period, the MCs were clearly of mixed types but predominantly of the South-to-North type, whereas those in the recent period are almost exclusively the North-to-South type; this change is consistent with global solar field changes predicted by Bothmer and Rust (Geophys. Monogr. Ser. 99, 139, 1997). As we have argued in earlier work (Lepping and Wu, J. Geophys. Res. 112, A10103, 2007), this change should make it possible to carry out (accurate short-term) magnetic storm forecasting by predicting the latter part of an MC from the earlier part, using a good MC parameter-fitting model with real-time data from a spacecraft at L1, for example. The recent set s average duration is 15.2 hours, which is a 27% decrease compared to that of the earlier set, which had an average duration of 20.9 hours. In fact, all physical aspects of the recent MC set are shown to drop with respect to the earlier set; e.g., as well as the average internal magnetic field drop, the recent set had a somewhat low average speed of 379 km/s (5% drop), and the average diameter had a 24% drop. Hence, compared to the earlier set, the recent set consists of events that are smaller, slightly slower, and weaker in every respect (and fewer in number), but in a relative sense the two three-year sets have similar frequency-of-occurrence profiles. It is also interesting that the two sets have almost the same average axial inclinations, i.e., axial latitude approx. = 31deg (in GSE). These MC characteristics are compared to relevant solar features and their changes. A preliminary assessment of the statistics on possible shocks and pressure pulses upstream of these recent MCs yields the following: About 28% of the MCs, at most, had shocks, and 33% had shocks and/or pressure pulses. These are low values, since typically the percentage of cases with shocks is about 50%, and the percentage with shocks and/or pressure pulses is usually about 75%.

Lepping. R. P.↗

NASA NTRS1980

The stability properties of cylindrical force-free fields - Effect of an external potential field

A large-scale potential field with an embedded smaller-scale force-free structure gradient x B equals alpha B is studied in cylindrical geometry. Cases in which alpha goes continuously from a constant value alpha 0 on the axis to zero at large r are considered. Such a choice of alpha (r) produces fields which are realistic (few field reversals) but not completely stable. The MHD-unstable wavenumber regime is found. Since the considered equilibrium field exhibits a certain amount of magnetic shear, resistive instabilities can arise. The growth rates of the tearing mode in the limited MHD-stable region of k space are calculated, showing time-scales much shorter than the resistive decay time.

Chiuderi, C.↗

NASA NTRS2003

Observed Helicity of Active Regions in Solar Cycle 21

We report the results of a study of helicity in solar active regions during the peak of activity in solar cycle 21 from observations with the Marshall Space Flight Center's solar vector magnetograph. Using the force-free parameter alpha as the proxy for helicity, we calculated an average value of alpha for each of 60 active regions from a total of 449 vector magnetograms that were obtained during the period 1980 March to November. The signs of these average values of alpha were correlated with the latitude of the active regions to test the hemispheric rule of helicity that has been proposed for solar magnetic fields: negative helicity predominant in northern latitudes, positive in the southern ones. We have found that of the 60 regions that were observed, 30 obey the hemispheric rule and 30 do not.

Hagyard, M. J.↗

NASA NTRS2012

Particle Acceleration in Dissipative Pulsar Magnetospheres

Pulsar magnetospheres represent unipolar inductor-type electrical circuits at which an EM potential across the polar cap (due to the rotation of their magnetic field) drives currents that run in and out of the polar cap and close at infinity. An estimate ofthe magnitude of this current can be obtained by dividing the potential induced across the polar cap V approx = B(sub O) R(sub O)(Omega R(sub O)/c)(exp 2) by the impedance of free space Z approx eq 4 pi/c; the resulting polar cap current density is close to $n {GJ} c$ where $n_{GJ}$ is the Goldreich-Julian (GJ) charge density. This argument suggests that even at current densities close to the GJ one, pulsar magnetospheres have a significant component of electric field $E_{parallel}$, parallel to the magnetic field, a condition necessary for particle acceleration and the production of radiation. We present the magnetic and electric field structures as well as the currents, charge densities, spin down rates and potential drops along the magnetic field lines of pulsar magnetospheres which do not obey the ideal MHD condition $E cdot B = 0$. By relating the current density along the poloidal field lines to the parallel electric field via a kind of Ohm's law $J = sigma E_{parallel}$ we study the structure of these magnetospheres as a function of the conductivity $sigma$. We find that for $sigma gg OmegaS the solution tends to the (ideal) Force-Free one and to the Vacuum one for $sigma 11 OmegaS. Finally, we present dissipative magnetospheric solutions with spatially variable $sigma$ that supports various microphysical properties and are compatible with the observations.

Kazanas, Z.↗

NASA NTRS2003

Observed Helicity of Active Regions in Solar Cycle 21

We report the results of a study of magnetic helicity in solar active regions during solar cycles 21 and 22 from observations with the Marshall Space Flight Center's solar vector magnetograph. Using the force-free parameter a as the proxy for helicity, we calculated an average value of a for each of 91 active regions from a total of 683 vector magnetograms that were obtained during the period March 1980 to July 1993. The signs of these average values of a were correlated with the latitude of the active regions to test the hemispherical rule of helicity that has been proposed for solar magnetic fields: negative helicity predominant in northern latitudes, positive in the southern ones. We have found that of the 65 regions that were observed in cycle 21,49% obey the hemispherical rule and 51% do not. On the other hand, for the 26 regions in cycle 22,65% do exhibit this correlation. In addition, our results seem to support the findings of Hagino and Sakurai (2002) which indicate evidence for a time variability in the hemispheric rule.

Hagyard, M. J.↗

NASA NTRS2003

Using an MHD simulation to interpret the global context of a coronal mass ejection observed by two spacecraft

In late February 1999 the ACE spacecraft observed a coronal mass ejection (CME) at 1 AU, in the ecliptic plane. Thirteen days later, Ulysses observed a CME at 5 AU and 22"s. We present a detailed analysis of the plasma, magnetic field, and composition signatures of these two events. On the basis of this comparison alone, it is not clear that the two spacecraft observed the same solar event. However, using a generic MHD simulation of a fast CME initiated at the Sun by magnetic flux cancellation and propagated out into the solar wind, together with additional evidence, we argue that indeed the same CME was observed by both spacecraft. Although force-free models appear to fit the observed events well, our simulation results suggest that the ejecta underwent significant distortion during its passage through the solar wind, indicating that care should be taken when interpreting the results of force-he models. Comparison of composition measurements at the two spacecraft suggests that significant spatial inhomogeneities can exist within a single CME.

Riley, Pete↗

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.↗

NASA NTRS2014

Statistical Mechanics of Turbulent Dynamos

Incompressible magnetohydrodynamic (MHD) turbulence and magnetic dynamos, which occur in magnetofluids with large fluid and magnetic Reynolds numbers, will be discussed. When Reynolds numbers are large and energy decays slowly, the distribution of energy with respect to length scale becomes quasi-stationary and MHD turbulence can be described statistically. In the limit of infinite Reynolds numbers, viscosity and resistivity become zero and if these values are used in the MHD equations ab initio, a model system called ideal MHD turbulence results. This model system is typically confined in simple geometries with some form of homogeneous boundary conditions, allowing for velocity and magnetic field to be represented by orthogonal function expansions. One advantage to this is that the coefficients of the expansions form a set of nonlinearly interacting variables whose behavior can be described by equilibrium statistical mechanics, i.e., by a canonical ensemble theory based on the global invariants (energy, cross helicity and magnetic helicity) of ideal MHD turbulence. Another advantage is that truncated expansions provide a finite dynamical system whose time evolution can be numerically simulated to test the predictions of the associated statistical mechanics. If ensemble predictions are the same as time averages, then the system is said to be ergodic; if not, the system is nonergodic. Although it had been implicitly assumed in the early days of ideal MHD statistical theory development that these finite dynamical systems were ergodic, numerical simulations provided sufficient evidence that they were, in fact, nonergodic. Specifically, while canonical ensemble theory predicted that expansion coefficients would be (i) zero-mean random variables with (ii) energy that decreased with length scale, it was found that although (ii) was correct, (i) was not and the expected ergodicity was broken. The exact cause of this broken ergodicity was explained, after much investigation, by greatly extending the statistical theory of ideal MHD turbulence. The mathematical details of broken ergodicity, in fact, give a quantitative explanation of how coherent structure, dynamic alignment and force-free states appear in turbulent magnetofluids. The relevance of these ideal results to real MHD turbulence occurs because broken ergodicity is most manifest in the ideal case at the largest length scales and it is in these largest scales that a real magnetofluid has the least dissipation, i.e., most closely approaches the behavior of an ideal magnetofluid. Furthermore, the effects grow stronger when cross and magnetic helicities grow large with respect to energy, and this is exactly what occurs with time in a real magnetofluid, where it is called selective decay. The relevance of these results found in ideal MHD turbulence theory to the real world is that they provide at least a qualitative explanation of why confined turbulent magnetofluids, such as the liquid iron that fills the Earth's outer core, produce stationary, large-scale magnetic fields, i.e., the geomagnetic field. These results should also apply to other planets as well as to plasma confinement devices on Earth and in space, and the effects should be manifest if Reynolds numbers are high enough and there is enough time for stationarity to occur, at least approximately. In the presentation, details will be given for both theoretical and numerical results, and references will be provided.

Shebalin, John V.↗

NASA NTRS2014

MHD Turbulence and Magnetic Dynamos

Incompressible magnetohydrodynamic (MHD) turbulence and magnetic dynamos, which occur in magnetofluids with large fluid and magnetic Reynolds numbers, will be discussed. When Reynolds numbers are large and energy decays slowly, the distribution of energy with respect to length scale becomes quasi-stationary and MHD turbulence can be described statistically. In the limit of infinite Reynolds numbers, viscosity and resistivity become zero and if these values are used in the MHD equations ab initio, a model system called ideal MHD turbulence results. This model system is typically confined in simple geometries with some form of homogeneous boundary conditions, allowing for velocity and magnetic field to be represented by orthogonal function expansions. One advantage to this is that the coefficients of the expansions form a set of nonlinearly interacting variables whose behavior can be described by equilibrium statistical mechanics, i.e., by a canonical ensemble theory based on the global invariants (energy, cross helicity and magnetic helicity) of ideal MHD turbulence. Another advantage is that truncated expansions provide a finite dynamical system whose time evolution can be numerically simulated to test the predictions of the associated statistical mechanics. If ensemble predictions are the same as time averages, then the system is said to be ergodic; if not, the system is nonergodic. Although it had been implicitly assumed in the early days of ideal MHD statistical theory development that these finite dynamical systems were ergodic, numerical simulations provided sufficient evidence that they were, in fact, nonergodic. Specifically, while canonical ensemble theory predicted that expansion coefficients would be (i) zero-mean random variables with (ii) energy that decreased with length scale, it was found that although (ii) was correct, (i) was not and the expected ergodicity was broken. The exact cause of this broken ergodicity was explained, after much investigation, by greatly extending the statistical theory of ideal MHD turbulence. The mathematical details of broken ergodicity, in fact, give a quantitative explanation of how coherent structure, dynamic alignment and force-free states appear in turbulent magnetofluids. The relevance of these ideal results to real MHD turbulence occurs because broken ergodicity is most manifest in the ideal case at the largest length scales and it is in these largest scales that a real magnetofluid has the least dissipation, i.e., most closely approaches the behavior of an ideal magnetofluid. Furthermore, the effects grow stronger when cross and magnetic helicities grow large with respect to energy, and this is exactly what occurs with time in a real magnetofluid, where it is called selective decay. The relevance of these results found in ideal MHD turbulence theory to the real world is that they provide at least a qualitative explanation of why confined turbulent magnetofluids, such as the liquid iron that fills the Earth's outer core, produce stationary, large-scale magnetic fields, i.e., the geomagnetic field. These results should also apply to other planets as well as to plasma confinement devices on Earth and in space, and the effects should be manifest if Reynolds numbers are high enough and there is enough time for stationarity to occur, at least approximately. In the presentation, details will be given for both theoretical and numerical results, and references will be provided.

Shebalin, John V↗

NASA NTRS1978

Measurement and analysis of magnetic field variation during a class 2b flare

A digital analysis of high-time-resolution videomagnetograms taken during a class 2b flare that occurred at 60 deg east is reported. The data were obtained at the Big Bear Observatory and calibrated by a Mt. Wilson magnetogram. Changes of weak magnetic fields (less than 100 G) with an amplitude of 30% to 100% have been detected over 55% of the optical flare region, apparently taking place at the initial phase of the flare. Statistical considerations suggest a real flare association with most of these changes. H-alpha observations show that large changes took place over the footpoints of heavily inclined structures like penumbral fibrils, while smaller changes took place over the plage region. An apparent polarity reversal was found at the feet of erupted fibrils. Based on force-free field calculations these changes can be reasonably explained as a transformation of the current-carrying fields to potential fields which produced large changes in the field-line inclination and rotation.

Tanaka, K.↗

DOE OSTI2025

Anisotropic Heating and Parallel Heat Flux in Electron-only Magnetic Reconnection with Intense Guide Fields

Electron-only reconnection (E-REC) is a process recently observed in the Earth’s magnetosheath, where magnetic reconnection occurs at electron kinetic scales, and ions do not couple to the reconnection process. Electron-only reconnection is likely to have a significant impact on the energy conversion and dissipation of turbulence cascades at kinetic scales in some settings. This paper investigates E-REC under different intensities of strong guide fields (the ratio between the guide field and the in-plane asymptotic field strength is 5, 10 and 20, respectively) via two-dimensional fully kinetic particle-in-cell simulations, focusing on electron heating. The simulations are initialized with a force-free current sheet equilibrium under various intensities of strong guide fields. Similarly to previous experimental studies, electron temperature anisotropy along separatrices is observed, which is found to be mainly caused by the variations of parallel temperature. Both regions of anisotropy and parallel temperature increase/decrease along separatrices become thinner with increasing guide fields. Besides, we find a transition from a quadrupolar to a hexapolar (six-polar) to an octopolar (eight-polar) structure in temperature anisotropy and parallel temperature as the guide field intensifies. Non-Maxwellian electron velocity distribution functions (EVDFs) at different locations in the three simulations are observed. Our results show that parallel electron velocity varies notably with different guide field intensities and finite parallel electron heat flux density is observed. The three simulations exhibit features of the Chew–Goldberger–Low theory, with the level of consistency increasing as the guide field strength increases. This explains the electron parallel temperature variations and the shape of the EVDFs observed along the separatrices. This work may provide insights into the understanding of electron heating and parallel heat flux density in E-REC observed in the turbulent magnetosheath.

79 ASTRONOMY AND ASTROPHYSICS↗

NASA NTRS2012

Alignment and Scaling of Large-Scale Fluctuations in Solar Wind Turbulence

We investigate the dependence of solar wind fluctuations measured by the Wind spacecraft on scale and on the degree of alignment between oppositely directed Elsasser fields. This alignment controls the strength of the non-linear interactions and, therefore, the turbulence. We find that at scales larger than the outer scale of the turbulence the Elsasser fluctuations become on average more anti-aligned as the outer scale is approached from above. Conditioning their structure functions using the alignment angle reveals turbulent scaling of unaligned fluctuations at scales previously believed to lie outside the turbulent cascade in the '1/f range'. We argue that the 1/f range contains a mixture of non-interacting anti-aligned population of Alfven waves and magnetic force-free structures plus a subdominant population of unaligned cascading turbulent fluctuations.

Wicks, R. T.↗

NASA NTRS2012

Survival of Flux Transfer Event (FTE) Flux Ropes Far Along the Tail Magnetopause

During intervals of southward IMF, magnetic reconnection can result in the formation of flux transfer events (FTEs) on the dayside magnetopause which travel along the magnetopause in the anti-sunward direction. Of particular interest is their fate and the role they play transporting solar wind plasma into the magnetosphere. We present the discovery of FTEs far along the distant tail magnetopause (x = 67 Earth radii) using data from ARTEMIS on the dusk flank magnetopause under southward/duskward IMF conditions. The identification of several events is further supported by excellent fits to a force-free flux rope model. The axis of each structure is principally north-south, i.e., perpendicular to the Sun-Earth line. Simultaneous observations by THEMIS on the dayside magnetopause indicate that FTEs are being produced there, although perhaps 2-4 times smaller in size. The convection time from the dayside magnetopause to ARTEMIS is 30 min, and the FTEs have a flux content comparable to those typically observed on the dayside magnetopause, indicating that these features are in quasi-equilibrium as they are convected downtail. By considering the relative orientations of the FTEs observed by THEMIS and ARTEMIS, the magnetic field geometry is consistent with the FTEs being produced on the dayside magnetopause along an extended X-line in the presence of IMF By and bending as they are convected to the flanks.

ARTEMIS↗

NASA NTRS2010

3D Relativistic Magnetohydrodynamic Simulations of Current-Driven Instability: Instability of a Static Column - 1

We have investigated the development of current-driven (CD) kink instability through three-dimensional relativistic MHD simulations. A static force-free equilibrium helical magnetic configuration is considered in order to study the influence of the initial configuration on the linear and nonlinear evolution of the instability. We found that the initial configuration is strongly distorted but not disrupted by the kink instability. The instability develops as predicted by linear theory. In the non-linear regime the kink amplitude continues to increase up to the terminal simulation time, albeit at different rates, for all but one simulation. The growth rate and nonlinear evolution of the CD kink instability depends moderately on the density profile and strongly on the magnetic pitch profile. The growth rate of the kink mode is reduced in the linear regime by an increase in the magnetic pitch with radius and the non-linear regime is reached at a later time than for constant helical pitch. On the other hand, the growth rate of the kink mode is increased in the linear regime by a decrease in the magnetic pitch with radius and reaches the non-linear regime sooner than the case with constant magnetic pitch. Kink amplitude growth in the non-linear regime for decreasing magnetic pitch leads to a slender helically twisted column wrapped by magnetic field. On the other hand, kink amplitude growth in the non-linear regime nearly ceases for increasing magnetic pitch.

Mizuno, Yosuke↗

NASA NTRS2013

Current Flow and Pair Creation at Low Altitude in Rotation-Powered Pulsars' Force-Free Magnetospheres: Space Charge Limited Flow

We report the results of an investigation of particle acceleration and electron-positron plasma generation at low altitude in the polar magnetic flux tubes of rotation-powered pulsars, when the stellar surface is free to emit whatever charges and currents are demanded by the force-free magnetosphere. We apply a new 1D hybrid plasma simulation code to the dynamical problem, using Particle-in-Cell methods for the dynamics of the charged particles, including a determination of the collective electrostatic fluctuations in the plasma, combined with a Monte Carlo treatment of the high-energy gamma-rays that mediate the formation of the electron-positron pairs.We assume the electric current flowing through the pair creation zone is fixed by the much higher inductance magnetosphere, and adopt the results of force-free magnetosphere models to provide the currents which must be carried by the accelerator. The models are spatially one dimensional, and designed to explore the physics, although of practical relevance to young, high-voltage pulsars. We observe novel behaviour (a) When the current density j is less than the Goldreich-Julian value (0 < j/j(sub GJ) < 1), space charge limited acceleration of the current carrying beam is mild, with the full Goldreich-Julian charge density comprising the charge densities of the beam and a cloud of electrically trapped particles with the same sign of charge as the beam. The voltage drops are of the order of mc(sup 2)/e, and pair creation is absent. (b) When the current density exceeds the Goldreich-Julian value (j/j(sub GJ) > 1), the system develops high voltage drops (TV or greater), causing emission of curvature gamma-rays and intense bursts of pair creation. The bursts exhibit limit cycle behaviour, with characteristic time-scales somewhat longer than the relativistic fly-by time over distances comparable to the polar cap diameter (microseconds). (c) In return current regions, where j/j(sub GJ) < 0, the system develops similar bursts of pair creation. These discharges are similar to those encountered in previous calculations by Timokhin of pair creation when the surface has a high work function and cannot freely emit charge. In cases (b) and (c), the intermittently generated pairs allow the system to simultaneously carry the magnetospherically prescribed currents and adjust the charge density and average electric field to force-free conditions. We also elucidate the conditions for pair creating beam flow to be steady (stationary with small fluctuations in the rotating frame), finding that such steady flows can occupy only a small fraction of the current density parameter space exhibited by the force-free magnetospheric model. The generic polar flow dynamics and pair creation are strongly time dependent. The model has an essential difference from almost all previous quantitative studies, in that we sought the accelerating voltage (with pair creation, when the voltage drops are sufficiently large; without, when they are small) as a function of the applied current.

Timokhin, A. N.↗