A mathematical magnetospheric field model with independent physical parameters

A quantitative magnetospheric magnetic field model has been calculated in three dimensions. The model is based on an analytical solution of the Chapman-Ferraro problem. For this solution, the magnetopause was assumed to be an infinitesimally thin discontinuity with given geometry. The shape of the dayside magnetopause is in agreement with measurements derived from spacecraft boundary crossings.The magnetic field of the magnetopause currents can be derived from scalar potentials. The scalar potentials result from solutions of Laplace's equation with Neumann's boundary conditions. The boundary values and the magnetic flux through the magnetopause are determined by all magnetic sources which are located inside and outside the magnetospheric cavity. They include the Earth's dipole field, the fields of the equatorial ring current and tail current systems, and the homogeneous interplanetary magnetic field. In addition, the flux through the magnetopause depends on two constants of interconnection which provide the possibility of calculating static interconnection between magnetospheric and interplanetary field lines. Realistic numerical values for both constants have been derived empirically from observed displacements of the polar cusps which are due to changes in the orientation of the interplanetary field. The transition from a closed to an open magnetosphere and vice versa can be computed in terms of a change of the magnetic boundary conditions on the magnetopause. The magnetic field configuration of the closed magnetosphere is independent of the amount and orientation of the interplanetary field. In contrast, the configuration of the open magnetosphere confirms the observational finding that field line interconnection occurs primarily in the polar cusp and high latitude tail regions.The tail current system reflects explicitly the effect of dayside magnetospheric compression which is caused by the solar wind. In addition, the position of the plasma sheet relative to the ecliptic plane depends explicitly on the tilt angle of the Earth's dipole. Near the tail axis, the tail field is approximately in a self-consistent equilibrium with the tail currents and the isotropic thermal plasma.The models for the equatorial ring current depend on the D_st-parameter. They are self-consistent with respect to measured energy distributions of ring current protons and the axially symmetric part of the magnetospheric field.     

Quantitative magnetospheric magnetic field modelling with toroidal and poloidal vector fields

Toroidal and poloidal vector fields allow divergence free magnetic field representations in regions where currents flow. We derive general magnetospheric magnetic fields using combinations of spherical harmonic expansions of the toroidal and poloidal fields. Adding restrictive conditions like the field line topology symmetry or the magnetic field measurements, more specific magnetospheric magnetic field models can be derived. Two examples of this technique are given : an axisymmetric model with a ring current in the equatorial region and a time-dependent model of the Earth's magnetosphere. Our results are compared with the Olson-Pfitzer model.     

Induced magnetic field effects at planet Mercury

At Mercury's surface external magnetic field contributions caused by magnetospheric current systems play a much more important role than at Earth. They are subjected to temporal variations and therefore will induce currents in the large conductive iron core. These currents give rise to an additional magnetic field superposing the planetary field. We present a model to estimate the size of the induced fields using a magnetospheric magnetic field model with time-varying magnetopause position. For the Hermean interior we assume a two-layer conductivity distribution. We found out that about half of the surface magnetic field is due to magnetospheric or induced currents. The induced fields achieve 7-12% of the mean surface magnetic intensity of the internal planetary field, depending on the core size. The magnetic field was also modeled for a satellite moving along a polar orbit in the Hermean magnetosphere, showing the importance of a careful separation of the magnetic field measurements.     

The reason for magnetospheric substorms and solar flares

It has been proposed that magnetospheric substorms and solar flares are a result of the same mechanism. In our view this mechanism is connected with the escape, or attempted escape, of energized plasma from a region of closed magnetic field lines bounded by a magnetic bottle. In the case of the Earth, it must be plasma that is able to maintain a discrete auroral arc, and we propose that the cross-tail current connected to the arc is filamentary in nature to provide the field-aligned current sheet above the arc. A localized meander of such an intense current filament could be caused by a tearing instability in the neutral sheet. Such a meander will cause an inductive electric field opposing the current change everywhere. In trying to reduce the component of the induction electric field parallel to the magnetic field lines, the plasma must enhance the transverse or cross-tail component; this action leads to eruptive behavior, in agreement with tearing theories. This enhanced induction electric field will cause a discharge along the magnetic neutral line at the apex of the magnetic arches, constituting an impulsive acceleration of all charged particles originally near the neutral line. The products of this phase then undergo betatron acceleration for a second phase. This discharge eventually reduces the electric field along the neutral line, and thereafter the enclosed magnetic flux through the neutral line remains nearly constant. The result is a plasmoid that has definite identity; its buoyancy leads to its escape. The auroral breakup (and solar flare) is the complex plasma response to the changing electromagnetic field.     

Effect of finite parallel conductivity on the magnetospheric convection

The magnetospheric plasma convection is studied, taking into account the finite conductivity along magnetic field lines. Field-aligned currents flowing at the inner boundary of the magnetospheric plasma sheet give rise to parallel electric fields which insignificantly affect the convection on the ionospheric level but change drastically the convection system in the magnetosphere. Intense azimuthal convective streams arise along both sides of the plasma sheet boundary. A part of convection lines appears to be completely closed in the inner magnetosphere.     

A magnetospheric field model incorporating the OGO 3 and 5 magnetic field observations

A magnetospheric field model is presented in which the usually assumed toroidal ring current is replaced by a circular disk current of finite thickness that extends from the tail to geocentric distances less than 3R _E. The drastic departure of this model from the concept of the conventional ring current lies in that the current is continuous from the tail to the inner magnetosphere. This conceptual change was required to account for the recent results of analysis of the OGO 3 and 5 magnetic field observations. In the present model the cross-tail current flows along circular arcs concentric with the Earth and completes circuit via surface currents on the magnetopause. Apart from these return currents in the tail magnetopause, Mead's (1964) model is used for the field from the magnetopause current. The difference scalar field, ΔB, defined as the difference between the scalar field calculated from the present model and the magnitude of the dipole field is found to be in gross agreement with the observed ΔB (i.e. the observed scalar field minus a scalar reference geomagnetic field). An updated version of the ΔB contours from the OGO 3 and 5 observations, which is used for the comparison, is presented in this paper. Significant differences in details exist, however, between the model and the observed results. These differences will provide a guide for making modifications in the equatorial current system in future models.     

Observations of Magnetic Fields on T Tauri Stars

We present measurements of magnetic field strength and geometry on the surfaces of T Tauri stars (TTS) with and without circumstellar disks. We use these measurements to argue that magnetospheric accretion models should not assume that a fixed fraction of the stellar surface contains magnetic field lines that couple with the disk. We predict the fractional area of accretion footpoints, using magnetospheric accretion models and assuming field strength is roughly constant for all TTS. Analysis of Zeeman broadened infrared line profiles shows that individual TTS each have a distribution of surface magnetic field strengths extending up to 6 kG. Averaging over this distribution yields mean magnetic field strengths of 1-3 kG for all TTS, regardless of whether the star is surrounded by a disk. These strong magnetic fields suggest that magnetic pressure dominates gas pressure in TTS photospheres, indicating the need for new model atmospheres. The He I 5876 Å emission line in TTS can be strongly polarized, so that magnetic field lines at the footpoints of accretion have uniform polarity. The circular polarization signal appears to be rotationally modulated, implying that accretion and perhaps the magnetosphere are not axisymmetric. Time series spectropolarimetry is fitted reasonably well by a simple model with one magnetic spot on the surface of a rotating star. On the other hand, spectropolarimetry of photospheric absorption lines rules out a global dipolar field at the stellar surface for at least some TTS.     

The Aurora and the magnetosphere: The Chapman memorial lecture

Magnetospheric physics owes its beginnings to the seventeenth- and eighteenth-century scientists who were fascinated by one of the most spectacular natural phenomena, the aurora. In the first section, a brief historical account of the growth of magnetospheric physics and solar-terrestrial physics is given.The main part of the paper reviews recent progress in magnetospheric physics, in particular, in understanding the magnetospheric substorm. A number of magnetospheric phenomena can now be understood by viewing the solar wind-magnetosphere interaction as an MHD dynamo; auroral phenomena are powered by the dynamo. We have also succeeded in identifying magnetospheric responses to variations of the north-south and east-west components of the interplanetary magnetic field.The magnetospheric substorm is entirely different from the responses of the magnetosphere to the southward component of the interplanetary magnetic field. It may be associated with the formation of a neutral line within the plasma sheet and with an enhanced reconnection along the line. A number of substorm-associated phenomena can be understood by noting that the new neutral line formation is caused by a short-circuiting of a part of the magnetotail current.     

Can a slowly rotating neutron star be a radio pulsar?

It is shown that the radius of curvature of magnetic field lines in the polar region of a rotating magnetized neutron star can be significantly less than the usual radius of curvature of the dipole magnetic field. The magnetic field in the polar cap is distorted by toroidal electric currents flowing in the neutron star crust. These currents close up the magnetospheric currents driven by the electron–positron plasma generation process in the pulsar magnetosphere. Owing to the decrease in the radius of curvature, electron–positron plasma generation becomes possible even for slowly rotating neutron stars, with   PB ^−2/3₁₂ < 10 s  , where P is the period of star rotation and   B ₁₂= B /10¹² G  is the magnitude of the magnetic field on the star surface.     

Possible relations between the rotational axis of the Earth's inner core and the magnetic dipole axis

The geomagnetic field is maintained by amagnetohydrodynamic dynamo process within the liquid outer core. The distribution of the associated electric currents is modified if the outer core is bounded by electrically conducting material. Then, eddy currents and the related magnetic fields are generated within these regions. In particular, the relative rigid rotation of the inner core produces a secondary magnetic field, which is superimposed on the dynamo field. The angle between the dipole axis of the total field and the rotational axis of the inner core is an important quantity needed for the theory of polar motion of the Earth. This angle is investigated for a broad spectrum of angular velocities of the inner core. To simplify the mathematical procedure, we model the dynamo field using an axisymmetric field generated by a system of electric currents within the outer core. The conductivity of the mantle is neglected. We find that the position of the dipole axis depends on the angular velocity of the inner core as well as on the distribution of the current system within the outer core. Coincidence of both axes can be reached if the angular velocity is high enough and if the current system is concentrated within a thin sheet near the outer core-inner core boundary.     

Currents over the auroral arc

The three-dimensional current system over an enhanced conductivity strip identified with an auroral arc is calculated for the case of the magnetospheric plasma convection across this strip. The strip produces a stationary Alfvén wave which propagates along magnetic field lines and is carried simultaneously by the convecting plasma. The Alfvén wave generation corresponds to an appearance of field-aligned currents over the arc. The three-dimensional current system generated over the arc is studied, taking into account reflection of the waves from the ionosphere of the opposite hemisphere. The correspondence of the theory with the experimental results is found.     

Mercury’s magnetospheric magnetic field after the first two MESSENGER flybys

The “paraboloid” model of Mercury’s magnetospheric magnetic field is used to determine the best-fit magnetospheric current system and internal dipole parameters from magnetic field measurements taken during the first and second MESSENGER flybys of Mercury on 14 January and 6 October 2008. Together with magnetic field measurements taken during the Mariner 10 flybys on 29 March 1974 and 16 March 1975, there exist three low-latitude traversals separated in longitude and one high-latitude encounter. From our model formulation and fitting procedure a Mercury dipole moment of 196 nT ·  (where R_M is Mercury’s radius) was determined. The dipole is offset from Mercury’s center by 405 km in the northward direction. The dipole inclination to Mercury’s rotation axis is relatively small, ∼4°, with an eastern longitude of 193° for the dipole northern pole. Our model is based on the a priori assumption that the dipole position and the moment orientation and strength do not change in time. The root mean square (rms) deviation between the Mariner 10 and MESSENGER magnetic field measurements and the predictions of our model for all four flybys is 10.7 nT. For each magnetic field component the rms residual is ∼6 nT or about 1.5% of the maximum measured magnetic field, ∼400 nT. This level of agreement is possible only because the magnetospheric current system parameters have been determined separately for each flyby. The magnetospheric stand-off distance, the distance from the planet’s center to the inner edge of the tail current sheet, the tail lobe magnetic flux, and the displacement of the tail current sheet relative to the Mercury solar-magnetospheric equatorial plane have been determined independently for each flyby. The magnetic flux in the tail lobes varied from 3.8 to 5.9 MWb; the subsolar magnetopause stand-off distance from 1.28 to 1.43 R_M; and the distance to the inner edge of the current sheet from 1.23 to 1.32 R_M. The differences in the current systems between the first and second MESSENGER flybys are attributed to the effects of strong magnetic reconnection driven by southward interplanetary magnetic field during the latter flyby.     

Model experiment for the interaction of solar plasma stream and geomagnetic field

A laboratory experiment is designed to study the interaction of the solar wind with the geomagnetic field. Time-exposure and time-resolved photographs are taken when plasma hits a model Earth, and direct measurements are made of the magnetic field change, plasma density and electric current distribution. The shape of the magnetic cavity formed on the upstream side of the model Earth is almost the same as that calculated for the geomagnetic cavity. The charged particles, which penetrate the magnetic cavity formed on the upstream side of the model Earth with east-west asymmetry from the neutral points on the cavity surface, appear to concentrate towards the equator on the rear side of the model, forming a westward electric current belt within the magnetosphere. When the dipole axis is not perpendicular to the plasma gun—magnetic dipole line, the invasion of plasma is more pronounced at the cusp of the cavity nearer to the gun. Charged particles appear to penetrate to a greater extent if a uniform external magnetic field is applied parallel to the magnetic dipole than if one is applied antiparallel.     

Electric fields and currents in the ionosphere generated by field-aligned currents observed by TRIAD

On the basis of the experimental data on the ionospheric conductivities and field-aligned currents the electric fields and currents in the ionosphere generated by the field-aligned currents were computated for various magnetic activity conditions. The model of the ionospheric conductivities by Vanyan and Osipova (1975) was used taking into account the influence of the universal time seasons and magnetic activity. The field-aligned current patterns and their change with magnetic activity was set on the basis of the TRIAD data. It is shown that the calculated patterns of the ionospheric electric fields and currents are in agreement with the measured electric fields and the equivalent current systems of the magnetic disturbances in high latitudes. The conclusion is made that the magnetospheric field-aligned currents are the main sources of the presently known polar magnetic disturbances.     

Modelling of magnetic field measurements at Mercury

Mapping Mercury's internal magnetic field with a magnetometer in closed orbit around the planet will provide valuable information about its internal structure. By measuring magnetic field multipoles of order higher than the dipole we could, in principle, determine some properties, such as size and location, of the internal source. Here we try to quantify these expectations. Using conceptual models, we simulate the actual measurement during the BepiColombo mission, and then we analyze the simulated data in order to estimate the measurement errors due to the limited spatial sampling. We also investigate our ability to locate the field generating current system within the planet. Finally, we address the main limitation of our model, due to the presence of time-varying external magnetospheric currents.     

Model for the onset of a magnetospheric substorm

In view of observations which show that a substorm often begins in a small local time sector, a model is assumed in which the neutral sheet current is diverted around a small region we call a bubble. The simplest assumption is that of a linear variation of current with distance from the centre of the bubble in the x-direction in a SM coordinate system, with the diverted current being channelled within narrow paths of width δ_y on the dawn and dusk sides of the bubble. This assumption leads to vector potential integrals that can be evaluated analytically. The addition of this current loop into the magnetotail results in a magnetic field structure where new neutral lines of X- and 0-type can be observed; these are connected to each other as a continuous neutral ring in the xy equatorial plane. The magnetic and electric field components around the neutral regions are calculated, and the time dependent evolution of the neutral ring is studied. Comparison with some published satellite observations shows good agreement. Taking typical values for the various quantities on the basis of actual observations within the magnetotail, we show that the induced electric field is at least comparable to the average cross-tail electrostatic field, and it may well be one or two orders of magnitude greater. The response of the plasma to the induction field is discussed qualitatively. It is concluded that field aligned currents may be produced due to inertial forces of the expanding disturbance. Interpretation of the ground based precipitation patterns of energized particles during auroral breakup is given.     

On the cause of northward magnetic field along the negative X axis during magnetospheric substorms

The magnetic fields produced by a three-dimensional current system, consisting of a flow into the morning part of the auroral oval along tail-like field lines, along the auroral oval and out from the evening part of the oval along tail-like field lines, are computed. It is demonstrated that the major parts of the well-known ‘positive bay’ in low latitudes on the Earth's surface, the positive H variation at the synchronous distance and the positive B_s variation along the magnetotail during magnetospheric substorms can be caused by the proposed current system.     

A model of the heliospheric magnetic field configuration

A three-dimensional model of the magnetic field configuration in the heliosphere is constructed by assuming that the interplanetary magnetic field consists of four components, (i) the solar dipole, (ii) a large number of small spherical dipoles located along an equatorial circle just inside the Sun (representing the magnetic field line arcade), (iii) the field of the poloidal current system generated by the solar unipolar induction and (iv) the field of an extensive current disc around the Sun lying in the ecliptic plane. The magnetic field intensity at a distance of 1 A.U. (about 20 R⊙ above the ecliptic plane) is normalized to fit the observed spiral configuration.     

Model pulsar magnetospheres: the perpendicular rotator

The simplest model illustrating the effect of the magnetospheric charge-current field on the structure of a pulsar magnetic field has the region within the light-cylinder filled with the GoldreichJulian charge density which corotates with the neutron star, but has no electric currents along the magnetic field lines. This model has previously been studied for the axisymmetric case, with the rotation and magnetic dipolar axes aligned. The analogous problem is now solved with the two axes mutually perpendicular, so that not only the material current arising from the rotating charges but also the displacement current contributes. Again, the constructed magnetic field B ₀ crosses the light-cylinder normally, and there is no energy flux to infinity. However, in a more realistic model there is a flow of current along B ₀, generating a field B ₁ which has a non-vanishing toroidal component at the light-cylinder, so yielding a finite integrated Poynting flux.     

Plasma-sheet dynamics and magnetospheric substorms

We present a conceptual model of the formation of the plasma sheet and of its dynamical behavior in association with magnetospheric substorms. We assume that plasma mantle particles $\text{E}\text{～}\text{×}\text{B}\text{～}$ drift toward the current sheet in the center of the tail where they are accelerated by magnetic-field annihilation to form the plasma sheet. Because of the velocity-dependent access of mantle particles to the current sheet, we argue that the convection electric field and the corresponding rate of field annihilation decrease with increasing radial distance. As a consequence, there exists no steady-state configuration for the plasma sheet, which must instead shrink continuously in thickness until the near-earth portion of the current sheet is disrupted by the formation of a magnetic neutral line. The current-sheet disruption launches a large-amplitude hydromagnetic wave which is largely reflected from the ionosphere. The reflected wave sets the neutral line in motion away from the earth; the neutral line comes to rest at a distance (which we estimate to be a few hundred earth radii) where the incoming mantle particles enter the current sheet at the local Alfvén velocity. At this “Alfvén point” reconnection ceases and the thinning of the plasma sheet begins again. Within this model, the magnetospheric substorm (which is associated with the current-sheet disruption) is a cyclical phenomenon whose frequency is proportional to the rate of convection in the magnetospheric tail.