ALBERT

Description: A comprehensive review is presented of the mathematical models used to represent magnetic fields in the Earth's magnetosphere, of the way existing data-based models use these methods and of the associated problems and concepts. The magnetic field has five main components: the internal field, the magnetopause, the ring current, the tail and Birkeland currents. Methods of representing separately each of these are discussed, as is the deformation of magnetic fields; Appendix B traces the connection between deformations and the Cauchy integral. A summary section lists the uses of data-based models and their likely future evolution, and Appendix A supplements the text with a set of problems.

Description: A search was conducted for the signatures of Birkeland currents in the Earth's magnetic tail, using observed values of B(sub x) and B(sub y) from large sets of spacecraft data. The data were binned by x and y for -10 greater than x(sub GSM) greater than -35 and absolute value of y(sub GSM) less than or equal to 20 R(sub E) (less than or equal to 30 R(sub E) for x(sub GSM) less than or equal to -25 R(sub E)) and in each bin their distribution in the (B(sub x), B(sub y)) plane was fitted by least squares to a piecewise linear function. That gave average x-y distributions of the flaring angle between B(sub xy) and the x direction, as well as that angle's variation across the thickness of the plasma sheet. Angles obtained in the central plasma sheet differed from those derived near the lobe boundary. That is the expected signature if earthward or tailward Birkeland current sheets are embedded in the plasma sheet, and from this dfiference we derived the dawn-dusk profiles of the tail Birkeland currents for several x(sub GSM) intervals. It was found that (1) the Birkeland currents have the sense of region 1 currents, when mapped to the ionosphere; (2) both the linear current density (kiloamperes/R(sub E)) and the net magnitude of the field-aligned currents decrease rapidly down the tail; (3) the total Birkeland current at x approximately equals -10 R(sub E) equals approximately equals 500-700 kA, which is approx. 30% of the net region 1 current observed at ionospheric altitudes, in agreement with model mapping results; and (4) the B(sub z) and B(sub y) components of the interplanetary magnetic field influence the distribution of Birkeland currents in the tail.

Description: The Tsyganenko models contain 'modules' for two external current systems, the tail and ring currents; the former is the more detailed. No specific modules, however, cover the magnetopause and the Birkeland current system. The effects of these currents are represented by an all-purpose 'polynomial', and the resulting formulas involve about 30 parameters that specify the model. The dependence on magnetospheric indices is also simplified.

Description: The paper develops a simple representation of the global circuit of Birkeland currents on the basis of a representation of the current density (j) in terms of Euler potentials (alpha, chi). The underlying magnetic field, which shares the potential alpha with j, is assumed to be dipolar, making the model applicable chiefly to region 2 Birkeland currents. A form of j is chosen that gives a current sheet with peak outflow at dawn and peak inflow at dusk (or vice versa), connected across a flat polar cap sheet. The superposition of harmonics of the same type, centered at different 'foci', are found to provide a flexible and powerful representation of harmonic functions, accurate within less than 1 percent. An interpolation formula by which current sheets of finite width could be consistently represented is developed.

Description: A new method is developed for representing the magnetospheric field B as a distorted dipole field. Because Delta-B = 0 must be maintained, such a distortion may be viewed as a transformation of the vector potential A. The simplest form is a one-dimensional 'stretch transformation' along the x axis, concisely represented by the 'stretch function' f(x), which is also a convenient tool for representing features of the substorm cycle. One-dimensional stretch transformations are extended to spherical, cylindrical, and parabolic coordinates and then to arbitrary coordinates. It is shown that distortion transformations can be viewed as mappings of field lines from one pattern to another; the final result only requires knowledge of the field and not of the potentials. General transformations in Cartesian and arbitrary coordinates are derived, and applications to field modeling, field line motion, MHD modeling, and incompressible fluid dynamics are considered.

Description: The derivation of the magnetosphere configuration is discussed, with consideration given to both descriptive models, which represent the observed magnetic field B and perhaps the electric field E, and physical models, which also include plasma effects. Consideration is given to global field models, static physical magnetohydrostatic models, slowly variable E, and MHD tail models, and to the type of data used for their derivation. It is emphasized that the current trend in derivation of magnetospheric models is for incorporating more physics( e.g., V x B) to match regions of observed current flow, plasma pressure obeying equilibrium conditions, E linked to interplanetary sources, and time-dependent convective flows.

Description: A model of the earth's tail field with no current density in its high latitude lobes is obtained as a modification of Tsyganenko's tail model. This is achieved by replacing Tsyganenko's current element with one in which the current density ends abruptly at a distance D from the central axis.

Description: The distribution of the average north-south magnetic field component Bz in the vicinity of the neutral sheet has been investigated. This component is crucial for mappings between the nightside polar ionosphere and the equatorial magnetosphere. Data sets consisting of about 0.5 R(E) averages of magnetic field observations by the IMP/HEOS and ISEE spacecraft have been compared to the field predicted by the Tsyganenko (1987, 1989) models T87 and T89. It was found that both T87 and T89 underestimate Bz in the near tail region by as much as a factor of 2. Modified versions of the T87 model, incorporating plasma sheet warping, were obtained by fitting the model parameters via nonlinear least squares to the ISEE data set and yielded Bz values in agreement with the ISEE data. The study reveals an enormous scatter among the observed baseline values of Bz (on a time scale of 10-20 min), as well as intrinsic biases imposed by the mathematical structure of tail models, and these two factors (especially the first one) greatly limit the accuracy of model predictions of tail Bz.

Description: The present one-dimensional model analysis of substorm electrodynamics proceeds from the standard scenario in which the plasma sheet collapses into a neutral sheet, and magnetic merging occurs between the two tail lobes; plasma flows into the neutral sheet from the lobes and the sides, undergoing acceleration in the dawn-dusk direction. The process is modified by the tendency of the accelerated plasma to unbalance charge neutrality, leading to an exchange of electrons with the ionosphere in order to maintain neutrality. The cross-tail current is weakened by the diversion: this reduces the adjacent lobe-field intensity, but without notable effects apart from a slight expansion of the tail boundary.

Description: Early research on the earth's magnetic environment is reviewed, with attention given to the period when only ground-based observations were possible. Early work on geomagnetism is discussed as well as the sunspot cycle, solar fares, the possibility of electron beams from the sun, and the Chapman-Ferraro cavity. Consideration is also given to the ring current, Alfvens theory and electric fields, interplanetary plasma, and polar magnetic storms.