A note on the vector potential of Connerney et al.'s model of the equatorial current sheet in Jupiter's magnetosphere
Introduction
Although the model of Jupiter's equatorial current sheet proposed by Connerney et al. (1981) was formulated some 20 years ago in response to the fly-by field observations of the Pioneer and Voyager spacecraft, it is still much in use today in various studies of the jovian plasma environment. In recent years it has been used, for example, to study the distribution of plasma along realistic field lines in the Io torus (Maurice et al., 1997), to interpret ground-based images of the torus plasma (Schneider and Trauger, 1995), to order and interpret jovian energetic ion observations (Cowley et al., 1996; Laxton 1997, Laxton 1999; Anglin et al., 1997), to model the latitude dependence of the middle magnetosphere field (Bunce and Cowley, 2001), to estimate the contribution of external currents in studies of the internal planetary field (Dougherty et al., 1996; Connerney et al., 1996), to investigate the mapping of jovian auroras into the magnetosphere (Gérard et al., 1994; Satoh et al., 1996; Clarke et al., 1998; Connerney et al., 1998; Prangé et al., 1998; Vasavada et al., 1999), and to map individual wave, particle, and field features between the magnetosphere and ionosphere (e.g. Ladreiter et al., 1994; Zhang et al., 1995; Prangé et al., 1997; Dougherty et al., 1998). In this model the equatorial current sheet is taken to be a semi-infinite annular disc of constant half-thickness, in which the azimuthal current varies inversely with the distance from the axis of symmetry. The field of a finite current sheet, extending between two fixed radii, can then be found by adding the field of a second similar semi-infinite sheet in which the current is reversed in sense, which extends to infinity beyond the larger distance. The plane of the current sheet is taken either to be parallel to the magnetic equatorial plane, as in the original formulation, or may be found by fitting to data, as in the study by Connerney et al. (1982). Although this model current system is no doubt highly simplified relative to reality (see e.g. the discussion by Vasyliunas, 1983), it has nevertheless been found to provide a good initial framework within which to understand magnetic effects and magnetic mapping issues in the jovian magnetosphere.
Connerney et al. (1981) obtained an exact solution for the vector potential of a current sheet of the above form in terms of an integral over Bessel functions, from which integral expressions for the field components were found by taking the curl. These integrals were then evaluated numerically to provide the basis of their modelling results. Connerney et al. (1981) also presented simple analytic expressions for the field components based on approximations to the integrals (corrected versions of which appear in the Appendix of Acuña et al., 1983), and these have formed the basis of most subsequent applications of the model. However, direct information and related approximations for the vector potential of this model have not been presented to date, and it is the principal purpose of this note to do so. Knowledge of the vector potential is useful for three main purposes. First, the magnetic field derived from a vector potential is guaranteed to be divergence-free. Successive approximations to the vector potential thus yield successive approximations to the magnetic field which are guaranteed to be divergence-free at each stage. In applying the approximate forms to physical modelling it is an advantage to employ fields which obey the fundamental laws of physics exactly. Second, in the axisymmetric case, the vector potential is simply related to the flux function F. The magnetic field lines are then obtained from the contours F=constant, without the need for numerical integration. Third, in potential form, the field is easy to generalise to the non-axisymmetric case, such that, for example, departures of the current sheet from planarity due to propagation delay and tail-hinging effects can be incorporated (e.g. Khurana 1992, Khurana 1997). For these reasons we believe that an explicit presentation of the approximate forms for the vector potential of Connerney et al.'s model will be a useful contribution to the literature.
In this note we thus derive approximate forms for the vector potential of Connerney et al.'s model, and compare these with numerical values obtained from the exact integrals. In addition, we also present the corresponding divergence-free approximate forms for the magnetic fields, and compare them both with the numerical values from the corresponding integrals, and with the forms previously suggested by Connerney et al. (1981) and Acuña et al. (1983).
Section snippets
Integral formulas
In cylindrical co-ordinates (ρ,ϕ,z), the vector potential of an axisymmetric field is given by , where the field components, given by areThe flux function for such a field is given by F=ρA, such that a field line is given by F=constant, and the magnetic flux, dΦ, per radian of azimuth between the field lines F and F+dF is dΦ=dF. Connerney et al. (1981) solved Ampère's law for the vector potential of a semi-infinite axisymmetric current sheet,
Comparison of numerical and approximate analytic values
In this section we will display some results for the vector potential and field components obtained from the approximate formulas derived above, and will compare them with values obtained directly by numerical evaluation of the corresponding integrals. In Fig. 1 the solid lines show the normalised vector potential A′ obtained from the integral formulas , , plotted versus normalised ρ′ at various values of normalised z′. The vector potential has been normalised to (μ0I0/2)RJ, while the distances
Summary
The principal purpose of this note has been to provide simple analytic approximations for the vector potential, , of Connerney et al.'s (1981) model of the jovian equatorial current sheet. Two approximations for A have been derived, valid for small and large radial distances, respectively, relative to the radius of the inner edge of the current sheet. These have been compared with numerical values derived from exact integral expressions, and have been shown to provide very accurate estimates
Acknowledgements
We would like to thank Dr. J.E.P. Connerney for supplying us with his original version of the model current sheet field software, and for useful comments on an earlier version of this report. TME acknowledges the financial support provided by the Direccion General de Estudios de Posgrado, UNAM. EJB was supported by a PPARC Quota Studentship.
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