A mathematical formalism for modeling the poloidal magnetic field and current density at satellite altitudes is presented

Backus, G. E.

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MODELING CURRENTS AT SATELLITE ALTITUDES
G.E. Backus
IGPP, UCSD, A-025
La Jolla, CA 92093
In a conducting medium, the magnetic field cannot be written
B---V_ because _ J--VxB_O. However. B is still solenoidal, and any
0 "+ "_ _ -_ _
solenoidal field can be written B--_xAp+Aq where A=rx_. Let S(r) be
the spherical surface of radius r centered on the origin, and let <f>
be the average of the function f on S(r). For each r the poloidal ard
toroidal fields, _x_p and Aq, are uniquely determined on S(r) by B on
S(r), and p and q are determined up to arbitrary additive constants.
These can always be chosen uniquely by demanding <p> =<q> =0. A
toroidal field is a solenoidal field without radial component. In the
solid _pherical shell between S(a) and S(b), J--O if and only if q=O
and V_p--O. Thus a vacuum field is poloidal, and its poloidal scalar _s
harmonic. In a source-free shell, the poloidal scalar p represents B
as economically as does the magnetic potential @; and p has the
advantage that it continues to be physically meaningful where J_O.
Being solenoidal, the current density J.can be analyzed into
poloidal and toroidal parts, and in fact __ J--_xB=_xAq-_(V p). Thus
U ° °
the toroidal currents are the source of the polozdal magnetic fzeld and
the poloidal currents are the source of the toroidal magnetic field.
For each r4 the radial component of J on S(r) determines the toroidal
part of B on S(r). If B is known on S(r) when Jr is determined
there, as are the poloidal magnetic fields produced by the toroidal
currents inside S(r) and by those outside S(r). The sources of the
poloidal magnetic field on S(r) are inside and outside S(r), while the
s_urces of the toroidal magnetic field on S(r) are on S(r) itself. If
B is known on S(a) and S(b) then the radial averages in a<r<b of the
toroidal current and the tangential component of the poloidal current
can be determined.
Analysis of p and q into surface spherical harmonics can replace
the conventional Gaussian analysis of the vacuum field. However, the
radial dependence of the spherical harmonic coefficients for p and q is
arbitrary in current-carrying regions unless some further physical
hypothesis is introduced. At MAGSAT altitudes, a reasonable hypothesis
is JxB=O (field-aligned currents, or a force-free plasma). This
hypothesis greatly reduces the space of field-models to be considered,
and at MAGSAT altitudes it can be implemented by linear iteration with a
vacuum_f_e_d as the first step. One must recognize, however, that even
with JxB=O the magnetic effects of J are non-local. Polar currents
produce equatorial magnetic fields.
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https://ntrs.nasa.gov/search.jsp?R=19860003421 2020-03-20T15:46:51+00:00Z


Modeling currents at satellite altitudes

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