When studying magnetospheric convection, it is often necessary to map the steady-state electric field, measured at some point on a magnetic field line, to a magnetically conjugate point in the other hemisphere, or the equatorial plane, or at the position of a satellite. Such mapping is relatively easy in a dipole field although the appropriate formulae are not easily accessible. They are derived and reviewed here with some examples. It is not possible to derive such formulae in more realistic geomagnetic field models. A new method is described in this paper for accurate mapping of electric fields along field lines, which can be used for any field model in which the magnetic field and its spatial derivatives can be computed. From the spatial derivatives of the magnetic field three first order differential equations are derived for the components of the normalized element of separation of two closely spaced field lines. These can be integrated along with the magnetic field tracing equations and Faraday's law used to obtain the electric field as a function of distance measured along the magnetic field line. The method is tested in a simple model consisting of a dipole field plus a magnetotail model. The method is shown to be accurate, convenient, and suitable for use with more realistic geomagnetic field models.

In studies of magnetospheric convection, such as those by the SuperDARN network

Most mapping has been done by concentrating on the electric potential. For an
electrostatic field the magnetic field lines are equipotentials. If the field
is required several magnetic field lines are traced and their separation
calculated. From their separation the gradient of the electric potential is
estimated to give the electric field. Examples of this approach are in

This paper presents a new technique that can directly compute the electric field mapping in any geomagnetic field model provided that the magnetic field and its spatial derivatives are given at all points of interest. Provided that the magnetosphere is in a steady state, the electric field mapping is a straightforward application of Faraday's law. If the element of separation of two field lines can be calculated as a function of position on the field line, then, since the field lines are equipotentials, the electric field can be calculated. Since the separation is calculated by integrating an analytic formula along with the field line, its accuracy is the same as that of the field line position.

It is, however, also very useful to have as a reference the expected mapping
for a purely dipole field. The derivation of expressions for mapping in a dipole
field is straightforward, but, surprisingly, no convenient collection of the
relevant formulae seems to be available.

We therefore first provide an easily accessible derivation of a number of
relevant formulae for
a dipole field. They are useful for tutorial purposes and provide
analytic expressions for validating the new tensor field mapping techniques.
These mappings are used for some illustrative examples
which give rough estimates of how electric fields and convective drifts vary
with altitude.
For example, we make comparisons between convective drifts measured by the
DMSP satellites at about 840 km altitude and SuperDARN F-region
measurements in the 250–325 km altitude range, for

Finally we introduce a new method in which a second rank tensor is analytically
derived which satisfies a set of first order differential equations. This
tensor provides a
measure of the divergence and convergence of the field lines and can be found
by a step-by-step integration of the differential equations simultaneously with
the tracing of
the magnetic field line. This can be used to deduce the electric field without
the inaccuracies inherent in the method used, inter alia, by

All mapping techniques depend on the same principle. Except where there are
electric fields parallel to

We shall map the electric field components in a simple centred magnetic dipole
field. Expressions describing
the dipole field and its geometry are given, for example, by

The field is given
as a function of position by

The equation of a dipole field line is

The unit outward normal vector in the meridian plane is given by

Dipole field: directions of unit vectors.

We complete the right-handed system with a unit vector

The figure also shows the unit vectors

The element of perpendicular distance between two field lines that lie in the
meridian plane is

The perpendicular distance between two closely spaced field lines that lie in
the same

Sometimes we need to consider a horizontal path between the field lines

If we assume that there are no potential drops along the magnetic field lines,
then there can be no component of the electric field parallel to
the magnetic field, and therefore can only have components along

Since

The mapping of the perpendicular components of the electric field is now
straightforward. We specify a point on the field
line by its latitude

There are two closed paths we shall use for Eq. (

We can eliminate

Dipole field: meridian plane.

The total field can then be written

The convective or Hall drift is given by

We can take the cross product of

Equations (

Consider as an example the mapping from the ionosphere to a DMSP satellite.
Table 1 gives the ratios of the velocity components for mapping along field
lines of varying

Mapping of the convective drift from the ionosphere at height

Since

Mathematical models of the geomagnetic field are now available that not only
provide for a good fit to the Earth's interior field, but also
allow for the various exterior current systems arising from the interaction of
the solar wind and the magnetosphere.
The IGRF is a spherical harmonic representation of the interior field

We discuss the mapping of electric fields along field lines in such a model. It is not the intention of this paper to perform detailed calculations in the various realistic models; this will be left to a future paper. We discuss only the principles with illustrative calculations in a simplified model.

In what follows we use vector notation and Cartesian tensor notation

The unit vector parallel to the field line is

Now consider a field line passing through some point in space that is taken as
the origin as shown in Fig.

If we now advance a small distance

Definition of

The elements of

The differential width

The initial value for integrating (Eq.

Equations (

The mapped electric field component in the direction of

The procedure can be carried out for two initial values of

Covariant and contravariant components of

Although the two initial values of

We illustrate the use of the first order differential Eqs. (

The derivatives of the components of

For our simple model we use what are effectively GSM coordinates with origin at
the centre of the Earth,

The initial value of

The starting value for

Element of field line separation in the meridian plane for a dipole magnetic field, normalized to the value at the Earth's surface. The full line shows the values obtained by integrating the first order differential equations, while the crosses are obtained from the analytic expression.

An initial test of the method is to use Eqs. (

We have tested the validity of the method of field line mapping that is
described in Sect.

The integration process is the same as described above for a dipole. The
results for the field line coordinates

Field line
trace (blue) and electric field mapping
(red) in a Harris model with

Figure

Electric field
normalized to the value at
the starting point, for the field line traced and electric field mapped in Fig.

Because there is strong dependence of the electric field magnitude on radius
the field vectors are barely discernable at larger radii.
The diagram is not, however, intended to provide a quantitative picture of the
electric field mapping but to give a feel for the geometry.
A better idea of the variation of

It is not the intention of this paper to provide the results of a large number of computations in a model that is only intended to be illustrative. The purpose is to validate the technique of computation in a relatively simple model. We describe below the various checks that we have made on the accuracy of the integration technique.

Element of field
line separation in the electric field direction,
for the field line traced in Fig.

We have already described how, in a dipole model, we can use the integration
technique and compare the results with the explicit formulae presented in the
first part of the paper. The agreement is, in all cases, very good. With
integration steps of 1.5 to 2.0

In the Harris model the curvature of the field lines near the plasma sheet can
be quite large. If the step length is comparable with the radius of curvature,
truncation error becomes important. In this case the orthogonality of

Another check that will be particularly useful in more elaborate models is to
take two initial values of

In this paper we have introduced a new method of mapping electric fields along geomagnetic field lines. A set of three differential equations for the components of the normalized separation of two field lines has been obtained. These can be integrated simultaneously with the equations that trace the field line. Since the magnetic field lines are equipotentials, this allows the calculation of a component of the electric field as a function of distance measured along the field line. Two values of the normalized separation are required to give two field line components resulting in a total of nine first order differential equations that must be simultaneously integrated.

The computational effort in integrating the set of nine equations is the same as that for tracing three field lines. The accuracy of the process is better, however, since finding the electric field by finding the difference between the end positions of the two field lines requires taking small differences of large quantities. Such numerical differentiation is notoriously inaccurate.

The viability of the method has been carefully tested. The analytic expressions
for a number of relevant properties of a magnetic dipole field, while easily
derived, are not readily available in the literature. We have provided a
derivation of these for convenient reference and compared the results
calculated from them with those obtained by the integration method. We have also
tested the method in a qualitatively more realistic model of the night side
magnetosphere. All the tests show that the method is accurate and suitable for
mapping in more realistic models. In the accompanying paper

This work was supported by the South African National Research Foundation under grant 93068 (SANAE HF Radar), and by the University of KwaZulu-Natal through a research incentive grant. The topical editor, E. Roussos, thanks J. M. Ruohoniemi and one anonymous referee for help in evaluating this paper.