A method of mapping electric fields along geomagnetic field lines is applied to the IGRF (International Geomagnetic Reference Field) model. The method involves integrating additional sets of first order differential equations simultaneously with those for tracing a magnetic field line. These provide a measure of the rate of change of the separation of two magnetic field lines separated by an infinitesimal amount. From the results of the integration Faraday's law is used to compute the electric field as a function of position along the field line. Examples of computations from a software package developed to implement the method are presented. This is expected to be of use in conjugate studies of magnetospheric phenomena such as SuperDARN (Super Dual Auroral Radar) observations of convection in conjugate hemispheres, or comparison of satellite electric field observations with fields measured in the ionosphere.

In an accompanying paper

Electric field mapping is important for conjugate studies of ionospheric and
magnetospheric convection. In this paper we discuss its application to the
IGRF (International Geomagnetic Reference Field)

Because we have limited this paper only to the Earth's internal field, the
results will only be suitable for use at fairly low altitude or at lower
latitudes, where the external fields are small. Future work will incorporate
the external fields in the form of the

The method of calculating the electric field mapping is described in detail in
Paper 1. It involves integrating a set of nine simultaneous
first order differential equations:

The IGRF is fixed in a frame rotating with the Earth. The coordinate system in
which it is expressed is the

In the

The transformation of

The transformation of the operator

For transformations between Cartesian systems

In Eqs. (

The electric field components are found using Faraday's law as described in
paper 1. The more elementary discussion there means that if

The internal magnetic field of the earth

The International Geomagnetic Reference Field (IGRF) is characterized by the
coefficients

The magnetic field is given by

The elements of the tensor

We require

The method of calculation is essentially the same as that described in Paper 1,
except that the magnetic field and the gradient tensor are found from the
IGRF rather than the Harris model. The integration is carried out in the GEO
system. The initial values for the integration are the coordinates in the GEO
system of
the starting point on the field line, and the components of the unit vectors
perpendicular to the magnetic field in the azimuthal and meridional directions.
The normalization of the field line separations is in terms of the initial
infinitesimal separation element, hence the initial values have magnitude unity.
At each step of the integration the magnetic field and gradient tensor are
computed in the

We present here some examples of mapping in the IGRF. It should be borne in mind that, because the external field is not included, the techniques are only valid for mid to low invariant latitudes, or for low altitudes.

In order to map to the ionosphere, either from the conjugate ionosphere in the
opposite hemisphere, or from a spacecraft position, after each step in the
integration we evaluate the difference between the calculated height and the
height at which to terminate the integration:

When

Normalized field line separation for a field line originating from
Sanae Antarctica (72.0

To illustrate the procedure we select a starting point at Sanae Antarctica
(

Figure

Different aspects of the geomagnetic field line originating from Sanae
Antarctica (Geographic coordinates

The electric field can be found at points along the field line by defining its
value at the starting point, by using Eqs. (52) and (53) of Paper 1. As an
example, let
its component in the meridian direction be 40 mV m

Electric field as a function of distance measured along the field line.

The components and total electric field as a function of distance measured
along the magnetic field line are shown in Fig.

Projection of electric field on the coordinate planes for the case
shown in Fig.

It should be noted that, while the electric field components are inversely
proportional to the field line separation, the magnetic field is inversely
proportional to the cross sectional area of the flux tube, which is of the
order of the square of the field line separation. The convection velocity,
with magnitude

Another representation is shown in Fig.

Comparison of electric fields at conjugate points. Left-hand panel: conjugate points in opposite hemispheres. Right-hand panel: conjugate points at Sanae and the magnetic equator.

We define the intersection with the magnetic equatorial surface to be the point
on the field line where

We need not show any graphical information for this case since it is the same as the initial half of the trace for the previous case.

The purpose of this paper has been to implement the method of electric field
mapping described by

The results of calculations show that the method is viable and generally useful. Currently the limitations are that it can only be used where external magnetic fields are small enough to be neglected, that is not between points on field lines that extend into a region where the external fields are important. This limitation will be removed by applying the method to the Tsyganenko field model.

Of course, the method only applies to electrostatic fields. In practice this means that scale of time variation of the electric field must be long compared with the Alfvén transit time along the field line. So long as we are only considering conditions where the IGRF is a good representation of the total magnetic field this is reasonable. Extending the method to greater radii and more active conditions will require implementation using the Tsyganenko models. In such cases it will be necessary to proceed with caution, ensuring that the electric fields induced by the changing magnetic fields are small compared with the electrostatic field.

A software package is being developed for general use and will be released under an open source licence.

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.