P. A. Webb and E. A. Essex

Cooperative Research Centre for Satellite Systems

Department of Physics, La Trobe University

Bundoora, Melbourne VIC 3083,

Australia

TEL: (03) 9479-2469

FAX: (03) 9479-1552

Email: p.webb@latrobe.edu.au,

e.essex@latrobe.edu.au

Abstract

With the recent advent of Global Positioning System (GPS) satellites as a research tool in studying the ionized regions that surround the Earth a need has arisen for a simple yet accurate model for plasmasphere, the region above the ionosphere. A model based on diffuse equilibrium has been developed, which is combined with the International Reference Ionosphere model to produce a global ionosphere-plasmasphere model. Some preliminary results from the model are presented and discussed.

The ionosphere is a layer of ionization
that surrounds the Earth, starting from an altitude of ~60km and with a
median altitude of approximately 400 km. The extension of the upper ionosphere
along the Earth's closed magnetic field lines is known as the plasmasphere.
A radio signal transmitted from a satellite to a ground receiver travels
through this ionized medium; the effects on the propagating signal depend
on the frequency and on the plasma density along the ray path. The effect
of the plasma is measured in Total Electron Content (TEC), which is defined
as the total number of electrons in a column with a cross sectional area
1 m^{2} along the ray path. TEC is measured in Total Electron Content
Units (TECU), where 1 TECU = 10^{16} electrons/m^{2}. By
studying the TEC derived from satellite signals it is possible to study
the ionized medium through which they have propagated.

The GPS satellites orbit at an altitude of 20,200 km with an orbital plane inclination of 55 degrees. Radio signals received from GPS satellites must propagate through some 19,500 km of plasmasphere and the underlying ionosphere. The plasmasphere's electron density is on average one to four orders of magnitudes less than the ionosphere's. However, as a result of the greater distance travelled through the plasmasphere, the total amount of the plasmasphere's plasma traversed by the signal is usually appreciable compared with that in the ionosphere. Consequently the plasmasphere's effect on a GPS signal needs to be considered, especially around solar cycle minimum and at night.

An increasing amount of the ionospheric electron density research being conducted at La Trobe University is based on GPS satellite data. It has therefore become important to understand the effect the plasmasphere's electron content has on GPS measurements, so that they can be correctly interpreted. To study this effect, a global electron density model of the ionosphere and plasmasphere has been developed.

In the past many computer-based models of the plasmasphere have been created, the FLIP model (Torr et al. 1990) and the Sheffield model (Bailey and Balan 1996) being two current examples. The primary aim of these models is to provide an understanding of, and to model the complex physical processes of the plasma flows along the magnetic field lines. In general these models required a computer workstation, because of the large number of calculations required. The primary aim of our research, however, is to produce a simple yet accurate global electron density model which can be run on a desktop personal computer (PC). The major goal of the model is to allow a user to specify a ray path between any two locations and have the model quickly and accurately determine the electron density at each point along the ray path. With this information, parameters such as TEC can easily be calculated.

The ionosphere is created by photoionization
of the neutral upper atmosphere by extreme ultraviolet and ultraviolet
solar radiation. The ionosphere typically has a peak around 300 km altitude,
which can vary between 200 km and 600 km. The peak ionospheric electron
density is of the order 10^{10} to 10^{12} electrons/m^{3}.
The peak height and the peak electron density depend on solar activity,
geographical location, time and season.

In the 1950's, through studies of the propagation of electromagnetic waves along the Earth's magnetic field lines, it was found that the region beyond the ionosphere, now known as the plasmasphere, had unexpectedly high electron densities. It was realised that the source of this plasma was the reaction:

(1)

which is accidentally energetically resonant and proceeds rapidly in both directions.

Chemical equilibrium as controlled
by Equation (1) is established at altitudes where the oxygen ions (O^{+})
are the dominant species. Consequently, the lighter hydrogen ions (H^{+})
experience an outward force due to the ambipolar electric field created
between the oxygen dominated ions and the electrons. Being charged, the
H^{+} are constrained to move parallel to the Earth's magnetic
field lines. At lower latitudes where the Earth's field lines are closed,
the upward flowing H^{+} are trapped in the region centred on the
magnetic equator. In the polar regions where the field lines are open,
the H^{+} do not become trapped and flow out into space, producing
what is called the 'polar wind'. The trapped plasma forms the plasmasphere,
which generally extends out to 3 to 6 Earth radii (some 20,000 km to 40,000
km), with plasma densities of the order of 10^{9} electrons/m^{3}.
The shape and density of the plasmasphere is constantly changing, primarily
due to changes in the underlying ionosphere (the source and sink of the
plasmasphere) and the effects of magnetic storms.

The global electron density model of the ionosphere/plasmasphere that has been developed is based on diffusive equilibrium, with chemical equilibrium used at low altitudes. The model requires three outside major sources of data:

- Electron and ion temperatures.
- The neutral temperature, oxygen and hydrogen densities.
- Ionospheric electron density.

The required neutral parameters are obtained from the MSIS-90 model (Hedin 1991, 1986). The ionospheric electron densities are obtained from the IRI95 model (Bilitza 1995). Both these models can be down loaded as FORTRAN programmes from the NSSDS Web Site (NSSDC 1999). The ion and electron temperatures are discussed below.

The electron and ion temperatures are calculated using a modified version of the upper ionosphere and plasmasphere temperature model published by Titheridge (1998). While it is beyond the scope of this paper to discuss these modifications in detail, in summary they are:

- The separation of the geographic and geomagnetic co-ordinates.
- The modifications to the sunrise and sunset shaping parameters.
- Re-calculation of the base temperature and gradient functions, using a reduced and modified satellite data set.
- The modification of the base temperatures to remove effects of averaging over a seven hour time period during both day and night.

The reader should refer to Titheridge (1998) for a description of these terms and their relevance to the temperature model.

Two examples of electron temperature profiles obtained from several models are shown in Figure 1. The plots include profiles from the FLIP model, Titheridge's original model, the modified version of Titheridge's model, and the IRI95 model. Note that the IRI95 temperatures are not smoothly varying with altitude, compared with the other models, which is due to the IRI95 profiles being calculated from independent satellite data sets (Brace andTheis (1981)).

While there is generally good agreement between the different temperature models during the day, there are large differences at night. One of the reasons for this is the different bases of the respective models. The FLIP model is theoretical, the IRI95 is an empirical model, and the Titheridge model is a combination of theory and empirical data. This is a small example of one of the major problems with research in the upper ionosphere and plasmasphere, namely the lack of consistency between models and observed temperatures. A modified version of Titheridge's model is used because no other " stand-alone" temperature model for this region exists (the IRI95 model does not extend above 3,000 km altitude).

The model that has been developed is based on diffusive equilibrium. Because the plasma is constrained to move along magnetic field lines, the diffusive profiles must be calculated along the field lines rather than simply in the vertical direction as is the case with the neutral.

atmosphere. The diffusive equilibrium equations used are given by Titheridge (1972) and are reproduced in equations (2) and (3). The scale height is given by:

(2)

where H_{j} is the j^{th} ion
scale height, k is the Boltzmann constant, T_{i} is the ion temperature,
m_{j}is the j^{th} ion mass, g is the gravitational acceleration,
m_{a} is the mean ion mass, T_{e} is the electron temperature
and the total temperature is T_{t} = T_{i} + T_{e}.
Taking n_{j0} as the ion number density at height h_{0}
where the total temperature T_{t0}, the ion number density n_{j}
at the height h_{0} + Δ h where the total temperature T_{t}
is given by:

(3)

As Titheridge notes, "the form given above is particularly suitable for numerical integration". If the density of a given ion species at a certain base height is known, then by using incremental steps, its density can be calculated at some greater height. This is only possible if the temperatures are known at each step and no chemical ion production or loss is occurring. Because of the large amount of production and loss, diffusive equilibrium profiles are not always accurate in the ionosphere, and chemical equilibrium profiles need to be considered.

The simplest approach to producing
an upper ionosphere/plasmasphere model is to consider O^{+} and
H^{+} ions only. Since O^{+} is the dominant species above
~150 km, its profiles can accurately be modelled using a diffusive profile
at altitudes above the ionosphere's peak density. To model the H^{+},
chemical equilibrium is used in the ionosphere and diffusive equilibrium
is used for the upper ionosphere and plasmasphere. The ion density profiles
produced by this approach agree with the observed profiles when a magnetic
field line (more correctly the magnetic flux tube) is "saturated", which
results in zero net H^{+} flow between the ionosphere and the plasmasphere.
The major reasons for this agreement is the approximate equality of the
diffusion and chemical scale heights (Richards and Torr 1985) for the H^{+}
when they are the minority ion species. The close equality of the scale
heights alleviates the need to determine a boundary at which to convert
from chemical to diffusive control of H^{+} profile, since in this
case diffusive equilibrium can be used from the starting altitude.

However, the plasmasphere is rarely
saturated, except at lower latitudes where the magnetic field line (flux
tube) volumes are small, and so changes in ion density initiate a quick
response. This lack of saturation is primarily due to solar magnetic storms
that periodically empty the outer plasmasphere, after which it starts to
refill. Diurnal plasmaspheric variations also occur due to changes in ionospheric
and plasmaspheric temperatures, changes in the ionospheric electron densities
and variations in the neutral atomic densities. The result is a highly
dynamic plasmasphere with plasma consisting predominantly of H^{+}
flowing up into the plasmasphere or flowing down into the ionosphere. Unless
the field line is saturated, a simple model that "projects up" (meaning
to use numerical integration in a series of small altitude "steps") using
chemical and diffusive equilibrium from the ionosphere will generally over
estimate the true density in the plasmasphere by several orders of magnitude.
This is because a field line that is in the process of refilling after
a storm will have H^{+} densities less than its saturated values,
which are equal to the values obtained by projecting up the field line
with chemical and diffusive equilibrium. This approach also has the problem
that H^{+} profiles projected up from northern and southern hemispheres
along a given field line will rarely have the same densities at the magnetic
equator, which in reality are the same.

A solution to these problems was suggested
by P. G. Richards (private communication, author of FLIP model mentioned
previously) ands forms the basic principle behind the developed model.
Assuming that the total H^{+} content of a field line is known,
then to a good approximation the equatorial density of a field line is
given by the total content divided by the total volume of the field line.
This is due to most of the volume been centred on the magnetic equator,
where the density is only slowly varying. This approximation is generally
correct to within 5%. Starting from this equatorial density, the H^{+}
density profile is projected down the field line into both hemispheres
using diffusive equilibrium (equations (2) and (3)). This automatically
solves the problem of "equatorial mismatch" between the H^{+} densities,
which occurs when the profiles were projected up the field lines from the
two hemispheres. A lower initial H^{+} equatorial density, which
occurs if the H^{+} tube content is less than the saturated content,
then represents a field line depleted by magnetic storm activity. The O^{+}
profile is still obtained by projecting up from the underlying ionosphere.
This is appropriate since the ionospheric O^{+} densities are primarily
controlled by chemical processes and the solar flux interactions with the
neutral atmosphere, and consequently recover very quickly after a magnetic
storm.

This approach creates the problem that
as the diffusive H^{+} profiles reach down into the ionosphere
they need to be matched with the chemical equilibrium H^{+} profiles.
A desire to have close agreement between the model under development and
the
FLIP model gives some guidance as to the desired profile that the H^{+}
should have in this region. A method has been developed which smoothly
joins the two profiles and is in good agreement with the FLIP model. While
the details are also beyond the scope this paper, the method is based upon
an equation derived by Richards and Torr (1985), which estimates of the
height where chemical loss is equal to diffusive loss. The corrected equation
(Rasmussen et al. 1993) is given below:

(4)

where z_{0} is the height where
chemical and diffusive height are equal, z_{r} is a reference height,
H is the diffusion scale height, H_{1} is the O^{+} scale
height, H_{2} is the O scale height, n_{r}(O^{+})
is the O^{+} number density and n_{r}(O) is the O number
density, both at the reference height r. All heights are in km, and densities
are in m^{-3}. Based on this equation a weighting function has
been derived which allows the two hydrogen ion profiles to be smoothly
joined.

A procedure for the accurate production
of the H^{+} profiles along a given field line has been presented
above. The final consideration is the time evolution of the H^{+}
content of the field line. As indicated earlier, the source of H^{+}
in the plasmasphere is represented in equation (1). The production rate
of H^{+} in m^{-3}s^{-1} is given by (see for example
Richards and Torr 1985):

(5)

and the loss rate by:

(6)

where T_{n} is the neutral
temperature, and n(#) is the number density of the given ion or neutral
atom species (denoted by #). Most of the H^{+} production and loss
occurs in the upper ionosphere and lower plasmasphere, due to the rapid
reduction in the neutral oxygen and oxygen ion densities with altitude,
which retards both production and loss at higher altitudes. Since at any
time the neutral densities and ion densities are known, both the production
and loss of H^{+} for a given field line can be calculated. Assuming
the density profiles do not change over a small time increment, the net
change in the H^{+} content of the field line can be calculated.
Advancing time by the given time increment, a new equatorial H^{+}
density can be calculated as before and the process repeated. This allows
the time development of the H^{+} content of a magnetic field line
to be modelled.

The other dynamic aspect of the plasmasphere that needs to be considered is the motion through space of the individual magnetic field lines, or more correctly the motion of the plasma aligned with a magnetic field line. At low magnetic latitudes the plasma remains attached to the field lines and co-rotates with the Earth. At higher latitudes the electric fields in the Earth's magnetosphere caused the plasma to move from one magnetic field line to another, with the magnitude of this motion being directly related to the solar activity. During periods of large solar magnetic storms the outer plasmasphere is torn away from the Earth by the increased strengths of these electric fields and lost into the outer magnetosphere.

In the first version of the model under
development it had been assumed that the plasma co-rotates with the Earth
magnetic field lines at all latitudes. The effects of magnetic storms are
included in this case by assuming that a field line is emptied of its plasma
if it is between 1200 magnetic local time (MLT) and 1800 MLT, and is currently
outside the plasmapause, the outer boundary of the plasmasphere. Ignoring
the plasmasphere's experimentally observed evening bugle, the location
of the plasmapause (the outer plasmasphere's boundary) in the magnetic
equatorial plane is determined by using equation (7) (Gallagher et al.
1995), where L_{pp} is in Earth radii.

(7)

Equation 7 shows that as the magnetic
storm activity increases (higher K_{p}) the plasmapause moves inwards.
The choice of the time interval 1200 MLT to 1800 MLT for the flux tubes
to be emptied is a first order approximation to the results obtained when
a magnetospheric electric field is included, which is the next level of
sophistication in modelling this process. At other times the plasma generally
co-rotates with the Earth, even if outside the plasmapause.

Up to this point, this paper has discussed
a general method to obtain the H^{+} and O^{+} densities
along a field line. However, many observations such as TEC require the
electron density and not the ion densities. This is obtained by assuming
the plasmasphere and ionosphere has a neutral charge at each point, thus
the electron density is taken to be equal to the sum of the H^{+}
and O^{+} densities at any given point. The electron density at
a required point can be determined by calculating the electron densities
along the field line that the point of interest lies, as described previously.
The field line is broken up into pre-determined distance intervals and
interpolation is used to determine the electron density at the require
point of interest. The electron density along a ray path is obtained by
breaking the path into a series of points, at which the electron densities
are separately calculated using interpolation along their respective field
lines.

Because the model has only been recently completed, the results that will be presented here are preliminary. They do however indicate the type of results that the model is capable of producing, noting that its primary goal is to calculate the electron density at any point surrounding the Earth up to a radius of 5 Earth radii.

With the ability to calculate electron densities along a ray path, it is straightforward to calculate TEC. Figure 2 shows an example where the vertical TEC has been calculated between the North and South magnetic poles at150° magnetic longitude.

Unlike direct satellite observations where only the total TEC can be measured, the contributions to the TEC from various altitude bands can be calculated. The examples shown in Figure 2 are for near solar minimum, at midday and midnight respectively. The height of 20,200 km is chosen as it corresponds to the altitude of the GPS satellites. It should be noted that the TEC above 20,200 km predicted by the model is at the most 0.15 TECU, which is insignificant.

Taking 800 km as being the transition from the upper ionosphere to the plasmasphere, the following features can be noted. Figure 2 clearly shows the symmetrical nature of the plasmasphere resulting from the magnetic field lines that control its shape. This leads to the offset from the geographic equator of the plasmasphere and the ionospheric anomaly by 10° in this case. The plasmasphere exists mainly between magnetic latitudes of ± 50°, and hence there is normally little contribution to the TEC from above 800 km in the polar regions. In the equatorial region, the plasmasphere contribution to the TEC is the order of 30%. Note that during the night in the southern hemisphere this contribution is over 50% and one consequence of this will be discussed below.

Knowledge of the electron densities along a ray path allows another ionospheric parameter, the Median Ionospheric Height to be calculated. This is the altitude at which half the TEC is above and half is below along a ray path. It is required for example when converting from slant TEC to vertical TEC, and is normally assumed to be a constant altitude of 400 km. In Figure 3 the Median Ionospheric Height has been calculated along the same magnetic longitude as Figure 2.

Figure 3 shows that a constant Median Ionospheric Height of 400 km is, in this case, generally a good approximation, except during the night in the southern middle latitudes where the correct value is over 1000 km. This is caused by the plasmasphere contributing approximately 50% to the TEC, as shown in Figure 2. Consequently any vertical TEC calculated from slant TEC using the 400 km approximation will be in error due to the actual incident angle of the ray path of the satellite signal being slightly smaller than the assumed incident angle.

Other results that can be obtained from the model include studying the plasmasphere's recovery after magnetic storms, and the effects on observed TEC measurements from GPS satellites. The length of time taken by the plasmasphere to recover can be estimated by comparing the refilling of the magnetic field lines at various times during the recovery phase. With the ability to calculate the plasmaspheric contribution to the observed TEC from a GPS satellite, it is possible to remove it so that the ionospheric contribution can be separately determined. This correction will allow more accurate studies of the ionosphere to be undertaken using GPS satellite signals. The model will be used to study, for example, the expected TEC along ray paths between two satellites, such as the Australian FedSat satellite and the GPS constellation.

A global electron density model of the ionosphere/plasmasphere has been developed at La Trobe University, which allows the calculation of electron densities at any point up to 5 Earth radii. This allows the densities along a given ray path to be calculated, from which parameters such as the Total Electron Content and Median Ionospheric Height can be determined. Knowledge of these parameters is important in allowing the correct interpretation of GPS satellite signals received by ground stations after passing through the ionosphere and plasmasphere.

This research is supported by an Australian Research Council (ARC) grant. P. Webb is supported by a La Trobe University Postgraduate Award (LUPA) and the Cooperative Research Centre for Satellite Systems (CRCSS). The authors wish to thank P. G. Richards of the University of Alabama and J. Titheridge from The University of Auckland for advice in the development of the model.

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