Institute of Geophysics and Planetary Physics, University of California, Los Angeles,
CA 90095-1567, USA
Originally published in:
Adv. Space Res., 20, 327-334, 1997.
The Dessler-Parker-Sckopke relationship states that the fractional depression in the magnetic field averaged over the surface of a non-conducting Earth is 2/3 of the ratio of the energy of the charged particles trapped in the Earth's magnetic field to the energy of that magnetic field above the surface of the Earth. When the interior conductivity of the Earth is included in the calculation, a moderate storm with a 100 nT depression corresponds to a ring current whose total energy is 2.8 x 1015J. Nevertheless deducing the ring current strength from ground magnetograms is not quite as simple as it would seem from the DPS relationship. The ring current is often not symmetric and the observing stations are not uniformly situated, nor are there enough of them. Accurate baselines are required and the strength of the quiet day ring current is unknown.
Despite these uncertainties, disturbances in the Dst index can be predicted quite well when accurate measurements of the solar wind and interplanetary magnetic field that strike the magnetosphere are available, by use of the formula of Burton et al. (1975). In this model the field on the surface of the Earth has two components: the magnetopause current flowing on the surface of the magnetopause and the ring current in the equatorial regions. The magnetopause current is taken to be proportional to the square root of the solar wind dynamic pressure. The ring current is energized by the solar wind east-west electric field and de-energized through an exponential decay of the energy into the upper atmosphere. This simple formula works quite well despite the fact that the solar wind input to the magnetosphere must be controlled by additional factors such as the dynamic pressure of the solar wind and the electrical conductivity of the ionosphere.
Improvements are needed in order to increase the advance warning time for solar wind conditions reaching the Earth. To achieve such an improvement requires multipoint measurements near the Earth-sun line. These measurements should be both continuous and significantly closer to the Sun. Such measurements are technically feasible today and affordable using solar-sailing "sentinels" carrying a small payload consisting of a solar wind analyzer and magnetometer, and possibly energetic particles.
The first realistic model of the magnetosphere grew out of an attempt to provide a physically realistic explanation of the magnetic storm, and the ring current that formed during the main phase of the storm (Chapman and Ferraro, 1931). Once rockets and spacecraft were able to probe the distant reaches of the magnetosphere, they found it was full of energetic plasma, so full in the outer magnetosphere that it could not contain any additional energetic plasma. The plasma beta was much greater than unity. When the energy content of the manetosphere plasma does increase, it does so by storing the energy in the innermost portions of the magnetosphere where the magnetic field has not reached its saturation level of plasma energy density, and beta is less than unity. The stronger the storm the deeper in the magnetosphere this energy storage takes place (Russell and Thorne, 1970). Once the energetic particles enter the magnetosphere they begin to be lost, eventually to the ionosphere if they penetrate very deep within the magnetosphere. This loss can be due to charge exchange with neutrals, wave-particle interactions, and classical collisions. The rate of decay will depend on the particle species, the process controlling the loss of the particles, and the specific magnetospheric conditions at the time of storm. It is surprising that a single loss rate is sufficiently accurate for most purposes.
The magnetosphere is also subject to a smaller disturbance, the substorm. Substorms do occur in conjunction with geomagnetic storms but they also occur at other times as well. They seem mainly associated with the dissipation of energy in the auroral regions. Both storms and substorms are controlled by the east-west component of the solar wind electric field. However the integrated solar wind electric field required for a substorm is less than that for a storm. Moreover, the storm requires a persistent strong electric field for the order of hours to build up to the intensities seen (Russell et al. 1974).
In this paper we review first how the ring current is deduced from ground based measurements and how that deduction can be improved with additional measurements. Then we examine how the ring current can be predicted from in situ measurements of the solar wind in front of the Earth. Finally, we discuss what improvements can be made both in the prediction formula and in the acquisition of the requisite solar wind input.
The energy of the particles stored in the ring current is simply related to the depression in the magnetic field on the surface of the Earth. This theoretical relationship was first derived by Dessler and Parker (1959) and then by Sckopke (1966) for a non-conducting Earth. The relationship can be most simply derived by considering a single energetic particle gyrating around the magnetic field with no motion parallel to the magnetic field while gradient drifting in longitude.
The guiding center drift of a charged particle gradient-drifting in the Earth's dipole equator is:
where W is the particle kinetic energy.
The magnetic field on the surface of the Earth due to this gradient drift is:
Bgd = -3/(4) oW/(RE3Bo) ez
This particle also has a magnetic moment
µ = -ez W/(BoL3)
This magnetic moment produces a field at the surface of the Earth
Bmm = o/(4) Wez/(RE3Bo)
Adding and summing over all particles
Btot = -o/(2) Wtot/(BoRE3) ez
But the total energy in the dipole field above the surface of the Earth is
Wmag = 4/(3o) Bo2 RE3
Btot/Bo = -(2/3) (Wtot/Wmag) ez
Btot/Bo = -(2/3) (Wtot/Wmag) ez
Accounting for exclusion of ring current field by the conducting Earth
Btot[nT] = 3.6 x 10-14 Wtot [J]
The magnetosphere is replete with current systems. On the outside of the magnetosphere the magnetopause current flows cross the front of the magnetosphere in a vortex about the polar cusps. This current also flows around the tail lobes and through the plasma sheet. The current across the dayside magnetopause acts to increase the field on the surface of the Earth. The sense of the current in the magnetotail is to reduce the current on the surface of the Earth. Currents also flow to transmit stress to the ionosphere from the magnetosphere that is being pushed and pulled by the solar wind and by magnetic and plasma forces within the magnetosphere. These currents flow along magnetic field lines and close in the ionosphere across the magnetic field. Field aligned current systems arise from interactions at the magnetopause, due to asymmetries in the plasma distribution in the ring current and due to substorm dynamics. The sketch in Figure 1 shows where some of these currents are thought to flow (Clauer and McPherron, 1980).
|Fig. 1. Sketch of the current systems thought to be responsible for the ring current and the partial ring current.|
To calculate the ring current in the midst of these myriad other currents is not easy, as has been recently emphasized by Campbell (1996). First, a set of ground magnetometer stations is selected that is out of the influence of both the equatorial electrojet and the auroral electrojet. The latitude range from 20o to 30o geomagnetic is generally believed to be most ideal for this purpose but often compromises have to be made between the optimum latitude and the need for equi-spaced longitudinal coverage. At present only 4 stations are being used: Kakioka, Honolulu, San Juan and Hermanus. Their locations are shown in Figure 2.
|Fig. 2. Location of existing magnetometer stations from which the Dst index is derived (circles) and proposed future observatories for this purpose (squares). The polar angle on this plot is geographic longitude the radial distance is the geomagnetic longitude. Closed symbols denote northern hemisphere stations.|
Since the ring current appears to be asymmetric from both terrestrial and in situ studies, it is important to have good longitudinal coverage. This would allow the deduction of an accurate measure of the symmetric part of the ring current as well as allow the asymmetric part to be found. T. Iyemori (personal communication, 1996) has suggested that Lanzhou, Tashkent, Easter Island and Los Acacias would provide a suitable complement to the present four stations. These sites are also shown in Figure 2. We note that 8 stations is still far less than used in such studies previously. For instance Clauer and McPherron (1980) used 19 low and midlatitude stations.
In order to calculate a Dst index that is truly representative of the energy in the ring current, it is necessary to take special care in deducing the baselines from which the deviations of the magnetic field are measured. These baselines vary because of internal field changes and possibly due to instrumental effects. They are best determined on quiet days but quiet days may be of different absolute quietude. One way to address this problem is to determine the asymptotic "baseline" (Araki et al., 1993).
In order to easily manipulate the data and calculate the index rapidly, digital data should be promptly made available at the data center responsible for Dst. Initially the calculation of Dst was under the supervision of M. Sugiura but more recently the work has been carried out by T. Araki, T. Iyemori and T. Kamei. Finally, we note that in situ studies with the CRRES mission near the equatorial plane shows that the ring current is generally quite asymmetric in local time (Hughes and Singer, 1996). While the CRRES satellite operated for a shorter period than is optimum for such studies we note that data do exist from ISEE and the ongoing POLAR mission that could supplement this study.
The algorithm for predicting the Dst index or ring current strength, Dsto, is based upon predicting the rate of change of the energy trapped in the charged particles in the magnetosphere. This consists of two terms, a rate of injection that is a function of the interplanetary electric field, E, and a decay of a constant fraction of that energy per unit time. Thus
d/dt Dsto = F(E) -a Dsto
The observed Dst index includes the magnetopause currents, the quiet day ring current as well as the disturbed ring current. To obtain a measure of the ring current, Dsto, free of these effects we use the formula:
Dsto = Dst - b(Pd)2 + c
where Pd is the solar wind dynamic pressure measured in nano Pascals.
The magnetosphere does not appear to extract energy from the solar wind for northward IMF or dusk-to-dawn electric fields. Thus Burton et al.  used the following formula for the injection term
F(E) = 0; Ey < 0.5 mV/m
F(E) = d(Ey - 0.5); Ey > 0.5 mV/m
Other constants were:
a = 3.6 x 10-5 s-1 b = 15.8 nT/(nPa)2
c = 20 nT d = -1.5 x 10-3 nT/(mV/m)/s
b = 15.8 nT/(nPa)2
c = 20 nT
d = -1.5 x 10-3 nT/(mV/m)/s
|Fig. 3. Dst predictions and observations using the Burton et al.  formula and ISEE-3 measurements of the solar wind and IMF.|
The Burton et al. formula in its original form has been used successfully for predicting Dst over the last 20 years. Figure 3 shows a sample of its predictive powers using ISEE-3 measurement as input to the algorithm [Lindsay 1996]. Generally the formula is excellent, although on occasion the formula underpredicts or overpredicts. These differences may be because we do not have the optimum measurement of the Dst index, because magnetospheric conditions altered the decay constant of the ring current, or because the injection parameter is dependent on more than just the rectified interplanetary electric field. For example, Scurry and Russell  suggested from an analysis of the Am index that high solar wind dynamic pressure increased the rate of reconnection and therefore the energy injection. High solar wind Mach numbers also seemed to affect the reconnection rate but in the opposite sense possibly due to a weakening of the magnetic field in the magnetosheath due to the high beta conditions behind the high Mach number shock. Such refinements still need to be made to possibly further improve the prediction of the Dst index.
Measurements of the solar wind just upstream of the Earth provide at most a few minutes warning of impending disturbances of the magnetosphere. Observations from the forward Lagrangian point provide about 45 minutes to 1 hour warning. However, operators of power and communication systems would like warnings several hours in advance. It may be possible to obtain such warnings.
Less than one day:
Predictions of the Dst index based on measurements made much closer to the sun when Pioneer Venus and Helios were near the Earth sun line have been made by Lindsay (1996). These predictions were useful to the extent that they provided advance warning of impending geomagnetic activity and its approximate magnitude, but the simple prediction of arrival time based on the solar wind velocity and the radial separation was often in error. We attribute this inaccuracy to the unknown geometry of the structures. In order to determine this structure we need multipoint measurements on scale sizes somewhat smaller than the scale of the overall structure of the solar wind disturbance. Since the largest disturbances are due to coronal mass ejections that have scale sizes up to 0.3 AU, we recommend a separation distance of about 0.1 AU for these multipoint measurements. Those who would benefit from accurate space weather predictions often request an advanced warning time of about 4 hours. This warning could be provided by spacecraft sitting at 0.95 AU heliocentric distance near the Earth-Sun line. If we were to deploy a string of spacecraft at 0.95 AU so that there was always a spacecraft within 0.05 AU of the Earth sun line many spacecraft would be required. The orbital period of spacecraft is shorter at 0.95 AU than at 1 AU, so that a spacecraft at 0.95 AU would be constantly drifting off the Earth-Sun line. However, there is a way to change the orbital period of a 0.95 AU spacecraft by erecting a large, ~100 m diameter, sail that reflects the sunlight, thereby effectively weakening the force pulling toward the Sun. Such a spacecraft configuration is shown in Figure 4. Figure 5 illustrates how these "solar sentinels" would be deployed to monitor the incoming solar disturbances.
|Fig. 4. Solar sail spacecraft suitable for a "sentinel" mission to measure the incoming solar wind.|
|Fig. 5. Schematic illustration (not to scale) of the proposed solar "sentinel" spacecraft configuration and a coronal "scout" spacecraft to forecast solar wind disturbances.|
Greater than one day:
The disturbances coming from the sun can be detected as they are ejected using orbiting coronographs. The material coming toward the Earth is best viewed at right angles to the Earth-sun line. The easiest way to obtain such a view is in the ecliptic plane at 90o in front of or behind the Earth. An equally advantageous view point is from high latitudes well above the ecliptic plane. Both views are in fact needed to reconstruct a stereoscopic picture of how the material is ejected. Energetically, an out-of-the-ecliptic mission is difficult. Solar sails, however, can again help us and over the course of a half dozen years reach a polar orbit that can be phased to provide a view of the sun constantly at a large angle (60o-90o) to the Earth-Sun line. With knowledge of the three-dimensional structure of the CMEs and a better understanding of the interaction of the CMEs with the ambient solar wind than we have today, we should be able to more accurately predict the time of arrival of the disturbance at Earth and its resulting magnitude than we can at present.
The underlying theory of why trapped energetic particles produce a ring current is well understood. However, measuring the ring current strength by the depression of the magnetic field observed on the surface of the Earth is still difficult. Much care needs to be taken and improved station coverage is needed. We recommend the installation and use of a 8-station Dst chain instead of the 4-station chain presently in use. This will not only improve the measurement of the symmetric ring current but also permit the measurement of the asymmetric part of the ring current.
Use of measurements of the solar wind near 1 AU and the Burton et al. (1975) formula provides an accurate prediction of disturbances in the ring current, limited mainly by one's ability to predict the solar wind and IMF parameters intersecting the Earth. Improvements can also be made in the solar wind coupling term to take account of the expected dependence of reconnection on the dynamic pressure and the Mach number of the solar wind.
Finally, we can increase the lead time of our predictions with novel advanced propulsion techniques that allow us to station our monitors at particularly advantageous locations. Solar sails would allow us to place spacecraft on the Earth-sun line at least 0.05 AU closer to the sun while maintaining a 1-year period. Three of these solar sail "sentinels" would allow us to determine the orientation of the disturbance in the solar wind and improve our predictions of both the size of the disturbance and the time of arrival. To obtain longer-term predictions, several days in advance, we need to monitor the corona. For coronal mass ejections, the solar wind structure responsible for the largest terrestrial disturbances, this is best done at right angles to the Earth-sun line with coronographs. For both locations in and out of the ecliptic plane, solar sails can provide the acceleration needed to change orbital inclination or phase. Thus we recommend continued development of solar sail spacecraft as well as improvement in our ground based monitoring and our prediction algorithms.
We would like to thank George Sprague of JPL for many helpful discussions on the design of solar sail missions. This research was supported by the National Science Foundation under research grant ATM 94-13081.
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