The Alfvén Velocity in the Magnetosphere and Its Relationship to ELF Emissions

RANDE K. BURTON AND CHRISTOPHER T. RUSSELL

Summary. Ogo 5 measurements of the ion density and magnetic field strength near the equator in the dawn quadrant of the magnetosphere have been used to determine the Alfvén velocity. The Alfvén velocity in the outer magnetosphere ranges between 1000 and 2000 km/sec. At decreasing radial distances the Alfvén velocity increases to a maximum (average, 4800 km/sec) and drops to a minimum (average, 490 km/sec) just inside the plasmasphere. ELF emissions are well ordered by the Alfvén velocity profiles. ELF chorus from 100 to 1000 Hz exists in the outer magnetosphere only for Alfvén velocities below about 3000 km/sec. ELF hiss is found immediately inside the plasmapause. This relationship between ELF emissions and the Alfvén velocity is further support for unstable wave generation by Doppler-shifted cyclotron resonance.

A knowledge of the Alfvén velocity throughout the magnetosphere or, equivalently, the magnetic energy density per particle is important in many areas of magnetospheric physics, such as studies of transient magnetic phenomena, explanations of ULF magnetospheric resonances, and studies of wave-particle interactions via ion and electron cyclotron resonance. In this note we present Ogo 5 measurements of the Alfvén velocity near the equator in the dawn quadrant of the magnetosphere. We then show the relationship between the velocity profiles and the measurements of ELF emissions on the same spacecraft.

The Alfvén velocity presented here is simply the field strength divided by the square root of 4 times the proton mass density. We have used 1-min averages of the magnetic field strength, as measured by the UCLA fluxgate magnetometer, and of the ion density, as measured by the Lockheed ion mass spectrometer. A detailed description of the magnetometer has been given by Aubry et al. [1970] and Snare and Benjamin [1966] and of the spectrometer by Harris and Sharp [1969].

Figure 1 shows the profile obtained for an inbound pass on April 7, 1968. The left scale shows the Alfvén velocity; the right scale gives the magnetic energy density per particle (per ion or per electron). The satellite position during the pass is given in terms of L value, magnetic latitude , and local time LT.

Fig. 1. The Alfvén velocity and magnetic energy density per particle as measured on the inbound pass on April 7, 1968.

The Lockheed ion mass spectrometer measures energies only up to 600 ev. The outer magnetosphere, however, contains energetic particles above 600 ev with number densities of about 1/cm³ [Vasyliunas, 1968; Schield and Frank, 1969]. Because the Alfvén velocity depends on the total number density, we have arbitrarily assigned a number density of 1 particle/cm³ whenever the measured density fell below this limit. These points are shown by open, circles on the plot. The solid circles represent measured number densities above 1 particle/cm³. We have plotted points only every 5 or 10 min as changes warranted. Within the plasmasphere the number density should be roughly constant along field lines within approximately ±30^o of the magnetic equator [Angerami and Carpenter, 1966]; accordingly, we have indicated by a dashed line the equatorial Alfvén velocity, using the expected field strength at the equator on the same field line and local number density. The field at the equator was obtained from the multipole expansion provided on the Ogo 5 orbit tapes. Confidence in this technique was gained by comparing predicted and measured fields at the satellite. These agreed to better than 5% in the region in which the extrapolations were made. The Alfvén velocity at the satellite in the outer magnetosphere rim from 2000 to 5500 km/sec just outside the plasmapause and falls to 1500 km/sec within the plasmasphere. Correspondingly, the magnetic energy per particle (ion or electron)E_c rises from 20 to almost 200 kev and falls to about 10 kev within the plasmasphere. The minimum Alfvén velocity extrapolated to the equator is 800 km/sec and the minimum E_c is 3.4 kev.

At the top of the figure the amplitude of the ELF signals is shown, as measured in the spectrum analyzer Phannels of the JPL-UCLA search coil magnetometer on this pass. Frandsen et al. [1969] have described this instrument. We refer to the signals in the outer magnetosphere as chorus because these signals usually have discrete structures which may or may not be accompanied by a band of structureless noise. The ELF signals at low L values, when examined with high-resolution observations, do not have discrete structures, and we refer to them as hiss.

The chorus in Figure 1 stops when the Alfvén velocity reaches 3400 km/sec or E_c reaches 60 kev. The hiss starts just inside the plasmasphere. Because of the rapid change in the plasma parameters it is not possible to assign a specific velocity when this occurs.

Figure 2 shows the velocity profile on the perigee pass of April 15, 1968. Here the Alfvén velocity reaches 9700 km/sec corresponding to an E_c of over 400 kev. Within the plasmasphere, however, the velocity drops to almost 100 km/ sec or an E_c of 6 kev. The minimum velocity extrapolated to the equator is 400 km/sec, and the minimum E_c is 0.9 kev. The chorus in the outer, magnetosphere stops when the Alfvén velocity rises above 3000 km/sec, and the hiss starts immediately inside the plasmasphere.

Fig. 2. The Alfvén velocity and magnetic energy density per particle as measured on the inbound pass on April 15, 1968.

Figure 3 shows the profile on April 22, 1968, again rising from about 1000 km/sec in the outer magnetosphere to 8300 km/sec just outside the plasmasphere. Inside the plasmasphere the local velocity drops to 600 km/sec. The minimum equatorial velocity is 350 km/sec. Chorus is present for E_c less than 50 kev in the outer magnetosphere, and hiss begins at the plasmapause and extends to lower radial distances.

Fig. 3. The Alfvén velocity and magnetic energy density per particle as measured on the inbound pass on April 22, 1968.

Figure 4 shows the profile on April 17, 1968. Here the maximum Alfvén velocity reached just outside the plasmapause is only 4200 km/sec, but the minimum inside the plasmasphere is about the same as the previous figure, 410 km/sec. Where the ELF chorus stops there are some missing data points for the Alfvén velocity, but apparently the velocity is close to 3000 km/sec. We note, however, that the chorus actually stops here. In the previous examples it may be argued that the chorus frequency rose above the passband of the search coil. However, in this case the signals are well within the search coil passband when they terminate.

Fig. 4. The Alfvén velocity and magnetic energy density per particle as measured on the inbound pass on April 17, 1968.

We have examined 14 near-equatorial passes in the dawn quadrant. All passes with adequate data coverage were used. There were large oscillations in the ion density in the outer magnetosphere on a few passes. The four passes used as illustrations in this paper contained no such large oscillations. The average Alfvén velocity in the far outer magnetosphere (beyond L=8) is 1400 km/sec. This number, however, is dependent on our arbitrary choice of density because the ion mass spectrometer rarely measures more than 1 ion/cm³ in this region. The average maximum Alfvén velocity reached just outside the plasmapause was 4800 km/sec. The average minimum Alfvén velocity at the equator was calculated to be 490 km/sec.

The chorus in the outer magnetosphere stopped or left the search coil-passband when the Alfvén velocity reached an average of 2300 km/sec, corresponding to an E_c of 34.5 kev. The hiss began just inside the plasmapause in every case.

From this brief examination of the Alfvén velocity in one region of space we see several things. First, at the plasmapause the Alfvén velocity changes quite rapidly. Just inside the plasmapause the Alfvén velocity reaches a magnetospheric minimum; just outside it reaches its magnetospheric maximum (excluding the tail). Second, the values of these velocity maximums and minimums depend critically on the position of the plasmapause and the density in the outer plasmasphere. Thus, the measured plasmapause position and outer plasmaspheric density are desirable indices to have at all times for quantitative study of both VLF and ULF wave phenomena.

The correlation of ELF emissions with the Alfvén velocity is expected in any theory that involves any resonance of particles with any wave describable by the cold plasma dispersion relation, because the resonant energy must then be proportional to the magnetic energy per particle purely from dimensional considerations. (This would include theories using cyclotron resonance, Landau resonance, perpendicularly propagating whistlers, and so forth.) However, the observed, increase in frequency with decreasing distance is consistent with a cyclotron resonance.

For example, we can show how our observations are consistent with the mechanism of Kennel and Petschek [1966] that employs a cyclotron resonance in combination with an anisotropic electron pitch angle distribution to generate ELF waves. The resonant energy for this mechanism is equal to the magnetic energy per particle times a function of / such that the resonant energy increases with increasing / .

For typical chorus frequencies from 0.25 to 0.5 _ the resonant parallel energy varies from 1.69 to 0.25 E_c. Thus if waves are generated by resonance with the same energy electrons, the wave frequency must become closer to the electron gyrofrequency as E_c increases on an inbound pass. In addition, the gyrofrequency increases, and hence the wave frequency must increase absolutely; it is reasonable, therefore, to assume that some of the time the chorus simply rises above the search coil passband on an inbound pass, thus accounting for the apparent cutoff. On the other hand, the instability is possible only if the electron pitch angle anisotropy exceeds a critical value that decreases with increasing / .

Thus as the wave frequency approaches the electron gyrofrequency the anisotropy required for wave generation increases. Therefore, there is a limit to the unstable wave frequencies that is defined by the pitch angle anisotropy. As one proceeds radially inward, E_c increases, and there is some point at which an initially unstable electron flux will stabilize to wave generation. This situation is presumably true for Figure 4: the waves simply stopped without rising out of the search coil passband. The reappearance of ELF noise within the plasmasphere is not surprising. When E_c is small, as in the plasma sphere, the wave frequency that resonates with 40- to 100-kev electrons is a small fraction of the electron gyrofrequency. Thus, the critical anisotropy for wave growth is small, and even electrons with only a weak pitch angle anisotropy destabilize to ELF wave generation.

We should also comment on the variability of the parameters we have measured. In the outer magnetosphere densities measured by the Lockheed spectrometer are rarely above 1 cm^-3. Thus, most of our densities are assumed in this region. At the point at which the chorus stopped, the average Alfvén velocity was 2300 km/sec, and the standard deviation was 1100 km/sec. Part of this spread in Alfvén velocity, or equivalently, E_c, at the chorus cutoff is probably due to variation in plasma density, because many chorus cutoffs occurred where we assumed a density of 1 cm^-3. Another part could be due to variations in the pitch angle anisotropy from orbit to orbit. Using analog data from the UCLA search coil magnetometers, we can determine completely the electron's parallel resonant energy under the assumption of cyclotron resonance. We are currently investigating this and will report on it in the near future.

The average Alfvén velocity maximum outside the plasmapause was 4800 km/sec, with a standard deviation of 2700 km/sec. This variability is real and meaningful because the maximum occurred at a measurable density. Similarly the average minimum equatorial Alfvén velocity of 490 km/sec with a standard deviation of 190 km/see is accurate.

The Alfvén velocity in the outer magnetosphere lies between 1000 and 2000 km/sec. At decreasing radial distances it rises to a maximum and falls to a minimum just inside the plasmasphere. The values of the maximum and minimum vary from day to day and are best defined by actual measurement. However, in the dawn quadrant in the period studied, the average maximum was 4800 km/sec and the average minimum was 490 km/sec. ELF emissions correlate with these profiles. ELF chorus from 100 to 1000 Hz is found only for Alfvén velocities below about 3000 km/sec in the outer magnetosphere. We note that the definition of chorus used here does not preclude an associated continuous band of noise. ELF hiss is always found immediately inside the plasmapause at these local times. The correlation of the ELF emissions with the Alfvén velocity supports the theory of wave-particle resonance. The increase in frequency of the ELF emissions with decreasing distance requires that the resonance be cyclotron. We have compared our results with the results of the Kennel-Petschek mechanism and found them consistent. However, such consistency does not eliminate other possible cyclotron resonance mechanisms.

Acknowledgments. We would like to thank the referee for many helpful suggestions. The principal investigators responsible for the Ogo 5 search coil magnetometer were Drs. R. E. Holzer and E. J. Smith; for the fluxgate magnetometer, Drs. P. J. Coleman, Jr., T. A. Farley, and D. Judge; and for the ion mass spectrometer, Dr. G. W. Sharp.

This report represents one aspect of research done by the Jet Propulsion Laboratory for the National Aeronautics and Space Administration under NASA contract 7-100, GSFC-623- S-70-21. Financial support for the work at the University of California was provided by the Jet Propulsion Laboratory under contract 950- 403 and National Aeronautics and Space Administration contracts NGR-05-007-235 and NAS- 5-9098 and also at Lockheed by NASA contract NAS-5-9092.

The Editor wishes to thank V. M. Vasyliunas for his assistance in evaluating this paper.

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Aubry, M., M. Kivelson, and C. T. Russell, The motion and structure of the magnetopause, to be submitted to J. Geophys. Res., 1970.

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