Bouncing Wave Packet

The concept of the bouncing wave packet evolved over several years during the period when the structure of the magnetosphere and its relation to the high-speed solar wind was just beginning to be understood. Several decades of ground-based observations had confirmed the pervasiveness of wave modes with repeating structures, and the period of repetition appeared to align with the theoretical travel times of waves and particles along magnetic field lines.

The earliest explanation for the repetition period relied on the travel time of discrete bunches of trapped particles. Jacobs and Watanabe[1962] andJacobs and Watanabe[1963] proposed a two- step generation mechanism, in which a stochastic process of Alfvén wave generation in the outer magnetosphere would isolate wave disturbances through their connection to the “frozen-in” bunch of trapped ions. This bunch would travel along field lines carrying the alfvénic disturbance until they encountered the ionospheric boundary. Here specific frequencies resonant with the Alfvén cavity within the ionosphere centered at the Alfvén speed minimum near the F2 ionization peak undergo conversion to the electromagnetic wave mode [Piddington, 1959]. The bunch of trapped particles and the alfvénic disturbance would reflect along the field lines to travel to the opposite hemisphere, while the electromagnetic waves would propagate to their points of observation at lower altitudes. The particles and disturbance would then continue to reflect, generating bursts of Pc1 activity at regular intervals. The travel time of the particles between ionospheres would depend upon the length of the field lines, so if these populations were originally distributed in L, they would undergo a dispersive signature. Jacobs and Watanabe [1963] describes this method as a way to explain the “fan-type” spectral signatures seen in some events.

This concept was later refined by Obayashi [1965] to examine the bounce motion of individual wave packets rather than particle bunches. These isolated wave packets would be made up of EMIC wave activity generated in the now standard source region at the magnetic equator3_{. These}

waves would travel along the magnetic field lines where they would be reflected at the conducting boundary that is the ionosphere. These reflected emissions would then travel back towards the conjugate ionosphere, passing through the wave generation region to become reenergized, repeating their motion. Assuming a dipole field configuration and a cold plasma density that falls off with an inverse cube law asn=n0(a/r)3, the transit time for this wave packet is given by

τ = 4aL 5/2 B0/ √ 4πn0M · ∫ ϕ0 0 cos4ϕ { 1±₂ω_ω c cos6ϕ √ 1 + 3sin2_ϕ } { 1±_ωω c cos6_ϕ √ 1 + 3sin2_ϕ }3/2 dϕ (4.20)

where r/a = L cos2ϕ traces the shape of a dipolar field line by the magnetic latitude ϕ and

the dimensionless parameter L defined as the multiple of the Earth’s radius a = 6371 km at the magnetic equator,ω/ωcis the ratio of the wave excitation to the equatorial ion cyclotron frequency,

B0 = 0.31Gauss is the Earth’s magnetic moment, and M is the ion mass. This initial simplistic

model assumed a quasineutral plasma composed entirely of protons, but the wave travel velocity can be computed with different mass loads along the field line in regimes of heavy element occurrence.

Obayashi [1965] additionally showed that the frequency-dependent phase velocity given byV⃗ph= ω_⃗k

would explain the rising tones often observed in sonograms, and that repeated bounces would result in the “fan-type” dispersive signatures observed in hydromagnetic (HM) chorus. Glangeaud and

Lacoume[1971] discuss the improved guiding of a bouncing wave packet along field lines by steep

density gradients, implying that density boundaries such as the well-defined plasmapause in the late morning sector are ideal regions to confine reflecting packets.

3_{This is due to the concentration of particle populations with pitch angles closer to}

90◦at magnetic field minima,

Figure 4-7: Conjugate ground-based Pc1 pearl pulsation observations from Gendrin and Troitskaya

[1965] (Reprinted courtesy of the National Institute of Standards and Technology, U.S. Department of Commerce).

The symmetry of the Earths magnetic field across the magnetic equator and reflective iono- spheres in both hemispheres would imply that not only will a wave packet be repeatedly reflected between these two hemispheres, but that observations of penetrating wave power should alternately be observable on the ground at magnetically conjugate stations. Observations have been made that show this alternation of wave power in the Pc1 band at conjugate sites [Jacobs and Watanabe, 1963;

Gendrin and Troitskaya, 1965] such as observed in Figure 4-7. As this example shows, however, a

time shift of 80 seconds showed similar waveform signatures in each hemisphere, but this packet was then proceeded after 56 seconds by a different signature. While at first these observations appear to satisfy the conditions of a bouncing wave packet, closer inspection reveals that there is more likely a wave source region transmitting wave power simultaneously towards both stations, but that there is either a difference between the distances from this source region to each observatory, or there exists some medium which delays one of these signals. While the two stations recorded byGendrin

and Troitskaya [1965] are generally conjugate, it may be that one is only observing a signal that

has been ducted across the ionospheric resonator, a common complication of ground-based EMIC observations.

The most convincing evidence to prove this method would of course be in situ observations of the ionospherically reflecting packets. Spacecraft in the magnetosphere equipped with sufficiently sensitive fluxgate magnetometers can measure EMIC wave activity, and modern magnetic field models such as that developed by Tsyganenko and Sitnov [2005] can provide accurate predictions of magnetic field line conjugacy with ground-based observatories. Such conjugate observations have

been performed using the Viking spacecraft [Mursula and Rasinkangas, 1997], the Time History of Events and Macroscale Interactions during Substorms mission [Usanova et al., 2008], and the Van Allen Probes mission [Paulson et al., 2014]. However, each of these conjugate observations of wave activity demonstrated the same modulation periodicity in situ as on the ground, meaning that any reflected wave packets did not return to the magnetospheric point of observation. Additionally, large studies of EMIC activity by spacecraft missions have demonstrated unidirectional Poynting flux measurements towards the ionosphere once outside the equatorial generation region [Fraser

et al., 1996; Loto’aniu, 2005], again implying that reflected wave packets did not return to the

spacecraft.

In chapter 5 we will discuss how the measurements we made for this study not only show similar wave behavior patterns, but in fact we find additional qualities of this wave mode which disagree with this generation mechanism.