Volume 125, Issue 3 e2019JA027370
Research Article
Free Access

Multiharmonic Toroidal Standing Alfvén Waves in the Midnight Sector Observed During a Geomagnetically Quiet Period

Kazue Takahashi

Corresponding Author

Kazue Takahashi

The Johns Hopkins University Applied Physics Laboratory, Laurel, MD, USA

Correspondence to: K. Takahashi,

[email protected]

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Massimo Vellante

Massimo Vellante

Department of Physical and Chemical Sciences, University of L'Aquila, L'Aquila, Italy

Consorzio Area di Ricerca in Astrogeofisica, L'Aquila, Italy

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Alfredo Del Corpo

Alfredo Del Corpo

Department of Physical and Chemical Sciences, University of L'Aquila, L'Aquila, Italy

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Seth G. Claudepierre

Seth G. Claudepierre

Space Sciences Department, The Aerospace Corporation, Los Angeles, CA, USA

Department of Atmospheric and Oceanic Sciences, University of California, Los Angeles, CA, USA

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Craig Kletzing

Craig Kletzing

Department of Physics and Astronomy, University of Iowa, Iowa City, IA, USA

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John Wygant

John Wygant

School of Physics and Astronomy, University of Minnesota, Twin Cities, Minneapolis, MN, USA

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Kiyokazu Koga

Kiyokazu Koga

Japan Aerospace Exploration Agency, Tsukuba, Japan

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First published: 28 December 2019
Citations: 8


Excitation of toroidal mode standing Alfvén waves in the midnight sector of the inner magnetosphere in association with substorms is well documented, but studies are sparse on dayside sources for the waves. This paper reports observation of midnight toroidal waves by the Van Allen Probe B spacecraft during a geomagnetically quiet period on 12–13 May 2013. The spacecraft detected toroidal waves excited at odd harmonics below 30 mHz as it moved within the plasmasphere from urn:x-wiley:jgra:media:jgra55471:jgra55471-math-00012100 magnetic local time to urn:x-wiley:jgra:media:jgra55471:jgra55471-math-00020030 magnetic local time through midnight in the dipole urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0003 range 4.2–6.1. The frequencies and the relationship between the electric and magnetic field components of the waves are consistent with theoretical toroidal waves for a reflecting ionosphere. At the time of the nightside toroidal waves, compressional waves were observed by geostationary satellites located on the dayside, and the amplitudes of both types of waves varied with the cone angle of the interplanetary magnetic field. The nightside toroidal waves were likely driven by fast mode waves that resulted from transmission of upstream ultralow frequency waves into the magnetosphere. Ground magnetometers located near the footprint of the spacecraft did not detect toroidal waves.

Key Points

  • Multiharmonic toroidal standing Alfvén waves were detected in the midnight sector of the plasmasphere
  • Interplanetary magnetic field cone angle was small, which suggests that foreshock ULF waves were the energy source
  • The toroidal waves were not detected on the ground at stations located in the midnight sector

1 Introduction

Toroidal mode standing Alfvén waves, hereinafter referred to as toroidal waves, are characterized by discrete frequencies that change with urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0004 (the magnetic field shell parameter) and magnetic local time (MLT) and by magnetic field perturbations in the azimuthal direction. These waves are routinely observed in the magnetosphere (Engebretson et al., 1986; Lin et al., 1986; Takahashi et al., 2015) and are the source of ground magnetic pulsations detected primarily in the north-south component (Hughes, 1974; Inoue, 1973). Although the basic properties of observed toroidal waves are theoretically well understood (Cummings et al., 1969; Radoski & Carovillano, 1966), the waves remain an important research topic because of their relevance to particle acceleration and energy and momentum transport in the magnetosphere (Elkington et al., 1999; Ukhorskiy et al., 2005; Zong et al., 2017). Toroidal waves also play an important role in magnetosphere-ionosphere coupling (Greenwald & Walker, 1980; Samson et al., 1992). Finally, the waves are used in magnetoseismic evaluation of the mass density and ion composition in the magnetosphere (Menk & Waters, 2013). On the ground, the rate of detection of toroidal waves with magnetometers is very low in the midnight sector (Del Corpo et al., 2019; Wharton et al., 2019), making magnetoseismology least effective in that region. We are interested in finding out whether spacecraft detect nightside toroidal waves when ground magnetometers do not.

In this study, we investigate the source mechanism of toroidal waves observed in the midnight sector of the inner magnetosphere, which we define to be the region extending to the distances covered by geostationary satellites. Toroidal waves are, in general, considered to be excited by coupling to driver fast mode waves through the field line resonance (FLR) mechanism (Hasegawa et al., 1983; Tamao, 1965). Statistical studies using satellite data indicate that toroidal waves are observed mainly on the dayside (Anderson et al., 1990; Junginger et al., 1984), suggesting that the driver waves are generated on the dayside. Possible dayside sources capable of generating the driver waves include variation of the solar wind dynamic pressure (Southwood & Kivelson, 1990), upstream ultralow frequency (ULF) waves (Yumoto & Saito, 1983), transient foreshock structures (Zhao et al., 2017), and the magnetopause Kelvin-Helmholtz instability (Chen & Hasegawa, 1974; Southwood, 1974).

Toroidal waves have been observed in the midnight sector, which poses an interesting question of whether the driver waves originate from sources in the solar wind or in the magnetotail. There is no question that some toroidal waves are excited by nightside sources. For example, sudden reconfiguration (dipolarization) of the near-Earth magnetototail at substorm onset excites nightside toroidal waves (Keiling et al., 2003; Nosé, Iyemori, et al., 2014; Saka et al., 1996; Takahashi et al., 2018). It remains to be seen, however, whether solar wind sources also commonly excite nightside toroidal waves. A multispacecraft study of fast mode waves originating from the ion foreshock (Takahashi et al., 2016) noted coexisting multiharmonic toroidal waves in the midnight sector, providing evidence for a dayside source. In that study, the nightside waves were observed near the magnetic equator, and the magnetic field spectral power was stronger in the compressional component than in the azimuthal component.

In the present study, we examine similar nightside toroidal waves observed at higher magnetic latitudes (MLATs) and provide further evidence for the dayside source. New to this study are a mode structure analysis of the toroidal waves and magnetic field observations on the ground near the northern footprint of a nightside spacecraft. We suggest that toroidal waves of dayside origin are routinely excited in the midnight sector but that spacecraft detection of the waves is limited because their amplitudes are only of the order of 0.1 nT in the magnetic field and 0.1 mV/m in the electric field. Ground magnetometers did not detect toroidal waves at midnight.

The remainder of the paper is organized as follows. Section 2 describes experiments. Sections 3 and 4 present data analysis. Section 5 presents theoretical modeling of toroidal waves. Section 6 presents discussion, and section 7 concludes the study.

2 Experiments and Data

This study uses data acquired by spacecraft and ground experiments. The spacecraft data are ion bulk velocity (McFadden et al., 2008) and magnetic field (Auster et al., 2008) measured by Time History of Events and Macroscale Interactions during Substorms (THEMIS)-B; magnetic field (Kletzing et al., 2013), electric field (Wygant et al., 2013), energetic particle fluxes (Blake et al., 2013), and electron density (Kurth et al., 2015) measured by Van Allen Probe B; and magnetic field measured by Geostationary Operational Environmental Satellite (GOES)-15 (Singer et al., 1996) and Engineering Test Satellite (ETS)-VIII (Koga & Obara, 2008; Nosé, Takahashi, et al., 2014) on geostationary orbits. The ground data are magnetic fields measured with European quasi-Meridional Magnetometer Array (EMMA) (Lichtenberger et al., 2013). In addition, we use the geomagnetic Dst, AL, and AU indices.

The electric field ( urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0005) 3-D vector samples from Van Allen Probe B are constructed from two components measured in the spacecraft spin plane and a third component derived using the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0006 = 0 assumption, where urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0007 is the magnetic field. This technique produces reliable results if the elevation angle of the magnetic field from the spin plane ( urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0008) is larger than a value. According to Ali et al. (2016), this value is 6°.

We express the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0010 and urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0011 fields in the magnetosphere in a local coordinate system that uses a model magnetic field ( urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0012) and the spacecraft geocentric position vector ( urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0013) as the reference. In this system, the coordinate axis urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0014 is parallel to urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0015, urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0016 (eastward) is parallel to urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0017, and urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0018 (= urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0019) is directed outward. We define urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0020 by combining the International Geomagnetic Reference Field model (Thébault et al., 2015) for the internal field and the T89c model (Tsyganenko, 1989) for the external field. The T89c model has an input parameter called IOPT, which specifies the geomagnetic activity level in terms of Kp. We set IOPT = 1, which corresponds to the lowest Kp level urn:x-wiley:jgra:media:jgra55471:jgra55471-math-00210+. This IOPT value was selected because it makes the model closest to the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0022 field observed by Van Allen Probe B.

Spectral parameters presented in this paper are computed using the standard Fourier transform method (e.g., Bendat & Piersol, 1971). Time series data subjected to the transform are detrended by removing a quadratic function of time obtained by the least squares method. The spectral parameters are smoothed by taking averages over three neighboring Fourier components.

3 Observation Overview

We have selected a 5 hr interval from 2100 UT on 12 May to 0200 UT on 13 May 2013 for data analysis. Figure 1 presents an overview of ULF wave activity during this interval at GOES-15, ETS-VIII, and Van Allen Probe B. Figure 1a shows the spacecraft locations projected to the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0031- urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0032 plane of solar magnetic (SM) coordinates. The two geostationary satellites were on the dayside, whereas Van Allen Probe B was located in the midnight sector, reaching the apogee at urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0033 = 6.1 (in dipole coordinates) near geomagnetic midnight. In addition to the three magnetospheric spacecraft, we have THEMIS-B, which was in the solar wind at (56, 33, urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0034urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0035) in geocentric solar ecliptic Cartesian coordinates. Figure 1b shows the spacecraft locations in the SM meridional plane. The two geostationary satellites were close to the equator with MLAT =  urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0036 (ETS-VIII) and urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0037 (GOES-15), where MLAT is defined using the centered dipole. Van Allen Probe B covered an MLAT range of 6.6– urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0038. The Dst (Figure 1c) and the auroral electrojet AL and AU indices (Figure 1d) indicate that the geomagnetic activity was low during the selected 5 hr interval (highlighted in green).

Details are in the caption following the image
Overview of the selected wave event. (a) Location of three spacecraft in the SM urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0023- urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0024 plane. THEMIS-B was located in the solar wind at (56, 33, urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0025urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0026) in geocentric solar ecliptic (GSE) coordinates. The dashed line is the bow shock model by Fairfield (1971), and the solid line is the magnetopause model by Shue et al. (1998) for the solar wind dynamic pressure of 0.7 Pa and urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0027 = 0, representative of the selected interval. (b) Locations of the magnetospheric spacecraft in the dipole meridian plane of each spacecraft. (c) Dst index. (d) Auroral electrojet indices. (e) Solar wind velocity. (f) IMF magnitude. (g) IMF cone angle. (h–m) Dynamic spectra of the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0028 and urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0029 components at the three spacecraft. The white or black lines indicate the frequency given by equation 1. The strong peak in (h), labeled P2, is due to a second harmonic poloidal wave with a finite urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0030 component. The white areas in (h) and (i) indicate bad data points.
The column on the right shows the solar wind velocity ( urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0039, Figure 1e), the magnitude ( urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0040, Figure 1f), and cone angle ( urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0041, Figure 1g) of the interplanetary magnetic field (IMF) as well as the power spectral density (PSD) of the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0042 and urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0043 components at the three spacecraft in the magnetosphere (Figures 1h–1m). The solar wind velocity is low (340 km/s on average) and steady, so it cannot be the major factor for the modulation of the amplitude of magnetospheric ULF waves. The cone angle is defined by urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0044 = cos urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0045( urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0046), where urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0047 is the component along the Earth-Sun line. The white or black lines superposed on the dynamic spectra indicate a theoretically predicted frequency of the upstream waves ( urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0048), given by Takahashi, McPherron, and Terasawa (1984) as

The cone angle and upstream wave frequency are calculated using the time-shifted THEMIS-B data.

The cone angle varied between 14° and 90° with large values ( urn:x-wiley:jgra:media:jgra55471:jgra55471-math-005270 urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0053) occurring at urn:x-wiley:jgra:media:jgra55471:jgra55471-math-00542150–2210 UT, urn:x-wiley:jgra:media:jgra55471:jgra55471-math-00552240–2340 UT, and urn:x-wiley:jgra:media:jgra55471:jgra55471-math-00560110–0140 UT. When urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0057 is low, an ion foreshock develops in a large volume upstream of the bow shock, where upstream waves are generated (Fairfield, 1969). The waves generate tailward propagating fast mode waves in the magnetosphere upon impact on the magnetopause (Clausen et al., 2009). The waves are effective in driving ULF waves observed in space (Takahashi, McPherron, & Terasawa, 1984; Yumoto et al., 1985) and magnetic pulsations observed on the ground (Greenstadt & Olson, 1976; Russell et al., 1983) in the Pc3–4 band (7–100 mHz). When urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0058 is high, a large foreshock region is not formed near the bow shock nose, and magnetospheric Pc3–4 ULF waves are suppressed. This scenario for magnetospheric ULF waves is supported by the dynamic spectra shown in Figure 1. For instance, the two dayside spacecraft (ETS-VIII and GOES-15) detected power enhancement in both urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0059 and urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0060 in a wide range of frequencies from urn:x-wiley:jgra:media:jgra55471:jgra55471-math-00612340 UT to urn:x-wiley:jgra:media:jgra55471:jgra55471-math-00620100 UT, when urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0063 was mostly urn:x-wiley:jgra:media:jgra55471:jgra55471-math-006460 urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0065(Figures 1g–1k). The wave power was substantially suppressed after urn:x-wiley:jgra:media:jgra55471:jgra55471-math-00660110 UT, following a urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0067 increase to urn:x-wiley:jgra:media:jgra55471:jgra55471-math-006890 urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0069.

Despite the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0070 control of the wave power, the dynamic spectra do not show enhancement at urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0071. At ETS-VIII, the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0072 power is elevated essentially at all frequencies below 25 mHz, implying that the ULF waves propagating into the magnetosphere also had a large bandwidth. By contrast, the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0073 power shows a strong peak at urn:x-wiley:jgra:media:jgra55471:jgra55471-math-007410 mHz. By examining the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0075 spectra (not shown), we find that the peak is associated with second harmonic poloidal standing Alfvén waves (denoted P2 waves), which can be attributed to instabilities involving bounce resonance of ring current ions (Liu et al., 2013; Southwood et al., 1969). The urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0076 spectra exhibit multiple peaks at urn:x-wiley:jgra:media:jgra55471:jgra55471-math-007710 mHz, suggesting the presence of commonly detected mulitiharmonic toroidal waves (Takahashi, McPherron, & Terasawa, 1984). However, peaks at similar frequencies also occur in the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0078 and urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0079 spectra, implying that incoming fast mode waves contain multiple spectral components. At GOES-15, the spectral features are similar to those found at ETS-VIII. The strong urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0080 power detected at this spacecraft at 15 mHz also appears to be associated with P2 waves. Interestingly, the power level at GOES-15 is generally low compared to ETS-VIII. It appears that there is a local time variation of power, with the power in the prenoon sector (ETS-VIII) exceeding that in the postnoon sector (GOES-15).

Statistical studies have shown that similar local time dependence of wave power is common to ULF waves in the Pc3–4 band (Orr & Webb, 1975; Saito, 1969). It is possible that this dependence is the result of forenoon/afternoon asymmetry of the source waves. For example, the asymmetry is consistent with excitation of upstream ULF waves in the prenoon region under the IMF orientation following the average Parker spiral. It is also possible that this IMF orientation makes the magnetopause in the prenoon sector more susceptible to the Kelvin-Helmholtz instability (Nosé et al., 1995). The forenoon/afternoon asymmetry might also result from a forenoon/afternoon asymmetry of the plasma mass density. Recent numerical studies (Degeling et al., 2018; Wright et al., 2018) indicated that the amplitude of ULF waves excited in a magnetosphere exhibits a forenoon/afternoon asymmetry when there is a forenoon/afternoon asymmetry in the mass density. In the Wright et al. (2018) study, which examined the plasmatrough region assuming higher mass density on the afternoon side, higher flux tube energy density of MHD waves was found on the morning side.

At the nightside spacecraft (Van Allen Probe B), enhancement of the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0081 power is visible at urn:x-wiley:jgra:media:jgra55471:jgra55471-math-00822140–0110 UT at discrete frequencies attributed to multiharmonic toroidal waves (Figure 1l). The disappearance of the waves after 0110 UT coincides with a reduction of the dayside urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0083 power. This suggests that ULF waves originating from the ion foreshock excited magnetospheric ULF waves even in the midnight sector.

Details are in the caption following the image
(a) IMF cone angle at THEMIS-B, time-shifted by 750 s to the bow shock nose. The vertical axis is reversed for comparison with wave amplitudes in the magnetosphere. (b) Root-mean-square amplitudes of urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0084 oscillations at ETS-VIII and GOES-15, obtained by integrating the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0085 power spectra over 5–40 mHz. (c) Root-mean-square amplitude of urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0086 at Van Allen Probe B.

We generated Figure 2 to further examine the cone angle control of ULF wave power. Figure 2a is the same as Figure 1g with the vertical axis reversed. Figures 2b and 2c show the amplitudes of magnetic field oscillations obtained by first integrating the PSD of field components in a 10 min moving data window and then taking its square root. The integration is done over 5–40 mHz. Figure 2b shows that the temporal variation of the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0087 amplitude is very similar between ETS-VIII and GOES-15 but with the amplitude at the former persistently higher as noted above. The similarity means that the magnetosphere was responding to external compressional disturbances or waves that have a large local time extent, at least 5 hr (the local time separation between the two spacecraft). It is also evident that the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0088 amplitude is generally anticorrelated with urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0089, most clearly in the second half of the interval displayed. In the first half, urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0090 changed stepwise in 20–30 min intervals. This behavior is not seen in the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0091 amplitudes. A possible explanation for this observation is that the time-shifted IMF is not an accurate representation of the actual IMF at the bow shock nose. Another possible explanation is that unless urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0092 is very close to 90 urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0093, quasiparallel shock is formed on the dayside at some MLT, and fast mode waves can propagate from there into the magnetosheath and spread out within the magnetosphere.

The nightside urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0094 amplitude (Figure 2c) behaves somewhat differently. The amplitude does not track either urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0095 or the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0096 amplitude very closely. This is not unexpected because the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0097 component was measured by a spacecraft moving in both urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0098 and MLAT. The motion means that the frequency and amplitude of a toroidal harmonic change continuously even when the wave field is time stationary, because the frequency changes with urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0099 and the amplitude changes with MLAT. When band integration of the PSD of the toroidal wave fields is computed, the result contains both spatial and temporal variations. Spatial effect will be much less significant for propagating fast mode waves observed at geosynchronous orbit, because the spacecraft urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0100 and MLAT do not change and the waves do not have standing wave structures.

Despite this complication, there are indications that the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0101 amplitude is also controlled by urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0102. For example, a urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0103 amplitude minimum occurs at urn:x-wiley:jgra:media:jgra55471:jgra55471-math-01042130 UT, urn:x-wiley:jgra:media:jgra55471:jgra55471-math-01052330 UT, and urn:x-wiley:jgra:media:jgra55471:jgra55471-math-01060130 UT, nearly simultaneously with a urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0107 amplitude minimum and a urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0108 maximum, implying global suppression of ULF waves when urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0109 approaches 90 urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0110. Also, the overall maximum of the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0111 amplitude occurs at urn:x-wiley:jgra:media:jgra55471:jgra55471-math-01120050 UT, at the time of the overall urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0113 amplitude maximum at the dayside spacecraft. These features support the foreshock source mechanism for the nightside toroidal waves.

Details are in the caption following the image
(a) Magnetic field data from Van Allen Probe B with 1 s resolution, expressed in the local coordinate system described in the text. The magnitude of the T89c model field is shown in green. (b) Proton fluxes measured by the Magnetic Electron Ion Spectrometer (MagEIS) instrument on Van Allen Probe B at 90 urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0114 pitch angle, plotted with the time resolution of the spacecraft spin period ( urn:x-wiley:jgra:media:jgra55471:jgra55471-math-011511 s).

It is unlikely that the nightside toroidal waves were excited by disturbances originating in the magnetotail. The quiescence of the nightside magnetosphere is demonstrated in Figure 3, using the magnetic field (Figure 3a) and proton fluxes (Figure 3b) observed by Van Allen Probe B. The magnitude of the model magnetic field ( urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0116) is very close to the measured urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0117, and the measured urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0118 and urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0119 are very close to zero. This indicates that the quiet-time T89c model field fits the observed field very well. Most importantly, there is no sign of substorms in the measured field, such as Pi2 pulsations (Takahashi et al., 2018), dipolarization (Nosé et al., 2016), or development of field-aligned currents (Nagai, 1982). The data shown in Figure 3b are proton fluxes evaluated in measurement sectors covering the 90 urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0120 pitch angle. No ion injections were detected. We also looked at the Magnetic Electron Ion Spectrometer energetic electron data from Van Allen Probes A and B. We found no evidence of injection activity around this time in the electron data. There were drift echoes, but they appear to be related to a substorm injection that occurred around 0400 UT on 12 May. Based on these observations, we exclude disturbances of magnetotail origin as the source of the nightside toroidal waves detected at Van Allen Probe B.

4 Toroidal Waves on the Nightside

This section describes properties of the nightside toroidal waves in some detail.

4.1 Dynamic Spectra

Figure 4 shows spectral parameters computed from the toroidal ( urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0128 and urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0129) components at Van Allen Probe B (Figures 4a–4c) along with urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0130 (Figure 4d) and the electron density ( urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0131, Figure 3e). With urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0132 exceeding 15 urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0133, we have confidence in the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0134 samples derived from the spin plane components. The electron density varied smoothly and had values higher than 130 cm urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0135. The spacecraft remained within the plasmasphere and did not encounter structures such as a drainage plume.

Details are in the caption following the image
Dynamic spectra of the toroidal components and related parameters at Van Allen Probe B for the selected 5 hr interval on Orbit 680. (a) Dynamic spectra of urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0121. The white area indicates a data gap. (b) Dynamic spectra of urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0122, repeated from Figure 1k. (c) urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0123- urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0124 cross spectra. The color-coded cross-phase value is shown against the white background if the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0125- urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0126 coherence is urn:x-wiley:jgra:media:jgra55471:jgra55471-math-01270.5. The black horizontal bar indicates the time interval shown in Figure 5. (d) Angle between the magnetic field and the spacecraft spin plane. (e) Electron number density derived from plasma wave spectra.

The dynamic power spectra for both urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0136 and urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0137 exhibit peaks arising from multiharmonic toroidal waves. These peaks are labeled T1, T3, T5, and T7, corresponding to the fundamental, third, fifth, and seventh harmonics based on the result of a model calculation described in section 5. We will hereinafter use the above shorthand notation (e.g., T1) for toroidal harmonics. The absence of even harmonics is attributed to the structure of the driver fast mode waves about the magnetic equator. We argue that the driver waves are symmetric about the magnetic equator and couple only to odd harmonics of toroidal waves, which have an antinode of field line displacement at the equator.

The dynamic cross-phase spectra (Figure 4c) exhibit multiple bands with alternating bluish and reddish colors. The cross-phase is defined to be positive if urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0138 leads urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0139. This pattern can be explained by the wave mode structure along the ambient magnetic field line. For example, the cross-phase of the T5 wave appears in blue (approximately urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0140) at 2200 UT (MLAT = 8.9 urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0141 and in orange ( urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0142) at 0100 UT (MLAT = 14.5 urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0143). This means that urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0144 and urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0145 of the T5 wave oscillated in quadrature and that the spacecraft crossed a urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0146 node of the T5 wave as it moved to higher MLAT. The urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0147 node is inferred from the absence of the T5 spectral peak in the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0148 dynamic spectra at 2240–2340 UT. We find that the cross-phase of the T3 and T7 waves at 0100 UT is approximately urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0149 (blue), in contrast to the T5 waves showing urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0150 (orange). This is also explained by the location of the spacecraft relative to the nodes of these harmonics. We will present a quantitative description of the nodal structures in section 5.

4.2 Relationship Between urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0151 and urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0152

In the dynamic spectra, toroidal waves at 0040–0110 UT exhibit the highest intensity and regularly separated spectral peaks. We examine this time interval in detail. Figure 5a shows the detrended urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0153 and urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0154 time series. Despite the unambiguous appearance of the waves in the dynamic spectra, their amplitudes are quite small. The peak-to-peak amplitudes are urn:x-wiley:jgra:media:jgra55471:jgra55471-math-01550.1 mV/m for urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0156 and urn:x-wiley:jgra:media:jgra55471:jgra55471-math-01570.4 nT for urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0158. The oscillations appear irregular, and it is difficult to infer the presence of multiple harmonics.

Details are in the caption following the image
(a) Detrended toroidal components urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0159 and urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0160 observed by Van Allen Probe B. (b–d) Spectral parameters computed from the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0161 and urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0162 time series. The cross-phase data points are shown only when the coherence is high enough to define the 95% confidence interval (Bendat & Piersol, 1971) shown by the vertical bar. The green vertical dashed lines indicate the theoretical frequencies described in section 5.

The lower panels of Figure 5 show spectral parameters computed from the time series data. In the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0163 power spectrum (Figure 5b), we find regularly spaced peaks at 2, 8, 15, and 22 mHz. These are attributed to the T1, T3, T5, and T7 waves as we stated above. The peaks are located near theoretically predicted frequencies marked by green dashed lines, which we describe in section 5. In the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0164 power spectrum, a weak peak appears near 2 mHz, and strong peaks appear near 15 and 22 mHz. At the frequencies of the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0165 spectral peaks, the coherence (Figure 5c) is elevated. This allows us to evaluate the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0166- urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0167 cross-phase (Figure 5d) with not too large 95% confidence intervals (Bendat & Piersol, 1971), shown by vertical bars. The cross-phase is in the positive domain in the T1 and T5 bands and in the negative domain in the T3 and T7 bands. These alternating cross-phase values are similar to those reported in a study of kinetic-scale FLRs (Chaston et al., 2014).

4.3 Mode Frequencies

Figure 6 shows the frequencies of the toroidal waves detected by Van Allen Probe B during the time period shown in Figure 4. The frequencies are determined by searching for peaks in the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0172 and urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0173 spectra as described by Takahashi et al. (2015) and are plotted versus UT (Figure 6a), MLT (Figure 6b), and urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0174 (Figure 6c). The spectra are computed in a moving 20 min data window shifted in 5 min steps. The waves span a MLT range of 20.8 to 0.4 hr through midnight and an urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0175 range of 4.3–6.1. The frequencies fall with urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0176, which is consistent with the monotonic decrease of the electron density with urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0177 (Figure 6d). The urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0178 profile of the mode frequencies is qualitatively quite similar to those found on the dayside (e.g., Takahashi et al., 2015) and indicates that the nightside toroidal waves are not different from dayside toroidal waves as long as the basic standing wave properties are concerned.

Details are in the caption following the image
(a) Frequencies of toroidal waves detected by Van Allen Probe B on Orbit 680, plotted versus UT. The color of the data points indicates the source field component: urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0168 (red) or urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0169 (blue). The vertical dotted line indicates MLT midnight. (b) The same frequencies plotted versus MLT. (c) The same frequencies plotted versus urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0170. (d) Electron number density plotted versus urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0171.

4.4 Ground Observations

We have examined EMMA data for signatures of nightside toroidal waves. Figure 7 shows the locations of the EMMA magnetometers in geographic coordinates along with the northern magnetic field footprint of Van Allen Probe B for 2100 UT on 12 May to 0200 UT on 13 May. We used the T89c model to determine the footprint. The EMMA magnetometers completely cover the latitudes of the footprint but were located urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0179 east of the footprint.

Details are in the caption following the image
Geographic locations of the EMMA magnetometers and the Van Allen Probe B magnetic field footprint for 12 May 2013 2000 UT to 13 May 2013 0130 UT. The footprint is determined using the T89c model with IOPT = 1.

We determined the presence and frequency of toroidal waves on the ground using the cross-phase analysis technique of Waters et al. (1991). The technique has been widely applied to data from various magnetometer arrays (Berube, 2003; Chi et al., 2013; Del Corpo et al., 2019; Dent et al., 2006; Lichtenberger et al., 2013; Vellante et al., 2007; Wharton et al., 2019). Cross-phase analysis of the EMMA data often yields a few frequencies, which we attribute to multiple toroidal harmonics. Identification of the harmonic modes for these frequencies is not necessarily straightforward because the frequencies are not evenly spaced in frequency and some harmonics may be missing. However, it is usually easy to identify the frequency of the T1 mode (denoted urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0182). Once urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0183 is determined, we use theoretical urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0184/ urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0185 ratios to assign the harmonic mode numbers to the remaining frequencies. Here, urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0186 denotes the frequency of the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0187th harmonic. We obtain the theoretical urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0188 values by solving the Singer et al. (1996) equation using the T01 magnetic field model (Tsyganenko, 1981) and the mass density model described in section 5. In the quiet-time inner magnetosphere, the theoretical urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0189 values depend little on the choice of the magnetic field model.

Details are in the caption following the image
Frequencies of toroidal waves estimated using EMMA data. The key on the right indicates the midpoint urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0180 values for station pairs, where urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0181 is defined using IGRF. The shading indicates the time interval during which Van Allen Probe B detected toroidal waves. (a–c) Fundamental, second, and third harmonic frequencies plotted as a function of UT. (d–f) Same as (a)–(c) except the frequencies are plotted as a function of MLT that is defined using the centered dipole for the geographic midpoint of each station pair.

Figure 8 shows the results of the cross-phase analysis. The selected 24 hr interval covers the 5 hr interval of toroidal wave activity observed by Van Allen Probe B (shaded in gray). We use a 2 hr ( urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0190) or 3 hr ( urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0191) data window in calculating the cross spectra and move the window in 30 min steps, where urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0192 is defined using International Geomagnetic Reference Field (Del Corpo et al., 2019). The three panels show urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0193, urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0194, and urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0195. It is evident that urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0196 and urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0197 are far more easily detected than urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0198, consistent with the Van Allen Probe B observations. As we will discuss in section 6, we attribute the prominence of odd harmonics to source disturbances that have a maximum amplitude at the magnetic equator. The most important feature in Figure 8 is the absence of toroidal waves from urn:x-wiley:jgra:media:jgra55471:jgra55471-math-01991700 UT ( urn:x-wiley:jgra:media:jgra55471:jgra55471-math-020020 hr MLT) on 12 May to urn:x-wiley:jgra:media:jgra55471:jgra55471-math-02010200 UT ( urn:x-wiley:jgra:media:jgra55471:jgra55471-math-020205 hr MLT) on 13 May, despite the fact that Van Allen Probe B detected toroidal waves within this time interval in the midnight sector.

Details are in the caption following the image
(a) T1 wave frequency versus the magnetic shell parameter defined using the T89c model. The open and filled circles indicate the values determined using EMMA data sampled at dawn and dusk, respectively. The other data points are taken from the Van Allen Probe B results shown in Figure 6. The red crosses indicate urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0210 values obtained using urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0211. The blue crosses indicate values estimated assuming urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0212 =  urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0213/4 from the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0214 values obtained using urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0215. (b) T3 wave frequency displayed in the same format. The blue crosses indicate the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0216 values taken directly from Figure 6.

Figure 9 compares urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0203 and urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0204 at EMMA and Van Allen Probe B. In this figure, we label field lines using the geocentric distance to the point of field line crossing of the dipole equator. This mapping is done using the T89c model, and the distance is denoted urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0205.

In Figure 9a, the EMMA urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0206 samples are shown for two epochs, 1500 UT on 12 May (dusk) and 0500 UT on 13 May (dawn). The dusk and dawn urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0207 samples follow similar decreasing trends versus urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0208, but the dawn samples show higher values. This behavior is compatible with the plasmasphere depletion occurring overnight due to plasma flow from the plasmasphere to the ionosphere. As shown in Figure 6, urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0209 at Van Allen Probe B was directly determined only in urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0217 in a short time interval corresponding to urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0218 6. Consequently, we estimated urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0219 values at other distances assuming a constant urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0220 ratio. The blue crosses indicate values obtained by assuming urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0221 =  urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0222/4, where the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0223 values are determined using urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0224 as shown in Figure 6. The relationship between urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0225 and urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0226 is based on a statistical study of multiharmonic toroidal waves detected at geostationary orbit (Takahashi et al., 2004). The Van Allen Probe B samples cover the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0227 range 4.3–6.2. In this range, the spacecraft and the dawn EMMA urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0228 values agree within an error of 50% in linear scale. At urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0229 4.2, the spacecraft and dusk EMMA urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0230 values agree very well.

In Figure 9b, the EMMA urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0231 samples are shown for two epochs, 1500 UT on 12 May (dusk) and 0530 UT on 13 May (dawn), and the Van Allen Probe B urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0232 samples are taken directly from Figure 6. The agreement between the spacecraft and ground results is very similar to that found in Figure 9a. We conclude that magnetospheric toroidal waves were present in the midnight sector but were not detected on the ground. The absence of ground signatures of midnight toroidal waves or FLRs is not a new finding. A statistical analysis of EMMA data (Del Corpo et al., 2019) showed that the detection rate of midnight FLRs at urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0233 = 2.4–5.5 is very low ( urn:x-wiley:jgra:media:jgra55471:jgra55471-math-023410%). Similar results were obtained using 1 year of observations at urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0235 = 1.4–3.4 with the Mid-continent Magnetoseismic Chain magnetometers (Chi et al., 2013).

5 Theoretical Toroidal Waves

To verify the mode identification presented above, we compare the observational results with theoretical models of toroidal waves. Considering the departure of the nightside magnetic field from a dipole, we obtain the frequencies and mode structures of toroidal waves as the eigenmode solutions of the wave equation derived by Singer et al. (1981). The equation allows one to use various models for the magnetospheric magnetic field and mass density. Our target for the modeling is the toroidal waves shown in Figure 5.

5.1 Magnetic Field and Mass Density Models

To model the waves, we use the T89c model with IOPT = 1. We specify the model field at the epoch time of 0055 UT of 13 May 2013 and focus on the field line that passes the location of Van Allen Probe B at this epoch: ( urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0238, 1.011, 2.307  urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0239) in geocentric solar ecliptic coordinates. Figure 10 illustrates the T89c model along with the dipole model that passes the same Van Allen Probe B location. Figure 10a shows that the T89c field line (solid line) intersects the dipole equator ( urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0240 = 0) at 5.98  urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0241, slightly (0.14  urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0242) outward of the corresponding dipole field line (dashed line). The two field lines exhibit similar shapes in this figure. However, Figure 10b shows that the field magnitude urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0243 in the equatorial region differs significantly between the two models. At the equator, urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0244 is 108 nT for T89c and 150 nT for the dipole. This difference is important, because the region of slowest Alfvén velocity makes the greatest contribution to the toroidal wave frequencies. The velocity is proportional to urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0245. With urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0246 usually showing a minimum at the equator, the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0247 value in the equatorial regions has the largest influence on the eigenmode frequencies if the density is held constant.

Details are in the caption following the image
(a) SM coordinates meridional plots of the dipole and T89c field lines that pass the Van Allen Probe B location at 0055 UT (red circle). (b) Magnitude of the same model fields evaluated along the field lines shown in (a) and plotted versus dipole MLAT. The red circle indicates the value observed at the spacecraft.
For the mass density ( urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0248), we adopt a power law model
where urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0250 is the equatorial mass density, urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0251 is geocentric distance to the field line, urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0252 is the equatorial value of urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0253, and urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0254 is a positive constant. This model has been widely used in theoretical studies (Cummings et al., 1969; Kabin et al., 2007; Orr & Matthew, 1971) and in estimating the mass density from observed toroidal wave frequencies (Dent et al., 2006).

5.2 Solutions of the Wave Equation

In the Singer et al. (1981) approach, the polarization of Alfvén waves is specified by selecting two field lines that define the direction of the wave magnetic field. In applying this approach to the observed toroidal waves, we use a field line that is illustrated in Figure 10 and another that is adjacent to it, with a purely azimuthal separation at the magnetic equator. Perfect reflection is assumed at the ionosphere.

Details are in the caption following the image
Eigenmode structures of the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0236 component of the T5 mode on the dipole and T89c field lines shown in Figure 10. The mass density variation along the field lines is given by urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0237 = 1.

The importance of using a realistic magnetic field model is illustrated in Figure 11, by contrasting urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0255 eigenmode structures of the T5 mode obtained for the dipole and T89c models. For both field models, the mass density is given by urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0256 = 156 amu  urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0257 (see section 5.3) and urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0258. The urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0259 value is reasonable in consideration of theory for the plasmasphere (Angerami & Carpenter, 1966; Vellante & Förster, 2006) and observations made in that region (Takahashi et al., 2004). In this trial, we find the frequency to be 17.0 mHz for the dipole and 14.5 mHz for T89c.

The off-equatorial nodes are located closer to the equator for the T89c model. This is a consequence of the lower equatorial Alfvén velocity on the T89c magnetic field line. In the MLAT urn:x-wiley:jgra:media:jgra55471:jgra55471-math-02600 domain, we find that the nodes are located at MLAT = 10.1 urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0261 and 30.6 urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0262 for the T89c model and MLAT = 12.7 urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0263 and 32.8 urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0264 for the dipole model. In the MLAT  urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0265 0 domain, the nodes are located at very similar distances from the magnetic equator. There are small north-south asymmetries of the mode structures, however, because there is a small asymmetry of the magnetic field model due to a finite dipole tilt angle. Note that the shape of the mode functions and the location of the nodes depend on urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0266 but not on urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0267.

Details are in the caption following the image
Dependence of toroidal wave eigenmode solutions on the mass density model parameter urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0280 for the T89c magnetic field model illustrated in Figure 10. The solutions are obtained at integer values of urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0281 (filled or open circles) and linearly interpolated. (a) Frequencies of the T1 through T7 modes. (b) Locations of the nodes of the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0282 (solid line) and urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0283 (dashed line) components of the T3 mode. (c) Same as (b) but for the T5 mode. (d) Same as (b) but for the T7 mode.

Figure 12 shows how the frequencies and the mode structure of toroidal waves depend on urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0268. We illustrate the dependencies for the T89c model by varying urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0269 at integer values from 0 to 6 as was done by Cummings et al. (1969) for the dipole model. All results shown are for urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0270 = 156 amu  urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0271 in reference to section 5.3. Only the odd harmonics T1, T3, T5, and T7 are considered.

Figure 12a shows the mode frequencies. The frequencies all decrease when urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0272 increases. This is simply because the mass loading over the entire field line is higher for larger urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0273 when urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0274 is held constant. The important fact is, because the Alfvén velocity is not constant along the field line, the degree of the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0275 dependence of the mode frequency varies among the harmonics. If we take the T7 mode as an example, the frequency changes from 22.1 mHz for urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0276 = 0 to 11.3 mHz for urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0277 = 6, a 49 urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0278 reduction. The reduction is less significant for the T1 mode: a 21 urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0279 reduction from 2.14 to 1.69 mHz. This means that the frequency ratio between different harmonics depends on urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0284 and that we can estimate urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0285 from the frequency ratios obtained from observation (Takahashi et al., 2004).

Figures 12b–12d show the locations of the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0286 (solid line) and urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0287 (dashed line) nodes. Only the MLAT  urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0288 0 domain is considered. The nodes all move to higher MLAT as urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0289 increases. For example, a urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0290 node of the T5 wave moves from MLAT = 10.2 urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0291 for urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0292 = 0 to MLAT = 18.3 urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0293 for urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0294 = 6. This urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0295 dependence also arises from the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0296 dependence of the Alfvén velocity and is potentially useful in constraining urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0297 from observationally determined locations of the nodes.

5.3 Comparison With Observation

Figure 13 displays the theoretical mode structures in a format that facilitates comparison with the spectral properties of observed waves shown in Figure 5. The four panels show the eigenmode structures of the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0298 (solid line) and urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0299 (dashed line) components of the T1, T3, T5, and T7 waves obtained using the T89c model illustrated in Figure 10 and the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0300 = 1 mass density model. The nodes are marked by filled ( urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0301) or open ( urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0302) circles. Because we solve the wave equation assuming perfect reflection at the ionosphere, urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0303 diminishes at the ionosphere (MLAT = 69 urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0304), and urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0305 and urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0306 oscillate in quadrature everywhere along the field line so that the time-averaged Poynting flux along the field line vanishes. The illustrated mode structures correspond to snapshots taken a quarter wave period apart between urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0307 and urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0308. In the MLAT domains shaded orange, urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0309 leads urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0310 by 90 urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0311. In the MLAT domains shaded blue, the phase relation is reversed. The vertical dotted line at MLAT = 14.4 urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0312 indicates the location of Van Allen Probe B at the middle of the 30 min interval shown in Figure 5.

Details are in the caption following the image
Toroidal wave eigenmode structures of the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0313 (solid line) and urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0314 (dashed line) components for the fundamental (T1), third (T3), fifth (T5), and seventh (T7) harmonics. The structures are obtained using the T89c model illustrated in Figure 10 and mass density parameter urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0315 = 1. The amplitudes are normalized to their maximum values. The filled and open circles indicate the nodes of urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0316 and urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0317, respectively. The orange rectangles indicate the MLAT domains in which urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0318 leads urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0319 by 90 urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0320. The blue rectangles indicates the MLAT domains in which urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0321 leads urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0322 by 90 urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0323. The vertical dotted line indicates the location of Van Allen Probe B at 0055 UT.

Following the vertical dotted line, we find that the model predicts the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0324- urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0325 cross-phase to be 90 urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0326 for the T1 and T5 modes and urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0327 for the T3 and T7 modes. This is in good qualitative agreement with the cross spectrum shown in Figure 5d. It is important to note that as the harmonic mode number increases, the MLAT spacing between the nodes decreases. As a consequence, the sign of the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0328- urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0329 cross-phase switches over small MLAT distances. If we take the T7 mode (Figure 13d) as an example, a sign switch occurs only a few degrees above and below the MLAT of the spacecraft. The accuracy with which we can determine the node latitude is limited by our observational ability to detect small amplitude oscillations in the presence of natural or instrumental noise. As the spacecraft approaches a node of a field component, the amplitude of that component approaches zero. This means that there is a small MLAT domain around the node where we cannot define the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0330- urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0331 cross-phase. This blank MLAT domain is estimated to be 1–2 urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0332 wide.

The eigenmode solutions also give a reasonable explanation to the observed toroidal wave frequencies. To obtain the theoretical frequencies, we use the measured urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0333 value of 165 cm urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0334 at Van Allen Probe B for the epoch of 0055 UT. Assuming that urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0335 changes along the magnetic field in the same manner ( urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0336 = 1) as the mass density, we find urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0337 = 155 cm urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0338 at the magnetic equator. We then assume that the ions are all protons, that is, urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0339 = 156 amu cm urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0340 (proton mass = 1.0073 amu). The proton dominance in the quiet-time plasmasphere has been inferred in a Van Allen Probes study of dayside toroidal waves (Takahashi et al., 2015). For the selected urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0341 value, the theoretical T1, T3, T5, and T7 frequencies are 2.1, 8.3, 14.5, and 20.7 mHz, respectively. These frequencies are marked by green vertical dashed lines in Figure 5b and match the peaks in the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0342 and urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0343 spectra. We conclude that the observed multifrequency oscillations in urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0344 and urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0345 are a manifestation of multiharmonic toroidal waves excited on the local field line.

Finally, we can use the theoretical mode structures shown in Figure 13 to estimate the wave amplitude at the ionospheric height ( urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0346120 km). For the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0347 component, the ratios between the amplitudes at the ionospheric end point of the field line and that at the Van Allen Probe B location are 14, 6.3, 5.6, and 6.7 for the T1, T3, T5, and T7 modes, respectively. Meanwhile, by integrating the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0348 spectrum shown in Figure 5b in the band occupied by each toroidal harmonic, we obtain the peak-to-peak amplitudes of 0.091, 0.076, 0.076, and 0.048 nT for the T1, T3, T5, and T7 modes, respectively. From these values, we estimate the peak-to-peak urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0349 amplitudes at the ionosphere to be 1.3, 0.5, 0.4, and 0.3 nT for the T1, T3, T5, and T7 modes, respectively. Magnetic pulsations of such amplitudes can readily be detected by the EMMA magnetometers. The fact that the cross-phase technique is unable to determine nighttime toroidal wave frequencies (Figure 8) implies that either there were other magnetic field disturbances that masked the toroidal waves on the ground or the toroidal wave signals were strongly attenuated below the ionosphere.

Concerning the latter possibility, we note that the ground to ionosphere amplitude ratio is given by urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0350, where urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0351 and urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0352 are the height-integrated Hall and Pedersen conductivities, urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0353 is the height of the ionosphere, and urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0354 is the magnitude of the horizontal wave number (Hughes & Southwood, 1976a). Model calculations indicate that during the solar maximum periods, which is relevant to our wave event, the damping is strong on the nightside with the amplitude ratio becoming as low as urn:x-wiley:jgra:media:jgra55471:jgra55471-math-03550.2 (Hughes & Southwood, 1976b) assuming particle precipitation does not contribute to the conductivity. In addition, stronger damping means larger latitudinal width of FLR (Hughes & Southwood, 1976a), making the cross-phase technique ineffective in identifying the local toroidal wave frequencies. These ionospheric effects are an adequate explanation for the ineffectiveness of the cross-phase technique on the nightside. We caution, however, that this explanation may not apply to our observations in the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0356 = 4–6 region (Figure 6c), where urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0357 was apparently high enough to sustain midnight toroidal waves excited in the fixed-end mode as discussed in section 6.2.

6 Discussion

6.1 Driving Mechanism of Midnight Toroidal Waves

We believe that the midnight toroidal waves observed by Van Allen Probe B were driven by fast mode waves that resulted from transmission of upstream ULF waves into the magnetosphere. Evidence for this scenario includes the relationship between urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0358 and the wave amplitudes in the magnetosphere shown in Figure 2.

More direct evidence of fast mode propagation to the midnight sector would be presence of clear urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0359 oscillations in that region, as was the case in the ULF wave event reported in a multisatellite study by Takahashi et al. (2016). However, a weak urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0360 signature does not necessarily mean absence of fast mode waves. The reason is that we can expect the fast mode amplitude in the magnetosphere to depend on MLAT. According to Lee (1996), fast mode waves are stronger at lower MLAT because of a cutoff effect arising from the variation of fast mode speed with MLAT. In the Takahashi et al. (2016) study, the nightside spacecraft (Van Allen Probe B and ETS-VIII) were located closer to the magnetic equator ( urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0361MLAT urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0362) than the nightside spacecraft in the present study (Van Allen Probe B, urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0363MLAT urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0364). This MLAT difference can explain the different urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0365 to urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0366 power ratios between the two studies. We argue that fast mode waves of foreshock origin propagated to the nightside magnetosphere in the equatorial region and coupled to toroidal waves through the broadband FLR mechanism (Hasegawa et al., 1983). This scenario also explains the preferential excitation of odd harmonics. Equatorial fast mode waves would mean maximum field line displacement at the equator, which, in turn, would mean excitation of standing Alfvén waves with an equatorial antinode of field line displacement and an equatorial node of urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0367.

Not every aspect of our observations fits the existing scenario for the relationship between upstream waves and magnetospheric ULF waves. Figure 1 shows that nightside toroidal waves were excited at multiple harmonics spanning a wide frequency range of 2–25 mHz. The same figure also shows that the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0368 power is enhanced below the predicted upstream wave frequency. These features disagree with those reported in previous studies that indicate urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0369 oscillations in space have spectral power concentrated in a band consistent with predicted upstream wave frequencies (Clausen et al., 2009; Heilig et al., 2007).

There are other studies that reported dayside observations similar to those in the present study. Takahashi, McPherron, and Hughes (1984) reported that the amplitudes of several toroidal harmonics ranging in frequency from 20 to 80 mHz were simultaneously controlled by the IMF cone angle. More recently, Takahashi et al. (2015) reported a similar observation by Van Allen Probes in the frequency range 5–40 mHz. The latter study noted that the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0370 power in the same frequency range was also similarly controlled by the cone angle.

There are possible explanations for the mismatch between the predicted upstream wave frequency and the spectrum of magnetospheric waves. One explanation is that upstream waves are excited at multiple frequencies. In a recent simulation study, Turc et al. (2018) demonstrated that upstream waves are excited at multiple frequencies because the velocity of ions backstreaming into the solar wind varies spatially. In the run that assumed an IMF magnitude of 5 nT, the upstream wave frequencies ranged from urn:x-wiley:jgra:media:jgra55471:jgra55471-math-037920 to urn:x-wiley:jgra:media:jgra55471:jgra55471-math-038070 mHz, although the average frequency was close to the prediction urn:x-wiley:jgra:media:jgra55471:jgra55471-math-038140 mHz given by equation 1. It should be noted that this analysis examined the frequency at the peak of a spectrum. Because the spectrum has a finite width, the wave power extended below 20 mHz. Such source spectrum can lead to excitation of toroidal harmonics at frequencies below 20 mHz.

Another explanation is the magnetospheric filtering effect. In general, the penetration depth of fast mode waves into the magnetosphere depends on the wavelength along the magnetopause. If the wavelength is short, the wave may become evanescent (Lee, 1996) and may not reach the spacecraft located in the inner magnetosphere. Fast mode waves with lower frequencies (longer wavelengths) may suffer less attenuation as they propagate into the magnetosphere.

6.2 Ionospheric Damping

In both the present and previous studies (e.g., Anderson et al., 1990), it is evident that nightside toroidal waves are weaker than their daytime counterparts. A possible reason for the day-night asymmetry is the low nighttime urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0382 due to lack of solar illumination (Allan & Knox, 1979; Bulusu et al., 2016; Newton et al., 1978). However, in section 5, we showed that theoretical toroidal waves obtained by imposing perfect reflection at the ionosphere explain the observed toroidal waves well. This implies that urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0383 was sufficiently high at the footprints of Van Allen Probe B to sustain standing Alfvén waves for several wave periods even after the energy source for the waves was removed.

We examined the ionospheric damping effect quantitatively. Following Takahashi et al. (1996), we numerically solved the Cummings et al. (1969) toroidal wave equation for the dipole field, with the ionospheric boundary condition given by
where urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0385 is the permeability of free space and the symbol urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0386 denotes the wave fields (Allan & Knox, 1979; Newton et al., 1978). Use of the dipole field instead of the T89c model simplifies the computation and is justified because the eigenmode solutions do not differ significantly between the two, as shown in Figure 11. For comparison with the spacecraft data shown in Figure 5, we solved the equation for urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0387 = 5.84, urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0388 = 156 amu cm urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0389, and urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0390 = 1. When urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0391 is infinitely high, the solutions reduce to those of the ideal toroidal waves, such as the fifth harmonic solution shown in Figure 11. When urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0392 is finite, urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0393 does not vanish at the ionosphere, and the mode frequencies become complex, with the imaginary part giving the damping rate. Figure 14a shows the real and imaginary parts of the complex frequencies of the fundamental and second harmonics as a function of urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0394. For simplicity, we assumed that urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0395 has the same value at both ends of the field line. The results shown in Figure 14a are essentially the same as those in Figure 12 of Newton et al. (1978) except for the model parameters chosen.
Details are in the caption following the image
(a) Theoretical frequencies of toroidal waves with ionospheric damping. The real (solid line) and imaginary (dashed line) parts of the fundamental (red line) and second (blue line) harmonic frequencies are plotted as a function of urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0371. Across urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0372 0.5 S, the waves can be classified as free-end mode or fixed-end mode. (b–d) Structure of the T2 mode for urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0373 = 0.3 S. The urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0374 amplitude is set to 1 nT at the ionosphere. The green vertical dashed line indicates the position of Van Allen Probe B at 0055 UT on 12 May. (e–g) Structure of the T1 mode for urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0375 = 4 S (solid line) and urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0376 =  urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0377 (dotted line). The urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0378 structure is indistinguishable between the two conductivity values.

The two vertical dashed lines in Figure 14a represent two different urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0396 models. The lower urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0397 value (0.3 S, labeled “IRI”) comes from the International Reference Ionospheric Model 2016 (Bilitza et al., 2017), with the height integration done online courtesy of the World Data Center for Geomagnetism, Kyoto (http://wdc.kugi.kyoto-u.ac.jp). The value is the average of those evaluated at the northern and southern field line footprints. The higher urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0398 value (4 S, labeled “Precipitation”) comes from ionospheric conductivity models that incorporate satellite measurements of precipitating particles. The value is an approximation of those listed in Table A4 of Wallis and Budzinski (1981) and Table A2 of Spiro et al. (1982) for the MLAT of 65 urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0399( urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0400 = 5.6), magnetic midnight, and low geomagnetic activity. The IRI conductivity is lower because the model does not fully incorporate conductivity enhancements caused by particle precipitation, in the auroral zone in particular. As noted by Newton et al. (1978), the fixed-end T1 mode cannot exist if urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0401 becomes lower than a certain value, urn:x-wiley:jgra:media:jgra55471:jgra55471-math-04020.6 S in the current example. As urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0403 becomes low, what is identified as the fixed-end T2 mode at the high urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0404 limit becomes the lowest-frequency free-end mode. The free-end T2 mode has a frequency lower than the fixed-end T2 mode but is higher than the fixed-end T1 mode.

We argue that the toroidal waves observed by Van Allen Probe B were excited in the fixed-end regime, corresponding to urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0405 0.6 S because of the following reasons. First, the IRI model predicts that the imaginary part of urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0406 is urn:x-wiley:jgra:media:jgra55471:jgra55471-math-04071 mHz, which is close to the peak value at urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0408 = 0.5 S and implies strong damping of the T2 mode. Second, in order for the free-end mode to produce a spectral peak at urn:x-wiley:jgra:media:jgra55471:jgra55471-math-04092 mHz as shown in Figures 4-6, urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0410 must be increased from 156 amu cm urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0411 by a factor of urn:x-wiley:jgra:media:jgra55471:jgra55471-math-04124, because the frequency is proportional to urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0413 and the lowest-frequency mode in the free-end regime has a frequency of 4.6 mHz (Figure 14a). This means a substantial presence of heavy ions in the plasmasphere, which is questionable at quiet times (Takahashi et al., 2006). Third, Figure 14d shows that the T2 mode in the free-end regime has a urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0414- urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0415 cross-phase close to urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0416 at the location of Van Allen Probe B (MLAT = 14 urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0417, green vertical dashed line), which is inconsistent with the urn:x-wiley:jgra:media:jgra55471:jgra55471-math-041890 urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0419 cross-phase observed (Figure 5d) and predicted for the T1 mode for high urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0420 (Figure 14g).

The question remains as to whether sufficient precipitation occurred in the outer plasmasphere to raise urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0421 well above 0.6 S. A close association between the equatorward edge of auroral precipitation and the plasmapause has been reported (Horwitz et al., 1982), but neither IRI nor the other conductivity models (Spiro et al., 1982; Wallis & Budzinski, 1981) provide information on the plasmapause location. This point is very important and needs to be addressed in the future.

7 Conclusions

We have studied multiharmonic toroidal waves in the midnight sector detected on a Van Allen Probe B orbit during a geomagnetically quiet period. Midnight toroidal waves were not detected by ground magnetometers located close to the field line footprint of the spacecraft. The IMF cone angle exhibited small ( urn:x-wiley:jgra:media:jgra55471:jgra55471-math-042245 urn:x-wiley:jgra:media:jgra55471:jgra55471-math-0423) values, and two dayside geostationary spacecraft detected elevated spectral power in the compressional magnetic field component. From these observations, we interpret that the nightside toroidal waves were driven by broadband fast mode waves that are transmitted from the foreshock region into the magnetosphere.

There are two remaining questions to be addressed in the future. One concerns the spectral content of magnetospheric waves. Both on the dayside and on the nightside, the wave spectra do not fit the known relationship between the IMF magnitude and the wave frequency. The other concerns the nightside ionospheric conductivity. The nightside toroidal waves were observed in the plasmasphere, where particle precipitation is not expected to be high enough to elevate the conductivity. Nevertheless, the toroidal waves were better explained by fixed-end modes (high conductivity) than free-end modes (low conductivity).


Work at JHU/APL was supported by NASA Grant NNX17AD34G and NSF Grant 1840970. Work at The Aerospace Corporation was supported by RBSP-ECT funding provided by JHU/APL Contract 967399 under NASA's Prime Contract NAS501072. International Space Science Institute, Bern, facilitated collaboration of K. T., M. V., and A. D. C. by hosting Magnetoseismology Team meetings (Peter Chi, lead). Data used in this study are available from the following sources: NASA/GSFC Space Physics Data Facility Coordinated Data Analysis Web (https://cdaweb.gsfc.nasa.gov) for Van Allen Probes; Zenodo (https://doi.org/10.5281/zenodo.3376790) for EMMA; NOAA National Geophysical Data Center (http://satdat.ngdc.noaa.gov) for GOES; Zenodo (https://doi.org/10.5281/zenodo.3385024) for ETS-VIII; Space Sciences Laboratory, University of California, Berkeley (http://themis.ssl.berkeley.edu/index.shtml) for THEMIS; and World Data Center for Geomagnetism, Kyoto (http://wdc.kugi.kyoto-u.ac.jp) for geomagnetic indices.