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 2100 magnetic local time to 0030 magnetic local time through midnight in the dipole 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.
- 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
Toroidal mode standing Alfvén waves, hereinafter referred to as toroidal waves, are characterized by discrete frequencies that change with (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 ( ) 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 = 0 assumption, where is the magnetic field. This technique produces reliable results if the elevation angle of the magnetic field from the spin plane ( ) is larger than a value. According to Ali et al. (2016), this value is 6°.
We express the and fields in the magnetosphere in a local coordinate system that uses a model magnetic field ( ) and the spacecraft geocentric position vector ( ) as the reference. In this system, the coordinate axis is parallel to , (eastward) is parallel to , and (= ) is directed outward. We define 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 0+. This IOPT value was selected because it makes the model closest to the 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 - 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 = 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, 3 ) 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 = (ETS-VIII) and (GOES-15), where MLAT is defined using the centered dipole. Van Allen Probe B covered an MLAT range of 6.6– . 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).
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 ( 70 ) occurring at 2150–2210 UT, 2240–2340 UT, and 0110–0140 UT. When 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 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 and in a wide range of frequencies from 2340 UT to 0100 UT, when was mostly 60 (Figures 1g–1k). The wave power was substantially suppressed after 0110 UT, following a increase to 90 .
Despite the control of the wave power, the dynamic spectra do not show enhancement at . At ETS-VIII, the 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 power shows a strong peak at 10 mHz. By examining the 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 spectra exhibit multiple peaks at 10 mHz, suggesting the presence of commonly detected mulitiharmonic toroidal waves (Takahashi, McPherron, & Terasawa, 1984). However, peaks at similar frequencies also occur in the and 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 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 power is visible at 2140–0110 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 power. This suggests that ULF waves originating from the ion foreshock excited magnetospheric ULF waves even in the midnight sector.
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 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 amplitude is generally anticorrelated with , most clearly in the second half of the interval displayed. In the first half, changed stepwise in 20–30 min intervals. This behavior is not seen in the 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 is very close to 90 , 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 amplitude (Figure 2c) behaves somewhat differently. The amplitude does not track either or the amplitude very closely. This is not unexpected because the component was measured by a spacecraft moving in both 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 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 and MLAT do not change and the waves do not have standing wave structures.
Despite this complication, there are indications that the amplitude is also controlled by . For example, a amplitude minimum occurs at 2130 UT, 2330 UT, and 0130 UT, nearly simultaneously with a amplitude minimum and a maximum, implying global suppression of ULF waves when approaches 90 . Also, the overall maximum of the amplitude occurs at 0050 UT, at the time of the overall amplitude maximum at the dayside spacecraft. These features support the foreshock source mechanism for the nightside toroidal waves.
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 ( ) is very close to the measured , and the measured and 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 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 ( and ) components at Van Allen Probe B (Figures 4a–4c) along with (Figure 4d) and the electron density ( , Figure 3e). With exceeding 15 , we have confidence in the samples derived from the spin plane components. The electron density varied smoothly and had values higher than 130 cm . The spacecraft remained within the plasmasphere and did not encounter structures such as a drainage plume.
The dynamic power spectra for both and 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 leads . 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 ) at 2200 UT (MLAT = 8.9 and in orange ( ) at 0100 UT (MLAT = 14.5 ). This means that and of the T5 wave oscillated in quadrature and that the spacecraft crossed a node of the T5 wave as it moved to higher MLAT. The node is inferred from the absence of the T5 spectral peak in the dynamic spectra at 2240–2340 UT. We find that the cross-phase of the T3 and T7 waves at 0100 UT is approximately (blue), in contrast to the T5 waves showing (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 and
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 and time series. Despite the unambiguous appearance of the waves in the dynamic spectra, their amplitudes are quite small. The peak-to-peak amplitudes are 0.1 mV/m for and 0.4 nT for . The oscillations appear irregular, and it is difficult to infer the presence of multiple harmonics.
The lower panels of Figure 5 show spectral parameters computed from the time series data. In the 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 power spectrum, a weak peak appears near 2 mHz, and strong peaks appear near 15 and 22 mHz. At the frequencies of the spectral peaks, the coherence (Figure 5c) is elevated. This allows us to evaluate the - 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 and spectra as described by Takahashi et al. (2015) and are plotted versus UT (Figure 6a), MLT (Figure 6b), and (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 range of 4.3–6.1. The frequencies fall with , which is consistent with the monotonic decrease of the electron density with (Figure 6d). The 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.
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 east of the footprint.
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 ). Once is determined, we use theoretical / ratios to assign the harmonic mode numbers to the remaining frequencies. Here, denotes the frequency of the th harmonic. We obtain the theoretical 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 values depend little on the choice of the magnetic field model.
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 ( ) or 3 hr ( ) data window in calculating the cross spectra and move the window in 30 min steps, where is defined using International Geomagnetic Reference Field (Del Corpo et al., 2019). The three panels show , , and . It is evident that and are far more easily detected than , 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 1700 UT ( 20 hr MLT) on 12 May to 0200 UT ( 05 hr MLT) on 13 May, despite the fact that Van Allen Probe B detected toroidal waves within this time interval in the midnight sector.
Figure 9 compares and 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 .
In Figure 9a, the EMMA samples are shown for two epochs, 1500 UT on 12 May (dusk) and 0500 UT on 13 May (dawn). The dusk and dawn samples follow similar decreasing trends versus , 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, at Van Allen Probe B was directly determined only in in a short time interval corresponding to 6. Consequently, we estimated values at other distances assuming a constant ratio. The blue crosses indicate values obtained by assuming = /4, where the values are determined using as shown in Figure 6. The relationship between and 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 range 4.3–6.2. In this range, the spacecraft and the dawn EMMA values agree within an error of 50% in linear scale. At 4.2, the spacecraft and dusk EMMA values agree very well.
In Figure 9b, the EMMA 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 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 = 2.4–5.5 is very low ( 10%). Similar results were obtained using 1 year of observations at = 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: ( , 1.011, 2.307 ) 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 ( = 0) at 5.98 , slightly (0.14 ) 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 in the equatorial region differs significantly between the two models. At the equator, 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 . With usually showing a minimum at the equator, the value in the equatorial regions has the largest influence on the eigenmode frequencies if the density is held constant.
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.
The importance of using a realistic magnetic field model is illustrated in Figure 11, by contrasting eigenmode structures of the T5 mode obtained for the dipole and T89c models. For both field models, the mass density is given by = 156 amu (see section 5.3) and . The 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 0 domain, we find that the nodes are located at MLAT = 10.1 and 30.6 for the T89c model and MLAT = 12.7 and 32.8 for the dipole model. In the MLAT 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 but not on .
Figure 12 shows how the frequencies and the mode structure of toroidal waves depend on . We illustrate the dependencies for the T89c model by varying at integer values from 0 to 6 as was done by Cummings et al. (1969) for the dipole model. All results shown are for = 156 amu 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 increases. This is simply because the mass loading over the entire field line is higher for larger when is held constant. The important fact is, because the Alfvén velocity is not constant along the field line, the degree of the 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 = 0 to 11.3 mHz for = 6, a 49 reduction. The reduction is less significant for the T1 mode: a 21 reduction from 2.14 to 1.69 mHz. This means that the frequency ratio between different harmonics depends on and that we can estimate from the frequency ratios obtained from observation (Takahashi et al., 2004).
Figures 12b–12d show the locations of the (solid line) and (dashed line) nodes. Only the MLAT 0 domain is considered. The nodes all move to higher MLAT as increases. For example, a node of the T5 wave moves from MLAT = 10.2 for = 0 to MLAT = 18.3 for = 6. This dependence also arises from the dependence of the Alfvén velocity and is potentially useful in constraining 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 (solid line) and (dashed line) components of the T1, T3, T5, and T7 waves obtained using the T89c model illustrated in Figure 10 and the = 1 mass density model. The nodes are marked by filled ( ) or open ( ) circles. Because we solve the wave equation assuming perfect reflection at the ionosphere, diminishes at the ionosphere (MLAT = 69 ), and and 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 and . In the MLAT domains shaded orange, leads by 90 . In the MLAT domains shaded blue, the phase relation is reversed. The vertical dotted line at MLAT = 14.4 indicates the location of Van Allen Probe B at the middle of the 30 min interval shown in Figure 5.
Following the vertical dotted line, we find that the model predicts the - cross-phase to be 90 for the T1 and T5 modes and 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 - 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 - cross-phase. This blank MLAT domain is estimated to be 1–2 wide.
The eigenmode solutions also give a reasonable explanation to the observed toroidal wave frequencies. To obtain the theoretical frequencies, we use the measured value of 165 cm at Van Allen Probe B for the epoch of 0055 UT. Assuming that changes along the magnetic field in the same manner ( = 1) as the mass density, we find = 155 cm at the magnetic equator. We then assume that the ions are all protons, that is, = 156 amu cm (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 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 and spectra. We conclude that the observed multifrequency oscillations in and 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 ( 120 km). For the 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 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 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 , where and are the height-integrated Hall and Pedersen conductivities, is the height of the ionosphere, and 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 0.2 (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 = 4–6 region (Figure 6c), where was apparently high enough to sustain midnight toroidal waves excited in the fixed-end mode as discussed in section 6.2.
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 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 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 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 ( MLAT ) than the nightside spacecraft in the present study (Van Allen Probe B, MLAT ). This MLAT difference can explain the different to 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 .
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 power is enhanced below the predicted upstream wave frequency. These features disagree with those reported in previous studies that indicate 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 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 20 to 70 mHz, although the average frequency was close to the prediction 40 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 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 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.
The two vertical dashed lines in Figure 14a represent two different models. The lower 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 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 ( = 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 becomes lower than a certain value, 0.6 S in the current example. As becomes low, what is identified as the fixed-end T2 mode at the high 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 0.6 S because of the following reasons. First, the IRI model predicts that the imaginary part of is 1 mHz, which is close to the peak value at = 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 2 mHz as shown in Figures 4-6, must be increased from 156 amu cm by a factor of 4, because the frequency is proportional to 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 - cross-phase close to at the location of Van Allen Probe B (MLAT = 14 , green vertical dashed line), which is inconsistent with the 90 cross-phase observed (Figure 5d) and predicted for the T1 mode for high (Figure 14g).
The question remains as to whether sufficient precipitation occurred in the outer plasmasphere to raise 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.
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 ( 45 ) 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.
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