Modulation of Ion and Electron Pitch Angle in the Presence of Large-amplitude, Low-frequency, Left-hand Circularly Polarized Electromagnetic Waves Observed by MMS

Low-frequency electromagnetic cyclotron waves (i.e., ion cyclotron waves) are usually observed in different solar–terrestrial environments, i.e., the Earth's magnetosphere, the magnetosheath, and the solar wind (e.g., Anderson & Fuselier 1993, 1994; Dunlop et al. 2002; Jian et al. 2014; Remya et al. 2014; Wicks et al. 2016; Zhao et al. 2018). These waves play an important role in the dynamics of charged particles. Linear and quasi-linear theories have shown that the resonance interaction of charged particles with ion cyclotron waves will result in the loss of relativistic electrons in the Earth's radiation belt (e.g., Summers & Thorne 2003; Summers et al. 2007; Jordanova et al. 2008) and the precipitation of ring current ions (e.g., Jordanova et al. 2001). Nonlinear mechanisms, including phase bunching and phase trapping of ions by ion cyclotron waves, can lead to a change in the ion pitch angle (e.g., Bortnik et al. 2010; Omidi et al. 2010; Zhu et al. 2012), i.e., the enhancement of low-energy protons and helium particles at the pitch angle ∼90° observed by the AST-6, GOES1 and 2, and AMPTE/CCE spacecraft (e.g., Mauk et al. 1981; Roux et al. 1982; Anderson & Fuselier 1994). However, in comparison with studies relating to small-amplitude waves, research on large-amplitude, low-frequency electromagnetic cyclotron waves has not attracted much attention.

Large-amplitude, circularly polarized electromagnetic proton cyclotron waves have been observed in the Earth's dusk-flank magnetosheath by Wind and Cassini measurements (Tsurutani et al. 2002; Remya et al. 2014), which find the magnetic field fluctuation to be ∼14 nT in the ambient magnetic field of ∼55 nT. Using Magnetospheric Multiscale (MMS; Burch et al. 2016) measurements, this study provides other observational evidence for the existence of large-amplitude, low-frequency electromagnetic cyclotron waves in the dusk-flank side of the Earth's magnetosheath. We show that the wave amplitude is about 1–2 nT, where the ambient magnetic field is ∼15 nT. The relative amplitude is of the order of 0.1. Since the plasma measurement in the MMS has high time resolution, i.e., 0.15 s for ions and 0.03 s for electrons, it provides an excellent opportunity to study ion and electron dynamics in the presence of large-amplitude electromagnetic waves.

Since we investigate both wave and particle behavior, we analyze MMS data from instruments including electric field double probes (Ergun et al. 2016; Lindqvist et al. 2016), fast plasma investigation (FPI; Pollock et al. 2016), and fluxgate magnetometers (Russell et al. 2016).

Figure 1 shows an overview of the observed wave during the time interval from 14:00:00 UT to 14:06:00UT on 2017 October 7. The MMS is located at nearly (−3.9, 23.7, 5.7) Earth radii in Geocentric Solar Magnetospheric coordinates, corresponding to the dusk-flank magnetosheath. The magnetic fluctuations are distinct over the whole time interval. At ${t}_{s\to n}\sim {\rm{14:03:37.37}}$ UT, which is denoted by the red vertical dashed line in Figure 1, the direction of B_z changes from southward to northward. The plasma density and ion streaming velocity are relatively steady. The averaged velocities, i.e., V_ix ∼ −265 km s⁻¹, V_iy ∼ 124 km s⁻¹, and V_iz ∼ 16 km s⁻¹, indicate ions streaming anti-parallel to the magnetic field. Polarization analysis in panels (d)–(g) show that the magnetic fluctuations in the frequency range ∼[0.1, 0.5] Hz have the following properties: (1) degree of polarization larger than 0.8; (2) ellipticity smaller than −0.8; (3) the normal angle smaller than 20° or larger than 160°. A 180° uncertainty for the normal angle comes from the wavelet (Morlet) analysis used in panel (e). Because of the inverse Poynting flux (not shown in Figure 1), and the positive correlation between velocity and magnetic fluctuations (shown in Figure 2), we conclude the normal angle is larger than 160°. This wave information indicates that the mode corresponds to a nearly anti-parallel propagating, left-hand circularly polarized wave, usually called the electromagnetic ion cyclotron wave in studies of the planetary magnetosphere.

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Figure 2 gives electromagnetic and velocity fluctuations in field-aligned coordinates. Since an abrupt change of the magnetic field configuration, not the wave fluctuation, occurs at nearly ${t}_{s\to n}$, we choose the time interval of 14:02:35.00UT–14:03:17.00UT before ${t}_{s\to n}$ to explore the general properties of the electromagnetic and velocity fluctuations. Full and filtered magnetic field components are shown in panels (a) and (b). Referring to Figure 1, coherent waves concentrate in the frequency range 0.1 Hz ≤ f ≤ 0.5 Hz; therefore, this frequency range is selected as the filtered frequency. The magnetic fluctuation is of the order of 1 nT, and the largest amplitude can reach ∼2 nT, where the ambient magnetic field is nearly 15 nT. The phase of B_⊥1 is π/2 ahead of B_⊥2, indicating the left-hand polarization mode. Also, left-hand polarization is seen from the phase relation between E_⊥1 and E_⊥2 shown in panel (c); the considerable ${B}_{| | }$ and ${E}_{| | }$ arise. Panels (d) and (e) present the phase relations between magnetic and velocity components, that is, V_i⊥/V_A ≃ B_⊥/B₀, indicating the wave propagating anti-parallel to the ambient magnetic field.

Figure 3 shows the ion energy distribution during the time interval from 14:02:35.00UT to 14:03:35.00UT. From panels (a), and (b), the ion omnidirectional flux mainly concentrates in the energy range 100 eV ≲ E_i ≲ 1000 eV, and the ion pitch angle θ is normally larger than 90°. Panels (c)–(g) give the ion pitch angle distribution (IPD) in different energy ranges. A chaotic IPD appears in the low-energy range 2.16 eV ≤ E_i ≤ 58.89 eV, indicating very low ion density therein. It is interesting to see a periodic variation of IPD in the energy ranges 77.98 eV ≤ E_i ≤ 239.63 eV, 317.28 eV ≤ E_i ≤ 736.45 eV, and 975.08 eV ≤ E_i ≤ 3967.62 eV. The change of pitch angle is nearly consistent with the variation of the angle ${\theta }_{{{BV}}_{i}}$ between the ion streaming velocity and magnetic field, defined as

$$\begin{eqnarray*}&&{\theta }_{{{BV}}_{i}}=\arccos \left(\displaystyle \frac{{\boldsymbol{B}}\cdot {{\boldsymbol{V}}}_{i}}{| B| | {V}_{i}| }\right).\end{eqnarray*}$$

Moreover, both θ and ${\theta }_{{{BV}}_{i}}$ variations are closely related to the magnetic fluctuations shown in panel (i). Therefore, Figure 3 provides observational evidence for modulation of the ion pitch angle by large-amplitude cyclotron waves. For high-energy ions, 5253.24 eV ≤ E_i ≤ 28301.89 eV, there is no modulation feature in the IPD.

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Figure 4 shows the electron energy distribution during the time interval from 14:02:35.00UT to 14:03:35.00UT. From the omnidirectional electron differential energy flux (panel (a)), the electron energy is normally smaller than 1 keV. The electron pitch angle distribution (EPD) in panels (b)–(h) exhibits different dynamics behaviors of electrons with different energy. For low-energy electrons, 6.52 eV ≤ E_e ≤ 19.15 eV, the pitch angle θ is always larger than 135° before ${t}_{s\to n}$ , and larger than 90° after ${t}_{s\to n}$. This indicates low-energy electrons mainly streaming anti-parallel to the magnetic field. In the energy range 25.07 eV ≤ E_e ≤ 42.95 eV, most electrons move along the magnetic field, i.e., θ ≲ 45°. We also find the appearance of cooling for electrons at ${\theta }_{p}\lt \theta \lt 180-{\theta }_{p}$, where the critical pitch angle θ_p is defined as ${\theta }_{p}=\arcsin \sqrt{B/{B}_{\max }}$, B is the magnitude of the magnetic field, and B_max is the maximal B during 14:02:35.00UT–14:03:35.00UT. Moreover, the cooling of trapped electrons arises in panel (e) where 56.23 eV ≤ E_e ≤ 73.60 eV. For ${E}_{e}\gtrsim 216.11$ eV, trapped electrons are slightly heating, shown in panels (g) and (h). On the other hand, there exist many localized EPDs for electrons with 56.23 eV ≲ E_e ≲ 830.63 eV. An example of localized EPD is shown as the band between the two vertical dashed lines in panels (e)–(g). The timescale of these localized EPDs approximates the period of the observed wave, implying the role of the large-amplitude cyclotron wave on the emergence of these confined electrons.

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Since the ion cyclotron anisotropic instability is widely believed to be the source of ion cyclotron waves in the Earth's magnetosphere (i.e., Cornwall 1965; Kennel & Petschek 1966), here we check the possibility of local excitation of the observed waves by this kind of instability. We adopt an instability threshold condition $({T}_{i\perp }/{T}_{i\parallel }-1)=0.35/{\beta }_{\parallel }^{0.42}$ (corresponding to $\gamma /{{\rm{\Omega }}}_{{ci}}={10}^{-4}$), assumed in plasmas only having electron and ion components (Gary & Lee 1994). Using the plasma parameters averaged from FPI during 14:02:35UT–14:03:17UT, n_i = 12 cm⁻³, ${T}_{i| | }=91\,\mathrm{eV},{T}_{i\perp }=111\,\mathrm{eV}$, ${T}_{e| | }=32$ eV, T_e⊥ = 34 eV, and B₀ = 15 nT, we find that $({T}_{i\perp }/{T}_{i\parallel }-1)\,=0.22\lt 0.35/{\beta }_{\parallel }^{0.42}=0.27$. Therefore, there is no ion temperature anisotropy instability in our event. Also, we use WHAMP (a dispersion equation solver based on linear plasma kinetic theory, Rönnmark 1982), and find that the wave is damping, i.e., $\gamma \sim -0.003{{\rm{\Omega }}}_{{ci}}$ at ω ∼ 0.2Ω_ci. On the other hand, Narita et al. (2004) found that some of the low-frequency, left-hand-polarized Alfvén waves in the Earth's foreshock propagate downstream. Therefore, our observed waves may be generated nonlocally in the foreshock.

Since the observed waves propagate at $| \theta | \sim 0^\circ$, the wave frequency at the plasma frame is ${\omega }_{\mathrm{plasma}}/{{\rm{\Omega }}}_{{ci}}\,\approx ({\omega }_{\mathrm{obs}}/{{\rm{\Omega }}}_{{ci}}){(1+{V}_{\parallel }/{V}_{A})}^{-1}\approx 0.15-0.6$, where the parallel ion streaming velocity is ${V}_{\parallel }\simeq 219$ km s⁻¹, the Alfvén velocity is ${V}_{A}\simeq 97$ km s⁻¹ during 14:02:35UT–14:03:17UT, and the observed wave frequency is ${\omega }_{\mathrm{obs}}\sim 0.5{{\rm{\Omega }}}_{{ci}}-2{{\rm{\Omega }}}_{{ci}}$. From linear plasma theory we have linear responses among perpendicular electromagnetic components for the left-hand polarized cyclotron wave: ${E}_{\perp 2}=-{{iE}}_{\perp 1}$, ${B}_{\perp 2}=-{{iB}}_{\perp 1}=({k}_{| | }/\omega ){E}_{\perp 1}$, and ${V}_{i\perp 2}=-{{iV}}_{i\perp 1}=-(\omega /{k}_{| | })({B}_{\perp 2}/{B}_{0})/(1-\omega /{{\rm{\Omega }}}_{{ci}})$ (i.e., Zhao 2015). When the wave is propagating anti-parallel to the ambient magnetic field, i.e., ${k}_{| | }$ is negative, E_⊥1 and B_⊥2 are out of phase, ${E}_{\perp 2}$ and B_⊥1 are in phase, and V_i⊥1,2 and ${B}_{\perp \mathrm{1,2}}$ are in phase. These theoretical predictions are consistent with our observations. Furthermore, when the wave is obliquely propagating, even at a small normal angle, the parallel electromagnetic fields ${E}_{| | }$ and ${B}_{| | }$ will arise; for example, ${E}_{| | }\sim 0.015{E}_{\perp 1}$ and ${B}_{| | }\sim 0.16{B}_{\perp 2}$ at ω ∼ 0.2Ω_ci as $| \theta | =10^\circ$. Since the observed ${E}_{| | }$ and ${B}_{| | }$, i.e., ${E}_{| | }\sim 0.2{E}_{\perp }$ and ${B}_{| | }\sim 0.5{B}_{\perp }$, are stronger than corresponding linear responses, this is in favor of ${E}_{| | }$ and ${B}_{| | }$ produced through the nonlinear mechanisms.

The most interesting finding in the study is the modulation of the ion and electron pitch angles by the observed large-amplitude electromagnetic cyclotron waves. The periodic variation of the pitch angle arises for ions with energy from E_i = 77.98 eV to E_i = 3967.62 eV. When ions stream into the dusk-flank side of the magnetosheath, owing to large magnetic fluctuations, they induce a variation of the angle between the ion streaming velocity and the magnetic field, which results in the ion pitch angle having a periodic variation. On the other hand, some electrons will be trapped in the inhomogeneous magnetic field due to large magnetic fluctuations. Cooling occurs for trapped electrons with 25.07 eV ≲ E_e ≲  73.60 eV, and heating appears for trapped electrons with E_e ≳ 216.11 eV. Moreover, for electrons having 56.23 eV ≲ E_e ≲ 830.63 eV, their pitch angles exhibit an intermittent distribution feature, and the timescale is nearly the same as the wave period. This implies that electrons may be trapped by the large parallel electric field fluctuations. Therefore, the observed large-amplitude electromagnetic wave can affect both the dynamics of ions and electrons.

In summary, using MMS, this study observes the periodic variation of ion and electron pitch angles in the presence of large-amplitude, low-frequency, left-hand circularly polarized electromagnetic waves in the Earth's dusk-flank magnetosheath. Modulation of the ion pitch angle results from the change in the ion streaming velocity relative to the magnetic field including the ambient and fluctuation fields. Modulation of the electron pitch angle is induced by the inhomogeneous magnetic field structure, which is partly caused by the magnetic field fluctuation, and also probably by electrons trapped in the electromagnetic field of the large-amplitude wave.

This work was supported by the NNSFC 11673069, 41531071, by NSF of Jiangsu Province under grant No. BK20161617, and the Youth Innovation Promotion Association CAS. M.W.D. is supported by NNSFC 41874193 and 41574155, NERC grant NE/H004076/1, and by STFC in-house research grant. T.Y.W. is supported by the Marie Skłodowska-Curie grant No. 665593 from the European Union's Horizon 2020 research and innovation programme. J.S.H. is supported by NNSFC 41574168 and 41874200. The data are available from the MMS Science Data Center (https://lasp.colorado.edu/mms/sdc/public/).