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Volume 65 Supplement 5

Physical Processes in Non-Uniform Finite Magnetospheric Systems—50 Years of Tamao's Resonant Mode Coupling Theory

Field-line resonance structures in Mercury’s multi-ion magnetosphere


Recently, MESSENGER spacecraft detected transverse waves at Mercury’s inner magnetosphere. The magnetic field fluctuations of theses waves are approximately perpendicular to the gradient of magnetic field magnitude which is similar to the field-line resonance characteristics predicted by numerical simulations in two-ion plasmas. In this paper, we perform a wave simulation in a three-ion plasma to consider the effect of multiple heavy ions at Mercury. Because recently observed wave frequencies are near the He+, He2+ and H+, we adopt multi-ion plasmas that contain H+, He2+, and He+. The simulation results show that several resonant waves between the ion gyrofrequencies can occur at the same location and also show the modulation of amplitude in time histories. Therefore, the simulation results suggest that it could be possible to observe two or three different resonant frequencies at certain locations in Mercury’s magnetosphere.

1. Introduction

Ultra-low frequency (ULF) waves near the ion gyrofrequencies are often observed by MESSENGER spacecraft inside Mercury’s magnetosphere (Boardsen et al., 2009a, b, 2012). Since the Mariner 10 spacecraft detected ULF waves for the first time (Russell, 1989), the identification of ULF waves at Mercury has been a topic of interest. The wave frequency detected by Mariner 10 spacecraft was 0.5 Hz at 38% of the proton gyrofrequency (ωcH) and wave frequencies from MESSENGER are near and . Because observations showed that the region around Mercury is filled with heavy ions, such as Na+, O+, K+ and He+ (Zurbuchen et al., 2008, 2011), waves at Mercury require a treatment that includes multiple ions with gyrofrequency effects (Othmer et al., 1999; Glassmeier et al., 2003; Klimushkin et al., 2006; Kim et al., 2008, 2011).

The first observed ULF waves at Mercury were believed to be a field-line resonance (FLR) standing mode along the magnetic field line in single fluid plasma (Russell, 1989). With a field line length of about 4,000 km based on a dipole model, Russell (1989) estimated that the observed waves could be the fourth harmonic of the fundamental wave. Later, Othmer et al. (1999) suggested that the crossover frequency is the preferred frequency. Kim and Lee (2003) examined the behavior of the Alfven resonance when ω ~ ωcH and discussed the implications for waves at Mercury in a single-ion plasma. However, Southwood (1997) and Blomberg (1997) both argued that the observed waves cannot be pure standing Alfven waves because they have a compressional component and polarization aligned in the north-radial meridian rather than the east meridian. More recently, Klimushkin et al. (2006) and Kim et al. (2008) suggested that FLRs at Mercury occur at the ion-ion hybrid and/or Alfvén resonance conditions and Kim et al. (2008) showed that the observed waves from the Mariner 10 are not FLRs.

When ω ωce,ωpe, where ωce and ωpe are the electron gyro- and plasma frequencies, respectively, the basic description of the plasma wave is given by the approximate cold plasma dispersion relation


where nǁ and n are refractive indices parallel and perpendicular to the background magnetic field (B0), respectively. R, L and S are the Stix’s tensor elements for two ions (Stix, 1992). This approximate dispersion relation has a resonance when n → ∞ at


and the cutoffs at


When wave frequencies (ωii) satisfy the resonance condition in Eq. (2), the ion-ion hybrid (IIH) resonance conditions lie at frequencies between each neighboring ion gyro-frequency pair. In two-fluid plasmas or below the heaviest ion gyrofrequency, this resonance corresponds to the Alfvén resonance (ω a ). Therefore, if the plasma contains m ion species, there are m − 1 ion-ion hybrid resonances and one Alfvén resonance. The wave simulation by Kim et al. (2008) showed that mode-converted waves at the IIH resonance location oscillate linearly, and that the wave energy is strongly guided by B0. They also showed that the coupling is stronger at the IIH resonance compared to the Alfvén resonance. Furthermore, Kim et al. (2011) showed that wave absorption coefficients at the IIH resonance are sensitive to the azimuthal and field-aligned wave numbers as well as heavy ion concentration ratio, thus the FLRs at Mercury can have various radial structures depending on heavy ion density and azimuthal wave numbers.

A recent survey of ULF waves at Mercury showed that the waves are highly transverse at large magnetic latitudes but are more compressional near the equator (Boardsen et al., 2012). They also found that for transverse dominant modes, the magnetic field fluctuations tend to be oriented perpendicular to the estimated field gradient. Previous studies of wave mode conversion at Mercury focused on Na+-H+ hybrid modes; however the statistical study by Boardsen et al. (2012) shows peaks just below and . In order to consider more recent observed ULF waves at Mercury, we perform a wave simulation in a multi-ion plasma contains H+, He2+, and He+ ions. Our results show that, under the right conditions, the field-aligned waves generated by mode conversion can have wave amplitude beating of several mode-converted waves at the IIH and/or Alfvén resonances.

2. Numerical Simulation

We employ the fluid wave simulation model which has been developed by Kim and Lee (2003). Similar to previous wave simulations (Kim et al., 2008), the background magnetic field B0 and the electron density Ne are assumed to be constants with Ne = 3 cm−3, and B0 = 86 nT at Mercury. The ambient magnetic field, B0, lies in the z direction and the inhomogeneity is introduced in the x direction. Since the Mercury’s magnetopause is located near 1.4 R m (Anderson et al., 2011), we assumed a shorter radial distance of 1 R m than the magnetopause location in x direction, where R m is the Mercury’s radius. To save computing time, electron mass is assumed to be me = m H /100. We limit ourselves to harmonic variations in the y and z and all waves are proportional to exp(ik y y+ik z z), where k y and k z are the given y and z direction wavenumbers. For simplicity, k y is assumed to be 0 and k z = 2π/L z , where L z = 1 R m , which is similar to the field line length of L z = 0.93 at L = 1.5 in dipole coordinate. The simulation is driven by imposing an impulse in E y at X =0 during the interval 0 ≤ τ ≤ 2, where X = x/L x , τ = t/tci, and tci = 2π/ωci. The simulation is run from τ = 0–60 and the boundaries become perfect reflectors after the impulsive stimulus ends (τ = 2), thus the total energy in the box model will remain constant in time after this interval.

Because H+ and He+ are major ions at Mercury (Zurbuchen et al., 2008, 2011) and waves near the He2+ gy-rofrequency are observed (Boardsen et al., 2009a, b), we adopt an electron-H+-He2+-He+ plasma and the electron density is also assumed to be sum of the ion densities. For simplicity, the He2+ density ratio to electron density (ηHe2 = NHe2/Ne) is assumed to be 0.025, however, in order to see the multi-ion effects easily, we assume decreasing He+ density ratio in space. Figure 1(a) shows the ion density ratio to electron density (ηion = Nion/Ne) profile. We assumed the H+ density ratio (ηH) increases from 0 at X = 0 to 0.95 at X = 1 while He+ density (ηHe) decreases from 0.95 to 0. Using this profile, we calculate the normalized critical frequencies to ωci (Ω = ω/ωci), such as cutoffs ΩL(R) (where ) and ion-ion hybrid (and Alfvén) resonances (Ωii and Ω A ) frequencies. These frequencies are plotted in Fig. 1(b) and shaded regions represent frequency stop-bands where waves are evanescent. Different from the two-ion case predicted by Kim et al. (2008), the IIH resonance modes in between the proton and heaviest ion gyrofrequencies split into the two branches of Ωii−1 (where ) and Ωii −2 (where ). Here, Ωii −1 decreases from 1 to 0.527 and Ωii −2 decreases from 0.486 to 0.312. When the impulsive input is excited at X =0, most waves in these resonance frequency ranges encounter the resonance location without cutoffs. In this case, the maximum absorption can increase up to 100% (Lee et al., 2008; Kim et al., 2011). However, below most waves except between the local cutoff of ΩR = 0.124 and resonance frequency of Ω = 0.143 at X = 0 cannot propagate toward the resonances. Therefore, if the wave power is not enough large to penetrate the wave stop gap, the mode conversion at the Alfvén resonance does not occur.

Fig. 1.
figure 1

(a) Ion density ratio (η = Nion/Ne) of H+ (dashed), He+ (solid), and He2+ (dotted-dashed). Here we assume ηHe2 = 2.5%. (b) The critical frequencies: cut offs ΩL (R) and ion-ion hybrid (and Alfvén) resonance (Ωii(A)) frequencies. The horizontal and vertical axes are normalized distance X = x/L x and normalized frequency Ω = ω/ωci. Here shaded regions represent frequency stop-bands where the wave is evanescent.

We store time histories of the electromagnetic fields at each grid point in X during the simulation running time of 0 < τ < 60 and obtain the wave power spectra through the fast Fourier transform. Figure 2 shows the wave spectra of (a) B x (radial component), which is and (b) B y (azimuthal component).

Fig. 2.
figure 2

The wave spectra of the perturbed electric field: (a) the radial component of B x and (b) the azimuthal component of B y . The horizontal axis is the direction of inhomogeneity, X, and the vertical axis is a normalized frequency that ranges from 0 to 1. The white solid, dashed, and dashed-dotted lines in (a) are Ωii(A), Ω L , and Ω, R , respectively, and yellow dashed lines in (b) represent Ωii(A) . Here, there are two IIH resonances in 0.5 < Ωii−1 < 1 and 0.25 < Ωii−2 < 0.5, and one Alfvén resonance in Ω A < 0.25. The open circles mark the point chosen for time histories.

In this figure, harmonics of the global cavity modes appear in both components near Ω = 0.14 which is in between Ω L < Ω < Ω A , and Ω = 0.32, 0.38, and 0.51 which are larger than the local Ω R . In contrast with the global cavity wave modes, the continuous spectrum appears only in azimuthal component of B y . For , there are two continuous bands whose frequencies decrease with increasing X. These continuous spectrum are impulsively excited by the broadband compressional source at X = 0 and corresponds to the IIH resonant conditions of Ωii−1 and Ωii−2. For , power enhancement of B y only occurs at Ω = 0.14 near X = 0. This mode is associated with Alfvén resonance. Different to Ωii−1 and Ωii−2, most waves in the frequency range of Ω A at X = 0 cannot propagate into the simulation domain as shown in Fig. 1(b) but some wave energy can directly reach Ω A near X = 0.

Figure 2 also shows that each resonant wave mode branches have different power at different locations. Resonant waves at Ωii−1 have strong power in wide range 0 ≤ X ≤ 0.95, while waves at Ωii−2 are strong for 0.6 ≤ X ≤ 0.85 but weak for 0 ≤ X < 0.6. Moreover, the Alfvén resonance mode (Ω A ) occurs only near X ≈ 0. Therefore, single or multiple frequencies of mode-converted wave can be detected at certain locations. For instance, two mode-converted waves can be detected at X = 0.06 (Ωhigh = Ωii−1 ≈ 0.93 and Ωlow = Ω A ≈ 0.16) and X = 0.7 (Ωhigh = Ωii−1 ≈ 0.57 and Ωlow = Ωii−2 ≈ 0.44) while a single frequency of mode-converted wave (Ωhigh = Ωii−1 ≈ 0.8) can be observed at X = 0.2 (these frequencies are marked as open circles in Fig. 2(b)). In order to examine the coupling properties, we plot the time histories of B x and B y in Fig. 3, which are obtained using the inverse Fourier transforms. In this figure, the first and second panels are wave time histories of higher (Ωhigh) and lower frequencies (Ωlow) of magnetic azimuthal component (B y ) at (a) X = 0.06, (b) 0.2, and (c) 0.7, respectively. We also plot the wave time histories of the azimuthal (B y ) and radial (B x ) components in the third and fourth panels of Fig. 3 that show superposition of two resonant waves of Ωhigh and Ωlow. However, because only one resonant wave mode is dominant at X = 0.2, the second panel is remained as blank.

Fig. 3.
figure 3

Time histories of electric fields at the IIH resonance marked as circles at (a) X = 0.06, (b) X = 0.2, and (c) X = 0.7. The first and second rows are time histories of the azimuthal component for Ωhigh and Ωlow, respectively. Here, for Ωhigh = 0.8 at X =0.2, for Ωhigh = 0.93 and Ωlow = 0.16 at X = 0.06, and for Ωhigh = 0.57 and Ωlow = 0.44 at X = 0.7, respectively. The third and fourth rows are wave modulations of azimuthal and radial components.

Similar to the previous multi-ion simulation study by Kim et al. (2008), wave azimuthal components with single frequency grow in time as shown in the first and second panels of Fig. 3. When mode-converted waves contain two different frequencies, waves can show modulation of the amplitude in time. For X = 0.06, because frequency ratio of two resonant waves Ωhigh/ Ωlow is 6.27 and the amplitude of Ωlow is greater than Ωhigh, wave amplitude beating is not significantly appeared in the third panel. Different to case of X = 0.06, for X = 0.7 in Fig. 3(c), Ωhighlow = 1.31 and the wave amplitudes are similar for both frequencies, therefore, the amplitude beating in time is clearly shown in the third panel of Fig. 3(c). In this case, when two wave amplitudes are assumed to be the same and constant in time, the beat period Tbeat can be approximated to be 1/(ΔΩ) ≈ 7.7, which is consistent with the simulation results. In contrast to the increase of the azimuthal component (B y ), the radial component (B x ) damps for two mode-converted waves (fourth panels in Figs. 3(a) and (c)) and for a single mode (Fig. 3(b)). When there are two mode-converted waves appeared at the same location in Fig. 3(a) and (c), B x also show wave amplitude beating with damping in time.

3. Discussion and Summary

In this paper, we perform a wave simulation in three-ion plasmas to consider multiple heavy ion effects at Mercury. The simulation results show that several resonant waves between the ion gyrofrequencies can occur at certain locations and show wave amplitude beating in time histories.

Mercury’s magnetosphere contains heavier (e.g., O+, K+, Na+) and lighter ions (e.g., He+, He2+) and MESSENGER spacecraft have recently detected ULF wave events near and (Boardsen et al., 2009a, b, 2012). The observed magnetic component of transverse waves tends to be strongly linearly polarized and oriented approximately perpendicular to the estimated field gradient (which is plasma inhomogeneity). This is consistent with wave simulation results: In Figs. 2 and 3, the B y component which is perpendicular to the plasma inhomogeneity is dominant at the mode-converted waves. Previous two-ion simulations demonstrated the IIH resonance occurs in H+-Na+ plasmas but did not include He+ or He2+ ions. In this paper, we show how small amount He2+ change the FLRs structure. The simulation results suggest that several resonant waves between the ion gyrofrequencies can occur at a certain location and show wave amplitude beating in time histories. Therefore, it could possible to observe two or three different resonant frequencies in Mercury’s magnetosphere. In addition, because the IIH resonance is the result of ion-ion effects and those frequencies are affected only by the relative ion density ratio rather than the absolute number density, the wave frequency could be used to estimate the local heavy ion density if a spacecraft detects multiple resonances (Kim et al., 2008).

In this study, a cold plasma model is adopted. However, because of the small dimensions of Mercury’s magnetosphere (for instance, magnetopause is located near 1.4 R m ~ 3400 km and the large gyro radii of ions ~100–1000 km, kinetic effects can play an important role for the generation and propagation of ULF waves (Glassmeier et al., 2004; Glassmeier and Espley, 2006; Slavin et al., 2009). When the kinetic effects are included, the compres-sional wave can be converted into the short wavelength ion Bernstein and/or ion cyclotron waves at the IIH resonance. These mode converted waves can be strongly damped in the vicinity of the mode-conversion surface via electron Landau damping or damped by ions at the Doppler broadened ion cyclotron resonances (e.g., Phillips et al., 1995; Intrator et al., 1986; Jaeger et al., 2003, 2006). Therefore, although the cold plasma wave simulation in this study shows the resonant absorption characteristics near the IIH and Alfvén resonance clearly, more detailed investigations on mode-converted waves including kinetic effects at Mercury’s multi-ion plasmas are necessary.


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The work at the Princeton University was supported by NASA grants (NNH09AM53I, NNH09AK63I, and NNH11AQ46I), NSF grant ATM0902730, and DOE contract DE-AC02-09CH11466. The work at the Kyung Hee University was supported by KRF-2008-31-C00375 and the WCU program through NRF funded by MEST of Korea (R31-10016). Eun-Hwa Kim thanks to Peter Damiano for useful discussions.

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Kim, EH., Johnson, J.R., Lee, DH. et al. Field-line resonance structures in Mercury’s multi-ion magnetosphere. Earth Planet Sp 65, 447–451 (2013).

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