Distribution of water-group ion cyclotron waves in Saturn’s magnetosphere

A fast Fourier transform (FFT) is performed after the magnetic field data have been detrended. The selected bandwidth is ±0.1 Hz with respect to the peak frequency. Signals with frequencies close to the local water-group ion cyclotron frequency f _W+ (within 40%, in the spacecraft moving frame) are analyzed in this section. Then, for these selected data, the magnetic perturbations with dominant transverse components (\(\delta B_{ \bot } /\delta B_{||} > 2\)) and left-hand polarization are identified as ICWs.

First, the Cassini spacecraft’s orbits are projected in a Saturn-centered rectangular (X, Y, Z) coordinate system. This is defined with the X–Y plane being Saturn’s equatorial plane, with the Z-axis along Saturn’s magnetic axis, the Sun in the X–Z plane, and the positive Y-axis pointing toward the dusk. Note that we do not differentiate the rotational axis from the magnetic axis because their angles differ by less than 1°. We thus assume that they are in the Z-direction. Figure 1a–e shows a projection of the Cassini orbits in the X–Y plane in the left panels and the projection in the R–Z plane of the cylindrical coordinate in the right panels. This is plotted for each year from January 2005 to December 2009. The Cassini orbit is shown in red when the wave frequency obtained from the magnetic field data is near the local W⁺ gyrofrequencies. The Cassini orbit projection in the X–Y equatorial plane (left panels) indicates the local time coverage. The orbit projection in the cylindrical R–Z plane (right panels) indicates the orbit inclination.

In the 2005 data (Fig. 1a), the W⁺ cyclotron waves are detected in the noon, dusk and midnight regions, with the magnetic latitude θ between −10° and 8° and the radial distance R between 3.5 R _S and 7.8 R _S . In the 2006 data (Fig. 1b), the W⁺ cyclotron waves are detected mostly in the noon and dusk sectors, with θ between −7° and 7° and R between 4 R _S and 7.5 R _S . In the 2007 data (Fig. 1c), the W⁺ cyclotron waves are detected in the midnight and dawn regions, with θ between −8° and 20° and R between 3.6 R _S and 7.2 R _S . In the 2008 data (Fig. 1d), wave signals are found in the midnight region with θ~ −40°–30° and R~ 3.7–6 R _S . In the 2009 data (Fig. 1e), the W⁺ cyclotron waves are detected in the midnight, dawn and noon regions with θ between −12° and 18° and R between 3.6 R _S and 7.7 R _S . The 5 years of data show that most of the ICW signals are found inside θ ~±5° except for the 2008 data and part of the 2007 data, when Cassini was in the midnight sector. The latitudinal and local time dependence of ICWs will be further discussed in section “Ion cyclotron wave amplitude versus latitude.”

The ratio of \(\delta B_{ \bot }\) to \(\delta B_{||}\) helps to determine the wave type, where the subscripts ⊥ and || indicate the components perpendicular and parallel to the ambient magnetic field, respectively. If \(\delta B_{ \bot } /\delta B_{||}\) is larger (smaller) than one, the magnetic fluctuations can be regarded as transverse (compressional) waves. To obtain the magnetic field perturbations in the perpendicular and parallel directions to the ambient magnetic field, the KRTP coordinates are locally rotated such that \(\hat{\theta }\) aligns with the ambient magnetic field direction. The averaged value of \(\delta B_{ \bot }\) is calculated by \(\delta B_{ \bot } = \left\langle {\sqrt {\left( {\delta B_{r} } \right)^{2} + \left( {\delta B_{\varphi } } \right)^{2} } } \right\rangle\), where the bracket means the average over a time period much longer than the wave period or an orbital distance. For example, the data series in the form \(\delta B_{\phi } = {\text{sin(}}\omega \,t ) { }\) and \(\delta B_{r} = {\text{cos(}}\omega \,t ) { }\) results in \(\delta B_{ \bot } = 1\).

The average value of \(\delta B_{||}\) is calculated by \(\delta B_{||} = \sqrt {2\left\langle {\left( {\delta B_{\theta } } \right)^{2} } \right\rangle }\), where the average bracket value is multiplied by 2 inside the square root to give the correct wave amplitude for a sinusoidal wave. Figure 2 shows the ratio of \(\delta B_{ \bot }\) to \(\delta B_{||}\) versus the cylindrical radial distance R, in the low-latitude (±5°) region. The results for the noon, dawn, dusk and midnight sectors are shown in panels (a), (b), (c) and (d), respectively. For each data point, \(\delta B_{ \bot }\) and \(\delta B_{||}\) are averaged over an orbital distance with ΔR = 0.2 R _S each time Cassini passes through the waves.

These 5 years of data have the same characteristics: Transverse waves are mostly inside 6 R _S . The data points with \(\delta B_{ \bot } /\delta B_{||} \gg 1\) outside 6 R _S are mainly in the midnight sector (see Fig. 2d). Some data points with \(\delta B_{ \bot } /\delta B_{||} \gg 1\) outside 6 R _S also occur in the dusk sector (see 2007 data points in Fig. 2c) although they are found at the very edge of the dusk sector and close to the midnight sector. The data points with small \(\delta B_{ \bot } /\delta B_{||}\) values represent compressional-type waves. The data points with large \(\delta B_{ \bot } /\delta B_{||}\) values are identified as ICWs if they have left-hand polarization. Some examples of ICW events are given in the next section.

In the analysis of the dayside and nightside segments of Rev. 19, Russell et al. (2006) reported that mirror modes can emerge outside 5 R _S and that ICWs appear inside 7 R _S . This is in general agreement with the analysis results from the five-year data, covering all local time sectors. For R < 6 R _S , we found that waves at low latitudes are mostly left-hand circularly polarized. They are identified as ion cyclotron waves. For R > 6 R _S , waves are generally in a compressional mode (though some mixed polarization cases can be found, see “Waves with mixed polarization” section), but waves near the midnight region are transverse, left-hand circularly polarized.

Left-hand circularly polarized waves: ion cyclotron waves

Figure 3 shows a typical left-hand circularly polarized wave event for R < 6 R _S . The perturbed magnetic field components in KRTP coordinates are shown in Fig. 3a, for a time interval of 3 min during 00:50 to 00:53 UT on July 24, 2006. The perturbed magnetic field components are filtered with a band width of ±0.08 Hz with respect to the peak frequency, and the distance between Cassini and Saturn is 4.7 R _S ; the latitude is −1°; the background magnetic field mean value is ~200 nT; and the observed wave frequency is ~0.17 Hz (\(f_{{O^{ + } }}\) is ~0.19 Hz). Figure 3b shows left-hand polarization for the perturbed magnetic field. We can estimate the wavelength and the phase velocity of the ion cyclotron waves by assuming that they propagate mainly along the ambient magnetic field line in the low-latitude equatorial region. The observed Doppler shift frequency can be written as \(\omega '\approx \omega { - }k_{\theta } v_{s\theta }\), where \(v_{s\theta }\) is the spacecraft velocity in the θ direction. For the case shown in Fig. 3, \(\omega '\) = 1.05 rad/s and \(v_{s\theta }\) = −3.77 km/s. If the ion cyclotron wave frequency is the same as the O⁺ gyrofrequency (given by ω = 1.23 rad/s) in the plasma moving frame, then \(k_{\theta } \approx (\omega - \omega ')/v_{S\theta } =\) −0.033 km⁻¹ (the wavelength is λ = 2π/|k _θ | = 188.7 km) and the phase velocity in the θ direction is V _ph = ω/k _θ  = −36.2 km/s (i.e., northward propagation). Note that our calculation is for a single event, which is different from the statistical average over different events performed by Leisner et al. (2011).

Figure 4 shows an ion cyclotron wave event for R > 6 R _S in the low-latitude region. The perturbed magnetic field components in KRTP coordinates are shown in Fig. 4a, for a time interval of 3 min during 02:45 to 02:48 UT on September 30, 2007. The left-hand circular polarization for the perturbed magnetic fields is shown in Fig. 4b. The perturbed magnetic field components are filtered with a band width of ±0.02 Hz, with respect to the peak frequency, and the distance between Cassini and Saturn is 7.1 R _S ; the latitude is −2°; the longitude is 147.6° with respect to the X-axis (counterclockwise, in the midnight region); the background magnetic field mean value is 50 nT; and the observed wave frequency is ~0.039 Hz, which is about 20% below the O⁺ cyclotron frequency \(f_{{O^{ + } }}\) of ~0.047 Hz. Thus, based on the wave frequency, the dominant transverse perturbation and left-hand circular polarization of the ICWs are identified inside 6 R _S at all local times. For ICW events outside 6 R _S , 42% of events with a frequency close to f _W+ were identified as ICWs in the 2007 data (with Cassini orbits ranged from pre-midnight to dawn). 7% were identified in the 2009 data (with Cassini orbits ranging from midnight to noon), and all the detected ICWs were near the midnight region.

Waves with mixed polarization

A typical wave event with a mixed polarization, outside 6 R _S in the low latitudes, is shown in Fig. 5. Figure 5a shows the perturbed magnetic field for a time interval of 3 min during 20:53 to 20:56 UT on October 13, 2009. The distance between Cassini and Saturn is 6.3 R _S ; the latitude is −0.308°; and the background magnetic field mean value is ~58 nT. Figure 5b shows the left-hand circular polarization in the wave’s transverse components (δB _r and δB _ϕ ) with a frequency of 0.04 Hz which is about 25% below the O⁺ cyclotron frequency (\(f_{{O^{ + } }}\) of 0.05 Hz). Figure 5c shows a mixture of linear and left-hand circular polarizations in δB _θ and δB _ϕ . However, from Fig. 5a the dominant frequency of δB _θ can be seen to be about a factor of 2 smaller than that of the transverse components. Thus, the magnetic perturbations are considered to be a linear superposition of the compressional mirror mode (Cowee et al. 2009) and transverse ion cyclotron waves.

Linearly polarized waves: mirror modes

A typical compressional wave event, with linear polarization outside 6 R _S in the low-latitude region, is shown in Fig. 6. Figure 6a shows the perturbed magnetic field for a time interval of 20 min during the period from 02:20 to 02:40 UT on March 9, 2005, when Cassini was 7 R _S from Saturn and its latitude was −0.2°. The amplitude of the δB _θ component, which is mainly along the ambient magnetic field, is much larger than the amplitudes of the other two perturbed magnetic field components. Thus, the wave is of a compressional type and has a linear polarization as shown in Fig. 6b. The observed wave frequency (0.01 Hz) is much smaller than the O⁺ cyclotron frequency, which is \(f_{{O^{ + } }}\) of 0.045 Hz for the background magnetic field with a mean value of 48 nT. Thus, the wave event is considered as a mirror mode due to its compressional nature and characteristic waveform (Russell and Blanco-Cano 2007).

Based on the magnetohydrodynamics (MHD) theory, the mirror mode is stationary (with zero frequency) in the plasma moving frame. However, when particle kinetic effects are considered, the mirror mode has a small real frequency associated with the diamagnetic drift frequency of the plasma density gradient (Hasegawa 1969). If the plasma is moving, it will also move with the plasma (e.g., Johnson and Cheng 1997). Thus, the Doppler frequency shift is caused by both the plasma flow velocity and the satellite moving velocity. For mirror modes, we assume that the wave vector is mainly perpendicular to the ambient magnetic field. The plasma flow is dominated by both the corotation velocity (\(v_{cor}\) ~39 km/s at Enceladus orbit), which is mainly in the azimuthal direction and is much larger than the satellite velocity (\(v_{s}\) ~10 km/s), as well as the radial flow velocity (up to 2 km/s, Wilson et al. 2008). This means that the Doppler shift frequency is given by \(\omega '\approx k_{\varphi } v_{\text{cor}} + k_{\varphi } v_{s\varphi } + k_{r} v_{sr}\), where \(v_{s\varphi }\) and \(v_{sr}\) are the spacecraft velocity in the \(\varphi\)-direction and r-direction, respectively. The mirror-mode free energy is associated with anisotropy in the pickup ion velocity space and is localized in the r-direction but is much broader in the \(\varphi\)-direction. This means that the wave vector is assumed to be mainly in the r-direction in order to carry the wave energy out of the unstable region for saturation. Thus, \(k_{r} \gg k_{\varphi }\), and the Doppler shift is caused mainly by the satellite velocity and is given by \(\omega '\approx k_{r} v_{sr}\). For the mirror mode shown in Fig. 6, the Doppler shift frequency is \(\omega ' \approx k_{r} v_{sr}\) = 0.01 rad/s, and for \(v_{sr}\) = 10 km/s, we obtain a value for k _r  = 10⁻³ km⁻¹, and the radial wavelength λ = 2π/k _r  = 6283.2 km.

In comparison with other years, the Cassini orbits in 2008 are more often in the high-latitude region of the inner magnetosphere. The upper panel of Fig. 7 shows a Cassini orbit (the red line) moving across the L-shell field lines from the northern high-latitude region (A) to the southern high-latitude region (B). The lower panel of Fig. 7 shows the power spectrum of δB for an ion cyclotron wave event observed on August 4, 2008, and Cassini passes from point A (r = 4.26; θ = 27.3) in the Northern Hemisphere to point B (r = 4.23; θ = −40.7) in the Southern Hemisphere, where r is the radial distance of Cassini from Saturn in a spherical coordinate system and θ is the magnetic latitude. The blue dashed line shows the local W⁺ gyrofrequencies f _local, and the black line shows the W⁺ gyrofrequency f _eq at the dipole L-shell equator of the Cassini orbit field lines. To calculate f _eq, we assume that in Saturn’s inner magnetosphere the magnetic field can be well described by the dipole field model. The dipole field line is determined by \(r = LR_{s} \cos^{2} \theta\), where L is the L-shell value. The dipole magnetic field at the equator is given by \(B_{eq} = B_{0} /L^{3}\), where \(B_{0} = 2 \times 10^{4}\) nT is the magnetic field strength at the equator on Saturn’s surface. The corresponding equatorial gyrofrequency for the L-shell field line going through point A is f _eqA = 0.095 Hz for \(L = 4.26/\cos^{2} (27.3^\circ ) = 5.4\), and the equatorial magnetic field strength is B _eqA = B ₀(5.4)⁻³ = 10 nT. Similarly, the corresponding equatorial W⁺ gyrofrequency for the L = 7.4 field line going through point B is f _eqB = 0.0475 Hz and the equatorial magnetic field strength is B _eqB = 50 nT.

From the lower panel of Fig. 7, we note that the observed wave frequencies are lower than the local W⁺ gyrofrequency (blue dotted line), except near the equator, and in the higher-latitude region where they are at least two times smaller than the local W⁺ gyrofrequency. However, the observed wave frequency matches well with the calculated equatorial W⁺ gyrofrequency f _eq (black line) of the corresponding L-shell field line. This supports the postulation that the ion cyclotron waves observed in the high-latitude region have their source farther away in the low-latitude equatorial region, from which the ion cyclotron waves propagate along magnetic field lines to the higher-latitude region. Note that this is also consistent with the Doppler shift analysis done by Leisner et al. (2011) on the 2006 MAG data and the hybrid simulation near Enceladus’ orbit done by Cowee et al. (2009). The wave sources are near the equator, and the waves then propagate both northward and southward along the field lines to the high-latitude regions. However, Cassini moves southward, and thus, the observed wave frequency is downward shifted in the Southern Hemisphere as opposed to an upward frequency shift in the Northern Hemisphere.

The magnetic components are obtained by a band-pass filter with a bandwidth of ±0.1 Hz with respect to the peak frequencies. Each point for δB shown in Figs. 8, 9 and 10 is averaged within a small region of the magnetic latitude range Δθ = 0.1° with an equatorial radial distance range of ΔR = 1 Rs when Cassini passed through.

Figure 8 plots the 2005 data. It shows the latitude dependence of the perturbed magnetic field amplitude δB, averaged at a radial distance range of 3.5–4.5 R _S in the dusk sector. The average δB amplitude versus the latitude is shown by the red curve, and it is clear that the peaks of δB are off the equator with maxima at about 1.8° and −1.2° latitude in the dusk sector.

Figure 9 plots data collected from 2005, 2006, 2007 and 2009. It shows the latitude dependence of the perturbed magnetic field amplitude δB averaged over the data at a radial distance range of 4.5–5.5 R _S for the noon sector. The black circles represent δB values on July 26, 2009, only, and the pink dots represent 2009 data without the data for July 26, 2009. In general, the off-equator peaks are found to be at about 2° and −2° latitude in the noon sector. However, the off-equator peak became broader and moved radically to about 10° on July 26, 2009.

Figure 10 plots data collected from 2006, 2007, 2008 and 2009. It shows the latitude dependence of the perturbed magnetic field amplitude δB averaged over the data in the radial distance ranges of (a) 3.5–4.5 R _S , (b) 4.5–5.5 R _S and (c) 5.5–6.5 R _S for the midnight sector. The average δB amplitude versus latitude is shown by the red curve. The peak of the average wave amplitude increases with increasing distance from Saturn. In Panel (a), the average δB peaks are at 4° and −3.5° latitude, with an average δB of ~0.7 nT. In Panel (b), the average δB peaks are at 4.5° and −3° latitude with an average δB of ~1.2 nT. In Panel (c), the average δB peaks are at 5° and −4° latitude with an average δB of ~1.7 nT. For the dawn sector, however, Cassini had no good latitudinal coverage from which to draw conclusions.

From Figs. 8, 9 and 10, the latitude position of maximum δB can be seen to be at a higher latitude in the midnight sector. It generally moves to a lower latitude in the noon sector and an even lower one in the dusk sector. Also, the latitudinal width of the δB peaks (the average δB intensity is shown by the red lines) is found to be broader in the midnight sector and becomes narrower in the noon and dusk sectors.

Figure 11 shows a projection of the δB intensity distribution in the equatorial plane (in rectangular X–Y coordinates) for the data from 2005 to 2009. The data are selected in between ±1° latitude from the magnetic equator to avoid the latitudinal effect. The ICW amplitude is relatively large in this region between the midnight and dawn, and the maximum amplitude occurs in the midnight region.

Note that the growth rate increases with increasing \(v_{ \bot 0}\) and \(n_{\text{ring}}\). For a fixed ratio of \(n_{\text{thermal}} /n_{\text{ring}} = R_{n}\), from Eq. (5), the ICW is unstable for \(n_{tot} < k_{||}^{2} c^{2} m_{i} /4\pi e^{2} [ ( 1 { + }R_{n} )\omega_{c}^{2} \, / 2\,k_{||}^{2} v_{ \bot 0}^{2} - 1 ]\). Because the total plasma density decreases along the field line away from the equator, due to the plasma rotation effect, the growth rate will vary along the field line. As the total ion density n _tot decreases to \(n_{0} = k_{||}^{2} c^{2} m_{i} /4\pi e^{2} [ ( 1 { + }R_{n} )\omega_{c}^{2} /k_{||}^{2} v_{ \bot 0}^{2} - 1 ]\), the growth rate increases and reaches the maximum value given by \({\gamma_{hbox{max}}}/\omega_{c} = k_{||}^{2} v_{\bot 0|^{2}}/2 (R_{n} + 1 )\omega_{c}^{2}\). When the total ion density n _tot decreases further with \(n_{\text{tot}} < n_{0}\), the growth rate starts to decrease. These analytical results are verified by the numerical solutions of Eq. (2).

To obtain numerical solutions of the dispersion relation (Eq. (2)) along the field lines, the B field in the inner magnetosphere of Saturn is approximated as a dipole field. The total ion density variation along the field lines is estimated from the 2D axisymmetric MHD force-balanced equilibrium model for azimuthally rotating plasmas, with small anisotropy of the total pressure (e.g., Chou and Cheng 2010). This is written as \(n_{W} = n_{\text{eq}} {\text{exp[}} - m_{i} {{\Omega }}^{ 2} (R_{\text{eq}}^{2} - R^{2} )/2T_{||} ]\) due to the centrifugal force. \(n_{\text{eq}}\) is the equatorial ion density, Ω is the plasma rotation frequency, T| is the ion temperature along the field line, R _eq = LR _S is the dipole L-shell distance, and R is the equatorial distance of the field line position. Tokar et al. (2008) reported that n _thermal = 33 cm⁻³ and the density of the pickup ions is n _ring = 20.6 cm⁻³ around \(R = 4 - 4.5R_{S}\) in the equatorial region (on October 11, 2005, from 21:08:03 to 22:06:43 UT) though these values are for a specific time. We set the density ratio as \(R_{n} = n_{\text{thermal}} /n_{\text{ring}} = 3/2\) at the equator and calculated the \(R_{n}\) variation along B in order to study the growth rate. Figure 12a shows the dependence of growth rate on latitude along the L = 4.5 field line for different values of \(k_{ | |}\), where the equatorial density is \(n_{\text{eq}} = 50\) cm⁻³ and the plasma rotation frequency is \(\Omega = 1. 6 4\times 1 0^{ - 4}\)/s.

The ion density along the L = 4.5 field line is concentrated at the equator as shown in Fig. 12b, due to the plasma rotation in Saturn’s magnetosphere. The damping effect becomes strong around the equator and results in a local growth rate minimum at the equator. This gives a reasonable explanation for the observed latitudinal dependence of the ICW amplitude δB, as shown in Figs. 8, 9 and 10, such that the ICW amplitude has a local minimum at the equator. The maximum growth rate \({\gamma_{\hbox{max}}} /\omega_{c}\) is sensitive to \(k_{||}\) as shown in the analytical results (Eq. (5)). The latitudinal position of the peak growth rate (or the corresponding latitude at which n _tot = n ₀) is sensitive to \(T_{||}\) because the ion density variation scale length depends on \(\sqrt {2T_{||} /m_{i} \Omega^{2} }\) in the Gaussian density model. By tuning \(k_{||}\) and \(T_{||}\), the latitudes of peak growth rate can be located at 2° and 10°. This is shown in Fig. 12a as a comparison to the observed \(\delta B\) peak in the noon sector (Fig. 9). The model implies that a higher temperature causes the off-equator growth rate peak to move to higher latitudes, which can explain the July 26, 2009, data. Thus, there could be a sudden increase in the plasma temperature on that day, and one possibility for the phenomena is that they occurred when Cassini was immersed in an energetic neutral atom (ENA) event.

The location of the peak growth rate moves to higher latitude for larger \(T_{ | |}\) values, due to the broader density profiles along the field line. This can be used to infer the parallel temperature distribution over the local time, by comparing the latitudinal distribution of ICW amplitude with the observational data over the different local time sectors (as given in “Ion cyclotron wave amplitude versus latitude” section). The peak ICW amplitude is located at about ±4° for the midnight sector, about ±2° for the noon sector and about ±1.5° for the dusk sector. Because of this, and the trend of the δB peak width, we propose that the averaged parallel ion temperature decreases from a few eVs in the midnight sector, to lower values toward the dawn sector. This decrease continues through the noon sector and then drops to below ~1 eV in the dusk sector (although there was no good latitudinal coverage of data for the dawn sector from 2005 to 2009). The decrement in temperature implies that there are extra ICW driving sources on the nightside, in addition to the pickup ion mechanism near Enceladus’ orbit.

Our local stability analysis of ICWs provides their local growth rate for each point along the field line. Ideally a nonlocal stability analysis along the field line should be performed to obtain the wave structure along it, as well as the global wave growth rate. As the wave grows in amplitude in the unstable region, it will propagate away along the field line to a higher-latitude location where its energy is absorbed by plasma via Landau or collisional damping processes. If the overall wave driving is stronger than the wave damping, the ICW will form an unstable eigenmode solution with the wave structure along the field line. From our experience on the nonlocal stability analysis, the wave amplitude is generally larger in the stronger local growth rate location. Because the observed wave frequency is rather coherent, in the saturation stage the ICW is expected to maintain its wave structure. The main nonlinear effect is due to a quasi-linear wave–particle interaction and velocity space spreading. Moreover, from the initial value simulations of ICWs, due to pickup ions in Saturn’s magnetosphere (e.g., Cowee et al. 2009), the growth time of ICWs is more than 100 times longer than the ion cyclotron wave period. This means that the wave saturates, and its structure is maintained, so long as there is a constant pickup ion production rate. Thus, our conclusion on the ICW amplitude variation along the field line, based on the local growth rate, remains qualitatively valid.