To understand why the maximum ICW wave amplitude is located off the equator, we investigate the stability of the left-hand circularly polarized ion cyclotron wave based on kinetic theory. The W^{+} ion is assumed to be composed of two distributions: Firstly, the cold-ring velocity distribution \(f_{i0} = 1/(2\pi v_{ \bot 0} )\delta (v_{ \bot } - v_{ \bot 0} )\delta (v_{||} )\) for the pickup ions, which are responsible for destabilizing the ion cyclotron waves. Secondly, the background plasma has an isotropic Maxwellian distribution with temperature *T*, which contributes to the wave stabilization. The H^{+} component becomes important at high latitude, but we concentrate our discussion on the low-latitude equatorial region. Then, based on linear kinetic theory the local dispersion relation for the left-hand polarized waves can be written as:

$$\omega^{2} = k_{||}^{2} c^{2} + \omega_{\text{pr}}^{2} \left[ {\frac{\omega }{{\omega - \omega_{c} }} + \frac{{k_{||}^{2} v_{ \bot 0}^{2} }}{{2 (\omega - \omega_{c} )^{ 2} }}} \right] - \frac{{\omega_{\text{pi}}^{2} \omega }}{{k_{||} v_{\text{th}} }}Z (\zeta )$$

(2)

where \(\omega_{\text{pr}}\) and \(\omega_{\text{pi}}\) are the plasma frequency of the ring and thermal components, respectively. \(\omega_{c}\) is the ion gyrofrequency and \(v_{\text{th}}\) is the thermal speed of the thermal component. *Z* is the plasma dispersion function with the argument \(\zeta = (\omega - \omega_{c} )/k_{||} v_{\text{th}}\). The pickup ion gyrovelocity can be estimated from the difference between the neutral cloud velocity and the plasma corotation velocity as \(v_{ \bot 0} = v_{\text{corotate}} - v_{\text{neutral}}\). At the distance of Enceladus’ orbit (*R* = 3.95 *R*
_{
S
}), the neutral cloud velocity is \(v_{\text{neutral}} = \sqrt {GM/R} \approx\) 12.6 km/s, the plasma corotation velocity is \(v_{\text{corotate}} = R\Omega \approx 39\, {\text{km/s}}\), and thus, \(v_{ \bot 0} \approx 26.4 {\text{km/s}}\). In Eq. (2), the second term represents the ring particle contribution and the third term is contributed by the thermal ions. Let \(X \equiv (\omega - \omega_{c} )/\omega_{c}\) and assume |*X*| ≪ 1 and \(|\zeta |\, > > 1\), then Eq. (2) can be simplified to

$$\left[ {1 + \frac{{k_{||}^{2} c^{2} }}{{\omega_{\text{ptot}}^{2} }} - \frac{{\omega_{c}^{2} }}{{\omega_{\text{ptot}}^{2} }}} \right]X^{2} + X + \left( {\frac{{n_{\text{ring}} }}{{n_{\text{tot}} }}} \right)\frac{{k_{||}^{2} v_{ \bot 0}^{2} }}{{2\omega_{c}^{2} }} \cong 0$$

(3)

by dropping the higher order terms, where \(\omega_{\text{ptot}}\) is the total plasma frequency associated with the total ion density \(n_{\text{tot}} \equiv n_{\text{ring}} + n_{\text{thermal}}\). For a typical pickup W^{+} ion density of *n*
_{
i
}~ O(10) cm^{−3} with the equatorial magnetic field intensity of *B* ~(20,000/*L*
^{3}) nT, we have \(\omega_{c}^{2} < < \omega_{ptot}^{2}\), and the solution of Eq. (3) is given by:

$$\frac{\omega }{{\omega_{c} }} \cong 1 - \frac{1}{{2 ( 1+ k_{||}^{2} c^{2} /\omega_{\text{ptot}}^{2} )}} \pm i\frac{{\sqrt {2 (n_{\text{ring}} /n_{\text{tot}} ) (k_{||}^{2} v_{ \bot 0}^{2} /\omega_{c}^{2} ) ( 1+ k_{||}^{2} c^{2} /\omega_{\text{ptot}}^{2} )- 1} }}{{2 (1 + k_{||}^{2} c^{2} /\omega_{\text{ptot}}^{2} )}}$$

(4)

The condition for an unstable ICWs is then \(2 (n_{\text{ring}} /n_{\text{tot}} ) (k_{||}^{2} v_{ \bot 0}^{2} /\omega_{c}^{2} ) ( 1+ k_{||}^{2} c^{2} /\omega_{\text{ptot}}^{2} )> 1\), and the growth rate is expressed as:

$$\frac{\gamma }{{\omega_{c} }} \cong \frac{{\sqrt {2 (n_{\text{ring}} /n_{tot} ) (k_{||}^{2} v_{ \bot 0}^{2} /\omega_{c}^{2} ) ( 1+ k_{||}^{2} c^{2} m_{i} /4\pi n_{\text{tot}} e^{2} )- 1} }}{{2 (1 + k_{||}^{2} c^{2} m_{i} /4\pi n_{\text{tot}} e^{2} )}}$$

(5)

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.