As regards the modeling of BGK modes, there are two main theoretical approaches: the integral solution or BGK methodology and the differential (or Schamel) technique. In the former method (BGK), one assumes that the initial particle distribution function and the electrostatic potential profiles are known, so these are substituted into the Poisson equation and the integral equation is solved to obtain the trapped particle distribution function (Bernstein et al. 1957; Aravindakshan et al. 2018a, 2018b, and the references therein). In Schamel’s approach, the form of the trapped particle distribution function and of the passing (i.e., free, nontrapped) particle distribution function is assumed and substituted in Poisson’s equation, leading to a differential equation that is then solved to obtain the form of the potential (Schamel 1986; Luque & Schamel 2005, and the references therein). A distinguishing factor in the former (BGK) approach is that it involves a condition in the form of an inequality to be satisfied by the potential parameters (width and amplitude) in order for a BGK mode to be sustained. The BGK approach will be adopted in this work. The above models tacitly assume a collisionless electron-ion plasma. These assumptions are acceptable in the Earth’s magnetosphere. However, as we move farther from near-Earth plasma environments, the presence of charged dust in the plasma cannot be neglected. In the case of Saturn, there are observations of streaming ions by the Cassini spacecraft (Badman et al. 2012a, 2012b). We know that these streaming ion flows can lead to the generation of ion holes. Electrostatic solitary waves have been observed in Saturn’s magnetosphere (Williams et al. 2006) and in the dusty environment near its moon Enceladus (Pickett et al. 2015). Williams et al. (2006) reported observations of solitary structures in the vicinity of Saturn’s magnetosphere. They detected a series of bipolar pulses and speculated that these could be either electron holes or ion holes (Williams et al. 2006). Later on, Pickett et al. (2015) observed solitary wave pulses within 10 Rs (Rs is the Saturn radius) and near Enceladus. Near the Enceladus plume, they discussed how dust impacts affected the observed solitary waves. In fact, Pickett et al. (2015) pointed out that some of the bipolar electric field pulses associated with the solitary waves observed had an inverse polarity (i.e., a positive pulse first, followed by a negative pulse in a short time period) and suggested that this might be due to either an inverse direction of propagation or to a true inverse potential pulse polarity (sign). Moreover, Farrell et al. (2017) examined the conditions that allow low-energy ions, such as those produced in the Enceladus plume, to be attracted and trapped within the sheath of negatively charged dust grains. Using particle-in-cell simulations, they showed that with dust in the system, the large electric field from the grain charge disrupts pickup and leads to ion trapping. Their simulation results also reveal that the bipolar pulses reported in the Enceladus plume by Williams et al. (2006) and Pickett et al. (2015) could most probably be ion holes. In the light of the above information, we may suggest that the formation of ion holes is highly likely in the dusty plasma of environments such as the one found in Saturn. Importantly, the instrument on board Cassini, the Radio Plasma Wave Science instrument, does not have the ability to determine the polarity of the electric fields associated with the ESWs observed with 100% certainty as it lacks the ability to perform interferometry (Williams et al. 2006).
Citation: Williams et al. (2006); Williams et al. 2006
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